· RELATIVE ENERGY FOR THE KORTEWEG THEORY AND RELATED HAMILTONIAN FLOWS IN GAS DYNAMICS JAN GIESSELMANN, CORRADO LATTANZIO, AND ATHANASIOS E. TZAVARAS Abstract. For an Euler system,

RELATIVE ENERGY FOR THE KORTEWEG THEORY AND RELATED

HAMILTONIAN FLOWS IN GAS DYNAMICS

JAN GIESSELMANN, CORRADO LATTANZIO, AND ATHANASIOS E. TZAVARAS

Abstract. For an Euler system, with dynamics generated by a potential energy functional,we propose a functional format for the relative energy and derive a relative energy iden-tity. The latter, when applied to specific energies, yields relative energy identities for theEuler-Korteweg, the Euler-Poisson, the Quantum Hydrodynamics system, and low orderapproximations of the Euler-Korteweg system. For the Euler-Korteweg system we provea stability theorem between a weak and a strong solution and an associated weak-stronguniqueness theorem. In the second part we focus on the Navier-Stokes-Korteweg system(NSK) with nonmonotone pressure laws: we prove stability for the NSK system via a modi-fied relative entropy approach. We prove continuous dependence of solutions on initial dataand convergence of solutions of a low order model to solutions of the NSK system. The lasttwo results provide physically meaningful examples of how higher order regularization termsenable the use of the relative entropy framework for models with entropies which are notpoly- or quasi-convex, but compensating via higher-order gradients.

Contents

1. Introduction 22. Hamiltonian flow and the relative energy 62.1. The relative potential energy 82.2. The relative kinetic energy 92.3. The functional form of the relative energy formula 102.4. The Euler-Korteweg system 102.5. The Euler-Poisson system 152.6. Order parameter model of lower order 172.7. Remarks on the Hamiltonian structure of the problem 193. Weak-strong stability for the Euler-Korteweg system 203.1. Energy estimate 213.2. Relative energy estimate for weak solutions 213.3. Stability estimates for weak solutions in energy norms and constant capillarity 273.4. Stability estimates for L∞ weak solutions and general capillarity 284. Stability estimates for non-convex energies 304.1. Assumptions 304.2. Continuous dependence on initial data 315. Model convergence 345.1. Assumptions on well-posedness and uniform bounds 35

JG partially supported by the German Research Foundation (DFG) via SFB TRR 75 ‘TropfendynamischeProzesse unter extremen Umgebungsbedingungen’.

AET acknowledges the support of the King Abdullah University of Science and Technology (KAUST) andof the Aristeia program of the Greek Secretariat for Research through the project DIKICOMA.

1

2 JAN GIESSELMANN, CORRADO LATTANZIO, AND ATHANASIOS E. TZAVARAS

5.2. Elliptic approximation 375.3. Relative energy 37Appendix A. Noether’s theorem and the Korteweg stress 43Appendix B. The relative energy transport identity for the Euler-Korteweg system 43Appendix C. Proof of Lemma 5.9 48References 50

1. Introduction

We study the system of partial differential equations

funsys-introfunsys-intro (1.1)

∂ρ

∂t+ divx(ρu) = 0

Du

Dt:=

∂u

∂t+ (u · ∇x)u = −∇x

δEδρ

(ρ)x ∈ Rd , t > 0 ,

where ρ ≥ 0 is a density obeying the conservation of mass equation, u is the fluid velocityand m = ρu the momentum flux. The dynamical equation determining the evolution of u isgenerated by a functional E(ρ) on the density and δE

δρ stands for the generator of the directional

derivative of that functional. The dynamics (1.1) formally satisfies the conservation of energyequation

toteintrototeintro (1.2)d

dt

(∫12ρ|u|

2 dx+ E(ρ)

)= 0 .

Depending on the selection of the functional E(ρ) several models of interest fit under thisframework (see Section 2). These include the equations of isentropic gas dynamics, theEuler-Poisson system (e.g. [31, 28, 22]) the system of quantum hydrodynamics (e.g. [1]), theEuler-Korteweg system (e.g. [15, 3]), and order-parameter models for the study of phasetransitions (e.g. [4, 32, 33]). The objective here is to review the relation of these problemswith the formal structure (1.1) and to use the structure in order to obtain a relative entropyidentity. The idea is quite simple: most (but not all) of the problems above are generatedby convex functionals. It is natural to use the quadratic part of the Taylor expansion of thefunctional E(ρ),

repotedef-introrepotedef-intro (1.3) E(ρ|ρ) := E(ρ)− E(ρ)−⟨δEδρ

(ρ), ρ− ρ⟩,

as a measure for comparing two states ρ(t, ·) and ρ(t, ·). This definition involves the directionalderivative of E(ρ) in the direction (ρ−ρ) and provides a functional called here relative potentialenergy. This is used along with the relative kinetic energy

rekinedef-introrekinedef-intro (1.4) K(ρ,m|ρ, m) =

∫ρ∣∣∣mρ− m

ρ

∣∣∣2dxas a yardstick for comparing the distance between two solutions (ρ, u) and (ρ, u). An addi-tional ingredient is needed: we postulate the existence of a stress tensor (functional) S(ρ)such that

stresshyp-introstresshyp-intro (1.5) −ρ∇xδEδρ

= ∇x · S .

RELATIVE ENERGY FOR HAMILTONIAN FLOWS 3

Hypothesis (1.5) holds for all the above examples, it gives a meaning to the notion of weaksolutions for (1.1) as it induces a conservative form

wkfunsys-introwkfunsys-intro (1.6)

∂ρ

∂t+ divx(ρu) = 0

∂

∂t(ρu) + divx(ρu⊗ u) = ∇x · S ,

and plays an instrumental role in devising a relative entropy identity. For the Euler-Kortewegsystem it is a consequence of the invariance under translations of the generating functionaland Noether’s theorem (see [3] and Appendix A). The relative energy identity takes theabstract form

reltote-introreltote-intro (1.7)

d

dt

(E(ρ|ρ) +

∫12ρ|u− u|

2 dx

)=

∫∇xu : S(ρ|ρ) dx−

∫ρ∇xu : (u− u)⊗ (u− u) dx ,

where E(ρ|ρ) is defined in (1.3), while the relative stress functional is

relstress-introrelstress-intro (1.8) S(ρ|ρ) := S(ρ)− S(ρ)−⟨δSδρ

(ρ), ρ− ρ⟩.

Formula (1.7) is the main result of this article. It should be compared to the well knownrelative entropy formulas initiated in the works Dafermos [9, 10], DiPerna[13] and analogs thathave been succesfully used in many contexts (e.g. [24, 27, 17, 6, 25, 19]). It has however adifferent origin from all these calculations: while the latter are based on the thermodynamicalstructure induced by the Clausius-Duhem inequality, the formula (1.7) is based on the abstractHamiltonian flow structure in (1.1). The reader familiar with the ramifications of continuumthermodynamics will observe that, as noted by Dunn and Serrin [15], constitutive theories withhigher-order gradients (unless trivial) are inconsistent with the Clausius-Duhem structure,and in order to make them compatible one has to introduce the interstitial work term in theenergy equation. Nevertheless, as shown in Sections 2.1-2.3, the structure (1.1), (1.5) inducesa relative entropy identity provided E(ρ|ρ) is defined through the Taylor expansion of thegenerating functional.

Despite this difference, there is also striking similarity between the formulas obtained in[10, 25] and the formula (1.7) in connection to the mechanical interpretations for the relativemechanical stress and the relative convective stress. This similarity is conceptually quiteappealing, yet the simplicity of the formula (1.7) is somewhat misleading. In fact, the actualformulas in specific examples are cumbersome, as can be noticed in Section 2.4 in formula(2.31) derived for the Euler-Korteweg system, or in Section 2.5 in the relative energy formula(2.52) for the Euler-Poisson system. In addition, the simplicity of the derivation using thefunctional framework presented in Sections 2.1, 2.2 should be contrasted to the lengthy directderivation in Appendix B for the Euler-Korteweg system.

Our work is closely related to the observations of Benzoni-Gavage et al. [2, 3] that theEuler-Korteweg system can be formally expressed for an irrotational flow velocity fields asa Hamiltonian system. While for general flows the structure of the problem fails to beHamiltonian (see (2.65)), this discrepancy is consistent with energy conservation and as shownhere induces with (1.5) the relative energy identity.


Gradient flows generated by functionals E(ρ) in Wasserstein distance give rise to parabolicequations

limitdifflimitdiff (1.9) ∂tρ = ∇x · ρ∇xδEδρ

and have received ample attention in multiple contexts (e.g. [30, 7]). Such systems can beseen as describing the long-time response of an Euler equation with friction (see (2.1) wherethe effect of a frictional term are added). At long times the friction term tends to equilibratewith the gradient of the energy, ρu ∼ −ρ∇x δEδρ , and produces the diffusion equation (1.9). A

justification of this process using the relative energy framework is undertaken in a companionarticle [26], see also [25] for an earlier special instance of that limit.

In the second part of this work, we establish some applications of (1.7). We consider firstthe Euler-Korteweg system

Kortcap-introKortcap-intro (1.10)

ρt + divx(ρu) = 0

(ρu)t + divx(ρu⊗ u) = −ρ∇x(Fρ(ρ,∇xρ)− divx Fq(ρ,∇xρ)

),

generated by a convex energy

kortfunc-introkortfunc-intro (1.11) E(ρ) =

∫F (ρ,∇xρ) dx =

∫h(ρ) +

1

2κ(ρ)|∇xρ|2 dx

and proceed to compare a conservative (or dissipative) weak solution (ρ,m) to a strong solu-tion (ρ, m) of (1.10). First, we establish the relative entropy transport formula for comparingsolutions of this regularity classes in Theorem 3.2. This is in turn used to establish twostability results between weak and strong solutions of (1.10):

(i) Theorem 3.9 valid for convex energies (1.11) under the hypothesis that the density ρof the weak solution is bounded while the density ρ of the strong solution does notinclude vacuum regions; this result generalizes to the class of energies (1.11) the resultin [14] valid for energies with κ(ρ) = Cκ, and in particular our result applies to thequantum hydrodynamics system (2.42).

(ii) Theorem 3.5 which restricts to convex energies (1.11) with constant capillarity (κ(ρ) =Cκ) and to strong solutions with density ρ bounded away from vacuum, but in returndoes not place any boundedness assumptions on the density ρ except for the naturalenergy norm bound.

Augmenting the Euler-Korteweg system with viscosity leads to the isothermal Navier-Stokes-Korteweg (NSK) model. When the capillarity coefficient κ(ρ) = Cκ > 0 is constant in(1.11) and the pressure function p(ρ) is non-monotone, the NSK system

ρt + divx(ρu) = 0

(ρu)t + divx(ρu⊗ u) +∇xp(ρ) = divx(σ[u]) + Cκρ∇x∆xρ ,NSK-introNSK-intro (1.12)

is a well-known model for compressible liquid-vapor flows undergoing phase transitions ofdiffuse interface type (e.g. [15, 3]). The term

def:nss_intdef:nss_int (1.13) σ[u] := λdivx(u)I + µ(∇xu+ (∇xu)T )

is the Navier-Stokes stress with coefficients λ, µ satisfying µ ≥ 0 and λ+ 2dµ ≥ 0, and I ∈ Rd×d

is the unit matrix and d the spatial dimension. To describe multi-phase flows the energy


density h = h(ρ) is non-convex and the associated pressure p′(ρ) = ρh′′(ρ) is non-monotone,so that the first order part of (1.12) is a system of mixed hyperbolic-elliptic type.

The relative entropy technique usually applies to situations where the energy is convex orat least quasi-convex or poly-convex [11, 24, 12]. Here we provide some examples on how – ina physically meaningful and multi-dimensional situation – the higher-order (second gradient)regularization mechanism compensates for the non-convexity of the energy in such a way thatthe relative entropy technique still provides stability estimates. This extends results from[19], valid in a one-dimensional Lagrangean setting. To highlight the use of a stability theoryfor (1.12) derived via a modified relative entropy approach, we prove two results:

(a) we show stability of smooth solutions to (1.12) or to the associated Korteweg-Eulersystem with non-monotone pressures for initial-data with equal initial mass.

(b) we show that solutions of a lower order approximation to the Navier-Stokes-Kortewegsystem given by (1.14) below converge to solutions of (1.12) in the limit α→∞.

To place (b) in the relevant context, we note that the system

∂

∂tρ+ divx(ρu) = 0

∂

∂t(ρu) + divx(ρu⊗ u) +∇x(p(ρ) + Cκ

α

2ρ2) = divx(σ[u]) + Cκαρ∇xc

c− 1

α∆xc = ρ ,

lo-introlo-intro (1.14)

where Cκ is as in (2.38), and α > 0 is a parameter, was introduced in [33] with the goalto approximate the NSK system. It has motivated very efficient numerical schemes for thenumerical treatment of diffuse interface systems for the description of phase transitions (cf.the discussion in the beginning of section 5). The model convergence from (1.14) to (1.12)as α → ∞ was investigated in simple cases by [8] (using Fourier methods) and [19]. Here,we exploit the fact that the system (1.14) fits into the functional framework of (1.1), (1.5)and is thus equipped with a relative energy identity (see section 2.6). Using the relativeenergy framework, as modified in section 4 pertaining to non-monotone pressures, we provein Theorem 5.14 convergence from (1.14) to (1.12) as α →∞ for initial data of equal initialmass.

The structure of the paper is as follows: in Section 2 we discuss the derivation of theformal relative energy estimate (1.7) at the level of the abstract equation (1.1), and then weimplement its application to various models: the Euler-Korteweg system in Section 2.4, theEuler-Poisson system in Section 2.5, and the order parameter model of lower order in Section2.6. The relation between (1.1) and a Hamiltonian structure is discussed in Section 2.7. InSection 3 we consider the Euler-Korteweg system (1.10) and we establish the relative energyestimate for solutions of limited smoothness, that is between a dissipative (or conservative)weak solution and a strong solution of (1.10). We then derive the two weak-strong stabilityTheorems 3.5 and 3.9. In Section 4 we consider the Navier-Stokes-Korteweg system (1.12)with non-monotone pressure p(ρ), we present a relative energy calculation and show thatsolutions of (1.12) depend continuously on their initial data in Theorem 4.5. In Section 5 webriefly introduce the lower order model (5.1) and derive an estimate for the difference betweensolutions of (1.14) and (1.12). In the appendices we present: a remark on the relation of (1.5)to the invariance under spatial-translations of the energy functional and Noether’s theorem inAppendix A; a direct derivation of the relative energy identity for the Euler-Korteweg system


(1.10) in Appendix B; the derivation of the relative energy identity for the lower order modelin Appendix C.

2. Hamiltonian flow and the relative energysec:ham

In this section we consider a system of equations consisting of a conservation of mass anda functional momentum equation

funsysfunsys (2.1)

∂ρ

∂t+ divx(ρu) = 0

ρDu

Dt= −ρ∇x

δEδρ− ζρu ,

where ρ ≥ 0 is the density, u the velocity, and DDt = ∂

∂t+u·∇x stands for the material derivative

operator. In (2.1)2, δEδρ stands for the generator of the first variation of the functional E(ρ)

(see the discussion below). The term (−ζρu) in (2.1) corresponds to a damping force withfrictional coefficient ζ > 0; the particular frictionless case ζ = 0, leading to (1.1), is alsoallowed. The objective of the section is to derive the main relative entropy calculation of thiswork and to apply it to certain specific systems. The derivation is formal in nature and hasto be validated via alternative methods for solutions of limited smoothness (e.g. for weak orfor measure-valued solutions). Nevertheless, the formal derivation uses in an essential waythe functional structure of (2.1) and it is fairly simple to achieve via this formalism.

In the sequel E [ρ] will stand for a functional on the density ρ(·, t). The directional derivative(Gateaux derivative) of the functional E : U ⊂ X → R, where U is an open subset of X, alocally convex togological vector space, is defined by

dE(ρ;ψ) = limτ→0

E(ρ+ τψ)− E(ρ)

τ=

d

dτE(ρ+ τψ)

∣∣∣τ=0

,

with the limit taken over the reals. When the limit exists the functional is Gateaux differen-tiable at ρ and dE(ρ;ψ) is the directional derivative. The function dE(ρ, ·) is homogeneous ofdegree one, but may in general fail to be linear (in ψ). We will assume that dE(ρ;ψ) is linearin ψ and can be represented via a duality bracket

funchyp1funchyp1 (2.2) dE(ρ;ψ) =d

dτE(ρ+ τψ)

∣∣∣τ=0

=⟨δEδρ

(ρ), ψ⟩,

with δEδρ (ρ) standing for the generator of the bracket.

Further, we define the second variation via

d2E(ρ;ψ,ϕ) = limε→0

⟨δEδρ (ρ+ εϕ), ψ

⟩−⟨δEδρ (ρ), ψ

⟩ε

(whenever the limit exists) and we assume that this can be respresented as a bilinear functionalin the form

funchyp2funchyp2 (2.3) d2E(ρ;ψ,ϕ) = limε→0

⟨δEδρ (ρ+ εϕ), ψ


⟩ε

=

⟨⟨δ2Eδρ2

(ρ), (ψ,ϕ)

⟩⟩.

We note that (2.2) and (2.3) hold when the functional E [ρ] is Frechet differentiable on aBanach space X and with sufficient smoothness. Moreover, in a framework when X is aFrechet space (locally convex topological vector space that is metrizable) there are availabletheorems that show that continuity of dE(ρ, ψ) : X × X → R guarantees the linearity ofdE(ρ, ·); corresponding theorems also hold for the second variation; see [21, Sec I.3]. In the


following formal calculations we will place (2.2) and (2.3) as hypotheses and will validatethem for two examples: the Euler-Korteweg and the Euler-Poisson system.

By standard calculations the system (2.1) can be expressed in the form

wkfunsyswkfunsys (2.4)

∂ρ

∂t+ divx(ρu) = 0

∂

∂t(ρu) + divx(ρu⊗ u) = −ρ∇x

δEδρ− ζρu .

The left part of (2.4) is in conservation form, however this is not generally true for the termρ∇x δEδρ . Nevertheless, for all examples of interest here it will turn out that

stresshypstresshyp (2.5) −ρ∇xδEδρ

= ∇x · S ,

where S = S(ρ) will be a tensor-valued functional on ρ that plays the role of a stress tensorand has components Sij(ρ) with i, j = 1, ..., d.

At this stage (2.5) is placed as a hypothesis. This hypothesis is validated in the sequel forvarious specific models. It also turns out that it is a fairly general consequence of the invarianceof the functional E(ρ) under space translations via Noether’s theorem (see appendix). Notethat (2.5) gives a meaning to weak solutions for (2.4) and is instrumental in the forthcomingcalculations for the potential, kinetic and total energy. It is assumed that no work is doneat the boundaries and thus integrations by parts do not result to boundary contributions. Inparticular they are valid for periodic boundary conditions (x ∈ Td the torus) or on the entirespace (x ∈ Rd).

The potential energy is computed via

potepote (2.6)d

dtE(ρ)

(2.2)= 〈δE

δρ(ρ), ρt〉

(2.1)= −〈δE

δρ(ρ), divx(ρu)〉 (2.5)

=

∫S : ∇xu dx .

Taking the inner product of (2.1)2 by u gives

12ρ

D

Dt|u|2 = −u · ρ∇x

δEδρ− ζρ|u|2 (2.5)

= u · divx S − ζρ|u|2 .

Then, using (2.1)1, we obtain

12∂t(ρ|u|

2) + divx(12ρ|u|

2u) = u · divx S − ζρ|u|2 ,

which leads to the evolution of the kinetic energy

kinekine (2.7)d

dt

∫12ρ|u|

2 dx = −∫ (

S : ∇xu+ ζρ|u|2)dx .

Combining (2.6) and (2.7) provides the balance of total energy

totetote (2.8)d

dt

(∫12ρ|u|

2 dx+ E(ρ)

)= −ζ

∫ρ|u|2 dx .

In the frictionless case ζ = 0 the total energy is conserved, while for ζ > 0 there is dissipationdue to friction.


sec-funcrel-pote

2.1. The relative potential energy. Consider now the system (2.1) generated by the func-tional E(ρ) and assume that the functional is convex. A natural quantity to monitor isprovided by the quadratic part of the Taylor series expansion of the functional with respectto a reference solution ρ(x, t); this quantity is called here relative potential energy and isdefined by

relendefrelendef (2.9) E(ρ|ρ) := E(ρ)− E(ρ)−⟨δEδρ

(ρ), ρ− ρ⟩.

We next develop the relative potential energy calculation, which is based on the hypotheses(2.2), (2.3), (2.5) and (2.11) below. For ϕi a vector valued test function, i = 1, ..., d, the weakform of (2.5) reads

varder1varder1 (2.10)⟨δEδρ

(ρ),∂

∂xj(ρϕj)

⟩= −

∫Sij(ρ)

∂ϕi∂xj

dx ,

where we employ the summation convention over repeated indices.Recall that the stress S(ρ) is a functional on the density ρ. We assume that the directional

derivative of S is expressed as a linear functional via a duality bracket,

funchyp3funchyp3 (2.11) dS(ρ;ψ) =d

dτS(ρ+ τψ)

∣∣∣τ=0

=⟨δSδρ

(ρ), ψ⟩,

in terms of the generator δSδρ (ρ) (in complete analogy to (2.2)).

We now take the directional derivative of (2.10) – viewed as a functional in ρ – along adirection ψ, with ψ a smooth test function, and use (2.3), (2.11) to arrive at the formula

varder2varder2 (2.12)

⟨⟨δ2Eδρ2

(ρ),(ψ,

∂

∂xj(ρϕj)

)⟩⟩+

⟨δEδρ

(ρ),∂

∂xj(ψϕj)

⟩= −

∫ ⟨δSijδρ

(ρ), ψ

⟩∂ϕi∂xj

dx .

Let now (ρ, u) and (ρ, u) be two smooth solutions of (2.1). Using (2.6), (2.1), we compute

∂t

(E(ρ)− E(ρ)−

⟨δEδρ

(ρ), ρ− ρ⟩)

=⟨δEδρ

(ρ),−divx(ρu)⟩−⟨δEδρ

(ρ),−divx(ρu)⟩

−⟨⟨δ2Eδρ2

(ρ), (ρt, ρ− ρ)⟩⟩

+⟨δEδρ

(ρ), divx(ρu− ρu)⟩

=⟨δEδρ

(ρ),−divx(ρu)⟩−⟨δEδρ

(ρ),−divx(ρu)⟩

+⟨δEδρ

(ρ), divx(ρ(u− u)

)⟩−⟨⟨δ2Eδρ2

(ρ), (divx(ρu), ρ− ρ)⟩⟩

+⟨δEδρ

(ρ),divx((ρ− ρ)u

)⟩(2.10),(2.12)

=⟨Sij(ρ),

∂ui∂xj

⟩−⟨Sij(ρ),

∂ui∂xj

⟩−⟨δEδρ

(ρ)− δEδρ

(ρ),divx(ρ(u− u)

)⟩−⟨Sij(ρ),

∂

∂xj(ui − ui)

⟩−∫ ⟨

δSijδρ

(ρ), ρ− ρ⟩∂ui∂xj

dx

=

∫ (Sij(ρ)− Sij(ρ)−

⟨δSijδρ

(ρ), ρ− ρ⟩)∂ui∂xj

dx−⟨δEδρ

(ρ)− δEδρ

(ρ),divx(ρ(u− u)

)⟩


We next define the relative stress tensor

relstressrelstress (2.13) S(ρ|ρ) := S(ρ)− S(ρ)−⟨δSδρ

(ρ), ρ− ρ⟩

and conclude with the relative potential energy balance

relpoterelpote (2.14)d

dtE(ρ|ρ) =

∫Sij(ρ|ρ)

∂ui∂xj

dx−⟨δEδρ

(ρ)− δEδρ


)⟩.

sec-funcrel-kine2.2. The relative kinetic energy. Next consider the kinetic energy functional

kinedefkinedef (2.15) K(ρ,m) =

∫1

2

|m|2

ρdx

viewed as a functional on the density ρ and the momentum m = ρu. The integrand of this

functional, k(ρ,m) = 12|m|2ρ , has Hessian the (d+ 1)× (d+ 1) matrix

∇2(ρ,m)k =

|m|2

ρ3−m

T

ρ2

−mρ2

1

ρId×d

,

which has eigenvalues

λ1 = 0, λ2 = ... = λd =1

ρ> 0, λd+1 =

1

ρ+|m|2

ρ3> 0 ,

and is positive semidefinite for ρ > 0. The kinetic energy functional K(ρ,m) is convex (thoughnot strictly convex) as a functional in (ρ,m).

The relative kinetic energy is easily expressed in the form

relkinenergyrelkinenergy (2.16)

K(ρ,m|ρ, m) :=

∫k(ρ,m)− k(ρ, m)−∇k(ρ, m) · (ρ− ρ,m− m) dx

=

∫1

2

|m|2

ρ− 1

2

|m|2

ρ−(− 1

2

|m|2

ρ2,m

ρ

)· (ρ− ρ,m− m) dx

=

∫1

2ρ|u− u|2 dx .

To compute the evolution of the relative kinetic energy, we consider the difference of thetwo equations (2.1)2 satisfied by (ρ, u) and (ρ, u),

∂t(u− u) + (u · ∇x)(u− u) +((u− u) · ∇x)u = −∇x

(δEδρ

(ρ)− δEδρ

(ρ))− ζ(u− u)

and take the inner product with (u− u) to deduce

1

2

D

Dt|u− u|2 +∇xu : (u− u)⊗ (u− u) = −(u− u) · ∇x

(δEδρ

(ρ)− δEδρ

(ρ))− ζ|u− u|2 ,

which, using (2.1)1, is expressed in the form

∂t(

12ρ|u− u|

2)

+ divx(

12ρu|u− u|

2)

= −ρ∇xu : (u− u)⊗ (u− u)

−ρ(u− u) · ∇x(δEδρ

(ρ)− δEδρ

(ρ))− ζρ|u− u|2 .


Integrating over space leads to the balance of the relative kinetic energy

relkinerelkine (2.17)

d

dt

∫1

2ρ|u− u|2 dx+ ζ

∫ρ|u− u|2 dx

= −∫ρ∇xu : (u− u)⊗ (u− u) dx+

⟨δEδρ

(ρ)− δEδρ


)⟩.

sec-funcrel-tote2.3. The functional form of the relative energy formula. Combining (2.14) with (2.17)we obtain the equation for the evolution of the (total) relative energy

reltotereltote (2.18)

d

dt

(E(ρ|ρ) +

∫12ρ|u− u|

2 dx

)+ ζ

∫ρ|u− u|2 dx

=

∫∇xu : S(ρ|ρ) dx−

∫ρ∇xu : (u− u)⊗ (u− u) dx ,

where E(ρ|ρ) and S(ρ|ρ) stand for the relative potential energy and relative stress functionalsdefined in (2.9) and (2.13), respectively.

An interesting feature of these calculation is how the contributions of the term

D =⟨δEδρ

(ρ)− δEδρ


)⟩in (2.14) and (2.17) offset each other, in complete analogy to the workings of the derivationof the total energy (2.8) from the potential (2.6) and kinetic (2.7) energies. The errors areagain formally quadratic in nature as in the corresponding calculations of [9] for the systemof thermoelasticity, but now functionals are involved in the final formulas.

While this abstract derivation has some elegance and ease of derivation, it has the drawbackthat it requires smoothness for the fields (ρ, u) and (ρ, u). In applications the relative energyis often used to compare a weak to a strong solution; in such cases the calculation has to berederived by alternate means for solutions of limited smoothness. Indeed, this is done for theEuler-Korteweg system in Section 3.2 and Appendix B.

The calculation is next applied to two specific examples: the Euler-Korteweg and theEuler-Poisson system.

sec-EK2.4. The Euler-Korteweg system. The functional

kortfunckortfunc (2.19) E(ρ) =

∫F (ρ,∇xρ) dx

generated by the smooth function F = F (ρ, q) : R+ × Rd → R is associated to the Kortewegtheory of capillarity; the dynamics of (2.1) or (2.4) along the functional (2.19) generates theEuler-Korteweg system.


For the functional (2.19) we next explore the precise meaning of the formulas (2.2) and(2.3). Note that, for a test function ψ,

firstvarfirstvar (2.20)

dE(ρ;ψ) =d

dτE(ρ+ τψ)

∣∣∣τ=0

=d

dτ

∣∣∣τ=0

∫F (ρ+ τψ,∇xρ+ τ∇xψ) dx

=

∫ (Fρ(ρ,∇xρ)ψ + Fq(ρ,∇xρ) · ∇xψ

)dx

=:⟨δEδρ

(ρ), ψ⟩.

This defines the meaning of the bracket 〈·, ψ〉 as the duality bracket between H−1 and H10

and identifies the generator δEδρ as

(2.21)δEδρ

(ρ) = Fρ(ρ,∇xρ)−∇x · Fq(ρ,∇xρ) .

Next, we computesecondvarsecondvar (2.22)

d2E(ρ;ψ, φ) = limε→0

⟨δEδρ (ρ+ εφ), ψ


⟩ε

=d

dε

∣∣∣ε=0

∫ (Fρ(ρ+ εφ,∇xρ+ ε∇xφ)ψ + Fq(ρ+ εφ,∇xρ+ ε∇xφ) · ∇xψ

)dx

=

∫ (Fρρ φψ + ψFρq · ∇xφ+ φFρq · ∇xψ + Fqq : (∇xφ⊗∇xψ)

)dx

=

∫(φ,∇xφ) ·

(Fρρ FρqFρq Fqq

)(ρ,∇xρ)

(ψ∇xψ

)dx

=:

⟨⟨δ2Eδρ2

(ρ), (ψ, φ)

⟩⟩,

where the last equation defines the meaning of the bracket 〈〈·, (φ, ψ)〉〉 and we have used thenotations Fq = ∇qF and Fqq = ∇2

qF . It is clear that d2E(ρ;ψ,ϕ) is a bilinear form and alsothat the convexity of F implies that the second variation of E is positive.

Using (2.20) (for the test function ψ = ρ− ρ) we compute the relative potential energy in(2.9) and express it in the form

relpotekortrelpotekort (2.23)

E(ρ|ρ) =

∫ (F (ρ,∇xρ)− F (ρ,∇xρ)− Fρ(ρ,∇xρ)(ρ− ρ)− Fq(ρ,∇xρ) · (∇xρ−∇xρ)

)dx

=

∫F (ρ,∇xρ | ρ,∇xρ) dx ,

where F (ρ, q|ρ, q) stands for the quadratic Taylor polynomial of F (ρ, q) around (ρ, q).


The Euler-Korteweg system (with friction when ζ > 0) of the form (2.1) generated by theKorteweg functional (2.19) takes the form

euler-korteuler-kort (2.24)

∂ρ

∂t+ divx(ρu) = 0

ρ[∂u∂t

+ (u · ∇x)u]

= −ρ∇x(Fρ −∇x · Fq

)− ζρu .

By direct computation, one checks the formula

formulaformula (2.25)

−ρ ∂

∂xi

δEδρ

= −ρ ∂

∂xi

(Fρ −

∂

∂xkFqk)

= − ∂

∂xi

(ρFρ − ρ

∂

∂xkFqk)

+∂ρ

∂xiFρ +

∂2ρ

∂xi∂xkFqk −

∂

∂xk

( ∂ρ∂xi

Fqk)

=∂

∂xj

((F − ρFρ + ρ

∂

∂xkFqk)δij −

∂ρ

∂xiFqj

)=

∂

∂xjSij ,

where the Korteweg stress tensor is defined by

kortstreskortstres (2.26)

S = (F − ρFρ + ρ divx Fq)I−∇xρ⊗ Fq

=[(F − ρFρ − (∇xρ) · Fq) + divx(ρFq)

]I−∇xρ⊗ Fq ,

with I the identity matrix and P = ρFρ−F the pressure. For the stress tensor to be symmetricit is often required that Fq = a(ρ,∇xρ)∇xρ with a a scalar valued function. The followingexpression of S(ρ) will be convenient for expressing the relative stress:

stressreexpstressreexp (2.27)

Sij(ρ) =(F − ρFρ −

∂ρ

∂xkFqk)δij +

∂

∂xk

(ρFqk

)δij −

∂ρ

∂xiFqj

= −s(ρ,∇xρ)δij +( ∂

∂xkrk(ρ,∇xρ)

)δij −Hij(ρ,∇xρ) ,

where the functions s : R+×Rd → R, r : R+×Rd → Rd and H : R+×Rd → Rd×d are definedby

stressdeffuncstressdeffunc (2.28)

s(ρ, q) = ρFρ(ρ, q) + q · Fq(ρ, q)− F (ρ, q) ,

r(ρ, q) = ρFq(ρ, q) ,

H(ρ, q) = q ⊗ Fq(ρ, q) .

We proceed to compute the relative stress tensor defined in (2.13). S(ρ) is viewed as afunctional and using (2.27) we compute, for a test function φ, the first variation of Sij(ρ) viathe formula

d

dτ

∣∣∣∣∣τ=0

Sij(ρ+ τφ)

= −(sρφ+ sq · ∇xφ)δij +∂

∂xk

(∂rk∂ρ

φ+∂rk∂ql

∂φ

∂xl

)δij −

(∂Hij

∂ρφ+

∂Hij

∂ql

∂φ

∂xl

)= 〈δSij

δρ, φ〉 .stressvarkortstressvarkort (2.29)


Note that (2.29) gives a meaning to the bracket defining the first variation of the Kortewegstress tensor.

In turn, using (2.27), (2.13) and (2.29) (with ρ → ρ and for the choice of φ = ρ − ρ) weobtain an expression for the relative stress Korteweg tensor:

relstresskortrelstresskort (2.30)

Sij(ρ|ρ) = Sij(ρ)− Sij(ρ)−⟨δSijδρ

(ρ), ρ− ρ⟩

= −s(ρ,∇xρ|ρ,∇xρ)δij +∂

∂xk

(rk(ρ,∇xρ|ρ,∇xρ)

)δij −Hij(ρ,∇xρ|ρ,∇xρ) ,

where, as usual,

s(ρ, q|ρ, q) = s(ρ, q)− s(ρ, q)− sρ(ρ, q)(ρ− ρ)− sq(ρ, q) · (q − q)

stands for the quadratic part of the Taylor expansion for the function s(ρ, q) and similarlyfor the functions r(ρ, q) and H(ρ, q).

We conclude by stating the relative energy formula induced by (2.18) for the specific caseof the Euler-Korteweg system (2.24). Using (2.23) and (2.30), we end up with

reltotekortreltotekort (2.31)

d

dt

(∫F (ρ,∇xρ | ρ,∇xρ) + 1

2ρ|u− u|2 dx

)+ ζ

∫ρ|u− u|2 dx

= −∫ [

(divx u) s(ρ,∇xρ|ρ,∇xρ) +( ∂

∂xkdivx u

)rk(ρ,∇xρ|ρ,∇xρ)

+∇xu : H((ρ,∇xρ|ρ,∇xρ)]dx

−∫ρ∇xu : (u− u)⊗ (u− u) dx .

Special choices of the energy functional (2.19) lead to some frequently occuring systems influid dynamics. Some of them are reviewed below:

2.4.1. The Euler system of isentropic gas flow. The choice

E(ρ) =

∫h(ρ) dx , S = −p(ρ)I ,gasdynfuncgasdynfunc (2.32)

p(ρ) = ρh′(ρ)− h(ρ) ,Gibbs-DuhemGibbs-Duhem (2.33)

produces the Euler system of compressible isentropic gas dynamics. The resulting relativeenergy formula coincides with the one computed in [10] and [25].

2.4.2. Special instances of the Euler-Korteweg system. An often used functional, within theframework of the Korteweg theory, is

specialkortspecialkort (2.34) E(ρ) =

∫ (h(ρ) + 1

2κ(ρ)|∇xρ|2)dx .

The formulas (2.25) and (2.27) now become

− ρ∇x(h′(ρ) + 1

2κ′(ρ)|∇xρ|2 − divx(κ(ρ)∇ρ)

)= ∇x · S ,specialformulaspecialformula (2.35)

S =[− p(ρ)− 1

2

(ρκ′(ρ) + κ(ρ)

)|∇xρ|2 +∇x ·

(ρκ(ρ)∇xρ

)]I− κ(ρ)∇xρ⊗∇xρ ,specialkortstressspecialkortstress (2.36)


and the system (2.4) takes the form

specialkortsysspecialkortsys (2.37)

∂ρ

∂t+ divx(ρu) = 0

∂

∂t(ρu) + divx(ρu⊗ u) = ∇x

(− p(ρ)− 1

2(ρκ′(ρ) + κ(ρ))|∇xρ|2 + divx(ρκ(ρ)∇xρ

))− divx(κ(ρ)∇xρ⊗∇xρ)− ζρu ,

where p = ρh′ − h. As shown later in Lemma 3.7 the convexity of the functional (2.34) isequivalent to the hypotheses

h′′(ρ) =p′(ρ)

ρ> 0 , κ(ρ) > 0 , κ(ρ)κ′′(ρ)− 2(κ′(ρ))2 ≥ 0 .

2.4.3. The Navier-Stokes-Korteweg system. The Euler-Korteweg model can be augmentedby viscosity leading to the isothermal Navier-Stokes-Korteweg (NSK) system which is a well-known model for compressible liquid-vapor flows undergoing phase transitions. It is a so-calleddiffuse interface model in which the fields are not discontinuous at the phase boundary, butundergo change smoothly from states in the one phase to states in the other, though usuallysteep gradients do occur. For the choice κ(ρ) = Cκ = const. the NSK model reads

ρt + divx(ρu) = 0

(ρu)t + divx(ρu⊗ u) +∇xp(ρ) = divx(σ[u]) + Cκρ∇x∆xρ ,NSK-secNSK-sec (2.38)

where

def:nss-secdef:nss-sec (2.39) σ[u] := λdivx(u)I + µ(∇xu+ (∇xu)T )

is the Navier-Stokes stress with coefficients λ, µ satisfying µ ≥ 0 and λ + 2dµ ≥ 0. Note that

following (2.26) the third order term in the momentum balance can be written in divergenceform. The potential energy for NSK is the same as for Euler-Korteweg. Adding in viscosityintroduces a dissipative mechanism which actually helps with our relative energy calculationsand which increases regularity of solutions.

The first order part of (2.38) is hyperbolic when p is monotone. The non-monotone pressuremakes the first order part of (2.38) a system of mixed hyperbolic-elliptic type. It should beemphasized, that for (2.38) to describe multi-phase flows it is mandatory that the energydensity h = h(ρ), related to the pressure by (2.33), is non-convex, which makes the pressurenon-monotone. We refer to [15] for a discussion of the thermodynamic structure of (2.38) andits relation to higher-order gradient theories. Also, to [3, 2] for a discussion of the generalstructure of Euler-Korteweg and Navier-Stikes-Korteweg models that relates to several aspectsof our work.

subsec:quantum

2.4.4. The Quantum Hydrodynamics system. Another special case arises when in (2.34) we

set κ(ρ) = 14ε2

ρ , where ε is a constant (the Planck constant). This leads to the energy

qhdenergyqhdenergy (2.40) E(ρ) =

∫ (h(ρ) +

1

8ε2 1

ρ|∇xρ|2

)dx =

∫ (h(ρ) +

1

2ε2|∇x

√ρ|2)dx .

In that case we have the identities

(2.41)

1

2ρ∇x

( 1√ρ

∆x√ρ)

= ρ∇x( 1

8ρ2|∇xρ|2 + divx

( 1

4ρ∇xρ

))(2.35),(2.36)

= ∇x ·(1

4∆xρ I−

1

4ρ∇xρ⊗∇xρ

)


and (2.37) becomes (for ζ = 0) the quantum hydrodynamics system (QHD)

qhdqhd (2.42)

∂ρ

∂t+ divx(ρu) = 0 ,

∂ρu

∂t+ divx(ρu⊗ u) +∇xp(ρ) = ε2

2 ρ∇x

(∆x√ρ

√ρ

).

We refer to [1] for details on the existence theory of finite energy weak solutions of the QHDsystem and detailed references on this interesting subject.

sec-EP2.5. The Euler-Poisson system. As a second application we consider the Euler-Poissonsystem,

euler-poissoneuler-poisson (2.43)

∂ρ

∂t+ divx(ρu) = 0 ,

ρ(∂u∂t

+ (u · ∇x)u)

= −∇xp(ρ) + ρ∇xc ,

−∆xc+ βc = ρ − < ρ > ,

which is often used for describing charged gases in semiconductor devices or under the influ-ence of a gravitational field. Here, p(ρ) is the pressure of the gas of charged particles and∇xc is the electrostatic force induced by the charged particles. The constant β ≥ 0 is oftenreferred as screening constant while < ρ > stands for the average charge. The problem can beset either on the torus Td or on Rd. Here, we restrict on the case of the torus Td and leave tothe reader to provide the necessary modifications for Rd. Also, we restrict to the frictionlesscase (ζ = 0), it is straightforward to adapt and include frictional effects. Our objective isto recast (2.43) under the framework of (2.1); as a byproduct we will infer a relative energycalculation.

Consider the functional

epfunc1epfunc1 (2.44)E(ρ) =

∫ (h(ρ)− 1

2ρc)dx ,

where c is the solution of −∆xc+ βc = ρ − < ρ > ,

with < ρ >=∫ρdx, β ≥ 0. The elliptic equation (2.44)2 is solvable on Td. It has a unique

solution for β > 0 while the solution is given in terms of an arbitrary constant for β = 0.This constant plays no role in determining the electrostatic force ∇xc and might be precisedby requiring

∫c = 0 for β = 0. With these remarks one may express the solution operator in

terms of the Green’s function using the convolution

c(x) = (K ∗ ρ)(x) =

∫K(x− y)ρ(y) dy ,

where the kernel K is a symmetric function. Then E is expressed in the equivalent form

epfuncepfunc (2.45) E(ρ) =

∫ (h(ρ)− 1

2ρ(K ∗ ρ))dx .


We now compute, using the symmetry of K, the directional derivative of E ,

firstvarepfirstvarep (2.46)

dE(ρ;ψ) =d

dτE(ρ+ τψ)

∣∣∣τ=0

=

∫ (h′(ρ)ψ − 1

2ψ(K ∗ ρ)− 12ρ(K ∗ ψ)

)dx

=

∫(h′(ρ)−K ∗ ρ)ψ dx

=⟨δEδρ

(ρ), ψ⟩.

We conclude that δEδρ (ρ) = h′(ρ) − K ∗ ρ and thus, if the pressure function p(ρ) is connected

with the energy (per unit volume) h(ρ) through the usual thermodynamic relation h′′ = p′

ρ ,

then the Euler-Poisson system is expressed in the form of the Hamiltonian flow (2.1) for theenergy functional (2.45) (or equivalently (2.44)).

Next, we prove

−ρ∇xδEδρ

(ρ) = −ρ∇x(h′(ρ)− c) = divx S ,hamflowephamflowep (2.47)

where

S = −(p(ρ)− 1

2 |∇xc|2 − β

2 c2− < ρ > c

)I−∇xc⊗∇xc .stressepstressep (2.48)

Note that (2.47) validates the hypothesis (2.5) and its weak form (2.10) for the case of theEuler-Poisson system with the stress functional S(ρ) defined by (2.48). To prove (2.47), notethat if (2.43)3 is multiplied by ∇xc then after some rearrangement of terms we obtain

ρ∇xc = ∇x(

12 |∇xc|

2 + β2 c

2+ < ρ > c)− divx(∇xc⊗∇xc) ,

which readily provides, for p′ = ρh′′, the formula

−ρ∇x(h′(ρ)− c) = divx

(−[p(ρ)− 1

2 |∇xc|2 − β

2 c2− < ρ > c

]I−∇xc⊗∇xc

)and expresses the stress in the form (2.48).

Suppose now that (ρ, u) with c = K ∗ ρ and (ρ, u) with c = K ∗ ρ are two solutions ofthe Euler-Poisson system. Our goal is to identify the form that the abstract relative entropyidentity takes for the specific case of the Euler-Poisson functional (2.45). First, using (2.9),(2.45) and (2.46) (for ψ = ρ− ρ) we compute

E(ρ|ρ) =

∫ (h(ρ)− 1

2ρc− h(ρ) + 12 ρc− (h′(ρ)− c)(ρ− ρ)

)dx .

Next, use the formulas

h(ρ|ρ) := h(ρ)− h(ρ)− h′(ρ)(ρ− ρ) ,∫cρ =

∫cρ , since K is symmetric and c = K ∗ ρ, c = K ∗ ρ ,

to recast E(ρ|ρ) in its final form

relpoteeprelpoteep (2.49) E(ρ|ρ) =

∫ (h(ρ|ρ)− 1

2(ρ− ρ)K ∗ (ρ− ρ))dx .


Note that for the case considered here, of an electrically attracting fluid, the relative en-ergy consists of two competing terms, and to exploit the relative energy E(ρ|ρ) additionalconsiderations will be needed. These are undertaken in a companion work [26].

The relative stress functional S(ρ|ρ) defined in (2.13) is computed as follows for the Euler-Poisson case. First, we compute the directional derivative of the stress functional (2.48): forψ a test function, c = K ∗ ρ and C = K ∗ ψ, the variation of S(ρ) is given by

firstvarstressepfirstvarstressep (2.50)

dS[ρ, ψ] =d

dτ

∣∣∣∣∣τ=0

S(ρ+ εψ)

=d

dτ

∣∣∣∣∣τ=0

([− p(ρ+ τψ) + 1

2 |∇xc+ τ∇xC|2 + β2 (c+ τC)2+ < ρ+ τψ > (c+ τC)

]I

− (∇xc+ τ∇xC)⊗ (∇xc+ τ∇xC))

=[− p′(ρ)ψ +∇x(K ∗ ρ) · ∇x(K ∗ ψ) + β(K ∗ ρ)(K ∗ ψ)+ < ψ > K ∗ ρ+ < ρ > K ∗ ψ

]I

−∇x(K ∗ ψ)⊗∇x(K ∗ ρ)−∇x(K ∗ ρ)⊗∇x(K ∗ ψ)

=: 〈δSδρ

(ρ), ψ〉 .

Using (2.48) and (2.50) in (2.13) we obtain, after rearranging the termsrelstresseprelstressep (2.51)

S(ρ|ρ) = S(ρ)− S(ρ)−⟨δSδρ

(ρ), ρ− ρ⟩

=(− p(ρ|ρ) + 1

2 |∇x(c− c)|2 + β2 (c− c)2+ < ρ− ρ > (c− c)

)I−∇x(c− c)⊗∇x(c− c) .

Using (2.49), (2.51) we express the relative energy identity (2.18) for the Euler-Poissonsystem,

reltoteepreltoteep (2.52)

d

dt

(∫ (h(ρ|ρ)− 1

2(ρ− ρ)K ∗ (ρ− ρ) + 12ρ|u− u|

2)dx

)=

∫divx u

(− p(ρ|ρ) + 1

2 |∇x(c− c)|2 + β2 (c− c)2+ < ρ− ρ > (c− c)

)dx

−∫∇xu : ∇x(c− c)⊗∇x(c− c) dx−

∫ρ∇xu : (u− u)⊗ (u− u) dx .

sec-LOA2.6. Order parameter model of lower order. Another example of a system fitting intoour framework is the following model introduced in [33]:

∂

∂tρ+ divx(ρu) = 0

∂

∂t(ρu) + divx(ρu⊗ u) +∇x(p(ρ) + Cκ

α

2ρ2) = Cκαρ∇xc

c− 1

α∆xc = ρ ,

lonlon (2.53)

where Cκ is as in (2.38), and α > 0 is a parameter. We will show in Section 5 that forα → ∞ classical solutions of (2.53) endowed with viscosity converge to solutions of (2.38).The motivation for introducing (5.1) in [33] was due to numerical considerations. We will


comment on this briefly at the beginning of Section 5. It should be noted that similar modelswere derived in [4, 32] as models in their own right without reference to NSK.

In [33] the potential energy

loe1loe1 (2.54) E(ρ, c,∇c) :=

∫h(ρ) +

Cκα

2(ρ− c)2 +

Cκ2|∇xc|2,

with p(ρ) = ρh′(ρ) − h(ρ) was considered. It is important to note that in this sense c is avariable which is independent of ρ and without an immediate physical interpretation. Using(2.54) we may write (2.53) as

ρt + divx(ρu) = 0

ρDu

D t= −ρ∇x

δEδρ

δEδc

= 0 .

lo2lo2 (2.55)

However, if we understand (2.53)3 not as an energy minimization condition but as thedefinition of c, similar to what we did in Section 2.5, we may rewrite the energy in thefollowing way which is in the spirit of our presentation of the Euler-Poisson system:

E(ρ) :=

∫h(ρ) +

Cκα

2ρ2 − Cκα

2ρc

with c solving c− 1

α∆xc = ρ ,

loe2loe2 (2.56)

since (2.53)3 implies

(2.57)

∫Cκα

2ρ2 − Cκα

2ρc =

∫Cκα

2(ρ− c)(ρ− c) + c(ρ− c)

=

∫Cκα

2(ρ− c)2 + c(− 1

α∆xc) =

∫Cκα

2(ρ− c)2 +

Cκ2|∇xc|2.

Note that (2.56)2 is nearly the same as (2.44)2.In the representation (2.56) of the potential energy we may express c, if the domain is Rd

or Td, using the Green’s function as c = G ∗ ρ, where G is a symmetric kernel. Using thesymmetry of G we can compute the directional derivative of E ,

firstvarRofirstvarRo (2.58)

dE(ρ;ψ) =d

dτE(ρ+ τψ)

∣∣∣τ=0

=

∫ (h′(ρ)ψ + Cκαρψ − Cκα

2 ψ(G ∗ ρ)− Cκα2 ρ(G ∗ ψ)

)dx

=

∫(h′(ρ) + Cκαρ− CκαG ∗ ρ)ψ dx

=⟨δEδρ

(ρ), ψ⟩.

From (2.58) we obtain

(2.59)δEδρ

= h′(ρ) + Cκα(ρ− c) ,


such that we may rewrite (2.53) as

ρt + divx(ρu) = 0

ρDu

D t= −ρ∇x

δEδρ

c− 1

α∆xc = ρ .

lo3lo3 (2.60)

In order to see that (2.53) fits into our framework it remains to show

−ρ∇xδEδρ

= ∇x · S

with

RohstressRohstress (2.61) S =(− p(ρ)− Cκα

2ρ2 +

Cκα

2c2 +

Cκ2|∇xc|2

)I− Cκ∇xc⊗∇xc .

This is based on the following observation:

αρ∇xc = αc∇xc−∇xc∆xc = ∇x(12αc

2)− divx(∇xc⊗∇xc

)+∇x

(12 |∇xc|

2).

Finally, we like to point out that the potential energy (2.54) can be rewritten as

E(ρ) :=

∫h(ρ) +

Cκ2∇xρ · ∇xc

with c solving c− 1

α∆xc = ρ ,

loe3loe3 (2.62)

since ∫α(ρ2 − ρc) =

∫ρ(−∆xc) =

∫∇xρ · ∇xc .

In (2.62) it is apparent that c converges to ρ, at least for sufficiently smooth ρ, and, thus, theenergy of the model at hand converges to the energy of the Euler-Korteweg model. Indeed,it was shown in [34] that the energy functional of this model Γ-converges to that of theEuler-Korteweg model for α→∞.

sec-hamrem2.7. Remarks on the Hamiltonian structure of the problem. Here we consider thesystem (1.1) and briefly outline an idea, adapted from [2], on the relation of (1.1) and Hamil-tonian systems. Consider the case of three space dimension, d = 3, and using the vectorcalculus formula

(u · ∇x)u =1

2∇x|u|2 − u× curlxu

rewrite (1.1) in the form

funsys-modfunsys-mod (2.63)

∂ρ

∂t= −∇x · (ρu)

∂u

∂t= −∇x

(12 |u|

2 +δEδρ

)+ u× curlxu .

Define the Hamiltonian

hamdefhamdef (2.64) H(ρ, u) = E(ρ) + K(ρ, u) = E(ρ) +

∫1

2ρ|u|2 dx .


Note that the kinetic energy K is viewed here as a functional on (ρ, u); this should becontrasted to (2.15). Then we easily compute the generators of the Gateaux derivatives

δHδρ

=δEδρ

+1

2|u|2 , δH

δu= ρu

and rewrite (2.63) as

hamiltonianformhamiltonianform (2.65)∂

∂t

(ρ

u

)=

(0 −divx−∇x 0

)( δHδρδHδu

)+

(0

1ρδHδu × curlx(1

ρδHδu )

).

The operator

J =

(0 −divx−∇x 0

)is skew adjoint and the system (2.65) is Hamiltonian whenever curlxu = 0, but it has anadditional term when curlu 6= 0. This additional term does not affect the conservation ofenergy as can be seen starting from (2.65) via the following formal calculation

(2.66)

d

dtH(ρ, u) =

⟨δHδρ, ρt⟩

+⟨δHδu

, ut⟩

= −⟨δHδρ,divx

δHδu

⟩−⟨δHδu

,∇xδHδρ

⟩= 0 .

We note that the kinetic energy functional K(ρ, u) is not convex as a functional in (ρ, u), butit becomes convex when viewed as a functional in the variables (ρ,m) in (2.15).

3. Weak-strong stability for the Euler-Korteweg systemsec:kort

In this section we consider the Euler-Korteweg system

eq:Kortcap2eq:Kortcap2 (3.1)

ρt + divx(ρu) = 0

(ρu)t + divx(ρu⊗ u) = −ρ∇x(Fρ(ρ,∇xρ)− divx Fq(ρ,∇xρ)

),

where ρ ≥ 0 is the density, u the velocity, m = ρu stands for the momentum variable, andF (ρ, q) a smooth function standing for the potential energy. As noted in (2.25) we have

−ρ∇x(Fρ − divx Fq

)= divx S , whereformulakortewegformulakorteweg (3.2)

S = (F − ρFρ + ρdivx Fq)I−∇xρ⊗ Fq

=[(F − ρFρ −∇xρ · Fq) + divx

(ρFq)]I−∇xρ⊗ Fq

stresskortstresskort (3.3)

is the stress tensor of the Korteweg fluid.The formula (3.2) enables to express the Euler-Korteweg system in conservative form

Euler-KortEuler-Kort (3.4)

∂tρ+ divxm = 0 ,

∂tm+ divx

(m⊗mρ

)= divx S ,

where S is given by (3.3), m = ρu, and to define weak solutions (see Definition 3.1 beow).


subsec:energyKort3.1. Energy estimate. We briefly review here the pointwise balance of the total (internaland kinetic) energy

∂t

(1

2

|m|2

ρ+ F (ρ,∇xρ)

)+ divx

(1

2m|m|2

ρ2+m

(Fρ(ρ,∇xρ)− divx

(Fq(ρ,∇xρ)

))+ Fq(ρ,∇xρ) divxm

)= 0eq:entrKortGen2eq:entrKortGen2 (3.5)

for smooth solutions of (3.1).To do that, we proceed along the lines of the derivation of (2.6), (2.7) and (2.8), but

proceeding now for the concrete right hand side of (3.1) and keeping track of the relevantfluxes. Indeed, a simple computation shows that the potential energy satisfies the equation

∂tF (ρ,∇xρ) + divx

(m(Fρ − divx Fq)− Fqρt

)= m · ∇x(Fρ − divx Fq) ,

the kinetic energy satisfies the equation

∂t(12ρ|u|

2) + divx(u1

2ρ|u|2)

= −ρu · ∇x(Fρ − divx Fq) ,

and adding the two leads to (3.5).In the sequel we often refer to the simplest structure for a capillary fluid, arising from

selecting in (2.19) the internal energy density

F (ρ, q) = h(ρ) +1

2κ(ρ)|q|2,

and associated to the stress tensor

S =

(−p(ρ)− 1

2

(ρκ′(ρ)− κ(ρ)

)|∇xρ|2 + ρ divx

(κ(ρ)∇xρ

))I− κ(ρ)∇xρ⊗∇xρeq:SigmaGeneq:SigmaGen (3.6)

with p(ρ) = ρh′(ρ)− h(ρ). The energy equation (3.5) then takes the form:(3.7)

∂t

(12

|m|2

ρ+ h(ρ) + 1

2κ(ρ)|∇xρ|2)

+ divx

(12m|m|2

ρ2+m

(h′(ρ) + 1

2κ′(ρ)|∇xρ|2 − divx(κ(ρ)∇xρ)

)+ κ(ρ)∇xρdivxm

)= 0 .

subsec:weakdissKort3.2. Relative energy estimate for weak solutions. Next, we consider the Euler-Kortewegsystem (3.4) and proceed to compare a weak solution (ρ,m) with a strong solution (ρ, m) viaa relative energy computation. For simplicity, we will focus on periodic solutions, defined inTd the d-dimensional torus. A similar analysis can be performed for solutions in Rd, but isomitted here. We recall:

def:wksol Definition 3.1. (i) A function (ρ,m) with ρ ∈ C([0,∞);L1(Td)), m ∈ C(([0,∞);

(L1(Td)

)d),

ρ ≥ 0, is a weak solution of (3.4), if m⊗mρ , S ∈ L1

loc

(((0,∞)× Td)

)d×d, and (ρ,m) satisfy

(3.8)

−∫∫

ρψt +m · ∇xψdxdτ =

∫ρ(x, 0)ψ(x, 0)dx , ∀ψ ∈ C1

c

([0,∞);C1

p(Td))

;

−∫∫

m · ϕt +m⊗mρ

: ∇xϕ− S : ∇xϕdxdt =

∫m(x, 0) · ϕ(x, 0)dx ,

∀ϕ ∈ C1c

([0,∞);

(C1p(Td)

)d).


(ii) If, in addition, 12|m|2ρ + F (ρ,∇xρ) ∈ C([0,∞);L1(Td)) and it satisfies

eq:Kortcap2Integreq:Kortcap2Integr (3.9)−∫∫ (

1

2

|m|2

ρ+ F (ρ,∇xρ)

)θ(t) ≤

∫ (1

2

|m|2

ρ+ F (ρ,∇xρ)

) ∣∣∣t=0

θ(0)dx ,

for any non-negative θ ∈W 1,∞[0,∞) compactly supported on [0,∞),

then (ρ,m) is called a dissipative weak solution.

(iii) By contrast, if 12|m|2ρ +F (ρ,∇xρ) ∈ C([0,∞);L1(Td)) and it satisfies (3.9) as an equality,

then (ρ,m) is called a conservative weak solution.

There is no complete agreement in the scientific community about the nature of weaksolutions for (3.1), that is whether one should refer to conservative or dissipative weak so-lutions. It appears the answer might depend on the method of construction of solutions(whether they are obtained by a limiting viscosity mechanism or arise from the nonlinearSchroedinger equation as in the case of the QHD system). In any case, here we will allow forboth eventualities.

We place the following assumptions:

(H1) (ρ,m) is a dissipative (or conservative) weak periodic solution of (3.1) with ρ ≥ 0 inthe sense of Definition 3.1, and have finite total energy according to

supt∈(0,T )

∫Tdρdx ≤ K1 <∞ ,hypCauchyK1hypCauchyK1 (3.10)

supt∈(0,T )

∫Td

1

2

|m|2

ρ+ F (ρ,∇xρ) dx ≤ K2 <∞ ,hypCauchyK2hypCauchyK2 (3.11)

(H2) (ρ, u) : (0, T ) × Td → Rd+1 is a strong conservative periodic solution of (3.1) withρ ≥ 0 and m = ρu. Note that the regularity “strong” refers to the requirement that the

derivatives ∂ρ∂t ,

∂2ρ∂xi∂xj

, ∂3ρ∂xi∂xj∂xk

as well as ∂ui∂t , ∂ui

∂xjand ∂2ui

∂xi∂xjare in L∞

((0, T )×Td

).

Note that for (3.11) to induce useful bounds a convexity condition is imposed on thefunction F (ρ, q) and the associated energy. The same condition is also required for therelative energy identity to be exploited later. Instead of specifyng hypotheses on F , we willassume at this stage that the weak solution enjoys the regularity

smoothnessasssmoothnessass (A) ρ ∈ C([0, T ];L1(Td)) and ∇xρ ∈ C([0, T ];L1(Td))

and proceed to establish the relative energy identity under (H1), (H2) and (A).

theo:relenweak Theorem 3.2. Assume that hypotheses (H1), (H2) and (A) hold. Then,

∫Td

(1

2ρ∣∣∣mρ− m

ρ

∣∣∣2 + F (ρ,∇xρ| ρ,∇xρ))dx∣∣∣t≤∫Td

(1

2ρ∣∣∣mρ− m

ρ

∣∣∣2 + F (ρ,∇xρ| ρ,∇xρ)

)dx∣∣∣0

eq:RelEnKorGenFinalweakeq:RelEnKorGenFinalweak (3.12)

−∫∫

[0,t)×Td

[ρ

(m

ρ− m

ρ

)⊗(m

ρ− m

ρ

): ∇x

(m

ρ

)+ divx

(m

ρ

)s(ρ,∇xρ |ρ,∇xρ)

]dxdt

−∫∫

[0,t)×Td

[∇x(m

ρ

): H(ρ,∇xρ |ρ,∇xρ) +∇x divx

(m

ρ

)· r(ρ,∇xρ |ρ,∇xρ)

]dxdt ,


where

s(ρ, q) = ρFρ(ρ, q) + q · Fq(ρ, q)− F (ρ, q) ;

r(ρ, q) = ρFq(ρ, q) ;

H(ρ, q) = q ⊗ Fq(ρ, q) .

A derivation of this formula via direct calculation valid for solutions of the Euler-Kortewegsystem that are both smooth is provided in Appendix B. Here, we relax the smoothness of oneof the two solutions by using an argument that validates the formal variational calculationsof Section 2 for the case of the Korteweg energy (2.19).

Proof. Let (ρ,m) be a weak dissipative (or conservative) solution and (ρ, u) with m = ρu astrong conservative solution. We introduce in (3.9) the choice of test function

testthetaStestthetaS (3.13) θ(τ) :=

1, for 0 ≤ τ < t,t−τε + 1, for t ≤ τ < t+ ε,

0, for τ ≥ t+ ε,

and let ε ↓ 0; we then obtain

lem:ret1lem:ret1 (3.14)

∫Td

(1

2

|m|2

ρ+ F (ρ,∇xρ)

)dx

∣∣∣∣tτ=0

≤ 0 .

The same argument applied now to the strong conservative solution (ρ, m) gives

lem:retlem:ret (3.15)

∫Td

(1

2

|m|2

ρ+ F (ρ,∇xρ)

)dx

∣∣∣∣tτ=0

= 0 .

Next, we justify the calculations that relate to the linear part of the relative energy, startingfrom the weak formulation for the equations satisfied by the differences (ρ− ρ,m− m):

−∫∫

[0,+∞)×Td

(ψt(ρ− ρ) + ψxi(mi − mi)

)dxdt−

∫Tdψ(ρ− ρ)

∣∣∣t=0

dx = 0 ,weakmassweakmass (3.16)

−∫∫

[0,+∞)×Td

(ϕt · (m− m) + ∂xiϕj

(mimj

ρ− mimj

ρ

)− ∂xiϕj

(Sij − Sij

))dxdt

−∫Tdϕ(x, 0) · (m− m)

∣∣∣t=0

dx = 0 ,weakmomentumweakmomentum (3.17)

where ϕ, ψ are Lipschitz test functions compactly supported in [0, T )× Td.In the above relations we introduce the test functions

ψ = θ(τ)

(Fρ − divx

(Fq)− 1

2

|m|2

ρ2

), ϕ = θ(τ)

m

ρ,


with θ(τ) as in (3.13). We first introduce the test function ψ above in (3.16) and let ε → 0to arrive at

∫Td

(Fρ(ρ− ρ) + Fq · ∇x(ρ− ρ)− 1

2

|m|2

ρ2(ρ− ρ)

)∣∣∣∣∣t

τ=0

dx

−∫∫

[0,t)×Td

[∂τ

(Fρ −

1

2

|m|2

ρ2

)(ρ− ρ) + ∂τ

(Fq)· ∇x(ρ− ρ)

]dxdτ

−∫∫

[0,t)×Td∇x(Fρ − divx

(Fq)− 1

2

|m|2

ρ2

)· (m− m)dxdτ = 0 .weakmass3weakmass3 (3.18)

Note that we can do that provided the weak solution ρ(·, t) ∈ W 1,1(Td) for each t ∈ [0,∞).This is the minimal regularity required to give meaning to the calculation, taking advantageof the fact that (ρ, u) is a strong solution. Similarly, from (3.17) we get

∫Td

m

ρ· (m− m)

∣∣∣tτ=0

dx =

∫∫[0,t)×Td

∂τ

(m

ρ

)· (m− m)dxdτ

+

∫∫[0,t)×Td

∂xi

(mj

ρ

)((mimj

ρ− mimj

ρ

)− (Sij − Sij)

)dxdτ .weakmomentum2weakmomentum2 (3.19)

Adding (3.18) and (3.19) and using the equation

∂tu+ (u · ∇x)u = −∇x(Fρ − divx Fq

)


and the chain rule for the Lipschitz solution (ρ, u), we obtain after some lengthy but straight-forward calculations

∫Td

[Fρ(ρ− ρ) + Fq · ∇x(ρ− ρ)− 1

2

|m|2

ρ2(ρ− ρ) +

m

ρ· (m− m)

]∣∣∣∣∣t

τ=0

dx

= −∫∫

[0,t)×Td

(Fρρ(divx m)(ρ− ρ) + Fρq · (∇x divx m)(ρ− ρ)

+ Fqρ · ∇x(ρ− ρ) divx m+ Fqjqi∂xj (ρ− ρ)∂xi(divx m))dxdτ

−∫∫

[0,t)×Td∂xi (uj) (Sij − Sij)dxdτ

+

∫∫[0,t)×Td

∇x(Fρ − divx Fq

)· (ρu− ρu)dxdτ

+

∫∫[0,t)×Td

[∂τ

(− 1

2 |u|2)

(ρ− ρ)−∇x(

12 |u|

2)· (ρu− ρu) + ∂τ (u) · (ρu− ρu)

+ ∂xi (uj) (ρuiuj − ρuiuj)]dxdτ

= −∫∫

[0,t)×Td

(Fρρ(divx m)(ρ− ρ) + Fρq · (∇x divx m)(ρ− ρ)

+ Fqρ · ∇x(ρ− ρ) divx m+ Fqjqi∂xj (ρ− ρ)∂xi(divx m))dxdτ

+

∫∫[0,t)×Td

∇x(Fρ − divx Fq

)· (ρ− ρ)u dxdτ

−∫∫

[0,t)×Td∇xu : (S − S)−∇xu : ρ(u− u)⊗ (u− u)dxdτ

=: J1 + J2 + J3 .eq:linearcorrweakeq:linearcorrweak (3.20)

Consider next the weak form of (3.2) for the strong solution ρ and for ϕ a vector-valuedtest function

−∫∇x(Fρ − divx Fq) · (ρϕ)dx =

∫Fρ divx(ρϕ) + Fq · ∇x divx(ρϕ)dx = −

∫Sij(ρ)

∂ϕi∂xj

dx .

We take the variational derivative of this formula along the direction of a smooth test functionψ. Using (2.22), (2.29) and recalling (2.28), we obtain

∫Fρρψ divx(ρϕ) + Fqρψ · ∇x divx(ρϕ) + divx(ρϕ)Fρq · ∇xψ +∇x divx(ρϕ) · Fqq∇xψ dx

−∫∇x(Fρ − divx Fq

)· (ψϕ) dx

= −∫ [− (sρψ + sq · ∇xψ)δij +

∂

∂xk

(∂rk∂ρ

ψ +∂rk∂ql

∂ψ

∂xl

)δij −

(∂Hij

∂ρψ +

∂Hij

∂ql

∂ψ

∂xl

)]∂ϕi∂xj

dx .


Now, set ϕ = u and ψ = ρ− ρ to obtainsecondvareksecondvarek (3.21)

J1 + J2 =

∫ [− (sρ(ρ− ρ) + sq · ∇x(ρ− ρ))δij +

∂

∂xk

(∂rk∂ρ

(ρ− ρ) +∂rk∂ql

∂(ρ− ρ)

∂xl

)δij

−(∂Hij

∂ρ(ρ− ρ) +

∂Hij

∂ql

∂(ρ− ρ)

∂xl

)]∂ui∂xj

dx .

Combining (2.23), (2.16), (3.14), (3.15), (3.20) and (3.21) and (2.30) leads to (3.12).

rem:weakjust Remark 3.3. In the above proof, the regularity assumption (A) is needed to justify therelative energy calculation in the framework of weak solutions (ρ,m). Let us also point outthat formal calculations like

eq:weakjusteq:weakjust (3.22) ρdivx(Fq)

= divx(ρFq)−∇xρ · Fq

(needed to justify the passage from (3.3)1 to (3.3)2) can be given sense in weak functionalframeworks in the following standard way: if one manages to justify one side of the identityin the given framework, then one gives meaning to the other side and the product rule byusing a density argument.

As an example of a possible framework, let us consider the case

F (ρ,∇xρ) = h(ρ) +1

2κ(ρ)|∇xρ|2,

and, for the sake of simplicity, let us temporarily confine ourselves to the usual γ–law pressures,which leads to h(ρ) = ργ . In this case, clearly

(∇xρ · Fq

)(t, ·) ∈ L1(Td) being bounded by

the energy and we are left to prove that(ρFq)(t, ·) ∈ L1(Td). Now, for κ(ρ) = const., or

more generally if κ(ρ) is uniformly bounded, let us assume the weak solution is continuousas a function of time with values in the appropriate spaces according to the bounded energycondition in (H1):

ρ ∈ C([0, T ];Lγ(Td)) and ∇xρ ∈ C([0, T ];L2(Td)) .Then,

(ρFq)(t, ·) ∈ L1(Td) is a consequence of ρ(t, ·) ∈ L2(Td) and this comes from ρ(t, ·) ∈

L1(Td) ∩ Lγ(Td) and, if γ < 2, the L2 integrability of ∇xρ and the Gagliardo–Niremberg–

Sobolev inequality. For the general case, clearly we need ρ(t, ·)√κ(ρ(t, ·)) ∈ L2(Td) and the

last relation is satisfied for instance if

κ(ρ) = ρs; s+ 2 ≤ γ .The right hand side of (3.22) is a well defined distribution, which defines in our frameworkalso the term ρdivx

(Fq)

weakly, establishing in turn the needed equality among the twoterms.

Starting from (3.12), we can obtain stability estimates provided the relative potential energyF (ρ,∇xρ |ρ,∇xρ) controls from above the terms s(ρ,∇xρ |ρ,∇xρ), H(ρ,∇xρ |ρ,∇xρ) andr(ρ,∇xρ |ρ,∇xρ). Keeping in mind the following framework of energies:

eq:generenlasteq:generenlast (3.23) F (ρ,∇xρ) = h(ρ) +1

2κ(ρ)|∇xρ|2,

in the sequel we shall perform this task in two different cases:

• κ(ρ) = Cκ constant and h(ρ) = ργ (more general functions can be also considered,provided a “γ-law behavior” is assumed; cfr. Lemma 3.4 and [25]); similar results inthis case have been also obtained in [14];


• general energies (3.23), under appropriate assumptions on h(ρ) and κ(ρ) which leadto uniform convexity of F , and in the framework of weak solutions uniformly boundedin L∞.

subsubsec:energysol

3.3. Stability estimates for weak solutions in energy norms and constant capillar-ity. In the case κ(ρ) = Cκ and for ρ ∈ C([0, T ];Lγ(Td)) and ∇xρ ∈ C([0, T ];L2(Td)) (seeRemark 3.3), the relative internal energy is

F (ρ,∇xρ |ρ,∇xρ) = h(ρ |ρ) +1

2Cκ|∇xρ−∇xρ|2,

and the quadratic remainder terms are given by:

H(ρ, q) = Cκq ⊗ q ;

s(ρ, q) = (γ − 1)ργ + Cκ|q|2;

r(ρ, q) = Cκρq .

Hence, we readily obtain∣∣H(ρ,∇xρ |ρ,∇xρ)∣∣ = Cκ

∣∣(∇xρ−∇xρ)⊗ (∇xρ−∇xρ)∣∣ ≤ CF (ρ,∇xρ |ρ,∇xρ) ;

s(ρ,∇xρ |ρ,∇xρ) = (γ − 1)h(ρ |ρ) + Cκ|∇xρ−∇xρ|2 ≤ CF (ρ,∇xρ |ρ,∇xρ) .

Moreover, to estimate

r(ρ,∇xρ |ρ,∇xρ) = Cκ(ρ− ρ)(∇xρ−∇xρ) ,

we recall the following result [25], clearly valid for our function h(ρ) = ργ .

lem:generalconvEuler3d Lemma 3.4. Let h ∈ C0[0,+∞) ∩ C2(0,+∞) satisfy h′′(ρ) > 0 for ρ > 0 and

eq:growthheq:growthh (3.24) h(ρ) =k

γ − 1ργ + o(ργ) , as ρ→ +∞

for some constant k > 0 and for γ > 1. If ρ ∈ Mρ = [δ, R] with δ > 0 and R < +∞, thenthere exist positive constants R0 (depending on Mρ) and C1, C2 (depending on Mρ and R0)such that

eq:hnormeq:hnorm (3.25) h(ρ |ρ) ≥

C1|ρ− ρ|2, for 0 ≤ ρ ≤ R0, ρ ∈Mρ ,

C2|ρ− ρ|γ , for ρ > R0, ρ ∈Mρ .

With Lemma 3.4 at our hands, enlarging if necessary R0 so that |ρ− ρ| ≥ 1 for ρ > R0 andρ ∈Mρ, we conclude h(ρ |ρ) ≥ c0|ρ− ρ|2 for any γ ≥ 2, ρ ≥ 0 and ρ ∈Mρ and hence∣∣r(ρ,∇xρ |ρ,∇xρ)

∣∣ ≤ CF (ρ,∇xρ |ρ,∇xρ) .

Finally, the first quadratic term on the right hand side of (3.12) is clearly bounded in termsof the relative kinetic energy and therefore, denoting

ϕ(t) =

∫Td

(1

2ρ

∣∣∣∣mρ − m

ρ

∣∣∣∣2 + F (ρ,∇xρ |ρ,∇xρ)

)dx

∣∣∣∣∣t

,

a straightforward application of the Gronwall Lemma gives the following result.


th:finalKortconst Theorem 3.5. Let T > 0 be fixed and assume hypotheses (H1) and (H2) hold, and con-sider the energy given in (3.23) with γ ≥ 2 and κ(ρ) = Cκ constant; in particular, ρ ∈C([0, T ];Lγ(Td)) and ∇xρ ∈ C([0, T ];L2(Td)) so that assumption (A) is also verified. More-over, assume that the smooth solution ρ is bounded and away from vacuum, that is ρ ∈ Mρ.Then, the stability estimate

eq:finalKortconsteq:finalKortconst (3.26) ϕ(t) ≤ Cϕ(0), t ∈ [0, T ] ,

holds, where C is a positive constant depending only on T , ρ and its derivatives. In particular,if ϕ(0) = 0 , then

eq:weakstronguniqueq:weakstronguniqu (3.27) supt∈[0,T ]

ϕ(t) = 0 ,

thus implying weak–strong uniqueness for the model and the framework of solutions underconsideration.

rem:gamma=2 Remark 3.6. It is worth to observe that the absence of vacuum for ρ required in the theoremabove is needed solely to apply Lemma 3.4, and thus to use the term h(ρ |ρ) to control the L2

distance between the two solutions. Hence, if we consider the special case γ = 2, for whichh(ρ |ρ) = |ρ − ρ|2, this extra condition is not needed anymore, and we obtain the stabilityestimate (3.26) even in the presence of vacuum for both solutions ρ and ρ. However, theregularity assumption (H2) is still needed, requiring that u is well defined and smooth, andthis might be inconsistent for certain models with the presence of vacuum.

subsubsec:linfsol3.4. Stability estimates for L∞ weak solutions and general capillarity. Let us nowpass to the study of energies (3.23) with non constant, smooth capillarities κ(ρ), and smooth,nonnegative functions h(ρ), that is

hyposphyposp (3.28) F (ρ, q) = h(ρ) +1

2κ(ρ)|q|2 with h(ρ) ≥ 0 and κ(ρ) > 0 .

The assumption κ(ρ) > 0 guarantees coercivity of the energy. It is also related to the convexityof the energy.

lem:Funifconvex Lemma 3.7. Let F (ρ, q) be defined by (3.28).

(i) If

hyp4chyp4c (H4c) h′′(ρ) > 0, κ(ρ) > 0 , κ(ρ)κ′′(ρ)− 2(κ′(ρ))2 ≥ 0 ,

then F is strictly convex for any (ρ, q) ∈ R× R3;(ii) If for some constants α1 > 0 and cκ > 0,

hyp4uhyp4u (H4uc) h′′(ρ) ≥ α1 , κ(ρ) ≥ cκ , κ(ρ)κ′′(ρ)− 2(κ′(ρ))2 ≥ 0 ,

then F is uniformly convex.

Proof. A direct calculation shows that the Hessian matrix of F (ρ, q) is given by

∇2(ρ,q)F (ρ, q) =

(h′′(ρ) + 1

2κ′′(ρ)|q|2 κ′(ρ)q

κ′(ρ)qT κ(ρ)I

)and its eigenvalues solve(

κ(ρ)− λ)d−1

[(κ(ρ)− λ

)(h′′(ρ) +

1

2κ′′(ρ)|q|2 − λ

)− |κ′(ρ)q|2

]= 0 .


Hence λj = κ(ρ) for j = 1, . . . , d− 1, while λd and λd+1 are given by

λ± =1

2

(κ(ρ) + h′′(ρ) +

1

2κ′′(ρ)|q|2

)

± 1

2

√(κ(ρ) + h′′(ρ) +

1

2κ′′(ρ)|q|2

)2

− 4κ(ρ)h′′(ρ)− 2(κ(ρ)κ′′(ρ)− 2|κ′(ρ)|2

)|q|2 .

Note that λi are all real, and that

λdλd+1 = κ(ρ)h′′(ρ) + 12

(κ(ρ)κ′′(ρ)− 2|κ′(ρ)|2

)|q|2.

Hence, the conclusion follows.

rem:forconvexity Remark 3.8. Note that (H4uc) in Lemma 3.7 is satisfied whenever we restrict in the follow-ing simple framework: the weak solution ρ is uniformly bounded, h(ρ) is a uniformly convexfunction, while κ(ρ) = ρs with s ∈ [−1, 0). The assumptions on h and κ include the QuantumHydrodynamics system discussed in Section 2.4.4.

We will obtain a stability estimate in terms of the following distance:

eq:normfineq:normfin (3.29) Φ(t) =

∫Td

(1

2ρ

∣∣∣∣mρ − m

ρ

∣∣∣∣2 + |ρ− ρ|2 + |∇xρ−∇xρ|2)dx

∣∣∣∣∣t

.

th:finalKortconst2 Theorem 3.9. Consider two solutions (ρ,m) and (ρ, m) satisfying the hypotheses (H1) and(H2) respectively for some T > 0 and suppose that (3.28) and (H4uc) hold. Furthermore,assume that ρ is uniformly bounded in L∞, that is 0 ≤ ρ(x, t) ≤ ρmax for a.e. (x, t), andthat the strong solution ρ is bounded away from vacuum, that is ρ ∈ Mρ. Then, the stabilityestimate

eq:finalKortconst2eq:finalKortconst2 (3.30) Φ(t) ≤ CΦ(0), t ∈ [0, T ] ,

holds, where C is a positive constant depending only on T , ρmax, ρ and its derivatives. Inparticular, if Φ(0) = 0 , then

eq:weakstronguniqu2eq:weakstronguniqu2 (3.31) supt∈[0,T ]

Φ(t) = 0 ,

which implies weak–strong uniqueness for the model and the framework of solutions underconsideration.

Proof. First, we check that the framework (3.28)–(H4uc) and ρ ∈ L∞ guarantee sufficientregularity to the weak solution (ρ,m) so that (A) holds and the arguments used in Theorem3.2 are valid for these energies. Since ρ is uniformly bounded in L∞, say 0 ≤ ρ(x, t) ≤ ρmaxfor a.e. (x, t), for capillarities bounded away from zero, namely κ(ρ) ≥ cκ > 0, the finiteenergy condition in (H2) guarantees ∇xρ ∈ L2 and that (A) holds. Moreover,

‖κ(ρ)‖∞ ≤ κ∞ = supξ∈[0,ρmax]

|κ(ξ)|

and thus ∇xρ ·Fq, ρFq ∈ L1 which leads also to (3.22), and the result of Theorem 3.2 is valid.Next, thanks to Lemma 3.7, there exists α > 0 such that∫

T3

(1

2ρ

∣∣∣∣mρ − m

ρ

∣∣∣∣2 + F (ρ,∇xρ |ρ,∇xρ)

)dx

∣∣∣∣∣t

≥ αΦ(t) ,

where Φ(t) is given in (3.29).


Hence, we need to bound the right hand side of the (weak form of the) relative energyrelation (3.12) in terms of Φ(t). To this end, we recall that

H(ρ, q) = κ(ρ)q ⊗ q ;

h(ρ, q) = ρh′(ρ)− h(ρ) +1

2(ρκ′(ρ) + κ(ρ))|q|2 = h(ρ) + κ(ρ)|q|2;

r(ρ, q) = ρκ(ρ)q = κ(ρ)q .

A direct calculation then shows

H(ρ,∇xρ |ρ,∇xρ) = κ(ρ |ρ)∇xρ⊗∇xρ+ κ(ρ)(∇xρ⊗∇xρ−∇xρ⊗∇xρ

)− κ(ρ)

(∇xρ⊗ (∇xρ−∇xρ) + (∇xρ−∇xρ)⊗∇xρ

)= κ(ρ |ρ)∇xρ⊗∇xρ+ κ(ρ)(∇xρ−∇xρ)⊗ (∇xρ−∇xρ)

+(κ(ρ)− κ(ρ)

)(∇xρ⊗ (∇xρ−∇xρ) + (∇xρ−∇xρ)⊗∇xρ

);

s(ρ,∇xρ |ρ,∇xρ) = h(ρ |ρ) + κ(ρ |ρ)|∇xρ|2 + κ(ρ)(|∇xρ|2 − |∇xρ|2

)− 2κ(ρ)∇xρ · (∇xρ−∇xρ)

= h(ρ |ρ) + κ(ρ |ρ)|∇xρ|2 + κ(ρ)|∇xρ−∇xρ|2

+ 2(κ(ρ)− κ(ρ)

)∇xρ · (∇xρ−∇xρ) ;

r(ρ,∇xρ |ρ,∇xρ) = κ(ρ |ρ)∇xρ+(κ(ρ)− κ(ρ)

)(∇xρ−∇xρ) .

Thus we can estimate the quadratic terms on the right hand side of (3.12) in terms of thedifferences |ρ− ρ|2 and |∇xρ−∇xρ|2:∣∣H(ρ,∇xρ |ρ,∇xρ)

∣∣, |s(ρ,∇xρ |ρ,∇xρ)|,∣∣r(ρ,∇xρ |ρ,∇xρ)

∣∣ ≤ C|ρ− ρ|2 + C|∇xρ−∇xρ|2,where the constant C depends solely on the smooth functions h and κ, the uniform boundsof ρ, that is ρmax, and ρ and its derivatives. Finally, as in the previous framework, the righthand side of (3.12) is bounded in terms of Φ and thus the Gronwall Lemma gives the desiredresult.

4. Stability estimates for non-convex energiessec:cd

In this section we study the model (3.1) on (0, T ) × T3 in case the local part h(ρ) ofthe internal energy is non-convex. Subsequently, we assume h ∈ C3((0,∞), [0,∞)), but noconvexity of h. We will see that the higher order terms compensate for the non-convex h, inthe sense that we are still able to use (a modified version of) the relative energy to obtaincontinuous dependence on initial data. To simplify the analysis we restrict ourselves to thecase that κ(ρ) = Cκ > 0. We are convinced that analogous results also hold for ρ dependentcapillarity in case K is bounded from above and below.

4.1. Assumptions. We are not (yet) able to carry out the subsequent analysis for weaksolutions and, thus, we will consider strong solutions of (3.1) with κ(ρ) = Cκ in the spaces

ρ ∈ C0([0, T ], C3(T3,R+)) ∩ C32 ((0, T ), C0(T3,R+)) ,

u ∈ C0([0, T ], C1(T3,R3)) ∩ C1((0, T ), C0(T3,R3)) .ass11ass11 (4.1)

We will compare solutions (ρ, u) and (ρ, u) corresponding to initial data (ρ0, u0) and (ρ0, u0),respectively, by relative energy.


To this end, we need to assume some uniform bounds. In particular, vacuum has to beavoided uniformly.

Assumption 4.1 (Uniform bounds). We assume that there are constants Cρ, cρ, cm > 0 suchthat

cρ ≤ ρ(t, x), ρ(t, x) ≤ Cρ ∀ t ∈ [0, T ], x ∈ T3,

cm ≥ max‖m‖L∞((0,T )×T3) , ‖m‖L∞((0,T )×T3)

,

ass12ass12 (4.2)

and define the following constants: pM := ‖p(ρ)‖C2([cρ,Cρ]) , hM := ‖h(ρ)‖C3([cρ,Cρ]) .

4.2. Continuous dependence on initial data. Let us state the particular form of therelative energy and relative energy flux between solutions (ρ,m) and (ρ, m) of (3.1) withthe choice F (ρ, q) = h(ρ) + Cκ|q|2. For notational convenience, we omit capillarity relatedcontributions in the relative energy flux:

η(ρ,m|ρ, m

)= h(ρ)− h(ρ)− h′(ρ)(ρ− ρ) +

Cκ2|∇x(ρ− ρ)|2 +

ρ

2|u− u|2,

q(ρ,m|ρ, m

)= mh′(ρ) +

|m|2

2ρ2m−mh′(ρ) +

|m|2

2ρ2m− m ·m

ρρm+

p(ρ)

ρm− p(ρ)

ρm .

def:re2NSKdef:re2NSK (4.3)

The main challenge we are faced with in this chapter is the non-convexity of h whichensues that the relative energy is not suitable for measuring the distance between solutions.However, if h vanished, the relative energy would be suitable for controlling the differencebetween solutions. To be more precise, the relative kinetic energy allows us to control u− uand

∫|∇x(ρ − ρ)|2 is equivalent to the squared H1 distance between ρ and ρ provided ρ, ρ

have the same mean value, due to Poincare’s inequality. We restrict ourselves to the case

meanmean (4.4)

∫T3

(ρ0 − ρ0) dx = 0

such that Poincare’s inequality is applicable. Therefore, we like to introduce the followingquantity, which is part of the relative energy:

def:rreNSKdef:rreNSK (4.5) ηR

(ρ,m|ρ, m

):=

Cκ2|∇x(ρ− ρ)|2 + k(ρ,m|ρ, m) ,

where k(ρ,m|ρ, m) is the density of the relative kinetic energy K(ρ,m|ρ, m) introduced in(2.16). We call ηR the reduced relative energy.

Due to the properties of the relative kinetic energy we obtain two estimates for the reducedrelative energy

ηR

(ρ,m|ρ, m

)≥ Cκ

2CP‖ρ− ρ‖2H1 +

cρ2‖u− u‖2L2 ,

ηR

(ρ,m|ρ, m

)≥ Cκ

4CP‖ρ− ρ‖2H1 +

Cκc2ρ

8CP c2m

‖m−m‖2L2 .

eq:er1eq:er1 (4.6)

where CP is the Poincare constant on T3.Based on the relative energy and the relative energy flux we can make the computations

for the general case, given in (2.31), more specific:

lem:freNSK Lemma 4.2 (Rate of the relative energy). Let T > 0 be given and let (ρ,m) and (ρ, m) bestrong solutions of (3.1), with κ(ρ) = Cκ > 0, corresponding to initial data (ρ0,m0) and


(ρ0, m0), respectively. Let (4.2) and (4.4) hold. Then, the rate (of change) of the relativeenergy defined in (4.3) satisfies

(4.7)d

d t

∫T3

η(ρ,m|ρ, m

)dx =

∫T3

4∑i=1

Ai dx ,

with

A1 := −Cκm

ρ· (ρ− ρ)

(∇x∆x(ρ− ρ)

),

A2 :=∇xρρ⊗ u :

((u− u)⊗ (ρu− ρu) +

1

3I(p(ρ)− p(ρ)− p′(ρ)(ρ− ρ)

)),

A3 :=1

ρ∇xm : (ρu− ρu)⊗ (u− u) ,

A4 := −divx(m)

ρ

(p(ρ)− p(ρ)− p′(ρ)(ρ− ρ)

).

(4.8)

Next, we determine the rate of the reduced relative energy based on Lemma 4.2.

lem:rreNSK Lemma 4.3 (Rate of the reduced relative energy). Let the assumptions of Lemma 4.2 be sat-isfied. Then, the rate of the reduced relative energy defined in (4.5) satisfies

(4.9)d

d t

∫T3

ηR

(ρ,m|ρ, m

)dx =

∫T3

6∑i=1

Ai dx ,

where A1, . . . , A4 are as in Lemma 4.2 and

A5 := divx(m)(h′(ρ)− h′(ρ)− h′′(ρ)(ρ− ρ)

),

A6 :=(h′(ρ)− h′(ρ)

)(divx(m)− divx(m)

).

(4.10)

Proof. By definition it holds

(4.11) (ηR − η)(ρ,m|ρ, m

)= −h(ρ) + h(ρ) + h′(ρ)(ρ− ρ) = −

∫ ρ

ρ(ρ− s)h′′(s) d s .

Therefore,

∂

∂t(ηR − η)

(ρ,m|ρ, m

)=ρt(ρ− ρ)h′′(ρ)−

∫ ρ

ρρth′′(s) d s

=− divx(m)(ρ− ρ)h′′(ρ) + divx(m)

∫ ρ

ρh′′(s) d s

=− divx(m)(ρ− ρ)h′′(ρ) + divx(m)(h′(ρ)− h′(ρ))

= divx(m)(h′(ρ)− h′(ρ)− (ρ− ρ)h′′(ρ))

+ (h′(ρ)− h′(ρ))(divx(m)− divx(m)) = A5 +A6.

eq:re4NSKeq:re4NSK (4.12)

The assertion of the lemma follows upon combining (4.12) and Lemma 4.2.

lem:destNSK Lemma 4.4 (Estimate of the reduced relative energy rate). Let the assumptions of Lemma4.2 be fulfilled and let the initial data ρ0, ρ0 satisfy (4.4). Then, there exists a constant C > 0


depending only on T,Cκ, ρ0, u0, cρ, Cρ, cm such that the rate of change of the reduced relativeenergy fulfills

grregrre (4.13)d

d t

∫T3

ηR

(ρ,m|ρ, m

)dx ≤ C

∫T3

ηR

(ρ,m|ρ, m

)dx .

Proof. The proof is based on Lemma 4.3 and estimates of the Ai. In order to keep the notationmanageable we suppress the t dependency of all quantities in the proof. For example ‖u‖C2

refers to ‖u‖C0([0,T ],C2(T3)) which is bounded and depends on ρ0,m0, T, Cκ, only. Let usestimate the Ai one by one: For A1 we conclude using (2.26) and integration by parts

∫T3

A1 dx

=

∫T3

−Cκu · (∇x∆xρ−∇x∆xρ)(ρ− ρ) dx

=

∫T3

Cκu · divx

(((ρ− ρ)∆x(ρ− ρ)− |∇x(ρ− ρ)|2

2

)I +∇x(ρ− ρ)⊗∇x(ρ− ρ)

)dx

=Cκ

∫T3

divx(u)(

(ρ− ρ)∆x(ρ− ρ) +|∇x(ρ− ρ)|2

2

)−∇xu : ∇x(ρ− ρ)⊗∇x(ρ− ρ) dx

≤Cκ ‖∇x divx(u)‖L∞ CP |ρ− ρ|2H1 +3Cκ

2‖∇xu‖L∞ |ρ− ρ|2H1

≤5 ‖u‖C2 CP

∫T3

ηR

(ρ,m|ρ, m

)dx .

est:a2Nest:a2N (4.14)

The summand A2 can be estimated as follows:

∫T3

A2 dx =

∫T3

∇xρρ⊗ u : (u− u)⊗ (ρu− ρu) +

1

3

∇xρρ· u(p(ρ)− p(ρ)− p′(ρ)(ρ− ρ)) dx

≤‖∇xρ⊗ u‖L∞

cρ

(∫T3

ρ|u− u|2 dx+ pM ‖ρ− ρ‖2L2

)≤‖∇xρ‖L∞ ‖u‖L∞

cρ

4pMCPCκ

∫T3

ηR

(ρ,m|ρ, m

)dx .


Concerning A3 we find

A3 =ρ

ρ∇xm : (u− u)⊗ (u− u)

such that

est:a4Nest:a4N (4.16)∣∣ ∫

T3

A3 dx∣∣ ≤ 2

∥∥∥∥∇xmcρ∥∥∥∥L∞

∫T3

ηR

(ρ,m|ρ, m

)dx .

The estimates for A4 and A5 are straightforward, i.e.,∣∣ ∫T3

A4 dx∣∣ ≤ 1

cρ‖divx m‖L∞ pM

CPCκ

∫T3

ηR

(ρ,m|ρ, m

)dx ,

∣∣ ∫T3

A5 dx∣∣ ≤ ‖divx m‖L∞ hM

CPCκ

∫T3

ηR

(ρ,m|ρ, m

)dx .



Using integration by parts we find for A6

est:a7N1est:a7N1 (4.18)

∫T3

A6 dx =

∫T3

(h′′(ρ)∇xρ− h′′(ρ)∇xρ

)·(m− m

)dx

=

∫T3

(h′′(ρ)∇xρ− h′′(ρ)∇xρ

)·(m− m

)dx+

∫T3

(h′′(ρ)∇xρ− h′′(ρ)∇xρ

)·(m− m

)dx

such that

est:a7N2est:a7N2 (4.19)∣∣ ∫

T3

A6 dx∣∣ ≤ hM ‖ρ‖C1 ‖ρ− ρ‖L2 ‖m− m‖L2 + hM |ρ− ρ|H1 ‖m− m‖L2 ,

which implies

est:a7N3est:a7N3 (4.20)∣∣ ∫

T3

A6 dx∣∣ ≤ hM (‖ρ‖C1 + 1)

CP c2m

8c2ρCκ

∫T3

ηR

(ρ,m|ρ, m

)dx ,

due to (4.6). The assertion of the Lemma follows upon combining (4.14), (4.15), (4.16), (4.17),and (4.20).

Now we are in position to state and prove the main result of this section: strong solutionsto (3.1) with κ(ρ) = Cκ depend continuously on their initial data, provided they satisfy theuniform bounds (4.2).

thrm:mt1 Theorem 4.5 (Stability). Let T,Cκ > 0 be given and let (ρ, u) and (ρ, u) be strong solutionsof (3.1), with κ(ρ) = Cκ > 0, corresponding to initial data (ρ0, u0) and (ρ0, u0), respectively.Let (4.2) and (4.4) hold. Then, there exists a constant C = C(T,Cκ, ρ0, u0, cρ, Cρ, cm) > 0such that the following estimate is satisfied

(4.21)Cκ2‖ρ− ρ‖2L∞(0,T ;H1(T3)) +

cρ2‖u− u‖2L∞(0,T ;L2(T3)) ≤ CηR

(ρ0,m0|ρ0, m0

)dx .

Proof. The assertion of the theorem follows by applying Gronwall’s Lemma to (4.13) andcombining the result with (4.6)1.

Remark 4.6 (Viscosity). Note that we could also add viscous terms into (3.1) such that weobtained (2.38). The arguments presented here can be extended easily to that case. In thecase with viscosity the results from [23] guarantee the existence of strong solutions for shorttimes.

rem:cap1 Remark 4.7 (Capillarity). The constant C in Theorem 4.5 depends on Cκ like exp(1/Cκ)at best, as can be seen from the estimates of the Ai in the proof of Lemma 4.4. In particular,the constant blows up for Cκ → 0.

5. Model convergencesec:mc

In this section we employ the relative energy framework to show that the isothermal Navier-Stokes-Korteweg model (2.38) is indeed approximated by the lower order model introducedin [33], which is given in (1.14). Note that (2.38) is the model investigated in the previoussection plus viscosity. Before we present our analysis let us digress a bit in order to justifyour interest in the relation between the two models. Numerical schemes for the isothermalNSK system (2.38) have been considered by several authors, see [20, 5, 35] and referencestherein. In these works the main effort was directed at overcoming stability issues, which aremainly caused by the non-convexity of the energy. Several of the approaches for constructingnumerical schemes were based on Runge-Kutta-discontinuous Galerkin type discretisations.


However, the number of numerical flux functions which may be used is severely restricted bythe non-hyperbolicity of the first order part of (2.38) which is caused by h being non-convex.

Moreover, discrete energy inequalities for explicit-in-time schemes cannot be proven bystandard arguments from hyperbolic theory. Indeed, a non-monotone behavior of the energyis observed in numerical experiments. To overcome these problems the following family ofapproximations, parametrized in α > 0, of the NSK system was introduced in [33]

ραt + divx(ραuα) = 0

(ραuα)t + divx(ραuα ⊗ uα) +∇x(p(ρα) + Cκα

2(ρα)2) = divx(σ[uα]) + Cκαρ

α∇xcα

cα − 1

α∆xc

α = ρα,

lolo (5.1)

where cα is an auxiliary variable without any immediate physical interpretation. It is astriking feature of (5.1) that the first order part of (5.1)1,2 forms a hyperbolic system forρα, mα, provided α is sufficiently large. Numerical studies showing that (5.1) offers numericaladvantages over (2.38) and that solutions of (5.1) are similar to those of (2.38) can be found in[29]. In particular, examples are presented in [29] which show that explicit-in-time schemes for(5.1) have far better stability properties than explicit-in-time schemes for (2.38). Variationalproblems related to minima of the energy functional of (5.1), see Lemma 5.6, were investigatedin [4, 34]. Based on formal arguments it was conjectured in [33] that for α → ∞ solutionsof (5.1) converge to solutions of (2.38). Results in this direction were obtained for similarmodels describing elastic solids in [16, 19]. The result in [16] is obtained using compactnessarguments, while [19] is based on a (technically simpler) version of the arguments presentedhere. The main result of this section, Theorem 5.14, is an estimate for the difference betweensolutions of (5.1) and (2.38).

5.1. Assumptions on well-posedness and uniform bounds. We complement (2.38),(5.1) with initial data

ρ(0, ·) = ρ0, u(0, ·) = u0 in T3,

ρα(0, ·) = ρα0 , uα(0, ·) = uα0 in T3,icic (5.2)

for given data ρα0 , ρ0 ∈ C3(T3, (0,∞)) and uα0 , u0 ∈ C2(T3,R3) which we assume to be relatedas follows ∫

T3

ρα0 − ρ0 dx = 0, ‖ρα0 − ρ0‖H1(T3) = O(α−1/2),

‖ρα0 ‖H3(T3) = O(1), ‖uα0 − u0‖L2(T3) = O(α−1/2).

ass:icass:ic (5.3)

Concerning the viscous part of the stress we will require that there is bulk viscosity, i.e.,

viscvisc (5.4) λ+2

3µ > 0.

The well-posedness of (5.1) was studied in [33] for two space dimensions on the whole ofR2. We will assume (local-in-time) existence of strong solutions to (5.1) and (2.38) posed onT3. In particular:


Assumption 5.1 (Regularity). We assume that there is some T > 0 such that strong solu-tions of (5.1) and (2.38) exist and satisfy

ρ ∈ C0([0, T ], C3(T3,R+)) ∩ C1((0, T ), C1(T3,R+)),

u ∈ C0([0, T ], C2(T3,R3)) ∩ C1((0, T ), C0(T3,R3)),

ρα ∈ C0([0, T ], C1(T3,R+)) ∩ C1((0, T ), C0(T3,R+)),

uα ∈ C0([0, T ], C2(T3,R3)) ∩ C1((0, T ), C0(T3,R3)),

cα ∈ C0([0, T ], C3(T3,R+)) ∩ C1((0, T ), C2(T3,R+)).

ass1ass1 (5.5)

rem:reg Remark 5.2 (Regularity). Note that the regularity assumed in (5.5) coincides with the reg-ularity asserted in [33] and [23], for T small enough. Therefore, for appropriate T , the onlyassumptions made here are that the change from natural to periodic boundary conditionsdoes not deteriorate the regularity of solutions and that the time of existence of solutions of(5.1) does not go to zero for α → ∞. The existence of a space derivative of ρt follows fromthe mass conservation equation and the regularity of ρ, u.

Assumption 5.3 (Uniform bounds). We assume that there are constants α0, Cρ, cρ, cm > 0such that

cρ ≤ ρ(t, x), ρα(t, x) ≤ Cρ ∀ t ∈ [0, T ], x ∈ T3, α > α0,

cm ≥ sup‖mα‖L∞((0,T )×T3) , ‖m‖L∞((0,T )×T3).ass2ass2 (5.6)

Remark 5.4 (Uniform a-priori estimates). The crucial assumption in (5.6) is that these esti-mates hold uniformly in α, while analogous estimates for fixed α are immediate for sufficientlysmall times and appropriate initial data. We are not aware that there is any mechanism in(5.1) which makes (5.6) unlikely to hold. While it would be desirable to prove (5.6) this isbeyond the scope of this work.

rem:max Remark 5.5 (A-priori estimate on cα). Note that the maximum principle applied to thescreened Poisson equation (5.1)3 immediately implies

cρ ≤ cα(t, x) ≤ Cρ ∀ t ∈ [0, T ], x ∈ T3, α > α0,

once we assume (5.6).

Note that, using (5.1)3, (5.1)2 can be rewritten as

(5.7) (ραuα)t + divx(ραuα ⊗ uα) +∇xp(ρα) = divx(σ[uα]) + Cκρα∇x∆xc

α.

Moreover, (5.1) conserves momentum as can be seen from (2.61).As already noted in [33] solutions of (5.1) satisfy an energy inequality. In order to keep

this paper self contained we state the energy inequality and sketch its proof.

lem:rel Lemma 5.6 (Energy balance for (5.1)). Let (ρα, uα, cα) be a strong solution of (5.1). Then,the following energy balance law is satisfied:

(5.8) 0 ≥ −σ[uα] : ∇xuα =(h(ρα) +

ρ

2|uα|2 +

Cκ2|∇xcα|2 +

αCκ2|ρα − cα|2

)t

+ divx(uα(ραh′(ρα) +

1

2ρα |uα|2−Cκρα∆xc

α)−Cκcαt ∇xcα +Cκρ

α∇xρα ·∇xuα−σ[uα]uα)


and

divvadivva (5.9) ‖divx uα‖2L2([0,T ]×T3) ≤

1

λ+ 23µ

∫T3

h(ρα0 ) +ρα02|uα0 |

2 +Cκα

2|ρα0 − cα0 |

2 +Cκ2|cα0 |

2 dx.

Proof. Equation (5.1)1 is multiplied by h′(ρα)− 12 |u

α|2 +αCκ(ρα−cα) and (5.1)2 is multipliedby uα. Then, both equations are added and integration by parts is used. The first assertionof the Lemma follows upon noting that

∇xcα · ∇xcαt − αCκcαt (ρα − cα)− divx(Cκcαt ∇xcα) = 0.

The second assertion of the Lemma is a result of the first assertion and the identity

dissipdissip (5.10) σ[u] : ∇xu = (λ+2

3µ)(divx u)2 + 2µ∇oxu : ∇oxu ≥ 0,

where ∇ox denotes the trace free part of the Jacobian.

5.2. Elliptic approximation. In this section we study properties of the screened Poissonoperator in (5.1)3. To quantify the approximation of ρα by cα cite the following result from[19]. The solution operator to Id− 1

α∆x on T3 is denoted by Gα :

lem:ea Lemma 5.7 (Elliptic approximation, [19]). The operator Gα has the following properties:

(a) For any f ∈ L2(T3) the following estimate is fulfilled

est:gest:g (5.11) ‖Gα[f ]‖L2(T3) ≤ ‖f‖L2(T3).

(b) For any k ∈ N and f ∈ Hk(T3) it is Gα[f ] ∈ Hk+2(T3).(c) For all f ∈ H1(T3) the following holds:

eq:diveq:div (5.12) Gα[fx] = (Gα[f ])x and ‖f −Gα[f ]‖2L2(T3) ≤2

α|f |2H1(T3).

(d) In case f ∈ H2(T3) the following (stronger) estimate is satisfied:

‖f −Gα[f ]‖2L2(T3) ≤1

α2|f |2H2(T3).

rem:divva Remark 5.8. As ‖ρα0 ‖H1 is bounded by (5.3), assertion (c) of Lemma 5.7 implies that theinitial energy of the lower order model is bounded independent of α. Due to (5.9) this impliesthat ‖divx u

α‖L2([0,T ]×T3) is bounded independent of α.

5.3. Relative energy. In this section we study the relative energy and relative energy fluxbetween a solution (ρ, v) of (2.38) and a solution (ρα, uα, cα) of (5.1). They are based on theenergies and energy fluxes of the systems (5.1) and (2.38) determined in Lemmas 3.15 and5.6, but we omit the higher order terms, i.e., those depending on Cκ, in the relative energyflux:

ηα := h(ρα) +Cκ2|∇xcα|2 +

αCκ2|ρα − cα|2 − h(ρ)− Cκ

2|∇xρ|2

− h′(ρ)(ρα − ρ)− Cκ∇xρ · ∇x(cα − ρ) +ρα

2|u− uα|2,

qα := mαh′(ρα) +|mα|2

2(ρα)2mα −mh′(ρ)− |m|

2

2ρ2m− h′(ρ)(mα −m)

+|m|2

2ρ2(mα −m)− m

ρ·(mα ⊗mα

ρα+ p(ρα)− m⊗m

ρ− p(ρ)

).

def:redef:re (5.13)


Several terms in (5.13) cancel out, such that

ηα = h(ρα)− h(ρ)− h′(ρ)(ρα − ρ) +Cκ2|∇x(cα − ρ)|2 +

αCκ2|ρα − cα|2 +

ρα

2|u− uα|2,

qα = mαh′(ρα) +|mα|2

2(ρα)2mα −mαh′(ρ) +

|m|2

2ρ2m− m ·mα

ραρmα − p(ρα)

ρm+

p(ρ)

ρm.

def:re2def:re2 (5.14)

As in Section 4 the relative energy is not suitable for measuring the distance betweensolutions and, again, we introduce the reduced relative energy and the relative kinetic energy:

ηαR :=Cκ2|∇xcα −∇xρ|2 +

αCκ2|ρα − cα|2 +Kα,

Kα :=ρα

2|u− uα|2.

def:rredef:rre (5.15)

Note that calculations analogous to the derivation of (4.6) imply∫T3

ηαR dx ≥ Cκ2|cα − ρ|2H1 +

αCκ2‖cα − ρα‖2L2 +

ρα

2‖u− uα‖2L2 ,∫

T3

ηαR dx ≥ Cκ4|cα − ρ|2H1 +

αCκ4‖cα − ρα‖2L2 +

Cκc2ρ

8CP c2m

‖m−mα‖2L2 .

eq:er12eq:er12 (5.16)

Based on the relative energy and the relative energy flux we can show the following estimatewhose proof is based on the same principles as the derivation of (B.10). However, an additionaldifficulty is presented by the fact that the functions which are compared solve different PDEs.

lem:fre Lemma 5.9 (Rate of the relative energy). Let T > 0 be given such that there are strongsolutions (ρ, u) and (ρα, uα, cα) of (2.38) and (5.1), respectively, satisfying (5.5) and (5.6).Then, the rate (of change) of the relative energy ηα defined in (5.13) fulfills

(5.17)d

d t

∫T3

ηα dx =

∫T3

6∑i=1

Aαi dx,

with

Aα1 :=divx σ[u]

ρ· (u− uα)(ρα − ρ) + (u− uα) · (divx σ[u]− divx σ[uα]),

Aα2 := −Cκm

ρ· ∇x∆x(ρ− cα)(ρ− ρα),

Aα3 :=∇xρρ⊗ u :

((u− uα)⊗ (ραu− ραuα) +

1

3I(p(ρα)− p(ρ)− p′(ρ)(ρα − ρ)

)),

Aα4 :=1

ρ∇xm : (ραuα − ραu)⊗ (u− uα),

Aα5 := −divx(m)

ρ

(p(ρα)− p(ρ)− p′(ρ)(ρα − ρ)

),

Aα6 := Cκ(ραt − cαt )∆xρ.

(5.18)

The proof of this Lemma is given in Appendix C.

Remark 5.10. Note that all but two of the Aαi in Lemma 5.9 correspond to terms in Lemma4.2. The term Aα1 is due to viscosity and Aα6 is due to the different regularizations. These arethe only terms having no counterparts in Lemma 4.2.


Our next step is to derive a representation of the rate of the reduced relative energy fromLemma 5.9.

lem:rre Lemma 5.11 (Rate of the reduced relative energy). Let the assumptions of Lemma 5.9 besatisfied. Then, the rate of the reduced relative energy ηαR defined in (5.15) satisfies

(5.19)d

d t

∫T3

ηαR dx =

∫T3

8∑i=1

Aαi dx,

where A1, . . . , A6 are as in Lemma 5.9 and

Aα7 := divx(m)(h′(ρα)− h′(ρ)− h′′(ρ)(ρα − ρ)

),

Aα8 :=(h′(ρα)− h′(ρ)

)(divx(mα)− divx(m)

).

(5.20)

The proof of Lemma 5.11 is analogous to the proof of Lemma 4.3.

lem:dest Lemma 5.12 (Estimate of the reduced relative energy rate). Let the assumptions of Lemma5.9 be satisfied. Then, there exist a constant C > 0 and a function E ∈ L1(0, T ), bothindependent of α > α0, such that the rate (of change) of the reduced relative energy satisfiesthe following estimate

grreNSKgrreNSK (5.21)d

d t

∫T3

ηαR dx ≤ C∫T3

ηαR dx+E

α.

Proof. The proof is based on Lemma 5.11 and estimates of the Aαi . For brevity we suppress thet dependency of all quantities in the proof. The terms Aα3 , A

α4 , A

α5 , A

α7 can be estimated anal-

ogous to the corresponding terms in the proof of Lemma 4.4. Let us estimate the remainingAαi one by one: We have, using integration by parts,

est:a1Nest:a1N (5.22)∫T3

Aα1 dx ≤‖u‖C2

cρ‖u− uα‖L2(T3) ‖ρ− ρ

α‖L2(T3) −∫T3

(∇xu−∇xuα

)(σ[u]− σ[uα]) dx

≤‖u‖C2

cρ‖u− uα‖L2(T3) ‖ρ− ρ‖L2(T3)

≤‖u‖C2

cρmax

1

cρ,4CPCκ

∫T3

ηαR


as, by the properties of the Lame coefficients, ∇x(uα−u) : (σ[uα]−σ[u]) ≥ 0. For Aα2 we findusing (2.26)

∫T3

Aα2 dx

=

∫T3

−Cκu · (∇x∆xρ−∇x∆xcα)(ρ− ρα) dx

=

∫T3

−Cκ(ρ− cα)u · ∇x∆x(ρ− cα)− Cκ(cα − ρα)u · ∇x∆x(ρ− cα) dx

=Cκ

∫T3

−u · divx

(((ρ− cα)∆x(ρ− cα) +

|∇x(ρ− cα)|2

2

)I−∇x(ρ− cα)⊗∇x(ρ− cα)

)− (cα − ρα)u · ∇x∆xρ+ (cα − ρα)u · ∇x∆xc

α dx

=Cκ

∫T3

divx(u)(

(ρ− cα)∆x(ρ− cα) +|∇x(ρ− cα)|2

2

)−∇xu : ∇x(ρ− cα)⊗∇x(ρ− cα)

− (cα − ρα)u · ∇x∆xρ+1

α∆xc

αu · ∇x∆xcα dx

≤Cκ ‖∇x divx(u)‖L∞ ‖ρ− cα‖2H1 +3Cκ

2‖∇xu‖L∞ ‖ρ− cα‖2H1

+ Cκ‖cα − ρα‖L2 ‖ρ‖C3 ‖u‖C0 −∫T3

Cκα

divx(u)(∆xcα)2 dx

≤3 ‖u‖C2 Cκ ‖ρ− cα‖2H1 + Cκ‖cα − ρα‖L2 ‖ρ‖C3 ‖u‖C0 + Cκα ‖u‖C1 ‖ρα − cα‖2L2

≤(6 ‖u‖C2 CP +

2

α‖ρ‖C3 ‖u‖C0

)ηαR.

est:a21est:a21 (5.23)

For Aα6 we find

est:a4est:a4 (5.24)

∫T3

Aα6 dx = Cκ

∫T3

∆xρ(cαt − ραt ) dx

= Cκ

∫T3

∆xρ(Gα[divx(mα)]− divx(mα)) dx = −Cκ∫T3

∇x∆xρ · (Gα[mα]−mα) dx

= −Cκ∫T3

∇x∆xρ · (Gα[m]−m) dx− Cκ∫T3

∇x∆xρ · (Gα[mα −m]− (mα −m)) dx

such that elliptic regularity for the operator Gα implies

est:a41est:a41 (5.25)∣∣ ∫

T3

Aα6 dx∣∣ ≤ ‖ρ‖C3

1

α|m|H2 + 2Cκ ‖ρ‖C3 ‖mα −m‖L2

≤ ‖ρ‖C3

1

α|m|H2 + 2Cκ ‖ρ‖C3

16CP c2m

c2ρ

ηαR,

see Lemma 5.7. In order to estimate Aα8 we decompose it as −Aα8 = Aα81 +Aα82 with

Aα81 =(h′(ρ)− h′(cα)


),

Aα82 =(h′(cα)− h′(ρα)


).

(5.26)


Using integration by parts we find

est:a81est:a81 (5.27)

∫T3

Aα81 dx =

∫T3

(h′′(ρ)∇xρ− h′′(cα)∇xcα

)·(m−mα

)dx

=

∫T3

(h′′(ρ)∇xρ−h′′(cα)∇xρ

)·(m−mα

)dx+

∫T3

(h′′(cα)∇xρ−h′′(cα)∇xcα

)·(m−mα

)dx

such that

est:a811est:a811 (5.28)∣∣ ∫

T3

Aα81 dx∣∣ ≤ hM ‖ρ‖C1 ‖ρ− cα‖L2 ‖mα −m‖L2 + hM |ρ− cα|H1 ‖mα −m‖L2 ,

because of Remark 5.5. We infer

est:a812est:a812 (5.29)∣∣ ∫

T3

Aα81 dx∣∣ ≤ hM (‖ρ‖C1 + 1)

CP c2m

4Cκc2ρ

ηαR,

due to (5.16). Finally, we turn to Aα82 and obtain

Aα82 = (h′(cα)− h′(ρα))ρα divx uα + (h′(cα)− h′(ρα))∇xρα · uα − (h′(cα)− h′(ρα)) divxm

such that, using Young’s inequality,

est:a821est:a821 (5.30)∣∣ ∫

T3

Aα82 dx∣∣ ≤ Cκα

2‖ρα − cα‖2L2(T3)

+2hMαCκ

(‖divxm‖2L2(T3) + Cρ ‖divx u

α‖2L2(T3)

)+∣∣ ∫

T3

Aα83 dx∣∣,

where Aα83 := (h′(cα) − h′(ρα))∇xρα · uα, for α sufficiently large and some constant Cρ > 0,according to (5.6). We have, using integration by parts,∣∣ ∫

T3

Aα83 dx∣∣ =

∣∣ ∫T3

h′(cα)∇x(ρα − cα) · uα −∇x(h(ρα)− h(cα)) · uα dx∣∣

≤ Cκα

2‖ρα − cα‖2L2(T3) +

(3h2M

Cκα‖∇xcα‖2L2(T3) +

2

Cκα‖divx u

α‖2L2(T3)

).

est:a83est:a83 (5.31)

Note that Cκα2 ‖ρ

α − cα‖2L2(T3) ≤ ηαR.The assertion of the Lemma follows upon combining (5.23), (5.25), (5.29),(5.30), and (5.31)

because of (5.5), (5.6), and Remark 5.8.

rem:bc Remark 5.13 (Boundary conditions). If we used natural boundary conditions instead of pe-riodic ones, we would obtain additional (non-vanishing) boundary terms in the estimate ofAα6 , and it is not clear how to estimate them properly.

thrm:mt2 Theorem 5.14 (Model convergence). Let T, µ, Cκ > 0 and λ ∈ R be fixed such that (5.4) issatisfied. Let initial data (ρ0, u0) ∈ H3(T3)×H2(T3) be given such that a strong solution (ρ, u)of (2.38) exists in the sense of (5.5). Let the sequence (ρα0 , u

α0 ) be such that (5.3) is satisfied

and that strong solutions (ρα, uα, cα) of (5.1) exist in the sense of (5.5). Let, in addition,(5.6) be satisfied. Then, there exists a constant C > 0 depending only on T, λ, µ, Cκ, ρ0, u0

such that for sufficiently large α the following estimate holds:

Cκ2|ρ− cα|2L∞(0,T ;H1(T3)) +

Cκ2‖ρα − cα‖2L∞(0,T ;L2(T3)) +

cρ2‖uα − u‖2L∞(0,T ;L2(T3)) ≤

C

α.


Proof. Integrating (5.21) in time we obtain

(5.32)

∫T3

ηαR(t, x) dx ≤ C∫ t

0

∫T3

ηαR(s, x) dx d s+

∫T3

ηαR(0, x) dx+

∫ t

0

|E(s)|α

d s

such that Gronwall’s inequality implies

GwGw (5.33)

∫T3

ηαR(t, x) dx ≤(∫

T3

ηαR(0, x) dx+

∫ t

0

|E(s)|α

d s)eCt.

In order to infer the assertion of the theorem from (5.33) we need to estimate the integral ofηαR(0, ·). Because of (5.2) we have

re0re0 (5.34)

∫T3

ηαR(0, x) dx ≤ αCκ2‖cα(0, ·)− ρα0 ‖

2L2(T3)

+Cκ2|cα(0, ·)− ρ0|2H1(T3) + Cρ ‖uα0 − u0‖2L2(T3) .

By definition cα(0, ·) = Gα[ρα0 ] such that

‖cα(0, ·)− ρα0 ‖2L2(T3) ≤

1

α2|ρα0 |

2H2(T3) ,

|cα(0, ·)− ρ0|2H1(T3) ≤2

α2|ρα0 |

2H3(T3) + ‖ρα0 − ρ0‖2H1(T3) .

eregereg (5.35)

Inserting (5.35) and (5.3) into (5.34) we obtain

re5re5 (5.36)

∫T3

ηαR(0, x) dx = O(α−1).

Upon using (5.36) in (5.33) we find

Gw1Gw1 (5.37)

∫T3

ηαR(t, x) dx ≤(Cα

+1

α‖E‖L1(0,t)

)eCt

for some constant C > 0, independent of α. Equation (5.37) implies the assertion of thetheorem as

Gw2Gw2 (5.38)Cκ2|ρ(t, ·)− cα(t, ·)|2H1(T3) +

Cκ2‖ρα(t, ·)− cα(t, ·)‖2L2(T3)

+cρ2‖uα(t, ·)− u(t, ·)‖2L2(T3) ≤

∫T3

ηαR(t, x) dx.

rem:cap2 Remark 5.15 (Parameter dependence). All the estimates derived in this section heavily relyon the capillary regularization terms which are scaled with Cκ. In particular, the estimatesTheorems 4.5 and 5.14 depend sensitively on Cκ. Indeed, the constants C in Theorems 4.5and 5.14 scale like exp(1/Cκ) for Cκ → 0. Thus, the results established here are only helpfulin the diffuse case Cκ > 0 and cannot be transferred to the sharp interface limit case Cκ → 0.


Appendix A. Noether’s theorem and the Korteweg stressapp-noether

In this section we show that the relation (2.5) (or (2.25)) for the Euler-Korteweg system(2.24) can be seen as a direct consequence of Noether’s theorem and the invariance of theKorteweg functional

kortfunctionalkortfunctional (A.1) E(ρ) =

∫ΩF (ρ,∇xρ) dρ

under translations. This relation is pointed out by Benzoni-Gavage [3] and is proved underthe hypothesis that ρ(·) is an extremum of E(ρ) under the constraint of prescribed mass

∫ρ.

Here, we will show that if Ω is a set that makes the functional invariant under translations,that is E(ρ(· + h)) = E(ρ(·)), then Noether’s theorem implies (2.25) without any further as-sumptions on ρ(·). Note that the invariance under translations holds if Ω is the d-dimensionaltorus Td or the entire space Rd.

Indeed, the proof of this is a direct application of Noether’s theorem on (A.1) (see Gelfandand Fomin [18, Sec 37]). Following the outline of Sec 37, we note that the functional (A.1) isinvariant under the transformation

specialvarspecialvar (A.2)

x∗i = xi +

∑k εkδik i = 1, ..., d

ρ∗ = ρ

The first variation of the functional E(ρ) along variations compatible with (A.2) is zero.Applying Theorem 1 in [18, Sec 37.4], which computes the first variation of E(ρ) on a generaldomain, we see that the invariance implies

− ∂ρ

∂xk

(Fρ −

∑i

∂

∂xiFqi

)−∑i

∂

∂xi

(Fqi

∂ρ

∂xk− Fδik

)= 0

After an integration by parts, this formula readily yields

formulaappformulaapp (A.3) −ρ ∂

∂xk

(Fρ −

∑i

∂

∂xiFqi

)=∑i

∂

∂xi

((F − ρFρ + ρdivx Fq

)δik − Fqi

∂ρ

∂xk

)which coincides with (2.25) or (2.5).

Appendix B. The relative energy transport identity for the Euler-Kortewegsystem

subsec:relenKort

We present here the relative entropy calculation for the Euler-Korteweg system when both(ρ, m) and (ρ, m) are smooth solutions of (3.1); the framework of weak solution (ρ, m) isdiscussed in Section 3.2. This calculation will determine explicitly the relative-energy flux (aterm omitted in the integral version valid for weak solutions).

In what follows, we often omit the dependence of the potential energy

F (ρ,∇xρ) = h(ρ) +1

2κ(ρ)|∇xρ|2,

the stress

S = −(ρFρ(ρ,∇xρ)− F (ρ,∇xρ)− ρdivx(Fq(ρ,∇xρ))

)I− Fq(ρ,∇xρ)⊗∇xρ

and their derivatives on the variables ρ and ∇xρ, and we denote with F and S these quantitieswhen evaluated for the solution ρ.


We start by analyzing the linear part of the potential energy about ρ, that is:

Fρ(ρ− ρ) + Fq · ∇x(ρ− ρ) =(Fρ − divx

(Fq))

(ρ− ρ) + divx(Fq(ρ− ρ)

).

Using the equation satisfied by the difference (ρ− ρ), after some calculation we obtain

∂t

(Fρ(ρ− ρ) + Fq · ∇x(ρ− ρ)

)+ divx

((Fρ − divx(Fq)

)(m− m) + Fq divx(m− m)

)= ∇x

(Fρ − divx(Fq)

)· (m− m)−

(Fρρ divx m+ Fρq · ∇x(divx m)

)(ρ− ρ)

− Fqρ · ∇x(ρ− ρ) divx m− Fqjqi∂xj (ρ− ρ)∂xi(divx m) .eq:linearKorGen1eq:linearKorGen1 (B.1)

Using

eq:potenKortGeneq:potenKortGen (B.2) ∂tF (ρ,∇xρ) + divx

(m(Fρ − divx(Fq)

)+ Fq divxm

)= m · ∇x

(Fρ − divx(Fq)

)for both (ρ, m) and (ρ, m) and (B.1), we get

∂tF (ρ,∇xρ |ρ,∇xρ) + divx

(m(Fρ − divx(Fq)− (Fρ − divx(Fq))

)+ (Fq − Fq) divxm

)= m · ∇x

(Fρ − divx(Fq)− (Fρ − divx(Fq))

)+(Fρρ divx m+ Fρq · ∇x(divx m)

)(ρ− ρ) + Fqρ · ∇x(ρ− ρ) divx m

+ Fqjqi∂xj (ρ− ρ)∂xi(divx m)

= ρ

(m

ρ− m

ρ

)· ∇x


)+m

ρ· ρ∇x


)+(Fρρ divx m+ Fρq · ∇x(divx m)


+ Fqjqi∂xj (ρ− ρ)∂xi(divx m)

= D + I1 + I2 ,

where

D = ρ

(m

ρ− m

ρ

)· ∇x


);

I1 =m

ρ· ρ∇x


);

I2 =(Fρρ divx m+ Fρq · ∇x(divx m)


+ Fqjqi∂xj (ρ− ρ)∂xi(divx m).

Recalling

eq:Sigmagenrewriteq:Sigmagenrewrit (B.3) ρ∂xj(Fρ − divx(Fq)

)= −∂xkSkj ,

I1 is rewriten as :

I1 =mj

ρ

(ρ∂xj

(Fρ − divx(Fq)

)− ρ∂xj

(Fρ − divx(Fq)

))− (ρ− ρ)

m

ρ· ∇x(Fρ − divx(Fq))

= −mj

ρ∂xk(Skj − Skj

)− (ρ− ρ)

m

ρ· ∇x(Fρ − divx(Fq)) ,


and therefore

∂tF (ρ,∇xρ |ρ,∇xρ)

+ divx

(m(Fρ − divx(Fq)− (Fρ − divx(Fq))

)+ (Fq − Fq) divxm +

(S − S

)mρ

)= D − ∂xk

(mj

ρ

)((ρFρ − F − ρdivx(Fq)− (ρFρ − F − ρdivx(Fq))

)δkj + Fqk∂xjρ− Fqk∂xj ρ

)− (ρ− ρ)

m

ρ· ∇x(Fρ − divx(Fq)) + I2

= D − I1a − I1b + I2 ,

eq:RelPotEnKorGen2eq:RelPotEnKorGen2 (B.4)

where

I1a = ∂xk

(mj

ρ

)((ρFρ − F − ρ divx(Fq)− (ρFρ − F − ρ divx(Fq))

)δkj + Fqk∂xjρ− Fqk∂xj ρ

);

I1b = (ρ− ρ)m

ρ· ∇x(Fρ − divx(Fq)) .

After some rearrangement of derivatives I2 becomes

I2 = divx

(m

ρ

)(ρ(Fρρ − ∂xi(Fρqi)

)(ρ− ρ) + ρ

(Fρqj − ∂xi(Fqjqi)

)∂xj (ρ− ρ)

− ρ(Fρqi∂xi(ρ− ρ) + Fqjqi∂xixj (ρ− ρ)

))+mj

ρ

(∂xj(Fρ)(ρ− ρ) + ∂xj

(Fql)∂xl(ρ− ρ)− ∂xl

(∂xj ρ

(Fρql(ρ− ρ) + Fqkql∂xk(ρ− ρ)

)))+ divx

((Fρq(ρ− ρ) + Fqq∇x(ρ− ρ)

)divx m

)= I2a + I2b − I2c + I2d ,

where

I2a = divx

(m

ρ

)(ρ(Fρρ − divx(Fρq)

)(ρ− ρ) + ρ

(Fρqj − divx(Fqjq)

)∂xj (ρ− ρ)

− ρ(Fρqi∂xi(ρ− ρ) + Fqjqi∂xixj (ρ− ρ)

));

I2b =mj

ρ

(∂xj(Fρ)(ρ− ρ) + ∂xj

(Fql)∂xl(ρ− ρ)

);

I2c =mj

ρ

(∂xl(∂xj ρ


)));

I2d = divx

((Fρq(ρ− ρ) + Fqq∇x(ρ− ρ)

)divx m

).


We shall now rearrange the various terms defined above as follows:

−I1b + I2b =mj

ρ∂xl∂xj

((ρ− ρ)Fql

)− ∂xl

(mj

ρ∂xj (ρ− ρ)Fql

)+ ∂xl

(mj

ρ

)(Fql∂xj (ρ− ρ)

);

−I2c = −∂xl(mj

ρ∂xj ρ


))+ ∂xl

(mj

ρ

)(∂xj ρFρql(ρ− ρ) + ∂xj ρFqkql∂xk(ρ− ρ)

).

Using the above relations in (B.4) we obtain

∂tF (ρ,∇xρ |ρ,∇xρ) + divx J1 = D

− divx

(m

ρ

)((ρFρ − F − ρdivx(Fq)− (ρFρ − F − ρdivx(Fq))

)− ρ(Fρρ − divx(Fρq)

)(ρ− ρ)− ρ


)∂xj (ρ− ρ)

+ ρFρqi∂xi(ρ− ρ) + ρFqjqi∂xixj (ρ− ρ))

− ∂xl(mj

ρ

)(Fql∂xjρ− Fql∂xj ρ− ∂xj ρFρql(ρ− ρ)− ∂xj ρFqkql∂xk(ρ− ρ)− Fqlδjk∂xk(ρ− ρ)

)+mj

ρ∂xl∂xj

((ρ− ρ)Fql

)− ∂xl

(mj


)= D −R1 −R2 +R3 ,

eq:RelPotEnKorGen3eq:RelPotEnKorGen3 (B.5)

where

J1 = m(Fρ − divx(Fq)− (Fρ − divx(Fq))

)+ (Fq − Fq) divxm

+(S − S

)mρ−(Fρq(ρ− ρ) + Fqq∇x(ρ− ρ)

)divx m

+

(m

ρ· ∇xρ

)(Fρq(ρ− ρ) + Fqq∇x(ρ− ρ)

);

R1 = divx

(m

ρ

)((ρFρ − F − ρ divx(Fq)− (ρFρ − F − ρdivx(Fq))

)− ρ(Fρρ − divx(Fρq)

)(ρ− ρ)− ρ


)∂xj (ρ− ρ)

+ ρFρqi∂xi(ρ− ρ) + ρFqjqi∂xixj (ρ− ρ))

;

R2 = ∂xl

(mj

ρ

)(Fql∂xjρ− Fql∂xj ρ− ∂xj ρFρql(ρ− ρ)− ∂xj ρFqkql∂xk(ρ− ρ)− Fqlδjk∂xk(ρ− ρ)

);

R3 =mj

ρ∂xl∂xj

((ρ− ρ)Fql

)− ∂xl

(mj


).


Now we recall the notations

H(ρ, q) = q ⊗ Fq(ρ, q) ;

s(ρ, q) = ρFρ(ρ, q) + q · Fq(ρ, q)− F (ρ, q) ;

r(ρ, q) = ρFq(ρ, q) .

and we readily obtain

R2 = ∇x(m

ρ

): H(ρ,∇xρ |ρ,∇xρ) ;

ρ(Fρρ − divx(Fρq)

)= sρ(ρ,∇xρ)− divx

(ρFρq

);

ρ(Fρqj − divx(Fqjq)

)= sqj (ρ,∇xρ)− divx

(ρFqjq

).

Therefore

−R1 +R3 = divx

(m

ρdivx

((ρ− ρ)Fq

)−(m

ρ· ∇x(ρ− ρ)

)Fq

)− divx

(m

ρ


+ divx

(m

ρ

)(divx

(ρFq)− divx

(ρFq)− divx

(ρFρq

)(ρ− ρ)− divx

(ρFqjq

)∂xj (ρ− ρ)

− ρFρqi∂xi(ρ− ρ)− ρFqjqi∂xixj (ρ− ρ)− divx((ρ− ρ)Fq

))= −divx J2 − divx

(m

ρ

)s(ρ,∇xρ |ρ,∇xρ)−∇x divx

(m

ρ

)· r(ρ,∇xρ |ρ,∇xρ) ,

where

J2 = −(m

ρdivx

((ρ− ρ)Fq

)−(m

ρ· ∇x(ρ− ρ)

)Fq

)− divx

(m

ρ

)r(ρ,∇xρ |ρ,∇xρ) .

Using these identities in (B.5) we finally express the relative potential energy in the form:

∂tF (ρ,∇xρ |ρ,∇xρ) + divx J = D − divx

(m

ρ


−∇x(m

ρ

): H(ρ,∇xρ |ρ,∇xρ)−∇x divx

(m

ρ

)· r(ρ,∇xρ |ρ,∇xρ) ,eq:RelPotEnKorGenFinaleq:RelPotEnKorGenFinal (B.6)

where

J = J1 + J2

= m(Fρ − divx(Fq)− (Fρ − divx(Fq))

)+ (Fq − Fq) divxmdefJdefJ (B.7)

+(S − S

)mρ−(Fρq(ρ− ρ) + Fqq∇x(ρ− ρ)

)divx m

+

(m

ρ· ∇xρ

)(Fρq(ρ− ρ) + Fqq∇x(ρ− ρ)

)−(m

ρdivx

((ρ− ρ)Fq

)−(m

ρ· ∇x(ρ− ρ)

)Fq

)− divx

(m

ρ

)r(ρ,∇xρ |ρ,∇xρ)

and

D = ρ

(m

ρ− m

ρ

)· ∇x


).


The density of the relative kinetic energy is given by

def:rkedef:rke (B.8)1

2ρ

∣∣∣∣mρ − m

ρ

∣∣∣∣2 .We recall (see also (B.3)) and thus the kinetic energy alone satisfies

∂t

(1

2

|m|2

ρ

)+ divx

(1

2m|m|2

ρ2

)= −m · ∇x

(Fρ − divx(Fq)

).

We write the difference for the equations satisfied by mρ and m

ρ ,

∂t

(m

ρ− m

ρ

)+

(m

ρ· ∇x

)(m

ρ− m

ρ

)+

((m

ρ− m

ρ

)· ∇x

)m

ρ

= −∇x(Fρ − divx(Fq)−

(Fρ − divx(Fq)

)).

and multiply by mρ −

mρ to obtain

1

2∂t

∣∣∣∣mρ − m

ρ

∣∣∣∣2 +

(m

ρ· ∇x

)(1

2

∣∣∣∣mρ − m

ρ

∣∣∣∣2)

+

(mi

ρ− mi

ρ

)(mj

ρ− mj

ρ

)∂xj

(mi

ρ

)= −

(m

ρ− m

ρ

)· ∇x

(Fρ − divx(Fq)−

(Fρ − divx(Fq)

)).

Using the conservation law (3.1)1,

eq:relkinKortGeneq:relkinKortGen (B.9)

∂t

(1

2ρ

∣∣∣∣mρ − m

ρ

∣∣∣∣2)

+ divx

(1

2m

∣∣∣∣mρ − m

ρ

∣∣∣∣2)

= −D − ρ(m

ρ− m

ρ

)⊗(m

ρ− m

ρ

): ∇x

(m

ρ

)Adding (B.6) and (B.9) leads to the identity for the transport of the relative energy,

∂t

(1

2ρ

∣∣∣∣mρ − m

ρ

∣∣∣∣2 + F (ρ,∇xρ |ρ,∇xρ)

)+ divx

(1

2m

∣∣∣∣mρ − m

ρ

∣∣∣∣2 + J

)

= −ρ(m

ρ− m

ρ

)⊗(m

ρ− m

ρ

): ∇x

(m

ρ

)− divx

(m

ρ


−∇x(m

ρ

): H(ρ,∇xρ |ρ,∇xρ)−∇x divx

(m

ρ

)· r(ρ,∇xρ |ρ,∇xρ) ,eq:RelEnKorGenFinaleq:RelEnKorGenFinal (B.10)

where the flux J is defined in (B.7).

Appendix C. Proof of Lemma 5.9sec:proofmc

The sole purpose of this Appendix is giving the details of the proof of Lemma 5.9.


Proof. We start with direct calculations

ηαt =h′(ρα)ραt +mα ·mα

t

ρα− |m

α|2

2(ρα)2ραt + αCκ(ρ− cα)(ραt − cαt )

+ Cκ∇xcα · ∇xcαt − h(ρ)ρt + Cκ∇xρ · ∇xρt − h′′(ρ)ρt(ρα − ρ)− h′(ρ)ραt

+ h′(ρ)ρt +m ·mt

ρ2ρα − |m|

2

ρ3ραρt +

|m|2

ρ2ραt

− Cκ∇xρt · ∇xcα − Cκ∇xρ · ∇xcαt −mt ·mα

ρ− m ·mα

t

ρ+m ·mα

ρ2ρt

eq:re1aeq:re1a (C.1)

and

divx qα = divx(mα)h′(ρα) +mα · ∇xh′(ρα) +

mα ⊗mα

(ρα)2: ∇xmα +

|mα|2

2(ρα)2divx(mα)

− |mα|2

(ρα)3mα · ∇xρα −∇xh′(ρ) ·mα − h′(ρ) divx(mα) +

m⊗mα

ρ2: ∇xm

+|m|2

2ρ2divx(mα)− |m|

2

ρ3mα · ∇xρ−

m⊗mα

ρρα: ∇xmα − mα ⊗mα

ρρα: ∇xm

− m ·mα

ρραdivx(mα) +

m ·mα

ρ2ραmα · ∇xρ+

m ·mα

ρ(ρα)2mα · ∇xρα −

divx(m)

ρp(ρα)

+m · ∇xρρ2

p(ρα)− m · ∇xp(ρα)

ρ+

divx(m)

ρp(ρ)− m · ∇xρ

ρ2p(ρ) +

m · ∇xp(ρ)

ρ.

eq:re1beq:re1b (C.2)

Inserting the evolution equations (2.38) and (5.1) in (C.1) we obtain

ηαt =|mα|2

(ρα)3∇xρ ·mα − mα ⊗mα

(ρα)2: ∇xmα − |m

α|2

(ρα)2divx(mα)− mα · ∇xp(ρα)

ρα+mα

ρα· divx(σ[uα])

+ Cκmα · ∇x∆xc

α +|mα|2

2(ρα)2divx(mα) + Cκ∆xc

α divx(mα) + Cκ divx(∇xcαcαt )

− Cκ∇xρ · ∇x divx(m) + h′′(ρ) divx(m)(ρα − ρ) + |m|2 ρα

ρ4m · ∇xρ−

ρα

ρ3m⊗m : ∇xm

− ρα

ρ3|m|2 divx(m)− ρα

ρ2m · ∇xp(ρ) +

ρα

ρ2m · divx(σ[u]) + Cκ

ρα

ρm · ∇x∆xρ+

ρα

ρ3|m|2 divx(m)

− |m|2

2ρ2divx(mα) + Cκ∇x divx(m) · ∇xcα + Cκ∆xρ(cαt − ραt )− Cκ divx(cαt ∇xρ)

+ Cκ∇xρ · ∇x divx(mα)− m ·mα

ρ3∇xρ ·m+

mα ⊗mρ2

: ∇xm+divx(m)

ρ2m ·mα +

mα

ρ· ∇xp(ρ)

− mα

ρ· divx(σ[u])− Cκmα · ∇x∆xρ−

mα ·mρ(ρα)2

∇xρα ·mα +m⊗mα

ρρα: ∇xmα

+m ·mα

ρραdivx(mα) +

m

ρ· ∇xp(ρα)− m

ρ· divx(σ[uα])− Cκ

ρα

ρm · ∇x∆xc

α − divx(m)

ρ2m ·mα.

eq:re2eq:re2 (C.3)


Adding (C.2) and (C.3) we observe that several terms cancel out and we obtain

ηαt + divx qα

=mα

ρα· divx(σ[uα]) +

ρα

ρ2m · divx(σ[u])− mα

ρ· divx(σ[u])− m

ρ· divx(σ[uα])

+ Cκ divx

((mα −m)∆x(cα − ρ) +∇x(cα − ρ)cαt +∇xρdivx(mα −m) + divx(m)∇xcα

)+ Cκ

(ραρ− 1)m · ∇x∆x(ρ− cα) + Cκ∇xρ · ∇x(ραt − cαt )

+1

ρ2m⊗∇xρ :

(mρ− mα

ρα

)⊗(mρα

ρ−mα

)+

1

ρ2m · ∇xρ

(p(ρα)− p(ρ)− p′(ρ)(ρα − ρ)

)+

1

ρ∇xm :

(mα −mρα

ρ

)⊗(mρ− mα

ρα

)− 1

ρdivx(m)

(p(ρα)− p(ρ)− p′(ρ)(ρα − ρ)

).

eq:re3eq:re3 (C.4)

Due to the periodic boundary conditions, (C.4) implies the assertion of the Lemma.

References

AM09 [1] Paolo Antonelli and Pierangelo Marcati. On the finite energy weak solutions to a system in quantumfluid dynamics. Comm. Math. Physics, 287 (2009), 657-686.

BDDJ05 [2] S. Benzoni-Gavage, R. Danchin, S. Descombes and D. Jamet, Structure of Korteweg models andstability of diffuse interfaces. Interfaces and free boundaries 7 (2005), 371-414.

B10 [3] Sylvie Benzoni-Gavage, Propagating phase boundaries and capillary fluids. Lecture Notes. 3emecycle. Levico, 2010, pp.57. 〈cel − 00582329〉

BLR95 [4] Deborah Brandon, Tao Lin, and Robert C. Rogers. Phase transitions and hysteresis in nonlocaland order-parameter models. Meccanica 30 (1995), 541–565. Microstructure and phase transitionsin solids (Udine, 1994).

BP13 [5] Malte Braack and Andreas Prohl. Stable discretization of a diffuse interface model for liquid-vaporflows with surface tension. ESAIM: Mathematical Modelling and Numerical Analysis 47 (2013),401–420

BT13 [6] Joshua Ballew and Konstantina Trivisa. Weakly dissipative solutions and weak-strong uniquenessfor the Navier-Stokes-Smoluchowski system. Nonlinear Anal. 91 (2013), 1–19.

Br14 [7] Yann Brenier. Topology-preserving diffusion of divergence-free vector fields and magnetic relax-ation. Comm. Math. Phys. 330 (2014), 757–770.

Cha14 [8] Frederic Charve. Local in time results for local and non-local capillary Navier–Stokes systems withlarge data. J. Differential Equations 256 (2014), 2152–2193.

Daf79 [9] C. M. Dafermos. The second law of thermodynamics and stability. Arch. Rational Mech. Anal. 70(1979), 167–179.

Daf79b [10] C. M. Dafermos. Stability of motions of thermoelastic fluids. J. Thermal Stresses 2 (1979), 127–134.Daf86 [11] C. M. Dafermos, Quasilinear hyperbolic systems with involutions, Arch. Rational Mech. Anal. 94

(1986), 373-389.Daf10 [12] C. M. Dafermos. Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren

der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 2010.

Dip79 [13] Ronald J. DiPerna. Uniqueness of solutions to hyperbolic conservation laws. Indiana Univ. Math.J. 28 (1979), 137–188.


DFM15 [14] D. Donatelli, E. Feireisl, and P. Marcati. Well/ill posedness for the Euler–Korteweg–Poisson systemand related problems. Comm. Partial Differential Equations 40 (2015), 1314–1335.

DS85 [15] J. E. Dunn and J. Serrin. On the thermomechanics of interstitial working. Arch. Rational Mech.Anal. 88 (1985), 95–133.

ERV14 [16] P. Engel, A. Viorel, and C. Rohde. A low-order approximation for viscous-capillary phase transitiondynamics. Port. Math. 70 (2014), 319–344.

FN12 [17] E. Feireisl and A. Novotny. Weak-strong uniqueness property for the full Navier-Stokes-Fouriersystem. Arch. Ration. Mech. Anal. 204 (2012), 683–706.

GF63 [18] I.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice Hall, 1963.Gie_14a [19] J. Giesselmann. A relative entropy approach to convergence of a low order approximation to a

nonlinear elasticity model with viscosity and capillarity. SIAM Journal on Mathematical Analysis,46 (2014), 3518–3539.

GMP_14 [20] J. Giesselmann, Ch. Makridakis, and T. Pryer. Energy consistent dg methods for the Navier-Stokes-Korteweg system. Mathematics of Computation 83 (2014), 2071–2099.

hamilton82 [21] R. S. Hamilton. The inverse function theorem of Nash and Moser. Bulletin of AMS 7, 65–222,1982.

JLLJ10 [22] Qiangchang Ju, Hailiang Li, Yong Li and Song Jiang. Quasi-neutral limit of the two-fluid Euler-Poisson system. Commun. Pure Appl. Anal. 9 (2010), 1577–1590.

Kot08 [23] M. Kotschote. Strong solutions for a compressible fluid model of Korteweg type. Ann. Inst. H.Poincare Anal. Non Lineaire 25 (2008), 679–696.

LT06 [24] Corrado Lattanzio and Athanasios E. Tzavaras. Structural properties of stress relaxation andconvergence from viscoelasticity to polyconvex elastodynamics. Arch. Rational Mech. Anal., 180(2006), 449–492.

LT13 [25] Corrado Lattanzio and Athanasios E. Tzavaras. Relative entropy in diffusive relaxation. SIAM J.Math. Anal. 45 (2013), 1563–1584.

LTforth [26] Corrado Lattanzio and Athanasios E. Tzavaras. From gas dynamics with large friction to gradientflows describing diffusion theories (forthcoming).

LV11 [27] Nicholas Leger and Alexis Vasseur. Relative entropy and the stability of shocks and contact discon-tinuities for systems of conservation laws with non-BV perturbations. Arch. Ration. Mech. Anal.201 (2011), 271–302.

LS09 [28] Tao Luo and Joel Smoller. Existence and non-linear stability of rotating star solutions of thecompressible Euler-Poisson equations. Arch. Ration. Mech. Anal. 191 (2009), 447–496.

NRS14 [29] Jochen Neusser, Christian Rohde, and Veronika Schleper. Relaxation of the Navier-Stokes-Korteweg equations for compressible two-phase flow with phase transition. to appear in Internat.J. Numer. Methods Fluids

Ot01 [30] Felix Otto. The geometry of dissipative evolution equations: the porous medium equation. Comm.Partial Differential Equations 26 (2001), 101–174.

PW05 [31] Y. Peng and Y. Wang, Convergence of compressible Euler-Poisson equations to incompressibletype Euler equations, Asymptotic Anal., 41 (2005), 141–160.

RT00 [32] X. Ren and L. Truskinovsky, Finite scale microstructures in nonlocal elasticity. J. Elasticity, 59(2000), 319–355.

Roh10 [33] Christian Rohde. A local and low-order Navier-Stokes-Korteweg system. In Nonlinear partial dif-ferential equations and hyperbolic wave phenomena, volume 526 of Contemp. Math., pages 315–337.Amer. Math. Soc., Providence, RI, 2010.

SV03 [34] Margherita Solci and Enrico Vitali. Variational models for phase separation. Interfaces Free Bound.5 (2003), 27–46.

TXKV14 [35] L. Tian, Y Xu, J.G.M. Kuerten, J.J.W. Van der Vegt, A local discontinuous Galerkin method forthe propagation of phase transition in solids and fluids. J. Sci. Comp. 9(3) (2014), 688–720.


(Jan Giesselmann)Institute of Applied Analysis and Numerical SimulationUniversity of StuttgartPfaffenwaldring 57D-70563 StuttgartGermany

E-mail address: [email protected]

(Corrado Lattanzio)Dipartimento di Matematica Pura ed ApplicataUniversita degli Studi dell’AquilaVia VetoioI 67010 Coppito (L’Aquila) AQItaly


(Athanasios E. Tzavaras)Computer, Electrical, Mathematical Sciences & Engineering DivisionKing Abdullah University of Science and Technology (KAUST)Thuwal, Saudi ArabiaandInstitute for Applied and Computational MathematicsFoundation for Research and TechnologyGR 70013 Heraklion, CreteGreece


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