An ALE formulation for a 3D eutrophication

model in a moving domain ?

Lino J. Alvarez-Vazquez a,∗, Francisco J. Fernandez b,Isabel Lopez a, Aurea Martınez a

aDepartamento de Matematica Aplicada II. ETSI Telecomunicacion. Universidadde Vigo. 36310 Vigo. Spain.

bDepartamento de Matematica Aplicada. Facultad de Matematicas. Universidad deSantiago de Compostela. 15706 Santiago. Spain.

Abstract

This work analyzes a realistic mathematical model simulating eutrophication (anecological process involving nutrients, phytoplankton, zooplankton, organic detri-tus and dissolved oxygen) into a moving aquatic domain. As a main result, weobtain existence-uniqueness results for the solution of the system within the gen-eral framework of non-cylindrical domains (based on studying the properties of ageneric parabolic problem). The fact of dealing with moving domains, and the lackof regularity, preclude the use of standard semigroup approach, forcing us towardsthe utilization of Arbitrary Lagrangian Eulerian techniques.

Key words: Eutrophication, Existence, Uniqueness, Non-cylindrical domain,Arbitrary Lagrangian Eulerian formulation

1 Introduction

Eutrophication is an environmental process whereby large water bodies (lakes,estuaries, slow-moving streams, and so on) receive excess nutrients (nitrogenand/or phosphorus) that stimulate excessive undesirable plant growth (mainly,

? The research contained in this work was supported by Project MTM2006-01177of Ministerio de Educacion y Ciencia (Spain).∗ Corresponding author. Tel: +34 986 812166. Fax: +34 986 812116.

Email addresses: lino@dma.uvigo.es (Lino J. Alvarez-Vazquez),franfdz@usc.es (Francisco J. Fernandez), isalo1@terra.es (Isabel Lopez),aurea@dma.uvigo.es (Aurea Martınez).

Preprint submitted to J. Math. Anal. Appl. 18 May 2009

algae). This enhanced plant growth, usually known as an algal bloom, reducesdissolved oxygen in the water when dead plant material decomposes and cancause other organisms (fish, shellfish, seabirds, and even small mammals) todie. Nutrients can come from many sources, such as fertilizers applied to agri-cultural fields, golf courses, and suburban lawns; deposition of nitrogen fromthe atmosphere; phosphate detergents; erosion of soil containing nutrients;and sewage treatment plant discharges. Water with a low concentration ofdissolved oxygen is called hypoxic (can derive anoxic by devoid of oxygen),often leading to changes in animal and plant populations and degradation ofwater and habitat quality. A detailed description of causes, symptoms andeffects of eutrophication can be found, for instance, in De Jonge and Elliott[12] or Clark [10].

It is commonly accepted that pristine aquatic ecosystems function in ap-proximate steady state in which primary production of new plant biomassis sustained by nutrients released as byproducts of microbial and animalmetabolism. This balanced state is disrupted by human activities that artifi-cially enrich water bodies with nitrogen and phosphorus, resulting in unnatu-rally high rates of plant production and accumulation of organic matter thatcan degrade water and habitat quality. The eutrophication problem illustrateshow human activities on land can degrade water quality, with potentiallylarge economic and ecological costs. The very different processes and effectsof coastal eutrophication are well known and documented (see, for instance,Cloern [11] and the references therein). Solutions to the coastal eutrophicationproblem require changes in all these human activities. Commitments to thesesolutions are now beginning (for instance, the European Union’s Water Frame-work Directive mandates strategies to reduce nitrogen-phosphorus delivery tocoastal waters, or a 2000 National Research Council report recommends aNational Coastal Nutrient Management Strategy for the USA).

Analyzing recent scientific literature, it can be noticed how eutrophicationhas been the subject of a wide range of biological and engineering researches.However, from a mathematical viewpoint, corresponding mathematical modelshave been much less addressed. Within the classical framework of ordinarydifferential systems, several models have been proposed and analyzed: Wemust point out here those works of Solidoro et al. [30], Jang et al., [21] andGarulli et al. [19] (interesting results on bifurcation and equilibria points canbe also found, for instance, in the recent works of Dimitrov and Kojouharov[13], or Shukla et al. [29]). Within the more realistic framework of partialdifferential systems, contributions are much more sparse: We can mentionhere the 1D model of Lunardini and Di Cola [24], the 2D (depth-averaged)models proposed by Arino et al. [4] and Cioffi and Gallerano [9], or the 3Dnumerical models introduced by Drago et al. [15], Yamashiki et al. [32] andPark et al. [26]. Unfortunately, no theoretical results of existence-uniquenessof solution are presented on them, only numerical simulations for particular

2

ecosystems. A first result of existence of (periodic) solutions for a simplifiedmodel has been given by Allegretto et al. [1]. However, the most general result- up to date - of existence-uniqueness of solution for a complete nutrient-phytoplankton-zooplankton-organic detritus-dissolved oxygen model can befound in the recent paper of the authors [3]. Unluckily, all above works havebeen developed for the case of fixed domains. We have been unable to find inthe mathematical literature, as far as we could, general existence-uniquenessresults for a eutrophication model, under the realistic hypothesis of a (non-cylindrical) moving domain (For a more complete state-of-art for the topic,interested readers are referred to the recent paper [3]).

Then, in Section 2 we present the mathematical formulation of the model,setting the system of partial differential equations for a complete set of fivespecies: nutrient, phytoplankton, zooplankton, organic detritus and dissolvedoxygen. In Section 3 we analyze an auxiliary generic parabolic equation, ob-taining existence-uniqueness-regularity results for it, which will be useful inthe analytical study of the eutrophication system. Finally, in Section 4, weprove our main result, related to existence and uniqueness of solution for theeutrophication system, for the realistic case of a non-cylindrical domain.

2 Mathematical formulation of the model

As above commented, eutrophication processes can be modelled by systemsof partial differential equations, usually presenting a high complexity due tothe great variety of phenomena appearing on them. In this paper we haveconsidered a realistic - but simple - model, where five biological species con-centrations appear (the formulation of biochemical interaction terms and theirmeaning are detailed, for instance, in Canale [7]). Then, we consider variableu = (u1, . . . , u5), where u1 stands for a generic nutrient concentration (nitro-gen in our case, but also phosphorus), u2 stands for phytoplankton concentra-tion, u3 stands for zooplankton concentration, u4 stands for organic detritusconcentration, and u5 stands for dissolved oxygen concentration.

The evolution of these five species into a moving water domain Ω(t) ⊂ R3

(with a smooth enough boundary ∂Ω(t)), for a time interval I = (0, T ), canbe described by the following system of coupled nonlinear partial differential

3

equations for advection-diffusion-reaction with Michaelis-Menten kinetics:

∂u1

∂t+ v · ∇u1 − µ1∆u1 + CncL

u1

KN +u1u2 − CncKru

2 − CncKrdΘθ−20u4 = g1,

∂u2

∂t+ v · ∇u2 − µ2∆u2 − L u1

KN +u1u2 +Kru

2 +Kmfu2 +Kz

u2

KF +u2u3 = g2,

∂u3

∂t+ v · ∇u3 − µ3∆u3 − CfzKz

u2

KF +u2u3 +Kmzu

3 = g3,

∂u4

∂t+ v · ∇u4 − µ4∆u4 −Kmfu

2 −Kmzu3 +KrdΘ

θ−20u4 +Wfd∂u4

∂x3= g4,

∂u5

∂t+ v · ∇u5 − µ5∆u5 − CocL u1

KN +u1u2 + CocKru

2 + CocKrdΘθ−20u4 = g5

in Ω(t), t ∈ I,

ui = hi, ∀i = 1, . . . , 5, on ∂Ω(t), t ∈ I,

ui = ui0, ∀i = 1, . . . , 5, in Ω(0), t = 0,

(1)

where v is the water velocity; µi, i = 1, . . . , 5, are the diffusion coefficientsof each species; Cnc is the nitrogen-carbon stoichiometric relation; L is theluminosity function, given by:

L(x, t, u2) = µCθ(x,t)−20t

I0

Ise−(φ1+φ2u2)x3 ,

with µ the maximum phytoplankton growth rate, Ct the phytoplankton growththermic constant, θ the water temperature, I0 the incident light intensity, Isthe light saturation, φ1 the light absorption by water, and φ2 the light absorp-tion by phytoplankton (for the sake of simplicity, in our model we will considerthat φ2 = 0 - i.e., light intensity only depends on water depth, not on phyto-plankton concentration - and that the water temperature θ(x, t) is bounded- hence, L will be also bounded by a constant Lmax > 0); KN is the nitro-gen half-saturation constant; Krd is the detritus regeneration rate; Θ is thedetritus regeneration thermic constant; Kr is the phytoplankton endogenousrespiration rate; Kmf is the phytoplankton death rate; Kz is the zooplanktonpredation (grazing); KF is the phytoplankton half-saturation constant; Cfzis the grazing efficiency factor; Kmz is the zooplankton death rate (includ-ing predation); Wfd is the falling velocity of organic detritus; and Coc is theoxygen-carbon stoichiometric relation.

All above coefficients will be assumed to be non-negative, except for half-saturation constants KN and KF that will be strictly positive. Finally, func-tions gi, hi, and ui0, for i = 1, . . . , 5, stand, respectively, for source termsdistributed in the domain, Dirichlet boundary conditions, and initial condi-tions.

Moreover, since water velocity v satisfies Navier-Stokes equations, it mustverify the incompressibility condition ∇ · v = 0 and, consequently, the terms

4

v · ∇ui in above system (1) can be written in the alternative equivalent way∇ · (vui), because of

∇ · (vui) = v · ∇ui + (∇ · v)ui = v · ∇ui.

As can be observed after a first investigation on the system, the equation fordissolved oxygen u5 plays a role completely different from the other species,since it can be solved once the other species u = (u1, . . . , u4) have been com-puted. Thus, the fundamental coupling of the system relies in the equationsmodelling the interaction of the generic nutrient u1, the phytoplankton u2, thezooplankton u3 and the organic detritus u4. This observation motivates thatwe separate the theoretical analysis of the dissolved oxygen equation from therest of the species.

So, in order to have a simpler expression for the system of equations (1) wewill consider, for u = (u, u5), the reaction mappings A = (A1, . . . , A4) and A5

given by:

A(x, t, u) =

Cnc(L(x, t) u1

KN +u1u2 −Kru

2)− CncKrdΘ

θ(x,t)−20u4

−(L(x, t) u1

KN +u1u2 −Kru

2)

+Kmfu2 +Kz

u2

KF +u2u3

−CfzKzu2

KF +u2u3 +Kmzu

3

−Kmfu2 −Kmzu

3 +KrdΘθ(x,t)−20u4

, (2)

A5(x, t, u) = −Coc(L(x, t)

u1

KN + u1u2 −Kru

2

)+ CocKrdΘ

θ(x,t)−20u4. (3)

The sedimentation term Wfd∂u4

∂x3will be incorporated into the convective term

∇ · (vu4), by redefining a new effective velocity v4 = v + (0, 0,Wfd). Aboutboundary conditions, we will assume homogeneous conditions (that is, hi =0, ∀i = 1, . . . , 5). The theoretical analysis of the problem with non-homogeneousconditions is analogous to the homogeneous case and, for the sake of simplicity,will not be detailed here.

Thus, taking into account previous considerations, the system (1) can berewritten in the following equivalent way:

∂ui

∂t+∇ · (viui)− µi∆ui + Ai(u) = gi in Ω(t), t ∈ I,

ui = 0 on ∂Ω(t), t ∈ I,

ui = ui0, ∀i = 1, . . . , 5, in Ω(0), t = 0,

(4)

where vi = v for i 6= 4, and v4 = v + (0, 0,Wfd).

5

In order to analyze above problem we will assume realistic regularity hypothe-ses on coefficients and data (mainly, the velocity v ∈ L∞(I;W 1,∞(Ω(t))3), thesource terms gi ∈ L2(I;L2(Ω(t))), and the initial conditions ui0 ∈ L2(Ω(0)),for all i = 1, . . . , 5). The main difficulties in order to find a solution of thesystem of equations (4) are due to the lack of regularity of the reaction termsin Ai(u) (since several of them are only L2 in time), and to the non-cylindricaldomain.

3 An auxiliary problem

The theoretical analysis of the system (4) needs a detailed study of the fol-lowing generic parabolic equation:

∂c∂t

+∇ · (vc)− µ∆c+ k1(x, t)c− k2(x, t)c = f in Ω(t), t ∈ I,

c = 0 on ∂Ω(t), t ∈ I,

c = c0 in Ω(0), t = 0,

(5)

with velocity v satisfying ∇ · v = 0, diffusion constant µ > 0, and reactioncoefficients k1(x, t), k2(x, t) ≥ 0 a.e. x ∈ Ω(t), t ∈ I. Smoothness of coefficientsk1 and k2 (mainly, L2 and L∞) will be detailed below. We will organize theseauxiliary results into four blocks:

• In the first block we will introduce the fundamentals of ALE techniques, pre-senting the conservative and non-conservative forms of the generic problem(5), also including the corresponding weak formulation.• In the second block we will impose the basic assumptions on the ALE map-

pings, giving several technical preliminary results in order to prove the ex-istence of solution for the equation (5).• In the third block we will demonstrate the existence and uniqueness of

solution for the equation (5), in the case of non-bounded coefficients, alsoincluding regularity results on the time derivative, and an a priori energyestimate for the solution.• Finally, in the fourth block, we will obtain several bounds for the solution

of the generic problem (5) - derived from boundedness of data - necessaryfor the theoretical analysis of the original system (4).

6

3.1 The ALE formulation

One of the most common techniques for the numerical simulation of partialdifferential equations on moving domains is the so-called Arbitrary LagrangianEulerian (ALE) formulation [20], widely used, for instance, for the analysis offluid-structure interaction problems: aeroelasticity, haemodynamics and so on(see, for instance, [14,16,17] and the references therein). Within this formula-tion, the partial time derivative is expressed with respect to a reference fixedconfiguration. A family of special homeomorphic maps Att∈I , (the so-calledALE mappings) associates - at each time t - a point in the current computa-tional domain Ω(t) to a point in the reference domain Ω(0).

So, let At be a family of ALE mappings which, at each t ∈ I, associates a pointy of the reference configuration Ω(0) to each point x on the current domainconfiguration Ω(t), that is, for t ∈ I,

At : y ∈ Ω(0) ⊂ R3 → At(y) = x(y, t) ∈ Ω(t) ⊂ R3 (6)

We assume ALE mappings At to be homeomorphisms, i.e. At ∈ C0(Ω(0))is invertible with continuous inverse A−1

t ∈ C0(Ω(t)), for each t ∈ I, and topreserve boundaries (that is, At(∂Ω(0)) = ∂Ω(t), for each t ∈ I). Furthermore,we assume the application taking t into x(., t) to be differentiable a.e. t ∈ I.Thus, we will name y ∈ Ω(0) the ALE coordinate, while x = x(y, t) willbe addressed as the spatial (Eulerian) coordinate. Finally, for the sake ofsimplicity, all along this Section we will denote Q =

⋃t∈I

Ω(t)× t.

Then, for any given function u : (x, t) ∈ Q → u(x, t) ∈ R, we will denoteu = u At the corresponding function on the ALE frame, that is,

u : (y, t) ∈ Ω(0)× I → u(y, t) = u(At(y), t) ∈ R

It is worthwhile mentioning here that the composition operator applies only tothe spatial variables, leaving the time variable t unchanged by the mapping.We will adopt the symbol ∂u

∂t|y to denote the ALE time derivative, defined as:

∂u

∂t

∣∣∣y

: (x, t) ∈ Q→ ∂u

∂t

∣∣∣y(x, t) =

∂u

∂t(y, t) ∈ R, for y = A−1

t (x). (7)

Finally, as a crucial point in the process, we define the domain velocity w asgiven by:

w(x, t) =∂x

∂t

∣∣∣y(x, t) (8)

7

Then, if we denote gradient, divergence, and Laplacian operators with respectto the Eulerian coordinate x by ∇x, ∇x· , and ∆x, respectively, we have that:

∂u

∂t

∣∣∣y

=∂u

∂t+∂x

∂t

∣∣∣y· ∇xu =

∂u

∂t+ w · ∇xu. (9)

Taking this equality into account, and bearing in mind that:

∇x · (wu) = w · ∇xu+ (∇x ·w)u (10)

we obtain the following non-conservative ALE formulation of the genericparabolic equation (5):

∂c∂t

∣∣∣y

+ (∇x ·w)c+∇x · ((v −w)c)− µ∆xc+ k1c− k2c = f in Ω(t), t ∈ I,

c = 0 on ∂Ω(t), t ∈ I,

c = c0 in Ω(0), t = 0.

(11)

Remark 1 In general, w is an arbitrary velocity. However, we may distin-guish two particular cases:

• If w = 0, the control domain is fixed, Ω(t) = Ω(0), so that we recover theEulerian description of the motion, and the ALE time derivative reduces tothe usual derivative: ∂u

∂t

∣∣∣y

= ∂u∂t

.

• If w = v, the moving control domain Ω(t) is the material domain, so thatwe recover the Lagrangian description of the motion, and the ALE timederivative is the total (or material) derivative: ∂u

∂t

∣∣∣y

= ∂u∂t

+ v · ∇xu.

These observations justify the denomination ALE (Arbitrary Lagrangian Eu-lerian) given to the formulation.

Whenever the conservative properties of the problem are important, a conser-vative formulation may be desirable. This can be easily obtained by the Eulerexpansion formula (see, for instance, [5]):

∂JAt

∂t

∣∣∣y

= JAt(∇x ·w), (12)

where JAt denotes the determinant of the Jacobian matrix of the ALE map-ping, that is,

JAt = det(∂x∂y

). (13)

8

Thus,

∂(JAtu)

∂t

∣∣∣y

= JAt

∂u

∂t

∣∣∣y

+∂JAt

∂t

∣∣∣yu = JAt

∂u

∂t

∣∣∣y

+ JAt(∇x ·w)u, (14)

and, consequently, the generic parabolic equation (5) can be rewritten in thefollowing conservative ALE form:

1JAt

∂(JAtc)

∂t

∣∣∣y

+∇x · ((v −w)c)− µ∆xc+ k1c− k2c = f in Ω(t), t ∈ I,

c = 0 on ∂Ω(t), t ∈ I,

c = c0 in Ω(0), t = 0.

(15)

Moreover, taking into account that, from equality (14):

d

dt

∫Ω(t)

u =d

dt

∫Ω(0)

JAtu =∫

Ω(0)

∂(JAtu)

∂t=∫

Ω(t)

(∂u

∂t

∣∣∣y

+ (∇x ·w)u

), (16)

we can derive the following weak formulation of above non-conservative form(see more details, for instance, in [25]):

d

dt

∫Ω(t)

cψ +∫

Ω(t)

∇x · ((v −w)c)ψ + µ∫

Ω(t)

∇xc · ∇xψ

+∫

Ω(t)

k1cψ −∫

Ω(t)

k2cψ =∫

Ω(t)

fψ, ∀ψ ∈ H10 (Ω(t)), a.e. t ∈ I.(17)

Finally, we can demonstrate the following properties for ALE functions:

Lemma 2 Let u : Q→ R be a function such that u(t) ∈ H10 (Ω(t)). Then,

(a)∫

Ω(t)

∂u

∂t

∣∣∣yu =

1

2

d

dt‖u‖2

L2(Ω(t)) −1

2

∫Ω(t)

(∇x ·w)u2

(b)∫

Ω(t)

∇x · ((v −w)u)u = −1

2

∫Ω(t)

(∇x ·w)u2

Proof The proof of property (a) is a direct consequence of equality (16)applied to u2, since

d

dt‖u‖2

L2(Ω(t)) =d

dt

∫Ω(t)

u2 =∫

Ω(t)

∂u2

∂t

∣∣∣y

+∫

Ω(t)

(∇x ·w)u2

9

= 2∫

Ω(t)

∂u

∂t

∣∣∣yu+

∫Ω(t)

(∇x ·w)u2

In order to demonstrate property (b) we have that, due to homogeneousboundary conditions,

∫Ω(t)

∇x · ((v −w)u)u = −∫

Ω(t)

(v −w)u · ∇xu

= −∫

Ω(t)

(v −w) · ∇x

(u2

2

)=

1

2

∫Ω(t)

(∇x · (v −w))u2

but, since ∇x · v = 0, we deduce:

∫Ω(t)

∇x · ((v −w)u)u = −1

2

∫Ω(t)

(∇x ·w)u2

3.2 General assumptions

All along this Section we will assume the following general assumptions forthe ALE mappings:

• For each t ∈ I, the domain Ω(t) = At(Ω(0)) is bounded, with boundary∂Ω(t) Lipschitz continuous.• The family of ALE mappings verifies At ∈ W 1,∞(I;W 1,∞(Ω(0))), with in-

verse A−1t ∈ W 1,∞(I;W 1,∞(Ω(t))).

Under these general assumptions, we have the following direct technical re-sults:

Lemma 3 The domain velocity satisfies

w ∈ [L∞(I;W 1,∞(Ω(t)))]3.

Lemma 4 The determinant of the Jacobian matrix of the ALE mapping sat-isfies

JAt ∈ W 1,∞(I;L∞(Ω(0))).

Lemma 5 The family of domains Ω(t)t∈I is uniformly bounded, that is,

10

there exists a constant mmax > 0 such that

meas(Ω(t)) ≤ mmax, ∀t ∈ I.

Lemma 6 For each function u ∈ H1(I;H1(Ω(0))), the corresponding func-tion u = u A−1

t ∈ H1(I;H1(Ω(t))). Moreover,

∂u

∂t

∣∣∣y∈ L2(I;H1(Ω(t))).

Lemma 7 The application mapping each function u into function u = u Atis an isomorphism

(a) from L2(I;H1(Ω(t))) onto L2(I;H1(Ω(0))),(b) from L∞(I;L2(Ω(t))) onto L∞(I;L2(Ω(0))),(c) and also from H1(I;H1(Ω(t))) onto H1(I;H1(Ω(0))).

3.3 Existence and uniqueness results

In this Subsection we will state that problem (5) has a unique solution ina suitable functional space, satisfying also an a priori energy estimate. Inorder to do this, we will assume realistic requirements on the regularity of thecoefficients v, k1 and k2, and of data f and c0.

Existence and uniqueness results of a weak solution for a linear advection-diffusion problem in non-cylindrical domains (that is, problem (5) withoutreaction term, i.e., k1 = k2 = 0) can be found in several works. For instance,in the pioneering paper of Lions [23] one can find a first proof of existence-uniqueness of a solution c ∈ L2(I;H1

0 (Ω(t))) with ∂c∂t∈ L2(I;H1

0 (Ω(t)))′ un-der strong regularity hypotheses (bounded coefficients and infinitely differ-entiable domains). Other theoretical results of existence-uniqueness for thesolution of advection-diffusion problems in non-cylindrical domains - but un-der weaker conditions on the domain regularity - can be found, for instance, in[8,22,6,31,18] where very different techniques (elliptic regularization, penaltymethod, abstract semigroup framework, ALE formulation. . . ) are applied.

The analysis of linear parabolic equations for a complete advection-diffusion-reaction problem has received much less attention in the literature. For thecylindrical case, the authors have previously obtained in [2,3] an existence-uniqueness result of a very weak solution for a linear advection-diffusion-reaction equation under very mild assumptions on coefficients (velocity v onlyL2 in time, and reaction coefficients k1 and k2 in L2 and L∞, respectively).Nevertheless, as far as we know, the analysis of a complete parabolic equationwith advection-diffusion-reaction terms and non-bounded reaction coefficients,

11

in non-cylindrical domains, has not been addressed in the mathematical liter-ature under the ALE framework.

In the following Theorem we will demonstrate the existence and uniquenessof solution for the generic parabolic equation (5) under non-restrictive hy-potheses, in such a way that can be applicable to the analysis of the originaleutrophication problem (1).

Theorem 8 Let us assume the following hypotheses on coefficients and data:

• v ∈ [L∞(I;W 1,∞(Ω(t)))]3 with ∇x · v = 0,• k1 ∈ L2(I;L2(Ω(t))) with k1(x, t) ≥ 0 a.e. x ∈ Ω(t), t ∈ I,• k2 ∈ L∞(I;L∞(Ω(t))) with 0 ≤ k2(x, t) ≤ K a.e. x ∈ Ω(t), t ∈ I,• f ∈ L2(I;L2(Ω(t))),• c0 ∈ L2(Ω(0)).

Then, problem (5) has a unique solution

c ∈ L2(I;H10 (Ω(t))) ∩ L∞(I;L2(Ω(t))).

Moreover, this solution satisfies the energy inequality:

‖c(t)‖2L2(Ω(t)) + 2µ

t∫0

‖∇xc(s)‖2L2(Ω(s))3ds (18)

≤ e(1+2K)t‖c0‖2L2(Ω(0)) +

t∫0

‖f(s)‖2L2(Ω(s))ds.

Finally, the “time derivative” of c also satisfies:

d

dt

∫Ω(t)

cψ ∈ L2(I), ∀ψ ∈ H10 (Ω(t)).

Proof We recall the weak formulation (17) of problem (5):

d

dt

∫Ω(t)

cψ + µ∫

Ω(t)

∇xc · ∇xψ +∫

Ω(t)

∇x · ((v −w)c)ψ +∫

Ω(t)

k1cψ

=∫

Ω(t)

k2cψ +∫

Ω(t)

fψ, ∀ψ ∈ H10 (Ω(t)), a.e. t ∈ I,

that, for c = c At and ψ = ψ At, can be rewritten in the reference configu-ration Ω(0) as:

12

d

dt

∫Ω(0)

cψJAt

︸ ︷︷ ︸m(t,c,ψ)

+µ∫

Ω(0)

3∑j=1

3∑k=1

(3∑

1=1

∂yj∂xi

∂yk∂xi

)∂c

∂yj

∂ψ

∂ykJAt

︸ ︷︷ ︸a(t,c,ψ)

+∫

Ω(0)

3∑j=1

3∑k=1

∂yj∂xk

∂

∂yj((vk − wk)c)ψJAt +

∫Ω(0)

k1cψJAt

︸ ︷︷ ︸b1(t,c,ψ)

(19)

=∫

Ω(0)

k2cψJAt

︸ ︷︷ ︸b2(t,c,ψ)

+∫

Ω(0)

f ψJAt

︸ ︷︷ ︸g(t,ψ)

, ∀ψ ∈ H10 (Ω(0)), a.e. t ∈ I.

In order to prove the existence of solution we will use a classical Faedo-Galerkinapproach. Let φj∞j=1 be a complete orthonormal basis of H1

0 (Ω(0)). For N ∈N, we denote V N = 〈φ1, . . . , φN〉, and consider the approximate problemcorresponding to (5), consisting of finding cN : t ∈ I → cN(t) ∈ V N solutionof:

ddtm(t, cN(t), φj) + a(t, cN(t), φj) + b1(t, cN(t), φj)

= b2(t, cN(t), φj) + g(t, φj), ∀j = 1, . . . , N, a.e. t ∈ I,

cN(0) = cN0 ,

(20)

where cN0 is the projection of c0 on V N in the sense of L2(Ω(0)).

We introduce the following notations:

• bN(t) = (bi(t))Ni=1 where bi(t) are the coordinates of cN(t) in the basis of

V N , that is, cN(t) =∑Ni=1 bi(t)φi ,

• M(t) = (Mij(t))Ni,j=1 with Mij(t) = m(t, φj, φi) ,

• A(t) = (Aij(t))Ni,j=1 with Aij(t) = a(t, φj, φi) ,

• B1(t) = (B1ij(t))

Ni,j=1 with B1

ij(t) = b1(t, φj, φi) ,

• B2(t) = (B2ij(t))

Ni,j=1 with B2

ij(t) = b2(t, φj, φi) ,

• G(t) = (Gi(t))Ni=1 with Gi(t) = g(t, φi) ,

• b0 = (b0,i)Ni=1 with b0,i =

∫Ω(0) c0φi .

13

Then, we have that (20) is equivalent to:ddt

(M(t)bN(t)) + A(t)bN(t) +B1(t)bN(t)

= B2(t)bN(t) +G(t), a.e. t ∈ I,

bN(0) = b0.

(21)

We must note here that matrix M ∈ W 1,∞(I)N×N is symmetric and positivedefinite (thus, there exists its inverse M−1 ∈ W 1,∞(I)N×N), and that matricesA,B2 ∈ L∞(I)N×N , B1 ∈ L2(I)N×N , and vector G ∈ L2(I)N .

So, for dN(t) = M(t)bN(t) (or, equivalently, for bN(t) = M−1(t)dN(t)) we canwrite (21) into the alternative way:

ddtdN(t) + A(t)M−1(t)dN(t) +B1(t)M−1(t)dN(t)

= B2(t)M−1(t)dN(t) +G(t), a.e. t ∈ I,

dN(0) = M(0)b0.

(22)

In order to demonstrate that problem (22) has a unique solution dN ∈ C(I)N ,we only need to prove that the mapping T : η ∈ C(I)N → T (η) ∈ C(I)N givenby:

T (η)(t) = M(0)b0 +

t∫0

G(s)ds−t∫

0

D(s)η(s)ds,

where D(s) = (A(s) +B1(s)−B2(s))M−1(s), has a unique fixed point. Then,for η ∈ C(I)N , we will consider the following norm:

‖η‖∗ = sup0≤t≤T

e−kt‖η(t)‖

,

with a suitable k > 0. As above remarked, due to the regularity of v, k1 andk2, we have that D = (A+B1 −B2)M−1 ∈ L2(I)N×N , and:

∣∣∣∣∣∣∣∣∣∣∣∣t∫

0

D(s)η(s)ds

∣∣∣∣∣∣∣∣∣∣∣∣ ≤ ekt√

2k‖D‖L2(I)N×N‖η‖∗.

So,

sup0≤t≤T

e−kt∣∣∣∣∣∣∣∣∣∣∣∣t∫

0

D(s)η(s)ds

∣∣∣∣∣∣∣∣∣∣∣∣ ≤ 1√

2k‖D‖L2(I)N×N‖η‖∗,

14

from which, taking k large enough so that‖D‖

L2(I)N×N√2k

< 1, we obtain that

mapping T es contractive in space C(I)N endowed with the norm ‖ · ‖∗, and,consequently, it has a unique fixed point dN ∈ C(I)N , solution of (22).

Thus, bN(t) = M−1(t)dN(t) is the unique solution of (21). Since M−1 ∈W 1,∞(I)N×N ⊂ C(I)N×N , then bN ∈ C(I)N and, consequently,

cN =N∑i=1

biφi ∈ C(I;H10 (Ω(0))).

On the other hand, we have that

bN(t)Td

dt(M(t)bN(t)) =

d

dt(bN(t)TM(t)bN(t))− d

dtbN(t)TM(t)bN(t)

=d

dt(bN(t)TM(t)bN(t))− bN(t)TM(t)

d

dtbN(t)

=d

dt(bN(t)TM(t)bN(t))− bN(t)T

d

dt(M(t)bN(t)) + bN(t)T

d

dtM(t)bN(t).

Thus,

bN(t)Td

dt(M(t)bN(t)) =

1

2 ddt

(bN(t)TM(t)bN(t)) + bN(t)Td

dtM(t)bN(t). (23)

We define C(t) = (Cij(t))Ni,j=1 with

Cij(t) =d

dtMij(t) =

d

dt

∫Ω(0)

φjφiJAt =∫

Ω(0)

φjφi3∑l=1

3∑k=1

∂yl∂xk

∂wk∂yl

JAt . (24)

Then, multiplying (21) by bN(t)T :

bN(t)Td

dt(M(t)bN(t)) + bN(t)TA(t)bN(t) + bN(t)TB1(t)bN(t)

= bN(t)TB2(t)bN(t) + bN(t)TG(t), ∀t ∈ I.

Taking into account expressions (23) and (24), above equality turns into:

1

2

d

dt(bN(t)TM(t)bN(t)) + bN(t)TA(t)bN(t) + bN(t)TB1(t)bN(t)

+1

2bN(t)TC(t)bN(t) = bN(t)TB2(t)bN(t) + bN(t)TG(t), ∀t ∈ I. (25)

15

which, returning to initial expression for cN(t), translates into

1

2

d

dt

∫Ω(t)

|cN(t)|2 + µ∫

Ω(t)

‖∇xcN(t)‖2 +

∫Ω(t)

∇x · ((v −w)cN(t))cN(t)

+∫

Ω(t)

k1(t)|cN(t)|2 +1

2

∫Ω(t)

(∇x ·w)|cN(t)|2 (26)

=∫

Ω(t)

k2(t)|cN(t)|2 +∫

Ω(t)

f(t)cN(t), ∀t ∈ I.

It is worthwhile remarking here that the only difficult point in above expres-sion is that concerning the existence of term

∫Ω(t) k1(t)|cN(t)|2, due to the low

regularity of functions involved. However, since cN(t) ∈ H10 (Ω(t)) ⊂ L6(Ω(t))

and k1(t) ∈ L2(Ω(t)), and from 12+ 1

6+ 1

6= 5

6< 1, we deduce that the following

integral makes sense: ∫Ω(t)

k1(t)|cN(t)|2 ≥ 0, ∀t ∈ I.

Moreover, since cN(t) ∈ H10 (Ω(t)) we have

∫Ω(t)

∇x · ((v −w)cN(t))cN(t) = −∫

Ω(t)

(v −w)cN(t) · ∇xcN(t)

= −∫

Ω(t)

(v −w) · ∇x

(|cN(t)|2

2

)=

1

2

∫Ω(t)

∇x · (v −w)|cN(t)|2

= −1

2

∫Ω(t)

(∇x ·w)|cN(t)|2.

Thus, we deduce:

d

dt

∫Ω(t)

|cN(t)|2 + 2µ∫

Ω(t)

‖∇xcN(t)‖2

≤ 2∫

Ω(t)

k2(t)|cN(t)|2 + 2∫

Ω(t)

f(t)cN(t), ∀t ∈ I.

Integrating now in [0, t], we achieve the following energy inequality for approx-imate solution cN :

16

‖cN(t)‖2L2(Ω(t)) + 2µ

t∫0

‖∇xcN(s)‖2

L2(Ω(s))3ds

≤ ‖cN(0)‖2L2(Ω(0)) + 2K

t∫0

‖cN(s)‖2L2(Ω(s))ds+ 2

t∫0

∫Ω(s)

f(s)cN(s)ds

≤ ‖cN(0)‖2L2(Ω(0)) + 2K

t∫0

‖cN(s)‖2L2(Ω(s))ds (27)

+

t∫0

∫Ω(s)

|f(s)|2ds+

t∫0

∫Ω(s)

|cN(s)|2ds

= ‖cN(0)‖2L2(Ω(0)) +

t∫0

‖f(s)‖2L2(Ω(s))ds+ (1 + 2K)

t∫0

‖cN(s)‖2L2(Ω(s))ds.

Finally, by applying the Gronwall Lemma (see, for instance, [27]) we obtain:

‖cN(t)‖2L2(Ω(t)) + 2µ

t∫0

‖∇xcN(s)‖2

L2(Ω(s))3ds

≤ e(1+2K)t‖cN(0)‖2L2(Ω(0)) +

t∫0

‖f(s)‖2L2(Ω(s))ds, ∀t ∈ I. (28)

Consequently, there exists a constant C > 0 such that:

‖cN‖2L∞(I;L2(Ω(t))) + ‖cN‖2

L2(I;H1(Ω(t))) ≤ C. (29)

From Lemma 7 (a)-(b) we can deduce the existence of a new constant C > 0such that:

‖cN‖2L∞(I;L2(Ω(0))) + ‖cN‖2

L2(I;H1(Ω(0))) ≤ C. (30)

Thanks to estimate (30), we know that cNN∈N is bounded in L2(I;H1(Ω(0)))∩L∞(I;L2(Ω(0))), thus, there exist a subsequence of cNN∈N, still denotedin the same way, such that, as N →∞:

cN c in L2(I;H1(Ω(0))),

cN ∗ c in L∞(I;L2(Ω(0))).

17

This convergence allows us to pass to the limit in the equation (20), obtainingthat c ∈ L2(I;H1(Ω(0))) ∩ L∞(I;L2(Ω(0))) is a solution of (19) and, con-sequently, c = c A−1

t ∈ L2(I;H1(Ω(t))) ∩ L∞(I;L2(Ω(t))) is a solution ofgeneric problem (5). Moreover, taking into account the regularity of all termsappearing in weak conservative formulation (17), we can deduce that c satis-fies:

d

dt

∫Ω(t)

cψ ∈ L2(I), ∀ψ ∈ H10 (Ω(t)). (31)

Also, since At is an isomorphism from L2(I;H1(Ω(t)))∩L∞(I;L2(Ω(t))) ontoL2(I;H1(Ω(0))) ∩ L∞(I;L2(Ω(0))) (see Lemma 7), we have convergence

cN c in L2(I;H1(Ω(t))),

cN ∗ c in L∞(I;L2(Ω(t))),

which allows us to pass to the limit in energy estimate (28) for cN , as N →∞,resulting:

‖c(t)‖2L2(Ω(t)) + 2µ

t∫0

‖∇xc(s)‖2L2(Ω(s))3ds

≤ e(1+2K)t‖c0‖2L2(Ω(0)) +

t∫0

‖f(s)‖2L2(Ω(s))ds. (32)

Finally, uniqueness of solution is a direct consequence of problem linearity andof above energy inequality (32).

Remark 9 From regularity (31) for the “time derivative” of c, and takinginto account that, from (16),

d

dt

∫Ω(t)

cψ =∫

Ω(t)

(∂c

∂t+ w · ∇xc+ (∇x ·w)c

)ψ

we deduce that ∂c∂t

+ w · ∇xc+ (∇x ·w)c ∈ L2(I;H−1(Ω(t))). Moreover, sincew ∈ [L∞(I;W 1,∞(Ω(t)))]3 and c ∈ L2(I;H1

0 (Ω(t)))∩L∞(I;L2(Ω(t))), we havethat

∂c

∂t∈ L2(I;H−1(Ω(t))).

18

Then, if we consider - for an abstract Banach space V - the Sobolev-Bochnerspace given by (see, for instance, [28]):

W 1,p,q(I;V, V ′) = u ∈ Lp(I;V ) :∂u

∂t∈ Lq(I;V ′),

we have that

c ∈ W 1,2,2(I;H10 (Ω(t)), H−1(Ω(t))).

It is also important to remark here that, as it is well known, the embeddingW 1,p,q(I;H1

0 (Ω(t)), H−1(Ω(t))) ⊂ C(I;L2(Ω(t))) is continuous for all q ≥ pp−1

.Thus, in our case,

c ∈ C(I;L2(Ω(t))).

Remark 10 It can be also demonstrated that the solution c of the weak con-servative formulation (17) does not depend on the choice of the family of ALEmappings At. In order to prove this, we can consider two possibly differentALE mappings At and At from which we obtain two solutions c and c. Sinceboth solutions satisfy (5), the difference c = c−c satisfies (5) with homogeneousdata. Moreover, c also satisfies the energy estimate (32). Then, necessarily,c = 0, and c = c.

Remark 11 Finally, taking into account - from Lemma 7 (c) - that At isan isomorphism from H1(I;H1(Ω(t))) onto H1(I;H1(Ω(0))) (thus, we couldconstruct an isomorphism for their dual spaces), and arguing in a similar wayto above Theorem 8 - but for a weak formulation both in time and space - wecould also state that c is the unique solution of the generic problem (5) in dualspace L2(I;H1

0 (Ω(t)))′.

3.4 Supplementary results

In this paragraph we will prove several technical results in order to obtainadditional properties for the solution of generic equation (5) under more re-strictive hypothesis on coefficients: mainly, positivity and boundedness.

First, assuming positivity of data, we can derive the positivity of solution:

Theorem 12 Under hypotheses of Theorem 8, if we also assume that:

• f(x, t) ≥ 0 a.e. x ∈ Ω(t), t ∈ I,• c0(x) ≥ 0 a.e. x ∈ Ω(0),

then, the unique solution c ∈ L2(I;H10 (Ω(t)))∩L∞(I;L2(Ω(t))) of (5) verifies

c(x, t) ≥ 0 a.e. x ∈ Ω(t), t ∈ I.

19

Proof The positivity of the weak solution will result from taking as test func-tion its negative part c− = minc, 0. We recall that the following integrationformula is satisfied:

d

dt

∫Ω(t)

c(t)c−(t) =d

dt

∫Ω(t)

|c−(t)|2 =d

dt‖c−(t)‖2

L2(Ω(t)).

From Remark 9 we know that c− ∈ L2(I;H10 (Ω(t))) ∩ C(I;L2(Ω(t))). Thus,

taking as a test function ψ = c−(t) in the weak formulation, integrating in[0, t], bearing in mind the results of Lemma 2, and using that, due to positivityof data, c−(0, x) = 0 a.e. x ∈ Ω(0), k1(x, t)|c−(x, t)|2 ≥ 0 a.e. x ∈ Ω(t), t ∈ I,and f(x, t)c−(x, t) ≤ 0 a.e. x ∈ Ω(t), t ∈ I, we obtain:

1

2‖c−(t)‖2

L2(Ω(t)) + µ

t∫0

‖∇xc−(s)‖2L2(Ω(s))ds+ ≤

t∫0

∫Ω(s)

k2(s)|c−(s)|2ds.

From previous inequality we deduce that:

‖c−(t)‖2L2(Ω(t)) ≤ 2K

t∫0

‖c−(s)‖2L2(Ω(s))ds,

what, thanks to Gronwall Lemma, implies that c−(x, t) = 0 a.e. x ∈ Ω(t), t ∈ Iand, consequently, c(x, t) ≥ 0 a.e. x ∈ Ω(t), t ∈ I.

We can also obtain, assuming bounded data, the boundedness of the solution:

Theorem 13 Under hypotheses of Theorem 8, if we also assume that:

• f(x, t) ≤M a.e. x ∈ Ω(t), t ∈ I,• c0(x) ≤M a.e. x ∈ Ω(0),

with M ≥ 0, then, the unique solution c ∈ L2(I;H10 (Ω(t))) ∩ L∞(I;L2(Ω(t)))

of (5) verifies

c(x, t) ≤M(1 + T )eKT a.e. x ∈ Ω(t), t ∈ I.

Proof We consider the change of variable c = eKtc1. Then, we have that c1

will be the solution of the following equation:

20

∂c1∂t

+∇x · (vc1)− µ∆xc1 + (k1(x, t)− k2(x, t) +K)c1 = e−Ktf in Ω(t), t ∈ I,

c1 = 0 on ∂Ω(t), t ∈ I,

c1 = c0 in Ω(0), t = 0.

Defining now k3(x, t) = k1(x, t)− k2(x, t) +K = k1(x, t) + (K − k2(x, t)), it isclear that k3(x, t) ≥ 0 a.e. x ∈ Ω(t), t ∈ I. If we consider the new change ofvariable c1 = c2 +Mt, the equation verified by c2 will read:

∂c2∂t

+∇x · (vc2)− µ∆xc2 + k3(x, t)c2 + k3(x, t)Mt = e−Ktf −M in Ω(t), t ∈ I,

c2 = 0 on ∂Ω(t), t ∈ I,

c2 = c0 in Ω(0), t = 0.

Taking now as a test function ψ = (c2 −M)+(t) = maxc2(t) −M, 0 in theweak formulation, integrating in [0, t], and using that, due to boundednessof data and positivity of coefficients, (c2 − M)+(0, x) = 0 a.e. x ∈ Ω(0),k3(x, t)|(c2−M)+(x, t)|2 ≥ 0 a.e. x ∈ Ω(t), t ∈ I, k3(x, t)Mt(c2−M)+(x, t) ≥ 0a.e. x ∈ Ω(t), t ∈ I, and (e−Ktf(x, t)−M)(c2−M)+(x, t) ≥ 0 a.e. x ∈ Ω(t), t ∈I, we obtain, from Lemma 2:

1

2‖(c2 −M)+(t)‖2

L2(Ω(t)) + µ

t∫0

‖∇(u2 −M)+(s)‖2L2(Ω)3ds ≤ 0,

which directly implies that c2(x, t) ≤ M a.e. x ∈ Ω(t), t ∈ I. Thus, takinginto account the previous changes of variable:

c2(x, t) ≤M ⇒ c1(x, t) ≤M(1 + T )

⇒ c(x, t) ≤M(1 + T )eKT a.e. x ∈ Ω(t), t ∈ I.

Finally, combining above results we can deduce the following Corollary:

Corollary 14 Under hypotheses of Theorem 8, if we also assume that:

• |f(x, t)| ≤M a.e. x ∈ Ω(t), t ∈ I,• |c0(x)| ≤M a.e. x ∈ Ω(0),

with M ≥ 0, then, the unique solution c ∈ L2(I;H10 (Ω(t))) ∩ L∞(I;L2(Ω(t)))

of (5) verifies

|c(x, t)| ≤M(1 + T )eKT a.e. x ∈ Ω(t), t ∈ I.

21

4 Existence and uniqueness of solution for the eutrophication model

We will demonstrate now that the eutrophication system (4) has a uniquebounded solution. In order to obtain the existence of solution, we will usethe Schauder’s Fixed Point Theorem on the subsystem formed by the fourequations corresponding to the species u = (u1, . . . , u4), and then we willanalyze the equation for u5, which is uncoupled from the rest. Boundedness ofsolution will be derived from supplementary results in above Section. Finally,uniqueness will be obtained by standard techniques, using the uniqueness ofsolution for the generic equation (5).

Theorem 15 Let us assume the following hypotheses on coefficients and data:

• v ∈ [L∞(I;W 1,∞(Ω(t)))]3 with ∇x · v = 0,• gi ∈ L2(I;L2(Ω(t))), ∀i = 1, . . . , 5,• ui0 ∈ L2(Ω(0)), ∀i = 1, . . . , 5.

Then, there exists a unique solution

u = (u1, . . . , u5) ∈ [W 1,2,2(I;H10 (Ω(t)), H−1(Ω(t))) ∩ L∞(I;L2(Ω(t)))]5

of the eutrophication problem (4) satisfying:

‖u‖[L2(I;H10 (Ω(t)))∩L∞(I;L2(Ω(t)))]5 ≤ C(T,M), (33)

where C(T,M) is a positive constant only depending on M and T . Moreover,

u ∈ [C(I;L2(Ω(t)))]5.

In addition, if we assume that problem data are bounded:

• 0 ≤ gi(x, t) ≤M a.e. x ∈ Ω(t), t ∈ I, ∀i = 1, . . . , 4,• |g5(x, t)| ≤M a.e. x ∈ Ω(t), t ∈ I,• 0 ≤ ui0(x) ≤M a.e. x ∈ Ω(0), ∀i = 1, . . . , 4,• |u5

0(x)| ≤M a.e. x ∈ Ω(0),

then, the unique solution is also bounded, u ∈ [L∞(I;L∞(Ω(t)))]5, since

0 ≤ ui(x, t) ≤ C(T,M) a.e. x ∈ Ω(t), t ∈ I, ∀i = 1, . . . , 4, (34)

|u5(x, t)| ≤ C(T,M) a.e. x ∈ Ω(t), t ∈ I. (35)

Proof The proof of the Theorem will be divided into two steps. As a firststep, we will demonstrate the existence-uniqueness of solution for the variableu = (u1, . . . , u4) via the Schauder’s Fixed Point Theorem, and, as a second

22

step, we will prove the existence-uniqueness of variable u5. Then, continuity ofu will be a direct consequence of Remark 9. Finally, boundedness of solutionwill be derived from supplementary results on above Subsection.

Step 1: Let us begin by establishing several notations related to the reactionand the source terms for the original system (1):

K(x, t, u, z) =

CncL(x, t) z2

KN +u1

Kr +Kmf +Kzu3

KF +u2 − L(x, t) u1

KN +u1

Kmz − CfzKzz2

KF +z2

KrdΘθ(x,t)−20

, (36)

F(x, t, z) =

g1(x, t) + CncKrz2 + CncKrdΘ

θ(x,t)−20z4

g2(x, t)

g3(x, t)

g4(x, t) +Kmfz2 +Kmzz

3

. (37)

Let us define now the mapping G : u ∈ B → G(u) = z ∈ B, where B is thefollowing bounded, closed, and convex subset of [L2(I;L2(Ω(t)))]4:

B = u ∈ [L2(I;L2(Ω(t)))]4 : u(x, t) ≥ 0 a.e. x ∈ Ω(t), t ∈ I

and the image of u by the mapping G, z = (z1, . . . , z4), is the solution ofthe following system, linear in each equation (not globally), with the followingorder in the resolution: firstly equation 2, then equation 3, then equation 4and, finally, equation 1:

∂zi

∂t+∇ · (vizi)− µi∆zi +Ki(x, t, u, z)zi = F i(x, t, z) in Ω(t), t ∈ I,

zi = 0 on ∂Ω(t), t ∈ I,

zi = ui0, ∀i = 1, . . . , 4, in Ω(0), t = 0.

(38)

Let us check now that the mapping G verifies the hypotheses (H1) and (H2)of the Schauder’s Theorem:

(H1) The mapping G : B→ B is well defined:

Reaction term K can be split into two parts K = K1 − K2:

23

K(x, t, u, z) =

CncL(x, t) z2

KN +u1

Kr +Kmf +Kzu3

KF +u2

Kmz

KrdΘθ(x,t)−20

︸ ︷︷ ︸

K1(x,t,u,z)

−

0

L(x, t) u1

KN +u1

CfzKzz2

KF +z2

0

︸ ︷︷ ︸

K2(x,t,u,z)

.

We will begin by solving the second equation of system (38), that is, theequation corresponding to z2. For this equation we have that, for any givenu ∈ B, K2(x, t, u, z) is independent on z. In fact, K2

1(x, t, u, z) ≡ K21(x, t, u) ∈

L2(I;L2(Ω(t))) (since u2 is non-negative), and K22(x, t, u, z) ≡ K2

2(x, t, u) ∈L∞(I;L∞(Ω(t))) (since L is bounded by Lmax, and u1 is non-negative, whichimplies u1

KN +u1 < 1). Moreover, F 2(x, t, z) = g2(x, t) ∈ L2(I;L2(Ω(t))). Then,we are under the hypotheses of Theorem 8 and, thus, there exists a uniquez2 ∈ W 1,2,2(I;H1

0 (Ω(t)), H−1(Ω(t))) ∩ L∞(I;L2(Ω(t))) solution of the secondequation of system (38), verifying the following estimate (direct consequenceof the energy inequality (18)):

‖z2(t)‖2L2(Ω(t)) + 2µ2

t∫0

‖∇z2(s)‖2L2(Ω(s))3ds

≤ e(1+2Lmax)t‖u20‖2L2(Ω(0)) +

t∫0

‖g2(s)‖2L2(Ω(s))ds

from which we obtain that:

‖z2‖L2(I;H10 (Ω(t)))∩L∞(I;L2(Ω(t))) ≤ C2(T,M). (39)

Finally, for non-negative, bounded data g2 and u20, by Theorems 12 and 13,

the solution z2 will be also non-negative, and bounded by a constant C2(T,M)only depending in M and T , i.e.

0 ≤ z2(x, t) ≤ C2(T,M) a.e. x ∈ Ω(t), t ∈ I. (40)

Let us pass now to solve the third equation of system (38), that is, the equa-tion for z3. We have that K3

1(x, t, u, z) = Kmz is constant, K32(x, t, u, z) ≡

K32(x, t, z) ∈ L∞(I;L2(Ω(t))) (since z2 is non-negative), and F 3(x, t, z) =

g3(x, t) ∈ L2(I;L2(Ω(t))). Thus, arguing in an analogous way to previous case,there exists a unique z3 ∈ W 1,2,2(I;H1

0 (Ω(t)), H−1(Ω(t)))∩L∞(I;L2(Ω(t))) so-lution of the third equation of system (38), which will be also non-negativeand bounded:

24

‖z3‖L2(I;H10 (Ω(t)))∩L∞(I;L2(Ω(t))) ≤ C3(T,M),

0 ≤ z3(x, t) ≤ C3(T,M) a.e. x ∈ Ω(t), t ∈ I.(41)

Once obtained the existence and uniqueness for the equation 2 and 3, we trynow to solve in a similar way the fourth equation. We only need to take intoaccount that, in this case, the already found solutions z2 and z3 only appearin the source term F 4(x, t, z), that will be in L2(I;L2(Ω(t))) (since z2 and z3

also are). Thus, there exists a unique z4 ∈ W 1,2,2(I;H10 (Ω(t)), H−1(Ω(t))) ∩

L∞(I;L2(Ω(t))) solution of the fourth equation of system (38), which will benon-negative and bounded:

‖z4‖L2(I;H10 (Ω(t)))∩L∞(I;L2(Ω(t))) ≤ C4(T,M),

0 ≤ z4(x, t) ≤ C4(T,M) a.e. x ∈ Ω(t), t ∈ I.(42)

Finally, for the last equation left to solve (which is actually the first equation ofsystem (38)), by putting together previous arguments, we can also obtain theexistence of a unique z1 ∈ W 1,2,2(I;H1

0 (Ω(t)), H−1(Ω(t))) ∩ L∞(I;L2(Ω(t)))solution of the first equation of system (38), also non-negative and bounded:

‖z1‖L2(I;H10 (Ω(t)))∩L∞(I;L2(Ω(t))) ≤ C1(T,M),

0 ≤ z1(x, t) ≤ C1(T,M) a.e. x ∈ Ω(t), t ∈ I.(43)

We must remark that the different constants Ci(T,M) appearing in the es-timates (39)-(43) are independent on u, which will play a crucial role in ourproof. So, denoting by C(T,M) the larger of previous constants, we have that,for all u ∈ B, the image G(u) verifies:

‖G(u)‖[L2(I;H10 (Ω(t)))∩L∞(I;L2(Ω(t)))]4 ≤ C(T,M),

0 ≤ G(u)(x, t) ≤ C(T,M) a.e. x ∈ Ω(t), t ∈ I.

If we define now the set B = u ∈ B : ‖u‖[L2(I;L2(Ω(t)))]4) ≤ C(T,M), from

above estimates we have that ∀u ∈ B, G(u) ∈ B. Thus, G|B : B→ B is welldefined.

(H2) The mapping G|B : B→ B is compact:

25

We will prove first that G is sequentially continuous. So, we consider a se-quence unn∈N ⊂ B convergent in [L2(I;L2(Ω(t)))]4 to an element u ∈ B.We only need to demonstrate that zn ≡ G(un)→ G(u) in [L2(I;L2(Ω(t)))]4.

It is worthwhile recalling here that, as a direct consequence of the generalizedAubin-Lions’ Lemma [28], the embedding of W 1,2,2(I;H1

0 (Ω(t)), H−1(Ω(t)))into L2(I;L2(Ω(t))) is compact.

Then, since G(un)n∈N is bounded in space [W 1,2,2(I;H10 (Ω(t)), H−1(Ω(t)))∩

L∞(I;L2(Ω(t)))]4 by a constant only dependent on data, we can find a subse-quence of znn∈N, still denoted in the same way, and an element z ∈ B suchthat:

zn → z in [L2(I;L2(Ω(t)))]4,

zn z in [L2(I;H10 (Ω(t)))4,

zn ∗ z in [L∞(I;L2(Ω(t)))]4.

(44)

We only have to demonstrate that z = G(u). From above definition (38) ofzn = G(un) we have that each element of the sequence znn∈N verifies:

4∑i=1

d

dt

∫Ω(t)

zinri +

∫Ω(t)

∇ · ((vi −w)zin)ri + µi

∫Ω(t)

∇zin · ∇ri

+∫

Ω(t)

Ki(x, t, un, zn)zinri

=4∑i=1

∫Ω(t)

F i(x, t, zn)ri,

∀r = (r1, r2, r3, r4) ∈ [H10 (Ω(t))]4, a.e. t ∈ I.

Passing to the limit in the above variational formulation is possible by conver-gence (44). The only difficulties in this pass to the limit appear in the terms

of the form∫

Ω(t)uk

n

K+ujnzjnr

i,∫Ω(t)

ukn

K+uknzjnr

i,∫

Ω(t)zkn

K+zknzjnr

i, or∫

Ω(t)zln

K+uknzjnr

i, but

in all these cases the boundedness and the pointwise convergence of ukn

K+ukn

touk

K+uk , and of 1K+uk

nto 1

K+uk a.e. x ∈ Ω(t), t ∈ I, together with the strong

convergence of zjn to zj in L2(I;L2(Ω(t))), allows us to overcome these diffi-culties. Thus, we conclude that z = G(u), which proves the continuity of themapping G.

Finally, the compactness of G is again a direct consequence of the gener-alized Aubin-Lions’ Lemma, which states the compactness of injection of[W 1,2,2(I;H1

0 (Ω(t)), H−1(Ω(t)))]4 into [L2(I;L2(Ω(t)))]4.

26

Thus, since hypotheses (H1) and (H2) are satisfied, Schauder’s Fixed PointTheorem allows us to obtain the existence of, at least, a fixed point u =(u1, . . . , u4) ∈ B such that u = G(u), that is, satisfying the system (4).

As a final point in this step, we are only left to prove the uniqueness of u.It is clear that any fixed point u of the mapping G will be bounded (sinceu = G(u), and we have demonstrated that all the images of G are bounded).Bearing this fact in mind, we will be able to obtain different estimates onseveral term, necessary in order to achieve the uniqueness of solution.

So, let us assume that the mapping G has two fixed points, u1 = (u11, . . . , u

41)

and u2 = (u12, . . . , u

42), solutions of the system (4), and denote by u12 =

(u112, . . . , u

412) = u1 − u2. In order to demonstrate that u12 ≡ 0 we will make

use of the following equality:

ui1K + ui1

uj1 −ui2

K + ui2uj2 = γ1(K, ui1)uj12 + γ2(K, ui1, u

j2)ui12 − γ3(K, ui1, u

j2, u

i2)ui12,

where the coefficients γi, i = 1, 2, 3, that are non-negative and bounded, aregiven by expressions:

γ1(K, ui1) =ui1

K + ui1,

γ2(K, ui1, uj2) =

uj2K + ui1

,

γ3(K, ui1, uj2, u

i2) =

ui2K + ui2

uj2K + ui1

.

Then, after tedious but simple computations, we have that u12 will satisfy thefollowing system:

∂ui12

∂t+∇ · (viui12)− µi∆ui12 + Ai1(u12)

+Ai3(u1, u2, u12) = Ai2(u12) + Ai4(u1, u2, u12) in Ω(t), t ∈ I,

ui12 = 0 on ∂Ω(t), t ∈ I,

ui12 = 0, ∀i = 1, . . . , 4, in Ω(0), t = 0,

(45)

where:

27

A1(u12) =

0

(Kr +Kmf )u212

Kmz u312

KrdΘθ−20 u4

12

,

A2(u12) =

KrCnc u212 + CncKrdΘ

θ−20 u412

0

0

Kmf u212 +Kmz u

312

,

A3(u1, u2, u12) =

CncLγ2(KN , u11, u

22)u1

12 + CncLγ1(KN , u11)u2

12

Lγ3(KN , u11, u

22, u

12)u1

12 +Kzγ2(KF , u21, u

32)u2

12 +Kzγ1(KF , u21)u3

12

CfzKzγ3(KF , u21, u

32, u

22)u2

12

0

,

A4(u1, u2, u12) =

CncLγ3(KN , u11, u

22, u

12)u1

12

Lγ2(KN , u11, u

22)u1

12 + [Kzγ3(KF , u21, u

32, u

22) + Lγ1(KN , u

11)]u2

12

CfzKzγ2(KF , u21, u

32)u2

12 + CfzKzγ1(KF , u21)u3

12

0

.

Taking u12(t) as a test function in the weak formulation of above problem, andregarding the boundedness and non-negativity of u1, u2 (thus, the bounded-ness and non-negativity of coefficients γi, i = 1, 2, 3), we have that:

1

2

4∑i=1

‖ui12(t)‖2L2(Ω(t)) ≤

4∑i=1

t∫

0

∫Ω(s)

Ai2(u12(s))ui12(s)ds

+

t∫0

∫Ω(s)

Ai4(u1(s), u2(s), u12(s))ui12(s)ds

≤ C4∑i=1

t∫0

‖ui12(s)‖2L2(Ω(s))ds.

Thus, by Gronwall’s Lemma, we obtain that, for i = 1, . . . , 4, ui12(x, t) = 0a.e. x ∈ Ω(t), t ∈ I, or, equivalently, that there exists a unique fixed pointu1 ≡ u2 of the mapping G.

Step 2: Once obtained existence and uniqueness for species u = (u1, . . . , u4),the existence, uniqueness and boundedness of species u5 is now a direct con-sequence of Theorem 8 and Corollary 14 in previous Section.

28

So, if we define, in an analogous way to step 1,

F 5(x, t) = g5(x, t) + CocL(x, t)u1(x, t)

KN + u1(x, t)u2(x, t)

−CocKru2(x, t)− CocKrdΘ

θ(x,t)−20u4(x, t),

the fifth equation of system (38) reads:∂z5

∂t+∇ · (v5z5)− µ5∆z5 = F 5(x, t) in Ω(t), t ∈ I,

z5 = 0 on ∂Ω(t), t ∈ I,

z5 = u50 in Ω(0), t = 0.

(46)

Since F 5 ∈ L2(I;L2(Ω(t))), then, by Theorem 8, we obtain the existence of aunique z5 ∈ W 1,2,2(I;H1

0 (Ω(t)), H−1(Ω(t))) ∩ L∞(I;L2(Ω(t))) solution of thefifth equation of system (38), satisfying:

‖z5‖L2(I;H10 (Ω(t)))∩L∞(I;L2(Ω(t))) ≤ C(T,M). (47)

Finally, from the boundedness of g5, L and u, we deduce that F 5 is bounded(but not necessarily non-negative) and, consequently, from Corollary 14 weobtain that:

|z5(x, t)| ≤ C(T,M) a.e. x ∈ Ω(t), t ∈ I, (48)

which concludes the proof.

5 Conclusions

In this paper we have analyzed a complex system simulating the interactions ofnutrient, phytoplankton, zooplankton, organic detritus and dissolved oxygeninto the eutrophication processes, posed in a realistic moving domain. Withour technique - based on the ALE formulation - we have obtained existenceand uniqueness of solution for the eutrophication system, along with regularityand boundedness results, and also an a priori energy estimate to be satisfiedby the solution. All these results are derived from the analysis of a genericparabolic equation in non-cylindrical domains, which has been extensivelystudied through the paper.

29

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