Some Mathematical Problems in Geophysical Fluid Dynamicsmypage.iu.edu/~temam/papers/315_PTZ2008.pdf · 1. Introduction The aim of this article is to address some mathematical aspects

CHAPTER 1

Some Mathematical Problems inGeophysical Fluid Dynamics

Madalina PetcuUniversité de Genève, Section de Mathématiques,

2-4 rue du Lièvre, CP64,Genève 4, Switzerland 1211

E-mail: [email protected]

Roger TemamThe Institute for Scientific Computing and Applied Mathematics, Indiana University,

Bloomington, IN 47405, USA, andLaboratoire d’Analyse Numérique, Université Paris-Sud, Bâtiment 425,

91405 Orsay, FranceE-mail: [email protected]

Mohammed ZianeUniversity of Southern California, DRB 155, 1042 W. 36 Place,

Los Angeles, CA 90049, USAE-mail: [email protected]

03.27.2008

Contents1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1. Physical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2. Mathematical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3. Content of this article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4. Summary of results for the physics oriented reader . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2. The primitive equations: Weak formulation, existence of weak solutions . . . . . . . . . . . . . . . . . . 92.1. The primitive equations of the ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2. Weak formulation of the PEs of the ocean. The stationary PEs . . . . . . . . . . . . . . . . . . . . 172.3. Existence of weak solutions for the PEs of the ocean . . . . . . . . . . . . . . . . . . . . . . . . . . 26

HANDBOOK OF NUMERICAL ANALYSIS, Special Volume on ComputationalMethods for the Oceans and the Atmosphere, R. Temam and J. Tribbia, guest editorsEdited by P.G. Ciarlet© 2008 Elsevier B.V. All rights reserved

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2 M. Petcu et al.

2.4. The primitive equations of the atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5. The coupled atmosphere and ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3. Strong solutions of the primitive equations in dimension two and three . . . . . . . . . . . . . . . . . . 423.1. Strong solutions in space dimension three . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2. Strong solutions in dimension 3 (global existence) . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3. Strong solutions of the two-dimensional primitive equations:

Physical boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.4. Uniqueness ofz-weak solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.5. The space periodic case in dimension two: Higher regularities . . . . . . . . . . . . . . . . . . . . 923.6. The space periodic case in dimension three: Higher Sobolev regularities and Gevrey regularity . . 1063.7. On the backward uniqueness of the primitive equations . . . . . . . . . . . . . . . . . . . . . . . . 126

4. Regularity for the elliptic linear problems in GFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374.1. Regularity of solutions of elliptic boundary value problems in cylinder type domains . . . . . . . . 1384.2. Regularity of solutions of a Dirichlet–Robin mixed boundary value problem . . . . . . . . . . . . . 1454.3. Regularity of solutions of a Neumann–Robin boundary value problem . . . . . . . . . . . . . . . . 1534.4. Regularity of the velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1644.5. Regularity of the coupled system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

AbstractThis article reviews the recently developed mathematical setting of the primitive equations

(PEs) of the atmosphere, the ocean and the coupled atmosphere and ocean. The mathematicalissues that are considered here are the existence, uniqueness and regularity of solutions for thetime dependent problems in space dimensions two and three, the primitive equations beingsupplemented by a variety of natural boundary conditions. The emphasis is on the case of theocean which encompasses most of the mathematical difficulties. This article is devoted to thePEs in the presence of viscosity, while the PEs without viscosity are considered in the articleby Rousseau, Temam and Tribbia in the same volume.

Whereas the theory of PEs without viscosity is just starting, the theory of PEs with viscosityhas developed since the early 1990s and has now reached a satisfactory level of completion.The theory of the PEs was initially developed by analogy with that of the incompressibleNavier Stokes equations, but the most recent developments reported in this article have shownthat, unlike the incompressible Navier-Stokes equations and the celebrated Millenium Clayproblem, the primitive equations with viscosity are well-posed in space dimension two andthree, when supplemented with fairly general boundary conditions. This article is essentiallyself-contained and all the mathematical issues related to these problems are developed.

A guide and summary of results for the physics oriented reader is provided at the end of theIntroduction (Section 1.4).

Some mathematical problems in GFD 3

1. Introduction

The aim of this article is to address some mathematical aspects of the equations of geo-physical fluid dynamics, namely existence, uniqueness and regularity of solutions.

The equations of geophysical fluid dynamics are the equations governing the motionof the atmosphere and the ocean, and are derived from the conservation equations fromphysics, namely conservation of mass, momentum, energy and some other componentssuch as salt for the ocean, humidity (or chemical pollutants) for the atmosphere. The basicequations of conservation of mass and momentum, that is the three-dimensional compress-ible Navier–Stokes equations contain however too much information and we cannot hopeto numerically solve these equations with enough accuracy in a foreseeable future. Owingto the difference of sizes of the vertical and horizontal dimensions, both in the atmosphereand in the ocean (10–20 km versus several thousands of kilometers), the most natural sim-plification leads to the so-calledprimitive equations(PEs) which we study in this article.

We continue this Introduction by briefly describing the physical and mathematical back-grounds of the PEs.

1.1. Physical background

The primitive equations are based on the so-called hydrostatic approximation, in which theconservation of momentum in the vertical direction is replaced by the simpler, hydrostaticequation (see, e.g., (2.25)).

As far as we know, the primitive equations were essentially introduced by Richardsonin 1922; when it appeared that they were still too complicated, they were abandonedand, instead, attention was focused on simpler models, such as the barotropic and thegeostrophic and quasi-geostrophic models, considered in the late 1940’s by von Neumannand his collaborators, in particular Charney. With the increase of computing power, interesteventually returned to the PEs, which are now the core of many Global Circulation Models(GCM) or Ocean Global Circulation Models (OGCM), available at the National Centerfor Atmospheric Research (NCAR) and elsewhere. GCMs and OGCMs are very complexmodels which contain many physical components including e.g. for the atmosphere, thechemistry (equations of concentration of pollutants), the physics of the cloud (radiationof solar energy, concentration of vapor), the vegetation, the topography, the albedo or forthe oceans, such phenomena as the sea ice or again the topography of the bottom of theoceans. Nevertheless, the PEs which describe the dynamic of the air or the water and thebalance of energy are the central components for the dynamics of the air or the water. Forsome phenomena there is need to give up the hydrostatic hypothesis and then nonhydro-static models are considered, such as in Laprise (1992) or Smolarkiewicz, Margolin andWyszogrodzki (2001); these models stand at an intermediate level of physical complexitybetween the full Navier–Stokes equations and the PEs-hydrostatic equations. Research onnonhydrostatic models is ongoing and, at this time, there is no agreement, in the physicalcommunity, for a specific model.

In this hierarchy of models for geophysical fluid dynamics, let us add also theShallow Water equation corresponding essentially to a vertically integrated form of the

4 M. Petcu et al.

Navier–Stokes equations; from the physical point of view they stand as an intermediatemodel between the primitive and the quasi-geostrophic equations.

In summary, in term of physical relevance and the level of complexity of the physicalphenomena they can account for, the hierarchy of models in geophysical fluid dynamics isas in the following Table:

Three-dimensional Navier–Stokes equations+

Nonhydrostatic models+

Primitive equations(hydrostatic equations)

+Shallow water equations

+Quasi-geostrophic models

+Two-dimensional barotropic equations.

Table 1. Level of physical complexity (richness)

We remark here also that much study is needed for the boundary conditions from boththe physical and mathematical points of views. As we said, our aim in this article is thestudy of mathematical properties of the PEs.

In the above, and in all of this article, the primitive equations that we consider are thePEs with viscosity; the primitive equations without viscosity are studied in the article ofRousseau, Temam and Tribbia (2008) in this volume. The PEs without viscosity raise ques-tions of a totally different nature. In particular whereas the PEs with viscosity bear somesimilarity with the incompressible Navier Stokes equations as we explain below, the PEswithout viscosity are different in many respects from the Euler equations of incompressibleinviscid flows; see the already quoted article of Rousseau, Temam and Tribbia.

1.2. Mathematical background

The level of mathematical complexity of the equations in Table 1 is not the same as thelevel of physical complexity: at both ends, the quasi-geostrophic models and barotropicequations are mathematically well understood (at least in the presence of viscosity; seeWang (1992 a,b), and despite its well-known limitations, the mathematical theory of theincompressible Navier–Stokes equations is also relatively well understood. On the otherhand, nonhydrostatic models are mathematically out of reach, and there are much lessmathematical results available for the shallow water equations than for the Navier–Stokesequations, even in space dimension two (see however Orenga (1995)).

The mathematical theory of the (viscous) primitive equations has developed in twostages. The first stage ranging from the article of Lions, Temam and Wang (1993 a,b) to


the review article by Temam and Ziane (2004) concentrated on the analogy of the primitiveequations with the three-dimensional incompressible Navier-Stokes equations. Indeed, andas we show below, the primitive equations although physically “poorer” than the Navier-Stokes equations are, in some sense, structurally more complicated than the incompressibleNavier–Stokes equations.

Indeed this is due to the fact that the nonlinear term in the Navier–Stokes equations, alsocalled inertial term, is of the form

velocity� first-order derivatives of velocity,

whereas, the nonlinear term for the primitive equations, is of the form

first-order derivatives of horizontal velocity� first-order derivatives of horizontal velocity.

The mathematical study of the primitive equations was initiated by Lions, Temam andWang (1992a,b). They produced a mathematical formulation of the PEs which resemblesthat of the Navier–Stokes due to Leray, and obtained the existence for all time of weak so-lutions; see Section 2, and the original articles Lions, Temam and Wang (1992 a, b, 1995)in the list of references. Further works, conducted during the 1990s and more especiallyduring the past few years, have improved and supplemented the early results of these au-thors by a set of results which, essentially, brings the mathematical theory of the PEs to thatof the three-dimensional incompressible Navier–Stokes equations, despite the added com-plexity mentioned above; this added complexity is overcome by a nonisotropic treatmentof the equations (of certain nonlinear terms), in which the horizontal and vertical directionsare treated differently.

In summary the following results have been obtained which were presented in the reviewarticle by Temam and Ziane (2006) and appear herein in Section 2 and 3:

(i) Existence of weak solutions for all time (dimension two and three) (See Section 2).(ii) In space dimension three, existence of a strong solution for a limited time (local in

time existence) (see Section 3.1).(iii) In space dimension two, existence and uniqueness for all time of a strong solution

(see Section 3.3).(iv) Uniqueness of a weak solution in space dimension two (see Section 3.4).In the above, the terminology is that normally used for Navier–Stokes equations: the

weak solutions are those with finite (fluid) kinematic energy (L1.L2/ andL2.H 1/), andthe strong solutions are those with finite (fluid) enstrophy (L1.H 1/ andL2.H 2/). Essen-tial in the most recent developments (ii)–(iv) above is theH 2 regularity result for a Stokestype problem appearing in the PEs, the analog of theH 2 regularity in the Cattabriga–Solonnikov results on the usual Stokes problem; the whole Section 4 is devoted to thisproblem.

The second stage of the mathematical theory of the (viscous) primitive equations is morerecent. It is based on the observation that the pressure like function (the surface pressure)is in fact a two-dimensional function (a function of the horizontal variables and time) andbecause of that the 3D primitive equations are also close to a 2D system. Technically,

6 M. Petcu et al.

by suitable estimates of the surface pressure, the difficulties related to the pressure areovercome. This approach was developed in the two independent articles (with differentproofs) by Cao and Titi (2007) and by Kobelkov (2006), for the case of an ocean with aflat bottom. The case of a varying bottom topography is studied in the subsequent article ofKukavica and Ziane (2007). These three articles combine the above mentioned results oflocal existence of a strong solution and the new a priori estimates to show that the strongsolution is defined for all time. These newest results appear in Section 3.2.

1.3. Content of this article

Because of space limitation it was not possible to consider all relevant cases here. Relevantcases include:

The Ocean, The Atmosphere and The Coupled Ocean and Atmosphere,

on the one hand, and, on the other hand, the study of global phenomena on the sphere(involving the writing of the equations in spherical coordinates), and the study of mid-latitude regional models in which the equations are projected on a space tangent to thesphere (the Earth), corresponding to the so-calledˇ-plane approximation: here0x is thewest–east axis,0y the south–north axis, and0z the ascending vertical.

In this article we have chosen to concentrate on the cases mathematically most signif-icant. Hence for each case, after a brief description of the equations on the sphere (inspherical coordinates), we concentrate our efforts on the correspondingˇ-plane case (inCartesian coordinates). Indeed, in general, going from theˇ-plane case in Cartesian coor-dinates to the spherical case necessitates only the proper handling of terms involving lowerorder derivatives; full details concerning the spherical case can be found also in the originalarticles Lions, Temam and Wang (1992 a,b, 1995).

In the Cartesian case of emphasis, generally we first concentrate our attention on theocean. Indeed, as we will see in Section 2, the domain occupied by the ocean containscorners (in dimension two) or wedges (in dimension three); some regularity issues occurin this case which must be handled using the theory of regularity of elliptic problems innonsmooth domains (Grisvard (1985), Kozlov, Mazya and Rossmann (1997), Mazya andRossmann (1994)). For the atmosphere or the coupled atmosphere–ocean, the difficultiesare similar or easier to handle – hence most of the mathematical efforts will be devoted tothe ocean in Cartesian coordinates.

In Section 2 we describe the governing equations and derive the result of existence ofweak solutions with a different method than in the original articles Lions, Temam andWang (1992 a,b, 1995), thus allowing more generality (for the ocean, the atmosphere andthe coupled atmosphere–ocean).

In Section 3 we study the existence of strong solutions in space dimension three and twoand a wealth of other mathematical results, regularity inHm–Higher Sobolev spaces,C1–regularity, Gevrey regularity,backward uniqueness. We establish in dimension three theexistence and uniqueness of strong solutions on a limited interval of time (Section 3.1) andthen for all time (see Section 3.2). In dimension two we prove the existence and uniqueness,


for all time, of such strong solutions (see Section 3.3). Section 3.4 contains a technicalresult. In Section 3.5 we consider the two-dimensional space-periodic case and prove theexistence of solutions for all time, in allHm;m > 2. In Section 3.6 we prove the Gevreyregularity of the solutions and in Section 3.7.2 the backward uniqueness result.

Section 4 is technically very important, and many results of Sections 2 and 3 rely on it:this section contains the proof of theH 2 regularity of elliptic problems which arise in theprimitive equations. This proof relies, as we said, on the theory of regularity of solutionsof elliptic problems in nonsmooth domains. It is shown there, that the solutions to certainelliptic problems enjoy certain regularity properties (H 2 regularity, that is the functionand their first and second derivatives are square integrable); the problems corresponding tothe (horizontal) velocity, the temperature and the salinity are successively considered. Thestudy in Section 4 contains many specific aspects which are explained in details in the longintroduction to that section.

More explanations and references will be given in the introduction of or within eachsection.

As mentioned earlier, the mathematical formulation of the equations of the atmosphere,of the ocean and of the coupled atmosphere ocean were derived in the articles Lions,Temam and Wang (1992 a,b, 1995). For each of these problems, these articles also containthe proof of existence of weak solutions for all time (in dimension three with a proof whicheasily extends to dimension two). An alternative slightly more general proof of this result,is given in Section 2. Concerning the strong solutions, the proof given here of the localexistence in dimension three is based on the article by Hu, Temam and Ziane (2003). Analternate proof of this result is due to Guillén-González, Masmoudi and Rodríguez-Bellido(2001). In dimension two, the proof of existence and uniqueness of strong solutions, for alltime, for the considered system of equations and boundary conditions is new, and based onan unpublished manuscript of Ziane (2000). This result is also established, for a simplersystem (without temperature and salinity), by Bresch, Kazhikhov and Lemoine (2004).Most of the results of Sections 3.4 to 3.7.2 are due to M. Petcu, alone or in collaborationwith D. Wirosoetisno.

1.4. Summary of results for the physics oriented reader

The physics oriented reader will recognize in (2.1)–(2.5) the basic conservation laws:conservations of momentum, mass, energy and salt for the ocean, equation of state.In (2.6) and (2.7) appears the simplification due to the Boussinesq approximation, andin (2.11)–(2.16) the simplifications resulting from the hydrostatic balance assumption.Hence (2.11)–(2.16) are the PEs of the ocean. The PEs of the atmosphere appear in(2.116)–(2.121), and those of the coupled atmosphere and ocean are described in Sec-tion 2.5. Concerning, to begin, the ocean, the first task is to write these equations, sup-plemented by the initial and boundary conditions, as an initial value problem in a phasespaceH of the form

dU

dtCAU CB.U;U /CE.U /D `; (1.1)

U.0/D U0; (1.2)

8 M. Petcu et al.

whereU is the set of prognostic variables of the problem, that is the horizontal velocityvD .u; v/, the temperatureT and the salinityS;U D .v; T;S/; see (2.66). The phase spaceH consists, for its fluid mechanics part, of (horizontal) vector fields with finite kinetic en-ergy. We then study the stationary solutions of (1.1) in Section 2.2.2 and, in Theorem 2.2,we prove the existence for all times of weak solutions of (1.1) and (1.2), which are so-lutions inL1.0; t1IL2/ andL2.0; t1IH 1/ (bounded kinetic energy and square integrableenstrophy for the fluid mechanics part). A parallel study is conducted for the atmosphereand the coupled atmosphere–ocean in Sections 2.4 and 2.5. Section 4 is mathematicallyvery important although technical.

For the physics oriented reader the most important results are those of Section 2 and 3.Section 2 contains the "weak" formulation of the primitive equations and show the exten-sive use of the balance of energy principles to prove them. The tools of balance of energyare also those needed for the study of stability of numerical results and they are thereforeboth physically and computationally revelant.

The main results of Section 3 are the existence and uniqueness of strong solutions forall time, now both available in space dimensions two and three. Noteworthy also in thissection are the results concerning the Gevrey regularity of the solutions which implies inparticular an exponential decay of the Fourier coefficients, results that have been used inthe recent article by Temam and Wirosoetisno (2007) to prove that the primitive equationscan be approximated by a finite-dimensional model up to an exponentially small error. Theresults of existence and uniqueness for all time of strong solutions are also important forthe study of the dynamical system generated by the primitive equation (attractors, etc....);see the first developments of this theory in the article by Ju (2007), and quoted thereinsome previous partial results.

The study presented in this chapter is only a small part of the mathematical problems ongeophysical flows, but we believe it is an important part. We did not try to produce herean exhaustive bibliography. Further mathematical references on geophysical flows will begiven in the text; see also the bibliography of the articles and books that we quote. Thereis also of course a very large literature in the physical context; we only mentioned some ofthe books which were very useful to us such as Haltiner and Williams (1980), Pedlovsky(1987), Trenberth (1992), Washington and Parkinson (1986), Zeng (1979).

The mathematical theory presented in this article focuses on questions of existence,uniqueness and regularity of solutions, the so-called issue of well-posedness. From thegeophysics point of view, these issues relate, according to J. von Neumann (1963) to theshort term forecasting. The other issues as described in von Neumann (1963) relate to thelong term climate and intermediate climate dynamics. Pertaining to the long term climatesare the questions ofattractors for the Primitive Equations which have been addressed ine.g. Lions, Temam and Wang (1992a), Lions, Temam and Wang (1992b), Lions, Temamand Wang (1993), Lions, Temam and Wang (1995), Ju (2007); see also the referencestherein. For intermediate climate dynamics the mathematical issues relate to successivebifurcations, transition and instabilities; see e.g. Ma and Wang (2005b), Ma and Wang(2005a), and the article Simonnet,Dijkstra and Ghil (2008) in this volume.

Beside the efforts of the authors, we mention in several places that this study is basedon joint works with Lions, Wang, Hu, Petcu, Ziane and others. Their help is gratefullyacknowledged and we pay tribute to the memory of Jacques-Louis Lions. The authors wish


to thank Denis Serre and Shouhong Wang for their careful reading of an earlier versionof this manuscript and for their numerous comments which significantly improved themanuscript. They extend also their gratitude to Daniele Le Meur and Teresa Bunge whotyped significant parts of the manuscript.

This article is an updated version of the article by Temam and Ziane (2004). It is includedin this volume by invitation of P.G. Ciarlet, editor of the Handbook of Numerical Analysis.The authors thank P.G. Ciarlet for his invitation and the Elsevier Company for endorsingit.

2. The primitive equations: Weak formulation, existence of weak solutions

As explained in the introduction to this chapter, our aim in this section is first to presentthe derivation of the PEs from the basic physical conservation laws. We then describe thenatural boundary conditions. Then, on the mathematical side, we introduce the functionspaces and derive the mathematical formulation of the PEs. Finally we derive the existencefor all time of weak solutions.

We successively consider the ocean, the atmosphere and the coupled atmosphere–ocean.

2.1. The primitive equations of the ocean

Our aim in this section is to describe the PEs of the ocean (see Section 2.1.1), we thendescribe the corresponding boundary conditions and the associated initial and boundaryvalue problems (Section 2.1.2).

2.1.1. The primitive equations. Generally speaking, it is considered that the ocean ismade up of a slightly compressible fluid with Coriolis force. The full set of equations of thelarge-scale ocean are the following: the conservation of momentum equation, the continuityequation (conservation of mass), the thermodynamics equation (that is the conservation ofenergy equation), the equation of state and the equation of diffusion for the salinityS :

�dV3dtC 2��V3Cr3pC �gDD; (2.1)

d�

dtC �div3V3 D 0; (2.2)

dT

dtDQT ; (2.3)

dS

dtDQS ; (2.4)

�D f .T;S;p/: (2.5)

HereV3 is the three-dimensional velocity vector,V3 D .u; v;w/, �,p, T are the density,pressure and temperature, andS is the concentration of salinity;gD .0; 0; g/ is the gravity

10 M. Petcu et al.

vector,D is the molecular dissipation,QT andQS are the heat and salinity diffusions.The analytic expressions ofD, QT andQS will be given below. We denote by�3, r3,div3, the three-dimensional Laplacian, gradient and divergence, leaving�, r; div to theirtwo-dimensional versions more frequently used.

The Boussinesq approximation.From both the theoretical and the computational pointsof view, the above systems of equations of the ocean seem to be too complicated to study.So it is necessary to simplify them according to some physical and mathematical con-siderations. The Mach number for the flow in the ocean is not large and therefore, as astarting point, we can make the so-calledBoussinesq approximationin which the densityis assumed constant,�D �0, except in the buoyancy term and in the equation of state.

This amounts to replacing (2.1) and (2.2) by

�0dV3dtC 2�0��V3Cr3pC �gDD; (2.6)

div3V3 D 0: (2.7)

Consider the spherical coordinate system.�;�; r/, where� (��=2 < � < �=2) standsfor the latitude on the Earth,� (06 � 6 2�) on the longitude of the Earth,r for the radialdistance, andz D r � a for the vertical coordinate with respect to the sea level, and lete� ;e� ;er be the unit vectors in the� -, �- andz-directions, respectively. Then we write thevelocity of the ocean in the form

V3 D v�e� C v�e� C vrer D vCw; (2.8)

wherevD v�e� C v�e� is the horizontal velocity field andw is the vertical velocity.Another common simplification is to replace, to first order,r by the radiusa of the Earth.

This is based on the fact that the depth of the ocean is small compared with the radius ofthe Earth. In particular,

d

dtD @

@tC v�

r

@

@�C v�

r cos�

@

@�C vr

@

@r(2.9)

becomes

d

dtD @

@tC v�

a

@

@�C v�

a cos�

@

@�C vz

@

@z; (2.10)

and, taking the viscosity into consideration, we obtain the equations of the large-scaleocean with Boussinesq approximation, which are simply calledBoussinesq equations ofthe ocean(BEs), i.e., equations (2.11)–(2.16) hereafter (for the equation of state (2.16),see Remark 2.1):

@v@tCrvvCw@v

@zC 1

�0rpC 2�sin� � v��v�v� �v

@2v@z2D 0; (2.11)


@w

@tCrvwCw

@w

@zC 1

�0

@p

@zC �

�0g ��v�w � �v

@2w

@z2D 0; (2.12)

divvC @w

@zD 0; (2.13)

@T

@tCrvT Cw

@T

@z��T�T � �T

@2T

@z2D 0; (2.14)

@S

@tCrvS Cw

@S

@z��S�S � �S

@2S

@z2D 0; (2.15)

�D �0�1� ˇT .T � Tr/C ˇS .S � Sr/

�; (2.16)

wherev is the horizontal velocity of the water,w is the vertical velocity, and,Tr; Sr are aver-aged (or reference) values ofT andS . The diffusion coefficients�v;�T ;�S and�v; �T ; �Sare different in the horizontal and vertical directions, accounting for some eddy diffusionsin the sense of Smagorinsky (1963).

The differential operators are defined as follows. The (horizontal) gradient operatorgradDr is defined by

gradpDrpD 1

a

@p

@�e� C

1

a cos�

@p

@�e� : (2.17)

The (horizontal) divergence operator divDr� is defined by

div.v�e� C v�e�/Dr � vD1

a cos�

�@.v� cos�/

@�C @v�

@�

�: (2.18)

The derivativesrv Qv andrveT of a vector functionQv and a scalar functioneT (covariantderivatives with respect tov) are

rv QvD�v�

a

@ Qv�@�C v�

a cos�

@ Qv�@�� v� Qv�

acot�

�e�

C�v�

a

@ Qv�@�C v�

a cos�

@ Qv�@�� Qv�v�

atan�

�e� ; (2.19)

rveT Dv�

a

@eT@�C v�

a cos�

@eT@�: (2.20)

Moreover, we have used the same notation� to denote the Laplace–Beltrami operatorsfor both scalar functions and vector fields onS2a , the two-dimensional sphere of radiusacentered at 0. More precisely, we have

�T D 1

a2 cos�

�@

@�

�cos�

@T

@�

�C 1

cos�

@2T

@�2

�; (2.21)

�vD�.v�e� C v�e�/

12 M. Petcu et al.

D��v� �

2sin�

a2 cos2 �

@v�

@�� v�

a2 cos2 �

�e�

C��v� �

2sin�

a2 cos2 �

@v�

@�� v�

a2 cos2 �

�e� ; (2.22)

where in (2.22),�v� ;�v� are defined by (2.21), and in (2.21),T is any given (smooth)function onS2a the two-dimensional sphere of radiusa.

REMARK 2.1. Generally speaking, the equation of state for the ocean is given by (2.5).Only empirical forms of the function� D f .T;S;p/ are known (see Washington andParkinson (1986), pp. 131–132). This equation of state is generally derived on a phe-nomenological basis. It is natural to expect that� decreases ifT increases and that�increases ifS increases.

The simplest law is (2.16) corresponding to a liberalization around average (or reference)values�0; Tr; Sr of the density, the temperature and the salinity,ˇT andˇS are positiveconstant expansion coefficients. Much of what follows extends to more general nonlinearequations of state.

REMARK 2.2. The replacement ofr by (2.10) in the differential operators implies achange of metric inR3, where the usual metric is replaced by that ofS2a � R, S2a thetwo-dimensional sphere of radiusa centered atO .

REMARK 2.3. In a classical manner, the Coriolis force2�� V3 produces the term2�sin�k � v and a horizontal gradient term which is combined with the pressure, so thatp in (2.11) is the so-calledaugmented pressure.

The hydrostatic approximation.It is known that for large-scale ocean, the horizontal scaleis much bigger than the vertical one (5–10 km versus a few thousands km’s). Therefore,the scale analysis (see Pedlovsky (1987)) shows that@p=@z and�g are the dominant termsin (2.12), leading to the hydrostatic approximation

@p

@zD��g; (2.23)

which then replaces (2.12). The approximate relation is highly accurate for the large-scaleocean and it is considered as a fundamental equation in oceanography. From the mathe-matical point of view, its justification relies on tools similar to those used in Section 4.1.

The rigorous mathematical justification of the hydrostatic approximation is given inAzérad and Guillén (2001). In this paper the authors studied the asymptotic behavior ofthe incompressible Navier-Stokes equations when the depth goes to zero and they provedthat the solutions of the Navier-Stokes equations converge to a weak solution of the primi-tive equations. The mathematical details will not be discussed in this chapter; see howeverAzérad and Guillén (2001) as well as Remark 4.1 in Section 4.1.


Using the hydrostatic approximation, we obtain the following equations called theprim-itive equations of the large-scale ocean(PEs):

@v@tCrvvCw@v

@zC 1

�0rpC 2�sin�k � v��v�v� �v

@2v@z2D Fv; (2.24)

@p

@zD��g; (2.25)

divvC @w

@zD 0; (2.26)

@T

@tCrvT Cw

@T

@z��T�T � �T

@2T

@z2D FT ; (2.27)

@S

@tCrvS Cw

@S

@z��S�S � �S

@2S

@z2D FS ; (2.28)

�D �0�1� ˇT .T � Tr/C ˇS .S � Sr/

�: (2.29)

Note thatFv, FT andFS corresponding to volumic sources (of horizontal momentum,heat and salt), vanish in reality; they are introduced here for mathematical generality. Wealso set� D �k, wherek is the unit vector in the direction of the poles (from south tonorth).

REMARK 2.4. At this stage the unknown functions can be divided into two sets. Thefirst one, called theprognostic variables, v; T;S (4 scalar functions); we aim to write thePEs as an initial (boundary value problem) for these unknowns, and we setU D .v; T;S/.The second set of variables comprisesp;�;w; they are called thediagnostic variables. InSection 2.1.2, we will see how, using the boundary condition, one can, at each instant oftime, express the diagnostic variables in terms of the prognostic variables (a fact which isalready transparent for� in (2.29).

REMARK 2.5. We integrate (2.28) over the domainM occupied by the fluid which isdescribed in Section 2.1.2. Using then the Stokes formula, and taking into account (2.26)and the boundary conditions (also described in Section 2.1.2) we arrive at

d

dt

Z

MS dMD

Z

MFS dMI (2.30)

henceZ

MS dM

ˇˇt

DZ

MS dM

ˇˇ0

CZ t

0

Z

MFS dMdt 0:

In practical applications,FS D 0 as we said, and the total amount of saltRM S dM is

conserved. In all cases we write

S 0 D S � 1

jMjZ

MS dM; F 0S D FS �

1

jMjZ

MFS dM; (2.31)

14 M. Petcu et al.

Fig. 1. The oceanM.

wherejMj is the volume ofM, and we see thatS 0 satisfies the same equation (2.15), withFS replaced byF 0S . From now on, dropping the primes, we consider (2.15) as the equationfor S 0 and we thus have

Z

MS dMD 0;

Z

MFS dMD 0: (2.32)

2.1.2. The initial and boundary value problems.We assume that the ocean fills a domainM of R3 which we describe as follows (see Figure 1):

The top of the ocean is a domain�i included in the surface of the EarthSa (spherecentered at0 of radiusa). The bottom�b of the ocean is defined by.z D x3 D r � a/

z D�h.�;'/;

whereh is a function of classC2 at least onx�i ; it is assumed also thath is bounded frombelow,

0 < h6 h.�;'/6 Nh; .�; '/ 2 �i : (2.33)

The lateral surface�` consists of the part of cylinder

.�; '/ 2 @�i ;�h.�;'/6 z 6 0: (2.34)

REMARK 2.6. Let us make two remarks concerning the geometry of the ocean; the firstone is that, for mathematical reasons, the depth is not allowed to be 0.h> h > 0/, and thus“beaches” are excluded. The second one is that the top of the ocean is flat (spherical), notallowing waves; this corresponds to the so-calledrigid lid assumptionin oceanography.The assumptionh > 0 can be relaxed for some of the following results, but this will notbe discussed here. The rigid lid assumption can be also relaxed by the introduction of anadditional equation for the free surface but this also will not be considered.

Boundary conditions. There are several sets of natural boundary conditions that one canassociate to the primitive equations; for instance the following:


On the top of the ocean�i .z D 0/:

�v@v@zC ˛v

�v� va

�D �v; wD 0;

�T@T

@zC ˛T

�T � T a

�D 0; (2.35)

@S

@zD 0:

At the bottom of the ocean�b .z D�h.�;'//:

vD 0; wD 0;(2.36)

@T

@nTD 0; @S

@nSD 0:

On the lateral boundary�` .�h.�;'/ < z < 0; .�;'/ 2 @�i/:

vD 0; wD 0; @T

@nTD 0; @S

@nSD 0: (2.37)

HerenD .nH ; nz/ is the unit outward normal on@M decomposed into its horizontaland vertical components; the co-normal derivatives@=@nT and@=@nS are those associatedwith the linear (temperature and salinity) operators, that is,

@

@nTD �T

1C jrhj2rrhC�

�T

1C jrhj2 C .�T ��T /�@

@z;

(2.38)@

@nSD �S

1C jrhj2rrhC�

�S

1C jrhj2 C .�S ��S /�@

@z;

whererrh is the (two-dimensional) covariant derivative in the direction ofrh (see e.g. inLions, Temam and Wang (1993) after (1.21) and after (3.27)).

REMARK 2.7. (i) The boundary conditions (which are the same) on�b and�` expressthe no-slip boundary conditions for the water and the absence of fluxes of heat or salt.For�i ;wD 0 is the geometrical (kinematical) boundary condition required by the rigid lidassumption; the Neumann boundary condition onS expresses the absence of salt flux.

(ii) In general, the boundary conditions onv andT on�i are not fully settled from thephysical point of view. These above correspond to some resolution of the viscous boundarylayers on the top of the ocean. Here˛v and˛T are given> 0;va andT a correspond to thevalues in the atmosphere and�v corresponds to the shear of the wind.

(iii) The first boundary condition (2.35) could be replaced byvD va expressing a no-slip condition between air and sea. However such a boundary condition necessitating anexact resolution of the boundary layer would not be practically (computationally) real-istic, and as indicated in (ii) we use instead some classical resolution of the boundarylayer (Schlichting (1979)).

16 M. Petcu et al.

(iv) As we said the boundary condition of�i are standard unless more involved inter-actions are taken into consideration. However for�b and�` different combinations of theDirichlet and Neumann boundary conditions can be (have been) considered; Lions, Temamand Wang (1992b).

Beta-plane approximation.For mid-latitude regional studies it is usual to consider thebeta-plane approximation of the equations in whichM is a domain in the spaceR3with Cartesian coordinates denotedx;y; z or x1; x2; x3. In the beta-plane approxima-tion, � D 2f k; f D f0 C ˇy, k the unit vector along the south to north poles,� D@2=@x2 C @2=@y2;r is the usual nabla vector.@=@x; @=@y/ andrv D u@=@x C v@[email protected] D .u; v//. With these notations, the equations (2.24)–(2.29) and the boundary condi-tions (2.35)–(2.38) keep the same form; here the depthhD h.x;y/ satisfies, like (2.33),

0 < h6 h.x;y/6 Nh; (2.39)

and the boundary ofM consists of�i ; �b; �`, defined as before.As indicated in the Introduction, we will emphasize in this chapter the regional model

which is slightly simpler, in particular because of the use of Cartesian coordinates. Usuallythe general model in spherical coordinates simply requires the treatment of lower-orderterms.

From now on we consider the regional (Cartesian coordinate) case.

The diagnostic variables.The first step in the mathematical formulation of the PEs con-sists in showing how to express the diagnostic variables in terms of the prognostic vari-ables, thanks to the equations and boundary conditions.

SincewD 0 on�i and�b, integration of (2.26) inz gives

wDw.v/DZ 0

z

divvdz0 (2.40)

and

Z 0

�hdivvdz D 0: (2.41)

Note that

divZ 0

�hvdz D

Z 0

�hdivvdzCrh � v

ˇˇzD�h

;

and sincev vanishes on�b, condition (2.41) is the same as

divZ 0

�hvdz D 0: (2.42)


Similarly, integration of (2.25) inz gives

pD psCP; P D P.T;S/D gZ 0

z

�dz0: (2.43)

Here� is expressed in terms ofT andS through (2.29) andpsD ps.x;y; t/D p.x;y; 0; t/is the pressure at the surface of the ocean.

Hence (2.40) and (2.43) provide an expression of the diagnostic variables in terms of theprognostic variables (and the surface pressure), and (2.42) is an additional equation which,we will see, is mathematically related to the surface pressure.

REMARK 2.8. The introduction of the nonlocal constraint (2.41) and of the surface pres-sureps was first carried out in Lions, Temam and Wang (1992 a,b). This new formulationhas played a crucial role in much of the mathematical analysis of the PEs in various cases.

2.2. Weak formulation of the PEs of the ocean. The stationary PEs

We denote byU the triplet.v; T;S/ (four scalar functions). In summary the equations thatwe consider for the subsequent mathematical theory (the PEs) are (2.24), (2.27) and (2.28),with w D w.v/ given by (2.40), andp given by (2.43) (� given by (2.29)); furthermorev satisfies (2.41); hence

@v@tCrvvCw@v

@zC 1

�0rpC 2f k � v��v�v� �v

@2v@z2D Fv; (2.44)

@T

@tCrvT Cw

@T

@z��T�T � �T

@2T

@z2D FT ; (2.45)

@S

@tCrvS Cw

@S

@z��S�S � �S

@2S

@z2D FS ; (2.46)

wDw.v/DZ 0

z

divvdz0; (2.47)

divZ 0

�hvdz D 0; (2.48)

pD psCP; P D P.T;S/D gZ 0

z

�dz0; (2.49)

�D �0�1� ˇT .T � Tr/C ˇS .S � Sr/

�; (2.50)

Z

MS dMD 0: (2.51)

The boundary conditions are (2.35)–(2.38).

18 M. Petcu et al.

2.2.1. Weak formulation and functional setting.For the weak formulation of this prob-lem, we introduce the following function spacesV andH :

V D V1 � V2 � V3; H DH1 �H2 �H3;

V1 D�

v 2H 1.M/2;divZ 0

�hv dz D 0;vD 0 on�b[ �`

�;

V2 DH 1.M/;

V3 D PH 1.M/D�S 2H 1.M/;

Z

MS dMD 0

�;

H1 D�

v 2L2.M/2;divZ 0

�hv dz D 0;nH �

Z 0

�hvdz D 0 on@�i .i.e., on�`/

�;

H2 DL2.M/;

H3 D PL2.M/D�S 2L2.M/;

Z

MS dMD 0

�:

These spaces are endowed with the following scalar products and norms:

��U;eU ��D ��v; Qv��

1CKT

��T;eT ��

2CKS

��S;eS ��

3;

��v; Qv��

1DZ

M

��vrv � r QvC �v

@v@z

@Qv@z

�dM;

��T;eT ��

2DZ

M

��TrT � reT C �T

@T

@z

@eT@z

�dMC

Z

�i

˛T TeT d�i ;

��S;eS ��

3DZ

M

��SrS � reS C �S

@S

@z

@eS@z

�dM;

�U;eU �

HDZ

M

�v � QvCKT TeT CKSSeS

�dM;

kU k D �.U;U /�1=2; jU jH D .U;U /1=2H :

HereKT andKS are suitable positive constants chosen below. The norm onH is ofcourse equivalent to theL2-norm and because of the Poincaré inequality,v vanishing on�b [ �`, and (2.51),k � ki D ..�; �//1=2i is a Hilbert norm onVi , andk � k is a Hilbert normonV ; more precisely we have, withc0 > 0 a suitable constant depending onM:

jU jH 6 c0kU k 8U 2 V: (2.52)


Let V1 be the space ofC1 (two-dimensional) vector functionsv which vanish in aneighborhood of�b[ �` and such that

divZ 0

�hvdz D 0:

ThenV1 � V1 and it has been shown in Lions, Temam and Wang (1992a) that

V1 is dense inV1: (2.53)

We also denote byV2 � V2 the set ofC1 functions on xM and byV3 � V3 the set ofC1functions on xM with zero average;V D V1 � V2 � V3 is dense inV .

To derive the weak formulation of this problem we consider a sufficiently regular testfunctioneU D .Qv;eT ;eS/ in V . We multiply (2.44) byQv, the second one byKTeT , the thirdone byKSeS , integrate overM and add the resulting equations;KT ;KS > 0 are twoconstants to be chosen later on.

The term involving gradpS vanishes; indeed, by the Stokes formula,

Z

MrpS � Qv dMD

Z

@MpSnH � Qvd.@M/�

Z

MpSr � Qv dM;

wherenD .nH ; nz/ is the unit outward normal on@M, andnH its horizontal component.The integral on@M vanishes becausenH � Qv vanishes on@M; the remaining integral onMvanishes too since by Fubini’s theorem, (2.48), andQvD 0 on�b:

Z

MpSr � Qv dMD

Z

�i

pS

Z 0

�hr � Qvdz d�i D

Z

�i

pS

�r �

Z 0

�hQvdz

�d�i D 0:

Using Stokes’ formula and the boundary conditions (2.35)–(2.38) we arrive after someeasy calculations at

�d

dtU;eU

�

H

C a�U;eU �C b�U;U;eU �C e�U;eU �D `�eU �: (2.54)

The notations are as follows:

�U;eU �

HDZ

M

�v � QvCKT TeT CKSSeS

�dM;

a�U;eU �D a1

�U;eU �CKT a2

�U;eU �CKSa3

�U;eU �;

a1�U;eU �D

Z

M


@v@z

@Qv@z

�dM

�Z

MxP .T;S/r � Qv dMC

Z

�i

˛vvQvd�i ;

20 M. Petcu et al.

xP .T;S/D gZ 0

z

.�ˇT T C ˇSS/dz0�see (2.49) and (2.50)

�;

a2�U;eU �D

Z

M


@T

@z

@eT@z

�dMC

Z

�i

˛T TeT d�i ;

a3�U;eU �D

Z

M

��SrS � reS C �S

@S

@z

@eS@z

�dM;

b D b1CKT b2CKSb3;

b1�U;eU ;U ]�D

Z

M

�v � r QvCw.v/@Qv

@z

�v] dM;

b2�U;eU ;U ]�D

Z

M

�v � reT Cw.v/@

eT@z

�T ] dM;

b3�U;eU ;U ]�D

Z

M

�v � reS Cw.v/@

eS@z

�S] dM;

e�U;eU �D 2

Z

M.f k � v/ � QvdM

and

`�eU �D

Z

M

�Fv QvCKTFTeT CKSFSeS

�dM

CZ

M

�g

Z 0

z

.1C ˇT Tr � ˇSSr/dz0�r � QvdM (2.55)

CZ

�i

�.gv/ � QvC gTeT

�d�i ;

where (see (2.35))

gv D �vC ˛vva; gT D ˛T T a:

For ` we observe that, ifTr andSr are constant, then

Z

M

�g

Z 0

z

.1C ˇT Tr � ˇSSr/dz0�r � vdM

DZ

@M

�g

Z 0

z

.1C ˇT Tr � ˇSSr/dz0�nH � vd.@M/

D 0: (2.56)


It is clear that eachai , and thusa, is a bilinear continuous form onV ; furthermore ifKTandKS are sufficiently large,a is coercive (a2; a3 are automatically coercive onV2; V3):

a.U;U /> c1kU k2 8U 2 V .c1 > 0/: (2.57)

Similarly e is bilinear continuous onV1 and evenH1, and

e.U;U /D 0 8U 2H: (2.58)

Before studying the properties of the formb, we introduce the spaceV.2/:

V.2/ is the closure ofV in�H 2.M/

�4: (2.59)

Then we have the following

LEMMA 2.1. The formb is trilinear continuous onV � V � V.2/ andV � V.2/ � V ,1

ˇb�U;eU ;U ]�

ˇ

6

8<:c2kU k

eU U ]

V.2/

; 8U;eU 2 V;U ] 2 V.2/;c2kU k

eU V.2/

U ] ; 8U;U ] 2 V;eU 2 V.2/;

(2.60)

or

ˇb�U;eU ;U ]�

ˇ

6 c2kU kˇeUˇ1=2H

eU 1=2 U ]

V.2/

; 8U;eU 2 V;U ] 2 V.2/: (2.61)

Furthermore,

b�U;eU ;eU �D 0 for U 2 V;eU 2 V.2/; (2.62)

and

b�U;eU ;U ]�D�b�U;U ];eU � (2.63)

for U;eU ;U # 2 V , andeU or U ] in V.2/.

1For (2.60) and (2.61), the specific form ofV andV.2/ is not important:b is as well trilinear continuous onH1.M/4 �H2.M/4 �H1.M/4 andH1.M/4 �H1.M/4 �H2.M/, and the estimates are similar, theH1 andH2 norms replacing theV andV.2/ norms.

22 M. Petcu et al.

PROOF. To show first thatb is defined onV �V �V.2/ let us consider the typical and mostproblematic term

Z

Mw.v/

@eT@zT ] dM: (2.64)

We have

Z

M

ˇˇw.v/@

eT@zT ]ˇˇdM6

ˇw.v/

ˇL2.M/

ˇˇ@eT@z

ˇˇL2.M/

ˇT ]ˇL1.M/

:

The first two terms in the right-hand side of this inequality are bounded byconst� kvk1(using (2.40)) andkeT k2. In dimension three,H 2.M/ � L1.M/ so that the third termis bounded byconst� kT ]kV.2/ , and hence the right-hand side of the last inequality isbounded by

ckU k eU

U ] V.2/

:

With similar (and easier) inequalities for the other integral, we conclude thatb is definedand trilinear continuous onV � V � V.2/.

For the continuity onV � V.2/ � V , the typical term above is bounded by

ˇw.v/

ˇL2.M/

ˇˇ@eT@z

ˇˇL4.M/

ˇT ]ˇL4.M/

;

which is bounded by

ckvk1 eT H2

T ] H16 ckU k

eU V.2/

U ] ;

sinceH 1.M/�L6.M/I hence the second bound (2.60).We easily prove (2.62) and (2.63) by integration by parts forU;eU ;U ] 2 V ; the relations

are then extended by continuity to the other cases, using (2.60).To establish the improvement (2.61) of the first inequality (2.60), we observe that

b.U;eU ;U ]/ D �b.U;U ];eU / and consider again the most typical termRMw.v/.@U ]=

@z/� eU dM, that we bound by

ˇw.v/

ˇL2

ˇˇ@U ]@z

ˇˇL6

ˇeUˇL3:

Remembering thatH 1 � L6 andH 1=2 � L3 in space dimension three, we bound thisterm by

ckvk U ]

V.2/

ˇeUˇ1=2 eU

1=2;

and (2.61) follows.


The operator form of the equation.We can write equation (2.54) in the form of an evolu-tion equation in the Hilbert spaceV 0

.2/. For that purpose we observe that we can associate

to the formsa; b; e above, the following operators:� A linear continuous fromV into V 0, defined by

˝AU;eU ˛D a�U;eU � 8U;eU 2 V;

� B bilinear continuous fromV � V into V 0.2/

defined by

˝B�U;eU �;U ]˛D b�U;eU ;U ]� 8U;eU 2 V; 8U ] 2 V.2/;

� E linear continuous fromH into itself, defined by

˝E.U /;eU ˛D e�U;eU � 8U;eU 2H:

SinceV.2/ � V � H , with continuous injections, each space being dense in the nextone, we also have the Gelfand–Lions inclusions

V.2/ � V �H � V 0 � V 0.2/: (2.65)

With this we see that (2.54) is equivalent to the following operator evolution equation

dU


understood inV 0.2/

and with` defined in (2.55). To this equation we will naturally add aninitial condition:

U.0/DU0: (2.67)�

2.2.2. The stationary PEs. We now establish the existence of solutions of the station-ary PEs. Beside its intrinsic interest, this result will be needed in the next section for thestudy of the time dependent case.

The equations to be considered are the same as (2.54), with the only difference that thederivatives@v=@t , @T=@t and@S=@t are removed, and that the source termsFv;FT ;FSare given independent of timet .

The weak formulation proceeds as before:

GivenF D .Fv;FT ;FS / in H�orL2.M/4

�, and

g D .gv; gT / in L2.�i/3, find U D .v; T;S/ 2 V; such that

a�U;eU �C b�U;U;eU �C e�U;eU �D `�eU �; for everyeU 2 V.2/I

(2.68)

a; b; e and` are the same as above.We have the following result.

24 M. Petcu et al.

THEOREM 2.1. We are givenF D .Fv;FT ;FS / in L2.M/4 (or in H ), andgD .gv; gT /

in L2.�i/3; then problem(2.68)possesses at least one solutionU 2 V such that

kU k6 1

c1k`kV 0 : (2.69)

PROOF. The proof of existence is done by Galerkin method, a priori estimates and passageto the limit. The proof is essentially standard, but we give the details because of somespecificities in this case.

We consider a family of elementsf j gj of V.2/ which is free and total inV (V.2/ isdense inV ); and for eachm 2 N, we look for an approximate solution of (2.68),Um DPmjD1 �jm j , such that

a.Um;ˆk/C b.Um;Um;ˆk/C e.Um;ˆk/D `.ˆk/; k D 1; : : : ;m: (2.70)

The existence ofUm is shown below. An a priori estimate onUm is obtained by multiply-ing each equation (2.70) by�km and summing fork D 1; : : : ;m. This amounts to replacingˆk byUm in (2.70); sinceb.Um;Um;Um/D 0 by Lemma 2.1, we obtain

a.Um;Um/D `.Um/

and, with (2.57),

c1kUmk2 6 k`kV 0kUmk;

kUmk61

c1k`kV 0 : (2.71)

From (2.71) we see that there existsU 2 V and a subsequenceUm0 , such thatUm0 ,converges weakly toU asm0!1. Since we cannot replaceeU byU in (2.68) it is usefulto notice that

kU k6 lim infm0!1

Um0 6 1

c1k`kV 0 ;

so that (2.69) is satisfied. Then we pass to the limit in (2.70) written withm0, andk fixedless than or equal tom0. We observe below that

b.Um0 ;Um0 ;ˆk/! b.U;U;ˆk/; (2.72)

so that, at the limit,U satisfies (2.68) foreU Dˆk ; k fixed arbitrary; hence (2.68) is validfor anyeU linear combination of k and, by continuity (Lemma 2.1), foreU 2 V.2/.

The proof is complete after we prove the results used above. �


Convergence of theb term. To prove (2.72) we first observe, with (2.63), thatb.Um0;Um0;ˆk/D�b.Um0;ˆk ;Um0/. We also observe that each component ofUm0 converges weaklyin H 1.M/ to the corresponding component ofˆk . Therefore, by compactness, the con-vergence takes place inH 3=4.M/ strongly; by Sobolev embedding,H 3=4.M/�L4.M/

in dimension three, and the convergence holds inL4.M/ strongly. Writingˆk D ˆ D.vˆ; Tˆ; Sˆ/, the typical most problematic term is

Z

Mw.vm0/

@vˆ@z

vm0 dM: (2.73)

Since divvm0 converges weakly to divv in L2.M/;w.vm0/ converges weakly inL2.M/

tow.v/Ivm0 converges strongly tov in L4.M/ as observed before, and since@vˆ=@z be-longs toL4.M/, the term above converges to the corresponding term wherevm0 is replacedby v.U D .v; T;S//. Hence (2.69).

Existence ofUm. Equations (2.70) amount to a system ofm nonlinear equations for thecomponents of the vector� D .�1; : : : ; �m/, where we have written�jm D �j for simplicity.Existence follows from the following consequence of the Brouwer fixed point theorem.(See Lions Lions (1969).)

LEMMA 2.2. LetF be a continuous mapping ofRm into itself such that

�F.�/; ��> 0 for Œ��D k; for somek > 0; (2.74)

whereŒ�; �� and Œ�� are the scalar product and norm inRm.Then there exists� 2Rm with Œ�� < k, such thatF.�/D 0.

PROOF. If F never vanishes, thenG D�kF.�/=ŒF.�/� is continuous onRm, and we canapply the Brouwer fixed point theorem toG which maps the ballC centered at0 of radiuskinto itself. ThenG has a fixed point�0 in C and we have

�G.�0/�D Œ�0�D k;

�G.�0/; �0�D�k ŒF.�0/; �0�

ŒF.�0/�D Œ�0�2:

This contradicts the hypothesis (2.74) onF ; the lemma is proven. �

We apply this lemma to (2.70) as follows:F D .F1; : : : ;Fm/, with

Fk.�/D ŒF.Um/;ˆk �D a.Um;ˆk/C b.Um;Um;ˆk/C e.Um;ˆk/� l.ˆk/:(2.75)

26 M. Petcu et al.

The spaceRm is equipped with the usual Euclidean scalar product, so that

�F.�/; ��DmX

kD1Fk.�/�k

D a.Um;Um/C b.Um;Um;Um/� l.Um/>�with (2.57), (2.58), (2.62) and Schwarz’ inequality

�

> c1kUmk2 � k`kV 0kUmk: (2.76)

Since the last expression converges toC1 askUmk � Œ�� converges toC1, there existsk > 0 such that (2.74) holds. The existence ofUm follows.

REMARK 2.9. A perusal of the proof of Theorem 2.1 shows that we proved the followingmore general result.

LEMMA 2.3. LetV;W be two Hilbert spaces withW � V , the injection being continuous.Assume thatNa is bilinear continuous coercive onV , and that Nb is trilinear continuous onV �W � V;V � V �W , and continuous onVw � Vw �W , whereVw is V equipped withthe weak topology. Furthermore

b�U;eU ;U ]�D�b�U;U ];eU � if U;eU ;U ] 2 V andeU or U ] 2W:

Then, forNl given inV 0, there exists at least one solutionU of

Na�U;eU �C Nb�U;U;eU �D Nl�eU � 8eU 2W; (2.77)

which satisfies

Na.U;U /6 N.U /: (2.78)

Lemma 2.3 will be useful in the next section.

2.3. Existence of weak solutions for the PEs of the ocean

In this section we establish the existence, for all time, of weak solutions for the equationsof the ocean. The main result is Theorem 2.2 given at the end of the section.

We consider the primitive equations in their formulation (2.54), that is, with the notationsof Section 2.2:

Given t1 > 0, U0 in H , F D .Fv;FT ;FS / in L2.0; t1IH/, andgD .gv; gT / in L2

�0; t1IL2.�i /

�3, to find

U 2L1.0; t1IH/\L2.0; t1IV /, such that�ddtU;

eU �C a�U;eU �C b�U;U;eU �C e�U;eU �D `�eU � 8eU 2 V.2/,(2.79)


U.0/DU0: (2.80)

Alternatively, and as explained in the previous section (see (2.66) and (2.67)), we canwrite (2.79) and (2.80) in the form of an operator evolution equation

dU


U.0/D U0: (2.82)

To establish the existence of weak solutions of this problem we proceed by finite differ-ences in time.2

Finite differences in time. Given t1 > 0 which is arbitrary, we considerN an arbi-trary integer and introduce the time stepk D �t D t1=N . By time discretization of(2.79) and (2.80), we are naturally led to define a sequence of elements ofV , U n,06 n6N , defined by

U 0 DU0; (2.83)

and then, recursively fornD 1; : : : ;N , by

1

�t

�U n �U n�1;eU �

HC a�U n;eU �C b�U n;U n;eU �C e�U n;eU �

D `n�eU � 8eU 2 V.2/: (2.84)

Here`n 2 V 0 is given by

`n�eU �D 1

�t

Z n�t

.n�1/�t`�t IeU �dt; (2.85)

where`.t IeU / is defined exactly as in (2.54), the dependence of` on t reflecting now thedependence ont of F;gv andgT .

The existence, for alln, of U n 2 V solution of (2.84) follows from Lemma 2.3, equa-tion (2.84) being the same as (2.77); the notations are obvious and the verification of thehypotheses of the lemma is easy; furthermore by (2.78), and after multiplication by2�t :

ˇU nˇ2H�ˇU n�1

ˇ2HCˇU n �U n�1

ˇ2HC 2�ta�U n;U n�

6 2�t`n�U n�: (2.86)

2At this level of generality it has not been possible to prove the existence of weak solutions to the PEs by anyother classical method for parabolic equations. In particular, the proofs in the articles Lions, Temam and Wang(1992 a,b, 1995) based on the Galerkin method, assume theH2 regularity of the solutions of the GFD–Stokesproblem, and this result is not available at this level of generality. We recall that the whole Section 4 is devoted tothis regularity question.

28 M. Petcu et al.

For (2.86), we also used (2.58) and the elementary relation

2�eU �U ];eU �

HDˇeUˇ2H�ˇU ]ˇ2HCˇeU �U ]

ˇ2H8eU ;U ] 2H: (2.87)

A priori estimates. We now proceed and derive a priori estimates for theU n and then forsome associated approximate functions.

Using (2.85), (2.54) and Schwarz’ inequality, we bound�t `n.U n/ by�t1=2.�n/1=2 �kU nk with

�n DZ n�t

.n�1/�t�.t/dt;

�.t/D c01�ˇF.t/

ˇ2HCZ

M

ˇ1C ˇT Tr.t/� ˇSSr.t/

ˇ2dM (2.88)

CZ

�i

�ˇgv.t/

ˇ2CˇgT .t/

ˇ2�d�i

�;

wherec01 is an absolute constant related toc0 (see (2.52)). Hence using also (2.57), weinfer from (2.86) that



ˇ2HC 2�tc1

U n 2

6 2�t1=2��n�1=2 U n

6�tc1 U n

2C c�11 �n:

Hence



ˇ2HC�tc1

U n 2

(2.89)6 c�11 �n for nD 1; : : : ;N:

Summing all these relations fornD 1; : : : ;N , we find

ˇUN

ˇ2HC

NX

nD1

�ˇU n �U n�1

ˇ2HC�tc1

U n 2�6 �1; (2.90)

with

�1 D jU0j2C1

c1

Z t1

0

�.t/dt:

Summing the relations (2.89) fornD 1; : : : ;m, with m fixed,16m6N , we obtain aswell

ˇUm

ˇ2H6 �1 8mD 0; : : : ;N: (2.91)


Approximate functions. The subsequent steps follow closely the proof in Temam (1977),Chapter 3, Section 4, for the Navier–Stokes equations, and we will skip many details.3

We first introduce the approximate functions defined as follows on.0; t1/ .k D�t/; forthe sake of simplicity we assume thatU0 2 V instead ofH; but this is not necessary:

Uk W .0; t1/‘ V; Uk.t/DU n; t 2 �.n� 1/k;nk�;`k W .0; t1/‘ V 0; `k.t/D `n; t 2 �.n� 1/k;nk�;eU k W .0; t1/‘ V;

eU k is continuous, linear on each interval�.n� 1/k;nk� and

eU k.nk/DU n; nD 0; : : : ;N:

An easy computation (see Temam (1977)) shows that:

ˇUk � eU k

ˇL2.0;t1IH/ 6

�k

3

�1=2 NX

nD1

ˇU n �U n�1

ˇ2H

!1=2; (2.92)

and we infer from (2.90) and (2.91) that

Uk andeU k are bounded independentlyof �t in L1.0; t1IH/ andL2.0; t1IV /. (2.93)

We infer from (2.92) and (2.93) that there existsU 2 L1.0; t1IH/\L2.0; t1IV /, anda subsequencek0! 0, such that, ask0! 0,

Uk0 andeU k0*U in L1.0; t1IH/weak star and inL2.0; t1IV / weakly,

(2.94)

Uk0 � eU k0! 0 in L2.0; t1IH/ strongly: (2.95)

Further a priori estimates and compactness.With the notations above and those used for(2.66) and (2.67) (or (2.81) and (2.82)), we see that the scheme (2.84) can be rewritten as

deU kdtCAUk CB.Uk ;Uk/CE.Uk/D `k ; 0 < t < t1; (2.96)

eU k.0/DU0: (2.97)

From (2.93) and (2.61) we see thatB.Uk ;Uk/ is bounded inL4=3.0; t1IV 0.2//; since the

other terms in (2.96) are bounded inL2.0; t1IV 0/ independently ofk, we conclude that

deU kdt

is bounded inL4=3�0; t1IV 0.2/

�: (2.98)

3The proof given here would apply to the Navier–Stokes equations in space dimensiond > 4; it extends theproof given in Temam (1977) which is only valid for the Navier–Stokes equations in dimensiond D 2 or 3.

30 M. Petcu et al.

We then infer from (2.93) and the Aubin compactness theorem (see, e.g., Temam(1977)), that ask0! 0,

eU k0!U in L2.0; t1IH/ strongly; (2.99)

and the same is true forUk because of (2.95).

Passage to the limit. The passage to the limitk0! 0 .k D�t/ follows now closely thatof Theorem 4.1, Chapter 2 in Temam (1977), we skip the details.

We considereU 2 V (which is dense inV.2/, see (2.59)) and a scalar function inC1.Œ0; t1�/, such that .t1/ D 0. We take the scalar product inH of (2.96) with eU ,integrate from 0 tot1 and integrate by parts the first term, we arrive at

�Z t1

0

�eU k ;eU�H 0 dt C

Z t1

0

�a�Uk ;eU

�C b�Uk ;Uk ;eU�C e�Uk ;eU

�� dt

D �U0;eU�H .0/C

Z t1

0

`k�eU � dt: (2.100)

We can pass to the limit in (2.100) for the sequencek0; for the b term we proceedsomehow as for (2.72). For the nonlinear term we write

Z t1

0

b�Uk ;Uk ;eU

� dt D�

Z t1

0

b�Uk ;eU ;Uk

� dt;

and, considering the typical most problematic term, we show that, ask0! 0,

Z t1

0

Z

Mw.vk/

@eT@zTk0 dMdt!

Z t1

0

Z

Mw.v/

@eT@zT dMdt I (2.101)

this follows from the fact that divvk0 converges to divv weakly inL2.M�.0; t1//, thatTk0converges toT strongly inL2.M� .0; t1//, and @eT =@z belongs toL1.M� .0; t1//.

From this we conclude thatU satisfies

�Z t1

0

�U;eU �

H 0 dt C

Z t1

0

�a�U;eU �C b�U;U;eU �C e�U;eU �� dt

D �U0;eU� .0/C

Z t1

0

`�eU � dt (2.102)

for all eU in V and all of the indicated type. Also, by continuity (Lemma 2.1), (2.101) isvalid as well for alleU in V.2/ sinceV is dense inV.2/ by (2.59).

It is then standard to infer from (2.101) thatU is solution of (2.79) and (2.80); this leadsus to the main result of this section.


THEOREM 2.2. The domainM is as before. We are givent1 > 0, U0 in H and F D.Fv;FT ;FS / in L2.0; t1IH/ or L2.0; t1IL2.M/4/; g D .gv; gT / is given inL2.0; t1IL2.�i /

3/. Then there exists

U 2L1.0; t1IH/\L2.0; t1IV /; (2.103)

which is solution of(2.79) and (2.80) (or (2.81) and (2.82)); furthermoreU is weaklycontinuous fromŒ0; t1� intoH .

2.4. The primitive equations of the atmosphere

In this section we briefly describe the PEs of the atmosphere, introduce their mathematical(weak) formulation and state without proof the existence of weak solutions; the proof isessentially the same as for the ocean.

We start from the conservation equations similar to (2.1)–(2.5). In fact, (2.1) and (2.2)are the same; the equation of energy conservation is slightly different from (2.3) becauseof the compressibility of the air; the state equation is that of perfect gas instead of (2.5);finally, instead of the concentration of salt in the water, we consider the amountq of waterin air. Hence, we have

�dV3dtC 2��V3Cr3pC �gDD; (2.104)

d�

dtC �div3V3 D 0; (2.105)

cpdT

dt� RT

p

dp

dtDQT ; (2.106)

dq

dtDQq; (2.107)

pDR�T: (2.108)

Here,D;QT andQq contain the dissipation terms. As we said the difference between(2.106) and (2.3) is due to the compressibility of the air; in (2.106),cp > 0 is the specificheat of the air at constant pressure andR is the specific gas constant for the air; (2.108) isthe equation of state for the air.

The hydrostatic approximation.We decomposeV3 into its horizontal and vertical com-ponents as in (2.8),V3 D .v;w/, and we use the approximation (2.10) of d=dt . Also, as forthe ocean, we use the hydrostatic approximation, replacing the equation of conservation ofvertical momentum (third equation (2.104)) by the hydrostatic equation (2.23). We find

@v@tCrvvCw@v

@zC 1

�rpC 2�sin� k � v��v�v� �v

@2v@z2D 0; (2.109)

32 M. Petcu et al.

@p

@zD��g; (2.110)

@�

@tC �

�rvC @w

@z

�C vr�Cw@�

@zD 0; (2.111)

@T

@tCrvT Cw

@T

@z��T�T � �T

@2T

@z2� RT

p

dp

dtDQT; (2.112)

@q

@tCrvqCw

@q

@z��q�q � �q

@2q

@z2D 0; (2.113)

pDR�T: (2.114)

The right-hand side of (2.112), which is different fromQT in (2.106) now representsthe solar heating.

Change of vertical coordinate.Since� does not vanish, the hydrostatic equation (2.110)implies thatp is a strictly decreasing function ofz, and we are thus allowed to usep as thevertical coordinate; hence in spherical geometry the independent variables are now', � ,p andt . By an abuse of notation we still denote byv; T; q; � these functions expressed inthe'; �;p; t variables. We also denote by! the vertical component of the wind, and onecan show (see, e.g., Haltiner and Williams (1980)) that

! D dp

dtD @p

@tCrvpCw

@p

@zI (2.115)

in (2.115),p is a dependent variable expressed as a function of'; �; z andt .In this context, the PEs of the atmosphere become

@v@tCrvvC! @v

@pC 2� sin� k � vCrˆ�LvvD Fv; (2.116)

@ˆ

@pC RT

pD 0; (2.117)

divvC @!

@pD 0; (2.118)

@T

@tCrvT C!

@T

@p� RxT

cpp! �LT T D FT ; (2.119)

@q

@tCrvqC!

@q

@p�Lqq D Fq ; (2.120)

pDR�T: (2.121)


We have denoted by D gz the geopotential (z is now a function of'; �;p; t/; Lv,LT andLq are the Laplace operators, with suitable eddy viscosity coefficients, expressedin the'; �;p variables. Hence, for example,

LvvD �v�vC �v@

@p

��gp

R xT

�2@v@p

�; (2.122)

with similar expressions forLT andLq . Note thatFT corresponds to the heating of thesun, whereasFv andFq which vanish in reality, are added here for mathematical gener-ality. In (2.119)T has been replaced byxT in the termRT!=cpp. See in Lions, Temamand Wang (1995) a better approximation ofRT!=cpp involving an additional term. Withadditional precautions, and using the maximum principle for the temperature as in Ewaldand Temam (2001), we could keep the exact termRT!=cpp.

The change of variable gives for@2v=@z2 a term different from the coefficient of�v. Theexpression above is a simplified form of this coefficient, the simplification is legitimatebecause�v is a very small coefficient; in particularT has been replaced byxT (known)which is an average value of the temperature.

Pseudo-geometrical domain.For physical and mathematical reasons, we do not allow thepressure to go to zero, and assume thatp > p0, with p0 > 0 “small”. Physically, in the veryhigh atmosphere (p very small), the air is ionized and the equations above are not validanymore; mathematically, withp > p0, we avoid the appearance of singular terms as, forexample, in the expressions ofLv;LT andLq . The pressure is then restricted to an intervalp0 < p < p1, wherep1 is a value of the pressure smaller in average than the pressure onEarth, so that the isobarp D p1 is slightly above the Earth and the isobarp D p0 is anisobar high in the sky. We study the motion of the air between these two isobars; as wesaid, forp < p0 we would need a different set of equations and for the “thin” portion ofair between the Earth and the isobarp D p1, another specific simplified model would benecessary.

For the whole atmosphere, the domain is

MD ˚.'; �;p/;p0 < p < p1;

and its boundary consists first of an upper part�u, pD p0 and a lower partpD p1 whichis divided into two parts:�i the part ofp D p1 at the interface with the ocean, and�e thepart ofpD p1 above the Earth.

Boundary conditions. Typically the boundary conditions are as follows:On the top of the atmosphere�u .pD p0/:

@v@pD 0; ! D 0; @T

@pD 0; @q

@pD 0: (2.123)

Above the Earth on�e:

vD 0; ! D 0;

34 M. Petcu et al.

�T@T

@pC ˛T .T � Te/D 0; (2.124)

@q

@pD gq :

Above the ocean on�i :

�v

�gp

R xT

�2@v@pC ˛v

�v� vs

�D �v; ! D 0;

�T

�gp

R xT

�2@T

@pC ˛T

�T � T s

�D 0; (2.125)

@q

@pD gq :

REMARK 2.10. The equations on�i (2.125) are similar to those on�i for the ocean (2.35),with different values of the coefficients�v; �T ; : : : I comparison between these two sets ofboundary conditions is made in Section 2.5 devoted to the coupled atmosphere and oceansystem. In (2.127) and (2.125)Te is the (given) temperature on the Earth, andvs; T s arethe (given) velocity and temperature of the sea. The boundary conditions (2.122) on�u

are physically reasonable boundary conditions; they can be replaced by other boundaryconditions (e.g.,vD 0), which can be treated mathematically in a similar manner.

Regional problems and beta-plane approximation.It is reasonable to study regional prob-lems, in particular at mid-latitudes and, in this case we use the beta-plane approxima-tion. In this case, as for the ocean, we use the Cartesian coordinates denotedx;y; z orx1; x2; x3, and�D .f0 C ˇy/k. The equations are exactly the same as (2.115)–(2.122),but now� D @2=@x2 C @2=@y2, r is the usual nabla vector.@=@x; @=@y/ andrv Du@=@xCv@=@y; vD .u; v/. The domainM is now some portion of the whole atmosphere:

MD ˚.'; �;p/; .�; '/ 2 �i [ �e; p0 < p < p1;

where�i [�e are only part of the isobarpD p1. The boundary ofM consists of�u; �i ; �e

defined as before and of a lateral boundary

�` D˚.'; �;p/;p0 < p < p1; .'; �/ 2 @�u

:

The boundary conditions are the same as before on�u, �i , �e, and, on�` the conditionswould be as follows:

Boundary conditions on�`.

vD 0; ! D 0; @T

@nTD 0; @q

@nqD 0: (2.126)


Here@=@nT and@=@nq are defined as in (2.37). Comparing to (2.37),vD 0; ! D 0, isnot a physically satisfactory boundary condition; we would rather assume that.v;!/ hasa nonzero prescribed value; however in the mathematical treatment of this boundary con-dition we would then recover.v;!/D 0 after removing a background flow; the necessarymodifications are minor.

Below we only discuss the regional case.

Prognostic and diagnostic variables.The unknown functions are regrouped in two sets:the prognostic variablesU D .v; T; q/ for which and initial value problem will be defined,and the diagnostic variables!;�;ˆ .D gz/ which can be defined, at each instant of timeas functions (functionals) of the prognostic variables, using the equations and boundaryconditions. In fact! is determined in terms ofv very much as in the case of the ocean:

! D !.v/D�Z p

p0

divvdp0; (2.127)

withZ p1

p0

divv dpD 0: (2.128)

Then� is determined by the equation of state (2.121) andˆ is a function ofp andTdetermined by integration of (2.118):

ˆDˆsCZ p1

p

RT.p0/p0

dp0I (2.129)

in (2.129),ˆsD ˆjpDp1 is the geopotential atp D p1, that isg times the height of theisobarpD p1. This is an auxiliary unknown, and its introduction has been a crucial step atthe basis of the new mathematical formulation of the Primitive Equations of the atmospherein Lions, Temam and Wang (1992a).

Weak formulation of the PEs.For the weak formulation of the PEs, we introduce functionspaces similar to those considered for the ocean, namely:

V D V1 � V2 � V3; H DH1 �H2 �H3;

V1 D�

v 2H 1.M/2;divZ p1

p0

vdpD 0;vD 0 on�e[ �`�;

V2 D V3 DH 1.M/;

H1 D�

v 2L2.M/2;divZ p1

p0

vdpD 0;

nH �Z p1

p0

vdpD 0 on@�u (i.e., on�`)

�;

36 M. Petcu et al.

H2 DH3 DL2.M/:

These spaces are endowed with scalar products similar to those for the ocean:

��U;eU ��D ��v; Qv��

1CKT

��T;eT ��

2CKq

��q; Qq��

3;

��Qv; Qv��1DZ

M

�rv � r QvC

�gp

R xT

�2@v@p

@Qv@p

�dM;

��T;eT ��

2DZ

M

�rT � reT C

�gp

R xT

�2@T

@p

@eT@p

�dMC

Z

�i

˛T TeT d�i ;

��q; Qq��

3DZ

M

�rq � r QqC

�gp

R xT

�2@q

@p

@ Qq@p

�dMCK 0q

Z

Mq Qq dM;

�U;eU �

HDZ

M

�v � QvCKT TeT CKqq Qq

�dM;

kU k D �.U;U /�1=2; jU jH D .U;U /1=2H :

HereKT ;Kq are suitable positive constants chosen below,K 0q > 0 is a constant of suit-able (physical) dimension. The norm onH is of course equivalent to theL2 norm and,thanks to the Poincaré inequality,k � k is a Hilbert norm onV ; more precisely, there existsa suitable constantc0 > 0 (different from that in (2.52)) such that

jU jH 6 c0kU k 8U 2 V: (2.130)

We denote byV1 the space ofC1 (R2 valued) vector functionsv which vanish in aneighborhood of�e[ �` and such that

divZ p1

p0

vdpD 0:

Let V2 D V3 be the space ofC1 functions onM, andV D V1 � V2 � V3; then, asin (2.53):

V1 is dense inV1; V is dense inV: (2.131)

We also introduceV.2/ the closure ofV in H 2.M/4.The weak formulation of the PEs of the atmosphere takes the form:

�dU

dt;eU�

H

C a�U;eU �C b�U;U;eU �C e�U;eU �

D `�eU � 8eU 2 V.2/: (2.132)


Hereb D b1CKT b2CKqb3, ande are essentially as for the ocean, replacing, forb, @=@zby @=@p andw.v/ by !.v/. ThenaD a1CKT a2CKqa3, with

a1�U;eU �D

Z

M


�gp

R xT

�2@v@p

@Qv@p

�dM

�Z

M

�Z p1

p

RT

p0dp0r Qv

�dMC

Z

�i

˛vvQv d�i ;

a2�U;eU �D

Z

M


�gp

R xT

�2@T

@p

@eT@p

�dM

�Z

M

R xT .p/cpp

!.v/eT dMCZ

�i

˛T TeT d�i ;

a3�U;eU �D

Z

M

��Srq � r QqC �S

�gp

R xT

�2@q

@p

@ Qq@p

�dM:

Finally,

`�eU �D

Z

M

�Fv QvCKTFTeT CKqFq Qq

�dM

CZ

�i

gv QvC gTeT d�i CZ

�e

gTeT d�e;

gv D �vC ˛vvs; gT D ˛T T s on�i ; gT D ˛T Te on�e:

We find that there existc1; c2 > 0 such that

a.U;U /C c2Z

Mq2 dM> c1kU k2 8U 2 V;

and that

e.U;U /D 0 8U 2 V:

Properties ofb are similar to those in Lemma 2.1, withV.2/ defined as the closure ofVin H 2.M/4.

The boundary and initial value problem.As for the ocean, the weak formulation reads:We are givent1 > 0,U0 inH , F D .Fv;FT ;Fq/ in L2.0; t1IH/ (or L2.0; t1IL2.M/4/,

gv in L2.0; t1IL2.�i/2/, gT in L2.0; t1I�i [ �e/). We look forU :

U 2L1.0; t1IH/\L2.0; t1IV /; (2.133)

such that

38 M. Petcu et al.

�dU

dt;eU�

H


D `�eU � 8eU 2 V.2/; (2.134)

U.0/DU0: (2.135)

REMARK 2.11. We can introduce the operatorsA;B;E and write (2.134) in an operatorform, as (2.66).

The analog of Theorem 2.2 can be proved in exactly the same way:

THEOREM 2.3. The domainM is a before. We are givent1 > 0, U0 in H , F D.Fv;FT ;Fq/ in L2.0; t1IH/ (or L2.0; t1IL2.M/4/, gv in L2.0; t1IL2.�i/

2/, gT inL2.0; t1I�i [ �e//. Then there existsU which satisfies(2.134)and (2.135).FurthermoreU is weakly continuous fromŒ0; t1� in H .

2.5. The coupled atmosphere and ocean

After considering the ocean and the atmosphere separately, we consider in this sectionthe coupled atmosphere and ocean (CAO in short). The model presented here was firstintroduced in Lions, Temam and Wang (1995); it is amenable to the mathematical andnumerical analysis and is physically sound. The model was derived by carefully examiningthe boundary layer near the interface�i between the ocean and the atmosphere. Althoughsome processes are still not fully understood from the physical point of view, the derivationof the boundary condition is based on the work of Gill (1982) and Haney (1971).

We will present the equations and boundary conditions, the variational formulation andarrive to a point where the mathematical treatment is the same as for the ocean and theatmosphere.

The pseudo-geometrical domain.Let us first introduce the pseudo-geometrical domain.Let h0 be a typical length (height) for the atmosphere; for harmonization with the oceanwe introduce the vertical variable�D z, for z < 0 (in the ocean) and

�D h0�p1 � pp1 � p0

�; 0 < � < h0; (2.136)

for the atmosphere. The pseudo-geometrical domain is

MDMa[Ms[ �i ;

whereMs is the ocean defined as in Section 2.1.2,Ma is the atmosphere,0 < � < h0, and�i is, as before the interface between the ocean and the atmosphere.


All quantities will now be defined as for the atmosphere alone, adding when needed,a superscript “a” or as for the ocean alone, adding a superscript “s”.4 Hence, with obviousnotations, the boundary ofM consists of

�s` [ �b[ �e[ �u: (2.137)

The governing equations.InMa, the variable isU aD .va; T a; q/ and inMs the variableisU sD .vs; T s; S/; we set alsoU D fU a;U sg, or alternativelyvD fva; vsg, T D fT a; T sg.

The equations forU s are exactly as in (2.24)–(2.29), introducing only a superscript “s”for w;p;�0;Fv;FT ;FS ; Tr; Sr, as well as the eddy viscosity coefficients�v; �v, etc.

The equations forU a are exactly as in (2.116)–(2.122), introducing again a superscript“a” for !;p;�;Fv;FT ;Fq , as well as the various coefficients.5 Of course the variablep isreplaced by� following (2.136), and the differential operators are changed accordingly.

Boundary conditions. Except for�i , the boundary conditions are the same as for theocean and the atmosphere taken separately. Hence we recover the conditions (2.122) on�u

and (2.127) on�e, adding the superscripts “a”.Let us now describe the boundary conditions on�i . These conditions were introduced

in Lions, Temam and Wang (1995); we refer the reader to this monograph for justificationand a detailed discussion.

We first have the geometrical (kinematical) condition:

wsD !aD 0 on�i ; (2.138)

which expresses that�D 0 .z D 0/ is indeed the upper limit of the ocean (under the rigidlid hypothesis) and�D 0 is the lower limit of the atmosphere (the isobarpD p1).

Then for the velocity we express the fact that the tangential shear-stresses exerted by theatmosphere on the ocean have opposite values and vice versa, and this value is expressed asa function of the differences of velocitiesva� vs using a classical empirical model of res-olution of boundary layers (see, e.g., Gill (1982), Haney (1971), Lions, Temam and Wang(1995); the boundary layer model is used to model the boundary layer of the atmospherethat is most significant). These conditions read:

�s0�

sv@vs

@zD� N�a�a

v@va

@zD N�aC a

D.˛/�va� vs

�ˇva� vs

ˇ˛; 6 (2.139)

Here˛ > 0 andC aD.˛/ > 0 are coefficients from boundary layer theory, andN�a> 0 is

an averaged value of the atmosphere density. Similar conditions hold for the temperatures,the salinityS in the ocean, and the humidityq in the atmosphere.

4“s” for sea, rather than “o” for ocean which could be confused with zero.5An additional term linear in!a with a coefficient depending onpa appears in the equation forT a in Lions,

Temam and Wang (1995); see Equation (1.11), p. 4, and Footnote 2, p.15 of Lions, Temam and Wang (1995).This term does not affect the discussion hereafter.

6The same equation appears in Lions, Temam and Wang (1995) withN�a replaced by�a. Replacing�a by N�a isa necessary simplification for the developments below.

40 M. Petcu et al.

For the sake of simplicity, to keep the boundary condition linear, we take˛D 0; we alsoneed to replacez by �.p/ in the atmosphere; see Lions, Temam and Wang (1995) for thedetails. In the end we arrive at the following conditions on�i

waD !sD 0;

�s0�

sv@vs

@zD� N�a�a

v

�gpa

R xT

�2@va

@paD ˛v

�va� vs

�;

(2.140)

csp�

s0�

sT

@T s

@zD�cap N�a�a

T

�gpa

R xT

�2@T a

@zD ˛T

�T a� T s

�;

@q

@paD @S

@zD 0:

Weak formulation of the PEs.For the sake of simplicity we restrict ourselves to a regionalproblem using the beta-plane approximation.

The function spaces that we introduce are similar to those used for the ocean and theatmosphere, hence

V D V1 � V2 � V3; H DH1 �H2 �H3;

whereVi D V ai � V s

i ,Hi DH ai �H s

i , the spacesV ai ,H a

i , V si ,H s

i being exactly like thoseof the atmosphere and the ocean, respectively. Alternatively we can write, with obviousnotations,V D V a� V s;H DH a�H s.

These spaces are endowed with the following scalar products:

��U;eU ��D ��U;eU ��aC

��U;eU ��s;��

U;eU ��aD��

va; Qva��

a;1CKT��T a;eT a

��a;2CKq

��q; Qq��a;3;

��U;eU ��sD

��vs; Qvs

��s;1CKT

��T s;eT s

��s;2CKS

��S;eS ��s;3;

�U;eU �

HD �U;eU �aC

�U;eU �s;

�U;eU �aD

Z

Ma

�va � QvaCKT T aeT aCKqq Qq

�dM;

�U;eU �sD

Z

Ms

�vs � QvsCKT T seT sCKSSeS

�dM:

Note thatKT is chosen the same in the atmosphere and the ocean. By the Poincaréinequality, there exists a constantc0 > 0 (different than those for the ocean and the atmo-sphere), such that

jU jH 6 c0kU k 8U 2 V: (2.141)

From this we conclude thatk �k is a Hilbert norm onV . We also introduce, in a very similarway the spacesV andV.2/.


With this, the weak formulation of the PEs of the coupled atmosphere and ocean takesthe form:

�dU

dt;eU�

H


D `�eU � 8eU 2 V.2/; (2.142)

U.0/D U0: (2.143)

Here

aD a1C a2C a3; b D b1C b2C b3; eD eaC es;

where

a1 D N�aaa1C �s

0as1C ˛v

Z

�i

ˇva� vs

ˇ2d�i ;

a2 DKT cap N�aaa

2CKT csp�

s0a

s2C ˛T

Z

�i

ˇT a� T s

ˇ2d�i ;

a3 DKqaa3CKsas

3;

b1 D N�aba1C �s

0bs1;

b2 DKT cap N�aba

2CKT csp�

s0b

s2;

b3 DKqba3CKSbs

3:

Here, of course, the formsaai , b

ai , e

a are those of the atmosphere, andasi , b

si , e

s are thoseof the ocean.

The form` is defined as for the ocean and the atmosphere, the terms concerning�i beingomitted. Hence

`D `aC `s;

`a�eU �D

Z

Ma

� N�aF av QvaCKT ca

p N�aF aTeT aCKqFq Qq

�dMaC

Z

�e

gaTeT ad�e;

`s�eU �D

Z

Ms

��s0F

sv QvsCKT cs

p�s0eT sCKSFSeS

�dMs

CZ

Ms.ˇT Tr � ˇSSr/r � QvsdMs:

With these definitions, the properties ofa; b; e; ` are exactly the same as for the oceanand atmosphere (separately) and we prove, exactly as before, the existence, for all time, ofweak solutions:

42 M. Petcu et al.

THEOREM 2.4. The domainM is as before. We are givent1 > 0, U0 in H and F inL2.0; t1IH/ (or L2.0; t1IL2.M/8/, andga

T in L2.0; t1I�e/.Then there existsU which satisfies(2.142)and (2.143),and

U 2L1.0; t1IH/\L2.0; t1IV /:

FurthermoreU is weakly continuous fromŒ0; t1� intoH .

3. Strong solutions of the primitive equations in dimension two and three

In this section we first show, in Section 3.1, the existence, local in time, of strong solutionsto the PEs in space dimension three, that is solutions whose norm inH 1 remains boundedfor a limited time. Then, in Section 3.2, we show the existence and uniqueness, global intime, of strong solutions to the PEs in space dimension three.

In Section 3.3 we consider the PEs in space dimension two in view of adapting to thiscase the results of Sections 2 and 3.1. The two-dimensional PEs are presented in Sec-tion 3.3.1 as well as their weak formulation (Section 3.3.2). Strong solutions are consideredin Section 3.3.3 and we show directly (unlike in dimension three) that the strong solutionsare now defined for all timet > 0.

Essential in all this section is the anisotropic treatment of the equations, the verticaldirection 0z playing a different role than the horizontal ones (0x in two dimensions,0x and0y in three dimensions).

3.1. Strong solutions in space dimension three

In this section we establish the local, in time, existence of strong solutions of the primitiveequations of the ocean. The result that we obtain is similar to that for the three-dimensionalNavier–Stokes equations. The analysis given in this section also applies to the primitiveequations of the atmosphere and the coupled atmosphere–ocean equations using the nota-tions and equations given in Section 2.

We first state the main result of Section 3.1, namely Theorem 3.1. We then prepare itsproof in several steps: in Step 1, we consider the linearized primitive equations and estab-lish the global existence of strong solutions. In Step 2 we use the solution of the linearizedequations in order to reduce the primitive equations to a nonlinear evolution equation withzero initial data and homogeneous boundary conditions. We also provide the necessarya priori estimates for this new problem with zero initial data and homogeneous boundaryconditions. Finally, in the last step, we actually prove Theorem 3.1; in particular, we showhow one can establish the existence of solutions for this problem using the Galerkin ap-proximation with basis consisting of the eigenvectors ofA (which are inH 2, thanks to theregularity results of Section 4). We use the previous estimates and then pass to the limit.

In rectangular coordinatesx;y; z or x1; x2; x3, the domain filled by the ocean is as in(2.33) and (2.38):

MD f.x;y; z/; .x; y/ 2 �i ; �h.x;y/ < z < 0g ; (3.1)


where�i is the surface of the ocean,�b its bottom,�` the lateral surface and0 < h �h.x;y/ < h.

The main result of this section is as follows:

THEOREM 3.1. We assume that�i is of classC3 and thath W x�i ! RC is of classC3; wealso assume(see(2.92))that

rh � n�i D 0 on@�i ; (3.2)

wheren�i is the unit outward normal on@�i (in the plane0xy). Furthermore, we aregivenU0 in V , F D .Fv;FT ;FS / in L2.0; t1IH/ with @F=@t in L2.0; t1IL2.M/4/ andg D .gv; gT / in L2.0; t1IH 1

0 .�i/3/ with @g=@t in L2.0; t1IH 1

0 .�i/3/.7 Then there exists

t� > 0, t� D t�.kU0k/, and there exists a unique solutionU D U.t/ of the primitive equa-tions(2.79)such that

U 2 C �Œ0; t��IV�\L2�0; t�IH 2.M/4

�: (3.3)

STEP 1. The first step in the proof of Theorem 3.1 is the study of the linear primitiveequations of the ocean.

Hence we consider the equations (to (2.44)–(2.51)):

@v�

@tCrp�C 2f k � v� ��v�v� � �v

@2v�

@z2D Fv; (3.4)

@p�

@zD��g; (3.5)

@T �

@t��T�T � � �T

@2T �

@z2D FT ; (3.6)

@S�

@t��S�S� � �S

@2S�

@z2D FS ; (3.7)

divZ 0

�hv� dz0 D 0; (3.8)

Z

MS� dMD 0; (3.9)

p� D p�s C gZ 0

z

�� dz0; (3.10)

�� D �0�1� ˇT

�T � � Tr

�C ˇS�S� � Sr

��; (3.11)

with the same initial and boundary conditions as for the full nonlinear problem, that is(see (2.35)–(2.38)):

�v@v�

@zC ˛v

�v� � va

�D 0;7The hypotheses on@F=@t and@g=@t can be weakened.

44 M. Petcu et al.

(3.12)

�T@T �

@zC ˛T

�T � � T a

�D 0; @S�

@zD 0 on�i ;

v� D 0;(3.13)

@T �

@nTD 0; @S�

@nSD 0 on�b[ �`;

v� D v0; T � D T0 and S� D S0 at t D 0: (3.14)

Comparing with the nonlinear problems (2.44)–(2.51), and using the same notationsas in (2.66), (2.67), we see thatU � D .v�; T �; S�/ is solution of the following equationwritten in functional form:

dU �

dtCAU �CE�U ��D `; (3.15)

U �.0/DU0: (3.16)

The right-hand side of (3.15) is exactly the same as in (2.66) (see (2.44)–(2.51) andthe expression of in (2.55)).

We also consider the solutionxU D . Nv; xT ; xS/ of the linear stationary problem, namely

��v�Nv� �v@2 Nv@z2Cr NpD Fv � 2f k � Nv;

@ Np@zD� N�g;

��T� xT � �T@ xT@z2D FT ;

��T S� xS � �S@2 xS@z2D FS ;

(3.17)

divZ 0

�hNvdz0 D 0;

Z

MxS dMD 0;

NpD NpsC gZ 0

z

N�dz0;

N�D �0�1� ˇT

� xT � Tr�C ˇS

� xS � Sr��;

with the boundary conditions

�v@Nv@zC ˛v

�Nv� va�D �v;


�T@ xT@zC ˛T

� xT � T a�D 0; @ xS

@zD 0 on�i ; (3.18)

NvD 0; @ xT@nTD 0; @ xS

@nSD 0 on�b[ �`:

Note that (as in (3.4)–(3.13)) the left- and right-hand sides in (3.17) and (3.18) depend onthe timet . The existence and uniqueness for (almost) every timet for (3.17) and (3.18), fol-lows from the Lax–Milgram theorem as explained, e.g., for the velocityNv, in Section 4.4.1and Proposition 4.1. Furthermore, the regularity results of Section 4 (in particular, Theo-rems 2.1–4.5) show that the solution belong toH 2.M/, and that

kNvk2H2.M/C xT 2H2.M/

CkNvk2H2.M/6 C0�1;

(3.19)�1 D �1.F; �v;va; Ta/D jF j2H Ck�vk2H1.�i/

Ckvak2H1.�i/CkTak2H1.�i/

:

Note that, again, each side of (3.19) depends ont , and (3.19) is valid for (almost) ev-ery t . The constantC0 depends only onM according to the results of Section 4; in particu-larC0 is independent oft . Hypothesis (3.2) is precisely what is needed for the utilization ofTheorem 4.5 used for (3.19). It is noteworthy that we have the same estimates as (3.19) forthe derivatives@Nv=@t , @ xT =@t , @ xS=@t , �1 being replaced by�01 which is defined similarlyin terms of the time derivatives@F=@t , etc.

Now let Qv D v� � Nv, Qp D p� � Np, Q� D �� N�, eT D T � � xT andeS D S� � xS . Theequations satisfied by. Qv;eT ;eS/ are the same as (3.4)–(3.14) but with.Fv;FT ;FS / D�dxU=dt;vaD �v D TaD Tr D TsD 0, and with initial data

QvˇtD0 D v0 � Nv.0/; eT

ˇtD0 D T0 � xT .0/ and eS

ˇtD0 D S0 � xS.0/:

(3.20)

Comparing with the nonlinear problem (2.44)–(2.51), and using the same notation as in(2.66) and (2.67), we see thateU D .Qv;eT ;eS/ is solution of the following equation writtenin functional form:

deUdtCAeU CE�eU �D�dxU

dt; (3.21)

eU.0/D eU 0 DU0 � xU.0/: (3.22)

Note that the contribution from vanishes (see the expression in the equation preced-ing (2.56)).

The existence for all time of a strong solutioneU to (3.21) and (3.22) is classical, and werecall the estimate obtained by multiplying (3.21) byAeU and integrating in time:

sup06t6t1

eU.t/ 2C

Z t1

0

ˇAeU.s/

ˇ2H

ds 6 c eU.0/

2C c�01:

46 M. Petcu et al.

Here we have used the analog of (3.19) for dxU=dt ; see the comments after (3.19). Fromthis we obtain

sup06t6t1

U �.t/ 2C

Z t1

0

ˇU �.s/

ˇ2H2.M/4

ds

6 c� U.0/

2C xU.0/

2�C cZ t1

0

ˇ xU.s/ˇ2H2.M/4

dsC cZ t1

0

�01.s/ds;

(3.23)

and, with (3.19), we bound the right-hand side of (3.23) by an expression�2 of the form:

�2 D ckU0k2C c0��1.0/C

Z t1

0

��1C �01

�.s/ds

�: (3.24)

STEP 2. We will now useU � D .v�; T �; S�/ and write the primitive equations of theocean using the decompositionU D U � C U 0; we note thatU 0.0/ D 0 and thatU 0 D.v0; T 0; S 0/ satisfies homogeneous boundary conditions of the same type asU . More pre-cisely, starting from the functional form (2.66) and (2.67) of the equation forU and using(3.15) and (3.16), we see thatU 0 satisfies

dU 0

dtCAU 0CB�U 0;U ��CB�U �;U 0�CB�U 0;U 0�CE�U 0�

(3.25)D�AU � �B�U �;U ��;

U 0.0/D 0: (3.26)

The existence of solution in Theorem 3.1 is obtained by proving the existence of solutionfor this system (on some interval of time.0; t�/). As usual this proof of existence is basedon a priori estimates for the solutionsU 0 of (3.25) and (3.26). Some a priori estimates canbe obtained by proceeding exactly as for Theorem 2.2, but additional estimates are neededhere. Essential for these new estimates is another estimate on the bilinear operatorB (orthe trilinear formb), which is obtained by ananisotropic treatment of certain integrals. Wehave the following result (cf. to Lemma 2.1):

LEMMA 3.1. In space dimension three, the formb is trilinear continuous onH 2.M/4 �H 2.M/4 �L2.M/4 and we have

ˇb�U;U [;U ]

�ˇ6 c3

�kU kH1 U [

1=2H1

U [ 1=2H2

CkU k1=2H1kU k1=2

H2

U [ 1=2H1

U [ 1=2H2

�ˇU ]ˇL2

(3.27)

for every.U;U [;U ]/ in this space.


The proof of this lemma is given below. Using Lemma 3.1, we obtain a priori estimateson the solutionU 0 of (3.25) and (3.26). We denote byA1=2 the square root ofA so that

�A1=2U;A1=2eU �

HD a�U;eU � 8U;eU 2 V:

Taking the scalar product of (3.25) withAU 0 in H , we obtain

1

2

d

dt

ˇA1=2U 0

ˇ2HCˇAU 0

ˇ2H

D�b�U 0;U �;AU 0�� b�U �;U 0;AU 0�� b�U 0;U 0;AU 0�

� b�U �;U �;AU 0�� AU �;AU 0�H� �E�U 0�;AU 0�

H: (3.28)

We bound each term in the right-hand side of (3.28) as follows, using Lemma 3.1 for theb-terms:

ˇ�AU �;AU 0

�H

ˇ6 1

12

ˇAU 0

ˇ2HC c

ˇU �ˇ2H2; 8

ˇ�E�U 0�;AU 0

�H

ˇ6 1

12

ˇAU 0

ˇ2HC c

ˇU 0ˇ2H;

ˇb�U 0;U �;AU 0

�ˇ6 c

U 0 U �

H2

ˇAU 0

ˇH

C c U 0

1=2 U � 1=2H1

U � 1=2H2

ˇAU 0

ˇ3=2H

6 1

12

ˇAU 0

ˇ2HC c

U 0 2 U �

2H2

�1C

U � 2H1

�;

ˇb�U �;U 0;AU 0

�ˇ6 c

U � 1=2H1

U � 1=2H2

U 0 1=2 ˇAU 0

ˇ3=2H

6 1

12

ˇAU 0

ˇ2HC c

U 0 2 U �

2H1

U � 2H2;

ˇb�U 0;U 0;AU 0

�ˇ6 c4

U 0 ˇAU 0

ˇ2H;

ˇb�U �;U �;AU 0

�ˇ6 c

U � H1

U � H2

ˇAU 0

ˇH

6 1

12

ˇAU 0

ˇ2HC c

U � 2H1

U � 2H2:

Here we used the fact that the normjAU 0jH is equivalent to the normjU 0jH2 , thanks tothe results of Section 4, and (this is easy), the fact that the normjA1=2U 0jH is equivalentto the normkU 0k D kU 0kH1 .

Taking all these bounds into account, we infer from (3.28) that

8U� is not inD.A/ andAU 0 2 V 0, becauseU� does not satisfy the homogeneous boundary conditions;however this bound is valid, see the details in Hu, Temam and Ziane (2003).

48 M. Petcu et al.

d

dt

ˇA1=2U 0

ˇ2HC �1� c4

ˇA1=2U 0

ˇH

�ˇAU 0

ˇ2H6 �.t/

ˇA1=2U 0

ˇ2HC�.t/;

(3.29)

with

�.t/D c U �

2H2C�.t/;

�.t/D c U �

2H1

U � 2H2:

We infer from (3.23) and (3.24) (and from the precise expression of�2 in (3.24)), that� and� are integrable on.0; t1/ and we set

�3 DZ t1

0

�.t/dt:

By Gronwall’s lemma and sinceU 0.0/D 0, we have, on some interval of time.0; t�/,and as long as1� c4jA1=2U 0jH > 0,

ˇA1=2U 0.t/

ˇ2H6�Z t

0

�.s/ds

�exp

�Z t

0

�.s/ds

�;

(3.30)ˇA1=2U 0.t/

ˇ2H6 exp

�Z t

0

�.s/ds

�:

In fact (3.29) is valid as long as0 < t < t� wheret� is the smaller oft1 andt�, wheret� is eitherC1 or the time at which

Z t�

0

�.s/ds D log

�1

4c24�3

�: (3.31)

We then have

ˇA1=U 0.t/

ˇ2H6 1

4c24for 0 < t < t�; (3.32)

and returning to (3.30) we find also a bound

Z t�

0

ˇAU 0.t/

ˇ2H

dt 6 const: (3.33)

STEP 3 (Proof of the existence in Theorem 3.1). As we said, the existence for Theorem 3.1is shown by proving the existence of a solutionU 0 of (3.25) and (3.26) inC.Œ0; t��IV / \L2.0; t�ID.A//. For that purpose we implement a Galerkin method using the eigenvectorsej of A:

Aej D �j ej ; j > 1; 0 < �1 6 �2 6 � � � :


The results of Section 4.1 show that theej belong toH 2.M/4 (since D.A/ �H 2.M/4). We look, for eachm> 0 fixed, for an approximate solution

U 0m DmX

jD1�jm.t/ej ;

satisfying (cf. (3.25) and (3.26))

�dU 0m

dt; ek

�

H

C a�U 0m; ek�C b�U 0m;U �; ek

�

C b�U �;U 0m; ek�C b�U 0m;U 0m; ek

�C e�U 0m; ek�

D�a�U �; ek�� b�U �;U �; ek

�; k D 1; : : : ;m; (3.34)

and

U 0m.0/D 0:

Multiplying (3.34) by�km.t/�k , and adding these equations fork D 1; : : : ;m, we obtainthe analogue of (3.28) forU 0m. The same calculations as above show thatU 0m satisfiesthe same estimates independent ofm as (3.32) and (3.33), with the same timet� (alsoindependent ofm).

It is then straightforward to pass to the limitm!1 and we obtain the existence.

STEP 4 (Proof of uniqueness in Theorem 3.1). The proof of uniqueness is easy. Considertwo solutionsU1;U2 of the primitive equations; letU D U1 � U2, and consider as abovethe associated functionsU 0i D Ui �U �, U 0 DU 01 �U 02. ThenU 0 satisfies

dU 0

dtCAU 0CB�U 0;U2

�CB�U2;U 0�CB�U 0;U 0�CE�U 0�D 0;

U 0.0/D 0:

Treating this equation exactly as (3.25) we obtain an equation similar to (3.29) but with�D 0 and a different�:

d

dt

ˇA1=2U 0

ˇ2HC �1� c4

ˇA0U

ˇH

�ˇAU 0

ˇ2H6 Q�.t/

ˇA1=2U 0

ˇ2H:

The uniqueness follows using Gronwall’s lemma.

To conclude this section and the proof of Theorem 3.1, we now prove Lemma 3.1.

PROOF OF LEMMA 3.1. We need only to show how the different integrals inb arebounded by the expressions appearing in the right-hand side of (3.27) and, in fact, werestrict ourselves to two typical terms, the other terms being treated in the same way.

50 M. Petcu et al.

The first one is bounded as follows:

ˇˇZ

M

�.v � r/v[�v] dM

ˇˇ6 jvjL6

ˇrv[

ˇL3

ˇv]ˇL2: (3.35)

By Sobolev embeddings and interpolation, we bound the right-hand side by

ckvkH1 v[

1=2H2

ˇv]ˇL2;

which corresponds to the first term on the right-hand side of (3.27).The second typical term which necessitates anisotropic estimates is bounded as follows:

ˇˇZ

Mw.v/

@v[

@zv] dM

ˇˇD

ˇˇZ

�i

Z 0

�hw.v/

@v[

@zv] dz d�i

ˇˇ

6Z

�i

ˇw.v/

ˇL1z

ˇˇ@v[@z

ˇˇL2z

ˇv]ˇL2z

d�i

6 Nh1=2Z

�i

jdivvjL2zˇˇ@v[@z

ˇˇL2z

ˇv]ˇL2z

d�i

6 (with Holder’s inequality)

6 Nh1=2jdivvjL4�iL2z

ˇˇ@v[@z

ˇˇL4�iL2z

ˇv]ˇL2�iL2z: (3.36)

For a (scalar or vector) function� defined onM, we have written.16 ˛;ˇ 61/:

j�jLˇzD j�j

Lˇz.x;y/D

�Z 0

�h.x;y/

ˇ�.x;y; z/

ˇˇdz

�1=ˇ;

j�jL˛�iLˇzDˇj�jLˇz

ˇL˛.�i/

D�Z

�i

j�j˛Lˇz

.x;y/d�i

�1=˛:

Notice also thatj�jL2�iL2zD j�jL2.M/ and that, from the expression (2.40) ofw.v/,

and (2.39),

ˇw.v/

ˇL1zDˇˇZ 0

z

divvdz0ˇˇL1z6 Nh1=2jdivvjL2z :

Now we remember that (in space dimension two) there exists a constantc D c.�i/ suchthat, for every function� in H 1.�i/,

j�jL4.�i/6 cj�j1=2

L2.�i/j�j1=2H1.�i/

; (3.37)


from which we infer, for a function� as above (setting� D j�jL2z /:

j�jL4�iL2z6 cj�j1=2

L2�iL2z

ˇj�jL2z

ˇ2H1.�i/

: (3.38)

As beforej�jL2�iL2zD j�jL2.M/, whereas

ˇj�jL2z

ˇ2H1.�i/

D j�j2L2�iL2zCˇrj�jL2z

ˇ2L2.�i/

D j�j2L2.M/

C jr� j2L2.�i/

; (3.39)

with

� D �.x;y/D j�jL2z .x;y/D�Z 0

�h

ˇ�.x;y; z/

ˇ2dz

�1=2:

We intend to show that for� in H 1.M/ (scalar or vector�):

j�jL4�iL2z6 cj�j1=2

L2.M/j�j1=2H1.M/

(3.40)

for some suitable constantc D c.M/. For that purpose we note that

r� D�Z 0

�hj�j2 dz

�1=2�Z 0

�h�r� dzC

ˇ�.�h/

ˇ2rh�;

where�.�h/ and below,�.z/, are simplified notations for�.x;y;�h.x;y// and�.x;y; z/respectively. With the Schwarz inequality we find that pointwise a.e. (for a.e..x;y/ 2 �i/.

jr� j6 jr�jL2z C cj�j�1L2z

ˇ�.�h/

ˇ2: (3.41)

We have classically:

ˇ�.�h/

ˇ2 Dˇ�.z/

ˇ2 � 2Z z

�h�@�

@zdz 6

ˇ�.z/

ˇ2C 2j�jL2zˇˇ@�@z

ˇˇL2z

;

and by integration inz from�h.x;y/ to 0:

hˇ�.�h/

ˇ2 6 j�j2L2zC 2 Nhj�jL2z

ˇˇ@�@z

ˇˇL2z

:

Hence (3.41) yields pointwise a.e.

jr� j6 jr�jL2z C cj�jL2z C cˇˇ@�@z

ˇˇL2z

:

52 M. Petcu et al.

By integration on�i , we find

jr� jL2.�i/6 cj�jH1.M/;

and then (3.38) and (3.39) yield (3.40).Having established (3.40), we return to (3.36): applying (3.40) with� D divv and� D

@Qv[=@z, we can bound the left-hand side of (3.36) by the second term on the right-handside of (3.27), thus concluding the proof of Lemma 3.1. �

3.2. Strong solutions in dimension 3 (global existence)

The aim of this section is to present very recent results on the global existence of a strongsolution for the Primitive Equations in space dimension 3. As pointed out in the generalintroduction of this chapter, this result was initially thought to be as difficult to prove asthe global existence of strong solutions for the incompressible three-dimensional Navier-Stokes equations. Recently, a series of three articles (Cao and Titi (2006), Kobelkov (2006),Kukavica and Ziane (2007)) prove, by different methods, that the Primitive Equationshave a two-dimensional intrinsic structure, and consequently prove the existence of globalstrong solutions.

The first articles are those, written independently, by Cao and Titi (2006) and Kobelkov(2006). These articles, using different methods, address the case of a cylindrical domainwith constant depth. The case of nonconstant depth is studied in the more recent article byKukavica and Ziane (2007). In this section we present the proof of Kobelkov (2006) andstate, without proof the result of Kukavica and Ziane (2007).

In Section 3.6 the method of Cao and Titi (2005) is described and used to prove theexistence, globally in time, of even more regular solutions in the space periodic case.

In what follows we do not work with both the salinity and the temperature, since theequation for the salinity does not bring any additional difficulty. And instead of the tem-peratureT we use the density� which is proportional toT:

As mentioned before, the domain we are working with is a cylinder until the end ofSection 3.2:

MD ˚x D .x1; x2; x3/I .x1; x2/ 2M0; x3 2 Œ0; 1�; (3.42)

whereM0 D �i is a two-dimensional domain with a boundary consisting of a finite numberof smooth arcs intersecting at nonzero angles;�i ; �` and�b are defined as before but thedepth here is constant,hD 1: We also use the following notation:

Mt DM� Œ0; t �:


The Primitive Equations then read (withz D x3 andrv as in (2.19)):

@v@tCrvvCw@v

@zC 1

�0rpC 2f k � v� �v�3vD Fv;

@p

@zD��g;

div vC @w

@zD 0;

@�

@tCrv�Cw

@�

@z� ��3�D F�;

(3.43)

The boundary conditions for (3:43) are:

v � nD @v@n� nD 0 on�l ;wD 0 on�i [ �b;

@v@nD 0 on�i [ �b

(3.44)

@�

@nD 0 on@M: (3.45)

We also provide the system with the following initial condition:

v.0; x/D v0.x/; �.0; x/D �0.x/; (3.46)

wherev0 has to satisfy the following compatibility condition:

Z 1

0

div v0 dz D 0: (3.47)

For simplicity, we consider the right-hand-sideFv of the first equation (3.43) to be zero,but the result stays valid for a nonzero forcing term (which is not of physical relevance ofcourse).

In order to prove the existence of the strong solution for (3.43), we need to prove theappropriate a priori estimates. We start by obtaining a priori estimates on the pressure andon the density; we proceed in a formal way, that is assuming enough regularity.

A priori estimates onv; � andp

LEMMA 3.2. The density� solution of the Primitive Equations satisfies the followingestimate:

sup06t6T

j�.t/j4L4C 12��

Z T

0

j�r3�j2L2dt 6 cj�0j4L4 : (3.48)

54 M. Petcu et al.

PROOF. The proof of the lemma is immediate; we take theL2-scalar product of (3.43)4with �3 and we find:

1

4

d

dtj�.t/j4

L4C 3��j�r3�j2L2 D 0: (3.49)

Integrating (3.49) int , we find (3.48). �

From the hydrostatic equation (3.43)2, the following inequality is then deduced:

sup06t6T

jpz.t/jL4 6 cj�0jL4 : (3.50)

We now estimate theL2-norm of the velocity field.

LEMMA 3.3. For a solution of the primitive equations, the following estimate holds:

sup06t6T

jv.t/j2L2C �v

Z T

L2jr3v.t/j2L2dt 6 c.jv0j2L2 C j�0j2L4/: (3.51)

PROOF. We take theL2-scalar product of (3.43)1, by v and using the conservation ofmass (3.43)3, and the boundary conditions, we find:

1

2

d

dtjv.t/j2

L2C �vjr3v.t/j2L2 �

1

�0.p; div v/L2 D 0: (3.52)

We need to estimate the pressure term from (3.52):

j.p; div v/L2 j D j.p;@w

@z/L2 j D j.pz ;w/L2 j

6 jpzjL2 jwjL2 6 cjpzjL2 jrvjL2

6 �v

2jrvj2

L2C c

�vjpzjL2 :

(3.53)

Using the estimates obtained above into (3.52), we find:

1

2

d

dtjv.t/j2

L2C �v

2jr3v.t/j2L2 6

c

�vjpz.t/j2L2 6

c

�vj�0j2L4 I (3.54)

we also took advantage of (3.50).The proof of Lemma 3.3 is completed by applying the Gronwall lemma to (3.51).�

An immediate consequence of this result is:


COROLLARY 3.1. For the vertical velocityw the following estimate holds:

Z T

0

.jw.t/j2L2C jwz.t/j2L2/dt 6 c: (3.55)

We can now estimate the pressure in terms of the velocity:

LEMMA 3.4. The pressurep from the primitive equations can be estimated as follows:

jpjL4 6 cŒ.jr3jvj2j1=2L2 C jvjL4 C 1/jvjL4 C 1�: (3.56)

PROOF. We write the pressurep aspD p1C p2, wherep1 is the average of the pressurein the vertical direction:

p1.x;y/DZ 1

0

p.x;y; z/dz; (3.57)

andp2 is an antiderivative ofpz in z, of zero average in the vertical direction. Since thepressurep is determined up to a constant, we can assume that.p31 ; 1/L2 D 0. Indeedp andp1 being defined up to an additive constantc, ..p1 C c/3; 1/L2 D 0 is an equation of thethird degree inc which always has a real solution.

Using (3.50), the pressurep2 is estimated as follows:

sup06t6T

jp2.t/jL4 6 [email protected]/jL4 D [email protected]/jL4 6 c: (3.58)

In order to estimatep1, we introduce the following boundary value problem inM0:

�q D p31 in M0;

@q

@nD 0 on@M0;

(3.59)

with q of zero average. As mentioned before,� stands for the two-dimensional Laplacian.We note that since we imposed.p31 ; 1/L2 D 0, the solvability of (3.59) is ensured.

Let v.x;y; t/D R 10

v.x;y; z; t/dz be the vertical average of the velocity field. We can

write thenvD vC Qv, and we note thatR 10Qvdz D 0. Due to the incompressibility equation

(3.43)3, v satisfies:

div vDZ 1

0

div vdz D�Z 1

0

@w

@zdz D 0;

v � nD @v@n� nD 0 on�l :

(3.60)

From (3.60) we deduce then thatv is of the formvDcurl .

56 M. Petcu et al.

We take theL2-scalar product of (3.43)1 with rq and integrating by parts we find:

.vt ;rq/L2 D�.div vt ; q/L2 D .wtz ; q/L2 D

DZ

M0q.

Z 1

0

wtzdz/dM0 D 0:(3.61)

Using the boundary condition@v=@nD 0 on�i [ �b , we also deduce:

.@2zv;rq/L2 D 0: (3.62)

By integration by parts, we also find:

I D�.�v;rq/L2 D�.�. curl C Qv/;rq/L2 . sinceR 10Qvdz D 0/

D�. curl� ;rq/L2.M0/ DZ

@M0�

�@q

@yn1 �

@q

@xn2

�d.@M0/:

(3.63)

We need to estimateI . Since� D curl v, we need to estimate curlv on @M0. For eachpointM0 2 @M0, we consider the basis.�;n/, wheren is the outward unit normal vectorto @M0, satisfying� � nD ez (whereez is the unitary vertical vector oriented upward).Let .x� ; xn/ be the coordinates of a pointM0 and.u� ; un/ the components of a vectoru inthe coordinate system.�;n/.

We know v � n D 0 on @M0, so this means that the normal component ofv is zero:vn D 0: This implies@vn=@� D 0; and so curlvD @vn=@� � @v�=@nD�@v�=@n:

We also know that.@v=@n/ � nD 0 on @M0, and this reduces to@v�=@nD 0. We cannow conclude that curlvD 0, which impliesI D 0.

Now, let us continue the estimation of the norm ofp. We have:

.rvv;rq/L2 C .w@v@z;rq/L2 C

1

�0.rp;rq/L2 C 2f .k � v;rq/L2 D 0:

(3.64)

We use the fact thatq D��1p31 , where��1 is an operator inverse to� with the Neumannboundary condition.

The pressure term from (3.64) becomes:

1

�0.rp;r��1p31/L2 D�

1

�0.p;p31/L2 D�

1

�0.p1; p

31/L2.M0/ D�

1

�0jp1j4L4.M0/:

(3.65)

For the Coriolis term we find:

2f j.k � v;rq/L2 j6 cjvjL4 jr��1p31 jL4=3

6 cjvjL4.Z

Mjp1j4dM/3=4 D cjvjL4 jp1j3L4 :

(3.66)


The scalar product containing the nonlinear terms is estimated as follows, using integrationby parts and Sobolev embeddings:

j.rvv;rq/L2 C .w@v@z;rq/L2 j D

D j.uv; @xrq/L2 C .vv; @yrq/L2 C .wv; @zrq/L2 jD j.uv; @xr��1p31/L2 C .vv; @yr��1p31/L2 j

6 cZ 1

0

jjvj2jL4.M0/jp1j3L4.M0/dz

6 cjp1j3L4.M0/Z 1

0

.jrjvj2j1=2L2.M0/C jjvj2j

1=2

L2.M0//jjvj2j1=2

L2.M0/dz

6 cjp1j3L4.jrjvj2j1=2

L2C jvjL4/jvjL4 :

(3.67)

Gathering all the above estimates, we find:

jp1jL4 6 cŒjr3jvj2j1=2L2 C jvjL4 C 1�jvjL4 ; (3.68)

and now Lemma 3.4 follows immediately. �

In the proof of the theorem, we used a generalization of the Sobolev embeddings, for thecase of a Lipschitz domain, namely:

LEMMA 3.5. If f 2H 1.M/, withM a three-dimensional Lipschitz domain, then:

jf jL4 6 c.jf jL2 C jr3f j3=4L2 jf j1=4

L2/: (3.69)

PROOF. Let us recall the proof of this well-known result. For a functiong 2H 10 .M/ (the

space of functions fromH 1.M/ vanishing on the boundary), the following inequality isvalid:

jgjL4 6 cjr3gj3=4jgj1=4: (3.70)

LetM�GIf can be extended fromM toG preserving the class and norm in such waythat the extended functionf vanishes on the boundary ofG. We also know that:

j Qf jH1.G/ 6 cjf jH1.M/; j Qf jL4.G/ 6 cjf jL4.M/: (3.71)

58 M. Petcu et al.

Then:

jf jL4.M/ 6 cj Qf jL4.G/ 6 cjr3 Qf j3=4L2.G/j Qf j1=4

L2.G/

6 cjf j1=4L2.M/

.jf j2L2.M/

C jr3f j2L2.M//3=8

6 c.jf jL2.M/C jr3f j3=4L2.M/jf j1=4

L2.M//:

(3.72)

The lemma is proved. �

A similar result works for the2D case which will be used later on:

LEMMA 3.6. If f 2H 1.M/, andM is a two-dimensional Lipschitz domain, then:

jf jL4.M/ 6 c.jf jL2.M/C jrf j1=2L2.M/jf j1=2

L2.M//: (3.73)

REMARK 3.1. We note here that Theorem 3.1 is not valid for the case of the Navier-Stokesequations. Indeed the fact that, here, the pressurep1 is only a function of the horizontalvariablesx;y is essential.

We now need to estimate theL4-norm ofv:

LEMMA 3.7. For v a solution of the primitive equations, the following estimate holds:

sup06t6T

jvj4L4C �v

Z T

0

jvjr3vjj2L2 6 c; (3.74)

wherec is a constant depending on the initial data.

PROOF. We take theL2-scalar product of (3.43)1 with vjvj2:

1

4

d

dtjvj4L4C .rvv;vjvj2/L2 C .w

@v@z;vjvj2/L2

C 1

�0.rp;vjvj2/L2 � �v.�3v;vjvj2/L2 D 0:

(3.75)

The scalar products from (3.75) containing the nonlinear terms are estimated as follows:


.rvv;vjvj2/L2 C .w@v@z;vjvj2/L2 D

1

4

Z

Mrvjvj4dMC

1

4

Z

Mw@

@zjvj4dM

D 1

4

Z

@Mv � n jvj4d.@M/C 1

4

Z

@Mwn3jvj4d.@M/

� 14

Z

M. div vC @w

@z/jvj4dM

D 0:(3.76)

We also have:

�.�3v;vjvj2/L2 D�Z

@M

@v@n� vjvj2d.@M/C .r3v;r3.vjvj2//L2

DZ

Mjr3vj2jvj2dMC

Z

Mjr3jvj2j2dM:

(3.77)

It now remains to estimate the pressure term from (3.75):

j.rp;vjvj2/L2 j D j.p; div .vjvj2//L2 j6 j.pv;rjvj2/L2 j C j.pjvj2; div v/L2 j6 cj.pjvj2; jrvj/L2 j6 cj.p1jvj2; jrvj/L2 j CCcj.p2jvj2; jrvj/L2 j:

(3.78)

For the term containing the pressurep1, we have:

j.p1jvj2; jrvj/L2 j6Z 1

0

jp1jL4.M0/jjvj2jL4.M0/jrvjL2.M0/dz

6 cjp1jL4.M0/Z 1

0

.jrjvj2j1=2L2.M0/C jjvj2j

1=2

L2.M0//jjvj2j1=2

L2.M0/jrvjL2.M0/dz

6 cjp1jL4.M0/.jrjvj2j1=2L2.M/C jvjL4.M//jvjL4.M/jrvjL2.M/

6 . by 3:68/ c.jrjvj2jL2.M/C jvj2L4.M/C 1/jvj2

L4.M/jrvjL2.M/

6 �v

4jrjvj2j2

L2C c

�vjvj4L4jrvj2

L2C c.jvj4

L4C 1/jrvjL2 :

(3.79)

60 M. Petcu et al.

Using the same reasoning as above, we also have:

j.p2jvj2; jrvj/L2 j6 jp2jL4 jjvj2jL4 jrvjL26 c.jr3jvj2j3=4L2 C jvj

3=2

L4/jvj1=2

L4jrvjL2

6 �v

4jr3jvj2j2L2 C cjvj4L4 C cjrvj2

L2:

(3.80)

Gathering all the estimates, we find:

d

dtjvj4L4C �v

Z

Mjr3vj2jvj2dMC �v

Z

Mjr3jvj2j2dM

6 c.jvj4L4C 1/.jr3vj2L2 C 1/:

(3.81)

Using the Gronwall lemma, (3.81) leads to:

sup06t6T

jvj4L4C �v

Z T

0

jr3jvj2j2L2.M/dt C �v

Z T

0

jjr3vjjvjj2L2.M/dt 6 c:

(3.82)�

COROLLARY 3.2. For the pressure function, the following estimate holds:

Z T

0

jpj4L4dt 6 c: (3.83)

PROOF. Using (3.56) and (3.74), the result follows immediately. �

Using all the estimates obtained till now, we can conclude that:

sup06t6T

.jvj4L4C j�j4

L4/C c0

Z T

0

.j�r3�j2L2

C jjr3vjjvjj2L2 C jr3jvj2j2L2 C j�j4L4/dt 6 c:(3.84)

A priori estimates onvz and�z


In order to estimatevz and�z , we differentiate (3.43)1 and (3.43)4 in z. DenotinguDvz , we find:

ut � �v�3uC 2f k � uC 1

�0rpz CrvuCw@u

@z

CruvCwz@v@zD 0;

div uD 0;

�zt Crv�z Cw@�z

@zCru�Cwz

@�

@z� ��3�D 0:

(3.85)

LEMMA 3.8. For uD vz , the following estimate holds:

sup06t6T

ju.t/j2L2C �v

Z T

0

jr3uj2L2dt 6 c: (3.86)

PROOF. We take theL2-scalar product of (3.85)1 with u:

1

2

d

dtju.t/j2

L2C �vjr3uj2L2 C

1

�0.rpz ;u/L2 C .ruvCwz

@v

@z;u/L2 D 0:

(3.87)

We note here that.rv;u;u/L2 C .w@v=@z;u/L2 D 0 because of the incompressibility con-dition.

The pressure term from (3.87) is estimated as follows:

j.rpz ;u/L2 j D j.pz ; div u/j6 cjpzjL2 jr3ujL2

6 cj�jL2 jr3ujL2 6 cjr3ujL2 6�v

4jr3uj2L2 C c:

(3.88)

The remaining nonlinear terms from (3.87) give us:

j.ruv;u/L2 � .. div v/@v@z;u/L2 j

6 cZ

MjvjjrujjujdMC j

Z

M. div vz/v � udMj

C jZ

M. div v/v � uzdMj

6 cjvjL4 jujL4 jrujL2 C j. div v/vjL2 juzjL26 cjvjL4 juj1=4L2 jr3uj

7=4

L2C j. div v/vjL2 juzj2L

6 �v

4jr3uj2L2 C cjvj8L4 juj2L2 C cj. div v/vj2

L2:

(3.89)

62 M. Petcu et al.

Gathering all these estimates, we find:

d

dtjuj2L2C �vjr3uj2L2 6 cjuj2L2 C cj.div v/vj2

L2: (3.90)

From (3.84) and using the Gronwall lemma, the required result is proved. �

We now estimate theL4-norm ofu:

LEMMA 3.9. For uD vz , the following estimate holds:

sup06t6T

juj4L4C �v

Z T

0

ˇjr3ujjuj

ˇ2L2dt C �v

Z T

0

jr3juj2j2L2dt 6 c: (3.91)

PROOF. We take theL2-scalar product of (3.85)1 with juj2u. After integration by parts,we find:

1

4

d

dtjuj4L4C �v

Z

Mjr3uj2juj2dMC

�v

2

Z

Mjr3juj2j2dM

C 1

�0.rpz ;ujuj2/L2 C .ruvCwz

@v@z; juj2u/L2 D 0:

(3.92)

The pressure term is estimated in the following way:9

j.rpz ;ujuj2/L2 j6 j.pz ; .div u/juj2/L2 jC j.pz ;u � r.juj2//L2 j6 cjpzjL4

ˇjrujjuj

ˇL2jujL4

6 cj�jL4ˇjrujjuj

ˇL2jr3uj3=4L2 juj

1=4

L2

6 �v

4

ˇjr3ujjuj

ˇ2L2C cjr3uj2L2 C cjuj2L2 :

(3.93)

Here we applied (foru) the fact that in a3D domain the estimatejwjL4 6 cjr3wj3=4L2 jwj1=4

L2

is valid for a functionw vanishing on part of the boundary.

9In the following equationsjuj2 stands for the pointwise value of the norm of the vectoru, and similarly forthe other quantities.


We also need to estimate the last term from the left-hand-side of (3.92):

I1 D j.ruv; juj2u/L2 j

D jZ

Mv.div u/juj2udMC

Z

Mvru.juj2u/dMj

� (by integration by parts)

6Z

Mjdiv ujjvjjuj3dMC

Z

Mjrujjvjjuj3dM

6 cˇjr3ujjuj

ˇL2jvjL4.

Z

Mjuj8dM/1=4

6 cˇjr3ujjuj

ˇL2jvjL4 jjuj2j1=4L2 jr3juj2j

3=4

L2

6 �v

8.ˇjr3ujjuj

ˇ2L2Cˇr3juj2

ˇ2L2/C cjvj8

L4juj4L4:

(3.94)

Integrating by parts, we also have:

I2 Dˇˇ�wz@v@z; juj2u

�

L2

ˇˇD j.div .v/ u; juj2u/L2 j

6Z

Mjvjjrjuj4jdM

6 cZ

Mjvjjuj2jrjuj2jdM

6 cjvjL4 jjuj2jL4 jrjuj2jL26 cjvjL4 jr3juj2j3=4L2 juj

1=2

L4jrjuj2jL2

6 �v

8jr3juj2j2L2 C cjvj8L4 juj4L4 :

(3.95)

Therefore, from all the above inequalities, we find:

d

dtjuj4L4C �v

Z T

0

Œˇjujjr3uj

ˇ2L2C jr3juj2j2L2 �dt 6 c.juj4L4 C jr3uj2L2 C juj2L2/:

(3.96)

Applying the Gronwall lemma to (3.96) and using (3.86), (3.91) follows immediately.�

We now need to deduce the estimates for�z :

64 M. Petcu et al.

LEMMA 3.10. For �z the following estimate holds:

sup06t6T

j�zj2L2 C ��Z T

0

jr3�zj2L2dt 6 c: (3.97)

PROOF. We take theL2-scalar product of (3.85)3 with �z and we find:

1

2

d

dtj�zj2L2 C ��jr3�zj2L2 C .ru�Cwz

@�

@z; �z/L2 D 0: (3.98)

We have:

I1 D jru�; �z/L2 j D j.div u; ��z/L2 C .�u;r�z/L2 j6 jdiv ujL2 j�jL4 j�zjL4 C j�jL4 jujL4 jr�zjL26 cjdiv ujL2 jr3�zj3=4L2 j�zj

1=4

L2C cjr�zjL2

6 ��4jr3�zj2L2 C c.jdiv uj2

L2C j�zj2L2/C c;

(3.99)

and also:

I2 D j.wz@�

@z; �z/L2 j D j.div v; �2z/L2 j D 2j.v; �zr�z/L2 j

6 cjvjL4 j�zjL4 jr�zjL26 cjvjL4 j�zj1=4L2 jr3�zj

7=4

L2

6 ��4jr3�zj2L2 C cjvj8L4 j�zj2L2 :

(3.100)

Using these estimates, (3.98) leads to:

d

dtj�zj2L2 C ��jr3�zj2L2 6 cjdiv uj2

L2C cj�zj2L2 C c; (3.101)

and using the Gronwall lemma (3.97) follows. �

A priori estimates onvt and�tFor obtaining a priori estimates onvt and�t , we differentiate equations (3.43)1 and

(3.43)4 in t and we obtain:


vt t � �v�3vt C 2f k � vt Crvvt Cw@vt@zCrvt vCwt

@v@zC 1

�0rpt D 0;

�t t � ��3�t Crv�t Cw@�t

@zCrvt�Cwt

@�

@zD 0:

(3.102)

We can then find:

LEMMA 3.11. For vt and�t the following estimate holds:

sup06t6T

.jvt .t/j2L2 C j�t .t/j2L2/C c1Z T

0

.jr3vt j2L2 C jr3�t j2L2/dt

6 c1.jvt .0/j2L2 C j�t .0/j2L2/:(3.103)

PROOF. We take theL2-scalar product of (3.102), withvt and of (3.102)2 with �t and weobtain:

1

2

d

dtjvt .t/j2L2 C �vjr3vt .t/j2L2 C

1

�0.rpt ;vt .t//L2 C .rvt vCwt

@v@z;vt /L2 D 0;

1

2

d

dtj�t .t/j2L2 C ��jr3�t .t/j2L2 C .rvt�Cwt

@�

@z; �t /L2 D 0:

(3.104)

We need to estimate the scalar products in (3.104):

j.rpt ;vt /L2 j D j.pt ;div vt /L2 j D j.pt ;wzt /L2 jD j.ptz ;wt /L2 j D gj.�t ;wt /L2 j

6 cj�t jL2 jrvt jL2 6�v

8jr3vt j2C cj�t j2L2 :

(3.105)

For the last term of (3.104)1, we have:

j.rvt v;vt /L2 j6 jrvt jL2 jvjL4 jvt jL46 cjvjL4 jvt j1=4L2 jr3vt j

7=4

L2

6 �v

8jr3vt j2L2 C cjvj8L4 jvt j2L2 ;

(3.106)

66 M. Petcu et al.

and

j.wt@v@z;vt /L2 j6 jwt jL2 jvzjL4 jvt jL4

6 cjrvt jL2 jvzjL4 jr3vt j3=4L2 jvt j1=4

L2

6 �v

8jr3vt j2L2 C cjvzj8L4 jvt j2L2 :

(3.107)

Gathering these estimates, the following inequality onvt is obtained:

d

dtjvt .t/j2L2 C

5

4�vjr3vt .t/j2L2 6 c.j�t j2L2 C jvt j2L2/: (3.108)

For the density, the last term from (3.104)2 is estimated as follows:

j.rvt�; �t /L2 j6 jrvt jL2 j�jL4 j�t jL46 cjrvt jL2 j�jL4.jr3�t j3=4L2 j�t j

1=4

L2C j�t jL2/

6 �v

8jr3vt j2L2 C

��

4jr3�t j2L2

C cj�t j2L2 j�j2L2 j�j2L4 C cj�j4L4 j�t j2L2 ;

(3.109)

and

j.wt@�

@z; �t /L2 j6 jwt jL2 j�zjL4 j�t jL4

6 cjrvt jL2 j�zjL4.jr3�t j3=4L2 j�t j1=4

L2C j�t jL2/

6 �v

8jr3vt j2L2 C

��

4jr3�t j2L2 C cj�t j2L2.j�zj2L4 C j�zj4L4/:

(3.110)

The estimates on� lead to:

1

2

d

dtj�t j2L2 C

��

2jr3�t j2L2 6

�v

4jr3vt j2L2 C cj�t j2L2 : (3.111)

By summing (3.108) and (3.111), we find:

d

dt.jvt .t/j2L2 C j�t .t/j2L2/C �vjr3vt j2L2 C ��jr3�t j2L2 6 c.jvt .t/j2L2 C j�t .t/j2L2/;

(3.112)


and from the Gronwall lemma, (3.103) follows. �

Note that the right-hand-side of (3.103) depends on values that are not input data of theproblem. So we need to estimate the right-hand-side of (3.103) in terms of the initial data.

From (3.43)4 we can deduce:

j�t .0/jL2 6 c.j�0jH2 C jv0jH2/: (3.113)

In order to boundvt .0/ we take the scalar product of (3.43)1 with vt :

jvt j2L2 D � �v.�3v;vt /L2 � 2f .k � v;vt /L2 � .rp;vt /L2

� .rvv;vt /L2 � .w@v@z;vt /L2 :

(3.114)

Then:

j.�3v;vt /L2 j6 j�3vjL2 jvt jL2 6 cjvjH2 jvt jL2 ;j2f .k � v;vt /L2 j6 cjvjL2 jvt jL2 :

(3.115)

For the pressure term, we write:

j.rp;vt /L2 j D j.rp1;vt /L2 C .rp2;vt /L2 j: (3.116)

Sincep1 is independent oft , one can easily see that:

j.rp1;vt /L2 j D j.p1; div vt /L2 j D j.p1;wtz/L2 j D 0: (3.117)

Forp2 we use the fact thatR 10p2dz D 0 and so the following inequality holds:

jrp2jL2 6 cjrp2;zjL2 ; (3.118)

and one can deduce the following:

j.rp2;vt /L2 j6 jrp2jL2 jvt jL2 6 cjrp2;zjL2 jvt jL2 6 cjr�jL2 jvt jL2 : (3.119)

The scalar products containing the nonlinear terms are bounded as follows:

j.rvv;vt /L2 j6 jrvvjL2 jvt jL2 6 cjvj2H2 jvt jL2 ; (3.120)

and

j.w @v@z;vt /L2 j6 jwjL6 j

@v@zjL3 jvt jL2 6 cjvj2H2 jvt jL2 : (3.121)

68 M. Petcu et al.

Taking into account these estimates, (3.114) implies:

jvt jL2 6 c.jvj2H2 C j�j2H1/; (3.122)

so we find:

jvt .0/jL2 C j�t .0/jL2 6 c.jv0jH2 C j�0jH2/: (3.123)

Using (3.103) we conclude that:

sup06t6T

.jvt .t/j2L2 C j�t .t/j2L2/C c1Z T

0

.jr3vt j2L2 C jr3�t j2L2/dt

6 cT .jv0j2H2 C j�0j2H2/:(3.124)

We can obtain a stronger estimate forjr3�jL2 : we take the scalar product of (3.43)4 with� and we find

��jr3�j2L2 D�.�t ; �/L2 6 j�t jL2 j�jL2 6 c; (3.125)

with c depending on theH 2 norms of the initial data.Similarly, we findjr3vj2L2 6 c.

We are now able to prove the existence and uniqueness of the strong solutions.

Existence and uniqueness of a strong solution

Our final result for this problem is the proof of existence and uniqueness of strong solu-tions of (3.43)—(3.47) which belong to the following spaces (in space dimension three):

V D fvD .u; v/ 2H 1.MT /Iv satisfies.3:44/;vz 2H 1.MT /;

Z 1

0

div v.t; x; y; z/dz D 0 g;(3.126)

RD f� 2H 1.MT /I�z 2H 1.MT / g:The uniqueness of such solutions can be proved classically, by estimating theL2 norm

of the difference of two solutions (see Section 3.1). The existence of such a solution followsfrom the existence of a local strong solution as in Section 3.1, which satisfies all the a prioriestimates proved before. We note that the a priori estimates are independent of the lengthof the time interval, hence the solution cannot blow up before the end of the consideredtime interval.

Thus we have proved the main result of this Section 3.2:

THEOREM 3.2. We assume thatM is cylindrical with constant depth as in (3.42). Letv0 2 H 2.M/; �0 2 H 2.M/ satisfying the boundary conditions (3.44), (3.45) and the


compatibility condition (3.47). Then, for anyT > 0; the problem (3.43) - (3.46) has aunique (strong) solutionv 2 V; � 2R onMT ; such thatv;vz ;r3v;r3vz ;vt ;vtz belong toL2.MT / and the normjr3vjL2 is continuous ont:

A new result, extending Theorem 3.2, was obtained in the very recent article by Kukav-ica and Ziane (2007). The authors proved the existence of global strong solutions of theprimitive equations of the ocean in the case of a non-flat bottom. The boundary conditionsconsidered by the authors are the Dirichlet conditions on the side and the bottom bound-aries. These conditions applied on a varying bottom topography are physically interestingand they were not covered by the previous works. They assume for simplicity that� D 0which is not mathematically restrictive. They then prove that ifFv is inL2.0; t1IL2.M/2/

for all t1 > 0 andv0 2 H 1.M/2, v0 D 0 on �l and�b , thenv, solution of (3.43)1 and(3.43)2 and v.0; x; y; z/ D v0.x;y; z/ exists and is unique inL1.0; t1IH 2.M/2/ andL2.0; t1IH 2.M/2/, for all t1 > 0.

3.3. Strong solutions of the two-dimensional primitive equations:Physical boundary conditions

In this section, we are concerned with the global existence and the uniqueness of strongsolutions of the two-dimensional primitive equations of the ocean. We will first derive theequations formally from the three-dimensional PEs under the assumption of invariancewith respect to they-variable, i.e., we will assume that the initial data, the forcing terms,as well as the depth functionh are independent of the variabley. The uniqueness of weaksolutions implies that the solution will be independent ofy.

In Sections 3.3.1 and 3.3.2 we introduce the two-dimensional PEs and present their weakformulation. In Sections 3.3.2 and 3.3.3 we show that the strong solutions provided by ananalog of Theorem 3.1 are in fact defined for allt > 0; this result is based on appropriatea priori estimates which are described hereafter; the approach is different and somehowsimpler than for dimension three in Section 3.2.

3.3.1. The two-dimensional primitive equations.We assume that the domain occupiedby the ocean is represented by

˚.x;y; z/ 2R3; x 2 .0;L/; y 2R;�h.x/ < z < 0;

and we denote byM its cross section:

MD ˚.x; z/; x 2 .0;L/;�h.x/ < z < 0: (3.127)

HereL is a positive number andh W Œ0;L�!R satisfiesh 2 C3.Œ0;L�/,

h.x/> h > 0 for x 2 .0;L/; h0.0/D h0.L/D 0: (3.128)

70 M. Petcu et al.

By dropping all the terms containing a derivative with respect toy in the three-dimensional primitive equations (2.44)–(2.50), we obtain the following system:

@u

@tC u@u

@xCw@u

@z��v

@2u

@x2� �v

@2u

@z2� f vC @ps

@xD g

Z 0

z

@�

@xdz0CFu;

(3.129)

@v

@tC u@v

@xCw@v

@z��v

@2v

@x2� �v

@2v

@z2C f uD Fv; (3.130)

@T

@tC u@T

@xCw@T

@z��T

@2T

@x2� �T

@2T

@z2D FT ; (3.131)

@S

@tC u@S

@xCw@S

@z��S

@2S

@x2� �S

@2S

@z2D FS ; (3.132)

@u

@xC @w

@zD 0; (3.133)

where

�D 1� ˇT .T � Tr/C ˇS .S � Sr/; (3.134)Z

MS dMD 0: (3.135)

Here u and v are the two components of the horizontal velocityv. Note that, despitey-invariance,v does not vanish in the problem of physical relevance (unlike the two-dimensional Navier–Stokes equations). The quantityps above is the same asp in (2.49),whereas the expressionP in (2.49) has been replaced byg

R 0z�dz0, with � being a function

of T andS through (3.10). Finally, as in the three-dimensional case,F D .Fu;Fv;FT ;FS /vanishes in the physical problem and it is added here for mathematical generality.

Boundary conditions. These equations are supplemented with the same set of boundaryconditions and initial data as in Section 2. On the top boundary ofM, denoted�i ; �i Df.x;y/Ix 2 .0;L/I z D 0g, we have (see also after (2.54)):

�v@u

@zC ˛vuD gu; �v

@v

@zC ˛vvD gv;

(3.136)

�T@T

@zC ˛T T D gT ;

@S

@zD 0:

On the remaining part of the boundary, we assume the Dirichlet boundary condition for thevelocity and the Neumann condition for the temperature and the salinity. That is,

.u; v;w/D .0; 0; 0/ on�` [ �b;(3.137)

@T

@nTD @S

@nSD 0 on�` [ �b;


where

�` D˚.x;y/Ix D 0 orL;�h.x/ < z < 0;

(3.138)�bD

˚.x; z/Ix 2 .0;L/; z D�h.x/:

We also have the initial data given by

ujtD0 D u0; vjtD0 D v0; T jtD0 D T0 and S jtD0 D S0: (3.139)

3.3.2. Weak formulation. The main result.We now proceed, as in Section 2, towards theweak and functional formulations of this problem (3.129)–(3.135) with some simplifica-tions due to the invariance with respect toy, and some other aspects which are specific todimension two.

We introduce, as in Section 2.2.1, the following spaces:

V D V1 � V2 � V3; H DH1 �H2 �H3;

V1 D�

vD .u; v/ 2H 1.M/2;@

@x

Z 0

�h.x/u.x; z/dz D 0;vD 0 on�` [ �b

�;

V2 DH 1.M/;

V3 D PH 1.M/D�S 2H 1.M/;

Z

MS dMD 0

�; (3.140)

H1 D�

vD .u; v/ 2L2.M/2;

Z 0

�h.x/u.x; z/dz D 0;uD 0 on�`

�;

H2 DL2.M/;

H3 D PL2.M/D�S 2L2.M/;

Z

MS dMD 0

�:

The scalar products are defined exactly as in Section 2.2.1,r being replaced by@=@x. Thecondition

R 0�h.x/ u.x; z/dz D 0 comes from the fact that the derivative inx of this quantity

vanishes (the two-dimensional analog of (2.48)) and that this quantity vanishes atx D 0andL (see Section 2.2.1 for the three-dimensional analog).

We also introduce the spacesV1;V2;V3 andV D V1 �V2 �V3 with a similar definition.Similarly we consider the formsa; b; c; e, defined exactly as in dimension three, just

deleting all quantities involving ay-derivative; the associated operatorsA;B;E are definedin the same way.

With these notations, the weak formulation is exactly as in dimension three (see (2.79)and (2.80) or, in operational form, (2.81) and (2.82)). There is no new difficulty in provingthe analogue of Theorem 2.4 giving the existence, for all time, of weak solutions.

Similarly we can prove, exactly as in Section (3.1), an analogue of Theorem 3.1. Ouraim in this section is to show thatt� D t1 in space dimension two, for thet� appearing inthe statement of Theorem 3.1.

72 M. Petcu et al.

More precisely we will prove the following (cf. Theorems 3.1 and 2.2):

THEOREM 3.3. We assume thatM is as in(3.127)and that(3.128)is satisfied.We are givent1 > 0;U0 2 V;F D .Fv;FT ;FS /, and g D .gv; gT / such thatF and

dF=dt are inL2.0; t1IH/ .or L2.0; t1IL2.M/4/ andg anddg=dt are inL2.0; t1IH 10 .�i/

3/.Then there exists a unique solutionU of the primitive equations(2.79)and (2.80)such

that

U 2 C�Œ0; t1�IV�\L2�0; t1IH 2.M/4

�: (3.141)

PROOF. The proof of uniqueness is easy and done as in Theorem 3.1 for dimension three.To prove the existence of solutions, we start from the strong solution given by the two-dimensional analogue of Theorem 3.1 and prove by contradiction thatt� D t1. Indeed letus denote byŒ0; t0� the maximal interval of existence of a strong solution, that is,10

U 2L1�0; t 0IV � (3.142)

for everyt 0 < t0 and (3.142) does not occur fort 0 D t0, which means, in particular, that

limsupt!t0�0

U.t/ DC1: (3.143)

We will show that (3.143) cannot occur: we will derive a finite bound forkU.t 0/k on Œ0; t0�,thus contradicting (3.143).

The bounds forkU.t/k will be derivedsequentially: we will show successively thatuz ; ux are inL1.0; t0IL2.M// andL2.0; t0IH 1.M//, where'x D @'=@x and'z D@'=@z; then we will prove at once thatv; T andS are inL1.0; t0IH 1/ andL2.0; t0IH 2/.In fact, we will give the proofs foruz ; ux ; T ; the other quantities being estimated in exactlythe same way. For the sake of simplicity, we assume hereafter thatgD .gv; gT /D 0. Wheng ¤ 0, we need to “homogenize” the boundary conditions by consideringU 0 D U � U �,with U � defined exactly as in Section (3.1), then perform the following calculationsfor U 0.11

Before we proceed, let us recall that we have already available the a priori estimatesfor U in L1.0; t1IL2/ andL2.0; t1IH 1/ used to prove the analog of Theorem 2.2 (thatis (2.93) in the discrete case). �

3.3.3. Vertical averaging. To derive the new a priori estimates, we need some operatorsrelated to vertical averaging that we now define.

For any function' defined and integrable onM, we set

xP'.x/DZ 0

�h.x/'.x; z/dz;

10It is easy to see that, ifU 2L1.0; t 0IV /, thenU 2L2.0; t 0IH2.M/4/ as well.11Note that, in (3.1),U� was chosen so that the initial and boundary conditions forU 0 vanish. Here we do not

need to homogenize the initial condition, but we can use the sameU�.


(3.144)

P' D 1

hxP'; Q' D ' �P':

We now establish some useful properties of these operators, some simple, some moreinvolved.

We first note thatPQD 0, so that,

Z 0

�h.x/Q'.x; z/dz D 0 8' 2L1.M/; (3.145)

andZ

M.P'/.x/.Q /.x; z/dx dz D 0 8'; 2L2.M/: (3.146)

Also, for all ' sufficiently regular,

xP @'

@xD @

@xxP' � h0.x/'�x;�h.x/�;

xP @2'

@x2D @2

@x2xP' � 2h0.x/@'

@x

�x;h.x/

�

C h0.x/2 @'@z

�x;�h.x/�� h00.x/'�x;�h.x/�: (3.147)

Now, if ' vanishes on�b; '.x;�h.x//D 0; 0 < x <L, then

@'

@x

�x;�h.x/�D h0.x/@'

@z

�x;�h.x/�; (3.148)

and hence

xP @'

@xD @

@xxP';

xP @2'

@x2D @2

@x2xP' � h0.x/@'

@x

�x;�h.x/�;

P@2'

@x2.x/D�h

0

h

@'

@x

�x;�h.x/�D�h

02

h

@'

@z

�x;�h.x/�: (3.149)

Finally, the following lemma will be needed.

LEMMA 3.12. For anyv 2H 2.M/ such that

�@v

@zC ˛vD 0 on�i ; vD 0 on�` [ �b;

74 M. Petcu et al.

we have

Z

M

@2v

@x2@2v

@z2dz dx D

Z

M

ˇˇ @2v@x@z

ˇˇ2

dz dxC ˛Z L

0

ˇˇ@[email protected]; 0/

ˇˇ2

dx

� 12

Z L

0

h00.x/ˇˇ@v@z

�x;�h.x/�

ˇˇ2

dx: (3.150)

PROOF. We give the proof forv smooth, sayv 2 C3.M/; the result extends then tov 2H 2.M/ using a density argument (that we skip).

We write

@2v

@x2@2v

@z2D @

@x

�@v

@x

@2v

@z2

�� @v@x

@3v

@x @z2

D @

@x

�@v

@x

@2v

@z2

�� @

@z

�@v

@x

@2v

@x @z

�Cˇˇ @2v@x @z

ˇˇ2

: (3.151)

We first integrate inz and, taking into account (3.147) we obtain

Z 0

�h

@2v

@x2@2v

@z2dz

D I � h0.x/ @v@x

�x;�h.x/�@

2v

@z2

�x;�h.x/�� @v

@x.x; 0/

@2v

@x @z.x; 0/

(3.152)

C @v

@x

�x;�h.x/� @

2v

@x @z

�x;�h.x/�C

Z 0

�h

ˇˇ @2v@x @z

ˇˇ2

dz;

I D @

@x

Z 0

�h

@v

@x

@2v

@z2dz:

We now integrate inx. The integral ofI D I.x/ vanishes becausev.0; z/D v.L; z/D 0for all z, so that

�@2v

@z2

�.0; z/D

�@2v

@z2

�.L; z/D 0 8z:

The third term on the right-hand side of (3.152) is equal to˛R L0j.@v=@x/.x; 0/j2 dx.

The sum of the second and fourth terms is equal to

Z L

0

@v

@x

�x;�h.x/�

�@2v

@x @z� h0 @

2v

@z2

��x;�h.x/�dx: (3.153)

Setting'.x/D .@v=@z/.x;�h.x//, we see that

'0.x/D @2v

@x @z

�x;�h.x/�� h0.x/@

2v

@z2

�x;�h.x/�;


and sincev.x;�h.x//D 0, we have

@v

@x

�x;�h.x/�� h0.x/@v

@z

�x;�h.x/�D 0;

and the integral in (3.153) is equal to

Z L

0

h0.x/'.x/'0.x/dx D�12

Z L

0

h00.x/'2.x/dxI

for the last relation we have used'.0/D '.L/D 0. The lemma is proved. �

3.3.4. Estimates foruz . To show thatuz 2 L1.0; t0IL2.M// \ L2.0; t0IH 1.M//,we multiply (3.129) byQuzz , integrate overM and integrate by parts, and rememberthatQuD u. Thus, for each term successively, omitting the variablet , we find:

Z

MutQ.�uzz/dMD�

Z

Mutuzz dM

D�Z L

0

utuz�0�h dxC

Z

Mutzuz dM

D ˛v

�v

Z L

0

ut .x; 0/u.x; 0/dxC1

2

d

dtjuzj2L2

D 1

2

d

dt

�juzj2L2 C

˛v

�v

ˇu.x; 0/

ˇ2L2.�i/

�;

Z

MuuxQ.�uzz/dMD

Z

Muuxuzz dMC

Z

MuuxPuzz dM

D�Z 0

L

uuxuz �0�h dxC

Z

Muuxzuz dM

CZ

Muxu

2z dMC

Z

MuuxPuzz dM

D ˛v

�v

Z L

0

u2.x; 0/ux.x; 0/dxC1

2

Z

@Munxu

2z d.@M/

C 1

2

Z

Muxu

2z dMC

Z

MuuzPuzz dM

D ˛v

3�vu3.x; 0/

�L0C 1

2

Z

Muxu

2z dMC

Z

MuuxPuzz dM

D 1

2

Z

Muxu

2z dMC

Z

MuuxPuzz dM:

76 M. Petcu et al.

In the relations above,nD .nx ; nz/ is the unit outward normal on@M, and we used thefact thatunx D 0 on@M, and thatu.0; 0/D u.L;0/D 0 becauseuD 0 on�`.

Z

MwuzQ.�uzz/dMD �

Z

Mwuzuzz dMC

Z

MwuzPuzz dM

D �12

Z L

0

wu2z�0�h dxC 1

2

Z

Mwzu

2z dM

CZ

MwuzPuzz dM

D .sincewD 0 on�i and�b andwz D�uz/

D �12

Z

Muxu

2z dMC

Z

MwuzPuzz dM:

SincePuzz is independent ofz andw vanishes on�i and�b, we have

Z

MwuzPwzz dMD

Z L

0

wu�0�hPuzz dx �

Z

MuwzPuzz dM

DZ

MuuzPuzz dM:

Finally the last two terms add up in the following way:Z

M.uux Cwuz/Q.�uzz/dMD 2

Z

MuuxPuzz dM

D 2Z

M

1

huux

�uz.x; 0/� uz.x; h/

�dM;

Z

MpsxQ.�uzz/dMD 0 (sinceps is independent ofz/;

��v

Z

MuzzQ.�uzz/dMD �vjQuzzj2L2 D �vjuzz j2L2 � �vjPuzz j2L2 ;

��v

Z

MuxxQ.�uzz/dMD �v

Z

Muxxuzz dM��v

Z

M.Puxx/uzz dM:

Using (3.149) and Lemma 3.1, we see that this expression is equal to

�vjuxzj2L2 C ˛v�v

Z L

0

ˇux.x; 0/

ˇ2dx

��v

Z L

0

�h0.x/2

h.x/C 1

2h00.x/

�ˇuz�x;�h.x/�

ˇ2dx

� ˛v

�v�v

Z L

0

uz.x;�h/u.x; 0/dx:


The other terms are left unchanged, then gathering all these terms we find

1

2

d

dt

�juzj2L2.M/

C ˛v

�v

ˇu.x; 0/

ˇ2L2.�i/

�C�vjuxzj2L2.M/

C ˛v�vˇux.x; 0/

ˇ2L2.�i/

C �vjuzzj2L2.M/

D �vjPuzz j2L2 C�v

Z L

0

�h02

hC 1

2h00�ˇuz�x;�h.x/�

ˇ2dx

C ˛v

�v�v

Z L

0

uz.x;�h/u.x; 0/dx

CZ

Mf vQuzz dM� g

Z

M

�Z 0

z

�x dz0�uzz dM: (3.154)

We estimate the right-hand side of (3.154) as follows,c denoting a constant dependingonly onM and on the coefficientsv;�v; �v,

Puzz D1

h

�uz.x; 0/� uz.x;�h/

�D 1

h

˛v

�vu.x; 0/� 1

huz�x;�h.x/�;

jPuzzjL2.M/ 6 cˇu.x; 0/

ˇL2.�i/

C cˇuz�x;�h.x/�

ˇL2.�i/

: (3.155)

The last two norms are bounded by the trace theorems:

jujL2.�i/6 cjuj1=2

L2.M/kuk1=2; (3.156)

ˇuz�x;�h.x/�

ˇL2.�i/

6 cjuzjL2.�b/

6 cjuzj1=2L2.M/jruzj1=2L2.M/

6 ckuk1=2�juzz j2L2.M/C juzxj2L2.M/

�1=4; (3.157)

jPuzzj2L2.M/6 cjujL2.M/kukC ckukkuzk:

We write alsoˇˇZ

Mf vQuzz dM

ˇˇ 6 cjvjL2.M/juzzjL2.M/

6 cjvjL2.M/kuzk:

Finally using again (3.156) and (3.157) for the other terms on the right-hand side of(3.154), we obtain

1

2

d

dt

�juzj2L2.M/

C ˛v

�v

ˇu.x; 0/

ˇ2L2.�i/

�C �kuzk2C ˛v�v

ˇux.x; 0/

ˇ2L2.�i/

78 M. Petcu et al.

6 cjujL2.M/kukC ckukkuzkC cjvjL2kuzkC c�jTxjL2 C jSxjL2

�juzz jL2 C jFujL2 juzzjL2

6 �2kuzk2C ckuk2C cjvj2L2 C ckU k2C cjFuj2L2 ;

where� Dmin.�v; �v/. Hence

d

dt

�juzj2L2.M/

C ˛v

�v

ˇux.x; 0/

ˇ2L2.�i/

�C �kuzk2 6 ckU k2C cjFuj2L2.M/

:

(3.158)

Taking into account the earlier estimates ofU in L1.0; t1IH/ andL2.0; t1IV /, weobtain an a priori bound ofuz in L1.0; t0IL2.M//, a first step in proving thatt0 can notbe less thant1.

REMARK 3.2. We recall that the estimates above were made under the simplifying as-sumption thatg D .gv; gT / D 0. When this is not the case, we explained that we oughtto considerU 0 D U � U �;U � defined as in Section 3.1. Then the calculations above aremade for the equation foru0. This equation will involve some additional terms such asu�@u0=@x;u0@u�=@x, etc.; these additional terms are estimated in a similar way, leadingto the same conclusions. The same remark applies for the estimates below concerningux ; vz , etc.

3.3.5. Estimates forux . To show thatux is bounded inL1.0; t0IL2.M// \ L2.0; t0IH 1.M//, we multiply (3.129) by�uxx , integrate overM and integrate by parts. At anyfixed timet , each term can be written as follows:

�Z

Mutuxx dMD�

Z

@Mutuxnx d.@M/C

Z

Mutxux dM

D 1

2

d

dt

Z

Mu2x dM;

�Z

Muuxuxx dMD�1

2

Z

@Muu2xnx d.@M/C 1

2

Z

Mu3x dM

D 1

2

Z

Mu3x dM;

�Z

Mwuzuxx dMD�

Z

@Mwuzuxnx d.@M/C

Z

Mwxuzux dM

CZ

Mwuzxux dM

DZ

Mwxuzux dMC 1

2

Z L

0

wu2x�0�h dx


� 12

Z

Mwzu

2x dM

DZ

Mwxuzux dMC 1

2

Z

Mu3x dM;

�v

Z

Muzzuxx dMD (thanks to Lemma 3.1)

D �vjuzxj2L2 C ˛v�v

Z L

0

ˇux.x; 0/

ˇ2dx

� 12

Z L

0

h00.x/ˇuz�x;�h.x/�

ˇ2dx:

The other terms are left unchanged and, with� Dmin.�v; �v/, we arrive at

1

2

d

dtjuxj2L2 C �kuxk2C ˛v�v

Z L

0

ˇux.x; 0/

ˇ2dx

D�Z

Mu3x dM�

Z

Mwxuzux dM

�Z

Mf vuxx dMC 1

2

Z L

0

h00.x/ˇuz�x;�h.x/�

ˇ2dx

� gZ

M

�Z 0

z

�x dz0�uxx dMC

Z

MFuuxx dM:

We writeˇˇZ

Mu3x dM

ˇˇ6 juxj3L3

6 (by Sobolev embedding and interpolation)

6 cjuxj3H1=3 6 cjuxj2L2 juxjH1

6 ckuk2kuxk

6 �

10kuxk2C ckuk4;

ˇˇZ

Mwxuzux dM

ˇˇ6 jwxjL2 juzjL4 juxjL4

6 (by Sobolev embedding and interpolation)

80 M. Petcu et al.

6 cjuxxjL2 juzj1=2L2 kuzk1=2juxj1=2

L2kuxk1=2

6 cjuzj1=2L2 kuzk1=2juxj1=2

L2kuxk3=2

6 �

10kuxk2C cjuzj2L2kuzk2juxj2L2

and

ˇˇZ

Mf vuxx dM

ˇˇ6 jf vjL2 juxx jL2 6

�v

10kuxk2C cjvj2L2 :

The next terms are bounded as before and the last term is easy. Hence

d

dtjuxj2L2.M/

C �kuxk2C ˛v�v

Z L

0

ˇux.x; 0/

ˇ2dx

6 ckuk4C cjvj2L2C cjuzj2L2.M/

kuzk2jwzj2L2.M/

C ckukkuzkC ckU k2C cjFuj2L2.M/: (3.159)

Remembering thatkuk2 D juxj2L2.M/C juzj2L2.M/

, we see that the right-hand side of

(3.159) is of the form�.t/C �.t/juxj2L2.M/, where�; � are inL1.0; t1/; for �D ckuk2C

cjuzj2L2.M/kuzk2, this follows from the previous estimates onU and onuz ; similarly

the contribution ofckukkuzk to � is in L1.0; t1/ due to the previous results onuz andU . Therefore the Gronwall lemma applied to (3.159) provides an a priori bound ofux inL1.0; t0IL2.M// and inL2.0; t0IH 1.M//.

3.3.6. Estimates forv;T andS . We now prove at once thatT is bounded inL1.0; t0IH 1.M// andL2.0; t0IH 2.M//; the proof is similar forv andS , and this will thus con-clude the proof of Theorem 3.3.

For this, we multiply each side of (3.131) byA2T D��T @2T=@x2 � �T @2T=@z2. Werecall thatgT D 0 here; see the end of Section 3.3.2. We have

�Z

MTt .�T Txx C �T Tzz/dMD�

Z

@MTt .�T Txnx C �T Tznz/d.@M/

C 1

2

Z

M.�T TtxTx C �T TtzTz/dM

D �see the notations in (2.38) and after (2.54)�

D�Z

@MTt@T

@nTd.@M/C 1

2

d

dta2.T;T /


D �with (2.35), (2.55) andgT D 0�

D 1

2

d

dta2.T;T /C

Z

�i

˛T T Tt d�i

D 1

2

d

dta2.T;T /C ˛T

ˇT .x; 0/

ˇ2L2.�i/

Hence, we find

1

2

d

dt

�ˇA1=22 T

ˇ2L2C ˛T

ˇT .x; 0/

ˇ2L2.�i/

�C jA2T j2L2

D�Z

M.uTx CwTz/A2T dMC

Z

MFTA2T dM: (3.160)

Each term on the right-hand side of (3.160) is bounded as follows:

ˇˇZ

MuTxA2T dM

ˇˇ6 jujL4 jTxjL4 jA2T jL2

6 cjuj1=2L2kuk1=2jTxj1=2L2 kTxk1=2jA2T jL2

6 cjuj1=2L2kuk1=2

ˇA1=22 T

ˇ1=2L2jA2T j3=2L2

6 16jA2T j2C cjuj2L2kuk2

ˇA1=22 T

ˇ2L2;

ˇˇZ

MwTzA2T dM

ˇˇ6 jwjL4 jTzjL4 jA2T jL2

6 cjuxjL4 jTzjL4 jA2T jL2

6 ckuk1=2kuxk1=2ˇA1=22 T

ˇ1=2L2jA2T j3=2

6 16jA2T j2C ckuk2kuxk2

ˇA1=22 T

ˇ2L2

and

ˇˇZ

MFTA2T dM

ˇˇ6 jFT jL2 jA2T jL2

6 16jA2T j2L2 C cjFT j2L2 :

82 M. Petcu et al.

Here we have used the fact (easy to prove) thatjA1=22 T jL2 is a norm equivalent tokT kin V2, and the much more involved result, proved in Section 4.3, thatjA2T jL2.M/ is,on D.A2/, a norm equivalent tojT jH2.M/; for the application of Theorem 4.3, we re-quired (3.128) which is the one-dimensional analog of (4.54). Note that, as explained inRemark 4.1, we believe that this purely technical hypothesis can be removed.

With this we infer from (3.160) that

d

dt

�ˇA1=22 T

ˇ2L2.M/

C ˛TˇT .x; 0/j2

L2.�i/

�C jA2T j2L2.M/

6 �.t/C �.t/ˇA1=22 T

ˇ2L2.M/

; (3.161)

with � D cjFT j2L2.M/and� D c.juj2

L2.M/C kuxk2/kuk2. By assumption� 2 L1.0; t0/

and the earlier estimates onU;ux anduz show that� 2L1.0; t0/. Then, Gronwall’s lemmaimplies thatjA1=22 T jL2.M/ is in L2.0; t0/, which means thatT is in L1.0; t0IH 1.M//

andL2.0; t0IH 2.M//. This concludes the proof of Theorem 3.3.

3.4. Uniqueness ofz-weak solutions for the space periodic case in dimension two

In this section and in Section 3.5, we (continue to) consider the2D Primitive Equations:all the functions are independent of thex2-variable but the velocityv is not zero, so westill model a three dimensional motion.

Our aim in these two sections is to present some additional existence, uniqueness andregularity results for the PEs of the ocean in space dimension two with periodic bound-ary conditions. In this section we prove the existence and uniqueness of the so-calledz-weak solutions and, in Section 3.5 we prove the existence and uniqueness of more regular(strong) solutions for the PEs, up toC1 regularity.

For the sake of simplicity, we do not consider the salinity; introducing the salinity wouldnot produce any additional technical difficulty. In this case� is a linear function ofT ,12

and, in what follows,� is the prognostic variable instead ofT .Because of the hydrostatic equation it is not possible to produce a solution that is

space periodic in all variables; for that reason�, p andT below represent the deviationfrom a stratified solutionN� for whichN 2 D �.g=�0/.d N�=dz/ is a constant, and, as usuald Np=dz D�g N� and N�D �0.1� ˛. xT � T0//, �0, T0 being reference values of� andT (ofthe same order asN� and xT ). Furthermore the periodic (disturbance) solutions that we con-sider present certain symmetries that are described below (see (3.164)). We refer the readerto Petcu, Temam andWirosoetisno (2004) for more details on the physical background.13

The equations (for the deviations) read:

@u

@tC u @u

@x1Cw @u

@x3� f vC 1

�0

@p

@x1D �v�uCFu; (3:163a)

12In fact� is an affine function ofT but the deviation from the density� considered below is a linear functionof the deviation from the temperatureT .

13Alternatively� is sometimes as in (2.16) and then (3:163e) is the combination of (2.14) and (2.15) when it isassumed thatT .�T � 1/D ˇS .�S � 1/ and�� D�ˇT�T C ˇS�S >0:


@v

@tC u @v

@x1Cw @v

@x3C f uD �v�vCFv; (3:163b)

@p

@x3D�g�; (3:163c)

@u

@x1C @w

@x3D 0; (3:163d)

@�

@tC u @�

@x1Cw @�

@x3� �0N

2

gwD ��CF�: (3:163e)

We notice easily that ifu;v; �;w;p are solutions of (3:163a)—(3:163e) forF D.Fu;Fv;F�/, then Qu, Qv, Q�, Qw, Qp are solutions of (3:163a)—(3:163e) foreF u, eF v , eF �,where

Qu.x; z; t/D u.x;�z; t/;Qv.x; z; t/D v.x;�z; t/;Qw.x; z; t/D�w.x;�z; t/;Qp.x; z; t/D p.x;�z; t/;

(3.164)Q�.x; z; t/D��.x;�z; t/;eF u.x; z; t/D Fu.x;�z; t/;eF v.x; z; t/D Fv.x;�z; t/;eF �.x; z; t/D�F�.x;�z; t/:

Therefore if we assume thatFu;Fv;F� are periodic andFu;Fv are even inz andF� is oddin z, then we can anticipate the existence of a solution of (3:163a)—(3:163e) such that

u;v;w;p; � are periodic inx andz with periodsL1 andL3, (3.165)

and14

u;v andp are even inz; w and� are odd inz, (3.166)

provided the initial conditions satisfy the same symmetry properties.One motivation for considering periodic boundary conditions is that they are needed in

numerical studies of rotating stratified turbulence (see e.g., Bartello (1995) and also for thestudy of the renormalized equations considered in Petcu, Temam andWirosoetisno (2004)).Our aim is to solve the problem (3:163a)—(3:163e) with initial data

uD u0; vD v0; �D �0 at t D 0: (3.167)14Alternatively� is sometimes as in (2.16) and then (3:163e) is the combination of (2.14) and (2.15) when it is

assumed thatT .�T � 1/D ˇS .�S � 1/ and�� D�ˇT�T C ˇS�S >0:

84 M. Petcu et al.

We introduce here the natural spaces for this problem:

V D�.u; v; �/ 2 � PH 1

per.M/�3; u; v even inz; � odd inz;

Z L3=2

�L3=2u�x; z0

�dz0 D 0

�; (3.168)

H D closure ofV in� PL2.M/

�3; (3.169)

V2 D the closure ofV \ . PH 2per.M//3 in . PH 2

per.M//3: (3.170)

HereM is the limited domain

MD .0;L1/� .�L3=2;L3=2/; (3.171)

and, as mentioned, we assume space periodicity with periodM, that is, all functions aretaken to satisfyf .xCL1; z; t/D f .x; z; t/D f .x; zCL3; t /when extended toR2. More-over, we assume that the symmetries (3.164) hold. The dot abovePH 1

per or PL2 denotes thefunctions with average inM equal to zero. These spaces are endowed with the followingHilbert scalar products; inH the scalar product is

�U;eU �

HD �u; Qu�

L2C �v; Qv�

L2C ��; Q��

L2; (3.172)

and in PH 1per andV the scalar product is (using the same notation when there is no ambigu-

ity):

��U;eU ��D ��u; Qu��C ��v; Qv��C ��; Q��; (3.173)

where we have written dM for dx dz, and

��; Q��D

Z

M

�@�

@x

@ Q�@xC @�

@z

@ Q�@z

�dM: (3.174)

The positive constant� is defined below. We have

jU jH 6 c0kU k 8U 2 V; (3.175)

wherec0 > 0 is a positive constant related to� and the Poincaré constant inPH 1per.M/.

More generally, theci , c0i , c00i will denote various positive constants. Inequality (3.175)

implies thatkU k D ..U;U //1=2 is indeed a norm onV .Let us show how we can express the diagnostic variablesw andp in terms of the prog-

nostic variablesu, v and�, the situation being slightly different here due to the boundary


conditions and the symmetries. For eachU D .u; v; �/ 2 V we can determine uniquelywDw.U / from (3.216),

w.U /Dw.x; z; t/D�Z z

0

ux�x; z0; t

�dz0; (3.176)

since w.x; 0/ D 0, w being odd in z. Furthermore, writing thatw.x;�L3=2; t/ Dw.x;L3=2; t/, we also have

Z L3=2

�L3=2ux�x; z0; t

�dz0 D 0: (3.177)

As for the pressure, we obtain from (3.215),

p.x; z; t/D ps.x; t/�Z z

0

��x; z0; t

�dz0; (3.178)

wherepsD p.x; 0; t/ is the surface pressure. Thus, we can uniquely determine the pres-surep in terms of� up tops.

It is appropriate to use Fourier series and we write, e.g., foru,

u.x; z; t/DX

.k1;k3/2Zuk1;k3.t/e

i.k01xCk0

3z/; (3.179)

where for notational conciseness we setk01 D 2�k1=L1 andk03 D 2�k3=L3. Sinceu isreal and even inz, we haveu�k1;�k3 D Nuk1;k3 D Nuk1;�k3 , where Nu denotes the complexconjugate ofu. Regarding the pressure, we obtain from (3.215)

p.x; z; t/D p.x; 0; t/�Z z

0

X

.k1;k3/

�k1;k3ei.k01xCk0

3z0/ dz0

DX

k1

psk1eik01x �

X

.k1;k3/;k3¤0

�k1;k3ik03

eik01x�eik0

3z � 1�

[using the fact that�k1;0 D 0, � being odd inz]

DX

k1

�psk1 C

X

k3¤0

�k1;k3ik03

�eik0

1x �

X

.k1;k3/;k3¤0

�k1;k3ik03

ei.k01xCk0

3z/

DX

k1

p?k1eik01x �

X

.k1;k3/;k3¤0

�k1;k3ik03

ei.k01xCk0

3z/;

where we denoted byps the surface pressure andp? DPk12Z p?k1e

ik01x , which is the

average ofp in the vertical direction, is defined by

p?;k1 D psk1 CX

k3¤0

�k1;k3ik03

:

86 M. Petcu et al.

Note thatp is fully determined by�, up to one of the termsps or p? which are connectedby the relation above.

The dots abovePH 1per and PL2 denote the functions with zero average overM.

The variational formulation of our problem is:

FindU W Œ0; t0�! V , such that,

d

dt.U;U [/H C a.U;U [/C b.U;U;U [/C e.U;U [/D .F;U [/H ; 8U [ 2 V;

U.0/DU0:(3.180)

In (3.180) we introduced the following forms:a W V � V !R bilinear, continuous, coercive:

a.U;U [/D �..u;u[//C �..v; v[//C ��..�; �[//; (3.181)

with � D g2=N 2�20,b W V � V � V2!R trilinear continuous (see Lemma 2.1):

b.U;U ];U [/DZ

M.u@u]

@xu[C v @u

]

@yu[Cw.U /@u

]

@zu[/dM

CZ

M.u@v]

@xv[C v @v

]

@yv[Cw.U /@v

]

@zv[/dM

C �Z

M.u@�]

@x�[C v @�

]

@y�[Cw.U /@�

]

@zQ�/dM;

(3.182)

e W V � V !R bilinear, continuous:

e.U;U [/D fZ

M.uv[ � vu[/dMC g

�0

Z

M�w.U [/dM

� g

�0

Z

M�[w.U /dM;

(3.183)

with e.U;U /D 0 for all U 2 V .Problem (3.180) can also be written as an operator evolution equation inV 02:

dU

dtCAU CB.U;U /CEU D F;

U.0/DU0;(3.184)


where we introduced the following operators:

A linear continuous fromV into V 0, defined by

hAU;U [i D a.U;U [/; 8U;U [ 2 V;(3.185)

B bilinear, continuous fromV � V into V 02, defined by

hB.U;U [/; U ]i D b.U;U [;U ]/ 8U;U [ 2 V; 8U ] 2 V2;(3.186)

E linear continuous fromV into V 0, defined by

hEU;U [i D e.U;U [/; 8U;U [ 2 V; with hEU;U i D 0:(3.187)

In the previous section we have proved the existence and uniqueness, globally in time,of strong solutions for the two dimensional Primitive Equations, but we could not prove theuniqueness of the weak solution (result that is available for the Navier-Stokes equations). Inthis section we prove an intermediate result, that is the existence and uniqueness, globallyin time, of solutions which are weak in the horizontal direction and strong in the verticaldirection (the so-calledz�weak solutions). We start by introducing the function spacesnecessary for this problem:

V D fU D .u; v; �/ 2 V; @U@x32 . PH 1

per.M//3g; (3.188)

which is a Hilbert space when endowed with the following norm:

jU j2V D kU k2C @U@x3

2: (3.189)

Another useful function space is:

HD fU D .u; v; �/ 2H; @U@x32 . PL2per.M//3g; (3.190)

which is a Hilbert space when endowed with the norm:

jU j2H D jU j2L2 Cˇ @U@x3

ˇ2L2: (3.191)

We now prove the existence and uniqueness, globally in time, of az-weak solution for(3.163).

THEOREM3.4. (z-weak solutions in dimension two)GivenU0 2H andF 2L1.0;T IH/,there exists a unique solutionU of problem (3.180) satisfying the initial conditionU.0/D U0 and:

U 2 C.Œ0; T �IH/\L2.0;T IV/: (3.192)

88 M. Petcu et al.

PROOF.In Section 2.3 we proved the existence of weak solutions for the Primitive Equations;

the proof have been done for the general case, both in space dimension two and three. Itremains to prove that starting with an initial data and a forcing more regular (satisfying thehypotheses of Theorem 3.4), the solution is strong in the vertical direction. In order to provethat, we need to obtain a priori estimates forUx3 D @U=@x3. We formally differentiate(3:163a), (3:163b) and (3:163e) inx3 and then multiply respectively byux3 , vx3 and�x3 ,add the resulting equation and integrate overM. We find:

1

2

d

dtjUx3 j2L2 C

Z

M.ux1 Cwx3/u2x3dMC

1

�0

Z

Mpx1x3ux3dM

CZ

M.ux3vx1 C vx3wx3/vx3dMC

Z

M.ux3�x1 Cwx3�x3/�x3dMC �kUx3k2

D .Fx3 ;Ux3/L2 :(3.193)

A term has been omitted in (3.193) which is zero because of the mass conservationequation. The pressure term can be estimated, using the hydrostatic equation (3:163c) andintegrating by parts:

ˇˇZ

Mpx1x3ux3dM

ˇˇD

ˇˇ�g

Z

M�x1ux3dM

ˇˇ6 gj�jkUx3k: (3.194)

We also estimate:

ˇˇZ

M.ux3vx1 C vx3wx3/vx3dM

ˇˇ 6 jux3 jL4 jvx1 jL2 jvx3 jL4 C jux1 jL2 jvx3 j2L4

6 cjUx3 jL2kUx3kkU k;(3.195)

and

ˇˇZ

M.ux3�x1 Cwx3�x3/�x3dM

ˇˇ 6 jux3 jL4 j�x1 jL2 j�x3 jL4 C jux1 jL2 j�x3 j2L4

6 cjUx3 jkUx3kkU k:(3.196)

Using the above estimates into (3.193), we find:

1

2

d

dtjUx3 j2L2 C �kUx3k2 6 jF jL2kUx3kC cj�jL2kUx3k

C cjUx3 jL2kUx3kkU k;(3.197)


which, by the Young inequality, implies:

d

dtjUx3 j2L2 C �kUx3k2 6 cjF j2L2 C cjUx3 j2L2kU k2C cjU j2L2 : (3.198)

Applying the Gronwall lemma to (3.198) and using the estimates valid for weak so-lutions (U in L2.0;T;V /; 8T ), we find a bound forUx3 in L1.0;T IL2.M/3/ andL2.0;T I PH 1

per.M/3/.Using all these estimates and the Galerkin method, we can prove the existence

of a z�weak solution that is withU and Ux3 belonging toL1.0;T IL2.M/3/ \L2.0;T I PH 1

per.M/3/.The forward uniqueness of az�weak solution is then proved classically: we suppose

thatU1 andU2 are twoz�weak solutions for (3.184), satisfying the same initial condition.Then, QU DU1 �U2 satisfies the following equation:

QU 0CA QU CE QU CB.U1; QU/CB. QU ;U2/D 0; (3.199)

with QU.0/D 0.We take theV 0–V duality product of (3.199) withQU . We find:

d

dtj QU j2H C c0k QU k2V C b.U1; QU ; QU/C b. QU ;U2; QU/6 0: (3.200)

From the orthogonality property we know thatb.U1; QU ; QU/D 0, under the hypothesesof Lemma 2.1. But we note here that in our caseU1 and QU do not satisfy the conditionsin Lemma 2.1; however, the same result can be easily obtained for the caseU1 2 V andQU 2 V , using the same kind of reasoning as before. It remains to estimateb. QU ;U2; QU/:

b. QU ;U2; QU/DZ

MQu@U2@x1QUdMC

Z

Mw. QU/@U2

@x3QUdM: (3.201)

The first term of (3.201) is estimated using the Holder inequality and the Sobolev em-beddings:

ˇˇZ

MQu@U2@x1QUdM

ˇˇ6 j QujL4

ˇˇ@U2@x1

ˇˇL2j QU jL4 6 cj QU jHk QU kkU2k: (3.202)

For the second term, we find:

ˇˇZ

Mw. QU/@U2

@x3QUdM

ˇˇ6 jw. QU/jL2

ˇˇ@U2@x3

ˇˇL4j QU jL4

6 cj QU j1=2H k QU k3=2ˇˇ@U2@x3

ˇˇ1=2

L2

@U2@x3

1=2

:

(3.203)

Using the above estimates into (3.200), we find:

d

dtj QU j2H C c0k QU k2V 6 g.t/j QU j2H ; (3.204)

90 M. Petcu et al.

where

g.t/D ckU2k2C cˇˇ@U2@x3

ˇˇ2 @U2@x3

2

:

SinceU2 is a z�weak solution, the functiong belongs toL1.0;T / for any T > 0. Soapplying the Gronwall lemma to (3.204), we find thatQU.t/D 0 for all t > 0.

It remains to prove that thez�weak solutionU belongs toC.Œ0;T �;H/. We start byproving thatB.U;U / belongs toL2.0;T;V 0/. Let QU be inV ; then:

<B.U;U /; QU >V 0;VD b.U;U; QU/DZ

Mu@U

@x1QUdMC

Z

Mw.U /

@U

@x3QUdM:

(3.205)

We estimateb.U;U; QU/ using Lemma 3.9 below which is the two-dimensional analogueof Lemma 3.1, and we find

kB.U;U /kV 0 6 cjU jHkU kV C cjUx1 jL2.M/

ˇˇ @U@x3

ˇˇL2.M/

; (3.206)

which, taking into account thatU 2L2.0;T;V/, implies thatB.U;U / 2L2.0;T;V 0/.Then one can easily conclude from (3.184) thatU 0 2 L2.0;T;V 0/. We know thatU 2

L2.0;T;V/ andV � V � H � V 0 � V 0 where each space is dense into the other andthe injections are continuous. We can then conclude, using a technical result (see Temam(1977) for more details), thatU belongs toC.Œ0;T �;H/, observing thatHD ŒV;V 0�1=2 isthe1=2�interpolate betweenV andV 0.

�

We now state and prove Lemma 3.13 which is the two-dimensional analogue of Lemma3.1.

LEMMA 3.13. In space dimension two, the formb is trilinear continuous onV � V � Vand we have

jb.U; NU ; QU/j6 c.jU jL2.M/jj NU jjV jj QU jjV C jjU jjj NU jHjj QU jj/: (3.207)

PROOF. In space dimension two we have

b.U; NU ; QU/DZ

Mu@ NU@x1QUdMC

Z

Mw.U /

@ NU@x3QUdM: (3.208)

The first term is estimated as follows


ˇˇZ

Mu@ NU@x1� QUdM

ˇˇD

ˇˇZ L1

0

Z �L3=2�L3=2

u@ NU@x1� QU dx3 dx1

ˇˇ6

Z L1

0

jujL2x3ˇˇ @U@x1

ˇˇL4x3

j QU jL4x3 dx1

6Z L1

0

jujL2x3ˇˇ @ NU@x1

ˇˇ1=2

L2x3

�ˇˇ @NU

@x1

ˇˇ1=2

L2x3

Cˇˇ @

2 NU@x1@x3

ˇˇ1=2

L2x3

�j QU j1=2

L2x3

�j QU j1=2

L2x3Cˇˇ @QU

@x3

ˇˇ1=2

L2x3

�:

(3.209)

Here and belowLqx1 isLq.0;L1/ andLqx3 isLq.�L3=2;L3=2/.The most difficult term of (3.209) is:

Z L1

0

jujL2x3ˇˇ @NU

@x1

ˇˇ1=2

L2x3

ˇˇ @

2 NU@x1@x3

ˇˇ1=2

L2x3

j QU j1=2L2x3

ˇˇ @QU

@x3

ˇˇ1=2

L2x3

dx1

6 cjU jL2.M/

ˇˇ @NU

@x1

ˇˇ1=2

L2.M/

ˇˇ @

2 NU@x1@x3

ˇˇ1=2

L2.M/jj QU jL2x3 j

1=2

L1x1

ˇˇˇˇ @QU

@x3

ˇˇL2x3

ˇˇ1=2

L1x1

6 cjU jL2.M/

ˇˇ @NU

@x1

ˇˇ1=2

L2.M/

ˇˇ @

2 NU@x1@x3

ˇˇ1=2

L2.M/k QU k1=2

@QU

@x3

1=2

6 cjU jL2.M/k NU kVk QU kV I(3.210)

we used the fact that, in dimension one, we have the Sobolev embeddingH 1x1� L1x1 ,

which implies that:

j QU jL1x1 .L2x3 / 6 cj QU jH1x1 .L2x3 / 6 ck QU k: (3.211)

The second term is estimated as follows

ˇˇZ

Mw.U /

@ NU@x3� QUdM

ˇˇ6

Z L1

0

jw.U /jL1x3ˇˇ @NU

@x3

ˇˇL2x3

j QU jL2x3 dx1

6 cZ L1

0

jUx1 jL2x3ˇˇ @NU

@x3

ˇˇL2x3

j QU jL2x3 dx1

6 cjUx1 jL2x1 .L2x3 /ˇˇ @NU

@x3

ˇˇL2x1 .L

2x3/j QU jL1x1 .L2x3 /

6 cjUx1 jL2.M/

ˇˇ @NU

@x3

ˇˇL2.M/

j QU jH1x1 .L2x3 /

6 cjUx1 jL2.M/

ˇˇ @NU

@x3

ˇˇL2.M/

k QU k:

6 cjjU jj j NU jHjj QU jj:

(3.212)

The lemma is proved. �

92 M. Petcu et al.

3.5. The space periodic case in dimension two: Higher regularities15

As announced, our aim in this section is to prove the existence and uniqueness of more reg-ular solutions, up toC1 regularity, in space dimension two. As in Section 3.4, we will use�

instead ofT but unlike the preceding sections (but this is not important), we consider herethe PEs written in nondimensional form, that is (see e.g. Petcu, Temam andWirosoetisno(2004)),

@u

@tC u@u

@xCw@u

@z� 1

R0vC 1

R0

@p

@xD �v�uCFu; (3.213)

@v

@tC u@v

@xCw@v

@zC 1

R0uD �v�vCFv; (3.214)

@p

@zD��; (3.215)

@u

@xC @w

@zD 0; (3.216)

@�

@tC u @�

@xCw@�

@z� N

2

R0wD ��CF�: (3.217)

Here .u; v;w/ are the three components of the velocity vector and, as usual, we denoteby p and� the pressure and density deviations, respectively, from the background statementioned above. The (dimensionless) parameters are the Rossby numberR0, the BurgersnumberN , and the inverse (eddy) Reynolds numbers�v and��.

In all the follows we assume that the solutions have the same symmetry properties as inthe previous section.

Our aim is to solve the problem (3.213)—(3.217) with the periodicity and symmetryproperties as in (3.164) and with initial data

uD u0; vD v0; �D �0 at t D 0:

The two spatial directions are0x and0z, corresponding to the west–east and verticaldirections in the so-calledf -plane approximation for geophysical flows;�D @2=@x2 C@2=@z2.

The rest of this section is organized as follows: We start in Section 3.5.1 by recallingthe variational formulation of problem (3.213)—(3.217) under suitable assumptions andwe say a few words about (the now standard) proof of existence of weak solutions forthe PEs. We continue in Section 3.5.2 by proving the existence and uniqueness of strongsolutions, giving another version of Theorem 3.3 for the case with periodic boundary con-ditions. Finally in Section 3.5.3 we prove the existence of more regular solutions, up toC1regularity.

15This section essentially reproduces the article Petcu, Temam andWirosoetisno (2004), with the authorizationof the publisher of the journal.


3.5.1. Existence of the weak solutions for the PEs.We now obtain the variational for-mulation of problem (3.213)–(3.217). For that purpose we consider a test functioneU D. Qu; Qv; Q�/ 2 V and we multiply (3.213), (3.214) and (3.217), respectively byQu, Qv and� Q�,where the constant� (which was already introduced in (3.172) and (3.173)) will be chosenlater. We add the resulting equations and integrate overM. We find

d

dt

�U;eU �

HC b�U;U;eU �C a�U;eU �C 1

R0e�U;eU �

D �F;eU �H8eU 2 V: (3.218)

Here we set (compare to (3.181) - (3.183)):

a�U;eU �D �v

��u; Qu��C �v

��v; Qv��C ��

��; Q��;

e�U;eU �D

Z

M

�u Qv � v Qu�dMC

Z

M

�� Qw � �N 2w Q��dM;

b�U;U ];eU �

DZ

M

�u@u]

@xCw.U /@u

]

@z

�QudMC

Z

M

�u@v]

@xCw.U /@v

]

@z

�Qv dM

CZ

M

�u@�]

@xCw.U /@�

]

@z

�Q�dM:

We now choose� D 1=N 2 and, in this way, we finde.U;U /D 0. Also it can be easilyseen that:

a WV � V !R is bilinear, continuous, coercive,a.U;U /> c1kU k2;e WV � V !R is bilinear, continuous,e.U;U /D 0;

(3.219)b is trilinear, continuous fromV � V2 � V intoR; and

from V � V � V2 intoR;

whereV2 is the closure ofV \ .H 2per.M//3 in .H 2

per.M//3. Furthermore,

b�U;eU ;U ]�D�b�U;U ];eU �;

(3.220)b�U;eU ;eU �D 0;

whenU;eU ;U ] 2 V with eU orU ] in V2. We also have the following:

LEMMA 3.14. There exists a constantc2 > 0 such that, for all U 2 V , eU 2 V2 andU ] 2V :

ˇb�U;U ];eU �

ˇ6 c2jU j1=2L2 kU k1=2

U ] ˇeU

ˇ1=2L2

eU 1=2

94 M. Petcu et al.

C c2kU k U ]

1=2 ˇU ]ˇ1=2V2

ˇeUˇ1=2L2

eU 1=2: (3.221)

PROOF. We only estimate two typical terms, the other terms are estimated exactly in thesame way. Using the Hölder, Sobolev and interpolation inequalities, we write

ˇˇZ

Mu@u]

@xQudM

ˇˇ6 jujL4

ˇˇ@u]@x

ˇˇL2

ˇQuˇL4

6 c01juj1=2L2 kuk1=2ˇˇ@u]@x

ˇˇL2

ˇQuˇ1=2L2

Qu 1=2;

ˇˇZ

Mw.U /

@u]

@zQudM

ˇˇ6

ˇw.U /

ˇL2

ˇˇ@u]@z

ˇˇL4

ˇQuˇL4

6 c02kukˇˇ@u]@z

ˇˇ1=2

L2

@u]

@z

1=2 ˇQuˇ1=2 Qu

1=2I

(3.221) follows from these estimates and the analogous estimates for the other terms.�

We now recall the result regarding the existence of weak solutions for the PEs of theocean; the proof is exactly the same as that of Theorem 2.2 in space dimension three (seealso Theorem 3.1 for the two-dimensional case with different boundary conditions).

THEOREM 3.5. GivenU0 2H andF 2L1.RCIH/, there exists at least one solutionUof (3.218),U 2L1.RCIH/\L2.0; t?IV / 8t? > 0, withU.0/DU0.

As for Theorem 2.2, the proof of this theorem is based on the a priori estimates givenbelow, which gives, as in Lions, Temam and Wang (1992b), thatU 2L1.0; t?IH/, 8t? >0; however, as shown below, we have in fact,16

U 2L1.RCIH/:

TakingeU DU in equation (3.218), after some simple computations and using (3.219), weobtain, assuming enough regularity:

d

dtjU j2H C c0c1jU j2H 6

d

dtjU j2H C c1kU k2 6 c01jF j21; (3.222)

wherejF j1 is the norm ofF in L1.RCIH/. Using the Gronwall inequality, we inferfrom (3.222) that:

ˇU.t/

ˇ2H6ˇU.0/

ˇ2H

e�c1c0t C c01c1c0

�1� e�c1c0t

�jF j21 8t > 0: (3.223)

16The same holds in the previous cases in dimension 3 and 2, although the result was not stated in this form. Atall orders, we present here results uniform in time,t 2RC.


Hence

limsupt!1

ˇU.t/

ˇ2H6 c01c1c0jF j21 DW r20 ;

and any ballB.0; r 00/ in H with r 00 > r0 is an absorbing ball; that is, for allU0, there existst0 D t0.jU0jH / depending increasingly onjU0jH (and depending also onr 00, jF j1 andother data), such thatjU.t/jH 6 r 00 8t > t0.jU0jH /. Furthermore, integrating (3.222) fromt to t C r , with r > 0 arbitrarily chosen, we find

Z tCr

t

U �t 0� 2 dt 0 6K1 for all t > t0

�jU0jH�; (3.224)

whereK1 denotes a constant depending on the data but not onU0. As mentioned before,(3.223) implies also that

U 2L1.RCIH/;ˇU.t/

ˇH6max

�jU0jH ; r0�:

REMARK 3.3. We notice that, in the inviscid case (�v D �� D 0 with F D 0), takingeU DU in (3.218), we find, at least formally,

d

dt

�juj2L2C jvj2

L2C 1

N 2j�j2L2

�D 0: (3.225)

The physical meaning of (3.225) is that the sum of the kinetic energy (given by12.juj2

L2C

jvj2L2/) and the available potential energy (given by1

2N2j�j2L2

) is conserved in time. Thisis the physical justification of the introduction of the constant� DN�2 in (3.172).

3.5.2. Existence and uniqueness of strong solutions for the PEs.The solutions given byTheorem 3.5 are called weak solutions as usual. We are now interested in strong solutions(and even more regular solutions in Section 3.5.3). We use here the same terminologyas before: weak solutions are those inL1.L2/ \ L2.H 1/, strong solutions are those inL1.H 1/ \ L2.H 2/. We notice that as for Theorem 3.1, we cannot obtain directly theglobal existence of strong solutions for the PEs from a single a priori estimate. Instead,we will proceed as for Theorem 3.1 and derive the necessary a priori estimates by steps:we successively derive estimates inL1.L2/ andL2.H 1/ for uz , ux , vz , vx , �z and�x(here the subscriptst , x, z denote differentiation). Notice that the order in which we obtainthese estimates cannot be changed in the calculations below.17

Firstly, using (3.178) we rewrite (3.213) as

@u

@tC u@u

@xCw@u

@z� 1

R0

@ps

@xC 1

R0

Z z

0

�x�x; z0; t

�dz0 D �v�uCFu:

(3.226)

17However, as for Theorem 3.3 we could, at once, obtain the estimates forvx andvz , and then for�x and�z .

96 M. Petcu et al.

We differentiate (3.226) with respect toz and we find, withwz D�ux ,

utz C uuxz Cwuzz �1

R0vz �

1

R0�x � �vuxxz � �vuzzz D Fu;z ;

whereFu;z D @zFu D @Fu=@z. After multiplying this equation byuz and integratingoverM, we find:

1

2

d

dtjuzj2L2 C �vkuzk2C

Z

Muuzuxz dMC

Z

Mwuzuzz dM

� 1

R0

Z

Mvzuz dM� 1

R0

Z

M�xuz dMD

Z

MuzFu;z dM:

Integrating by parts and taking into account the periodicity and the conservation of massequation (3.216) we obtain:

1

2

d

dtjuzj2L2 C �vkuzk2 �

1

R0

Z

Mvzuz dM� 1

R0

Z

M�xuz dM

DZ

MuzFu;z dM: (3.227)

In all that followsK,K 0,K 00; : : : ; denote constants depending on the data but not onU0;we use the same symbol for different constants. We easily obtain the following estimates:

1

R0

ˇˇZ

Mvzuz dM

ˇˇD 1

R0

ˇˇZ

Mvuzz dM

ˇˇ6Kjvj2

L2C �v

6kuzk2;

1

R0

ˇˇZ

M�xuz dM

ˇˇD 1

R0

ˇˇZ

M�uxz dM

ˇˇ6 �v

6kuzk2CKj�j2L2 ;

ˇˇZ

MFu;zuz dM

ˇˇD

ˇˇZ

MFuuzz dM

ˇˇ6 �v

6kuzk2C c01jFuj2L2 :

Applied to (3.227), these give

d

dtjuzj2L2 C �vkuzk2 6K

�jvj2L2C j�j2

L2

�C c01jFuj2L2 : (3.228)

We apply the Poincaré inequality (3.175) and we find

d

dtjuzj2L2 C c0�vjuzj2L2 6K

�jvj2L2C j�j2

L2

�C c01jFuj2L2 : (3.229)

Using the Gronwall lemma, we infer from (3.229) that

ˇuz.t/

ˇ2L26ˇuz.0/

ˇ2L2

e�c0�vt CKe�c0�vt

Z t

0

�ˇv�t 0�ˇ2L2Cˇ��t 0�ˇ2L2

�ec0�vt

0dt 0


C c02jFuj216ˇuz.0/

ˇ2L2

e�c0�vt CK 0�1� e�c0�vt��jvj21C j�j21

�C c02jFuj216ˇuz.0/

ˇ2L2

e�c0�vt CK 0�jvj21C j�j21�C c02jFuj21; (3.230)

wherejvj1 D jvjL1.RCIL2.M//, and similarly for� andFu. We obtain an explicit boundfor the norm ofuz in L1.RCIH/,

ˇuz.t/

ˇ2L26ˇuz.0/

ˇ2L2CK 0�jvj21C j�j21

�C c02jFuj21: (3.231)

For what follows, we recall here the uniform Gronwall lemma (see, e.g., Temam (1997)):If � , � andy are three positive locally integrable functions on.t1;1/ such thaty0 is

locally integrable on.t1;1/ and which satisfy

y0 6 �y C �;Z tCr

t

�.s/ds 6 a1;Z tCr

t

�.s/ds 6 a2; (3.232)

Z tCr

t

y.s/ds 6 a3 8t > t1;

wherer , a1, a2 anda3 are positive constants, then

y.t C r/6�a3

rC a2

�ea1 ; t > t1: (3.233)

The bound (3.231) depends on the initial dataU0. In order to obtain a bound independentof U0 we apply the uniform Gronwall lemma to the equation

d

dtjuzj2L2 6K

�jvj2L2C j�j2

L2

�C c01jFuj2L2 ; (3.234)

to obtain

ˇuz.t/

ˇ6K 0

�r; r 00

� 8t > t 01; (3.235)

wheret 01 D t0.jU0jL2/C r andr > 0 is fixed. Integrating equation (3.228) fromt to t C rwith r > 0 as before, we also find:

Z tCr

t

uz.s/ 2 ds 6K 00

�r; r 00

� 8t > t 01: (3.236)

We now derive the same kind of estimates forux : We differentiate (3.226) with respectto x and we obtain

utx C u2x C uuxx Cwuxz Cwxuz

98 M. Petcu et al.

� 1

R0vx C

1

R0ps;xx C

Z 0

z

�xx�z0�dz0 � �vuxxx � �vuzzx D Fu;x : (3.237)

Multiplying this equation byux and integrating overM we find, using (3.216):

1

2

d

dtjuxj2L2 C

Z

Mu3x dMC

Z

Mwxuzux dM

� 1

R0

Z

Mvxux dM� 1

R0

Z

Mps;xxux dM

CZ

M

�Z 0

z

�xx�z0�dz0�ux dMC �vkuxk2 D

Z

MuxFu;x dM: (3.238)

Based on the Hölder, Sobolev and interpolation inequalities, we derive the following esti-mates:

ˇˇZ

Mu3x dM

ˇˇ6 juxj3L3.M/

6 c04juxj3H1=3.M/

6 c05juxj2L2kuxk6�v

12kuxk2C c06juxj4L2 ;

ˇˇZ

Mwxuzux dM

ˇˇ6 c07jwxjL2 juzj1=2L2 kuzk1=2juxj

1=2

L2kuxk1=2

6 c08juxxjL2 juzj1=2L2 kuzk1=2juxj1=2

L2kuxk1=2

6 �v

12kuxk2C c09juzj2L2kuzk2juxj2L2 :

By the definition ofV , and sinceps is independent ofz, we find

1

R0

ˇˇZ

Mps;xxux dM

ˇˇD 1

R0

ˇˇZ L

0

ps;xx

Z L3=2

�L3=2ux dz dx

ˇˇD 0:

We can also prove the following estimates:

1

R0

ˇˇZ

Mvxux dM

ˇˇ6 �v

12kuxk2CK 0jvj2L2 ;

1

R0

ˇˇZ

M

�Z 0

z

�xx�z0�dz0�ux dM

ˇˇD 1

R0

ˇˇZ

M

�Z 0

z

�x�z0�dz0�uxx dM

ˇˇ

6 �v

12kuxk2CK 00j�xj2L2 ;

ˇˇZ

MuxFu;x dM

ˇˇ6 �v

12kuxk2C c010jFuj21:


With these relations (3.238) implies

d

dtjuxj2L2 C �vkuxk2 6 �juxj2L2 C �; (3.239)

where we denoted

� D �.t/D 2c06juxj2L2 C 2c09juzj2L2kuzk2;

and

�D �.t/D 2K 0jvj2L2C 2K 00j�xj2L2 C 2c010jFuj21:

We easily conclude from (3.239) that

ux 2L1�0; t?IL2

�\L2�0; t?IH 1� 8t? > 0: (3.240)

However, for later purposes, (3.240) is not sufficient, and we need estimates uniform intime.

We will apply the uniform Gronwall lemma to (3.239) witht1 D t 01 as in (3.235). Notingthat

Z tCr

t

��t 0�dt 0 D

Z tCr

t

�2c06juxj2L2 C 2c09

ˇuz�t 0�ˇ2L2

uz�t 0� 2�dt 0

6 2c06Z tCr

t

ˇux�t 0�ˇ2L2

dt 0C 2c09juzj21Z tCr

t

uz�t 0� 2 dt 0

6 a1 8t > t 01; (3.241)

Z tCr

t

��t 0�dt 0 D

Z tCr

t

�2K 0jvj2

L2C 2K 00j�xj2L2 C 2c010jFuj21

�dt 0

6K C 2c010r jFuj21

D a2 8t > t 01; (3.242)

Z tCr

t

ˇux�t 0�ˇ2L2

dt 0 6 a3 8t > t 01; (3.243)

(3.233) then yields

ˇux.t/

ˇ2L26�a3

rC a2

�ea1 8t > t 01C r; (3.244)

100 M. Petcu et al.

and thus

juxjL2 2L1.RC/: (3.245)

Note that in (3.241)–(3.243) we can use bounds onjuzj1 (and other similar terms) in-dependent ofU0, sincet > t0.jU0jL2/ C r . Integrating (3.239) from0 to t 01 C r wheret 01 D t 01.jU0jL2/, we obtain a bound forux in L2.0; t 01 C r IH 1/ which depends onkU0k.A bound independent ofU0 is obtained if we work witht > t 01 C r D t 001 D t 001 .jU0jL2/:Integrating (3.239) fromt to t C r with r as before, we find

Z tCr

t

ux.s/ 2 ds 6K 8t > t 001 : (3.246)

We perform similar computations forvz : We differentiate (3.214) with respect toz,multiply the resulting equation byvz and integrate overM. Using again the conservationof mass relation, we arrive at

1

2

d

dtjvzj2L2 C

Z

Muzvxvz dMC

Z

Mwzv

2z dM

C 1

R0

Z

Muzvz dMC �vkvzk2 D

Z

MvzFu;z dM: (3.247)

We notice the following estimate

ˇˇZ

Muzvxvz dM

ˇˇ 6 c011juzj1=2L2 kuzk1=2jvxjL2 jvzj

1=2

L2kvzk1=2

6 �v

8kvzk2C c012juzj2=3L2 kuzk2=3jvxj

4=3

L2jvzj2=3L2

6 �v

8kvzk2C c012juzj2=3L2 kuzk2=3jvxj

4=3

L2

�1C jvzj2L2

�:

We also see thatˇˇZ

Mwzvzvz dM

ˇˇD

ˇˇZ

Muxvzvz dM

ˇˇ

6 c013juxj1=2L2 kuxk1=2jvzj3=2

L2kvzk1=2

6 �v

8kvzk2C c014juxj2=3L2 kuxk2=3jvzj2L2 ;

1

R0

ˇˇZ

Muzvz dM

ˇˇD 1

R0

ˇˇZ

Muvzz dM

ˇˇ6 �v

8kvzk2CKjuj2L2 ;

ˇˇZ

MFv;zvz dM

ˇˇD

ˇˇZ

MFvvzz dM

ˇˇ6 �v

8kvzk2C c015jFvj21;


which gives

d

dtjvzj2L2 C �vkvzk2 6 �jvzj2C �; (3.248)

where we denoted

�D �.t/D 2c012juzj2=3L2 kuzk2=3jvxj4=3

L2C 2Kjuj2C 2c015jFvj21

and

� D �.t/D 2c012juzj2=3L2 kuzk2=3jvxj4=3

L2C 2c014juxj2=3L2 kuxk2=3:

From (3.248), using the estimates obtained before and applying the classical Gronwalllemma we obtain bounds depending on the initial data forvz in L1loc.0; t?IL2/ andL2loc.0; t?IH 1/, valid for any finite interval of time.0; t?/.

To obtain estimates valid for all time, we apply the uniform Gronwall lemma observingthat

Z tCr

t

��t 0�dt 0

6 2c012juzj2=31�Z tCr

t

uz�t 0� dt 0

�1=3�Z tCr

t

ˇvx�t 0�ˇ2L2

dt 0�2=3

C 2Kjuj21r C 2c015r jFvj216 a1 8t > t 001 ; (3.249)

Z tCr

t

��t 0�dt 0

6 2c012juzj2=31�Z tCr

t

uz�t 0� dt 0

�1=3�Z tCr

t

ˇvx�t 0�ˇ2L2

dt 0�2=3

C 2c014juxj2=31Z tCr

t

ux�t 0� 2=3 dt 0

6 a2 8t > t 001 ; (3.250)Z tCr

t

ˇvz�t 0�ˇ2

dt 0 6 a3 8t > t 001 : (3.251)

Then the uniform Gronwall lemma gives

ˇvz.t/

ˇ2L26�a3

rC a2

�ea1 8t > t 001 C r; (3.252)

102 M. Petcu et al.

with a1, a2, a3 as in (3.249), (3.250) and (3.251). Integrating (3.248) fromt to t C r withr > 0 as above andt > t 001 C r , we find

Z tCr

t

vz.s/ 2 ds 6K 8t > t 001 C r: (3.253)

The same methods apply tovx , �z and�x , noticing that at each step we precisely usethe estimates from the previous steps, so the order cannot be changed in this calculations.

With these estimates, the Galerkin method as used for the proof of Theorem 2.2 givesthe existence of strong solutions.

THEOREM 3.6. GivenU0 2 V andF 2 L1.RCIH/, there exists a unique solutionU ofequation(3.218)withU.0/DU0 such that

U 2L1.RCIV /\L2�0; t?I

� PH 2.M/�3� 8t? > 0: (3.254)

PROOF. As we said, the existence of strong solutions follows from the previous estimates.The proof for the uniqueness follows the same idea as of the Theorem 3.1 so we skip it.�

3.5.3. More regular solutions for the PEs.In this sub-section we show how to obtainestimates on the higher-order derivatives from which one can derive the existence of solu-tions of the PEs in. PHm.M//3 for all m 2 N, m > 2 (hence up toC1 regularity). In allthat follows we work withU0 in . PHm

per.M//3.

We setjU jm D .PŒ˛�Dm jD˛U j2

L2/1=2. We fixm> 2 and, proceeding by induction, we

assume that for all06 l 6m� 1, we have shown that

U 2L1�RCI� PH l .M/

�3�\L2�0; t?I� PH lC1.M/

�3� 8t? > 0; (3.255)

withZ tCr

t

ˇU�t 0�ˇ2lC1 dt 0 6 al 8t > tl .U0/; (3.256)

whereal is a constant depending on the data (andl) but not onU0, andr > 0 is fixed (thesame as before). We then want to establish the same results forl Dm.

In equation (3.218) we takeeU D �mU.t/ with m > 2 and t arbitrarily fixed, and weobtain:

�dU

dt;�mU

�

L2C a�U;�mU �C b�U;U;�mU �C 1

R0e�U;�mU

�

D �F;�mU �L2: (3.257)

Integrating by parts, using periodicity and the coercivity ofa and the fact thate.U;U /D 0,we find

1

2

d

dt

ˇU.t/

ˇ2mC c1jU j2mC1 6

ˇb�U;U;�mU

�ˇCˇ�F;�mU

�L2

ˇ: (3.258)


We need to estimate the terms on the right-hand side of (3.258). We first notice that

ˇ�F;�mU

�2L

ˇ6 cjF j2m�1C

c1

2.mC 3/ jU j2mC1; (3.259)

and it remains to estimatejb.U;U;�mU/j.By the definition ofb we have

b�U;U;�mU

�DZ

M

�uux Cw.U /uz

��mudM

CZ

M

�uvx Cw.U /vz

��mv dM

CZ

M

�u�x Cw.U /�z

��m�dM: (3.260)

The computations are similar for all the terms, and, for simplicity, we shall only estimatethe first integral on the right-hand side of (3.260).

We notice thatb.U;U;�mU/ is a sum of integrals of the type

Z

Mu@u

@xD2˛11 D

2˛33 udM;

Z

Mw.U /

@u

@zD2˛11 D

2˛33 udM;

where˛i 2N with ˛1 C ˛3 Dm. By Di we denoted the differential operator@=@xi . Inte-grating by parts and using periodicity, the integrals take the form

Z

MD˛

�u@u

@x

�D˛udM;

Z

MD˛

�w.U /

@u

@z

�D˛udM; (3.261)

whereD˛ DD˛11 D

˛33 . Using Leipzig formula, we see that the integrals are sums of inte-

grals of the form

Z

MuD˛ @u

@xD˛udM;

Z

Mw.U /D˛ @u

@zD˛udM; (3.262)

and of integrals of the form

Z

Mıkuım�k

@u

@xD˛udM;

Z

Mıkw.U /ım�k

@u

@zD˛udM; (3.263)

with k D 1; : : : ;m, whereık is some differential operatorD˛ with Œ˛� D ˛1 C ˛3 D k.For each , after integration by parts we see that the sum of the two integrals in (3.262) iszero because of the mass conservation equation (3.216). It remains to estimate the integralsof type (3.263). We use here the Sobolev and interpolation inequalities. For the first term

104 M. Petcu et al.

in (3.263) we write

ˇˇZ

Mıkuım�k

@u

@xD˛udM

ˇˇ

6ˇıku

ˇL4

ˇˇım�k @u

@x

ˇˇL4

ˇD˛u

ˇL2

6 c01ˇıku

ˇ1=2L2

ˇıku

ˇ1=2H1

ˇˇım�k @u

@x

ˇˇ1=2

L2

ˇˇım�k @u

@x

ˇˇ1=2

H1

ˇD˛u

ˇL2

6 c01jU j1=2k jU j1=2

kC1jU j1=2

m�kC1jU j1=2

m�kC2jU jm; (3.264)

wherek D 1; : : : ;m.The second term from (3.263) is estimated as follows

ˇˇZ

Mıkw.U /ım�k

@u

@zD˛udM

ˇˇ

6ˇıkw.U /

ˇL2

ˇˇım�k @u

@z

ˇˇL4

ˇD˛u

ˇL4

6 c02ˇıkw.U /

ˇL2

ˇˇım�k @u

@z

ˇˇ1=2

L2

ˇˇım�k @u

@z

ˇˇ1=2

H1

ˇD˛u

ˇ1=2L2

ˇD˛u

ˇ1=2H1

6 c03jU jkC1jU j1=2m�kC1jU j1=2

m�kC2jU j1=2m jU j1=2mC1; (3.265)

wherek D 1; : : : ;m.From (3.264) and (3.265) we obtain that

ˇb�U;U;�mU

�ˇ

6 c3mX

kD1jU j1=2

kjU j1=2

kC1jU j1=2

m�kC1jU j1=2

m�kC2jU jm

C c3mX

kD1jU jkC1jU j1=2m�kC1jU j

1=2

m�kC2jU j1=2m jU j1=2mC1: (3.266)

We now need to bound the terms on the right-hand side of (3.266). The terms corre-sponding tok D 2; : : : ;m� 1 in the first sum do not containjU jmC1 and we leave them asthey are. Fork D 1 andk Dm, we apply Young’s inequality and we obtain

c3jU j1=21 jU j1=22 jU j3=2m jU j1=2mC16 c1

2.mC 3/ jU j2mC1C c04jU j2=31 jU j2=32 jU j2m: (3.267)


For the terms in the second sum in (3.266) we distinguish betweenk D 1, k D m andk D 2; : : : ;m� 1. The term corresponding tok D 1 is bounded by

c3jU j2jU jmjU jmC1 6c1

2.mC 3/ jU j2mC1C c05jU j22jU j2m: (3.268)

Fork Dm we find

c3jU j1=21 jU j1=22 jU j1=2m jU j3=2mC1 6c1

2.mC 3/ jU j2mC1C c06jU j21jU j22jU j2m:

(3.269)

For the terms corresponding tok D 2; : : : ;m� 1, we apply Young’s inequality in the fol-lowing way:

c3jU jkC1jU j1=2m�kC1jU j1=2

m�kC2jU j1=2m jU j1=2mC1

6 c1

2.mC 3/ jU j2mC1C c07jU j4=3kC1jU j

2=3

m�kC1jU j2=3

m�kC2jU j2=3m : (3.270)

Gathering all the estimates above we find

d

dtjU j2mC c1jU j2mC1 6 � C �jU j2m;

where the expressions of� and � are easily derived from (3.258) and (3.267)–(3.270).Using the Gronwall lemma and the induction hypotheses (3.255) and (3.256) we ob-tain a bound forU in L1.0; t?IHm/ andL2.0; t?IHmC1/, for all fixed t? > 0, thisbound depending also onjU0jm. We also see that, because of the induction hypotheses(3.255) and (3.256), we can apply the uniform Gronwall lemma and we obtainU boundedin L1.RCIHm/ with a bound independent ofjU0jm whent > tm.U0/; we also obtain ananalogue of (3.256). The details regarding the way we apply the uniform Gronwall lemmaand derive these bounds are similar to the developments in Section 3.5.2.

In summary we have proven the following result:

THEOREM 3.7. Givenm 2 N, m > 1, U0 2 V \ . PHmper.M//3 and F 2 L1.RCIH \

. PHm�1per .M//3/, (3.218)has a unique solutionU such that

U 2L1�RCI� PHm

per.M/�3�\L2�0; t?I

� PHmC1per .M/

�3� 8t? > 0: (3.271)

REMARK 3.4. SinceTm>0 PHm

per.M/ D PC1per.M/, given U0 2 . PC1per.M//3 and F 2L1.RCI . PC1per.M//3/, (3.218) has a unique solutionU belonging to1.RC; . PHm

per.M//3/

for all m 2 N; that is,U is in L1.RCI . PC1per.M//3/. Regularity (differentiability) in timecan be also derived ifF is alsoC1 in time. However the arguments above do not providethe existence of an absorbing set in. PC1per.M//3.

106 M. Petcu et al.

3.6. The space periodic case in dimension three: Higher Sobolev regularities and Gevreyregularity

In this section we consider the primitive equations of the ocean, in a3D domain, withperiodic boundary conditions. Petcu and Wirosoetisno (2005) it was proved the short timeexistence and uniqueness of the strong solutions in space dimension three, and also thelocal existence of very regular solutions, up to theC1 regularity. They also studied theGevrey type regularity for the solutions of the primitive equations, the result being obtainedon a limited interval of time. In the present section we extend these results, proving thatthey remain true for all time. Of course this section is also related to Section 3.5 in whichsimilar results are proved in space dimension two, but the methods needed for dimensionthree are different.

This section is structured in two parts: in a first part, using the results of Cao and Titi(2006) and Kobelkov (2006) and improvements due to?, we prove the long time existenceof the solutions with values in the Sobolev spacesHm, for allm> 0, and derive from thistheC1 regularity of the solution. In the second part we study the Gevrey regularity for thesolutions (analyticity in space).

We start by recalling the model and the functional settings of the problem. The PrimitiveEquations in their dimensional form read:

@u

@tC u @u

@x1C v @u

@x2Cw @u

@x3� f vC 1

�0

@p

@x1D ��3uCFu;

@v

@tC u @v

@x1C v @v

@x2Cw @v

@x3C f uC 1

�0

@p

@x2D ��3vCFv;

@p

@x3D��g;

@u

@x1C @v

@x2C @w

@x3D 0;

@T

@tC u @T

@x1C v @T

@x2Cw @T

@x3D ��3T CFT :

(3.272)

In this model,.u; v; w/ are the three components of the velocity vector andp, � andT are respectively the perturbations of the pressure, of the density and of the temperaturefrom the reference (average) constant statesp0, �0, andT0. The relation between the fulltemperature and the full density is given by the equation of state, and we consider here aversion of this equation linearized around the reference state�0 andT0,

�full D �0.1� ˇT .Tfull � T0//; (3.273)

so that for the perturbations� andT :

�D�ˇT �0T: (3.274)


The constantg is the gravitational acceleration andf the Coriolis parameter,� and� arethe eddy diffusivity coefficients,.Fu;Fv/ represent body forces per unit of mass andFTrepresents a heating source.

We work in a bounded domain:

MD .0;L1/� .0;L2/� .�L3=2;�L3=2/; (3.275)

and we assume space periodicity with periodM. We also assume that the functions havethe same symmetries as in (3.164).

The variational formulation of the problem

We start by introducing the natural function spaces for this problem:

V D fU D .u; v; T / 2 . PH 1per.M//3; u; v even inx3; T odd inx3; (3.276)

R L3=2�L3=2.ux1.x1; x2; x

03/C vx2.x1; x2; x03//dx03 D 0g;

H D closure ofV in . PL2.M//3;

V2 D the closure ofV \ . PH 2per.M//3 in . PH 2

per.M//3: (3.277)

As before, we endow these spaces with the following scalar products:OnH we consider:

.U; QU/H D .u; Qu/L2 C .v; Qv/L2 C �.T; QT /L2 ; (3.278)

and onV :

..U; QU//V D ..u; Qu//C ..v; Qv//C �..T; QT //: (3.279)

Here the dots abovePH 1per and PL2 denote the functions with zero average overM. Since

we work with functions with zero average overM, we can use the generalized Poincaréinequality:

c0jU jH 6 kU kV ; 8U 2 V; (3.280)

so we know that the normk � kV defined above is equivalent to the usualH 1-norm, wherec0 is a constant related to the Poincaré constant.

We briefly recall that the unknown functions are of two types: the prognostic variablesu, v andT , for which the initial values are prescribed, and the diagnostic variables�, w,p which can be defined, at each instant of time, as functions of the prognostic variables.More details regarding the way the diagnostic variables can be determined, are available inPetcu and Wirosoetisno (2005).

108 M. Petcu et al.

The variational formulation of this problem:

FindU W Œ0; t0�! V , such that,

d

dt.U;U [/H C a.U;U [/C b.U;U;U [/C e.U;U [/D .F;U [/H ; 8U [ 2 V;

U.0/DU0:(3.281)

In (3.281) we introduced the following forms:

a W V � V !R bilinear, continuous:

a.U;U [/D �..u;u[//C �..v; v[//C ��..T;T [//� gˇTZ

MTw.U [/dM;

(3.282)

b W V � V � V2!R trilinear:

b.U;U ];U [/DZ

M.u@u]

@x1u[C v @u

]

@x2u[Cw.U /@u

]

@x3u[/dM

CZ

M.u@v]

@x1v[C v @v

]

@x2v[Cw.U / @v

]

@x3v[/dM

CZ

M.u@T ]

@x1T [C v @T

]

@x2T [Cw.U /@T

]

@x3QT /dM;

(3.283)

e W V � V !R bilinear, continuous:

e.U;U [/D fZ

M.uv[ � vu[/dM: (3.284)

We choose� sufficiently large, so thata is coercive onV .

The trilinear formb has the properties proved in Lemma 2.1.Problem (3.281) can also be written as an operator evolution equation inV 02:

dU

dtCAU CB.U;U /CEU D F;

U.0/DU0;(3.285)


where we introduced the following operators:

A linear continuous fromV into V 0, defined by

hAU;U [i D a.U;U [/; 8U;U [ 2 V;(3.286)

B bilinear, continuous fromV � V into V 02, defined by

hB.U;U [/; U ]i D b.U;U [;U ]/ 8U;U [ 2 V; 8U ] 2 V2;(3.287)

E linear continuous fromV into V 0, defined by

hEU;U [i D e.U;U [/; 8U;U [ 2 V:(3.288)

3.6.1. Existence of strong solutions for all timeThe classical result about the existenceof weak solutions can be found in Section 2. We briefly recall this result and the main linesof the proof, since the estimates obtained are used later.

THEOREM 3.8. GivenU0 2H andF 2L1.RCIH/, there exists at least one solutionUof problem (3.281) such that:

U 2L1.RCIH/\L2.0; t 0IV /; 8 t 0 > 0: (3.289)

PROOF.We start by estimatingU in L2.M/:

djU j2L2

dtC c1kU k2 6 c01jF j2L2 ; (3.290)

wherec1 is the constant of coercivity fora.Using the Poincaré inequality as well as the Gronwall lemma, we obtain:

jU.t/j2L26 e�c1c0t jU0j2L2 C

c01c1c0jF j21; (3.291)

wherej � j1 is the norm inL1.RCIH/. We then deduce:

limsupt!1

jU.t/j2L26 c01c0c1jF j21 WD r20 : (3.292)

From (3.292), we find that any ballB.0; r 00/ in H , with r 00 > r0, is an absorbing ball. Infact, for allU0 in H , there existst0 D t0.jU0jH / depending increasingly onjU0jH , such

110 M. Petcu et al.

that jU.t/jH 6 r 00 for all t > t0.jU0jH /. Furthermore, we integrate (3.290) over.t; t C r/an arbitrary interval of time, witht > t0.jU0jH /, and using (3.291) we find:

Z tCr

t

kU.s/k2 ds 6 jU.t/j2L2C rc01c0c1jF j21;

Z tCr

t

kU.s/k2 ds 6K1; 8 t > t0.jU0jH /;(3.293)

whereK1 is a constant depending on the data but not onU0.All the estimates above lead us to the existence, globally in time, of a weak solution, as

well as to the existence of an absorbing set forU D .v; T / in H . �

The long time existence of the strong solutions was established by Cao and Titi, and byKobelkov (see Kobelkov (2006) or Section 3.2 of this paper), and the result was improvedin ?, showing the existence of the global attractor proved for the strong solution of theprimitive equations.

For subsequent utilization in the proof of the higher order regularity of the solution, webriefly recall the main steps for proving the existence and uniqueness of a global strongsolution for the primitive equations, following Cao and Titi (2006) and?.

The main idea is to split the velocity profilev into two parts: the vertical averageNv andthe remaining part,QvD v� Nv. As a general notation, we have:

N�.x1; x2/D1

L3

Z L3=2

�L3=2�.x1; x2; x3/ dx3; Q� D � � N�: (3.294)

We can find the equation forNv by integrating the equations for the horizontal velocity inthe vertical direction:

@Nv@t� ��NvC .Nv � r/NvC .Qv � r/QvC .r � Qv/QvC f E� � Nv

Cr�ps.x1; x2; t /C ˇT �0g1

L3

Z L3=2

�L3=2

Z x3

�L3=2T .x1; x2; s; t/ ds dx3

�D 0;

r � NvD 0; in .0;L1/� .0;L2/:(3.295)


For Qv we have the following equation:

@Qv@t� ��3 QvC .Qv � r/Qv�

�Z x3

�L3r � Qv.x1; x2; s; t/ ds

� @Qv@x3

C .Qv � r/QvC .r � Qv/QvC .Qv � r/NvC .Qv � r/QvC f E� � Qv

Cr��1Z x3

�L3=2T .x1; x2; s; t/ds

� �11

L3

Z L3=2

�L3=2

Z x3

�L3=2T .x1; x2; s; t/ ds dx3

�D 0;

(3.296)

where�1 denotes T g.The idea of Cao and Titi (2006) is to notice that in equation (3.296), the term for the

surface pressure disappears and we can obtain an a priori estimate ofQv in L6.M/. Werather recall here the improved estimates from? for T and Qv in L6.M/: the estimateof T in L6.M/ is obtained by multiplying the equation for the temperature byT 5 andintegrating overM

d

dtkT k6

L6C 5

9�jr3.T 3/j2 6 jFT jjT 5j D jFT jkT 3k5=3L10=3

6 29�jr3.T 3/j2C c0jFT j2kT k4L6 C c00jFT jkT k5L6 ;

(3.297)

where we used the following inequality:

kf kL10=3 6 cjf j2=5jr3f j3=5C cjf j: (3.298)

Using the fact thatkT kL6 6 ckT kV in space dimension three, we can apply the UniformGronwall Lemma to (3.297) and we obtain, using (3.293), a time-uniform bound onkT kL6and also the existence of an absorbing ball forT in L6.M/.

For Qv, multiplying (3.296) byjQv.t/j4 Qv.t/ and integrating overM, we obtain:

d

dtkQvk6

L6C �jjr3 QvjjQvj2j2 6 .c1C c2kvk2/kQvk6L6 C c3jvj2kvk2kQvk2L6 C c4kT k6L6 :

(3.299)

From the previous estimates, we obtain, after applying the Uniform Gronwall Lemma, that:

kQvk6L66K2; 8 t > t0.jU0jH /C r; (3.300)

whereK2 is a constant independent of the initial data, andt0.jU0jH / andr were definedabove.

Using the estimates obtained above and the fact thatkQvkL6 6 ckvkH1 , we find:

112 M. Petcu et al.

Z tCr

t

jjQv.s/j2jr3 Qvjj2 ds 6 c1 suptkQvk6

L6C c2 sup

tjvj2kQvk6

L6

Z tCr

t

jr3v.s/j2 ds

C c3 suptkQvk6

L6

Z tCr

t

jr3 Qv.s/j2 dsC c4 suptkT k6

L6:

(3.301)

H 1 estimates

The next step in obtaining the a priori estimates for the strong solution, is to estimatetheH 1 norm of Nv. We notice that equation (3.295) mainly behaves as a2D Navier-Stokesequation with rotation, so the estimates are classical. Multiplying (3.295) by��Nv andintegrating over.0;L1/� .0;L2/, we obtain:

d

dtjr Nvj2C �j�Nvj2 6 c1jvj2jr3vj2jr2 Nvj2

C c2.jr3vj2C jvj2C jjQvj2jr3 Qvjj2/:(3.302)

Applying the Uniform Gronwall Lemma to (3.302) and using the previous estimates, wefind a time-uniform estimate forjr Nvj and also the existence of an absorbing ball forjr Nvjin L2. Since

kNvkL6 6 cjr Nvj;we also have the uniform boundedness ofv in L6.M/ as well as the existence of anabsorbing ball forv in L6.M/.

After this additional step, one can now obtainH 1 estimates forv, which are no longerlocal in time. We first estimatevx3 and thenr3vD .rv;vx3/. Forvx3 , we find:

d

dtjvx3 j2C �kvx3k2 6 c1kvk6L6 jvx3 j2C c2jT j2: (3.303)

As before, we obtain uniform estimates onvx3 . We can write:

d

dtjr3vj2C �j�3vj2 6 �.t/jr3vj2C �.t/; (3.304)

where

�.t/D c1.kv.t/k4L6 C jr3v.t/j2jvx3.t/j2/; �.t/D c2jr3T .t/j2:We find, by the Uniform Gronwall Lemma, an uniform bound forv in H 1.M/ and alsothe existence of an absorbing ball forv in H 1.M/.

In order to be able to use the Gronwall Uniform Lemma for higher order estimates,notice that we also deduce that:

Z tCr

t

j�3vj2 dt 6K1; 8 t > t1.U0/D t0.jU0jH /C 2r;


whereK1 is a constant depending on the data but not on the initial dataU0.TheH 1-estimates forT are now immediate. Using the results above we find the exis-

tence globally of a strong solutionU and also the existence of an absorbing ball forU inV .

The proof of Kobelkov (2006) presented in Section 3.2 is different and based on a suit-able a priori estimate of the surface pressure, but it leads mainly to the same existence anduniqueness result (as in Theorem 3.2).

THEOREM 3.9. GivenU0 2 V andF 2 L1.RCIH/, there exists a unique solutionU DU.t/ of (3.281) onRC such that:

U 2 C.RCIV /\L2.0; t 0I . PH 2per.M//3/; 8 t 0 > 0: (3.305)

REMARK 3.5. Using these a priori estimates, Ning Ju proved the existence of the globalattractor for the strong solutions of the primitive equations. In what follows, we prove theexistence of absorbing balls for the solutionU in all theHm.M/ Sobolev spaces, andthe existence of the global attractor inHm.M/, for all m > 1 will follow by the generaltheory of the global attractors (more details regarding the existence of global attractorscan be found in a general context for example in Temam (1997), Chepyzhov and Vishik(2002), Hale (2001), or in the context of the Navier-Stokes equations in Constantin, Foiasand Temam (1985)).

3.6.2. Higher order regularity results H 2 estimates

We are now interested in proving the long-time existence of a solution inH 2.M/. Inorder to simplify the computations, we introduce the following notations: we setjU jm D.PŒ˛�Dm jD˛U j2H /1=2, whereD˛ is the differential operatorD˛ DD˛1

1 D˛22 D

˛33 , Di D

@=@xi ; ˛ is a multi-index, D .˛1; ˛2; ˛3/, ˛i 2N andŒ˛�D ˛1C ˛2C ˛3.We multiply the equation for the velocity by.��3/2v and integrate overM. We find:

1

2

d

dtj�3vj2C

Z

M.v � r/v � .��3/2vdMC

Z

Mw.v/

@v@x3

.��3/2vdM

C fZ

M.� � v/ � .��3/2vdMC 1

�0

Z

Mrps � .��3/2vdM

C �1Z

Mr�Z x3

0

T .z/ dz� � .��3/2vdMC �j.��3/3=2vj2

DZ

MFv � .��3/2vdM;

(3.306)

where byFv we understand the function.Fu;Fv/.

114 M. Petcu et al.

We can easily check that

f

Z

M.� � v/ � .��3/2vdMD 0:

For the integral from (3.306) containing the surface termps , integrating by parts, usingthe boundary conditions and the conservation of mass equation, we find:

Z

Mrps � .��3/2vdMD

Z L1

0

Z L2

0

rps � Z L3=2

�L3=2.��3/2v dx3

!dx2 dx1

D�Z L1

0

Z L2

0

ps

Z L3=2

�L3=2.��3/2.r � v/ dx3

!dx2 dx1

DZ L1

0

Z L2

0

ps

Z L3=2

�L3=2.��3/2wx3 dx3

!dx2 dx1 D 0:

(3.307)

The integral containingT is also easy to estimate:

ˇ�1

Z

Mr�Z x3

0

T .z/ dz

�� .��3/2vdM

ˇ6 c

Z

Mj�3T jj.��3/3=2vjdM

6 �6j.��3/3=2vj2C cj�3T j2:

(3.308)

It remains to estimate the integrals obtained from the nonlinear terms. We note that infact we need to estimate integrals of the type:

Z

Mu@u

@x1D2˛11 D

2˛22 D

2˛33 udM;

Z

Mv@u

@x2D2˛11 D

2˛22 D

2˛33 udM;

Z

Mw.v/

@u

@x3D2˛11 D

2˛22 D

2˛33 udM;

(3.309)

where˛i 2N with Œ˛�D ˛1C ˛2C ˛3 D 2.Integrating by parts and using periodicity, the integrals become:

Z

MD˛

�u@u

@x1

�D˛udM;

Z

MD˛

�v@u

@x2

�D˛udM;

Z

MD˛

�w.U /

@u

@x3

�D˛udM:

(3.310)


Using Leibniz’ formula, we see that the integrals can be written in the form

Z

MuD˛ @u

@x1D˛udM;

Z

MvD˛ @u

@x2D˛udM;

Z

Mw.v/D˛ @u

@x3D˛udM;

(3.311)

or in the form

Z

Mıkuı2�k

@u

@x1D˛udM;

Z

Mıkvı2�k

@u

@x2D˛udM;

Z

Mıkw.v/ı2�k

@u

@x3D˛udM;

(3.312)

wherek D 1; 2 andık is some differential operatorD˛ with Œ˛�D k.Note that for each , after integration by parts, the sum of the integrals of type (3.311)

is zero because of the mass conservation equation.It remains to estimate the integrals of type (3.312). The first two integrals in (3.312) lead

to the same kind of estimates, so in fact we only need to estimate the first and last integrals.The first integrals are easiest to estimate: forŒk�D 1 we write:

ˇ Z

Mıkuı2�k

@u

@x1D˛udM

ˇ6 jıkujL6

ˇı2�k

@u

@x1

ˇL3jD˛ujL2

6 c1jvj5=22 jvj1=23 6�

6jvj23C c2jvj10=32 ;

(3.313)

while for Œk�D 2 we find:

ˇ Z

Mıku

@u

@x1D˛udM

ˇ6 jıkujL3

ˇ @u@x1

ˇL3jD˛ujL2

6 c1jvj5=22 jvj1=23 6�

6jvj23C c2jvj10=32 :

(3.314)

The last type of integrals from (3.312) is estimated as follows: forŒk�D 2 we find:

ˇ Z

Mıkw.v/

@u

@x3D˛udM

ˇ6 jıkw.v/jL2

ˇ @u@x3

ˇL6jD˛ujL3

6 c1ˇ @u@x3

ˇL6jvj1=22 jvj3=23 6

�

6jvj23C c2jvx3 j4L6 jvj22;

(3.315)

116 M. Petcu et al.

and forŒk�D 1, we have:

ˇ Z

Mıkw.v/ı2�k

@u

@x3D˛udM

ˇ6 jıkw.v/jL3

ˇ @u@x3

ˇL6jD˛ujL2

6 c1jvj3=22 jvj1=23 jvx3 jH1

6 �6jvj23C c2jvj22jvx3 j4=3H1 :

(3.316)

Gathering all the estimates above, equation (3.306) leads to:

1

2

d

dtjvj22C �jvj23 6

5�

6jvj23C ˛.t/jvj22C ˇ.t/; (3.317)

where by andˇ we understand the following functions:

˛.t/D c1jT j22C c2;

ˇ.t/D c3jvx3 j4L6 C jvx3 j4=3

H1:

Taking into account the estimates derived in the previous section, we can apply theUniform Gronwall Lemma to (3.317) and we find a uniform bound forv in H 2.M/ ifwe estimate the norm ofvx3 in L6. The same kind of estimates can be deduced for thetemperatureT .

We start deducing the required estimates onvx3 in L6. The equation forvx3 reads:

@vx3@t� ��3vx3 Crvvx3 �

�Z x3

0

r � v.�/d��@vx3@x3

Crvx3v� .r � v/vx3 C f �vx3 C �1rT D 0

(3.318)

We multiply (3.318) byjvx3 j4vx3 and integrate overM. We find:

1

6

d

dtjvx3 j6L6 C �

Z

M.jr3vx3 j2jvx3 j4C jr3jvx3 j2j2jvx3 j2/dM

C fZ

M� � vx3 � jvx3 j4vx3dMC

Z

Mrvvx3 � jvx3 j4vx3dM

�Z

M

�Z x3

0

r � v.�/d��@vx3@x3� jvx3 j4vx3dM

CZ

Mrvx3

v � jvx3 j4vx3dM�Z

M.r � v/vx3 � jvx3 j4vx3dM

C �1Z

MrT � jvx3 j4vx3dMD 0:


(3.319)

Integrating by parts, we find:

Z

Mrvvx3 � jvx3 j4vx3dM�

Z

M

�Z x3

0

r � v.�/d��@vx3@x3� jvx3 j4vx3dMD 0:

(3.320)

It is also immediate that

f

Z

M� � vx3 � jvx3 j4vx3dMD 0:

The following term is estimated using the integration by parts:

ˇ Z

Mrvx3

v � jvx3 j4vx3dMˇ6 c

Z

Mjrvx3 jjvx3 j5jvjdM

6 c�Z

Mjrvx3 j2jvx3 j4dM

�1=2�Z

Mjvx3 j6dM

�1=2jvjL1

6 �6

Z

Mjrvx3 j2jvx3 j4dMC cjvjL1

Z

Mjvx3 j6dM

(using Agmon’s inequality: see for example Temam (1997))

6 �6

Z

Mjrvx3 j2jvx3 j4dMC cjvjH2 jvx3 j6L6 :

(3.321)

Similarly, we find:

ˇˇZ

M.r � v/vx3 � jvx3 j4vx3dM

ˇˇ6 c

Z

Mjrvx3 jjvx3 j5jvjdM

6 �6

Z

Mjrvx3 j2jvx3 j4dMC cjvjH2 jvx3 j6L6 :

(3.322)

For the last integral in (3.319), we find by integration by parts:

118 M. Petcu et al.

ˇ�1

Z

MrT � jvx3 j4vx3dM

ˇ6 c

Z

MjT jjrvx3 jjvx3 j5dM

6 �6

Z

Mjrvx3 j2jvx3 j4dMC cjT jL1

Z

Mjvx3 j6dM

6 �6

Z

Mjrvx3 j2jvx3 j4dMC cjT jH2 jvx3 j6L6 :

(3.323)

Gathering all the above estimates, we find:

1

6

d

dtjvx3 j6L6 C

�

2

Z

Mjrvx3 j2jvx3 j4dM6 c.jvjH2 C jT jH2/jvx3 j6L6 ; (3.324)

and by the Uniform Gronwall lemma we obtain a uniform estimate forjvx3 jL6 . We cannow return to (3.317) and apply the Gronwall Lemma.

In order to be able to use the Uniform Gronwall Lemma for further a priori estimates,we also need to integrate equation (3.317) fromt to t C r . We thus obtain:

Z tCr

t

jv.t 0/j23 dt 0 6K2; 8 t > t2.U0/D t0.jU0jH /C 3r; (3.325)

whereK2 is a constant depending on the data but not on the initial dataU0, r > 0 is fixedas before andt2 is a time depending on the initial data.

The same computations can be done in order to estimate the temperatureT :

1

2

d

dtjT j22C�jT j23 6

�

2jT j23C c.jvj4=33 C jvj4=32 /jT j22C jFT j22: (3.326)

Using the uniform bounds obtained above, we prove the existence of an absorbing ballfor T in H 2.M/. Integrating fromt to t C r , we also compute an uniform bound for thetime average ofjT jH3 , which will be used later for theHm estimates, withm> 2.

Combining the results onv and T , we found uniform bounds forU D .u; v;T / inH 2.M/, and we also proved the existence of an absorbing ball forU in H 2.M/.

Estimates inHm, form> 2

In order to obtain the a priori estimates inHm with m> 2, we can work directly withthe variational formulation of the primitive equations. In (3.285) we take the test functionU [ D .��3/mU.t/ wheret is a fixed, arbitrary moment in time. The computations followexactly the same steps as in Petcu and Wirosoetisno (2005):

d

dt.U; .��3/mU/H C a.U; .��3/mU/C b.U;U; .��3/mU/C e.U; .��3/mU/

D .F; .��3/mU/H :


(3.327)

We also note that:

a.U; .��3/mU/D a..��3/m=2U; .��3/m=2U/> c1jU.t/j2mC1; (3.328)

where we used the coercivity ofa.Integrating by parts and using the periodicity, we find:

1

2

d

dtjU.t/j2mC c1jU.t/j2mC1 6 jb.U;U; .��3/mU/j C j.F; .��3/mU/H j:

(3.329)

We then need to estimate the terms in the right-hand-side of (3.329). Analysing thebehavior of the integrals of the type (3.312), we find:

jb.U;U; .��3/mU/j6 cmX

kD1jU jm�kC2jU j1=2k jU j

1=2

kC1jU jm

C cm�1X

kD1jU jkC1jU j1=2m�kC1jU j

1=2

m�kC2jU jmC1C cjU j2jU j1=2m jU j3=2mC1:

(3.330)

Gathering all the estimates above, we obtain the differential inequality:

d

dtjU j2mC c1jU j2mC1 6 ˛.t/C ˇ.t/jU j2m; (3.331)

where the expressions of the functions˛.t/ andˇ.t/ can be easily derived from the esti-mates above. The functionsandˇ are formed from sums involving the termsjU jk , withk 6 m. Taking into account the estimates from all the previous steps, we can apply theUniform Gronwall Lemma to (3.331) and we find a uniform bound forU in Hm.M/ aswell as the existence of an absorbing set forU in Hm.M/, for allm> 2.

We conclude with the following result:

THEOREM 3.10. Givenm 2 N, m > 2, U0 2 V \ . PHmper.M//3 and F 2 L1.RCIH \

. PHm�1per .M//3/, there exists a unique solutionU of equation (3.285) onRC such that:

U 2 C.RCI . PHmper.M//3/\L2.0; t 0I . PHmC1

per .M//3/; 8 t 0 > 0: (3.332)

Moreover, ifU0 2 V and F 2 L1.RCIH \ . PHm�1per .M//3/, then the solutionU of

equation (3.285) belongs toC.R?CI PHmper.M//3/.

120 M. Petcu et al.

PROOF. The proof for the existence of the solution is based on the a priori estimates onhigher orders, which were obtained above, and on the Galerkin method, using the Fourierseries for the Galerkin basis. The uniqueness of the solution can be easily obtained byclassical methods: we considerU1 andU2 two solutions of the primitive equations (3.281)and estimating theH 1-norm ofU DU1 �U2, we find that the solutions coincide.

In order to prove the second part of the theorem, regarding the regularity of the solu-tion starting with an initial condition inV , we use the same reasoning as in Petcu andWirosoetisno (2005):If U0 belongs toV , the solutionU of problem (3.285) belongs toL2.0; t I PH 2

per/, for all

t > 0. This means thatU.t/ 2 PH 2per.M/ almost everywhere onRC, so there exists at1

arbitrarily small such thatU.t1/ 2 PH 2per.M/. Using the first part of the theorem, we know

that the solutionU is such that:

U 2 C.Œt1;C1/I PH 2per.M//\L2.t1; t I PH 3

per.M//; 8 t > 0:

By the same argument as before, we find at2 arbitrarily close tot1, such thatU.t2/ 2PH 3

per.M/. Using the first part of the theorem, we obtain that the solutionU is such that:

U 2 C.Œt2;C1/I PH 3per.M//\L2.t2; t I PH 4

per.M//; 8 t > 0:

Recurrently, we find:

U 2 C.Œtm�1;C1/I PHmper.M//\L2.tm�1; t I PHmC1

per .M//; 8 t > 0:

wheretm�1 is arbitrarily close to zero. From this relation, the result follows immediately:

U 2 C..R?CI PHmper.M//:

�

3.6.3. Gevrey regularity results In Petcu and Wirosoetisno (2005) the authors proved thatthe solutions of the3D Primitive Equations are the restriction on a positive finite intervalin time of some complex analytic function in time with values in a Gevrey space. In viewof the above results of long time existence of strong solutions, we can actually prove thatthe solution is analytic in time on every interval of time.0; t/, with t > 0, that is on.0;1/:

This result follows the main steps as in Petcu and Wirosoetisno (2005), so we intro-duce the main notations and we briefly recall the methods used in Petcu and Wirosoetisno(2005).

We introduce the following notation:

ŒUk �2� D jukj2C jvkj2C �jTkj2: (3.333)


We define the Gevrey spaceD.e�.��3/1=2/, � > 0, as the set of functionsU in H satis-

fying

jMjX

k2Z3e2� jk

0jŒUk �2� D je�.��3/1=2

U j2H <1: (3.334)

The Hilbert norm ofD.e�.��3/1=2/ is given by

jU j� WD jU jD.e�.��3/1=2 / D je�.��3/1=2U jH ; for U 2D.e�.��3/1=2/; (3.335)

and the associated scalar product is

.U;V /� WD .U;V /D.e�.��3/1=2 / D .e�.��3/1=2U;e�.��3/

1=2

V /H ;

for U;V 2D.e�.��3/1=2/:(3.336)

Another Gevrey space that we will use isD..��3/m=2e�.��3/1=2/, m > 1 integer, whichis a Hilbert space when endowed with the inner product:

.U;V /D..��3/m=2e�.��3/1=2 /

D ..��3/m=2e�.��3/1=2U; .��3/m=2e�.��3/1=2V /H :(3.337)

In order to be able to estimate the norm of the solutionU into a Gevrey space, we firstneed to be able to estimate the nonlinear term. The result we give here was proved in Petcuand Wirosoetisno (2005).

LEMMA 3.15. Let U , U ] andU [ be given inD..��3/3=2e�.��3/1=2/, for � > 0. Thenthe following inequality holds:

j..��3/1=2B.U;U ]/; .��3/3=2U [/� j6 cj�3U j� j�3U ]j1=2� j.��3/3=2U ]j1=2� j.��3/3=2U [j�C cj�3U j1=2� j.��3/3=2U j1=2� j�3U ]j� j.��3/3=2U [j� :

(3.338)

PROOF. We first write the trilinear formb in Fourier modes. In order to simplify the writ-ing, we define, for eachj D .j1; j2; j3/ 2 Z3, ıj;n asj 0n=j

03 whenj 03 ¤ 0 and as0 when

j 03 D 0, for nD 1; 2. With obvious notations, the trilinear form is then written as:

122 M. Petcu et al.

b.U;U ];U [/DX

jClCkD0i.l 01uj C l 02vj C l 03wj /u]lu[k

CX

jClCkD0i.l 01uj C l 02vj C l 03wj /v]l v[k C

X

jClCkD0i.l 01uj C l 02vj C l 03wj /T ]l T [k

D .using the fact that, by definition,wj D 0 for j3 D 0) (w is odd inx3)

DX

jClCkD0i Œ.l 01 � ıj;1l 03/uj C .l 02 � ıj;2l 03/vj �.u]lu[k C v

]

lv[k C �T ]l T [k /:

(3.339)

We then compute:

..��3/1=2B.U;U ]/; .��3/3=2U [/�D

X

jClCkD0i Œ.l 01 � ıj;1l 03/uj C .l 02 � ıj;2l 03/vj �e2� jk0jjk0j4.u]lu[k C v

]

lv[k C �T ]l T [k /:

(3.340)

We associate to each functionu, a function Lu defined by:

LuDX

j2Z3Luj ei.j 01xCj 02yCj 03z/; where Luj D e� jj 0jjuj jI (3.341)

we also use similar notations for the other functions.Since all the terms are similar, we need only to estimate the first sum from (3.340),

denoted byI . We find:

jI j6 cX

jCkClD0jl 0jjj 0jjk0j4e2� jk0jjuj jju]l jju[kj; (3.342)

where we used the estimatejl 01 � ıj 0;1l 03j 6 .L3=2�/jj 0jjl 0j. Sincej C k C l D 0”j 0C k0C l 0 D 0, we findjk0j � jl 0j � jj 0j6 0 and we have:

jI j6X

jCkClD0jl 0jjj 0j.jl 0j C jj 0j/jk0j3 Luj Lu]l Lu[k

6X

jCkClD0jj 0jjl 0j2jk0j3 Luj Lu]l Lu[k C

X

jCkClD0jj 0j2jl 0jjk0j3 Luj Lu]l Lu[k

D 1

jMjZ

Mq1.x/q

]2.x/q

[3.x/dMC

1

jMjZ

Mq2.x/q

]1.x/q

[3.x/dM;

(3.343)


where we wrote:

q1.x/DX

j2Z3jj 0j Luj ei.j 01xCj 02yCj 03z/; q2.x/D

X

j2Z3jj 0j2 Luj ei.j 01xCj 02yCj 03z/;

q3.x/DX

j2Z3jj 0j3 Luj ei.j 01xCj 02yCj 03z/;

(3.344)

and the definitions forq]i andq[i for i D 1; 2; 3 are similar.Using the Hölder and the imbedding inequalities, we find:

jI j6 jq1jL6 jq]2jL3 jq[3jL2 C jq2jL3 jq]1jL6 jq[3jL26 cjq1jH1 jq]2j1=2L2 jq

]2j1=2H1 jq[3jL2 C cjq2j

1=2

L2jq2j1=2H1 jq

]1jH1 jq[3jL2

6 cj�3U j� j�3U ]j1=2� j.��3/3=2U ]j1=2� j.��3/3=2U [j�C cj�3U j1=2� j.��3/3=2U j1=2� j�3U ]j� j.��3/3=2U [j� :

(3.345)

Analogue estimates for the other terms yield Lemma 3.15. �

A priori estimates

As announced, we want to prove that the solution is analytic in time with values in acertain Gevrey space and that it coincides with the restriction of a complex function intime to the real axis. We thus consider equation (3.285) with a complex time� 2C, andUa complex function. We take the complexified spacesH andV denotedHC andVC 18, soequation (3.184) is rewritten as:

dU

d�CAU CB.U;U /CE.U /D F;

U.0/DU0;(3.346)

where� 2C is the complex time.We consider� of the form� D sei� , wheres > 0 and cos� > 0 so that the real part of�

is positive. We applye'.s cos�/.��3/1=2�3 to equation (3.346) and take the scalar productinHC with e'.s cos�/.��3/1=2�3U . We then multiply the resulting equation byei� and takethe real part. We find:

18For the scalar products and the norms we use the same notations as in the real case.

124 M. Petcu et al.

Reei� .e'.s cos�/.��3/1=2�3dU

d�;�3e

'.s cos�/.��3/1=2U/H

D 1

2

d

dsje'.s cos�/.��3/1=2�3U j2H

C '0.s cos�/cos� Reei� .�3e'.s cos�/.��3/3=2U;�3e'.s cos�/.��3/1=2U/H

> 12

d

dsj�3U j2'.s cos�/ � cos� j.��3/3=2U j'.s cos�/j�3U j'.s cos�/:

(3.347)

Sincea is coercive for our choice of� ande is antisymmetric, we also find:

Reei� .e'.s cos�/.��3/1=2�3AU;e'.s cos�/.��3/1=2�3U/H

CReei� .e'.s cos�/.��3/1=2�3EU;e'.s cos�/.��3/1=2�3U/H

> c1 cos� je'.s cos�/.��3/1=2.��3/3=2U j2HD c1 cos� j.��3/3=2U j2'.s cos�/:

(3.348)

For the forcing term, we write:

jReei� .e'.s cos�/.��3/1=2�3F; e'.s cos�/.��3/1=2�3U/H j6 j.��3/1=2F j'.s cos�/j.��3/3=2U j'.s cos�/

6 c16

cos� j.��3/3=2U j2'.s cos�/C1

c1 cos�j.��3/1=2F j2'.s cos�/:

(3.349)

For the bilinear termB we use Lemma 3.15 and the Young inequality:

jReei� .�3B.U;U /;�3U/'.s cos�/j6 c2j�3U j3=2'.s cos�/j.��3/3=2U j3=2

'.s cos�/

6 c16

cos� j.��3/3=2U j2'.s cos�/Cc3

.cos�/3j�3U j6'.s cos�/:

(3.350)

Gathering all the estimates above, we find the following differential inequality:

1

2

d

dsj�3U j2'.s cos�/C

c1

2cos� j.��3/3=2U j2'.s cos�/ 6

1

c1 cos�j.��3/1=2F j2'.s cos�/

C cos�

c1j�3U j2'.s cos�/C

c3

.cos�/3j�3U j6'.s cos�/:

(3.351)


We restrict� so thatp2=26 cos� 6 1 (in fact we can restrict� to any domain such that

cos� > c > 0). Writing

y.s/D 1C j�3U.s/j2'.s cos�/;

the differential inequality (3.351) becomes:

dy.s/

ds6 c.F /y3.s/; 0 < s < t1; (3.352)

where c.F / is a constant depending as before on the forcing termF . Therefore, bysimilar reasonings as for the real case, we find that there exists a timet 0?, 0 < t 0? 6 t1,t 0? D t 0?.F;U0/ such that:

je'.s cos�/.��3/1=2�3U.sei� /j2H 6 1C 2j�3U0j2H ; 806 s 6 t 0?.F;U0/:(3.353)

Considering the complex region

D.U0;F; �1/D f� D sei� ; j� j6 �=4; 0 < s < t 0?.F;U0/g; (3.354)

estimate (3.353) gives a bound forU.�/, when� 2D.U0;F; �1/.In the previous section we proved the existence of an absorbing set forU in PH 2

per, so theprevious arguments can be reiterated for all times, and we find:

je'.s cos�/.��3/1=2�3U.sei� /j2H 6 1C 2M; 8 s > 0; (3.355)

whereM is the uniform bound forU in H 2 and'.t/Dmin.t; �1; t 0?.F;M//.We can now state the main result of this section:

THEOREM 3.11. LetU0 be given in PH 2per.M/ and letF be a given function analytic in

time with values inD.e�1.��3/1=2.��3/1=2/ for some�1 > 0. Then the unique solution

U of (3.285) is analytic in time on.0; t?/ with values inD.e'.t/.��3/1=2.��3/1=2/ where

'.t/Dmin.t; �1; t 0?.F;M// and t? Dmin.�1; t 0?.F;M// and on.t?;C1/ with values in

D.e�1.��3/1=2.��3/1=2/.

PROOF. The proof is based on the a priori estimates obtained above and the use of theGalerkin–Fourier method; see e.g. Foias et al. (1989). The solutions of the Galerkin ap-proximation satisfy rigorously the estimates formally derived above, and the bounds areindependent of the orderj of the Galerkin approximation. We can then pass to the limitj !1, using classical results on the convergence of analytic functions. �

126 M. Petcu et al.

3.7. On the backward uniqueness of the primitive equations

In this section we consider the primitive equations of the ocean, in a two dimensionaland then a three dimensional domain, with periodic boundary conditions. The questionto which we want to respond is: for which kind of solutions can we prove the backwarduniqueness, that is, if two solutions of the primitive equations (with same forcingsF ) coin-cide at a given timet� 2R, then they coincide at all timest < t� for which they are defined.Lions and Malgrange (1960) treated the problem of the backward uniqueness in Lions andMalgrange (1960) for certain parabolic problems and later in? the authors proved in par-ticular that the weak solutions for the2D Navier-Stokes equations have this property. Inthis section we will prove that the2D primitive equations possess the backward unique-ness property for a special class of weak solutions, that we called thez-weak solutions (seeSection 3.4). We also prove that the strong solutions of the3D primitive equations possessthe backward uniqueness property as well. For more details, see also Petcu (2007).

We will first treat the backward uniqueness for the two dimensional primitive equationsand then we consider the three dimensional case. We start by proving the existence anduniqueness ofz-strong solutions (the strong solutions for which thez derivative is boundedin H 1 for all finite time). The result is needed in the proof of the backward uniqueness forthe two dimensional primitive equations but we give it here in general, both for the twoand three dimensional cases.

The equations we are working with read:

@u

@tC u @u

@x1C v @u

@x2Cw @u

@x3� f vC 1

�0

@p

@x1D ��3uCFu;

@v

@tC u @v

@x1C v @v

@x2Cw @v

@x3C f uC 1

�0

@p

@x2D ��vCFv;

@p

@x3D��g;

@u

@x1C @v

@x2C @w

@x3D 0;

@�

@tC u @�

@x1C v @�

@x2Cw @�

@x3� �0N

2

gwD ��3�CF�:

(3.356)

The function spaces for this problem are the same as in the previous section, as well as thesymmetry conditions imposed on the solutions.

3.7.1. Existence and uniqueness ofz�strong solutions in dimension 3We recall thatin Section 3.4 we proved the existence and uniqueness of what we called thez�weaksolutions of the Primitive Equations. In what follows, we also need the existence globallyin time as well as the uniqueness ofz�strong solutions in dimension 2 and 3, a conceptdefined below (see the statement of Theorem 3.12). We can prove the following result:


THEOREM 3.12. (z-strong solution in dimension two and three)GivenU0 2 V andF 2L1.0;T IV/, there exists a unique solutionU of problem (3.356), satisfying the initialconditionU.0/DU0 and:

U 2L1.Œ0;T �IV/\L2.0;T I PH 2per.M//;

@U

@x32L2.0;T I PH 2

per.M//:

(3.357)

PROOF. We start by mentioning that the following reasoning is related to dimension 3; thedimension 2 is similar and much easier.

In Section 3.1 the existence and uniqueness of a strong solution, locally in time, wasproved, using the Galerkin approximation. We are now interested in obtaining a priori esti-mates for thez-strong solution, so that, using the Galerkin method, to prove the existencelocally in time and the uniqueness of az-strong solution.

We assumeU is a smooth solution for the primitive equations and we first derive heresome a priori estimates onUx3 . At the end of the proof we explain how these estimatesprovide the existence of thez-strong solution, globally in time.

We start by differentiating the evolution equation (3.184) inx3; we find:

U 0x3 CAUx3 CEUx3 C .B.U;U //x3 D Fx3 : (3.358)

Multiplying (3.358) by��3Ux3 and integrating overM, we find:

1

2

d

dtkUx3k2C c0j�3Ux3 j2 6

ˇˇZ

Mux3Ux1 ��3Ux3dM

ˇˇC

ˇˇZ

Mvx3Ux2 ��3Ux3dM

ˇˇ

CˇˇZ

Mw.U /x3Ux3 ��3Ux3dM

ˇˇC

ˇˇZ

MuUx1x3 ��3Ux3dM

CZ

MvUx2x3 ��3Ux3dMC

Z

Mw.U /Ux3x3 ��3Ux3dM

ˇˇC

ˇˇZ

MFx3�3Ux3dM

ˇˇ:

(3.359)

We need to estimate the terms from the right-hand-side of (3.359). The first three termsare similar so we will estimate just one of them:

ˇˇZ

Mux3Ux1 ��3Ux3dM

ˇˇ6 jUx3 jL4 jUx1 jL4 j�3Ux3 jL2

6 cj�3Ux3 jjUx1 j1=4kUx1k3=4jUx3 j1=4kUx3k3=4

6 cj�3Ux3 jkU k1=4jU j3=4H2 jUx3 j1=4kUx3k3=4

6 c08j�3Ux3 j2C cjU j1=2H1 jU j

3=2

H2jUx3 j1=2kUx3k3=2:

(3.360)

128 M. Petcu et al.

By integration by parts we also find for the other terms:

Z

M.uUx1x3 C vUx2x3 Cw.U /Ux3x3/ ��3Ux3dMD�

Z

Mur3Ux1x3 � r3Ux3dM

�Z

Mvr3Ux2x3 � r3Ux3dM�

Z

Mw.U /r3Ux3x3 � r3Ux3dM

�Z

MŒ.r3u � r3/Ux3 � �Ux1x3dM�

Z

MŒ.r3v � r3/Ux3 � �Ux2x3dM

�Z

MŒ.r3w.U / � r3/Ux3 � �Ux3x3dM:

(3.361)

We first notice that by integration by parts and using the mass conservation, we find:

Z

Mur3Ux1x3 � r3Ux3dMC

Z

Mvr3Ux2x3 � r3Ux3dM

CZ

Mw.U /r3Ux3x3 � r3Ux3dMD 0:

(3.362)

We need to estimate the remaining terms, which are of two types: containing or notw.U /. We find:

ˇˇZ

MŒ.r3u � r3/Ux3 � �Ux1x3dM

ˇˇ6 jr3U jL2 jr3Ux3 j2L4 6 ckU kkUx3k1=2jUx3 j

3=2

H2

6 c08j�3Ux3 j2L2 C ckU k4kUx3k2;

(3.363)

and

Z

MŒ.r3w.U / � r3/Ux3 � �Ux3x3dMD

Z

M0

Z L3=2

�L3=2Œ.r3w.U / � r3/Ux3 � �Ux3x3 dx3dM0

6Z

M0jr3w.U /jL1x3 jr3Ux3 j

2

L2x3dM0 6 c

Z

M0j�3U jL2x3 jr3Ux3 j

2

L2x3dM0

6 cjj�3U jL2x3 jL2.M0/jjr3Ux3 jL2x3 j2L4.M0/

6 cj�3U jL2.M/jjr3Ux3 jL2x3 jL2.M0/jjr3Ux3 jL2x3 jH1.M0/;(3.364)

whereM0 D .0;L1/� .0;L2/.


One can easily show, by direct differentiation and classical estimates, that:

jjr3Ux3 jL2x3 j2H1.M0/ 6 c.jr3Ux3 j2L2.M/

C jUx3 j2H2.M//6 cjUx3 j2H2.M/

:

(3.365)

Using (3.365) into (3.364), we find:

Z

MŒ.r3w.U / � r3/Ux3 � �Ux3x3dM6 cj�3U jjr3Ux3 jj�3Ux3 j

6 c08j�3Ux3 j2C cj�3U j2jr3Ux3 j2:

(3.366)

The forcing term is easy to estimate, and gathering all the above estimates we find:

d

dtkUx3k2C c0j�3Ux3 j2 6 f .t/kUx3k2C g.t/; (3.367)

with

f .t/D c.kU k2C j�3U j2L2/; g.t/D jFx3 j2L2 :Using these a priori estimates and the Galerkin method, we prove that thez-weak solu-

tion exists on an interval.0; t?/, with t? 6 T . But the recent improvements from Cao andTiti (2006) and Kobelkov (2006) showed the existence of a global strong solution (mean-ing t? D T ) and since the estimates in (3.367) depend only on�U , we conclude that thez-strong solution exists globally in time.

�

3.7.2. Backward uniqueness for thez�weak solutions in dimension twoIn what followswe prove that thez�weak solutions for the2D primitive equations have the backwarduniqueness property. This means that if twoz�weak solutionsU1 andU2 defined on theinterval Œ0; T � coincide at a pointt? 2 .0;T /, then we can conclude that the solutionscoincide on the whole intervalŒ0; t?�. The arguments we use are similar to the case ofNavier-Stokes equations considered in?, ?.

In fact we can prove that:

THEOREM 3.13. (z-weak solutions in dimension two)Let F be inL2.0;T;V/ and letU1, U2 be twoz�weak solutions for the primitive equations (3.356),U1, U2 belongingto C.Œ0; T �IH/ \ L2.0;T;V/, such thatU1.t?/ D U2.t?/. ThenU1 D U2 on the intervalŒ0; t?�.

Before starting to prove the result announced, we give the following useful result:

130 M. Petcu et al.

PROPOSITION3.1. LetF be inL2.0;T IV/ andU0 in V . Let us also considerU solutionof the linear primitive equations:

U 0.t/CAU.t/CEU.t/D F;U.0/DU0:

(3.368)

For all time t such thatU.t/¤ 0, we define the following function:

�.t/D ..ACE/U.t/;U.t//HjU.t/j2H

: (3.369)

Then,� is differentiable for almost everyt where it is defined (meaning whereU.t/¤ 0)and

�0.t/6 jF.t/j2H

jU.t/j2H: (3.370)

PROOF. By classical methods, one can immediately show (compare to Theorem 3.12) thatthe solutionsU of the linear primitive equations satisfy:

U 2L1.0;T IV/\L2.0;T I PH 2per.M//;

@U

@x32L2.0;T I PH 2

per.M//; U 2 C.Œ0;T �;H/:

We first note that the function� is defined on the open subset of.0;T / wherejU.t/jH >0; the set wherejU.t/jH > 0 is open becauseU 2 C.Œ0; T �;H/.

Then, all the computations below, performed formally, can be fully justified by usinga Galerkin approximation. We first note that, sinceE is an skewsymmetric operator, wehave:


D .AU.t/;U.t//HjU.t/j2H

:

We find:

�0.t/D 2< AU0.t/;U.t/ >V 0;V C<AU 0x3.t/;Ux3.t/ >V 0;V

jU.t/j2H� 2.AU.t/;U.t//HjU.t/j4H

f<U 0.t/;U.t/ >V 0;V C<U 0x3.t/;Ux3.t/ >V 0;V g

D 2.F �AU.t/�EU.t/;AU.t//HjU.t/j2H� 2.AU.t/;U.t//HjU.t/j4H

.F �AU.t/�EU.t/;U.t//H

D 2.F;AU.t//HjU.t/j2H� 2 jAU.t/j

2H

jU.t/j2H� 2.AU.t/;U.t//HjU.t/j4H

.F;U.t//H

C 2 j.AU.t/;U.t//Hj2

jU.t/j4H;


(3.371)

where, in the computations above, we used the fact that:

<AU.t/;EU.t/ >V 0;VD 0:

The relation above can be formally checked as follows (rigorous justifications can bederived):

<AU.t/;EU.t/ >V 0;VD � fZ

M.u�v � v�u/dM

� g

�0

Z

M�w.�U /dMC g

�0

Z

M��w.U /dM

D � g

�0

X

lCmD0;l3¤0jl j2�l

m1

m3umC

g

�0

X

lCmD0;m3¤0�lm1

m3jmj2um

D 0;(3.372)

where we used the definition ofw.U / as�k1=k3uk for k3 ¤ 0, and0 whenk3 D 0.We have the following relation:

j.AU.t/;U.t//Hj2 � .AU.t/;U.t//H.F;U.t//HC1

4j.F;U.t//Hj2

D j.AU.t/�F=2;U.t//Hj2 6 jAU.t/�F=2j2HjU j2H:(3.373)

Continuing to estimate�0 in (3.371), we can conclude:

�0.t/6 2.F;AU.t//HjU.t/j2H� 2 jAU.t/j

2H

jU.t/j2HC 2 jAU.t/�F=2j

2H

jU.t/j2H� 12

j.F;U.t//Hj2jU.t/j4H

6 2.F;AU.t//HjU.t/j2H� 2 jAU.t/j

2H

jU.t/j2H� 12

j.F;U.t//Hj2jU.t/j4H

C 2

jU j2HfjAU j2H � .F;AU.t//HC

1

4jF j2Hg

6 jF.t/j2H

jU.t/j2H:

(3.374)�

We can now start to prove the main result of this section.

132 M. Petcu et al.

REMARK 3.6. A similar result is also true in dimension three but in other spaces. Moreexactly, letF be inL2.0;T IV / andU0 in V . Let us also considerU as the solution of thelinear primitive equations:

U 0.t/CAU.t/CEU.t/D F;U.0/DU0:

(3.375)

For all timet such thatU.t/¤ 0, we define the following function:


:

Then,� is differentiable for almost allt where it is defined and

�0.t/6jF.t/j2HjU.t/j2H

: (3.376)

Proof of Theorem 3.13.We notice that sinceU1 andU2 arez�weak solutions,U1 andU2 belong toL2.0;T;V/ and we can thus find aı arbitrarily small such thatU1.ı/ andU2.ı/ belong toV . Considering the primitive equations havingU1.ı/ andU2.ı/ as initialcondition att D ı, one obtains, using Theorem 3.12 (for the dimension 2), the existence ofz�strong solutionsQU1 and QU2.

By the uniqueness of the solution we conclude thatQU1 DU1 and QU2 DU2 on the intervalŒı; T �, soU1, U2 belong toL1.ı; T;V/ \ L2.ı; T; PH 2

per.M//, and@U1=@x3, @U2=@x3belong toL2.ı; T; PH 2

per.M// for ı > 0 arbitrarily small.

We writeU ] D U1 � U2 and QU D U1 C U2. Combining the equations forU1 andU2,we find thatU ] satisfies the following equation:

U ]0CAU ]CEU ]C 1

2B. QU ;U ]/C 1

2B.U ]; QU/D 0; (3.377)

with U ].t?/D 0.We define the following operator:

M.t/U ] D 1

2B. QU ;U ]/C 1

2B.U ]; QU/

D 1

2

� Qu@U]

@x1Cw. QU/@U

]

@x3

�C 1

2

�u]@ [email protected] ]/ @

QU@x3

�:

(3.378)

In what follows, the task is to prove thatkM.t/kL.V;H/ belongs toL2.ı; T /. We thuscompute:


ˇˇw. QU/@U

]

@x3

ˇˇL26 jw. QU/jL4

ˇˇ@U

]

@x3

ˇˇL46 cjw. QU/j1=2

L2kw. QU/k1=2

ˇˇ@U

]

@x3

ˇˇ1=2

L2

@U]

@x3

1=2

6 ck QU k1=2j QU j1=2H2kU ]kV ;

j Qu@U]

@x1jL2 6 j QU jL1kU ]k6 cj QU j1=2H2 k QU k1=2kU ]kV :

(3.379)

We also find:

ˇˇw.U ]/ @

QU@x3

ˇˇL26 jw.U ]/jL2

ˇˇ @QU

@x3

ˇˇH26 ckU ]kV

ˇˇ @QU

@x3

ˇˇH2;

ju] @QU

@x1jL2 6

ˇˇU ]

ˇˇL4

ˇˇ @QU

@x1

ˇˇL46 ckU ]kVk QU k1=2j QU j1=2H2 :

(3.380)

Gathering the above estimates, we find:

jM.t/U ]jL2 6 ck QU k1=2j QU j1=2H2 kU ]kV C cˇ @ QU@x3

ˇH2kU ]kV : (3.381)

We now need to estimate theL2�norm of thex3-derivative ofM.t/U ], in fact we needto estimate the following expression:

2.M.t/U ]/x3 D Qux3@U ]

@x1Cw. QU/x3

@U ]

@x3C Qu @2U ]

@x1@x3Cw. QU/@

2U ]

@x23

C u]x3@ [email protected] ]/x3

@ QU@x3C u] @2 QU

@[email protected] ]/@

2 QU@x23

;

(3.382)

and we separately bound each of the terms.We easily find:

ˇQux3

@U ]

@x1

ˇL26 j Qux3 jL1

ˇ@U ]@x1

ˇL26 cj Qux3 jH2kU ]kV ;

ˇw. QU/x3

@U ]

@x3

ˇL26 j Qux1 jL4

ˇ@U ]@x3

ˇL46 cj QUx1 j1=2L2 k QUx1k1=2kU ]kV ;

ˇQu @2U ]

@x1@x3

ˇL26 j QujL1

ˇ @2U ]@x1@x3

ˇL26 cj QujH2kU ]kV ;

(3.383)

134 M. Petcu et al.

as well as:

ˇu]

@2 QU@x1@x3

ˇL26 cjU ]jH1

ˇ @2 QU@x1@x3

ˇH16 ckU ]kV

ˇ @ QU@x3

ˇH2;

ˇu]x3

@ QU@x1

ˇL26 cjU ]x3 jH1 j QU jH2 6 ckU ]kV j QU jH2 ;

ˇw.U ]/x3

@ QU@x3

ˇL26 jU ]x1 jL2

ˇ @ QU@x3

ˇL1 6 ckU ]kV

ˇ @ QU@x3

ˇH2:

(3.384)

We remain with some more delicate terms to estimate, which need anisotropic estimates:

ˇw. QU/@

2U ]

@x23

ˇL26ˇjw. QU/jL1x3 j

@2U ]

@x23jL2x3

ˇL2x16 cjj QUx1 jL2x3 jL1x1

ˇ@2U ]@x23

ˇL2

6 cj QU jH2ˇ@2U ]@x23

ˇL26 cj QU jH2kU ]kV ;

ˇw.U ]/

@2 QU@x23

ˇL26ˇjw.U ]/jL1x3 j

@2 QU@x23jL2x3

ˇL2x16 cjjU ]x1 jL2x1 .L2x3 /

ˇˇ@2 QU@x23

ˇL1x1 .L

2x3/

ˇL2x1

6 ckU ]kV @2 QU@x23

H2:

(3.385)

From the computations above we can now conclude that:

jM.t/U ]jH 6 cfk QU k1=2j QU j1=2H2 Cˇ @ QU@x3

ˇH2C j QU jH2gkU ]kV : (3.386)

Thus kM.t/kL.V;H/ is bounded by the expression between brackets in (3.386) and, weconclude, taking into account the properties ofQU , thatkM.t/kL.V;H/ belongs toL2.ı; T /for ı > 0 arbitrarily small.

We now need to prove that ifjU.t?/jH D 0, thenjU.t/jH D 0 for all t 2 Œı; t?�, 0 < ı <t?. The equivalent relation that we prove is that if there exists a timet 2 .ı; t?/ such thatjU ].t/jH > 0, thenjU ].t?/jH > 0. Since we proved thatU ] 2 C.Œ0; T �;H/, it is enough toshow that logjU ].t/jH is bounded from below onŒı; t?�.

Writing (3.377) as:

U ]0CAU ]CEU ]CM.t/U ] D 0;

we can use Proposition 3.1 where� is defined as in (3.369) forU ]. We find:

�0.t/6 jM.t/U].t/j2H

jU ].t/j2H6 kM.t/k2L.V;H/

jU ].t/j2VjU ].t/j2H

6 1

c0kM.t/k2L.V;H/�.t/I


(3.387)

in (3.387) we used the fact that

..ACE/U ].t/;U ].t//H D .AU ].t/;U ].t//H > c0kU ]k2V :

SincekM.t/kL.V;H/ belongs toL2.ı; T /, we can apply the Gronwall lemma to (3.387)and find:

�.t/6 �.ı/exp.Z t

ı

c�10 kM.s/k2L.V;H/ ds/6K; (3.388)

with K a constant independent oft .Considering the function logjU ].t/j2H, we have:

d

dt.logjU ].t/j2H/D 2

.U ];U ]0/H

jU ].t/j2HD�2.U

]; .ACE/U ]/HjU ].t/j2H

� 2.U];M.t/U ]/HjU ].t/j2H

>�2�.t/� 2c0kM.t/kL.V;H/�.t/;(3.389)

since we can estimate:

.U ];M.t/U ]/H 6 jU ]jHjM.t/U ]jH6 jU ]jHkM.t/kL.V;H/kU ]kV6 c0jU ].t/j2HkM.t/kL.V;H/�.t/:

(3.390)

Using (3.388) into (3.389), we find that:

d

dt.logjU ].t/j2H/>�2K.1C c0kM.t/kL.V;H//; (3.391)

and sincekM.t/kL.V;H/ is inL1.ı; T /, we find that

logjU ].t/j2H >�2K.t? � t/C logjU ].ı/j2H >K1; 8t 2 Œı; t?�;

with K1 a constant independent oft . This gives thatjU ].t?/j2H ¤ 0, which implies that ifU ].t?/j2H D 0, thenjU ].t/j2H D 0 on the intervalŒı; t?�. But we know that this relation canbe proved for almost allı in Œ0; t?� and from the fact thatU ] 2 C.Œ0;T �;H/, the desired re-sult follows. �

3.7.3. Backward uniqueness for the strong solutions of the three dimensional primitiveequations The purpose of this section is to prove the backward uniqueness for the strongsolutions of the three dimensional primitive equations (3.272). In Section 3.6 we have

136 M. Petcu et al.

shown the existence and uniqueness of the strong solutions, as well as the existence and(forward) uniqueness of very strong solutions (solutions with values inHm,m> 2). Theseresults will be used in what follows (see also Petcu (2006)).

The result we will prove here is the following one:

THEOREM 3.14. LetF be inL2.0;T;V / and letU1, U2 be two strong solutions for theprimitive equations (3.356),U1, U2 belonging toC.Œ0;T �IV /\L2.0;T; PH 2

per.M//, suchthatU1.t?/D U2.t?/. ThenU1 DU2 on the intervalŒ0; t?�.

The proof of the theorem follows the main steps as in Theorem 3.13 so we only empha-size the points which are different.

Proof of Theorem 3.14:Let U1 andU2 be two strong solutions. We can then find aıarbitrarily small such thatU1.ı/ andU2.ı/ belong to PH 2

per.M/. This implies, with theresults of Petcu (2006), that:

U1;U2 2 C.ı; T; PH 2per.M//\L2.ı; T; PH 3

per.M//:

As in the previous section, we writeU ] D U1 � U2 and QU D U1 C U2. Combiningthe equations forU1 andU2, we find thatU ] satisfies the same equation as (3.377) withU ].t?/D 0.

We need again to prove that the operatorM.t/ defined by:

M.t/U ] D 1

2B. QU ;U ]/C 1

2B.U ]; QU/

D 1

2

� Qu@U]

@x1Cw. QU/@U

]

@x3

�C 1

2

�u]@ [email protected] ]/ @

QU@x3

�;

(3.392)

has the property thatjM.t/jL.V;H/ belongs toL2.ı; T /.Here we estimate each term of (3.392) as follows:

ˇQu@U

]

@x1

ˇH6 j QujL1

ˇ@U ]@x1

ˇH6 cj QU jH2 jU ]jV ;

ˇQv @U

]

@x2


ˇw. QU/@U

]

@x3

ˇH6 jw. QU/jL1

ˇ@U ]@x3


(3.393)


and also:

ˇu]@ QU@x1

ˇH6 ju]jL4

ˇ @ QU@x1

ˇL46 cj QU jH2 jU ]jV ;

ˇv]@ QU@x2


ˇw.U ]/

@ QU@x3

ˇH6 jw.U ]/jL2

ˇ @ QU@x3

ˇL1 6 cj QU jH3 jU ]jV :

(3.394)

Gathering the estimates above, we find:

jM.t/U ]jH 6 cj QU jH3 jU ]jV ; (3.395)

which implies thatjM.t/jL.V;H/ belongs toL2.ı; T /. We can now perform the same kindof reasoning as in Theorem 3.13 in order to prove the desired result. �

4. Regularity for the elliptic linear problems in GFD

We have used many times, in particular in Section 3 the result ofH 2 regularity of thesolutions to certain linear elliptic problems. Following the general results from Agmon,Douglis and Nirenberg (1959), and Agmon, Douglis and Nirenberg (1964), we know thatthe solutions of second-order elliptic problems are inHmC2, if the right-hand sides of theequations are inHm;m > 0, and the other data are in suitable spaces; see also Lions andMagenes (1972) form< 0. Results of this type are proved in this section.

There are several specific aspects and several specific difficulties which justify thelengthy and technical developments of this section which do not allow us to directly re-fer to the general results of Agmon, Douglis and Nirenberg (1959) and Agmon, Douglisand Nirenberg (1964):

For the whole atmosphere (not studied in detail here) and for the space periodic casestudied in Section 3.5 the domains are smooth, making the results of this section easy.

(i) The (linear, stationary) GFD–Stokes problem (see Section 4.4.1), involves an inte-gral equation (the second equation in (4.96)), which prevents from a purely local treatment,like for the classical Stokes problem of incompressible fluid mechanics.

(ii) The boundary conditions of the problem can be a combination of Dirichlet,Neumann and/or Robin boundary conditions.

(iii) The domains that we have considered and that we consider in this section are notsmooth, they have angles in two dimensions and edges in three dimensions. This is au-tomatically the case for the ocean and for regional atmosphere or ocean problems. Forthis reason, technique pertaining to the theory of elliptic problems in nonsmooth domains(see, e.g., Grisvard (1985), Kozlov, Mazya and Rossmann (1997), Mazya and Rossmann(1994)), are needed and used here.

(iv) Because the domain is not smooth, only theH 2 regularity is proved here,mD 0.TheH 3 regularity,m> 1 is not expected in general.

138 M. Petcu et al.

(v) Another aspect of the study in this section concerns the shape of the ocean or theatmosphere (shallowness). A “small” parameter" is introduced, the depth being called"hinstead ofh; 0 < " < 1, and we want to see how the regularity constants (which depend onthe domain) depend on".

The small depth hypothesis was considered in Hu, Temam and Ziane (2003) and is notconsidered in this chapter. Introducing the parameter" makes the proofs of this sectiongenerally more involved than needed for this chapter. However, these results usefully com-plement the article Hu, Temam and Ziane (2002) used in Hu, Temam and Ziane (2003).

Many of the results presented in this section are new although some related results ap-peared in Ziane (1995) and Ziane (1997). The results are fairly general, except for theorthogonality condition appearing in (4.54) (�b orthogonal to�`). This condition is notphysically desirable; and it is not either mathematically needed (most likely), as no suchcondition appears for the regularity theory of elliptic problems in angles or edges Grisvard(1985), Mazya and Rossmann (1994), Kozlov, Mazya and Rossmann (1997). We believethat it can be removed, but this problem is open. Let us recall also that all the results inSection 3 are valid whenever the necessaryH 2-regularity results can be proved.

For the notations, the basic domain under consideration isM":

M" D˚.x;y; z/; .x;y/ 2 �i ;�"h.x;y/ < z < 0

:

For " D 1, we recover the domainM.M1 DM/ used in Sections 2 and 3. Below, fortechnical reasons, we introduce auxiliary domainscM" andQ". A number of unspecifiedconstants independent of" are generically denoted byC0.

Note also that this section is closer to PDE theory than to geoscience and therefore westay closer to the PDE notations than to the geoscience notations. Hence the notations arenot necessarily the same as in the rest of the chapter; in particular, we do not use bold facesfor vectors and the current point ofR2 or R3 is denotedx D .x1; x2/ or x D .x1; x2; x3/instead of.x;y/ or .x;y; z/.

4.1. Regularity of solutions of elliptic boundary value problems in cylinder type domains

We study in this section theH 2 regularity of solutions of elliptic problems of the sec-ond order in a cylinder type domain; the boundary condition is either of Dirichlet or ofNeumann type on all the boundary.

Since the domain contains wedges, it is not smooth and we rely heavily on the results ofGrisvard (1985) about regularity for elliptic problems in nonsmooth domains. However aconvexity assumption of the domain is essential in Grisvard (1985), that we want to avoid:this section is mainly devoted to the implementation of a suitable technique, correspondingto a tubular (cylindrical) covering of the domain under consideration.

Let cM" D f.x1; x2; x3/ 2 R3I .x1; x2/ 2 �i ;�"h.x1; x2/ < x3 < "h.x1; x2/g; where�i

is a bounded open subset ofR2, with @�i aC 2-curve, andh W x�i ! RC is a positive func-


tion,h 2 C 4.x�i/ and there exist two positive constantsh and Nh such thath6 h.x1; x2/6 Nhfor all .x1; x2/ in x�i . Define the elliptic operatorA by

AuD�3X

k;`D1

@

@xk

�ak`.x/

@u

@x`

�C

3X

kD1bk.x/

@u

@xkC c.x/u; (4.1)

where the coefficientsak;`; k; `D 1; 2; 3 are of classC 2.cM"/; bk ; k D 1; 2; 3, are of class

C 1.cM"/ andc is of classC 0.cM"/. Furthermore, we assume thatA is uniformly stronglyelliptic, i.e., there exists a positive constant˛ independent ofx and" such that

3X

k;`D1ak`.x/�k�` > ˛j�j2 8x 2cM"; 8� 2R3: (4.2)

We also assume that the functionsak`; bk ; c; k; `D 1; 2; 3, are independent of". We aimto study the regularity and the dependence on" of the solutions to the Dirichlet problem

(AuD f in cM";

uD 0 in @cM";(4.3)

and the solutions of the Neumann problem,

(AuD f in cM";@u@nAD 0 on@ OM";

(4.4)

where @@nA

denotes the co-normal boundary operator defined by

@u

@nADX

k;`

ak`@u

@x`nk ; (4.5)

andnD .n1; n1; n3/ denotes the unit vector in the direction of the outward normal to@cM".Our goal is to prove theH 2 regularity of solutions to the Dirichlet problem (4.3) or theNeumann problem (4.4), and to obtain the dependence on" of the constantC" appearingin the inequality

X

k;`

ˇˇ @2u

@xk @x`

ˇˇ2

L2.bM"/

6 C"jAuj2L2.bM"/

: (4.6)

In fact we will show thatC" D C0 is independent of" and, more precisely, we will provethe following.

140 M. Petcu et al.

THEOREM 4.1. Let cM" D f.x1; x2; x3/ 2 R3I .x1; x2/ 2 �i , �"h.x1; x2/ < x3 <

"h.x1; x2/g, where�i is a bounded subset ofR2, with @�i a C 2-curve, andh 2 C 4.�i/,and there exists two positive constantsh; Nh such thath 6 h.x1; x2/ 6 Nh for all .x1; x2/in x�i . Let f 2 L2.cM"/ andu 2H 1

0 .cM"/, with kuk

H1.bM"/6 C0jf jL2.bM"/

with C0 in-dependent of".

If u satisfies

AuD�3X

k;`D1

@

@xk

�ak`.x/

@u

@x`

�C

3X

kD1bk.x/

@u

@xkC c.x/uD f;

whereaij ; bi are in C 2.cM"/; c in C 1.cM"/ andA is uniformly strongly elliptic in thesense of(4.2), thenu 2 H 2.cM"/ and there exists a constantC0 independent of" suchthat

X

i;jD1

ˇˇ @2u

@xi @xj

ˇˇ2

L2.bM"/

6 C0jf j2L2.bM"/

: (4.7)

PROOF. The proof of Theorem 2.1 is divided into four steps.

STEP 1 (Flattening the top and bottom boundaries). We straighten the bottom and topboundaries and transform the domaincM" into the cylinderQ" D �i � .�"; "/, and theoperatorA is transformed to an operatoreA of the same form and satisfying the sameassumptions asA. In fact let

‰ W .x1; x2; x3/‘ .y1; y2; y3/;(4.8)

y1 D x1; y2 D x2; y3 Dx3

h.x1; x2/:

Sinceh 2 C 4.x�i/, we have‰ 2 C 4.cM"/. Furthermore, we note thatAu may be writ-ten as

AuD�3X

k;`D1ak`.x/

@2u

@xk @x`C

3X

kD1dk.x/

@u

@xkC c.x/u; (4.9)

where

dk.x/D bk.x/�3X

`D1

@a`k.x/

@x`2 C 1�cM"

�: (4.10)

Now let Qu WQ"! R; Qu.y1; y2; y3/D u.x1; x2; x3/. Since‰ 2 C 4.cM"/ and‰ is inde-pendent of", theH 2-norm of Qu is equivalent to theH 2-norm ofu with constants indepen-


dent of". More precisely, there exists a constantC0 independent of" such that

C�10X

k;`

ˇˇ @2 Qu@yk @y`

ˇˇ2

L2.Q"/

6X

k;`

ˇˇ @2u

@xk @x`

ˇˇ2

L2.bM"/

6 C0X

k;`

ˇˇ @2 Qu@yk @y`

ˇˇ2

L2.Q"/

: (4.11)

Furthermore, we can easily check that

eA Qu.y/D�X

k;`

Qak`.y/ @2 Qu@yk @y`

CX

k

Qdk.y/ @ Qu@ykC Qc.y/ Qu.y/D Qf .y/; (4.12)

where8ˆ<ˆ:

Qak`.y/DPr;s@‰k@xr

@‰`@xsars�‰�1.y/

�;

Qdk.y/D�Pr;s@2‰k@xr@xs

asr�‰�1.y/

�CPr@‰k@xr

d r�‰�1.y/

�;

Qc.y/D c�‰�1.y/� and Qf .y/D f �‰�1.y/�:(4.13)

It is clear, since‰ 2 C 4.cM"/, that Qak;` 2 C 2. xQ/; Qdk 2 C 1. xQ/ and Qc 2 C 0. xQ"/. Fi-

nally, if uD 0 on @cM" then QuD 0 on @Q" and if @u=@nA D 0 on @cM" then@ Qu=@neA D 0on@Q", where, as in (4.5),

@ Qu@neADX

k;`

Qak` @ Qu@y`

nk : (4.14)

This is classical, but we include the verification here at the bottom boundaryx3 D�"hor equivalentlyy3 D�": By (4.5) we have

�1C "2jrhj2�1=2 @u

@nA

D "3X

jD1a1j

@u

@xj

@h

@x1C "

3X

jD1a2j

@u

@xj

@h

@x2C

3X

jD1a3j

@u

@xj; (4.15)

but @u@xjDP3

rD1@ Qu@yr

@‰r@xj

, and thus

�1C "2jrhj2�1=2 @u

@nA

D3X

rD1

3X

jD1

�a1j

�"@h

@x1

�C a2j

�"@h

@x2

�C a3j

�@‰r

@xj

@ Qu@yr

: (4.16)

142 M. Petcu et al.

On the other hand, by definition, we have

@ Qu@neAD�

3X

rD1Qa3r .�"/ @ Qu

@yr(the normal is in the direction ofy3 < 0); (4.17)

but

Qa3r DXm;n

@‰3

@xm

@‰r

@xnamn D

3X

nD1

@‰r

@xn

3X

mD1amn

@‰3

@xm

!; (4.18)

and since‰3.x1; x2; x3/D x3=h.x1; x2/, we have

Qa3r .�"/D 1

h

3X

jD1

�a1j

�"@h

@x1

�C a2j

�"@h

@x2

�C a3j

�@‰r

@xj: (4.19)

Hence

@ Qu@neAD 0 aty3 D�": (4.20)

A similar computation yields@ Qu=@neA D 0 aty3 D ". Now we check the Neumann condi-tion at the lateral boundary. First write

@u

@nAD

2X

kD1

3X

`D1ak`

@u

@x`nk D

2X

kD1

3X

`;rD1ak`

@ Qu@yr

@‰r

@x`nk (4.21)

and

@ Qu@neAD

2X

kD1

3X

`D1Qak` @ Qu

@y`nk D

2X

kD1

3X

`;r;sD1asr

@‰k

@xs

@‰`

@xr

@ Qu@y`

nk ; (4.22)

but since fork D 1; 2;‰k.x1; x2; x3/D xk , we have@‰k=@xs D ısk (the Kronecker sym-bol). Hence

@ Qu@neAD

3X

k;`;rD1akr

@‰`

@xr

@ Qu@y`

nk : (4.23)

Interchanging andr , we obtain@ Qu=@neA D @u=@nA D 0 on the lateral boundary. Fromnow on we concentrate on the Dirichlet boundary condition, the Neumann condition casefollows in the same manner.


STEP 2 (Interior regularity). LetBR be an open ball, withBR �� i ; without loss ofgenerality we assume thatBR is centered at0. By Step 1, we may assume thatcM" is aright cylinder, i.e.,cM" DQ" D �i � .�"; "/. Now let� 2 C10 .BR/ .� independent ofx3/with � � 1 in BR=2 and06 � 6 1. Then

eA�� Qu�D � Qf CE� Qu;

whereE� is a first-order differential operator, which implies thatjE� QujL2.Q"/ 6 C0jf j"whereC0 is independent on", andjf j" is an alternate notation forjf j

L2.bM"/. Hence

eA�� Qu�D floc; with jflocjL2.BR�.�";"// 6 C0jf j";

and, for either boundary condition (Dirichlet, Neumann),� QuD 0 on @.BR/� .�"; "/ and,in the Dirichlet case,� QuD 0 onBR � .�"; "/, and, in the Neumann case,@.� Qu/=@z D 0onBR � .�"; "/.

We quote the following theorems from Grisvard (1985), we start first with the case ofthe Dirichlet boundary condition.

THEOREM A (Grisvard (1985), Theorem 3.2.1.2).Let� be a convex, bounded and opensubset ofRn. Then for eachf 2 L2.�/, there exists a uniqueu 2H 2.�/, the solution ofAuD f in �;uD 0 on@�.

The proof of the theorem, given in Grisvard (1985), pp. 148–149, is based on a prioribounds for solutions inH 2.�/. These bounds depend in the case of general domains onthe curvature of@�; however in the case of a convex domain the curvature is negative andthe constants in the bounds are therefore independent on the domain.

Similarly, in the case of Neumann boundary condition, we have the following theorem.

THEOREM B (Grisvard (1985), Theorem 3.2.1.3).Let� be a convex, bounded and opensubset ofRn. Then for eachf 2 L2.�/ and for each� > 0, there exists a uniqueu 2H 2.�/, the solution of

�3X

k;`D1

Z

�

ak`.x/@u

@x`

@v

@xkdxC �

Z

�

uv dx DZ

�

f v dx (4.24)

for all v 2H 1.�/.

We note that (4.24) is the weak form of the Neumann problem for the equation

�3X

k;`D1

@

@xk

�ak`.x/

@u

@x`

�C �uD f in �

together with the boundary condition@u@nAD 0 on @�. Again, here, the convexity of the

domain implies that the curvature of the boundary of the domain is negative, and therefore

144 M. Petcu et al.

the constants in the bounds on theL2 norm of the mixed second derivatives in terms of theL2 norm off are independent on the domain. For more details, see Grisvard (1985).

Now we use Theorem A above by first rewritingeA in a divergence form and moving theextra terms to the right-hand side. As above, since� Qu 2H 1.BR � .�"; "// the extra termsare inL2.BR � .�"; "// and

X

i;j

ˇˇ @2.� Qu/@yi @yj

ˇˇ2

L2.BR�.�";"//6 C0jf j2

L2.bM"/:

Finally, since� � 1 in BR=2, we haveQu 2H 2.BR=2 � .�"; "// and

X

i;j

ˇˇ @2 Qu@yi @yj

ˇˇ2

L2.BR=2�.�";"//6 C0jf j2

L2.bM"/:

STEP 3 (Boundary regularity). LetR2C D fx 2 R2Ix2 > 0g and letBCr D fx 2 R2CI jxj<rg be the open half-ball with center at the origin and radiusr contained inR2. By theassumption on�i , for all x0 2 @�i , there exists a neighborhoodV of x0 in R2 and adiffeomorphisme‰ such that

e‰.V \ �i/DBCr ;e‰�x0�D 0:

Using the diffeomorphisme‰, we can construct a (tubular) diffeomorphism‰ in R3 suchthat

‰�V � .�"; "/\Q"

�DBCr � .�"; "/;

by setting‰i .y1; y2; y3/D e‰i .y1; y2/; i D 1; 2, and‰3.y1; y2; y3/D y3. Following thesame procedure as in Steps 1 and 2, letW be an open set ofR2 containingx0 such thatxW � V and let� 2 C10 .V / be such that06 � 6 1 and� � 1 in W . Then

eA�� Qu�D floc with jflocjL2.V�.�";"/\Q"/ 6 C0jf j";

and, in the case of the Dirichlet boundary condition� QuD 0 [email protected] � .�"; "/\Q"/. Nextwe use the transformation‰ which is independent of" and which transforms the domainV � .�"; "/ \Q" into BCr � .�"; "/; � Qu into u�, andeA into A� with A�u� given as inStep 1. Now,u� D 0 on @.BCr � .�"; "// andBCr � .�"; "/ is convex; hence rewritingA�

in a divergence form we obtain using Grisvard (1985),u� 2H 2.BCr � .�"; "// and

X

i;j

ˇˇ @2u�@zi @zj

ˇˇ2

L2.BCR�.�";"//

6 C0jf j2" :


Going back toV � .�"; "/\Q" using‰�1, we obtain

X

i;j

ˇˇ @2.� Qu/@yi @yj

ˇˇ2

L2.V�.�";"/\Q"/6 C0jf j2" :

Hence

X

i;j

ˇˇ @2 Qu@yi @yj

ˇˇ2

L2.V�.�";"/\Q"/6 C0jf j2" :

STEP 4 (Partition of unity and conclusion). LetV0; V1; : : : ; VN andW0; : : : ;WN be twofinite open coverings ofx�i satisfying xV0 � �i IVk ; k > 1 is contained in the domain of alocal mape‰.k/ such that

e‰.k/.Vk \ �i/DBCr ;W0 D V0;xWk � Vk for all k > 1:

Finally let f'kgk be a partition of unity subordinated to the coveringfWkgk of �i . ThenQuDPN

kD1 'k Qu and by Steps 1–3,'k Qu 2H 2.Q"/ and

NX

kD0

3X

i;jD1

ˇˇ@2.'k Qu/@yi @yj

ˇˇ2

L2.Q"/

6 C0jf j2" :

ThereforeQu 2H 2.Q"/ and

3X

i;jD1

ˇˇ @2 Qu@yi @yj

ˇˇ2

L2.Q"/

6 C0jf j2" :

Finally, we go back to the domaincM" and conclude thatu 2H 2.cM"/ and

3X

i;jD1

ˇˇ @2u

@xi @xj

ˇˇ2

L2.bM"/

6 C0jf j2" :

Theorem 2.1 is proved. �

4.2. Regularity of solutions of a Dirichlet–Robin mixed boundary value problem

We now want to derive a result similar to that of Section 4.1, for a boundary value problemwith mixed Dirichlet–Robin boundary conditions, the elliptic operator being the same as

146 M. Petcu et al.

in (4.1). The proof consists in reducing the boundary condition to a full Dirichlet boundarycondition and then use Theorem 2.1.

From now on, we will consider the actual domain

M" D˚.x1; x2; x3/ 2R3I .x1; x2/ 2 �i ;�"h.x1; x2/ < x3 < 0

I

j � j" will denote the norm inL2.M"/ (or product of such spaces) andj � ji will denote thenorm inL2.�i/ (or product of such spaces),kgki D jrgji .

We will prove the following theorem:

THEOREM 4.2. Assume that�i is an open bounded set ofR2, withC 3-boundary@�i andh 2 C 4.�i/. Then, forf 2 L2.M"/ andg 2H 1

0 .�i/, there exists a unique‰ 2H 2.M"/

solution of

8<:

��3‰D f in M";

@‰@x3C ˛‰D g on�i ;

‰D 0 on�b[ �`:(4.25)

Furthermore, there exists a constantC D C.h;�i ; ˛/ independent on" such that

3X

i;jD1

ˇˇ @2‰

@xi @xj

ˇˇ2

"

6 C.h;�i ; ˛/�jf j2" Ckgk2H1.�i/

�: (4.26)

PROOF. The proof is divided into several steps.First we construct a function‰� satisfying the boundary conditions in (4.25), and find

the precise dependence on" of theL2-norm of the second-order derivatives of‰� (seeLemma 4.1). Then we set

bD e˛x3�‰ �‰�� (4.27)

and verify thatb satisfies the homogeneous Neumann condition on�i .@b=@x3 D 0 on�i/

and the homogeneous Dirichlet boundary condition on�` [ �b .b D 0 on �` [ �b/. Bya reflection argument, we extendf to x3 > 0 to be an even function and consider a ho-mogeneous Dirichlet problem on the extended domaincM" D f.x1; x2; x3/I .x1; x2/ 2 �i ;

�"h.x1; x2/ < x3 < "h.x1; x2/g, the solution of which,bW , coincides withb onM". Fi-nally we invoke Theorem 2.1 to conclude theH 2 regularity ofbW and thus ofb, along withan estimate of the type (4.26) forb; we therefore obtain theH 2 regularity of‰ and (4.26)by simply using (4.27).

Thus the whole proof of Theorem 4.2 hinges on the following lifting lemma.


LEMMA 4.1. Let h 2 C 4.x�i/ and g 2 H 10 .�i/. There exists‰� 2 H 2.M"/ such that

@‰�@x3C ˛‰� D g on �i ;‰

� D 0 on �` [ �b, and there exists a constantC D C.h;�i/

independent on", such that for0 < "6 1,

X

i;j

ˇˇ @2‰�@xk @xj

ˇˇ2

"

6 C.h;�i/kgk2H1.�i/: (4.28)

PROOF. First we construct a functione‰ as a solution of the heat equation with�x3 cor-responding to time

8<:

@e‰@x3D��e‰ in �i � .�1; 0/;

e‰D 0 on@�i � .�1; 0/;e‰.x1; x2; 0/D g.x1; x2/ on�i :

(4.29)

Here�D�2 D .@2=@x21 C @2=@x22/ and, below,r Dr2 D .@=@x1; @=@x2/. The function‰� is then constructed as

‰�.x1; x2; x3/D e�˛x3Z x3

�"h.x1;x2/e‰.x1; x2; z/dz: (4.30)

It is clear that‰� � 0 on �` [ �b, and@‰�=@x3 C ˛‰� D e�˛x3e‰.x1; x2; x3/ in �i �.�1; 0/, which implies@‰�=@x3 C ˛‰� D g on �i . We only need to check that‰� 2H 2.M"/ and that the inequality (4.28) is valid. This will be done using the classical energyestimates on the solution of the heat equation which are recalled in Lemmas 4.2 and 4.3.

We note that fork D 1; 2,

e˛x3@‰�

@xkDZ x3

�"h.x1;x2/

@e‰@xk

.x1; x2; z/dzC "@h

@xke‰�x1; x2;�"h.x1; x2/

�

(4.31)

and, fork; j D 1; 2,

e˛x3@2‰�

@xk @xjDZ x3

�"h.x1;x2/

@2e‰@xk @xj

.x1; x2; z/dz

C " @h@xj

@e‰@xk

�x1; x2;�"h.x1; x2/

�

C " @h@xk

@e‰@xj

�x1; x2;�"h.x1; x2/

�

C " @2h

@xk @xje‰�x1; x2;�"h.x1; x2/

�

� "2 @h@xk

@h

@xj

@e‰@x3

�x1; x2;�"h.x1; x2/

�: (4.32)

148 M. Petcu et al.

Here we need bounds on theL2-norm (on�i) of .@e‰=@xk/.x1; x2, �"h.x1; x2// and(@e‰=@x3/.x1; x2;�"h.x1; x2// which are provided by Lemma 4.3. We have

Z

�i

ˇe‰�x1; x2;�"h.x1; x2/�ˇ2

dx1 dx2 6 C0kgk2i ;Z

�i

ˇre‰�x1; x2;�"h.x1; x2/

�ˇ2dx1 dx2 6 C0kgk2i ; (4.33)

Z

�i

ˇˇ @e‰@x3

�x1; x2;�"h.x1; x2/

�ˇˇ2

dx1 dx2 61

"C0kgk2i :

Now, using (4.30)–(4.32) and (4.33), we obtain

ˇˇ @2‰�@xk @xj

ˇˇ2

"

6 C0kgk2i ; k; j D 1; 2: (4.34)

Similar relations hold for‰� andr‰�, using (4.30) and (4.31). Furthermore, since@‰�=@x3 D�˛‰�C e�˛x3e‰, we have

ˇˇ@‰�@x3

ˇˇ2

"

6 2˛2ˇ‰�ˇ2"C 2e2˛ Nh

ˇe‰j2" 6 "2C.h/kgk2i ; (4.35)

ˇˇ @@x3r‰�

ˇˇ2

"

6 2˛2ˇr‰�

ˇ2"C 2e2˛ Nh

ˇre‰

ˇ2"6 "2C.h/kgk2i : (4.36)

Finally

@2‰�

@x23D�˛@‰

�

@x3� ˛e�˛x3e‰C e�˛x3

@e‰@x3

; (4.37)

impliesˇˇ@2‰�@x23

ˇˇ2

"

6 C.h/kgk2i : (4.38)

The proof of Lemma 4.1 is complete. �

The proof of Theorem 4.2 relied on estimates given by Lemmas 4.2 and 4.3 which wenow state and prove

LEMMA 4.2 (Estimates on solutions of the heat equation).Let ‰ be the solution of theheat equation

@‰

@x3D��‰ in �i � .�1; 0/;

(4.39)‰.x1; x2; 0/D g.x1; x2/ on�i ;


with either Dirichlet or Neumann boundary condition, and

g 2H 10 .�i/ and ‰D 0 on@�i � .�1; 0/

or

g 2H 1.�i/ and@‰

@n�i

D 0 on@�i � .�1; 0/:

Then

1

2jx3jk

ˇˇ@j‰@xj3

ˇˇ2

i

.x3/CZ 0

x3

jzjkˇˇr @

j‰

@xj3

ˇˇ2

i

.z/dz

6(12jgj2i for k D j D 0;C jx3jk�2jC1kgk2i for k > 2j � 1; j > 1;

(4.40)

and

1

2jx3jk

ˇˇr @

j‰

@xj3

ˇˇ2

i

.x3/CZ 0

x3

jzjkˇˇ@jC1‰@xjC13

ˇˇ2

i

.z/dz

6(12kgk2i for k D 0; j D 0;C jx3jk�2j kgk2i for k > 2j; j > 1:

(4.41)

In (4.40) and (4.41),C is a constant depending onk, j andh. As before,r D r2 D.@=@x1; @=@x2/.

PROOF OFLEMMA 4.2. Denote byek;j andfk;j the left-hand sides of (4.40) and (4.41).We differentiate (4.39)j times inx3, multiply the resulting equation byxk3 @

j‰=@xj3

and integrate over�i . Using Stokes formula and observing that@j‰=@xj3 satisfies the same

boundary condition on@�i as‰, we obtain after multiplication by.�1/k :

ek;j D

8<:

k2

R 0x3jzjk�1

ˇ@j‰

@xj3

ˇ2i.z/dz for k > 1;

12

ˇ@j‰

@xj3

ˇ2i.0/D 1

2

ˇ�j‰

ˇ2i.0/D 1

2

ˇ�jg

ˇ2i

for k D 0.(4.42)

Similarly, if we differentiate (4.39)j times in x3, multiply the resulting equation byxk3 @

jC1‰=@xjC13 and integrate over�i , we find

fk;j D

8<:

k2

R 0x3jzjk�1

ˇr @j‰@xj3

ˇ2i.z/dz for k > 1;

12

ˇr @j‰@xj3

ˇ2i.0/D 1

2

ˇr�jg

ˇ2i

for k D 0:(4.43)

150 M. Petcu et al.

Using (4.42), (4.43) withk D j D 0, (4.42) withk D j D 1 and (4.43) withk D 2; j D 1,we find some of the relations (4.40), (4.41), namely

1

2

ˇ‰.x3/

ˇ2iCZ 0

x3

jr‰j2i .z/dz 61

2jgj2i ;

1

2

ˇr‰.x3/

ˇ2iCZ 0

x3

ˇˇ @‰@x3

ˇˇ2

i

.z/dz 6 12kgk2i ;

(4.44)1

2jx3j

ˇˇ @‰@x3

ˇˇ2

i

.x3/CZ 0

x3

jzjˇˇr @‰@x3

ˇˇ2

i

.z/dz 6 14kgk2i ;

1

2jx3j2

ˇˇr @‰@x3

ˇˇ2

i

.x3/CZ 0

x3

z2ˇˇ@2‰@x23

ˇˇ2

i

.z/dz 6 14kgk2i :

To derive the other relations (4.40) and (4.41), we first integrate (4.42) fromx3 to 0,with x3 < 0;k > 1 andj > 0; we obtain

Z 0

x3

jzjkˇˇ@j‰@xj3

ˇˇ2

i

.z/dz 6 kZ 0

x3

Z 0

t

jzjk�1ˇˇ@j‰@xj3

ˇˇ2

i

.z/dz dt

6 kZ 0

x3

.z � x3/jzjk�1ˇˇ@j‰@xj3

ˇˇ2

i

.z/dz:

Thus, fork > 1; j > 0,Z 0

x3

jzjkˇˇ@j‰@xj3

ˇˇ2

i

.z/dz 6 k

kC 1 jx3jZ 0

x3

jzjk�1ˇˇ@j‰@xj3

ˇˇ2

i

.z/dz (4.45)

and, by induction onk and using (4.41),

Z 0

x3

jzjkˇˇ@j‰@xj3

ˇˇ2

i

.z/dz 6 jx3jk

kC 1Z 0

x3

ˇˇ@j‰@xj3

ˇˇ2

i

.z/dz;

6

8<:jx3jkC12.kC1/ jgj2i for j D 0;jx3jk2.kC1/kgk2i for j D 1 .k > 1/:

(4.46)

Now, combining (4.42) and (4.43), we find fork > 2; j > 1,

ek;j 6k.k � 1/22

ek�2;j�1:

Hence, by induction

ek;j 6kŠ

22r .k � 2r/Šek�2r;j�r ;


ek;j 6kŠ

22j�2.k � 2j C 2/Šes;1;

s D k � 2j C 2, for k > 2j � 2; j > 1. For s > 1, i.e.,k > 2j � 1 andj > 1, we have,thanks to (4.42) and (4.46),

es;1 D1

2

Z 0

x3

jzjs�1ˇˇ @‰@x3

ˇˇ2

i

.z/dz

6 12jx3js�1kgk2i ; (4.47)

ek;j 6kŠ

22j�1.k � 2j C 2/Š jx3jk�2jC1kgk2i ;

for k > 2j � 1; j > 1. The relations (4.40) are proven, the relations (4.41) followfrom (4.44) forj D 0 and fromfk;j 6 .k=2/ek�1;j for j > 1.

Lemma 4.2 is proved. �

From Lemma 4.2 we easily infer the following lemma:

LEMMA 4.3. Under the hypotheses of Lemma4.2,

Z

�i

ˇˇr @

j‰

@xj3

ˇˇ2�x1; x2;�"h.x1; x2/

�dx1 dx2 6 C"�2j kgk2H1.�i/

for j > 0;

(4.48)Z

�i

ˇˇ@j‰@xj3

ˇˇ2�x1; x2;�"h.x1; x2/

�dx1 dx2

6(Ckgk2

H1.�i/for j D 0;

C"�2jC1kgk2H1.�i/

for j > 1;(4.49)

Z

�i

ˇrj‰

ˇ2�x1; x2;�"h.x1; x2/

�dx1 dx2 6 C"�jC1kgk2H1.�i/

; j D 2; 3;(4.50)

Z

�i

ˇˇ @@x3r2‰

ˇˇ2�x1; x2;�"h.x1; x2/

�dx1 dx2 6 C"�3kgk2H1.�i/

; (4.51)

wherekgk2H1.�i/

D jgj2i Ckgk2i , andC is a constant depending onj andh and indepen-dent of".

PROOF. We write

�r @

j‰

@xj3

�2�x1; x2;�"h.x1; x2/

�

152 M. Petcu et al.

D�r @

j‰

@xj3

�2�x1; x2;�" Nh

�

C 2Z �"h.x1;x2/�" Nh

�r @

j‰

@xj3

��r @

jC1‰

@xjC13

�.x1; x2; z/dz:

Integrating inx1; x2 on�i we find

Z

�i

�r @

j‰

@xj3

�2�x1; x2;�"h.x1; x2/

�dx1 dx2

6ˇˇr @

j‰

@xj3

ˇˇ2

i

��" Nh�

C 2�Z �"h�" Nh

ˇˇr @

j‰

@xj3

ˇˇ2

i

.z/dz

�1=2�Z �"h�" Nh

ˇˇr @

jC1‰

@xjC13

ˇˇ2

i

.z/dz

�1=2:

By (4.41),

ˇˇr @

j‰

@xj3

ˇˇ2

i

��" Nh�6 C �" Nh��2j kgk2H1.�i/

.j > 0/

and by integration of (4.41),

Z �"h�" Nh

ˇˇr @

j‰

@xj3

ˇˇ2

i

.z/dz 6 C�" Nh��k�" Nh�k�2jC1kgk2

H1.�i/

6 C"�2jC1kgk2H1.�i/

.j > 0/;

(4.48) follows. The proof of (4.49) is similar.For (4.50) and (4.51) we observe that, by the regularity property for the Neumann prob-

lem in�i ,

2X

k;`D1

ˇˇ @2‰

@xk @x`

ˇˇ2

L2.�i/

.x3/ 6 C�j‰j2

L2.�i/.x3/C j�‰j2L2.�i/

.x3/�

6 C�j‰j2

L2.�i/.x3/C

ˇˇ @‰@x3

ˇˇ2

L2.�i/

.x3/

�: (4.52)

By repeating the argument above, it appears that the bounds for the left-hand-sides of(4.50) and (4.51) are the same as those of (4.48) and (4.49), forj D 1 and2 respectively;(4.50) and (4.51) are proved. �


4.3. Regularity of solutions of a Neumann–Robin boundary value problem

We now want to derive a result similar to that of Sections 4.1 and 4.2. for a mixedNeumann–Robin type boundary condition, that is for the problem (4.53) below. Our re-sult is quite general except for the restrictions (4.54) below.

We will prove the following theorem:

THEOREM4.3. Assume�i is an open bounded set ofR2, with aC 3-boundary@�i andh 2C 4.x�i/. For f 2L2.M"/ andg 2H 1.�i/, there exists a unique‰ 2H 2.M"/ solution of

��3‰D f;@‰

@x3C ˛‰D g on�i ; (4.53)

@‰

@nD 0 on�b[ �`:

Furthermore if

rh � n�i D 0 on@�i ; (4.54)

then there existsC D C.h;�i ; ˛/ such that

3X

k;`D1

ˇˇ @2‰

@xk @x`

ˇˇ2

"

6 C�jf j2" Ckgk2H1

�:

REMARK 4.1. Condition (4.54) means that�b and�` meet at a wedge angle of�=2. Thisis a technical condition needed in the method of proof used below, this condition is notrequired by the theory of regularity of elliptic problems in nonsmooth three-dimensionalproblems Grisvard (1985). It might be possible to remove this condition with a differentproof. Note that this condition is needed for the dependence on" and not for the soleH 2-regularity.

PROOF OF THEOREM 4.3. The proof is divided into several steps. First we reduce theproblem to the caseD 0, then we reduce the problem to the case wheregD 0 and˛D 0.Thus we obtain a homogeneous Neumann problem. Then by a reflection aroundx3 D 0,we can use Theorem 2.1 and conclude the proof.

STEP 1 (Reduction to the casegD 0). Our goal now is to find an explicit functionxT thatsatisfies the nonhomogeneous boundary conditions imposed on the temperature. Since weare interested in obtaining a sharp dependence on the thickness", we have to constructxTexplicitly instead of the classical method of lifting by localization and straightening theboundary which yields constants which do not have the right dependence on". We willcarry the computations only in the case whererh � n�i D 0 on@�i .

154 M. Petcu et al.

LEMMA 4.4. Leth 2 C 4.x�i/ withrh � n�i D 0 on @�i and letg 2H 1.�i/. There exists afunction xT 2H 2.M"/ such that

@ xT@x3C ˛ xT D g on�i ;

(4.55)@ xT@nD 0 on�` [ �b:

Furthermore

3X

k;lD1

ˇˇ @2 xT@xk @xl

ˇˇ2

"

6 Ckgk2H1.�i/

; (4.56)

whereC is a constant independent on".

PROOF. Let x‰ be a solution of the heat equation, where�x3 corresponds to time:

8ˆ<ˆ:

@x‰@x3D��x‰ in �i � .�1; 0/;

@x‰@n�iD 0 on@�i � .�1; 0/;

x‰.x1; x2; 0/D g.x1; x2/;(4.57)

and definexT by

xT .x1; x2; x3/D e�˛x3Z x3

�"h.x1;x2/x‰.x1; x2; z/dz

��x3 �

1

˛

��1.x1; x2/C x23.x3C "h/�2.x1; x2/; (4.58)

where

�1.x1; x2/D e˛"h.x1;x2/ x‰�x1; x2;�"h.x1; x2/��1C "2jrhj2� (4.59)

and

�2.x1; x2/D�."hC 1

˛/

"h2.1C "2jrhj2/r�1 � rh: (4.60)

Then

@ xT@x3D�˛e�˛x3

Z x3

�"hx‰.x1; x2; z/dzC e�˛x3 x‰.x1; x2; x3/

� �1.x1; x2/C 2x3.x3C "h/�2.x1; x2/C x23�2.x1; x2/; (4.61)


@ xT@x3C ˛ xT

ˇˇx3D0

D x‰.x1; x2; 0/D g; (4.62)

r xT D e�˛x3Z x3

�"hr x‰.x1; x2; z/dzC "e�˛x3 x‰.x1; x2;�"h/rh

��x3 �

1

˛

�r�1C x23.x3C "h/r�2C "x23�2rh; (4.63)

"r xT � rhˇx3D�"h D "

2e˛"h x‰.x1; x2;�"h/jrhj2

C "�"hC 1

˛

�r�1 � rhC "4h2�2jrhj2; (4.64)

@ xT@x3

ˇˇx3D�"h

D e˛"h x‰.x1; x2;�"h/� �1.x1; x2/C "2h2�2

and

@ xT@x3C "r xT � rh

ˇˇx3D�"h

D e˛"h x‰.x1; x2;�"h/�1C "2jrhj2�

C "2h2�2�1C "2jrhj2�� 1

C "�"hC 1

˛

�r�1 � rh:

Hence, with�1; �2 as in (4.59) and (4.60), we have

@ xT@x3C "r xT � rhD 0 on�b:

Now, we use the assumption (4.54) and prove that

r�1 � n�i D 0;r�2 � n�i D 0 on@�i ; (4.65)

which implies thatr xT � n�i D 0 on@�i .First, by working in local coordinates.s; t/ wheres is the coordinate in the normal

direction of@�i andt the coordinate in the tangential direction, the conditionrh � n�i D 0on�i implies (since@�i is smooth)

@h

@sD 0 and

@2h

@s @tD 0 on@�i : (4.66)

Therefore

@

@sjrhj2 D @

@s

�ˇˇ@h@s

ˇˇ2

Cˇˇ@h@t

ˇˇ2�D 2

�@h

@s

@2h

@s2C @h

@t

@2h

@s @t

�D 0: (4.67)

156 M. Petcu et al.

Now

r�1 � n�i D ˛"e˛"h x‰.x1; x2;�"h/�1C "2jrhj2�rh � n�i

C e˛"h�1C "2jrhj2�r x‰ � n�i

� "e˛"h @x‰

@x3.x1; x2;�"h/

�1C "2jrhj2�rh � n�i

C e˛"h x‰.x1; x2;�"h/r�jrhj2� � n�i (4.68)

and sincerh �n�i D 0;r x‰ �n�i D 0 andr.jrhj2/ �n�i D 0 on@�i , we haver�1 �n�i D 0on@�i .

Next we check thatr�2 � n�i D 0 on@�i . Here we only need to show thatr.r�1 � rh/ �n�i D 0 on@�i . Again, this can be done by working in local coordinates. We have

r�1 � rhD@�1

@s

@h

@sC @�1

@t

@h

@t(4.69)

and therefore

r.r�1 � rh/ � n�i D@

@n�i

.r�1 � rh/

D @�1

@s

@2h

@s2C @2�1

@s2@h

@sC @2�1

@t @s

@h

@tC @�1

@t

@2h

@s @t(4.70)

but since@h=@s D 0 and@�1=@s D 0 we have@2h=@s @t D 0 and@2�1=@s @t D 0. Thus

r.r�1 � rh/ � n�i D 0 on@�i : (4.71)

Finally, since�2 is the product of functions each of which has normal derivative to@�i

vanishing on@�i , we have

r�2 � n�i D 0 on@�i : (4.72)

This concludes the verification ofxT satisfying the boundary conditions. We now use es-timates on solutions of the heat equation and the explicit expression ofxT to establish theinequality (4.56) of the lemma. Here we fully rely on the estimates forx‰ provided byLemmas 4.2 and 4.3.

With @ xT =@x3 andr xT given by (4.61) and (4.63), we write

@2 xT@x23D ˛2e�˛x3

Z x3

�"hx‰.x1; x2; z/dz

� 2˛e�˛x3 x‰.x1; x2; x3/C e�˛x3@x‰@x3

.x1; x2; x3/

C 2.x3C "h/�2.x1; x2/C 4x3�2.x1; x2/; (4.73)


and, fork D 1; 2,

@2 xT@xk @x3

D�˛e�˛x3Z x3

�"h

@x‰@xk

.x1; x2; z/dz � ˛"e�˛x3 x‰.x1; x2;�"h/@h

@xk

C e�˛x3@x‰@xk� @�1@xkC 2"x3�2

@h

@xkC 2x3.x3C "h/

@�2

@xkC x23

@�2

@xk: (4.74)

Finally, for k; `D 1; 2,

@2 xT@xk @x`

D e�˛x3Z x3

�"h.x1;x2/

@2 x‰@xk @x`

.x1; x2; z/dz

DC"e�˛x3 @x‰

@x`

�x1; x2;�"h.x1; x2/

� @h@xk

C "e�˛x3 @x‰

@xk

�x1; x2;�"h.x1; x2/

� @h@x`

� "2e�˛x3 @x‰

@x3.x1; x2;�"h/

@h

@xk

@h

@x`

C "e�˛x3 x‰.x1; x2;�h/@2h

@xk @x`��x3 �

1

˛

�@2�1

@xk @x`

C x23.x3C "h/@2�2

@xk @x`C "x23

@h

@xk

@�2

@x`C "x23�2

@2h

@xk @x`: (4.75)

To estimate theL2-norms of the second derivatives ofxT , we need to bound theL2-normsof �1; �2 and their derivatives, which we do in Lemma 4.4.

Using Lemma 4.4, we estimate as follows, the norm, inL2.M"/, of @2 xT =@xk @x`,k; `D 1; 2, as given by (4.75).

The first term in the right-hand side of (4.75) is bounded by a constant times the normof @2 x‰=@xk @x` in Q" D �i � .�" Nh; 0/; using (4.51) and (4.41), this term is bounded asin (4.56). We then use Lemma 4.3 to estimate the four subsequent terms, and the boundsare consistent with (4.56). The remaining terms involve�1; �2 and their derivatives; theintegration over�i of these functions provide the bounds given by Lemma 4.4, and, foreach of these terms there is a factor ofC"m;m> 2, which is due to the integration inx3.The bound (4.56) follows.

We proceed similarly for@2 xT =@x23 given by (4.75) and for@2 xT =@xk @x3, k D 1; 2, givenby (4.74). Lemma 4.4 follows. �

We now conclude the proof of Lemma 4.4 by proving Lemma 4.4.

158 M. Petcu et al.

LEMMA 4.5. The functions�1 and �2 introduced in(4.59) and (4.60),are bounded as

follows:

j�1jL2.�i/C jr�1jL2.�i/

C "1=2ˇr2�1

ˇL2.�i/

6 Ckgk2H1.�i/

; (4.76)

j�2jL2.�i/C "1=2

ˇr�2

ˇL2.�i/

C "ˇr2�2

ˇL2.�i/

6 C"�1kgk2H1.�i/

; (4.77)

whereC is a constant independent of".


PROOF. The proof strongly relies on the definition (4.59) and (4.60) of�1 and�2, and onthe estimates onx‰ given by Lemmas 4.2 and 4.3.

We writee‰.x1; x2/D x‰.x1; x2;�"h.x1; x2//, and observe that, pointwise,

re‰Dr x‰C "� @x‰

@x3;

r2e‰Dr2 x‰C "� @rx‰

@x3C "� @

x‰@x3C "2� @

2 x‰@x23

;

(4.78)

r3e‰Dr3 x‰C "� @r2 x‰

@x3C "� @r

x‰@x3

C "� @2 x‰@x23C "� @

x‰@x3

C "2� @2r x‰@x23

C "3� @2 x‰@x33C "� @r

x‰@x3

C "2� @2 x‰@x23

:

Here the� are (different) continuous (scalar, vector or tensor) functions bounded on�i

independently of" ." 6 1/;e‰ and its derivatives are evaluated at.x1; x2/ 2 �i ; x‰ and itsderivatives are evaluated at.x1; x2;�"h.x1; x2//.

It follows then from Lemma 4.3 that

ˇe‰ˇL2.�i/

6 C;ˇre‰

ˇL2.�i/

6 C;(4.79)ˇ

r2e‰ˇL2.�i/

6 C"1=2;ˇr3e‰

ˇL2.�i/

6 C"�1:

Now, similarly,�1 and its first, second and third derivatives are of the following form:

�1 D �e‰; r�1 Dr� � e‰C �re‰;r2�1 Dr2� � e‰C 2r� ˝re‰C �r2e‰;r3�1 Dr3� � e‰C 3r2� ˝r Q C 3r� ˝r2e‰C �r3e‰;

where� and its first, second and third derivatives are uniformly bounded on�i (for "6 1/;hence (4.76) using (4.78). To obtain (4.77), we observe that, with a different�; �2 is of theform "�1� � r�1.

Lemma 4.4 is proved. �

STEP 2 (Reduction to the caseD 0 (andgD 0)). LeteT be the solution of

��3eT D f2 in M";

@eT@x3C ˛eT D 0 on�i ; (4.80)

@eT@nD 0 on�b[ �`:

160 M. Petcu et al.

We first note that

ˇreT

ˇ2"Cˇˇ @eT@x3

ˇˇ2

"

C ˛ˇeTˇ2i6 jf2j"

ˇeTˇ": (4.81)

Also by a density argument and, since

ˇeT .x1; x2; x3/ˇ6ˇeT .x1; x2; 0/

ˇCZ x3

0

ˇˇ @eT@x3

ˇˇdx03

6ˇeT .x1; x2; 0/

ˇCp" Nh�Z 0

�"h.x1;x2/

ˇˇ @eT@x3

ˇˇ2

dx3

�1=2; (4.82)

and

Z

M"

ˇeT .x1; x2; x3/ˇ2

dx1 dx2 dx3 6 2" NhˇeTˇ2iC 2" Nh

ˇˇ @eT@x3

ˇˇ2

"

: (4.83)

We infer from (4.81) that

ˇreT

ˇ2"Cˇˇ @eT@x3

ˇˇ2

"

C ˛ˇeTˇ2i6p2" Nhjf2j"

ˇeTˇiCp2" Nhjf2j"

ˇˇ @eT@x3

ˇˇ"

6 ˛2

ˇeTˇ2iC " Nh2˛jf2j2" C

1

2

ˇˇ @eT@x3

ˇˇ2

"

C " Nhjf2j2" : (4.84)

Therefore

ˇreT

ˇ2"C 1

2

ˇˇ @eT@x3

ˇˇ2

"

C ˛

2

ˇeTˇ2i6 " Nhjf2j2"

�1

2˛C 1

�: (4.85)

Hence

ˇeTˇ2i6 C"jf2j2" (4.86)

and

ˇˇ @eT@x3

ˇˇ2

"

6 C"jf2j2" ;

whereC is a constant independent of". Then by (4.82),

ˇeTˇ2"6 C"2jf2j2" : (4.87)


Next, transform (4.79) into a homogeneous Neumann condition. LetT � D �eT , where

�.x1; x2; x3/

D exp

�˛x3C

˛x232"h.x1; x2/

C ˛x23�x3C "h.x1; x2/

�'.x1; x2/

�(4.88)

with

' D jrhj22h2.1C "2jrhj2/ : (4.89)

Noting that

@T �

@x3D @�

@x3eT C � @

eT@x3

;@T �

@nD @�

@neT C �@

eT@n;

@�

@x3D �

�˛C ˛x3

"hC 2˛x3.x3C "h/' C ˛x23'

�; (4.90)

and atx3 D 0; @�=@x3 D ˛� and�D 1, which implies that

@T �

@x3D �

�@eT@x3C ˛eT

�D 0:

Furthermore, atx3 D�"h.x1; x2/, we have

@�

@x3D ˛�"2h2.x1; x2/'.x1; x2/: (4.91)

Now we computer�, wherer D .@=@x1; @=@x2/:

r�D �� ˛x

23

2"h2rhC "˛x23'rhC ˛x23.x3C "h/r'

�: (4.92)

Hence atx3 D�"h.x1; x2/;r�D �Œ�12˛"�rhC "3˛h2�rh� and

@�

@nD @�

@x3C "rhr�

D ��˛"2h2' � "

2˛

2jrhj2C "4˛h2'jrhj2

�

D "2�˛�h2'

�1C "2jrhj2�� 1

2jrhj2

�D 0: (4.93)

That is,@�=@nj�b D 0, and@T �=@nj�b D 0. Assume now (4.54), that is,

rh � n�i D 0 on@�i : (4.94)

162 M. Petcu et al.

One can easily check thatr' � n�i D 0: using a system.s; t/ of local coordinates on@�i ,with s normal to@�i andt tangential, and using (4.66) and (4.67). Hence

rh � n�i D 0 and r' � n�i D 0 on@�i ; (4.95)

and we have immediately

r� � n�i D 0 on@�i ;

and therefore

@T �

@n�i

D 0 on@�i :

The conclusion of these computations and of Step 2 is summarized by the followinglemma.

LEMMA 4.6. Assume that@�i is of classC 2; h W x�i!RC belong toC 4.x�i/ and

rh � n�i D 0 on@�i :

LeteT be a solution of(4.79)andT � D �eT , with

�D exp

�˛x3C

˛x232"h.x1; x2/

C ˛x23�x3C "h.x1; x2/

�'.x1; x2/

�;

where

'.x1; x2/Djrh.x1; x2/j2

2h2.x1; x2/.1C "2jrh.x1; x2/j2/:

Then

��T � D f �;@T �

@x3D 0 on�i ;

@T �

@nD 0 on�b[ �`;

wheref � D �f2 � 2r3� � r3eT �eT�3� and jf �j" 6 C0jf2j", with C0 independent on".

PROOF. It remains only to estimate theL2-norm off �. First note that there exists a con-stantC0 independent of" (depending on andh) such that

1

C06 �.x1; x2; x3/6 C0 for .x1; x2; x3/ 2M";


and using (4.90) and (4.92), there exists another constant still denotedC0 such that

ˇˇ @�@x3

ˇˇL1.M"/

6 C0 and jr�jL1.M"/ 6 C0:

Now we compute�� .�D @2=@x21 C @2=@x22/:

��D div.r�/

D ��˛x232"

�

�1

h

�C "˛x23 div.'rh/C ˛x23 div

�.x3C "h/r'

��;

and therefore, sinceh 2 C 4.x�i/, we have

j��jL1.M"/ 6 C0; with C0 independent on":

Finally, we compute@2�=@x23 :

@2�

@x23D @�

@x3

�˛C ˛x3

"hC 2˛x3.x3C "h/' C ˛x23'

�

C ��˛

"hC 2˛.x3C "h/' C 6˛x3'

�:

Therefore

ˇˇ @2�@x23

ˇˇL1.M"/

6 C";

whereC is independent on". Now

ˇf �ˇ"6 j�jL1 jf2j"C jr3�jL1

ˇr3eT

ˇ"C j��jL1

ˇeTˇ"Cˇˇ @2�@x23

ˇˇL1

ˇeTˇ";

and since by (4.84) and (4.87),

ˇreT

ˇ"6 C jf2j" and

ˇeTˇ"6 C"jf2j";

we have

ˇf �ˇ"6 C jf2j";

whereC is independent of". �

164 M. Petcu et al.

4.4. Regularity of the velocity

In this section, we study theH 2 regularity of the velocity, solution of the GFD–Stokesproblem (we use eitherx3 or z to denote the vertical variable):

8ˆˆ<ˆˆ:

��vC @2v

@x23

�CrpD fv in M";

divR 0�"h v dz D 0 in �i ;

vD 0 on�` [ �b;@v@x3C ˛vvD gv on�i :

(4.96)

TheH 2 regularity of problems similar to (4.96) are given in Ziane (1995) where"D 1 andgv D 0, and in Hu, Temam and Ziane (2002), in the case of constant depth function andunder a convexity condition ofM". We study here theH 2 regularity of solutions to (4.96),and give the dependence on" of the constant appearing in the Cattabriga–Solonnikov typeinequality associated to theH 2 regularity of solutions. By contrast with the articles quotedabove, our analysis here will be carried out in the case whereM" is not necessarily convex,and with a varying bottom topography. This regularity result is discussed in Section 4.4.2,and we start in Section 4.4.1 with a discussion of the weak formulation of the GFD–Stokesproblem (4.96).

4.4.1. Weak formulation of the GFD–Stokes problem.In this section we drop the index"which is irrelevant."D 1;M" DM/. For the weak formulation of (4.96) we consider thespace

V D�v 2H 1.M/2;div

Z 0

�hv dz D 0 on�i ; vD 0 on�` [ �b

�I

thanks to the Poincaré inequality, this space is Hilbert for the scalar product

��v; Qv��D

2X

iD1

3X

jD1

Z

M

@vi

@xj

@ Qvi@xj

dM:

To obtain the weak formulation, we multiply the first equation (4.96) by a test functionQv 2 V and integrate overM; assuming regularity forv;p and Qv, the term involvingp(independent ofx3) disappears:

Z

Mrp Qv dM

DZ

@Mp Qv � nd.@M/�

Z

Mp div Qv dM (by Stokes formula)

D�Z

Mp div Qv dM


D�Z

�i

p

�Z 0

�hdiv Qv dx3

�dx1 dx2

D�Z

�i

p

�div

Z 0

�hQv dx3

�dx1 dx2

D 0 .by the properties ofQv/:

We also have, with Stokes formula, and sinceQv vanishes on�b[ �`,

�Z

M

��vC @2v

@x23

�Qv dMD �

Z

�i

@v

@x3Qv d�i C

��v; Qv��

D ˛vZ

�i

v Qv d�i �Z

�i

gv Qv d�i C��v; Qv��:

Hence the weak formulation of (4.96):To findv 2 V such that

a�v; Qv�D `� Qv� 8Qv 2 V; (4.97)

with

a�v; Qv�D ��v; Qv��C ˛v

Z

�i

v Qv d�i ;

(4.98)

`� Qv�D �fv; Qv

�HCZ

�i

gv Qv d�i :

Existence and uniqueness of a solutionv 2 V of (4.97) follow promptly from the Lax–Milgram theorem. More delicate is the question of showing that, conversely,v is, in somesense, solution of (4.96). The second equation (4.96) andv D 0 on �` [ �b follow fromthe fact thatv 2 V ; hence we need to show that there exists a distributionp independentof x3 such that

��vC @2v

@x23

�CrpD fv in M; (4.99)

and also that

@v

@x3C ˛vvD gv on�i : (4.100)

For (4.99), consider a test function' 2 C10 .M/ .C1 with compact support inM),and observe thatQv D . Qv1; 0/; Qv1 D @'=@x3, belongs toV . Writing (4.97) with this Qv, weconclude that

@

@x3.�3v1C fv1/D 0:

166 M. Petcu et al.

In the same way we prove that

@

@x3.�3v2C fv2/D 0; (4.101)

showing that each component of�vC fv is a distribution onM independent ofx3.

Distributions independent ofx3. Now we can identify a distributionG onM indepen-dent of x3, with a distribution on�i as follows: let� be anyC1 scalar function withcompact support in.�h; 0/, and such that

Z 0

�h�.z/dz D 1: (4.102)

Then, if' 2 C10 .�i/ is aC1 scalar function with a compact support in�i ; '� 2 C10 .M"/,and we associate toG a distributioneG on�i by setting

˝eG;'˛�iD hG;'�iM: (4.103)

The right-hand side of (4.103) is independent of� ; indeed if�1 and�2 are two such func-tions thenhG;'�1i D hG;'�2i because

Z 0

�h.�1 � �2/.z/dz D 0;

so that�0.x3/DR 0x3.�1� �2/.z/dz is aC1 function with compact support in.�h; 0/, and

˝G;'.�1 � �2/

˛M D�

�G;

@

@x3.'�0/

�

MD�@G

@x3; '�0

�

MD 0:

It is then easy to see that (4.103) defineseG as a distribution on�i .Now, conversely, assume thateG is a distribution on�i , and let' 2 C10 .M/. It is clear

that Q' D R 0�h ' dx3 belongs toC10 .�i/ and we associate toeG a distributionG onM bysetting

hG;'iM D˝eG; Q'˛

�i8' 2 C10 .M/:

Introduction ofp. Thanks to the previous discussion, we can now consider�v C fv asa distribution on�i . As in the theory of Navier–Stokes equations, consider now a vectorfunction v� 2 V.�i/; that is, v� is (two-dimensional)C1 with compact support in�i ,


and divv� D 0. It is clear that Qv D v�� belongs toV , where� is a function as above(see (4.102)). Writing (4.97) with thisQv, we obtain

��v; v��

��D �fv; v��H;

˝�vC fv; v��

˛M D 0; (4.104)

˝�vC fv; v�

˛�iD 0 8v� 2 V.�i/:

The last equation, which is well known in the theory of Navier–Stokes equations (see, e.g.,Lions (1969), Temam (1977)), implies that there exists a distributionp on�i such that

�vC fv Drp in �i (orM/;

and (4.99) is proven.

A trace theorem. The proof of (4.100) necessitates establishing first a trace theorem: weneed to show that, for a functionv in V , such that (4.99) holds, one can define the traceof @v=@x3 on �i , as an element of.H 1=2

00 .�i//0, the dual ofH 1=2

00 .�i/ (that is the1=2interpolate betweenH 1

0 .�i/ andL2.�i//.Observe first that the trace on�i of any functionˆ in H 1.M/ which vanishes on�`

belongs toH 1=200 .�i/. Indeed by odd symmetry and truncation one can extend such aˆ as

a functionˆ� in H 10 .�i �R/, vanishing forjx3j sufficiently large, and the trace of such a

function on any planex3 D c0, belongs toH 1=200 .�i/. Conversely, if' belongs toH 1=2

00 .�i/,there exists � in H 1

0 .�i �R/ such that the trace of � on�i is ', the mapping'!ˆ�

being linear continuous (lifting operator). From the remark above we infer that the trace on�i of a function inV belongs toH 1=2

00 .�i/.

We then show that the traces on�i of the functions ofV are all inH 1=200 .�i/

2. Indeed

let ' 2H 1=200 .�i/

2. Using the previous lifting operator, there existse 2H 10 .�i �R/2 such

that ej�i D '; by truncation we can assume thate 2H 1.�i � .�h; 0//2 ande vanisheson@�i and atx3 D�h. Let

� D divZ 0

�hedx3;

and observe that� 2L2.�i/ and

Z

�i

� d�i DZ

Qh

div edMDZ

@Q"

e � nh d.@M/D 0; (4.105)

wherenh is the horizontal component of the unit outward normaln on @M. Because

168 M. Petcu et al.

of (4.105), we can solve inQh D �i � .�h; 0/, the (usual) Stokes problem

8<:

�ˆ�Cr� D 0 in Qh;

div3ˆ� D 1h� in Qh;

ˆ� D 0 on@Qh;

(4.106)

andˆ� 2H 1.Qh/3; � 2L2.Qh/. Now,

Z 0

�hdiv3ˆ

� dx3 D divZ 0

�h

��1 ; �

�2

�dx3 D �;

and it is easy to see that the functionˆD e�.ˆ�1 ;ˆ�2/ extended by0 inMnQh belongs toV , and its trace on�i is precisely'. We can furthermore observe that with the constructionabove, the mapping'‘ˆ is linear continuous fromH 1=2

00 .�i/ into V .Finally, (4.100) follows promptly from (4.97), (4.99) and the following proposition.

PROPOSITION4.1. Letv be a function inH 1.M/2 which vanishes on�b[�` and assumethat��vCrp 2L2.M/2, for some distributionp independent ofx3.

Then there exists 1v 2 .H 1=200 .�i//

2 such that

1vD@v

@x3

ˇˇ�i

if v 2 C2� xM�2; (4.107)

and 1v is defined by

h 1v;'i D�.v;ˆ/

��Z

M.��vCrp/ˆdM; (4.108)

where' is arbitrary inH 1=200 .�i/ andˆ is any function ofV such that j�i D '.

PROOF. We first show that the right-hand sideX.ˆ/ of (4.108) depends on' and not on .Indeed, let 1 andˆ2 be two functions ofV such that 1j�i D ˆ2j�i D '. Thenˆ� Dˆ1 � ˆ2 belongs toH 1

0 .M/2 and divR 0�hˆ

� dx3 D 0. It was shown in Lions, Temam

and Wang (1992a) that� is limit in H 10 .M/2 of C1 functionsˆ�n with compact support

in M such that divR 0�hˆ

�n dx3 D 0. It is easy to see thatX.ˆ�n/D 0 and, by continuity,

X.ˆ�/D 0, i.e.,X.ˆ1/DX.ˆ2/.After this observation we choose as constructed above, so that the mapping'! ˆ

is linear continuous fromH 1=200 .�i/

2 into V . It then appears that the right-hand side of

(4.108) is a linear form continuous onH 1=200 .�i/

2, and thus 1v is defined and belongs to

.H1=200 .�i//

2. Finally, (4.107) follows from the fact that (4.108) is easy whenv andˆ aresmooth and 1v is replaced by@v=@x3j�i . �

REMARK 4.2. We have shown the complete equivalence of (4.96) with its variationalformulation (4.97).


4.4.2. H 2 regularity for the GFD–Stokes problem.For convenience, we use hereafter theclassical notationL2;H1, etc., for spaces of vector functions with components inL2;H 1,etc.

The main result of this section is the following theorem:

THEOREM 4.4. Assume thath is a positive function inC4.x�i /; h > h > 0 and thatfv 2L2.M"/ and gv 2 H10.�i/. Let .v;p/ 2 H1.M"/ � L2.�i/ be a weak solution of(4.96).Then

.v;p/ 2H2.M"/�H 1.M"/: (4.109)

Moreover, the following inequality holds:

jvj2H2.M"/C "jpj2

H1.�i/6 C

�jfvj2" C jgvj2L2.�i/C "jrgvj2L2.�i/

�: (4.110)

The approach to the proof of theH 2 regularity in Theorem 4.4 is the same as in thearticles Ziane (1995, 1997) and Hu, Temam and Ziane (2002), it is based on the followingobservation: the weak solution of (4.96) satisfiesp 2L2.�i/; assume further that the solu-tion v of (4.96) satisfies@v

@x3j�i 2 L2.�i/, @v

@x3j�b 2 L2.�b/ and @v

@xkj�b 2 L2.�b/; k D 1; 2,

then an integration of the first equation in (4.96) with respect tox3 over .�"h; 0/ yieldsa two-dimensional Stokes problem on the smooth domain�i with a homogeneous bound-ary condition. By the classical regularity theory of the two-dimensional-Stokes problem insmooth domains, see, for instance, Ghidaglia (1984), Temam (1977) and Constantin andFoias (1988),p belongs toH 1.�i/. Then, by moving the pressure term to the right-handside, problem (4.96) reduces to an elliptic problem of the type studied in Section 4.2, andtheH 2 regularity ofv follows. The estimates on theL2-norms of the second derivativesare then obtained using the trace theorem and the estimates in Section 4.2.

We start this proof by showing that@v@x3j�i 2 L2.�i/, @v

@x3j�b 2 L2.�b/ and @v

@xkj�i 2

L2.�i/; k D 1; 2. The following lemma is just a rewriting of Theorem 4.2.

LEMMA 4.7. Assume thath 2 C2.x�i /. For f 2 L2.M"/ andg 2H 10 .�i/, there exists a

unique‰ 2H 2.M"/ solution of

8<:��3‰D f in M";@‰@x3C ˛‰D g on�i ;

‰D 0 on�b[ �l :

(4.111)

Furthermore, there exists a constantC.h;˛/ depending only on and h (and�i), suchthat

3X

k;jD1

ˇˇ @2‰

@xk @xj

ˇˇ2

"

6 C.h;˛/�jf j2" C jgj2i C jrgj2i

�:

As we said, this lemma is just a rewriting of Theorem 4.2. We will also need the follow-ing intermediate result simply obtained by interpolation betweenH 1 andH 2.

170 M. Petcu et al.

LEMMA 4.8. Under the assumptions of Lemma4.7,and withgD 0: if f 2H�1=2Cı.M"/

with�12< ı < 1

2; ı 6D 0, then‰ 2H 3=2Cı.M"/.

Before we start the proof of the main result of this section (Theorem 4.1), we first prove

LEMMA 4.9. Assume thath 2 C3.x�i /. For f 2 L2.M"/, g 2 H 10 .�i/, and i 2

H1C 0 .�b/, �12 < < 1

2; ¤ 0, there exists a unique‰i 2H 3=2C .M"/ solution of

8ˆ<ˆ:

�43‰i D f in M";@‰i@x3C ˛v‰i D g � ˛v i on�i ;

‰i D� i on�b;

‰i D 0 on�l :

(4.112)

PROOF. Using Lemma 4.7, we reduce the problem to the casef D 0 andg D 0, by re-placing‰i with ‰i �‰, where‰ is the function constructed in Lemma 4.7. Thus, withoutloss of generality, we will assume from now on thatf D 0 andg D 0. Our next step isto construct a functionQvp which agrees with‰i on @M". This will be done by first con-structing an auxiliary functionvp on a straight cylinder and then the explicit expression ofQvp will be given.

LetQ" be the cylinderQ" D �i � .�"; 0/, and letvp be the unique solution of

8ˆ<ˆ:

�3vp D 0 in Q";

vp D 0 on@�i � .�"; 0/;vp D� i on�i � f�"g;vp D "h˛v i on�i � f0g:

(4.113)

We will show thatvp 2H 3=2C .Q"/ for all �12< < 1

2; ¤ 0. Then, setting

Qvp.x1; x2; x3/

D� x3

"h.x1; x2/vp

�x1; x2;

x3

h.x1; x2/

�for .x1; x2; x3/ 2M"; (4.114)

it is obvious thatQvp 2H 3=2C .M"/; Qvp.x1; x2;�"h.x1; x2//D� i .x1; x2/, and @Qvp@x3C

˛v Qvp D�˛v i on�i . Therefore settingeV D‰i � Qvp , we have

�3eV D��3 Qvp 2H�1=2C .M"/;

eV D 0 on�l [ �b; (4.115)

@eV@x3C ˛veV D 0 on�i :

Hence, thanks to Lemmas 4.7 and 4.8, we see thateV and thus‰i are inH 3=2C .M"/ for�12< < 1

2; ¤ 0.


To complete the proof of Lemma 4.9, it remains only to show thatvp 2H 3=2C .Q"/for all �1

2< < 1

2; ¤ 0. To this end, letbQ" be anyC 2-domain containingQ" such that

�i �f�"; 0g � @bQ". Since i (resp.h˛v i ) is inH 1C 0 .�i �f�"g/ (respH 1C

0 .�i �f0g/),we can define a functionVi 2H 1.@bQ"/ by settingVi D� i on�i � f�"g, Vi D "h˛v ion�i�f0g, andVi D 0 on@bQ"n�i�f�"; 0g. Now letVp be the unique solution of�3Vp D0 in bQ" andVp D Vi on @bQ". Since@bQ" is of classC 2, the classical regularity resultsfor elliptic problems (see, e.g., Lions and Magenes (1972)) yieldVp 2 H 3=2C .bQ"/ for�12< < 1

2; ¤ 0. Now leteV i be the trace ofVp on @�i � .�"; 0/. It is easy to see that

eV i 2H 1C 0 .@�i � .�"; 0//. LeteVp D Vp � vp , we have

�3eVp D 0 in Q";

eVp D 0 on�i � f�"; 0g; (4.116)

eVp D eV i on@�i � .�"; 0/:

Using a reflection argument aroundx3 D 0 (resp.x3 D �") by extendingeV i in a “sym-metrically” odd function defined on@�i � .�"; "/ (resp.@�i � .�2"; 0/), and using theclassical local regularity theory (see, e.g., Lions and Magenes (1972)), we conclude thateVp 2H 3=2C .Q"/ for �1

2< < 1

2; ¤ 0. Therefore, sinceVp 2H 3=2C .Q"/, we have

vp D Vp � eVp 2H 3=2C .Q"/. �

LEMMA 4.10. Assume thath 2 C 3.x�i/, with h > h1 > 0 on x�i : Let .v;p/ be the weaksolution of (4.96),thenv 2H2�ı.M"/ for 0 < ı < 1

2and consequently,

@v

@x3

ˇˇ�i

2 L2.�i/; rvj�b and@v

@x3

ˇˇ�b

2 L2.�b/: (4.117)

PROOF. We saw in Section 4.1.1 that (4.97) has a unique solutionv 2H 1.M/2, and thatthere existsp such that.v;p/ satisfy (4.96). By (4.96),rp belongs toH�1.M"/ andthus toH�1.�i/ sincep is independent ofx3 (see Section 4.4.1). Letvi 2H 1

0 .�i/ be theunique solution of the two-dimensional Dirichlet problem on�i :

��vi Drp in �i ;

vi D 0 on@�i :(4.118)

Let QvD v � vi , then Qv satisfies

8ˆ<ˆ:

�3 QvD fv in M";

QvD 0 on�`;

QvD�vi on�b;@Qv@x3C ˛v QvD gv � ˛vvi on�i :

(4.119)

172 M. Petcu et al.

Thanks to Lemma 4.7, withg D gv; i D vi and D �ı for some0 < ı < 12, we have

Qv 2H 3=2�ı.M"/. Hence,

gi D�1

"h

Z 0

�"hdiv Qv dx3 2H 1=2�ı.�i/: (4.120)

Therefore, since divvi D gi , we rewrite the equation forvi in the form of a two-dimensional Stokes problem

8<:

��vi CrpD 0 in �i ;

divvi D gi 2H 1=2�ı.�i/;

vi D 0 on@�i ;

(4.121)

and thanks to the classical regularity result for the nonhomogeneous Stokes problemon�i , (see, e.g., Ghidaglia (1984), Temam (1977)), we havevi 2H 3=2�ı.�i/\H 1

0 .�i/DH3=2�ı0 .�i/. With this new information on the regularity ofvi , we return to problem

(4.119) and using Lemma 4.9 with i D vi and D 12� ı; 0 < < 1

2;, we conclude that

Qv 2H 2�ı.M"/: Thereforegi , given by (4.120), belongs toH 1�ı.�i/. This in turn impliesby the classical regularity of the two-dimensional Stokes problem, that the solutionvi of(4.121) is inH 2�ı.�i/. Thereforev D Qv C vi belongs toH 2�ı.M"/. Consequently thetrace on�i of the normal derivative@v=@x3j�i belongs toH 1=2�ı.�i/, hence toL2.�i/,taking, e.g.,ı D 1=4. Similarly the traces on�b of v and its normal derivative@v=@n be-long toH 3=2�ı.�b/ andH 1=2�ı.�b/ respectively, from which we infer thatrvj�b and@v=@x3j�b are inH 1=2�ı.�b/ and therefore inL2.�b/. The proof of the lemma is nowcomplete. �

PROOF OF THEOREM 4.4. The proof is divided into two steps. In Step 1, we provetheH 2 regularity of solutions, i.e.,v 2 H 2.M"/ andp 2 H 1.�i/. Then, in Step 2, weestablish the Cattabriga–Solonnikov type inequality on the solutions, i.e., establish thebounds (2.97) on theL2-norms of the second derivatives ofv and theH 1-norm on thepressure, in particular, we establish their (non)dependence on".

STEP 1 (TheH 2 regularity of solutions). LetNvD R 0�"h v dz; we have

@2

@x2k

Nv.x1; x2; x3/DZ 0

�"h

@2

@x2k

v.x1; x2; z/dzC Ik.v/; (4.122)

Ik.v/D 2"@h

@xk

@v

@x3

�x1; x2;�"h.x1; x2/

�

� "2�@h

@xk

�2v�x1; x2;�"h.x1; x2/

�; k D 1; 2: (4.123)


Therefore, by integrating the first equation in (4.141) with respect tox3 we obtain thetwo-dimensional Stokes problem:

�� NvCr."hp/D Nf in �i ;

div NvD 0 in �i ; vD 0 on@�i ;(4.124)

where

Nf DZ 0

�"hfv dzC @v

@x3

ˇˇx3D0

� @v

@x3

ˇˇx3D�"h

C I1.v/C I2.v/C "prh:(4.125)

Thanks to Lemma 4.10, each term on the right-hand side of (4.125) is inL2.�i/, whichimplies Nf 2 L2.�i/: this is stated in (4.117) for@v=@x3j�i and@v=@x3j�b; similarly eachterm in I1 and I2 belongs toL2.�b/ (and thusL2.�i/) becausev 2 H 2�ı.M"/, 0 <ı < 1=2; finally for p we recall from (4.118) thatrp 2 H�1.�i/ and thusp 2 L2.�i/.Therefore from the classical regularity theory of the two-dimensional Stokes problem, weconclude thatr.hp/ 2 L2.�i/, and thenrp 2 L2.�i/. We return to problem (4.96), andmove the gradient of the pressure to the right hand-side and obtain, thanks to Lemma 4.7,v 2H 2.M"/ and

3X

k;jD1

ˇˇ @2v

@xk @xj

ˇˇ2

"

6 C.h;˛v/�jfvj2" C jgvj2i C jrgvj2i

�CC.h;˛/"jrpj2i : (4.126)

Note that we have the pressure term on the right-hand side of (4.126). Removing that termis done in the second step below.

STEP 2 (The Cattabriga–Solonnikov type inequality). Our aim is now to boundjrpjiproperly and to derive (4.110) from (4.126). First we homogenize the boundary conditionin (4.96). Letv` D .‰1;‰2/ where‰1 and‰2 are constructed using Lemma 4.7, i.e.,

8<:

��3‰k D fv;k in M", k D 1; 2;@‰k@x3C ˛v‰k D gv;k on�i , k D 1; 2;

‰k D 0 on�b[ �`, k D 1; 2;

174 M. Petcu et al.

wherefv D .fv;1; fv;2/, gv D .gv;1; gv;2/. Thanks to Lemma 4.7,

3X

k;jD1

ˇˇ @2vl

@xk @xj

ˇˇ2

"

6 C.h;˛v/�jfvj2" C jgvj2i C jrgvj2i

�: (4.127)

Settingv� D v � v`, it suffices to establish (4.127) withvl replaced byv�. We have:

8ˆˆ<ˆˆ:

��v�C @2v�@x23

�CrpD 0 in M";

divR 0�"h v

� dz D g� on�i ;

v� D 0 on�` [ �b;@v�@x3C ˛vv� D 0 on�i ;

(4.128)

where

g� D�divZ 0

�"hvl dx3:

Note that inequality (4.127) together with the Cauchy–Schwarz inequality imply

g� 2H1.�i/

6 C.h;˛v/"�jf1j2" C jgvj2i C jrgvj2i

�: (4.129)

Define

V � DZ 0

�"hv� dx3I

V � is the solution of the two-dimensional Stokes problem

8<:��V �Cr."p/D F � in �i ;

divV � D g�;V � D 0 on@�i ;

(4.130)

where

F � D @v�

@x3

ˇˇx3D0

� @v�

@x3

ˇˇx3D�"h

C I1�v��C I2

�v��;

with I1 andI2 as in (4.123). Hence

ˇF �ˇ2L2.�i/

6 C.h/�ˇˇ@v�@x3

ˇˇ2

L2.�i/

Cˇˇ@v�@x3

ˇˇ2

L2.�b/

�: (4.131)


Now, sincev� D 0 on�b, we have@v�

@xkD " @h

@xk

@v�@x3

on�b and, by the Poincaré inequality

and the boundary condition satisfied byv� on �i , we havej @v�@x3j2L2.�i/

6 2˛2v" Nhj @v�

@x3j2" .

Furthermore, we can write

ˇˇ@v�@x3

ˇˇ2

L2.�b/

6ˇˇ@v�@x3

ˇˇ2

L2.�i/

C 2ˇˇ@v�@x3

ˇˇ"

ˇˇ@2v�@x23

ˇˇ"

6 2˛2v" Nhˇˇ@v�@x3

ˇˇ2

"

C �"ˇˇ@2v�@x23

ˇˇ2

"

C C"

�

ˇˇ@v�@x3

ˇˇ2

"

; (4.132)

where� is a positive constant independent of", that will be chosen below.Therefore

ˇF �ˇ2L2.�i/

6 Cˇˇ@v�@xk

ˇˇ2

"

CC.h/�"ˇˇ@2v�@x23

ˇˇ2

"

: (4.133)

We estimate theH 1-norm of v�, usingv� D v � vl and theH 1-estimates ofv andvl .Therefore we can easily obtain

ˇF �ˇ2L2.�i/

6 C.h/"�jf1j2" C jgvj2i C jrgvj2i

�CC.h/�"ˇˇ@2v�@x23

ˇˇ2

"

: (4.134)

Now using the Cattabriga–Solonnikov inequality for the two-dimensional Stokes prob-lem (4.130), there exists a constantC independent of" such that

ˇV �ˇ2H2.�i/

C "2ˇr.hp/

ˇ2L2.�i/

6 CˇF �ˇ2L2.�i/

: (4.135)

From this we obtain

"2ˇr.hp/

ˇ2L2.�i/

6 C.h; �/"�jfvj2" C jgvj2i C jrgvj2i

�CC.h/�"ˇˇ@2v�@x23

ˇˇ2

"

;

"2jrpj2L2.�i/

6 C.h; �/"2jpj2L2.�i/

(4.136)

CC.h; �/"�jfvj2" C jgvj2i C jrgvj2i�CC jhj�"

ˇˇ@2v�@x23

ˇˇ2

"

:

From (4.96) and the weak formulation (4.98) of (4.96), we see that

jpjL2.�i/=R 6 C jrpjH�1.�i/

6 CkvkH1.M"/6 C jfvj";

so that we actually have the same type of estimate (4.136) forrp as forr.hp/. Finally,since�3v� Drp, in M"; v

� D 0 on�b[ �` and @v�

@x3C ˛�v D 0 on�i , we have thanks

176 M. Petcu et al.

to Lemma 4.7,

3X

k;jD1

ˇˇ @2v�@xk @xj

ˇˇ2

"

6 C.h;˛v/"jrpj2L2.�i/

6 C.h;˛v/�jf1j2" C jgvj2i C jrgvj2i

�CC.h;˛v/�ˇˇ@2v�@x23

ˇˇ2

"

;

and therefore for� small enough, so thatC.h;˛v/� 6 12, we conclude that

3X

k;jD1

ˇˇ @2v�@xk @xj

ˇˇ2

"

6 C.h;˛/"jrpj2L2.�i/

6 C.h;˛/�jf1j2" C jgvj2i C jrgvj2i

�:

The proof of Theorem 4.4 is now complete. �

By interpolation, it is easy to derive from Theorem 4.4 the following result:

THEOREM 4.5. Assume thath is a positive function inC3.x�i /. Let .v;p/ 2 H1.M"/ �L2.�i/ be a weak solution of

8ˆˆ<ˆˆ:

��vC @2v

@x23

�CrpD fv in M";

divR 0�"h v dz D 0 on�i ;

vD 0 on�` [ �b;@v@x3C ˛vvD gv on�i :

(4.137)

Then, if fv 2 L2.M"/ andgv 2Hs.�i/, 06 s 6 1,

.v;p/ 2HsC1.M"/�H s.M"/: (4.138)

Moreover the following inequality holds:

jvj2H1Cs.M"/C "jpj2H s.�i/

6 C0�jfvj2" C "1�skgvk2H s.�i/

�; (4.139)

whereC0 is a constant depending on the data but not on".

4.5. Regularity of the coupled system

In this section we prove theH 2 regularity of the solution of a coupled system of equationscorresponding to the linear part of the primitive equations of the coupled atmosphere–ocean. We will concentrate on the velocity part; the temperature and salinity parts followin the same manner.


The unknown isv D .va; vs/, with va, vs corresponding to the horizontal velocities inthe air and in the ocean.19 These functions satisfy the following equations and boundaryconditions:

8ˆˆˆ<ˆˆˆ:

��va� @2va

@x23

CrpaD f av in Ma

";

divR L0va.x1; x2; z/dz D 0; .x1; x2/ 2R2;

vaD 0 on�u[ �a`;

@va

@x3C ˛v

�va� vs

�D gv on�i ;

@va

@x3C ˛vvaD gv on�e;

(4.140)

and8ˆˆ<ˆˆ:

��vs� @2vs

@x23

CrpsD f sv in Ms

";

divR 0�"h v

sdz D 0 in �i ;

vsD 0 on�s`[ �b;

� @vs

@x3C ˛va

�va� vs

�D gv on�i :

(4.141)

The domainMs" is the domain occupied by the ocean whileMa

" is the domain occupiedby the atmosphere andM" DMa

" [Ms":

Ms" D

˚.x1; x2; x3/ 2R3I .x1; x2/ 2 �i ;�"h.x1; x2/ < x3 < 0

;

Ma" D

˚.x1; x2; x3/ 2R3I .x1; x2/ 2 �;0 < x3 < "L

:

Here,� , which is a bounded domain in the planex3 D 0, is the lower boundary of theatmosphere; it consists of the interface�i with the ocean and�e, the interface with theEarth,� D �i [ �e .�i \ �eD;/ (see Section 2.5); furthermore, and as in Section 2.5,

�bD˚�x0; x3

�Ix0 2 �i ; x3 D�"h�x0�;

�a` D

˚�x0; x3

�; x0 2 @�;0 < x3 < "L

;

�s` D

˚�x0; x3

�Ix0 2 @�i ;�"h�x0�< x3 < 0

;

�uD˚�x0; x3

�Ix0 2 �i ; x3 D "L;

�eD �n�i ; � and �i as above.

The coefficient v is a positive number, andgv is a function defined on� .Problem (4.140)–(4.141) is the stationary linearized form of the primitive equations of

the coupled system atmosphere–ocean. Besides its intrinsic interest, the study of this prob-lem is needed for the study of the full nonlinear (stationary or time dependent) coupledatmosphere–ocean system.

19We recall that we use the superscript “s” as sea, instead of “o” as ocean which can be confused with a zero.

178 M. Petcu et al.

4.5.1. Weak formulation of the coupled system.As in Section 4.4.1 we start with theweak formulation of (4.140) and (4.141). In this section we drop the index" which isirrelevant ("D 1).

We are givenfv in L2.M/ andgv in H1=2.�/.For the weak formulation of (4.140) and (4.141) we consider the space

V D�vD �va; vs

� 2H1�Ma��H1�Ms

�;div

Z 0

�hvsdz D 0;

vaD 0 on�u[ �a`; v

sD 0 on�b[ �s`

�:

HerevaD vjMa andvsD vjMs; note that the traces ofva andvs on�i are not necessarilyequal, as explained in Remark 2.7(iii). We set, with obvious notations:

��v; Qv��D ��va; Qva

��aC

��vs; Qvs

��s

D2X

iD1

3X

jD1

Z

M

@vi

@xj

@ Qvi@xj

dMI

because of the Poincaré inequality,kvk D ..v; v//1=2 is a Hilbert norm onV .To obtain the weak formulation, we consider a test functionQv D . Qva; Qvs/ 2 V ; we mul-

tiply the first equation (4.140) byQva and the first equation in (4.141) byQvs. We integrateoverMa andMs respectively and add the resulting equations; we proceed exactly as inSection 4.4.1, using the boundary condition in (4.140) and (4.141) and we arrive at thefollowing: To findv 2 V such that

a�v; Qv�D `� Qv� 8Qv 2 V; (4.142)

with

a�v; Qv�D ��v; Qv��C

Z

�i

˛v�va� vs

�� Qva� Qvs�d�i C

Z

�e

˛vva Qvad�e;

`� Qv�D

Z

Maf av QvadMaC

Z

Msf av QvsdMs

CZ

�i

gv� Qva� Qvs

�d�i C

Z

�e

gv Qvad�e: (4.143)

The existence and uniqueness of a solutionv 2 V of (4.142) is elementary, and followsfrom the Lax–Milgram theorem. The more delicate question of showing thatv D .va; vs/

actually satisfies all the equations (4.140) and (4.141) is handled as follows: we findpa andps such that the first equation (4.140) and (4.141) are valid exactly as we did inSection 4.4.1, for the ocean and the atmosphere. Using also Proposition 4.1 for the oceanand the atmosphere, we obtain the boundary conditions on�i and�e; the other equationsand boundary conditions follow fromv 2 V .


REMARK 4.3. Setting QvD v in (4.142), we find

ˇrva

ˇ2L2.Ma

"/4 C

ˇrvs

ˇ2L2.Ms

"/4 C

Z

�i

˛vˇva� vs

ˇ2d�i C

Z

�e

˛vˇvaˇ2

d�e

DZ

Maf av v

adMa"C

Z

Msf sv v

sdMs"C

Z

�i

gv�va� vs

�d�i C

Z

�e

gvvad�e:

(4.144)

4.5.2. H 2 regularity for the coupled system.Having established the complete equiva-lence of (4.140)–(4.141) with (4.142), we now want to show that the solution of this systempossesses theH 2 regularity, namely

�va; pa

� 2H2�Ma��H 1

�Ma�

and�vs; ps

� 2H2�Ms��H 1

�Ms�;

(4.145)

wheneverf av 2 L2.Ma/, f s

v 2 L2.Ms/ andgv 2H1=2.�/.More precisely, we will prove the following theorem:

THEOREM 4.6. Assume thath is a positive function inC3.x�i /. Let .va; pa/ 2H1.Ma"/�

L2.�i [ �e/, and .vs; ps/ 2H1.Ms"/�L2.�i/ be a weak solution of(4.140)and (4.141)

(or (4.142)).If f av 2 L2.Ma

"/; fsv 2 L2.Ms

"/2 andgv 2H1.�/, gv D 0 on@�e, then

�va; pa

� 2H2�Ma"

��H 1.�i [ �e/ and�vs; ps

� 2H2�Ms"

��H 1.�i/:

(4.146)

Moreover, the following inequality holds:

ˇvaˇ2H2.Ma

"/Cˇvsˇ2H2.Ms

"/C "

ˇpaˇ2H1.�i/

C "ˇpsˇ2H1.�i/

6 C0�ˇf av

ˇ2"Cˇf sv

ˇ2"C jrgvj2L2.�/

�: (4.147)

PROOF. Sinceva 2H1.Ma"/ andvs 2H1.Ms

"/, vaj�i andvsj�i belong toH1=2�ı.�i/ for

all ı, 0 < ı < 1=2, and there exists a constantC0 independent of" such that

ˇvaˇ2H1=2�ı.�i/

Cˇvsˇ2H1=2�ı.�i/

6 C0� va

2H1.Ma

"/C vs

2H1.Ms

"/

�: (4.148)

Furthermore, (4.144) implies that the right-hand side of (4.148) can be bounded by anexpression identical to the right-hand side of (4.146).

The boundary conditions on�i imply then that

@va

@x3C ˛vva and � @v

s

@x3C ˛vvs

180 M. Petcu et al.

belong toH1=2�ı.�i/ and their norm in these spaces are bounded similarly.Therefore, by Theorem 4.5 applied separately toMa

" andMs", we conclude that

�va; pa

� 2H3=2�ı�M a"

��H 1=2�ı.�/;�vs; ps

� 2H3=2�ı�M s"

��H 1=2�ı.�i/

and

ˇvaˇ2H3=2�ı.Ma

"/Cˇvsˇ2H3=2�ı.Ms

"/C "

ˇpaˇ2H1=2�ı.�i/

C "ˇpsˇ2H1=2�ı.�i/

6 Q�;(4.149)

whereQ� is the right-hand side of (4.139) with a possibly different constantC0.Using the trace theorem again, we see that

@va

@x3C ˛vva and � @v

s

@x3C ˛vvs belong toH1�ı0 .�i/

2 8ı;

and there exists a constantC0 independent of" such that

va 2H1�ı.�/C

vs 2H1�ı.�i/

6 C0� va

2H3=2�ı.Ma

"/C vs

2H3=2�ı.Ms

"/

�:

(4.150)

Therefore, by Theorem 4.5, we conclude that

�va; pa

� 2H2�ı�Ma"

��H 1�ı.�/;�vs; ps

� 2H2�ı�Ms"

��H 1�ı.�i/

and

va 2H2�ı.Ma

"/C vs

2H2�ı.Ms

"/C "

ˇpaˇ2H1�ı.�/C "

ˇpsˇ2H1�ı.�i/

6 Q�;

Q� as above. A final application of the trace theorem and of Theorem 4.5 toMa" andMs

"

yields

�va; pa

� 2H2�ı�Ma"

��H 1�ı.�/;�vs; ps

� 2H2�ı�Ms"

��H 1�ı.�i/

and

ˇvaˇ2H2.Ma

"/Cˇvsˇ2H2.Ms

"/C "

ˇpaˇ2H1.�i/

C "ˇpsˇ2H1.�i/

6 Q�;

Q� as above. The proof is complete. �


Acknowledgments

This work was partially supported by the National Science Foundation under the grantsNSF-DMS-0074334, NSF-DMS-0204863 and NSF-DMS-0604235 and by the ResearchFund of Indiana University.

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Documents

Some Mathematical Problems in Geophysical Fluid Dynamicsmypage.iu.edu/~temam/papers/315_PTZ2008.pdf · 1. Introduction The aim of this article is to address some mathematical aspects