link.springer.com...Weak Solutions with Density-Dependent Viscosities D. Bresch and B. Desjardins...

Weak Solutions with Density-DependentViscosities

D. Bresch and B. Desjardins

Abstract

In this chapter, we focus on compressible Navier-Stokes equations with density-dependent viscosities in the multidimensional space case. The main objectiveof these notes is to present at the level of beginners an introduction to suchsystems showing the difference with the constant viscosity case. The guidelineis to show a nonlinear hypercoercivity property due to the density dependencyof the viscosities, to explain how it may be used to provide global existenceof weak solutions to the barotropic compressible Navier-Stokes equations, andto the heat-conducting Navier-Stokes equations with a total energy formulation.We will also focus on the relative entropy method for such systems showingthe difficulty coming from the density dependency. We hope to motivate bythis chapter young researchers to work on such difficult topic trying to fill thegap between the constant viscosities case and the density-dependent viscositiessatisfying the BD relation, trying to relax some modeling hypotheses and toextend the results.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Simple Fluid Flow Models and Historical Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 Why Degenerate Viscosities May Help and Why It Seems to be Necessary to

Consider Such Situation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

D. Bresch is partially supported by the ANR- 13-BS01-0003-01 project DYFICOLTI.

D. Bresch (�)LAMA UMR 5127 CNRS Batiment le Chablais, Université de Savoie Mont-Blanc, Le Bourgetdu Lac, Francee-mail: [email protected].

B. DesjardinsFondation Mathématique Jacques Hadamard, CMLA, ENS Cachan, CNRS and ModélisationMesures et Applications S.A., Paris, Francee-mail: [email protected]

© Springer International Publishing AG 2017Y. Giga, A. Novotny (eds.), Handbook of Mathematical Analysis in Mechanicsof Viscous Fluids, DOI 10.1007/978-3-319-10151-4_44-1

1

mailto:[email protected].

mailto:[email protected]

2 D. Bresch and B. Desjardins

3.1 Model Behavior When the Density Vanishes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Density and Velocity Fluctuations Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Basic a Priori Estimates, Identities, and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.1 Stability Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Construction of Approximate Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 The �-entropy and Two-Velocities Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.1 The Barotropic Compressible System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 The Heat-Conducting Compressible System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Relative Entropy and Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

AMS Classifications. 35Q30, 35D30, 54D30, 42B37, 35Q86, 92B05.

1 Introduction

The aim of this chapter is to describe some recent progress related to the math-ematical analysis of the Navier-Stokes equations for compressible viscous andheat-conducting fluids in the case of density-dependent viscosity coefficients.

Some basic notations of fluid dynamics variables are introduced together witharguments motivating the analysis of density-dependent viscosities. Next, propertiesof the so-called BD entropy are detailed in the framework of barotropic flows,i.e., in the case when pressure does not depend on temperature. Such results arethen extended to the full system including heat-conducting equations, for whichthe construction of suitable approximate solutions is described. More detailedintroduction on fluid dynamics models and related mathematical results of Navier-Stokes equation can be found in monographs [26,44,45,51] and papers [8,9,12,13]to name a few.

In order to avoid technicalities and additional mathematical difficulties (some ofthem being completely unsolved so far) associated to boundaries or infinite spaces,the space domain will be � D T

d where d D 2 or 3, i.e., a d-dimensional box withperiodic boundary conditions.

A compressible heat-conducting fluid in � governed by the Navier-Stokesequations satisfies the following system

8

<̂

:̂

@t�C div .�u/ D 0;

@t .�u/C div .�u˝ u/Crp D div � C �f;

@t .�E/C div .�uH/ D div .� � u/C div .�r�/C �f � u;

(1)

where u 2 Rd denotes the velocity field of the fluid, � the density, � the temperature,

� the thermal conductivity coefficient, � the viscous stress tensor, p the pressure

Weak Solutions with Density-Dependent Viscosities 3

field, e the specific internal energy, and h D e C p=� the specific enthalpy.The specific total energy and the associated specific total enthalpy are denoted,respectively, by

E D e Cjuj2

2and H D hC

juj2

2:

Equations (1) express the conservation of mass, momentum, and total energy,respectively. In order to close this system, two additional ingredients are necessary.First, the fluid is assumed to be Newtonian, i.e., there exist two viscosity coefficients�.�; �/ and �.�; �/ such that the stress tensor expresses as

D � � pIRn D 2�D.u/C .� div u � p/IRn

where D.u/ D .ruCrut /=2 denotes the strain rate tensor defined as thesymmetric part of the velocity gradient ru and IRn the identity tensor. As a secondcondition, a thermodynamical closure law defines the pressure p and the internalenergy e as functions of the density � and the temperature � :

p D P.�; �/; e D E.�; �/: (2)

For perfect gases, the pressure law and the internal energy read p D �R� ande D Cv� , R and Cv being positive constants. It is important to recall to thereaders that the difficult problem of global existence of weak solutions to theheat-conducting case with constant viscosities or viscosities depending on thetemperature has been firstly established by E. Feireisl [26] and then by E. Feireisland A. Novotný in [28]. Here we will focus on the case where� and � depend on thedensity. This will crudely change the situation and will ask for new mathematicaltools initiated by the two authors of this chapter and C.K. Lin for the Kortewegcompressible system, see [13].

2 Simple Fluid Flow Models and Historical Comments

Below we discuss mathematical properties of some particular cases of system (1).

I) Incompressible Navier-Stokes equations without temperature (homogeneouscase). First, if we assume that � is a constant, the continuity equation in (1)reduces to the incompressibility constraint div u D 0 with an associatedLagrangian multiplier. Forgetting the total energy equation, we obtain

(div u D 0;

�.@tuC div .u˝ u// D div C �f;(3)


where D 2�D.u/ � p IRn with p the Lagrangian multiplier associated tothe incompressibility constraint. Notice that if the viscosity � is constant, theviscous dissipation term rewrites div .2�D.u// D �u, so that the momentumequation can be rewritten as

@tuC div .u˝ u/ D�

�u � r

�p

�

�C f:

Defining the kinematic viscosity � D �=� and P D p=�; one can rewrite theincompressible equations (3) as

(div u D 0;

@tuC div .u˝ u/CrP D �uC f;(4)

which is the classical incompressible Navier-Stokes equations for homogeneousfluids. The first global existence of weak solutions result for such system is dueto [42]. Such system may be obtained from the barotropic compressible Navier-Stokes equations by letting the Mach number go to zero.

II) Incompressible Navier-Stokes equations without temperature (nonhomogeneouscase ). If we just assume that the fluid is incompressible and forget the totalenergy equation, we end up with the nonhomogeneous incompressible Navier-Stokes equations

8

<̂

:̂

div u D 0;

@t�C div .�u/ D 0;

@t .�u/C div .�u˝ u/ D div C �f

(5)

where D 2�.�/D.u/ � pId. Existence results of global weak solutions forsuch system are due to A. Kazhikhov [40] (initial density far from vacuumand constant viscosity), J. Simon [56] (initial density with possible vacuumand constant viscosity) and E. Fernández-Cara and F. Guillén-Gonzalez [31],P.L. Lions [44] (initial density with possible vacuum and density-dependentviscosity but strictly positive). Note that such a system may be obtained fromthe compressible Navier-Stokes equations with heat conductivity letting theMach number go to zero. The interested reader is referred to [1] and referencecited therein for such derivation. More general models may be encountered, forinstance, addressing large heat-release properties: see, for instance, [15] and [1].In that case, the divergence-free constraint may be replaced, for instance, by

divu D g.�/

with g a given function which encodes the heat-conducting pressure law.III) Barotropic compressible Navier-Stokes equations. If we assume that the

temperature � is constant, the pressure p only depends on the density �, whichcorresponds to the so-called “barotropic” case (also sometimes improperly


called “isentropic”: entropy production rate is eventually proportional to thework of viscous stresses, and must therefore be positive). In the framework ofbarotropic flows, the most commonly found equation of state is the so-called��law, i.e., p.�/ D a�� , where � � 1; a > 0 are constants. Thus, System (1)becomes

8

<̂

:̂

@t�C div .�u/ D 0;

@t .�u/C div .�u˝ u/Crp D div .2�D.u//Cr.� div u/C �f;

p D p.�/

(6)

supplemented with the initial conditions

�jtD0 D �0; .�u/jtD0 D m0: (7)

Several remarks should be given on the barotropic Navier-Stokes equations (6):

Remark 1. The first existence result on global weak solutions in the multidimen-sional in space case was obtained by P.L. Lions [45] in 1998 for large enoughexponents � � 3d=.d C 2/, d D 2; 3. This result has been later extended to thesomehow optimal case � > d=2 by E. Feireisl et al. [29] in 2001. Note the recentpaper by P. Plotnikov and W. Weigant [53] where the case � D 1 is covered in thetwo-dimensional space case.

Remark 2. Note that the pressure law can be generalized, a typical example being

p 2 C1.Œ0;C1//; p.0/ D 0 with

a��1 � b � p0.�/ �1

a��1 C b with � > d=2

for some constants a > 0I b � 0: See E. Feireisl [25], B. Ducomet, E. Feireisl, H.Petzeltova, I. Straskraba [24] for slightly more general assumptions. However it isalways required that p.�/ be increasing after a certain critical value of �. Note thata new global existence result without monotonicity assumptions on the pressure lawhas been recently proved by D. Bresch and P.E. Jabin in [17]: It requires a moreprecise analysis of the structure of the equations combined to a novel approach tothe compactness of the continuity equation.

Remark 3. In the result of P.L. Lions and E. Feireisl, the viscosity coefficients �and � need to be constants. The case when � and � depend on the density, i.e.,� D �.�/; and � D �.�/ is an open problem in its full generality. Only partialresults have been obtained, such as D. Bresch, B. Desjardins et al. [8, 12, 13], A.Mellet and A. Vasseur [46], J. Li and Z. Xin [43], and A. Vasseur and C. Yu [59]based on a new mathematical entropy discovered by D. Bresch and B. Desjardins ifthe relation �.�/ D 2.�0.�/��.�// is satisfied. See also the work by A. Kazhikhov


and W. Weigant [41] where they consider the case �.�/ D cste and �.�/ D �ˇ withˇ > 3 in a periodic box � D f.x; y/ 2 R

2 W 0 < x < 1; 0 < y < 1g: They getglobal existence of strong solution in 2D using this density dependency of �. Moreprecisely, they prove the following theorem

Theorem 1. Let

�.�/ D 1; �.�/ D �ˇ with ˇ > 3

and P .�/ D R�� with R > 0 and � � 0. Assuming

0 < m0 � �0.x; y/ �M0 < C1; �0 2 W 1;q.�/ with q > 2; u0 2 H2.�/

then there exists a unique strong solution to the compressible Navier-Stokessystem (6) satisfying the initial conditions with moreover

u 2 L2.0; T IH2.�// \H1.0; T IL2.�//;

� 2 L1.0; T IW 1;q.�// \W 1;1.0; T ILq.�//:

The solution exists for all time and the density is an L1 function and is boundedaway from vacuum.

Note that recently several authors succeeded to improve the hypothesis on ˇ(see B. Haspot, Z.P. Xin et al. and others) and to consider bounded domains (see,for instance, B. Ducomet and S. Necasova). See also the work by M. Perepetlisa(see [52]) regarding the existence of global weak solution with interesting featureregarding non-vacuum and boundedness for the density.

Remark 4. The essential difficulties of the incompressible Navier-Stokes equa-tions (5) are also present in the framework of barotropic flows for constantviscosities. More precisely, in (5), the main difficulty is how to pass to the limitin �nun and more precisely in the term div .�u˝u/. We will explain later on quicklyhow to handle this difficulty. For barotropic flow, compactness of the pressure law isalso necessary. Such issues will be discussed in detail in the next section. This is themain difficulty for compressible barotropic Navier-Stokes equations with constantviscosities and power law pressure state.

3 Why Degenerate Viscosities May Help and Why It Seems tobe Necessary to Consider Such Situation?

Let us explain why density-dependent viscosities may be important and whynew velocity field may be encountered for strongly heterogeneous flows. Wefocus on model behavior when the density vanishes, dimensional analysis, anddensity/velocity fluctuations modeling.


3.1 Model Behavior When the Density Vanishes

When the density � tends to zero, compressible fluid models are no longer valid froma physical viewpoint, rarefied gas models being in that case more relevant. Still,Navier-Stokes numerical codes need to be robust near regions where the density�! 0, which occurs in many industrial simulations of complex flows.

As a matter of fact, the compressible Navier-Stokes equations when the densitytends to vacuum behaves like the degenerate model

�div .2�D.u// � r.�div u/ D 0

where � D �.� D 0; �/ and � D �.� D 0; �/. Such an equation has no physicalmeaning. Using density-dependent viscosity coefficient .�; �/ vanishing when �!0 has the advantage to remove the above “ghost elliptic equation” in vacuum regions.The reader is referred to [37] and [54] for discussion on the problem of continuitywith respect to the initial data when the viscosities are assumed to be constant.

3.2 Dimensional Analysis

Once the idea of using viscosity coefficients vanishing as a function of the densitywhen � tends to zero has been introduced, dimensional analysis may help findrelevant velocities linked to density or viscosity variations.

First, the following gradient length associated with viscosity variations can bedefined

L� D�

kr�k

A (dimensionless) Reynolds number can then be introduced

Re D�VL�

�;

which defines a velocity scale V associated with viscosity variations,

V ��

�L�Dkr�.�/k

�:

Hence velocities such as r�=� seem to play a role in the dynamics of hetero-geneities. We will see that this is exactly this quantity which will play a crucialrole in the BD entropy and more generally in the global existence of weak solutionsfor compressible Navier-Stokes equations with degenerate viscosities.


3.3 Density and Velocity Fluctuations Modeling

Turbulence models can be derived by systematic use of averaging operators, leadingto correlations of products of variable fluctuations. Such statistical derivation ofturbulence models can be found, for instance, in Ishii [38] and Drew and Passman[23] in the context of multiphase flows. It leads to averaged models involvingvelocities associated with density or viscosity small-scale fluctuations.

3.3.1 Reynolds AverageReynolds average of a given variable is defined as the ensemble average ofthis quantity (mean over a large number of realizations of the same experiment)satisfying the following properties of any observable variables a and b

a D a; aC b D aC b

ab D ab

@ta D @ta @xi a D @xi a 8i D 1 : : : d

Reynolds fluctuations are then defined as the difference of observable a with itsaverage value a:

a D aC a0

Application to homogeneous flows � � constant leads to the derivation of the so-called RANS equations (Reynolds Averaged Navier-Stokes equations). Note thattime or combined space-times averages may also be used instead of set average.

3.3.2 Favre AverageFavre average is adapted to heterogeneous flows (compressibility/large densityvariations). Indeed, the density is used as a weight in Reynolds statistical averagefor a given variable a:

Qa D�a

�

Favre fluctuations are similarly defined by

a D QaC a00

Example 1. Taking the average of mass conservation equation leads to

@t�C div .�u/ D 0;

which simplifies using the properties of the Reynolds average


@t�C div .�u/ D 0:

Since by definition �u D � Qu, one has

@t�C div .� Qu/ D 0;

so that the averaged mass conservation equations is exactly preserved by thenonhomogeneous averaging process.

Averaging compressible fluid mechanics models leads to correlations of fluctua-tions of physical quantities a with velocity fluctuations u00

�u00a00:

Boussinesq-Reynolds closure consists in assuming that this correlation is propor-tional to the gradient of the Favre average of a

�u00a00 D ��t

ar Qa

where �t is the turbulent viscosity, a is a constant (Prandtl-Schmidt number). Thisclosure can be rigorously mathematically justified under some restrictive hypotheseson the fluctuations (see [2], for instance).

Example 2. Difference between Favre and Reynolds velocity

u D �uv where v D 1=�

D QuC �u00v00 D u � ��t

ar Qv

since Qv D �v=� D 1=�, one ends up with

u D Qu � ��t

ar1

�

D QuC�t

a

r�

�:

Consequently, velocities based on density gradients seem to play a role whenmodeling turbulent compressible flows with significant density heterogeneities.Note the importance of the quantity �tr log � which is a part in the velocitydedicated to counteract osmotic effect as mentioned by A. EINSTEIN: See the bookby NELSON [49].


4 Basic a Priori Estimates, Identities, and Comments

This section is devoted to basic estimates on solutions of fluid equations introducedin Sect. 1. Let us recall that the domain is assumed to be a torus � D T

d , i.e., a boxwith periodic boundary conditions. This part is for people who are not familiar withpartial differential equations arising in fluid mechanics modeling.

I) Incompressible Navier-Stokes equations without temperature (homogeneouscase). Taking the scalar product of u with the momentum equation in (4) andintegrating over the space domain �, one obtains

1

2

d

dt

Z

�

juj2 C �Z

�

jruj2 DZ

�

f � u: (8)

Here the fact thatR�rp � u D 0 has been used. In the case of vanishing bulk

forces f � 0, one deduces from (8) that for all positive time T

u 2 L1.0; T IL2.�// \ L2.0; T IH1.�//:

Similar bounds can be obtained for nonzero f if suitable regularity is assumedsuch as f 2 L2.0; T IH�1.�// for all T > 0.

II) Incompressible Navier-Stokes equations without temperature (nonhomoge-neous case).In this case, multiplying by u the momentum equation in (5) and the massequation by juj2=2, adding the results, and integrating with respect to x 2 �leads to

1

2

d

dt

Z

�

�juj2 C �Z

�

jruj2 DZ

�

�f � u: (9)

Similarly, if f � 0, one deduces from (9) that

p�u 2 L1.0; T IL2.�//; ru 2 L2.0; T IL2.�//:

Similar bounds can be obtained for nonzero f if f 2 L1.0; T IL2.�//. Thisuses the fact that � 2 L1..0; T / � �/ if the initially density �0 is bounded(note that u is divergence-free). We just have to bound the term coming fromthe external force as follows

Z

�

�u � f � kp�kL1.�/k

p�ukL2.�/kf kL2.�/:

III) Barotropic compressible Navier-Stokes equations. The same integration proce-dure as in the incompressible case of (5) gives the total energy balance

1

2

d

dt

Z

�

�juj2 C 2�Z

�

jD.u/j2 CZ

�

�jdiv uj2 CZ

�

u � rp DZ

�

�f � u:


From the mass conservation equation @t� C div .�u/ D 0 in (6), no information ondiv u can be extracted as in the incompressible case, since the term

R�

u � rp has tobe controlled. Let us define such that � 0.�/ � .�/ D p.�/, then the followingequation reads at least formally

@t .�/C div. .�/u/C p.�/divu D 0:

Integrating in space and time and adding to the previous relation, we get usingthe periodic boundary condition

1

2

d

dt

Z

�

�juj2dx Cd

dt

Z

�

.�/dx C 2�

Z

�

jD.u/j2dx C �Z

�

jdiv uj2dx

D

Z

�

�f � udx (10)

where is defined by � 0.�/� .�/ D p.�/: Choosing .�/ D �e.�/, we find thate is defined as

e.�/ D

Z �

�ref

p.s/=s2 ds

where �ref is a constant reference density. Assuming that f � 0 and �; � areconstants, one deduces that

p�u 2 L1.0; T IL2.�//I ru 2 L2.0; T IL2.�//I

.�/ 2 L1.0; T IL1.�//:

Assuming the pressure satisfies

C�1�� C � p.�/ � C�� C C

for some constant C > 0, similar bounds can be obtained for nonzero f if f 2L1.0; T IL2�=.��1/.�// noticing that

Z

�

�u � f � k�ukL2�=.�C1/kf kL2�=.��1/ � kp�kL2� .�/k

p�ukL2.�/kf kL2�=.��1/ :

Remark 5. If we assume a � -type law p.�/ D a�� ; � > 1 and �ref D 0, we findthat e.�/ D a��1=.� � 1/: Thus (10) implies that � 2 L1.0; T IL�.�//:

In order to get the global weak solutions of (6), three steps are necessary:

• Derive A priori estimates• Obtain stability properties of approximate solutions sequences• Construct approximate solutions


From the basic energy estimates, using the equations and Aubin-Lions-Simoncompactness lemma, it is possible to obtain the following results:

1. For homogeneous incompressible equations (4): Stability in u˝ u can be proved(See [42]).

2. For nonhomogeneous incompressible equations (5): Stability in �u, �u ˝ u canbe obtained if �.�/ � C > 0 (See [44]).

3. For compressible barotropic models (6):– Constant viscosities case: Stability in �u, �u˝ u can be proved and stability

in �� requires more work (See [45, 51]).– Density-dependent case: stability on �� can be proved with an extra estimate

and stability in �u˝ u asks for more work.

Assuming that .�n; un/ is a sequence of approximate solutions satisfying energyestimates for the barotropic system and the equation in a weak sense.

(I) Barotropic case:(I-1) The case � and � constants (nondegenerate case). In that case, the maindifficulty is to pass to the limit in the pressure term proportional to �

�n (the

compactness ofp�nun will be discussed later on). Additional information is

required on density oscillations: how to get compactness of �n in Lebesgue spaces.How to pass to the limit in .�/�n is the main difficulty in P.L. Lions’ and E. Feireisl’sresults. Their methodology uses strongly that viscosities are constants. Recall thatthe equations of (6) reads

(@t�C div .�u/ D 0;

@t .�u/C div .�u˝ u/ � �u � .�C �/rdiv uCrp D �f(11)

supplemented with the initial conditions

�jtD0 D �0; .�u/jtD0 D m0: (12)

Applying formally the divergence operator div to the momentum equation leads to

div .@t .�u/C div .�u˝ u// � �div u � .�C �/div uCp D div .�f /:

Remarking

��div u � .�C �/div uCp D �a�� .2�C �/ div u

�

where a�� .2�C �/ div u is called the effective flux. The equation above showsthat in some sense there exists some compactness on the effective flux: With onederivative of the equation, we find the Laplacian of this quantity. We use stronglythe fact that � and � are constants to commute the divergence operator with the


diffusion operator. Note that there exists several steps to prove global existenceof weak solutions for the barotropic compressible Navier-Stokes equations withconstant viscosities:

• Energy estimates:See calculation before

• Extra integrability on �:� 2 L�C� ..0; T / �� with � � 2d=� � 1): Use Bogovski operator.

• Effective flux property:�divu��divu D

�p.�/��p.�/�

�=.2�C�/ where � denotes the weak limit

• Compactness on the density:@t .� ln � � � ln �/ C div.� ln �u � � ln �u/ D �divu � �divu D

�p.�/� �

p.�/��=.2�C �/

Use monotonicity of p to have a sign of the right-hand side and the strictconvexity of s 7! s ln s to provide the propagation of compactness on �n ifinitially compact.

The compactness onp�nun in L2..0; T / ��/ follows in some sense what is done

for the incompressible nonhomogeneous Navier-Stokes equations: we will discussthat later on.

For reader’s convenience, let us give the existence result in the case withoutexternal force, namely, f D 0 and in a periodic box � D T

d :

Theorem 2. Let p.s/ D as� with � > d=2 and a > 0. Assume the initial datasatisfy

�0 2 L�.�/; �0 � 0;

Z

�

�0 DM0 > 0

and

m0 2 L1.�/; m0 D 0 if �0 D 0; jm0j

2=�0 2 L1.�/:

Then there exists a global weak solutions of the compressible Navier-Stokessystem (11) and (12) that means a solution satisfying the energy inequality andthe system in a weak sense.

With some modifications, the existence result may be extended to the case men-tioned in Remark 2 as proved in [25]. Note also the recent result by A. Plotnikovand W. Weigant related to � D 1 in the two dimensional in space case. For details,interested readers are referred to the other contributions in this handbook.

Remark 6. Note the recent work by D. Bresch and P.E. Jabin [17] where they relaxthe hypothesis on the pressure law in the constant viscosities case; more preciselythey assume p to be locally lipschitz on Œ0;C1/, with p.0/ D 0

C�1�� C � p.�/ � C�� C C


and for all s � 0, they only assume

jp0.s/j � s Q��1

for some Q� > 1. Then they prove the following result without external force, namely,f D 0 and in a periodic box � D T

d :

Theorem 3. Let .�0; u0/ such that

E.�0; u0/ DZ

Td

jm0j2

2�0C �0e.�0/ < C1

where e.s/ DR s0p.�/=�2 d� . Let p satisfying the previous hypothesis with

� >�max.2; Q�/C 1

�d=.d C 2/

then there exists a global weak solution to the compressible Navier-Stokes equations(11) and (12).

Note that the authors write in [18] an introduction of the memoir [17]. This allowsto explain, on a simpler but still relevant and important system, the tools recentlyintroduced by the authors and to discuss the important results that have beenobtained on the compressible Navier-Stokes equations. To get such global existenceof weak solutions result, the two authors have revisited the classical compactnesstheory on the density by obtaining precise quantitative regularity estimates: Thisrequires a more precise analysis of the structure of the equations combined to a novelapproach to the compactness of the continuity equation (by introducing appropriateweights).

(I-2) The case � and � not constants (degenerate case: density dependency). Herethe difficulty will be to pass to the limit �u ˝ u. The main difficulty is that lessinformation on the velocity compared to the constant viscosities case is known: noL2.0; T IH1.�// bound on u. What kind of additional estimate can be derived tohelp to conclude? Some results about nonconstant viscosities have been obtained:W. Vaı̌gant and A. Kazhikov [58], D. Bresch and B Desjardins [8,12], A. Mellet andA. Vasseur [46], A. Vasseur and C. Yu [59], and J. Li and Z. Xin [43]. In particular,under the relation �.�/ D 2.�0.�/� � �.�//, a mathematical entropy has beendiscovered by Bresch and Desjardins [8] (so-called BD entropy): a kind of nonlinearhypocoercivity property which is the cornerstone of all the previous works exceptthe one by W. Vaı̌gant and A. Kazhikov who considered the case �.�/ D cste and�.�/ D �ˇ with ˇ � 3. Using the BD entropy gives control of gradient of � but asksto work with degenerate parabolic behavior for the velocity. Extra work is thereforenecessary to ensure extra integrability on u if it is the case initially. Let us present theBD entropy, explain why it asks for degeneracy, and discuss the relation assumedbetween the two viscosities �.�/ D 2.�0.�/� � �.�//.


For the reader’s convenience, let us give the very nice existence result which isactually known: stability and construction of approximate solutions. The novelty isthe construction of approximate solutions (which is really more complicated thanfor the constant viscosities case) and is due independently to A. Vasseur, C. Yu in[59] and J. Li, Z.P. Xin [43]. We consider no external force, namely, f D 0 and aperiodic box � D T

d :

Theorem 4. Let �.�/ D 0 and �.�/ D � with P .�/ D a�� with � > 1 for d D 2

and 1 < � < 3 for d D 3. Assume

�0 2 L�.�/; �0 � 0; r

p�0 2 L

2.�/;

m0 2 L1.�/; m0 D 0 if �0 D 0; jm0j

2=�0 2 L1.�/:

Assume moreover thatZ

�

�0.1C ju0j2/ ln.1C ju0j

2/ dx < C1:

Then there exists a global weak solution .�; u/ of the degenerate compressibleNavier-Stokes equations that means satisfying the energy inequality, the BD entropy,the Mellet-Vasseur estimate, and the compressible Navier-Stokes equation in a weaksense.

Note that the more general case is still in progress in [21] with the Bresch-Desjardinsrelation �.�/ D 2.�0.�/� � �.�//. Partial results were obtained in [43] with anonphysical stress tensor S D �.�/ruC�.�/divuIRn instead of the symmetric oneS D 2�.�/D.u/ C �.�/divuIRn . Anyway this paper is really interesting becauseit helps to understand more the difficult problem of constructing approximatesolutions in the density-dependent viscosities case for compressible Navier-Stokesequations.(I-2-a) BD entropy: Barotropic case/nonlinear hypocoercivity – An extra esti-mate. In this subsection, we will introduce the BD entropy to the equations (6) indetail.

Lemma 1. Assume that �.�/ D 2.�0.�/� � �.�//. The formal equality associatedwith BD entropy can be written as

1

2

d

dt

Z

�

�juC 2r'j2 C 2Z

�

�.�/jA.u/j2

Cd

dt

Z

�

.�/C 2

Z

�

p0.�/�0.�/jr�j2

�D

Z

�

�f � .uC 2r'/; (13)


where A.u/ D .ru � rut /=2 denotes the skew-symmetric part of the velocitygradient.

Note that in the one dimensional in space case, such property has been noticedby Y. Kanel [39].

Proof. Multiplying the mass conservation equation by �0.�/, one deduces that

@t�.�/C div .u�.�//C .�0.�/� � �.�// div u D 0:

Applying the operator r to the above equation leads to

@tr�.�/C div .�.�/ru/C div .u˝r�.�//Cr.�0.�/� � �.�//div u D 0:

Defining �0.s/ D '0.s/=s; then the above equation becomes

@t .�r'.�//C div .�u'.�//C div .�.�/D.u//C div .�.�/A.u//

Cr.�0.�/� � �.�//div u D 0: (14)

The momentum equation reads

@t .�u/C div .�u˝ u/ � 2div .�.�/D.u// � r.�.�/div u/Crp D �f:

Multiplying Eq. (14) by 2 and adding it to the momentum equation, it gives

@t .�.uC 2r'//C div .�u˝ .uC 2r'//C 2div .�.�/A.u//Crp D �f:

Then one can multiply the above equation by u C 2r' and the mass equation byjuC 2r'j2=2. Summing up the results and integrating over �; one obtains that

1

2

d

dt

Z

�

�juC 2r'j2 C 2Z

�

�.�/jA.u/j2 CZ

�

rp � .uC 2r'/

D

Z

�

�f � .uC 2r'/:

Notice thatZ

�

rp � .uC 2r'/ DZ

�

rp � uC 2Z

�

rp � r'

Dd

dt

Z

�

.�/C 2

Z

�

p0.�/�0.�/jr�j2

�;

then we obtain the equality (13).


Assume that f � 0 and p0.�/ �0.�/ > 0; the new entropy estimates give thefollowing additional regularities:

p� .uC 2r'/ 2 L1.0; T IL2.�//I

p�.�/A.u/ 2 L2.0; T IL2.�//I

sp0.�/�0.�/

�r� 2 L2.0; T IL2.�//I .�/ 2 L1.0; T IL1.�//:

In view of the energy estimates, we already know that

p� u 2 L1.0; T IL2.�//I

p�.�/D.u/ 2 L2.0; T IL2.�//I

.�/ 2 L1.0; T IL1.�//:

Combining with the above results, one has

p�r' 2 L1.0; T IL2.�//I

p�.�/ru 2 L2.0; T IL2.�//:

Obviously, additional information has been obtained on � (depending on thebehavior of �).

Remark 7. Assume �.�/ D �; then '.�/ D log �: The new estimate is onp�.uC

2r log �/:

(I-2-b) The problem of degeneracy occurring with the BD entropy. The property�.�/ D 2.�0.�/� � �.�//; �.�/ � c > 0; and �.�/ � 0 are impossible to besatisfied simultaneously. Then �.�/C 2�.�/=d � 0; �.�/ � 0 is the only way thathas been done. Recall that

�.�/jD.u/j2 C �.�/jdiv uj2 D �.�/ˇˇD.u/ �

div u

dIRnˇˇ2 C

�

�C2�

d

�

jdiv uj2:

Assume �.�/ D �˛; then

d�.�/C 2�.�/ D 2.d˛ � d C 1/�˛ � 0

if ˛ � .d � 1/=d: We will see in the case of heat-conducting fluids by Bresch andDesjardins [9] that

�.�/

(�n with n � 2

3close to vacuum,

�m with m � 1 far to vacuum.

It means that the viscosities are degenerate. Thus more information is needed on usince a H1 bound in space is no longer available.


Remark 8. Hoff and Serre [37] obtained that it is a failure of continuous dependenceon initial data for the compressible Navier-Stokes equations if the viscosities areconstants in one dimension in space. Thus it is necessary to consider the degenerateviscosities in the compressible Navier-Stokes equations. It is a really interesting andchallenging work to perform the same study in the multidimensional case, see, forinstance, the paper by D. Serre [54].

(I-2-c) Comments on the relation between �.�/ and �.�/. It is really strange toask such relation between the two viscosities (namely, �.�/ D 2.�0.�/� � �.�//)and the authors do not know if there is some physical meaning for such relation.Anyway let us give an example where similar relation occurs directly. This exampleis linked to Korteweg compressible systems. Recall that the compressible Euler-Korteweg system reads

@t% C div.%u/ D 0; (15)

@t .%u/ C div.%u˝ u/Crp.%/ D div.K/; (16)

where % denotes the fluid density, u the fluid velocity, p.%/ the fluid pressure, andK the Korteweg stress tensor defined as

K D�

%div.K.%/r%/C1

2.K.%/ � %K 0.%//jr%j2

�

IRn �K.%/r%˝r%: (17)

with K.%/ the capillary coefficient. In [7], it has been observed that divK may bewritten

divK D 2div��1.%/rv1

Cr

��1.�/divv1

with �1.�/ D 2Œ�01.%/%��1.%/� and where v1 D r'1.�/withp%'01.%/ D

pK.%/,

�01.%/ DpK.%/ %=2. Remark that we find an interesting second-order operator on

v1 with coefficients satisfying the same structure than the viscosities �.�/ and �.�/.This is perhaps an algebraic coincidence, but this is interesting enough to be noticedin this chapter.

(II) The full heat-conducting system case (stability results)The main difficulties associated with weak solutions for heat-conducting fluids aredescribed now. Note that there are fundamental differences between the frameworkconsidered by E. Feireisl with constant viscosities (or temperature-dependent vis-cosities) and that of D. Bresch and B. Desjardins with density-dependent viscosities.More precisely, E. Feireisl studies pressure laws expressed as p.�; �/ D pe.�/ C

�p�.�/ where p�.�/ pe.�/. The assumptions considered by D. Bresch and B.Desjardins are different, since pressure is the form of p.�; �/ D pc.�/C r��; witha so-called cold pressure component pc.�/ (pressure at zero temperature) defined


close to vacuum such that it vanishes away from vacuum. In particular, perfect gaspressure laws can be considered far from vacuum. Remark that interesting otherpressure law states have been considered by E. Feireisl and A. Novotný [28] suchthat those with radiative part helping to get compactness on temperature.

One mathematical difficulty is that a priori estimates are not sufficient to defineweak solution �; u, and � . From the total energy estimates and BD entropy estimates,the following bounds hold

� log � 2 L1.0; T IL1.�//; �jr log � j2;jD.u/j2

�2 L1.0; T IL1.�//:

But fluxes such as �u.juj2=2Ce/Cpu in the total energy conservation equation arenot necessarily integrable. The second difficulty appears in the compactness of thetemperature. The first challenges are to prove the following properties:

(i) The control �juju2 2 Lp.0; T ILp.�// with p > 1(ii) The control �r� 2 Lp.0; T ILp.�// with p > 1

(iii) The compactness on � :We need more information on some negative power of � as in [9] or radiative

term as proposed in [47].

We will prove that the generalization of the BD entropy for the heat-conductingNavier-Stokes equations and the presence of a cold pressure degenerating close tovacuum will help to conclude of global existence of weak solutions via the totalenergy formulation. This cold pressure part is crucial to deduce some compactnesson the temperature and pass to the limit in the total energy formulation.

4.1 Stability Properties

4.1.1 The Barotropic CaseIn this subsection, we want to prove the weak stability of the approximate solutions.�n; un/ of the compressible barotropic Navier-Stokes equations. The main difficultyfor the density-dependent viscosity case is to pass to the limit in the term �nun˝un,which requires the strong convergence of

p�nun: We will see that passing to the

limit in the pressure quantity a��n is simple due to the BD entropy information.Let us explain, by hand, the difference with the constant viscosities case to pass tothe limit in �nun and �nun ˝ un. In the constant viscosities case, we can write thefollowing identities:

Z T

0

Z

�

�nun D< �n; un >H�1.�/�H1.�/

and

Z T

0

Z

�

�njunj2 D< �nunI un >H�1.�/�H1.�/


using that un 2 L2.0; T W H1.�//. Then use the strong convergence of �nun inL2.0; T IH�1.�//, �n in L2.0; T IH�1.�// [using, respectively, the momentumequation, the mass equation, and Aubin-Lions-Simon compactness lemma] and theweak convergence of un in L2.0; T IH1.�// to pass to the limit in the relationwritten above. This gives the convergence in norm of

p�nun in L2.0; T IL2.�//

and thus the announced strong convergence using weak convergence ofp�nun to

p�u in L2..0; T / � �/. The first strong convergence in negative Sobolev spaces

is obtained using the bounds given by the energy estimates, the extra integrabilityon ��n and the bound on @t .�nun/ through the momentum equation and the energyestimates (Aubin-Lions-Simon compactness). With density-dependent viscositieswhich is degenerated close to vacuum, we understand that we loose this lastestimate un 2 L2.0; T IH1.�// and therefore loose a priori the convergenceofp�nun in L2.0; T IL2.�//. The key ingredient to achieve is an additional

estimate which bounds �nu2n in a space better than L1.0; T IL1.�//; namely,L1.0; T IL logL.�//:We will give the sketch proof here, see [46] for more details.Remark that compared to the constant viscosities case, pass to the limit in thepressure term is quite easy in the density-dependent viscosities case using theBD entropy. Let us explain the various convergence for the degenerate density-dependent viscosity case. This is a summary of what may be found in [46] or in[11].

Let us show the following result

Theorem 5. Let us consider a sequence .�n; un/n2N satisfying uniformly the energyestimates, the BD entropy, and the compressible Navier-Stokes equations in a weaksense. Let us also assume that it satisfies the bound

Z T

0

Z

�

�nf .junj/ < C1

uniformly with respect to n with f an increasing function such that s2=f .s/ ! 0

when s ! C1: Then there exists a subsequence such that the weak limit satisfiesthe energy and BD entropy estimates and the compressible Navier-Stokes equationsin a weak sense.

Step 1: Convergence ofp�n

Using the mass conservation equation, it follows that for all t � 0

k�n.t; �/kL1.�/ D k�n;0kL1.�/:

Hence up to the extraction of a subsequence, one may assume that .�n/n�1 tends inthe sense of distributions to some nonnegative density �. The BD entropy introducedin (I-2-a) leads to the fact that

p�n is bounded inL1.0; T IH1.�//. Next, we notice

that


@tp�n D �

1

2

p�ndiv un � un � r

p�n

D1

2

p�ndiv un � div .un

p�n/:

which allows to conclude that @tp�n is bounded in L2..0; T / � �/ C

L1.0; T IH�1.�//. Then using Aubin-Lions lemma leads to the strongconvergence of

p�n in L2loc..0; T / ��/ to

p�.

Step 2: Convergence of pressure

Due to the BD entropy and the energy estimates, it follows that ��n 2L1.0; T IL3.�//: Since ��n is bounded in L1.0; T IL1.�//; Hölder inequalitygives

k��nkL53 ..0;T /��/

� k��nk25

L1.0;T IL1.�//k��nk

35

L1.0;T IL3.�//� C:

Hence ��n is bounded in L53 ..0; T / ��/. Since we already know that ��n converges

almost everywhere to �� , those bounds yield the strong convergence of ��n inL1loc..0; T / ��/:

Step 3: Convergence of the momentum

In this step, we will prove that �nun ! m in L2.0; T ILp/ with p 2 Œ1; 32/: In

particular, �nun ! m almost everywhere .x; t/ 2 � � .0; T /:

Proof. We just consider the case of d D 3. First, �nun Dp�n�p�n un. From the BD

entropy as in the previous subsection, we know thatp� r'.�/ 2 L1.0; T IL2.�//.

If � satisfies that �0 � c > 0; then we get rp� 2 L1.0; T IL2.�//: Similar to

[46], one deduces thatp� 2 L1.0; T IL6.�//: Besides, we know that

p�nun 2

L1.0; T IL2.�// from the energy estimates. Then �nun 2 L1.0; T IL32 .�// by

HRolder inequality.Next we consider @i .�nun/; i D 1; 2; 3;

@i .�n unj / D �n @iunj C @i�n unj Dp�np�n @iunj C 2

p�n unj @i

p�n:

Using the BD entropy estimates, the property of �0 � c > 0; and �.�/ � c�, weobtain that

@i .�nun/ 2 L2.0; T IL1.�//:

Finally, we want to derive the regularity of @t .�nun/: The momentum equation is

@t .�nun/C div .�nun ˝ un/

� 2 div .�.�n/D.un// � r.�.�n/div un/C ar��n D 0:


From the facts that

�n un ˝ un Dp�n un ˝

p�n un 2 L

1.0; T IL1.�//;

and ��n 2 L1.0; T IL1.�//; it follows that r��n 2 L1.0; T IW �1;1.�// and

div .�nun ˝ un/ 2 L1.0; T IW �1;1.�//.

Using the assumptions on � with respect to � in [46], it follows that�.�n/p�n2

L1.0; T IL6.�// andp�n r'.�n/ 2 L

1.0; T IL2.�//: One can refer to [46] fordetails. Then

�.�n/run D r.�.�n/un/ � un � r�.�n/

D r

��.�n/p�n�p�nun

�

�p�nun �

p�n r'.�n/:

It deduces that div .�.�n/run/ 2 L1.0; T IW �1;1.�//. It is the same procedure forr.�.�n/ div un/ .

Applying Aubin-Lions lemma, the proof is complete.

Step 4: Convergence ofp�nun

Next, we will prove the following factsp�nun !

p�u in L2loc.0; T ��/:

First of all, since mn=p�n is bounded in L1.0; T IL2.�//; and Fatou’s lemma

yields that

Z

lim infm2n

�ndx <1:

We define u such that

u D

8<

:

m

�when � ¤ 0;

0 when � D 0:

Then

Zm2

�dx D

Z

�juj2dx <1:

Moreover, from Step 3 and Fatou’s lemma, it yields that

Z

�f .juj/dx �Z

lim inf �nf .junj/dx

� lim infZ

�nf .junj/dx;


and thus �f .u/ is in L1.0; T IL1.�//: Next, since mn and �n converge almosteverywhere, it is readily seen that in f�.x; t/ ¤ 0g;

p�nun D mn=

p�n converges

almost everywhere top�u D m=

p�: Moreover

p�nun1junj�M �!

p�u1juj�M almost everywhere:

As a matter of fact, the convergence holds almost everywhere in f�.x; t/ ¤ 0g; andin f�.x; t/ D 0g; thus

p�nun1junj�M �M

p�n ! 0:

For M > 0; we cut the L2 norm as follows:

Z

jp�nun �

p�uj2dxdt �

Z

jp�nun1junj�M j �

p�u1juj�M j

2dxdt

C 2

Z

jp�nun1junj�Mdxdt j C 2

Z

jp�u1juj�Mdxdt j:

It is obvious thatp�nun1junj�M is bounded uniformly in L1.0; T I

L3.�//: Then the convergence of the above first holds, i.e.,

Z

jp�nun1junj�M �

p�u1juj�M j

2dxdt �! 0:

Finally, we write

Z

jp�nun1junj�M j

2dxdt �M2

f .M/

Z

�nf .junj/dxdt;

and

Z

jp�u1juj�M jdxdt �

M2

f .M/

Z

�f .juj/dxdt:

Summing the above results, we deduce that

lim supn!1

Z

jp�nun �

p�uj2dxdt �

CM2

f .M/

for all M > 0: Then the strong convergence ofp�u follows by taking M ! 1

using the required property on f .

Step 5: Convergence of the diffusion terms. In this step the following facts hold

�.�/run �! �.�/ru in D0;�.�/div un �! �.�/div u in D0;


as n ! 1: The proof is based on the energy estimates and compactnessarguments, see [46] for more details.

Remark 9. It is important to note that we don’t need Mellet-Vasseur estimate if weadd extra terms in the momentum equations which will help to get, for instance, thebound �juj2Cı 2 L1..0; T / � �/ for some ı > 0. This is the case, if we add dragterms such as �jujıu in the momentum equations or a singular pressure law close tovacuum. This has been discussed by D. Bresch, B. Desjardins, and coauthors severaltimes.

An interesting propagation property: Mellet-Vasseur property. It is interestingto note that it is not necessary to have extra terms such as drag terms or singularpressure laws to get extra integrability on un. Let us show that it is included inthe compressible Navier-Stokes equations with degenerate viscosities using energyand BD entropy. We consider the case �.�/ D �� and �.�/ D 0 for simplicity.For more general cases, the reader is referred to [46]. Multiplying the momentumequation by .1 C ln.1 C juj2//u, multiplying the mass equation by .1 C juj2/ ln.1C juj2/=2, integrating in space, and adding the result, we get

d

dt

Z

�

�1C juj2

2ln.1C juj2/C �

Z

�

� Œ1C ln.1C juj2//�jD.u/j2

� �

Z

�

Œ1C ln.1C ju2//�u � r�� C CZ

�

�jruj2:

It remains to bound the right-hand side. The second term is controlled using theenergy and BD entropy. Concerning the first term, it suffices to integrate by parts torewrite terms and put space derivative on u instead of � to get the bound

ˇˇˇ

Z

�

Œ1C ln.1C ju2//�u � r��ˇˇˇ �

Z

�

�jruj2 C�

2

Z

�

� Œ1C ln.1C juj2//�jD.u/j2

CC�

Z

�

Œ2C ln.1C juj2/j�2��1:

The first term is controlled using the energy and BD entropy estimates; the secondterm is absorbed in the left-hand side of the inequality written before. Let us remarkthat the last term is controlled through the estimate

Z

�

Œ2Cln.1Cjuj2/j�2��1 ��Z �

�2��ı=2�1�2=.2�ı/

.2�ı/=2.

Z

�Œ2Cln.1Cjuj2/�2=ı/ı=2

with ı 2 .0; 2/. The right-hand side is bounded in time using the energyand BD entropy controls. All the calculations show that �juj2 ln.1 C juj2/ 2L1.0; T IL1.�// if initially �0ju0j2 ln.1 C ju0j2/ 2 L1.�/. This plays a crucialrole in [59] to get existence of global weak solutions for barotropic compressible


Navier-Stokes equations with the degenerate viscosity �.�/ D ��, �.�/ D 0.This propagation of integrability is really due to the control obtained using the BDentropy and the energy estimates.

Remark 10. It is important to note that A. Mellet and A. Vasseur have proved in[46] the bound �njunj2 ln.1Cjunj2/ 2 L1.0; T IL1.�// if this quantity is integrableinitially using the energy estimate and the BD entropy. Remark that we can get alsosuch bound if an additional term is included in the momentum equations such asdrag term or singular close to vacuum pressure laws, see [14], and thus get thecompactness on

p�nun.

4.1.2 Heat-Conducting FluidsIn this subsection we consider the full Navier-Stokes system. We will focus onthe various energy estimates and compare the difference between the barotropiccase and the full system case. The full Navier-Stokes system consists of threeequations:

• Mass equation�! �

• Momentum equation �! P.�; �/• Total energy equality�! E D e C juj

2

2; e D E.�; �/

Energy estimate for heat-conducting flows: Integrating the total energy conserva-tion equation, we get

d

dt

Z

�

�

�

e Cjuj2

2

�

D

Z

�

�f � u:

Note that in this estimate, we loose any information on the gradient of the velocity.This is one of the difficulty for heat-conducting Navier-Stokes equations. Let’s tryto have some extra control: Multiplying the momentum equation by u and the massequation by juj2=2, summing the result and integration by part, we get that

1

2

d

dt

Z

�

�juj2 CZ

�

�2�jD.u/j2 C �jdiv uj2

�C

Z

�

rp � u DZ

�

�f � u:

We need to control the last term in the left-hand side. The pressure law is assumedto express as p.�; �/ D pc.�/C ph.�; �/, where pc.�/ (also called cold pressure,associated with pressure at zero temperature) will generate a positive term at the left-hand side of the equality, ph.�; �/ is a hot pressure. In order to be consistent withthe second principle of thermodynamics, the following compatibility condition, so-called “Maxwell equation” between P and E has to be satisfied:

P.�; �/ D �2 @E@�

ˇˇˇ�C �

@P@�

ˇˇˇ�: (18)


The specific entropy s D S.�; e/ is defined up to an additive constant by:

@S@e

ˇˇˇ�D1

�;@S@�

ˇˇˇ�D �

p

�2�:

Another important assumption on the entropy function is made:

the entropy S is a concave function of .��1; �/;

which ensures the nonnegativity of the so-called Cv coefficient given by

Cv D@E@�

ˇˇˇ�D �

1

�2@2S@e2

ˇˇˇ�1

�:

Finally, we assume that the equations of state (2) are of ideal polytropic gas type:

p D �r� C pc.�/; e D Cv� C ec.�/; (19)

(I) Assumptions.This part deals with assumptions regarding physical coefficients, such as viscosity,thermal conductivity, and equation of state and the assumptions on the initial data.

Viscosities, thermal conductivity, and equation of state.First of all, the viscosity coefficients � and � are assumed to be, respectively,C0.RC/ and C0.RC \ C

1.RC// functions of the density only, such that �.0/ D 0,and the following constraints are satisfied: there exists positive constants c0, c1, A,m � 1, and .d � 1/=d < n < 1 such that

for all s > 0; �.s/ D 2.s�0.s/ � �.s//;

for all s < A; �.s/ � c0sn and d�.s/C 2�.s/ � c0sn;

for all s � A; c1sm � �.s/ �sm

c0

and casm � d�.s/C 2�.s/ �

sm

c1:

(20)

We require that ec is a C2 nonnegative function on RC and the followingconstraint is satisfied in order to satisfy assumption (18)

pc.�/ D �2 dec

d�.�/:

We also require that there exists �� > 0; �� > 0; k > 1; ` > 1; C� > 0;C 0� > 0,and C�� > 0 such that for all � 2 .0; ��/;


��`�1

C�� p0c.�/ � C��

�`�1;��`�1

C 0�� e0c.�/ � C

0��`�1;

where ` �2n.3m � 2/

m � 1� 1;

(21)

and for all � > ��;

�1

�� p0c.�/ � C��

k�1; 0 � e0c.�/ � C0��

k�1;

where k �

�

m �1

2

�5.`C 1/ � 6n

`C 1 � n:

(22)

Initial data.Concerning the initial data, we assume that the functions �0, m0, and G0 satisfy

�0 � 0 a.e. on �; andjm0j

2

�0D 0 a.e. on fx 2 �=�0.x/ D 0g; (23)

and G0 has to be taken in such a way that

G0.x/ 2 �0.x/E.�0.x/;RC/ for a.e. x 2 �; (24)

which allows to define the initial temperature �0 on fx 2 �=�0.x/ ¤ 0g, which isassumed to be non negative

�0.x/DE.�0.x/; �/�1�nG0.x/=�0.x/

o�0 a.e. on fx 2 �=�0.x/¤0g: (25)

(II) Main result.Let us give the result

Theorem 6. Let us assume that the viscosity, thermal conduction, and equation ofstate satisfy the assumptions mentioned before. The initial data .�0;m0;G0/ aretaken in such a way that the assumptions mentioned before are satisfied and that

H.0/ DZ

�

�

G0 Cjm0j

2

2�0

�

dx < C1; (26)

that the initial density �0 satisfies

�0 2 L1.�/; and

r�.�0/p�0

2 L2.�/d ; (27)

and that the initial entropy density s0 D Cv log.�0=��0 / satisfies

�0s0 2 L1.�/: (28)


Then, there exists a global in time weak solution to heat-conducting Navier-Stokesequations (mass equation, momentum equation, and total energy satisfied + initialdata in the distribution sense) and weak regularity

�e and �juj2 2 L1.0; T IL1.�//;r�.�/p�2 L1.0; T IL2.�/d /; (29)

p�.�/ru 2 L2.0; T IL2.�/d�d /; (30)

.1Cp�/r�a=2 and .1C

p�/r�

�2 L2.0; T IL2.�/d /; (31)

for a � 2. Finally, one has for some large enough positive number

� and �E 2 C.RCIH� .�//; �u 2 C.RCIH

� .�/d /: (32)

(III) Physical energy.As it is pointed out above, the physical energy equality

d

dt

Z

�

�

�

e Cjuj2

2

�

dx D

Z

�

�f � u (33)

can be obtained in a classical way by multiplying the momentum equation by u andusing the energy equation.

Remark 11. For the barotropic case, we obtain the estimate

1

2

d

dt

Z

�

�juj2 C 2Z

�

�.�/jD.u/j2 Cd

dt

Z

�

.�/C

Z

�

�.�/jdiv uj2 DZ

�

�f � u;

which provides a H1 bound in space on the velocity u. But for the full system withheat conductive equation, the following facts hold:

1

2

d

dt

Z

�

�juj2 C 2Z

�

�.�/jD.u/j2 CZ

�

�.�/jdiv uj2

Cd

dt

Z

�

c.�/C

Z

�

rph.�; �/ � u DZ

�

�f � u;

where c.�/ D �ec.�/; ph.�; �/ D p.�; �/�pc.�/; and eh.�; �/ D e.�; �/�ec.�/.If we want to control on ru; we have to control

Z

�

rph.�; �/ � u D �Z

�

ph.�; �/ � div u:


In [26], E. Feireisl takes ph.�; �/ D �p�.�/; pc.�/ �� ; p� .�/ ��3 . Then it

holds that

ˇˇˇ

Z

�

�p�.�/div uˇˇˇ � k�kL6.�/kp�.�/kL3.�/kdiv ukL2.�/

� Ck�kH1.�/k�k�3

L� .�/kukH1.�/;

where we used the regularity of �; �, and u. For simplicity, we omit the details.

(IV) Thermal equations. Two different reformations of the internal energy equa-tions will lead to useful bounds on the temperature. The first one is written as

Cv�@t .��/C div .��/C r� � div u

�

D 2�D.u/ W D.u/C �jdiv uj2 C div .� � r�/

D 2�S.u/ W S.u/C

�

�C2�

d

�

jdiv uj2 C div .�r�/;

(34)

where the derivatotic part S.u/ of the strain rate tensor D.u/ is defined as the zerotrace component:

S.u/ D D.u/ � .div u/I=d:

Using the assumption that �0 � 0 a.e. on � and the fact that the first two terms ofthe right-hand side of (34) are nonnegative, the minimum principle formally appliesto the temperature.

The second form of the internal energy equation is the most physically relevant,since it involves the specific entropy s: Indeed, we deduce from the definition of sthat it satisfies formally:

�.@t .�s/C div .�us// D 2�D.u/ W D.u/C �jdiv uj2 C div .�r�/: (35)

Hence dividing (35) by � and integrating over �, we end up with

Z

�

�2�

�jD.u/j2 C

�

�jdiv uj2 C

�

�2jr� j2

�

Dd

dt

Z

�

�s:

Using the assumption that �0.s0/� 2 L1.�/; for t � 0; one has

Z t

0

Z

�

1

�

�2�D.u/ W D.u/C �jdiv uj2

�C

Z t

0

Z

�

�

�2jr� j2

�

Z

�

.�s.t; �/C �0.s0/�/ ;

(36)


where s0� D max.�s0; 0/: Obviously, the property on s will give the control of

Z T

0

Z

�

�

�2jr� j2 < C1: (37)

When the equation of state takes the form of (19), the physical entropy can be

written in terms of � and � as s D Cv log.�

��/; where � D � � 1: Introducing

�1 > 0 and recalling that s is defined up to an additive constant, we may write

�s D Cv� log � C �Cv� log.�1

�/

� Cv�� C �Cv� log.�1

�/:

Then it follows thatZ

�

�s �

Z

�

�Cv� C

Z

�

�Cv� log

��1

�

�

:

It remains to control the above term, which is done by using the mass conservationequation in a renormalized way with ˇ1.�/ D � log.�1=�/;

@tˇ1.�/C div .ˇ1.�/u/ � �div u D 0:

The right-hand side of (36) is estimated by

Z

�

�0.s0/� C

Z

�

�Cv�.t; �/C �Cv

�Z

�

ˇ1.�0/C

Z T

0

Z

�

�jdiv uj

�

:

The last term above can be estimated by the left-hand side of (36) as follows:

Z t

0

Z

�

�jdiv uj �Z t

0

Z

�

�12

.d�C 2�/12

�.d�C 2�/

12

�12

� jdiv ujp��;

where we use the Young inequality with the bound of �� in L1.0; T IL1.�// andthe assumption that s 7�! .d�.s/ C 2�.s// belongs to L1.RC/ since n � 1 andm � 1:

From above, it concludes that

p�

�jr� j 2 L2.0; T IL2.�//: From the properties

of �; i.e.,

�.�; �/ D �0.�; �/.1C �/.1C �˛/; ˛ � 2; (38)

where �0 is a C0.R2C/ function satisfying for all positive � and � ,

c3 � �0.�; �/ �1

c3;


for some positive constant c3; the following four quantities .d�C 2�/12 jdiv uj=

p�;

�12 S.u/=

p�; .p�C1/r�

a2 , and .1C

p�/r log � are a priori bounds inL2..0; T /�

�/ as soon as �0.s0/� and �0 log.�1=�0/ are bounded in L1.�/:(V) A priori estimates: Kinds of Energy and BD entropy. The additionalestimates are deduced as

1

2

d

dt

Z

�

�juj2 CZ

�

2�.�/D.u/ W D.u/CZ

�

�.�/jdiv uj2 DZ

�

p.�; �/div u;

and

1

2

d

dt

Z

�

�juC 2r'.�/j2 CZ

�

2�.�/A.u/ W A.u/

D �

Z

�

rp.�; �/.uC 2r'.�//;

where A.u/ D .ru � rut /=2 denotes the skew symmetric part of ru and ' isdefined up to a constant by '0.�/ D �0.�/=�: Adding the above two equalities, thefollowing terms have to be controlled :

Z

�

p div u; andZ

�

rp � r'.�/: (39)

Remark 12. For the barotropic case, the two terms in (39) are easily controlled by

Z

�

rp � u Dd

dt

Z

�

.�/;

and

Z

�

rp � r' D

Z

�

p0.�/�0.�/

�jr�j2:

(V-1) Control ofZ

�

rp �r'. In order to bound such an integral, the thermodynam-

ical properties of the fluid have to be used. Defining the “hot” pressure and energycomponents as ph.�; �/ D p.�; �/�pc.�/; and eh.�; �/ D e.�; �/�ec.�/; i.e., thepressure and energy associated with nonzero temperature effects. Notice that in thecase of an equation of state satisfying (19) with coefficients � > 1 andCv;where thepreceding five coefficients are constants and given by � D � � 1: Then tedious butstraightforward computations lead to the differential identity valid for any equationof state,

rp D rpc C r.�r�C �r�/:


Similar to the barotropic case, we obtain the following facts:

Z

�

rp � r' D

Z

�

p0c.�/�0.�/

�jr�j2 C

Z

�

rph � r':

Taking ph.�; �/ D r��; it follows that

Z

�

rph � r' D

Z

�

r��0.�/jr�j2

�C

Z

�

r�0.�/r� � r�;

where '0.�/ D �0.�/=�: Hence, the cold component of the pressure yields thefollowing integral for some positive constant c0 W

Z

�

p0c.�/�0.�/jr�j2

�� c0

Z

�

jr�.�/�.`C1�n/=2j2 �1

��

Z

�

jr�.�/j2

�;

the lower bound being deduced from assumption (22) and � being taken such that�.�/ D � for � � ��=2 and �.�/ D 0 for � � ��: On the other hand, it follows thatby using Cauchy-Schwarz-type inequalities

jr�0.�/r� � r�j � �.�; �/jr� j2

�2Cr2�2jr�.�/j2

�

� �.�; �/jr� j2

�2Cc

�0� �jr'.�/j2;

where (38) is used in the last inequality and c > 0 is a constant. Due to the

inequality (37) one can control the termZ

�

rp � r'. Note that integrability of

negative power of the density will be really important to get the compactness onthe temperature and to pass to the limit in the total energy formulation.

(V-2) Control ofZ

�

p div u. It remains to bound the term

Z

�

p � div u DZ

�

pc.�/div uCZ

�

ph.�/ � div u

D

Z

�

�2dec

d�div uC

Z

�

r�� div u

D �d

dt

Z

�

�ec.�/C

Z

�

r�� div u:

In the case of a perfect gas equation of state, we split the integrated expression intobounded and unbounded densities


ˇˇˇˇ

Z

�

r��div u

ˇˇˇˇ

�crk.d�C 2�/12 div ukL2.�/

��k�

25 �kL2.�/k�

.6�5n/=101�<AkL1.�/ C A� 12 k�1��AkL3.�/k�kL6.�/

�crk.d�C 2�/12 div ukL2.�/

�

�

k��k25

L1.�/k�k

35

L6.�/A

6�5n10 C A�

12 k�1�>AkL3.�/k�kL6.�/

�

��k.d�C 2�/12 div uk2

L2.�/Cc

�k��k2

L1.�/

Cc

�.1C k�k2

L6.�//.kr�.�/p�k2L2.�/

C k�k2L1.�/

/;

for all positive � and constant c:Using together with Sobolev embedding, we deducethat t 7�! k�.t; �/k2

L6.�/is a priori bounded in L1loc.RC/: Taking � small enough in

order to absorb �k.d�C2�/12 div uk2

L2.�/by the left-hand side coming from viscous

dissipation, observing that �� is already known to belong to L1loc.RCIL1.�//; the

third term will be estimated by a Gronwall-type lemma.With those estimates and some local integrability analysis of various energy

fluxes such as �ujuj2, �eu, pu, and �r� , we can study the compactness of sequencesof approximate solutions .�n; un; �n/ and pass to the limit in nonlinear terms toestablish the global existence of weak solutions, we omit it here and the reader canrefer [9] for the details. We will make only several comments. Compared to thebarotropic case and after getting the a priori estimates we have presented before, thedifficulty is of course the temperature dependency.

Remark that using the cold pressure term, we can get some bounds on the velocitywith weights depending on the density, for instance, on �1=3u in Lq with q > 3,we can also get some bounds on the temperature and the heat flux. To get strongcompactness for the internal energy and the temperature, the first step is to deriveuniform bounds on @t .�nEn/ using the bounds described before and then to getenough space compactness on

p�n�n taking advantage of the strong convergence

ofp�nun and that En D Cv�n C junj2=2. Using the strong compactness of ��1=2n

(obtained using the singular pressure close to vacuum) will give compactness on�n. Remark just that to pass to the limit in the total energy conservation equation,the only difficulty is to pass to the limit in the energy flux �nun.en C junj2=2/, theheat flux �.�n; �n/r�n, and the stress flux 2�.�n/D.un/ � un, �.�n/undivun. Forthe energy flux, �nun�n converges strongly to �u� in L1 as a product of

p�nun

andp�n�n which converges in L2. Moreover, �1=3n un converges to �1=3u in L3. For

the heat flux, it converges in L1 using the compactness on the temperature obtainedthrough the control of negative power of the density. Finally the stress flux converges


weakly in L1 using strong convergence on un, and on �n and weak convergenceon �.�n/1=2run. We refer to [9] for details and also mentioned [47] where heat-conducting compressible mixture with a degenerate viscosity with multicomponentdiffusion is considered.

4.2 Construction of Approximate Solutions

Compactness of the set of weak solutions has been proved. It remains to eventuallyconstruct solutions of approximate systems for which existence and uniqueness canbe easily obtained. The challenge is to add smoothing terms to the partial differentialequations that preserve the mathematical structure of the original equations, inparticular energy estimates and BD entropy estimates obtained, respectively, bymultiplying the momentum conservation equation either by u or by r'.�/, where'0.s/ D �0.s/=s. This issue is not straightforward for compressible Navier-Stokesequation with density dependent viscosities. Let us explain how it can be doneif a cold pressure is added for the heat-conducting Navier-Stokes equations: Thestability was proved in [10]. We will also explain the construction of approximatesolutions for the barotropic compressible Navier-Stokes equation with degenerateviscosities. It has been, for instance, fully described in [60] and [34] based onproposition made in [10].

(I) Barotropic case.Let us present the approximate procedure used in [60] and based on propositionswritten in [10]:

@t�C div .�u/ � "� D 0

@t .�u/C div .�u˝ u/Crp Crpc.�/ � 2div.�D.u//C "r� � ru (40)

C�2uC r0uC r1�juj˛u � ��r.

1p�p�/ � ı�r2sC1� D �f (41)

with s large enough and ˛ > 0. The interested reader is also referred to [48]where they design appropriate approximate solutions to a model of two componentcompressible reactive flows. Note that all terms �, pc.�/, 2u, and �r2sC1�

are important to build approximate solutions (the first one is also present in theconstant viscosities case, see [51]). Then it is possible to pass to the limit letting", ı, and � tend to zero. It is possible to build approximate solutions satisfyinguniformly the energy estimates and the BD entropy but which does not satisfy theMellet-Vasseur estimate uniformly (Galerkin method and fixed-point procedure arethe usual steps). Therefore, at this stage, we are just able to obtain the followingexistence results:


• If pc.�/ maintained, we can let the other quantities go to zero, namely, r0; r1; �:Global existence of compressible Navier-Stokes equations with singular pressure(cold pressure).

• If r1 is maintained, we can let the other quantities go to zero.Global existence of compressible Navier-Stokes equations with turbulent dragterm.

• We can prove the existence with r0; r1 > 0 and � > 0.Global existence of compressible quantum Navier-Stokes equations with dragterms.

It is important to note the last papers written recently by A. Vasseur andC. Yu [59] and by J. Li and Z. Xin [43] where they get global existenceof weak solutions of compressible Navier-Stokes equations with degenerateviscosities without any additional terms. In [59], they use the existence ofglobal weak solutions for compressible Navier-Stokes quantize equations withdrag terms (choosing ˛ D 2) and quantum term ��r.

p�=p�/ to perform

a kind of renormalization technic on juj to show the possibility to pass tothe limit with respect to the capillarity coefficient using truncation close tovacuum and for large density (a link between the truncated parameter andthe capillarity one is necessary). Then they pass to the limit with respectto the drag coefficients. In [43], they propose an appropriate regularizedsystem and show the limit passage. The interested reader is referred to theseworks.

(II) Heat-conducting case.Let us present the different quantities step by step in order to explain things forbeginners.

Step 1: Density control. The first step is to add to the right-hand side of themomentum conservation equation a high-order Korteweg-like term and anadditional pressure term in order to control the density:

@t�C div .�u/ � "� D 0

@t .�u/C div .�u˝ u/Crp Crpc.�/C "r� � ru � 2div.�D.u// D �f

C��r�2sC1�

�

@t .�E/C div .�uH C pau/ D div .� � u/C div .�r�/C �f � u

C��u � r�2sC1�

�: (42)

The additional cold pressure term is designed to get high integrability propertiesfor 1=�: one can choose, for instance, pc.�/ D �N�1�N where N D 6n. Thecorresponding cold internal energy therefore satisfies ec.�/ D ��N .

Let us observe that (42) has an additional term proportional to " associated withthe work of high regularity Korteweg force in the momentum conservation equation.


As a consequence, the entropy evolution equation remains identical to (35)

� .@t s C div .�us// D 2�D.u/ W D.u/C �jdiv uj2 C div .�r�/ ;

so that the estimates resulting from integration of the entropy equation remain valid.In particular, the estimates involving

Z

�

p div u

remain unchanged. In particular, one obtains L1.0; T IL1.�// bounds on "��N . Itremains to identify the contribution of the high regularity Korteweg force in the twoestimates obtained by multiplying the momentum conservation by u and ��1r�.�/.

On the one hand, multiplying the high regularity Korteweg force by the velocityu and integrating by parts using the mass conservation equation leads to

Z

�

�u � r�2sC1�

�D

Z

�

@t�2sC1�

D �1

2

d

dt

Z

�

jjDj2sC1�j2:

As a consequence of this bound, one may write

D

2 1

�

L2.�/

� C

D2�

L1.�/

1

�2

L2.�/

C kD�k2L1.�/

1

�3

L2.�/

!

:

Using the fact that �.s/ � c0sn, one concludes that for " > 0 given ��1 belongs toL1..0; T / ��/, so that the density is bounded from below by a positive constant.

On the other hand, multiplication of Korteweg-like force by ��1r� leads to

�

Z

�

�2sC1� D �

Z

�

jjDj2sC2�j2:

Step 2: Velocity smoothing. In order to smooth out the velocity field u, we add asmoothing term depending on parameter � > 0

@t�C div .�u/ D "�

@t .�u/C div .�u˝ u/Crp Crpa.�/ D div � C �f

C��2sC1.�u/C ��r2sC1� � "r� � ru

@t .�E/C div .�uH/ D div .� � u/C div .�r�/C �f � u

C��u �2sC1.�u/C ��u � r2sC1�: (43)


The contribution of this additional term �2sC1.�u/ to the estimates is the fol-lowing: scalar multiplication by u and integration over � of the momentumconservation equation leads to an additional dissipative term

Z

�

jjDj2sC1.�u/j2;

whereas its contribution to the entropy equation is zero because the work of suchforces has been added to the total energy conservation equation. We therefore obtainan addition bound in L2.0; T IH2m.�// on �u. On the other hand, the BD entropyestimate obtained by multiplying the momentum conservation equation by '.�/leads to

Z

�

�2sC1.�u/ � r log � DZ

�

r2sC1.�u/ � r2sC1�:

A Cauchy-Schwarz argument for the right-hand side then allows to conclude thatthis term is estimated by the L2.0; T IH2sC1.�// bound on �u, using the fact thathigh derivatives of functions of � are L2 integrable for given " > 0.

Step 3: Temperature smoothing and control. In order to smooth out the temperaturewhile preserving the structure of the entropy evolution equation, we add thefollowing perturbation to the temperature equation for some small parameter! > 0

Cv�@t .�� C ˇ�

4/C div ..�� C ˇ�4/u/C .r� � Cˇ

3�4/ div u

�� div.��r�/

D 2�D.u/ WD.u/C�

�2� ��5

(44)where �� is the regularized conductivity coefficient. Note that the two terms��2� ��5 are added as in [25] or [47] to ensure the temperature stays away from

zero and is bounded from above. The added radiative quantity in the pressureˇ�4 is to ensure compactness of density without playing with the cold pressure.The corresponding entropy evolution equation integrated by parts leads to

Z

�

�2�

�jD.u/j2 C

�

�2jr� j2

�

Dd

dt

Z

�

�s:

Step 4: Construction of smooth solutions. Given fixed positive parameters, the apriori estimates induced allow to build solutions by using fixed-point procedurevia Galerkin type methods: continuity equation and temperature equation(regularizing the equation in such a way that the classical theory for quasilinearparabolic equations could be applied). Thus uniform estimates provide global intime solvability. A fundamental estimate is obtained on the entropy dividing theinternal energy equation by � and integrating. Limit passage in the Galerkinapproximation. The strong convergence of the temperature is obtained by


stability process. It remains now to let the parameters go to zero using thestability process described before due to uniform bounds. We will not detailthings and refer to [47] for details in the case of heat-conducting, compressiblemixtures with multicomponent diffusion.

5 The �-entropy and Two-Velocities Hydrodynamics

In this section, we will discuss a new concept of solution introduced in [14]. Thisshows interesting phenomena with two-velocities hydrodynamic in the barotropiccompressible system and an arbitrary mixing parameter � 2 .0; 1/: This presenceof two velocities is due to the density dependency of the viscosities. This will givea nice � nonlinear hypocoercivity property which will be helpful for asymptoticanalysis through relative entropy techniques.

It is strange that some properties artificially introduced in the constant viscositiescase by H. Brenner appear in this situation, see [4–6]. For reader’s convenience, letus first present H. Brenner’s system studied mathematically in [30]:

8<

:

@t�C div.�um/ D 0;@t .%u/C div.�u˝ um/Crp D divS;@t��. 1

2jvj2 C e/

�C div

��. 1

2jvj2 C e/vm

�C div.pv/C divq D div.Sv/

(45)

with pressure laws p D pe.�/ C �pt .�/ where � is the temperature. The internalenergy splits into two parts

e.�; �/ D ee.�/C cv� with cv > 0

the two-velocity fields are related through the following constitutive relation

v � vm D Kr log �

and the Fourier’s law is given by

q CKPe.�/r log � D ��r�

where � is the heat conductivity coefficient and S is given through the relation

S D ��2D.u/ �

2

3divu IRn

C �divu IRn

with D.u/ D .ru C r tu/=2 is the strain tensor. Remark the two-velocitieshydrodynamic introduced by H. BRENNER artificially. The new model providesa relatively simple and rather transparent modification of the classical systemreplacing the Eulerian mass velocity um by its volume counterpart u in the viscousstress tensor and the specific momentum, where um and u are interrelated through


a kind of Fick formula. The two velocities coincide in the “incompressible” regimewhen � D const . We will see that such property (two-velocities framework) is infact included in the compressible Navier-Stokes equations when the two viscositiesdepend on the density, see [14] for discussions around heat-conducting flows.

5.1 The Barotropic Compressible System

Let us recall the compressible Navier-Stokes equations with the following viscosi-ties �.%/ D ��, �.�/ D 0:

�@t�C div.�u/ D 0;@t .%u/C div.�u˝ u/Crp.%/ � 2�div.�D.u// D 0

(46)

with D.u/ D .ru C .ru/T /=2. Recently in [14], it has been observed that suchcompressible Navier-Stokes system may be reformulated through an augmentedsystem. The main idea is to use a parameter � 2 .0; 1/ to mix energy and BDentropy, namely, we consider the following quantity in the kinetic energy

��juj2 C .1 � �/�juC 2�r log �j2:

This may be seen as a two-velocity hydrodynamic with �1 D ��, u1 D u and�2 D .1 � �/�, u2 D u C 2�r log �, see [32] and [55]. This motivates theintroduction of an intermediate velocity v D u C 2��r log �, a drift velocityw D 2

p�.1 � �/�r log �, and a mixture coefficient �, then the system reads as

a two-velocity hydrodynamic system:

8ˆ̂ˆ̂ˆ̂ˆ̂<

ˆ̂ˆ̂ˆ̂ˆ̂:

@t%C div.% .v � 2��r log �// D 0;@t .%v/C div.%v ˝ .v � 2��r log �/Crp.%/

D �div.2�.1 � �/D.v//

C�div.2�%A.v// � �div�2p�.1 � �/%rw

;

@t .�w/C div.�w˝ .v � 2��r log �//D �div.2�%rw/ � �div.2

p�.1 � �/%.rv/T /:

(47)

with A.v/ D .rv � .rv/T /=2. The associated �-entropy reads for all t 2 Œ0; T �:

sup�2Œ0;t �

Z

�

h�.jvj2

2Cjwj2

2/C .�/

i.�/dx

C2�R t0

R��jA.v/j2 C jD.

p1 � �v/ � r.

p�w/j2

dx ds

C2��

Z t

0

Z

�

p0.�/

�jr�j2 dx ds �

Z

�

h%.jvj2

2Cjwj2

2/C .%/

i.0/

(48)


with s 0.s/� .s/ D p.s/. This �-entropy is obtained by taking the scalar productof equation satisfied by v and w, respectively, with v and w, adding the results andusing the mass equation. Note that the more general case with �.%/ arbitrary and�.%/ D 2.�0.%/% � �.%// for the compressible Navier-Stokes equations is coveredin [14] (but with extra drag or cold pressure terms for the existence). This allows todefine appropriate global solutions named �-entropy solutions to the compressibleNavier-Stokes equation with degenerate viscosities. It works well also for generalviscosities �.�/ and �.�/ such that �.�/ D 2.�0.�/� � �.�//: See [14] for details.It is then possible to prove the following mathematical result.

Theorem 7. Let �.�/ D 2.�0.�/� � �.�// with P .�/ D a�� with � > 1. Assume

�0 2 L�.�/; �0 � 0; r�.�0/=

p�0 2 L

2.�/;

m0 2 L1.�/; m0 D 0 if �0 D 0; jm0j

2=�0 2 L1.�/:

Then there exists a global �-entropy solution .�; u/ of the degenerate compressibleNavier-Stokes equations with extra drag terms or cold pressure that means satisfy-ing a �-entropy similar to (48), and the compressible Navier-Stokes equation withits boundary conditions in a weak sense.

Remark 13. It is important to note that global existence of �-entropy solution hasbeen proved in [14] for the compressible Navier-Stokes with a drag term or singularcold pressure. Note that it is not difficult to prove that a �-entropy solution satisfiesalso the augmented formulation in a weak sense.

Let us just precise here the regularized system in the case �.�/ D �� and�.�/ D 0 which allows to build approximate solutions which will give after passingto the limit a global �-entropy solution:

8ˆ̂ˆ̂ˆ̂ˆ̂<

ˆ̂ˆ̂ˆ̂ˆ̂:

@t%C div.%vı/ � 2�� D 0;@t .%v/C div.v ˝ .%vı � 2��r�/Crp.%/C "2sv � �div..1C jrvj2/rv/

D �div.2�.1 � �/D.v//

C�div.2�%A.v// � �div�2p�.1 � �/%rw

;

@t .�w/C div.w˝ .�vı � 2��r�//D �div.2�%rw/ � �div.2

p�.1 � �/%.rv/T /

(49)

where the smoothing parameter ı designs standard mollification with respect to timeand space. The smoothing high-order derivative term 2s with s � 2, depending onsmall parameter � > 0, has to be introduced to control large spatial variations ofv, because div v is not a priori bounded in L1.0IT I .L1.�//. Such bound will berequired to be able to have bounds on the density. We will also need to show thatw D 2r log � at some point of the construction process and the second term in theregularization process will be helpful.


Remark 14. This is interesting to note that, by changing the definition of the veloc-ity, a diffusive term appears in the mass equation. No need to add it as previously:compressible Navier-Stokes equations with density-dependent viscosities ensurein some sense parabolicity property of the density. This property has been alsoobserved by B. Haspot, see [36]. Note also the cross-diffusion effect, we get,between the velocity v and the drift velocity w.

Remark 15. Note that this formulation will be very helpful to obtain global exis-tence of weak solutions for compressible Navier-Stokes equations with degenerateviscosities for quite large range of viscosities �.�/, �.�/ such that �.�/ D2.�0.�/� � �.�//. This is the purpose of the forthcoming paper [21]. It requiresto consider a third-order regularized term compatible with the viscous term.

5.2 The Heat-Conducting Compressible System

In this section, we present the equations of motions for the heat-conducting fluidwritten in terms of the two velocities u and u C 2r'.�/ with correspondingdensities .1 � �/� and ��: This presentation follows [14]. We do not aim atproving the existence result for such system but on showing that the two-velocityhydrodynamics in the spirit of the work by S.M. SHUGRIN [55] is consistent withthe study performed for the low Mach number system in the first part of this diptychin [15]. More precisely, we will show that the formal low Mach number limit forthe two-velocities system gives the augmented system used in [15] to constructthe approximate solution. An important observation is that the system presentedbelow does not coincide with the usual heat-conducting compressible Navier-Stokesequations. Indeed, the two-velocities description of the dynamics of the fluid lead todifferent energy equations with a generalized temperature, called the �-temperature.However, this is not a priori the usual temperature, unless the system reducesto the angle velocity one (i.e., the density �� is equal to 0). This property wasalso explained in the works [55] and [32] where the authors discuss the capillarytemperature. As mentioned these calculations may be found in [14].

We assume that � is a periodic box in R3, i.e., � D T

3, and we consider thefollowing two-velocity system

8ˆ̂ˆ̂ˆ̂ˆ̂<

ˆ̂ˆ̂ˆ̂ˆ̂:

@t�C div.�u/ D 0@t�uC div.�u˝ u/ � div.2�.�/D.u// � r.�.�/divu/Crp.�; e�/ D 0@t .�.uC 2r'.�//C div.�u˝ .uC 2r'.�/// � div.2�.�/A.u//

Crp.�; e�/ D 0

@tE� C div.E�u/C div.p Œ.1 � �/uC �.uC 2r'.�/�/C divQ�

�div�.1 � �/S1.u/C �S2.uC 2r'.�//

D 0;

(50)


where we denotedD.u/ D 12.ruCr tu/ and A.u/ D 1

2.ru � r tu/. The viscosity

coefficients �.�/, �.�/ satisfy the BRESCH-DESJARDINS relation

�.�/ D 2�0.�/� � 2�.�/:

The total �-energy E� is defined as follows

E� D ��e� C

.1 � �/

2juj2 C

�

2juC 2r'.�/j2

:

Remark 16. Note that E�u is expressed as a sum of two energies

.1 � �/�

�

e� Cjuj2

2

C ��

�

e� CjuC 2r'.�/j2

2

similarly to energy from [55]. Integrating the total �-energy equation with respectto space, we obtain

d

dt

Z

�

E� D 0:

Thus (5.2) and the identity

.1 � �/

2juj2C

�

2juC 2r'.�/j2 D

juC 2�r'.�/j2

2C .1� �/�

j2r'.�/j2

2(51)

yields the following conservation property

d

dt

Z

�

��e� C

1

2juC �r'.�/j2 C

.1 � �/�

2jr'.�/j2

D 0:

This quantity may be treated as a generalization of the �-entropy, found for thebarotropic case, to the heat-conducting case.

The viscous tensors S1 and S2 are given by

S1 D 2�.�/D.u/C �.�/div u Id

and

S2 D 2�.�/A.u/:

The heat flux Q� is given by standard Fourier’s law, i.e.,

Q� D �Kr��;


whereK is the positive heat-conductivity coefficient and �� denotes the generalizedtemperature (the �-temperature). Let us consider an ideal polytropic gas, namely,

p.�; e�/ D r�� C pc.�/; e� D Cv�� C ec.�/;

where r and Cv are two positive constant coefficients, see, for instance, [9].For convenience, we denote � D 1 C r=Cv . Moreover, the additional pressureand internal energy, pc and ec , respectively, are associated to the “zero Kelvinisothermal,” which roughly speaking means that

lim�!0C

pc.�/ D �1:

Further, we require that ec is aC2.0;1/ nonnegative function and that the followingconstraint is satisfied

pc.�/ D �2 dec

d�.�/:

Below we present two different formulations of the internal energy equationwhich lead to useful bounds on �-temperature similarly as in [9] for the usualtemperature. The first formulation reads

Cv

�@t .��/C div.�u��/C ��div w

D 2.1 � �/�.�/jD.u/j2 C 2��.�/jA.u/j2

C2.1 � �/.�0.�/� � �.�//jdiv uj2 C div.Kr��/; (52)

with � the Gruneisen parameter and where the mixing temperature �� becomes theusual temperature if � D 0. Note that for 0 � � � 1, the �-temperature remainsnonnegative in view of the maximum principle. The second formulation is based onthe notion of generalized �-entropy s� . It is the usual entropy in which the standardtemperature has been replaced by the �-temperature, i.e.,

s� D Cv log �� r log �;

thus, when �; �� is sufficiently regular we can derive the following equation

@t .�s�/C div.�us�/ � div.Kr log ��/ (53)

D 2.1 � �/�.�/jD.u/j2

��C 2�

�.�/jA.u/j2

��

C2.1 � �/.�0.�/� � �.�//jdiv uj2

�� 2��'.�/CK

jr�� j2

�2�:


Note that, recalling the relation

jD.u/j2 D

ˇˇˇˇD.u/ �

1

3divu Id

ˇˇˇˇ

2

C1

3jdivuj2;

the terms on the right-hand side, when integrated over space, give nonnegativecontribution using the assumption on 3�.�/C2�.�/ and �.�/ D 2.�0.�/��.�//.Indeed, it suffices to check that for the penultimate term reads

Z

�

��'.�/

��D

Z

�

�'0.�/jr�j2 � 0:

Using all this information, it could be possible to prove global existence of�-entropy solution for the heat-conducting compressible Navier-Stokes systemunder analogous assumptions as in [9], replacing the usual temperature by the �-temperature. The existence of the approximate solution could be then proven byusing the augmented system written in terms of w D uC2�r'.�/ and v D 2r'.�/as it was done in [55] or in [14] addressing barotropic flows

@t�C div.�Œw�ı/ � 2�div.Œ�0.�/�˛r�/ D 0; (54)

@t .�w/C div..�Œw�ı � 2�Œ�0.�/�˛r�/˝ w/ � r..�.�/

�2�.�0.�/� � �.�///div.w � �v// � 2.1 � �/div.�.�/D.w//

�2�div.�.�/A.w//C"2sw � "div..1C jrwj2/rw/Crp.�; e�/

D �2�.1 � �/div.�.�/rv/; (55)

@t .�v/C div..�Œw�ı � 2�Œ�0.�/�˛r�/˝ v/ � 2�div.�.�/rv/

C2r..�0.�/� � �.�//div.w � �v// D �2div.�.�/r tw/ (56)

with the �-total energy supplemented by the � correction corresponding to the �regularization of the momentum

@t .�E�/C div.��Œw�ı � 2�Œ�

0.�/�˛r��E�/C div.p w/C divQ (57)

�div�S1wC .1 � �/� S2v

C "sw � w � "div..1C jrwj2/jwj2 D 0 (58)

and the set of initial conditions. Above, the total �-energy E� is defined as

E� D e� C1

2jwj2 C

�.1 � �/

2jvj2:

Note, however, this construction would not lead to the usual heat-conductingcompressible Navier-Stokes system in the limit � ! 0. Indeed, the difference isagain due to the �-temperature that is not the usual one. But, performing a formal


low Mach number limit for this system, we would get p D 1, div w D 0 (comparingterms of the same order). In the equation on w, being now incompressible, thepressure gradient rp would be replaced by Lagrangian multiplier r . As aresult, we would get the augmented system defined in [15] in the part devoted toconstruction of solution.

6 Relative Entropy and Some Applications

(I) Relative entropy for barotropic Navier-Stokes equations withdensity-dependent viscosities.In this section, let us provide recent results obtained by D. Bresch, P. Noble, and J.P.Vila in [19] with details in [20]. Since the pioneering work by C. Dafermos and H.T.Yau, relative entropy methods have become a crucial and widely used tool in thestudy of asymptotic analysis (singular limits, long-time behavior). The very roughidea is work with an energy and make a Taylor expansion of this energy around astate written in terms of conservative quantities. In our case, linked to the �-entropyestimate, let us consider the relative energy functional, denoted E.�; v;w

ˇˇr; V;W /,

defined by

E.�; v;wˇˇr; V;W / D

1

2

Z

�

%.j w �W j2 C jv � V j2/dx (59)

C

Z

�

.F .%/ � F .r/ � F 0.r/.% � r//dx

which measures the distance between a �-entropic weak solution .%; v;w/ to anysmooth enough test function .r; V;W /. This is similar to the one that we can findin [27] for compressible Navier-Stokes with constant viscosities but here with two-velocity fields because we will play with the augmented system presented in theprevious section. In the following theorem, we consider the case where �.�/ D ��with � a strictly positive constant (for the general case, the reader is referredto [16]). We will explain later on for readers who don’t know how to find theappropriate relative entropy quantity in the compressible Navier-Stokes equationssetting. Consider the case where �.�/ D ��, �.�/ D 0, then we can prove:

Theorem 8. Any weak solution .�; v;w/ of the compressible Navier-Stokes equa-tions satisfies the following so-called relative entropy inequality: For all � 2 Œ0; T �and for any pair of test functions such that r 2 C1.Œ0; T � � �/ with r > 0, andV;W 2 C1.Œ0; T � ��/ W

E.�; v;wˇˇr; V;W /.�/ �E.�; v;w

ˇˇr; V;W /.0/

C2��

Z �

0

Z

�

%ŒA.v � V /j2 C 2�Z �

0

Z

�

%jD.p.1 � �/.v � V / �

p�.w �W //j2


C2��

Z �

0

Z

�

%hp0.%/r log % � p0.r/r log r

i�hr log % � r log r

i

�

Z �

0

Z

�

%�..v �

r�

.1 � �/w/ � rW / � .W � w/

C..v�r

�

.1 � �/w/ � rV / � .V � v/

C

Z �

0

Z

�

%�@tW � .W �w/C@tV � .V �v/

C

Z �

0

Z

�

@tF0.r/.r � %/ �

Z �

0

Z

�

rF 0.r/ �h%.v �

r�

.1 � �/w/

�r.V �

r�

.1 � �/W /

iC

Z �

0

Z

�

.p.r/ � p.%// div.V �r

�

.1 � �/W /

��

Z �

0

Z

�

p0.%/r% � Œ2�rr

r�

1p.1 � �/�

W �C 2�

Z �

0

Z

�

%�D�p.1 � �/V /

�r.p�W /

�W�D�p.1 � �/.V � v// � r.

p�.W � w/

�

C2��

Z �

0

Z

�

%A.V / W A.V � v/C 2��Z �

0

Z

�

%

rp0.r/rr � .

rr

r�r%

%/

C2p�.1 � �/�

Z �

0

Z

�

%hA.W / W A.v � V / � A.w �W / W A.V /

i: (60)

These are simple calculations and interested readers are referred to [19] (and [20]for details). By using a density argument, we can of course relax the regularity onthe test functions using the regularity of the �-entropy solutions as it was done in[27] for the constant viscosities. Remark that here, we do not assume W to be agradient. The third line is also original compared to [35] and allows to relax thestrong constraint imposed in [35] that the viscosity is proportional to the pressurelaw and covering now the physically founded case of the shallow-water equations.Note the recent work in [16], where they prove that the augmented system is reallyinteresting to define relative entropy estimates for Euler-Korteweg systems in asimplest way compared to recent works [22] and [33] with less hypothesis on thecapillarity coefficient. They also provide relative entropy tools for compressibleNavier-Stokes-Korteweg with density-dependent viscosities satisfying the Bresch-Desjardins relation �.�/ D 2.�0.�/� � �.�/. This algebraic relation plays again acrucial role to get the results.

(I-a) How to get relative entropy terms? Let us present for reader’s conveniencehow to get modulated quantity in the framework of the compressible Navier-Stokesequation. In the �-entropy, the kinetic quantity reads

1

2

Z

�

�.jvj2 C jwj2/CZ

�

F .�/:


These quantities correspond to �, �v, and �w in the time derivatives of mass andmomentum equations. We therefore linearized the kinetic energy with respect tothis variables. The quantity coming from the pressure F .�/ is modulated as follows(linearization around r)

I1 D F .�/ � F .r/ � F0.r/.� � v/

and the quantity related to �jvj2 or �jwj2 as follows (linearization around .r; rV / or.r; rW /)

I2 D j�vj2=� � jrV j2=r C

jrV j2

r2.� � r/ � 2

rV � .�v � rV /

r

D �jvj2 C �jV j2 � 2�V � v D �jv � V j2:

The same conclusion occurs for .r;W /. We therefore find the relative kineticquantity already present for compressible Navier-Stokes equations with constantviscosities: see [28].

Let us now modulate the quantities coming from the viscous quantities and thepressure term in the � entropy. Remark that in the viscous quantity the conservedquantity is �D.u/ or �A.u/ and remark that for the pressure term the quantity is �and �v.

Concerning the pressure term, it reads p0.�/�jvj2. Then after calculation, we getthe following modulated quantity

I3 Dp0.�/

�j�vj2 �

p0.r/

rjrV j2 �

�p0.r/

r

�0jrV j2.�� r/� 2

p0.r/

r.rV / � .�v � rV /:

Concerning the viscosity term j�D.u/j2=�, the calculation is the same than for thekinetic quantity and the modulated quantity reads

I4 D1

�j�D.u/j2 �

1

rjrD.U /j2 �

�1

r

�0jrD.U /j2.� � r/

�21

rrD.U / � .�D.u/ � rD.U // D �jD.u/ �D.U /j2 (61)

The expression is similar for �.�/A.u/. Remark that we can write

I3 D %Œp0.%/r log % � p0.r/r log r� � Œr log % � r log r�

CrŒp.�/ � p.r/ � p0.r/.� � r/� � r log r:

This explains the chosen modulated quantity for the term coming from the pressurenoticing that the last quantity is related to the relative kinetic entropy. For moregeneral viscosities, the reader is referred to [16]. The interesting feature is thatrelative entropy for compressible Navier-Stokes equation work using the BDrelation between � and �, namely, � D 2.�0.�/� � �.�//.


(I–b) Some applications. Let us now present several applications that we canconsider using this relative entropy tool: see [28] for the compressible Navier-Stokesequations with constant viscosities. Using the relative entropy (60) and the identity

%Œp0.%/r log % � p0.r/r log r� � Œr log % � r log r� (62)

D %p0.%/jr log % � r log r j2

CrŒp.%/ � p.r/ � p0.r/.% � r/� � r log r

��%.p0.%/ � p0.r// � p00.r/.% � r/r

�ˇˇr log r j2;

we can justify several mathematical results. The interested reader is referred to [20]for details of the proof. Let us mention two of them which extend to the density-dependent viscosities the well-known results for constant viscosities.

(I) Weak-strong uniqueness.Let us consider a �-entropy solution .%;u/ and recall v D u C 2��r log % andw D 2�

p�.1 � �/r log %. Assume that .r;W; V / satisfies the augmented system

with the regularity written before and assume thatW D 2�p�.1 � �/r log r . Then

we prove that .%; v;w/ D .r; V;W / that means weak-strong uniqueness property:this gives .%;u/ D .r; U / with U D V �

p�W =

p.1 � �/. More precisely the

following result holds.

Theorem 9. Let � be a periodic box. Suppose that p.%/ D a%� with � > 1.Let .%;u/ be a �-entropy solution to the compressible Navier-Stokes system (46).Assume that there exists a strong solution .r; U / of the compressible Navier-Stokes equations (46) such that the terms in (60) are defined with r > 0 andr 2 L2.0; T IW 1;1.�// \ L1.0; T IW 2;1.�//. Then the following weak-stronguniqueness result holds: .%;u/ D .r; U /.

(I-c) Inviscid limit: convergence to dissipative solution.Let us recall the definition of a dissipative solution of compressible Euler equations.Such concept has been introduced by P.L. LIONS in the incompressible setting: see,for instance, [45]. The reader is referred to [27] and [3] for the extension to thecompressible framework with constant viscosities. Here we deal with an example ofdensity-dependent viscosities with a dissipative solution target. Of course the targetcould be the local strong solution of the compressible Euler equations similarlyto [57].

Definition. The pair .%; u/ is a dissipative solution of the compressible Eulerequations if and only if .%; u/ satisfies the relative energy inequality

E.%; u; 0ˇˇr; U; 0/.t/ � E.%; u; 0

ˇˇr; U; 0/.0/ exp

�c0.r/

Z t

0

kdivU.�/kL1.�/d��

C

Z t

0

exphc0.r/

Z t

s

kdivU.�/kL1.�/d�i Z

�

%E.r; U / � .U � u/ dxds


for all smooth test functions .r; U / defined on Œ0; T ��/ so that r is bounded aboveand below away from zero and .r; U / satisfies

@t r C div.rU / D 0; @tU C U � rU CrF0.r/ D E.r; U /

for some residual E.r; U /. We prove the following result in [20]:

Theorem 10. Let .%";u"/ be any finite �-entropy solution to the compressibleNavier-Stokes equations (46) in the periodic setting replacing � by ". Then, anyweak limit .%; u/ of .%";u"/ in the sense

%" ! % weakly in L1.0; T IL�.�//;

%"jv"j2 ! % juj2 weakly in L1.0; T IL1.�//

%"jw"j2 ! 0 weakly in L1.0; T IL1.�//

with v" D u" C 2"�r log %" and w" D 2"p�.1 � �/ log %" as " tends to zero is a

dissipative solution to the compressible Euler equations.

As a by-product this justifies the limit between a viscous shallow-water system tothe inviscid shallow-water system. Using the relative entropy, it is possible to provethe convergence of the viscous shallow-water system to the incompressible Eulerequations: low Froude and inviscid limit using the mean velocity plus the oscillatingpart as target functions (see [25] for the constant viscosities case and [20] for thedensity-dependent viscosities case).

(II) The heat-conducting Navier-Stokes equations.Concerning relative entropy and heat-conducting Navier-Stokes equations, nothinghas been done for the density-dependent viscosities case. The only results known bythe authors are the nice works by E. Feireisl and A. Novotný et al. concerning theheat-conducting Navier-Stokes equations with constant viscosities: The interestedreader is referred to [50] and references cited therein.

7 Conclusion

In this chapter, we discussed the compressible Navier-Stokes equations withdensity-dependent viscosities. We hope to have shown that it is not straightforwardto extend to this framework the important properties proved in the constantviscosties case by P.L. Lions, E. FEIREISL, A. NOVOTNÝ et al., and more recentlyD. Bresch and P.E. Jabin.

The main challenging question is to understand how to make a link betweenthese two frameworks which seem to be so orthogonal perhaps playing with the� parameter. Concerning the relative entropy, it would be interesting to extend


what has been done for the barotropic case to the heat-conducting Navier-Stokesequations when density dependent viscosities are taken into account. It would bealso interesting to understand the two-velocities hydrodynamic problem for the heat-conducting Navier-Stokes equations and this new PDEs that we discussed with aperturbed temperature definition due to the heterogeneity.

Cross-References

�Equations and Various Concepts of Solutions in the Thermodynamics of Com-pressible Fluids

� Scale Analysis and Singular Limits in the Mathematical Theory of Compressible,Viscous, Heat Conducting and/or Rotating Fluids

�Weak Solutions for the Compressible Navier-Stokes Equations: Existence, Stabil-ity, Long Time Behavior

�Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in CriticalCases

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