Lecture Notes on the Navier-Stokes-Fourier system: weak

solutions, relative entropy inequality, weak strong uniqueness

Antonin Novotny

IMATH, Universite du Sud Toulon-Var, BP 132, 839 57 La Garde, France

Dedicated to the memory of Alexander Kazhikov

Abstract

We investigate three issues in the theory of the Navier-Stokes-Fourier system describing the motionof a compressible, viscous and heat conducting uid: weak compactness of the family of weak solutions(that is the main ingredient in the proof of the existence of weak solutions), relative entropy inequalityand weak-strong uniqueness principle.

1 Introduction

These Lecture Notes are devoted to some aspects of the theory of the Navier-Stokes-Fourier system. Weshall discuss 1) existence of weak solutions, 2) existence of suitable weak solutions and relative entropies,3) weak strong uniqueness property in the class of weak solutions. For physical reasons, we shall limitourselves to the three dimensional physical space, and for the sake of simplicity, to the ows in boundeddomains with no-slip boundary conditions.

1.1 Nonexhausting bibliographic remarks

1.1.1 Weak solutions

There are several ways to dene weak solutions for the complete Navier-Stokes-Fourier system. Here, weshall mention three of them: the convenience of each denition depends on the mathematical assumptionsthat one imposes on the constitutive laws for pressure (internal energy) on one hand, and on the transportcoecients on the other hand. First one and the third one are continuations of the theories based onthe so called eective viscous ux identity started by P.L. Lions [32] and the second one, due to Bresch,Desjardins [2] can be considered as a continuation of theories based on new a priory estimates in the linestarted by Kazhikov [28].

The rst approach due to Feireisl [14] based on a weak formulation of the continuity, momentum andinternal energy equations is convenient for the pressure laws

p(%; #) = pc(%) + #p#(%);

1

wherepc(%) % ; p#(%) %; =3:

Here, # denotes the temperature, % the density and is the adiabatic coecient of the uid. The heatconductivity in this approach has to be temperature dependent (with a convenient polynomial growth),and the viscosity coecients have to be constant.

The second approach due to Bresch, Desjardins [2], [3] (see also Mellet, Vasseur [36]) is convenientin the case when the shear viscosity and the bulk viscosity depend on the density and satisfy thedierential identity

( 23)0(%) = 2%0(%) 2(%);

and the pressure satisesp(%; #) = %#+ pc(%);

where pc(%) is singular at % ! 0 and behaves as % at % ! 1. The main ingredient in the prof inthis situation is the fact that the particular relation between viscosities stated above makes possible toestablish a new mathematical entropy identity, that provides estimates for the gradient of density. Thisestimate implies compactness of the sequence of approximating densities.

The third approach is due to [17]. It is based on the weak formulation of continuity and momentumequations and on the formulation of the conservation of energy in terms of the specic entropy thatinvolves explicitly the second law of thermodynamics with entropy production rate being a non negativemeasure. This approach is applicable for the pressure laws p(%; #) exhibiting the coercivity of type %

and #4 for large densities and temperatures. The viscosity coecients are temperature dependent andhave to behave as 1 + # (1 > 2=5), and the heat conductivity has to behave as 1 + #3. This settingincludes at least one physically reasonable case of a monoatomic gas in the situation when the radiationis not neglected and is given by the Stephan-Boltzman law.

The above formulation is suciently weak to allow existence of variational solutions for large data.On the other hand it is suciently robust to yield most of low Mach and low Reynolds number limits ofmathematical uid mechanics, and - more surprisingly - it obeys the weak-strong uniqueness principle.

In this Lecture Notes, we shall concentrate on the third approach. We shall discuss within this conceptthree issues: 1) existence of weak solutions (or, more precisely, in order to concentrate on the essenceof the problem, the compactness of the family of weak solutions); 2) Relative entropy inequality for thecomplete system; 3) Weak strong uniqueness property in the class of weak solutions. This text does notcontain any new results: it is a compilation of our recent works [17, Chapter3] and [18].

1.1.2 Lions' approach and Feireisl's approach

The concept of weak solutions in uid dynamics was introduced in 1934 by Leray [30] in the contextof incompressible newtonian uids. It has been extended more than 60 years later to the Newtoniancompressible uids in barotropic regime (meaning that p = p(%) %) by Lions [32].

The Lions' theory relies on two crucial observations:

1) A discovery of a certain weak continuity property of the quantity

p(%) + (4

3+ )divu

called eective viscous ux. This part is essential for the existence proof; it employs certain cance-lation properties that are available due to the structure of the equations, that are mathematicallyexpressed through a commutator involving density, momentum and the Riesz operator.

2

2) Theory of renormalized solutions to the transport equation that P.L. Lions introduced together withDiPerna in [7]. In the context of compressible Navier-Stokes equations, the DiPerna-Lions transporttheory applies to the continuity equation. The theory asserts among others that the limitingdensity is a renormalized solution to the continuity equation provided it is squared integrable. Thishypothesis is satised only provided 9=5. The condition on the squared integrability of thedensity is the principal obstacle to the improvement of the Lions result.

Notice that some indications on the particular importance of the eective viscous ux have been knownat about the same time to several authors and used in dierent problems dealing with small data (seeHo [26], Padula [38]) and that the suggestion to use the continuity equations to evaluate the oscillationsin the sequence of approximating densities has been formulated and performed in the one dimensionalcase by D. Serre [44].

All physical reasonable adiabatic coecients belong to the interval (1; 5=3], the value = 5=3 beingreserved for the monoatomic gas. This is the reason why it is interesting and important to relax thecondition on the adiabatic coecient in the Lions theory. This has been done Feireisl et al. in [19]. Thenew additional aspects of this extension are based on the previous observations by Feireisl in [13] and arethe following:

1) As suggested in [13], the authors have used the oscillations defect measure to evaluate the oscillationsin the sequence of approximating densities, and proved that it is bounded provided > 3=2.

2) The boundedness of the oscillations defect measure is a criterion that replaces the condition of thesquared integrability of the density in the DiPerna-Lions transport theory. Consequently if anyterm of the sequence of approximating densities satises the renormalized continuity equation, andif the oscillations defect measure of this sequence is bounded, then the weak limit of the sequenceis again a renormalized solution of the continuity equation.

1.1.3 Weak solutions for the complete Navier-Stokes-Fourier system

The existence theory for the complete Navier-Stokes-Fourier system (with temperature dependent vis-cosities) employs both Lions' and Feireisl's techniques. In addition, it presents the following diculties:

1) In order to reduce the investigation to a situation similar to the barotropic case, one has to proverst the strong convergence of the temperature sequence. This point involves the treatment of theentropy production rate as a Radon measure and a convenient use of the compensated compactness,namely of the Div-curl lemma in combination with the theory of parametrized Young measures.

2) Even after the strong convergence of temperature is known, the weak continuity of the eective viscous

ux is not an obvious issue. It requires to use another cancelation property that mathematicallyexpresses through another commutator including shear viscosity, symmetric velocity gradient andthe Riesz operator.

3) Once the weak continuity property of the eective viscous ux is known, the proof follows the linesof Lion's and Feireisl's approaches: a) one proves rst the boundedness of the oscillation defectmeasure for the sequence of densities; b) the boundedness of oscillations defect measure impliesthat the limiting density is a renormalized solution to the continuity equation; c) the renormalizedcontinuity equation is used to show that the oscillations in the density sequence do not increase intime. This means the strong convergence of density.

3

1.1.4 Relative entropies and weak strong uniqueness

Weak solutions are not known to be uniquely determined (cf. e.g. exposition of Feerman [12] dealingwith three dimensional incompressible Navier-Stokes equations) and may exhibit rather pathologicalproperties, see e. g. Ho and Serre [27]. So far, the best property that one may expect in the directionof an uniqueness result, is the weak-strong uniqueness, meaning that any weak solution coincides withthe strong solution emanating from the same initial data, as long as the latter exists. The weak-stronguniqueness principle is known for the incompressible Navier-Stokes equations since the works of Prodi[42],Serrin [45], 1959, 1962. About 50 years later, the weak-strong uniqueness problem has been revisitedby Desjardins [6] and Germain [22]) for the compressible Navier-Stokes equations. They obtained somepartial and conditional results. Finally, the unconditional weak strong uniqueness principle has beenproved in [16] (see also related paper [20]).

Only very recently the weak strong uniqueness property has been proved in [18] for weak solutions ofthe complete Navier-Stokes-Fourier system in the entropy formulation introduced in [17].

In all cases cited above, the weak strong uniqueness principle has been achieved by the method ofrelative entropies. Relative entropy is a functional whose role is to measure the distance between a weaksolution of the investigated equations and any arbitrary smooth function exhibiting some characteristicproperties of solutions of the investigated equations, as e.g. the sign or the boundary conditions. Thisfunctional must satisfy a convenient dierential inequality (called relative entropy inequality) that is thenused to evaluate the evolution of the relative entropy along the time line. There is no general algorithm toconstruct this functional and this fact makes the construction of a convenient relative entropy functionalfor any given specic problem quite dicult. The method of relative entropies has been used in variouscontext by dierent authors, see Dafermeos [5] and later on P.L. Lions [32], Saint-Raymond [43], Grenier[23], Masmoudi [33], Ukai [48], Wang and Jiang [50], among others. The notion of dissipative solutionsintroduced in Lions [31] for the incompressible Euler equations is very much related to the concept ofrelative entropies.

1.2 Navier-Stokes-Fourier system

Navier-Stokes-Fourier system describes the motion of a compressible, viscous and heat conducting uid.For simplicity, we suppose that the uid lls a xed bounded domain and we shall investigate itsevolution through an arbitrary large time interval (0; T ). The motion will be described by means ofthree basic state variables: the mass density % = %(t; x), the velocity eld u = u(t; x), and the absolutetemperature # = #(t; x), where t 2 (0; T ) is time variable and x 2 R3 is the space variable in theEulerian coordinate system. We shall investigate the time evolution of these quantities. It is describedby the following system of partial dierential equations.

@t%+ divx(%u) = 0; (1.1)

@t(%u) + divx(%u u) +rxp(%; #) = divxS(#;rxu); (1.2)

@t(%s(%; #)) + divx(%s(%; #)u) + divx

q(#;rx#)

#

= ; (1.3)

where p = p(%; #) is the pressure, s = s(%; #) is the (specic) entropy, is the entropy production rateS is the viscous stress tensor and q is the heat ux. The rst equation expresses the mathematicalformulation of the balance of mass, the second one, the conservation of momentum, and the third one,the conservation of energy in terms of the specic entropy.

4

We suppose that the viscous stress S is described by Newton's law

S(#;rxu) = (#)T(rxu) + (#)divxuI; T(rxu) = rxu+ (rxu)T 23divxuI; (1.4)

where I is the identity tensor, while q is the heat ux satisfying Fourier's law

q = (#)rx#: (1.5)

The entropy production rate, if all quantities are smooth, satises

=1

#

S(#;rxu) : rxu q(#;rx#) rx#

#

: (1.6)

The system of equations (1.1 - 1.3), with the constitutive relations (1.4), (1.5) is called Navier-Stokes-Fourier system. Equations (1.1 - 1.6) are supplemented with initial conditions

%(0; ) = %0; %u(0; ) = %0u0; %s(%; #)(0; ) = %0s(%0; #0); %0 0; #0 > 0; (1.7)

and no-slip boundary conditions for velocity and zero heat transfer conditions through the boundary

uj@ = 0; q nj@ = 0; (1.8)

where n denotes the external normal to the boundary @ of .For simplicity, we have neglected external forces and heat sources. We however consider large motions,

since the initial data may be arbitrary large.In [17], we have introduced a concept of weak solution to the Navier-Stokes-Fourier system (1.1 -

1.8). This concept postulates, in agreement with the Second law of thermodynamics, that the entropyproduction rate is a non-negative measure,

1#

S(#;rxu) : rxu q(#;rx#) rx#

#

: (1.9)

With this postulate, equation (1.3) becomes inequality. In order to compensate the loss of information,we may postulate, that the total energy of the system in the volume is conserved, namely

d

dt

Z

1

2%juj2 + %e(%; #)

dx = 0: (1.10)

We introduce the quantity H# = %e #%s (that is reminiscent of the Helmholtz free energy), wheree = e(%; #) denotes the specic energy of the gas. Adding equations (1.3) and (1.10), we obtain

d

dt

Z

1

2%juj2 +H#(%; #) @%H#(%; #)(% %)H#(%; #)

dx (1.11)

+

Z

#

#

S(#;rxu) : rxu q(#;rx#) rx#

#

dxdt 0;

where we have taken into account inequality (1.9)

5

On the other hand, if (%; #;u) % > 0, # > 0 is a trio of smooth functions satisfying (1.1{1.8) one mayderive, at least formally, the so called relative entropy identity,Z

1

2%juUj2 + E(%; #jr;)

(; ) dx (1.12)

+

Z 0

Z

S(#;rxu)

#: rxu dxdt

Z 0

Z

q(#;rx#)

#2: rx# dxdt

=

Z

1

2%0ju0 U(0; )j2 + E(%0; #0jr(0; );(0; ))

dx

+

Z 0

Z

S(#;rxu) : rxUdxdtZ 0

Z

q(#;rx#)#

rxdxdt

+

Z 0

Z

%@tU+ u rxU

(U u) dx dt

+

Z 0

Z

%(U u) rxp(r;)r

dx dt;

+

Z 0

Z

p(r;) p(%; #)

divxU dxdt

+

Z 0

Z

%s(r;) s(%; #)

@t+ u rx

dx dt

+

Z 0

Z

1 %

r

@tp(r;) +U rxp(r;)

dx dt;

where we have denoted

E(%; # jr;) = H(%; #) @%H(r;)(% r)H(r;);H(%; #) = %e(%; #)%s(%; #):

In (1.12), (r;) is a couple of positive suciently smooth functions on [0; T ] and U is a sucientlysmooth vector eld with compact support in [0; T ] .

Conformably to (1.9), for a weak solution (%; #;u), the identity (1.12) has to be replaced by aninequality with the inequality sign . This inequality is usually called the relative entropy inequality.Notice that the dissipation inequality (1.11) is a particular case of the relative entropy inequality, wherer = %, = #, U = 0.

1.3 Constitutive relations

We assume that the thermodynamic functions p, e, and s are interrelated through Gibbs' equation

#ds(%; #) = de(%; #) p(%; #)%2

d%: (1.13)

We suppose that the uid veries the thermodynamic stability conditions,

@p(%; #)

@%> 0;

@e(%; #)

@#> 0 for all %; # > 0: (1.14)

6

We easily verify by using (1.13), that

@H#@#

(%; #) = %# ##

@e

@#(%; #) and

@2H#@%2

(%; #) =1

%

@p

@%(%; #): (1.15)

Thus, the thermodynamic stability in terms of the function H#, can be reformulated as follows:

% 7! H#(%; #) is strictly convex, (1.16)

while# 7! H#(%; #) attains its global minimum at # = #: (1.17)

The above relations reect stability of the equilibrium solutions to the Navier-Stokes-Fourier system (seeBechtel, Rooney, and Forest [1]) and play a crucial role in the analysis of the Navier-Stokes-Fouriersystem.

Taking into account the eects of radiation expressed through the Stefan-Boltzman law, we shallassume that the pressure p = p(%; #) can be written in the form, in agreement with the above considera-tions

p(%; #) = #=(1)P %#1=(1)

+a

3#4; a > 0; > 3=2; (1.18)

whereP 2 C1[0;1); P (0) = 0; P 0(Z) > 0 for all Z 0: (1.19)

In agreement with Gibbs' relation (1.13), the (specic) internal energy can be taken as

e(%; #) =1

1#=(1)

%P %#1=(1)

+ a

#4

%: (1.20)

Furthermore, by virtue of the second inequality in thermodynamic stability hypothesis (1.14), we have

0 0: (1.21)

Relation (1.21) implies that the function Z 7! P (Z)=Z is decreasing, and we suppose that

limZ!1

P (Z)

Z= P1 > 0: (1.22)

Finally, the formula for (specic) entropy reads

s(%; #) = S %#1=(1)

+

4a

3

#3

%; (1.23)

where, in accordance with Third law of thermodynamics,

S0(Z) = 1

1

P (Z) P 0(Z)ZZ2

< 0; limZ!1

S(Z) = 0: (1.24)

From the point of view of statististical mechanics, the above hypotheses are physically reasonable atleast in two cases: if = 5=3 they modelize the monoatomic gas, if = 4=3 they modelize the so called

7

relativistic gas. The reader may consult Eliezer, Ghatak, and Hora [9] and [17, Chapter 3] for the physicalbackground and further discussion concerning the structural hypotheses (1.18 - 1.22).

For the sake of simplicity and clarity of presentation, we take the transport coecients in the form

; 2 C1[0;1) are globally Lipschitz and 0(1 + #) (#); 0 (#); 0 > 0; (1.25)

2 C1[0;1); 0(1 + #3) (#) 1(1 + #3); 0 < 0 1; (1.26)cf. [17, Chapter 3].

2 Denitions and main results

2.1 Weak and suitable weak solutions to the Navier-Stokes-Fourier system

Here and heareafter, we shall suppose that the initial data satisfy the following assumptions

0 < % %0 2 L(); %0u0 2 L2=(+1)(;R3); (2.1)0 < # < #0; %0s(%0; #0) 2 L1(); (2.2)Z

1

2%0ju0j2 +H#(%0; #0) @%H#(%; #)(%0 %)H#(%; #)

dx 0 for a.a. (t; x) 2 (0; T ) ,% 2 L(), %u 2 L1(0; T ;L2=(+1)(;R3)), %u2 2 L1(0; T ;L1()), # 2 L4())\L2(0; T ;W 1;2()),log # 2 L2(0; T ;W 1;2(;R3)) and u 2 L2(0; T ;W 1;20 (;R3));

(ii) % 2 Cweak([0; T ];L()) and equation (1.1) is replaced by a family of integral identitiesZ

%' dx0=

Z 0

Z

%@t'+ %u rx'

dx dt (2.4)

for all 2 [0; T ] and for any ' 2 C1([0; T ] );(iii) %u 2 Cweak([0; T ];L2=(+1)(;R3)) and momentum equation (1.2) is satised in the sense of

distributions, specically,Z

%u ' dx0=

Z 0

Z

%u @t'+ %u u : rx'+ p(%; #)divx' S(#;rxu) : rx'

dx dt (2.5)

for all 2 [0; T ] and for any ' 2 C1c ([0; T ] ;R3);

8

(iv) the entropy balance (1.3), (1.9) is replaced by a family of integral inequalities

Z

%s(%; #)' dx0+

Z 0

Z

'

#

S(#;rxu) : rxu q(#;rx#) rx#

#

dx dt (2.6)

Z 0

Z

%s(%; #)@t'+ %s(%; #)u rx'+ q(#;rx#) rx'

#

dx dt

for a.a. 2 (0; T ) and for any ' 2 C1([0; T ] ), ' 0;(v) the conservation of total energy in the volume is veriedZ

1

2%juj2 + %e(%; #)

(; ) dx =

Z

1

2%0ju0j2 + %0e(%0; #0)

dx (2.7)

for a.a. 2 (0; T ).

Here and hereafter, the symbolR

gdx

0means

R

g(x; )dx R

g0(x)dx.

RemarkAny weak solution satises the so called dissipation inequality:Z

1

2%juj2 +H#(%; #) @%H#(%; #)(% %)H#(%; #)

(; )dx (2.8)

+

Z 0

Z

#

#

S(#;rxu) : rxu q(#;rx#) rx#

#

dxdt Z

1

2%0ju0j2 +H#(%0; #0) @%H#(%; #)(%0 #)H#(%; #)

dx

for a.a. 2 (0; T ). Indeed, the dissipation inequality (2.8) is obtained from the sum of identity (2.7) andthe entropy balance (2.6) multiplied by #.

Denition 2We say that the triplet (%; #;u) is a renormalized weak solution to the Navier-Stokes-Fourier system (1.1- 1.8) if it is a very weak solution, if b(%) 2 Cweak([0; T ]; L1()) and if the couple (%;u) satises thecontinuity equation in the renormalized sense,Z

b(%)' dx0=

Z 0

Z

b(%)@t'+ u rx'

dxdt

Z 0

Z

%b0(%) b(%)

divu'dxdt (2.9)

for any 2 [0; T ], anyb 2 C[0;1) \ C1(0;1); zb0 2 L1(0; 1); b=z5=6; zb0=z=2 2 L1(1;1)

and' 2 C1([0; T ) ):

9

Denition 3We say that the triplet (%; #;u) is a suitable weak solution to the Navier-Stokes-Fourier system (1.1 -1.8) if it is a weak solution and if it satises the relative entropy inequalityZ

1

2%juUj2 + E(%; #jr;)

(; ) dx (2.10)

+

Z 0

Z

S(#;rxu)

#: rxu dxdt

Z 0

Z

q(#;rx#)

#2: rx# dxdt

Z

1

2%0ju0 U(0; )j2 + E(%0; #0jr(0; );(0; ))

dx

+

Z 0

Z

S(#;rxu) : rxUdxdtZ 0

Z

q(#;rx#)#

rxdxdt

+

Z 0

Z

%@tU+ u rxU

(U u) dx dt

+

Z 0

Z

%(U u) rxp(r;)r

dx dt;

+

Z 0

Z

p(r;) p(%; #)

divxU dxdt

+

Z 0

Z

%s(r;) s(%; #)

@t+ u rx

dx dt

+

Z 0

Z

1 %

r

@tp(r;) +U rxp(r;)

dx dt

for a.a. 2 (0; T ) with any

(r;;U) 2 C1([0; T ] ;R5); r > 0; > 0; Uj@ = 0; (2.11)

whereE(%; # jr;) = H(%; #) @%H(r;)(% r)H(r;); (2.12)

H(%; #) = %e(%; #)%s(%; #):

2.2 Main results

In this section, we shall formulate four theorems. First theorem asserts existence of weak solutions.

Theorem 2.1 Let R3 be a bounded Lipschitz domain. Suppose that the thermodynamic functions p,e, s satisfy hypotheses (1.18 - 1.24), and that the transport coecients , , and obey (1.25), (1.26).Finally assume that the initial data (1.7) verify (2.1{2.3). Then the complete Navier-Stokes-Fouriersystem (1.1{1.8) admits at least one renormalized weak solution.

10

The proof of this result can be found in [17, Chapter 3] for domains with regularity C2; , 2 (0; 1).The necessary modications to accommodate the Lipschitz domains are explained in Poul [40]. In thisLecture Notes, we shall prove solely the weak compactness property of the (hypothetical) set of weaksolutions. This property is usually considered to be the main ingredient in the existence proof. Notehowever, that the compressible models are very much "approximation sensitive". Therefore, the wayfrom the weak compactness to the real existence can still be very long and should not be underestimated.

Theorem 2.2 Let R3 be a bounded Lipschitz domain. Suppose that the thermodynamic functions p,e, s satisfy hypotheses (1.18 - 1.24), and that the transport coecients , , and obey (1.25), (1.26).Finally assume that the initial data (1.7) verify (2.1{2.3). Let (%n; #n;un) be a sequence of weak solutionsto the complete Navier-Stokes-Fourier system (1.1{1.8). Then there exists a subsequence (denoted again(%n; #n;un)) such that

%n * % in L1(0; T ;L();

#n * # in L2(0; T ;W 1;2());

un * u in L2(0; T ;W 1;20 (;R

3));

and the trio (%; #;u) is again a weak solution of the complete Navier-Stokes-Fourier system (1.1{1.8).

This compactness result will be proved in Section 3.The second result to be discussed in this Lecture Notes claims that any weak solution satises the

relative entropy inequality.

Theorem 2.3 Let all assumptions of Theorem 2.1 be satised and let the trio (%; #;u) be a weak solutionto the complete Navier-Stokes-Fourier system (1.1{1.8).

Then (%; #;u) is a suitable weak solution. In particular, it satises the relative entropy inequality(2.10) with test functions (r;;U) belonging to the class (2.11).

Theorem 2.3 has been proved in [18]. We shall reproduce this proof in Section 4.The third result to be dealt with in this Lecture Notes is the weak strong uniqueness principle. It

asserts that any weak solution coincides with the strong solution emanating from the same initial dataas long as the latter exists.

Theorem 2.4 Let assumptions of Theorem 2.1 be satised, where, moreover, the function P introducedin (1.19) is twice continuously dierentiable on (0;1). Let (%; #;u) be a weak solution of the Navier-Stokes-Fourier system (1.1{1.8). Assume that (~%; ~#; ~u) is a classical (strong) solution to the Navier-Stokes-Fourier system (1.1{1.8) in (0; T ) that satises dissipation inequality (2.8) and that belongsto the class

0 < # ~# #

The weak strong uniqueness property in the class of weak solutions has been proved in [18]. Thisproof will be reproduced in Section 5.

Remark(Relaxing regularity of test functions in Theorem 2.3.)Using a density argument we can extend the class of test functions r, , U appearing in the relativeentropy inequality (2.10), (2.11).

Sucient conditions for the left hand side of the relative entropy inequality to be well dened are,for example,

0 < % r %

hH#(%; #)H#(%; #)

i+hH#(%; #) @%H#(%; #)(% %)H#(%; #)

i:

To continue we recall the equivalence

P 0(Z) 1 + Z1; Z > 0 (3.2)

that can be derived from (1.21{1.22). Employing (1.15), (1.18), (1.20) and (3.2), we obtain

H#(%; #)H#(%; #) =Z ##

@#H#(%; z)dz =

Z ##

%

z(z #)@#e(%; z)dz 4a

Z ##

z2(z #)dx (3.3)

and

H#(%; #)@%H#(~r; #)(%~r)H#(~r; #) =Z %~r

Z z~r

@2%H(w; #)dwdz =

Z %~r

Z z~r

1

w@%p(w; #)dw

dz (3.4)

h% log(%=%)

% %

i+h% %1(% %) %

i:

With observations (3.3{3.4) at hand we deduce from (2.8) the following estimates

esssup(0;T )

Z

%nu2ndx c; (3.5)

esssup(0;T )

Z

%ndx c; (3.6)

esssup(0;T )

Z

#4ndx c (3.7)

By virtue of (3.5{3.6), we deduce for the momentum,

k%nunkL1(0;T ;L2=(+1)(;R3)) c: (3.8)

Step 2 (Estimates due to the entropy production rate)

Here and hereafter we shall need three specic inequalities that may be of general interest. We refer to[17, Chapter 2 and Appendix] for their proofs.

We recall rst the following Korn's type inequality

Lemma 3.1 Let be a bounded Lipschitz domain. There exists c > 0 such that for all w 2W 1;20 (;R3),there holds

krxwkL2(;R33) ckT(rxw)kL2(;R33): (3.9)Another inequality of general interest to be recalled at this place is the following Poincare's type inequality.

Lemma 3.2 Let be a bounded Lipschitz domain. For any 0 < M < jj there exists c(M) such thatfor any w 2W 1;2() and V , jV j M , we have

kwkL2(;R3) c(M)krxwkL2(;R33) +

ZV

w2dx: (3.10)

13

The last inequality includes the properties of the entropy function S.

Lemma 3.3 Let be a bounded Lipschitz domain and p; > 1. Let S 2 C(0;1) be a strictly decreasingfunction such that limZ!1 S(Z) = 0 and

lim supn!1

Zf%n#1=(1)n g

%nS %n#1=(1)n

dx = 0

whenever %n 0 is bounded in L() and 0 #n ! 0 in Lp().Then for any M0 > 0, 0 > 0, and S0 > 0 there exist = (M0;0; S0) > 0, # = #(M0;0; S0) > 0

such that for any non-negative functions %, #Z

%dx M0;Z

(% + #p)dx 0

and Z

%S %#1=(1)

dx S0

we have f# #g : (3.11)The "velocity part" of the entropy production yields bounds

kT(rxun)kL2(0;T ;L2(;R33)) c; T(rxw) = rxw +rTxw 23divw;R T0

R

1#njT(rxun)j2dxdt c;

(3.12)

whence employing rst Lemma 3.1 and then the standard Poincare inequality, we get

kunkL2(0;T ;W 1;2(;R3)) c: (3.13)The "temperature part" of the entropy production rate gives

krx#nkL2(0;T ;L2()) c; 2 [1; 3=2]

krx log #nkL2(0;T ;L2()) c;(3.14)

where c > 0 is independent of n. In agreement with (1.23{1.24),

j%s(%; #)j c(%+ %j log %j+ %(log #)+ + #3): (3.15)Consequently, recalling (1.23), we verify that assumptions of Lemma 3.3 are satised with some 3 < p < 4.(We recall that S0 is positive due to (2.6) with test function ' = 1.) Therefore, we deduce from (3.14)and Lemmas 3.2, 3.3.

k#n #kL2(0;T ;W 1;2() c; 2 [1; 3=2];

k log #n log #kL2(0;T ;W 1;2() c:(3.16)

Step 3 (Consequence of standard imbeddings)

14

We get by the Sobolev imbedding and by interpolation from (3.7) and (3.16) that

k#nkL3(0;T ;L9()) c; k#nkL17=3((0;T )) c (3.17)

Recalling (3.7), (3.14), (3.13) and having in mind (1.4), (1.5), (1.21), (1.26) we deduce that

kS(#n;rxun)kL2(0;T ;L4=3(;R33)) c: (3.18)

andkq(#n;rx#n)=#nkL2(0;T ;L8=7(;R3)) c: (3.19)

Returning to (3.15), we get

k%ns(%n; #n)kL1(0;T ;Lq()) c with some q > 1 (3.20)

as well ask%ns(%n; #n)unkLq((0;T );R3) c(n) with some q > 1: (3.21)

Step 4 (Estimates via the Bogovskii operator)

Under assumptions (1.18{1.22)jp(%; #)j c(%#+ % + #4): (3.22)

Consequently, we can deduce from (3.6{3.7) only kp(%n; #nkL1(0;T ;L1()) c. We however need for thepressure better estimate than an estimate in L1(). One way to get a better estimate is based on thefollowing lemma.

Lemma 3.4 Let Q be a bounded Lipschitz domain. There an operator BQ : Lp(Q) ! W 1;p0 (Q;R3),1 < p

Bounds (3.6{3.7), (3.13), (3.16{3.23) imply existence of a subsequence (denoted again (%n; #n;un)) andof a trio (%; #;u) such that

%n * % in L1(0; T ;L());

#n * # in L1(0; T ;L4());

un * u in L2(0; T ;W 1;2(;R3));

#n * # in L2(0; T ;W 1;2()):

(3.24)

Moreover, if we denote by g(%; #;u) a weak limit of the sequence g(%n; #n;un) in L1((0; T ))), we have

for the non linear quantities,

log #n * log # in L2(0; T ;W 1;2());

p(%n; #n)* p(%; #) in Lq((0; T ) n) with some q > 1,

S(#n;rxun)* S(#;rxu) in L4=3((0; T ) ;R33),

%ns(%n; #n)* %s(%; #) in Lq((0; T ) ) with some q > 1,

q(#n;rx#n)=#n * q(#;rx#)=# in L8=7((0; T ) ;R3):

(3.25)

Next, we can use continuity equation (2.4), renormalized continuity equation (2.9) and momentumequation (2.5) to show the equi-continuity of functions

t 7!Z

%n' dx; t 7!Z

h(%n)' dx; t 7!Z

%nun' dx;

on [0; T ], where ' 2 C1c (). This fact makes possible to use the Arzela-Ascoli compactness argumentwhich in combination with the density argument yields convergence of the sequences %n, h(%n) and %nunin Cweak([0; T ];L

q()) with some q > 6=5. Employing moreover the compact imbedding Lq() ,!,!W1;2(), we get the convergence of these quantities in L2(0; T ;W 1;2()). Summarizing the above, wehave

%n ! % in Cweak([0; T ];L()) and in L2(0; T ;W1;2());

h(%n)! h(%) in Cweak([0; T ];Lq()), q 2 [1;1) and in L2(0; T ;W1;2());

where h 2W 1;1((0;1));

%nun ! %u in Cweak([0; T ];L2=(+1)(;R3)) and in L2(0; T ;W1;2(; R3));

%nun un ! %u u in L2(0; T ;L6=(4+3)(; R33)):

(3.26)

Step 6 (Limiting equations)Now, we are ready to let n!1 in equations (2.4), (2.5) and (2.9) (written with (%n; #n;un)). We get,in particular,

Z

%0'(0; ) dx =Z T0

Z

%@t'+ %u rx'

dx dt (3.27)

16

for any ' 2 C1c ([0; T ) );Z

%0u0 '(0; ) dx (3.28)Z T0

Z

%u @t'+ %u u : rx'+ p(%; #)divx' S(#;rxu) : rx'

dx dt

for any ' 2 C1c ([0; T ) ;R3), 'j@ = 0;Z T0

Z

%Lk(%)@t'+ u rx'

dxdt =

Z T0

Z

Tk(%)divu'dxdtZ

%0Lk(%0)'(0; )dx (3.29)

andZ T0

Z

Tk(%)@t'+ u rx'

dxdt+

Z T0

Z

(%T 0k(%) Tk(%))divu'dxdtZ

Tk(%0)'(0; )dx; (3.30)

where ' 2 C1c ([0; T ) ):Tk(z) = kT (z=k); Lk(z) =

Z z1

Tk(w)

w2dw;

T (z) =

8>>>>>>>:z if z 2 [0; 1];

concave on [0;1);

2 if z 3:

3.2 Strong convergence of temperature

Step 1The entropy inequality (2.6) can be rewritten as identityZ

%0s(%0; #0)'(0; )dx+ < n; ' > (3.31)

= Z T0

Z

%ns(%n; #n)@t'+ %ns(%n; #n)un rx'+ q(#n;rx#n) rx'

#n

dx dt

where n is a positive continuous linear functional on the (metric) space C1c ([0; T ) ) dened by the

above equation. We therefore easily verify by the standard Schwartz argument that

j < n; ' > j c(K;n)k'kC(K) for any compact K [0; T ) ;

whence, in fact, n can be extended in a unique way to a positive linear functional on the (metric) spaceCc([0; T ) ). By virtue of the Riesz representation theorem, n can be identied with a non negativeregular measure n on the family of Borel subsets of [0; T ) such that,

n(K)

in the sense that

< n; ' >=

Z[0;T )

'dn ; ' 2 Cc([0; T ) ):

Moreover, taking for any xed compact interval [0; L] [0; T ),

L 2 C1c [0; T ); 0 L 1; L = 1 in [0; L]; 2

T L @tL 0

and using (3.31) with test function L(t)1(x), we get, due to (3.20),

j < n; L()1() > j c; (3.32)where c is independent on L and n.

Hence nally, the measure n (acting on the family of Borel subsets of [0; T ) ) can be extendedto the family of Borel subsets of [0; T ] by setting

n

fTg

= 0:

Consequently the linear functional n (continuous on the metric space Cc([0; T ) )) can be extendedto a linear functional on the Banach space Cc([0; T ] ) C([0; T ] ) via the formula

< n; ' >=

Z[0;T ]

'dn ; ' 2 C([0; T ] );

and we have

knk[C([0;T ])] = n([0; T ] ) =Z[0;T ]

dn < c;

where we have used (3.32) to get the last bound.

Step 2We start by recalling the Div-Curl lemma, that says:

Lemma 3.5 Let N be a positive integer and Q an open set in RN . If Xn * X in Lp(Q;RN ), Yn *

Y in Lq(Q;RN ), 1p +1q =

1r < 1 and the sequences divXn, curlYn are compact in W

1;s(Q) resp.W1;s(Q;RN RN ) with some s > 1 then

Xn Yn * X Y in Lr(Q):

We may now apply this lemma to the four dimensional vectors

Vn = (%ns(%n; #n); %ns(%n; #n)un + q(#n;rx#n)=#n); Wn = (Tk(#n); 0; 0; 0)Since divVn = n and since the imbedding [C([0; T ] )] ,! W1;q((0; T ) ) is compact for anyq 2 (1; 4=3), the assumptions of the lemma on (0; T ) are satised. Therefore,

Tk(#)%sM (%; #) + 43aTk(#)#3 = Tk(#) %sM (%; #) + 4

3aTk(#) #3; (3.33)

18

where sM (%; #) = S(%=#1

1 ).We shall rst prove that

Tk(#)%sM (%; #) Tk(#) %sM (%; #); (3.34)where

Tk(z) = kT (z=k); C[0;1) 3 T =

8>>>>>>>:z if z 2 [0; 1];

T strictly increasing on [0;1) ;

limz!1 T (z) = 2:

9>>>>=>>>>; :To this end we write

%ssM (%n; #n)Tk(#n) Tk(#)

=

%n

hsM

%n; T 1k

Tk(#n)

sM

%n; T 1k

Tk(#)

iTk(#n)Tk(#)

+%nsM

%n; T 1k

Tk(#)

Tk(#n)Tk(#)

:

Therefore, inequality (3.34) will be shown if we prove that

%nsM

%n; T 1k

Tk(#)

Tk(#n) Tk(#)

* 0 weakly in L1((0; T ) ) as n!1: (3.35)

The quantity

%nsM

%n; T 1k

Tk(#)

(t; x)) = (t; x; %n) (3.36)

can be regarded as a composition of a Caratheodory function with a weakly convergent sequence %n.

Step 3In what follows, we shall use the fundamental theorem on parametrized Young measures. It is recalledin the following Lemma.

Lemma 3.6 Let Q 2 RN be a measurable set. Then for any bounded sequence wn in L1(Q;RM ) thereexists a family of parametrized Young measures fygy2Q such that

(y;w) =

ZRM

(y; )dy();

for any Caratheodory function : QRM 7! R satisfying

(y;wn)* (y;w) weakly in L1(Q):

Since according to (3.24), (3.26)

h(%n)! h(%) in L2(0; T ;W1;2());

G(#n)! G(#) in L2(0; T ;W 1;2());we have

h(%)G(#) = h(%) G(#) (3.37)

for any h belonging to the class (2.9) and G 2 W 1;1((0;1)). This implies (3.35) by virtue of Lemma3.6.

19

Indeed, denote %;#(t;x), %(t;x) and

#(t;x) the parametrized Young measures corresponding, in accordance

with Lemma 3.6, to the sequences (%n; #n), %n and #n, respectively. Then we have, due to (3.37) and inagreement with Lemma 3.6,Z

R2h()G()d(%;#)(; ) =

ZR

h()d%()ZR

G()d#():

Consequently,

(t; x; %)G(#)(t; x) =

ZR2

(t; x; )G() d%(t;x)() d#(t;x)() =

(t; x; %) G(#)

(t; x):

This implies (3.35).

Step 4Now, we have to recall several general properties of monotone operators with respect to weak convergence.

Lemma 3.7 Let Q RN be a domain and (P;G) 2 C(R)C(R) two non decreasing functions. Let wnbe a sequence of functions in L1(Q) such that8>>>>>>>:

P (wn)* P (w);

G(wn)* G(w);

P (wn)G(%n)* P (w)G(w)

9>>>>=>>>>; in L1(Q):

(i) ThenP (w) G(w) P (w)G(w):

(ii) If G(z) = z and ifP (w) w = P (w)w;

thenP (w) = P (w):

In the last two formulas w denote weak limit of wn in L1(Q).

Lemma 3.7 implies, in particular,Tk(#)#3 Tk(#) #3

that in turn with (3.33{3.34)yieldsTk(#)#3 = Tk(#) #3

and nally, by monotone convergence, as k !1,

#4 = ##3: (3.38)

The last identity implies#n ! # a.e. in (0; T ) : (3.39)

20

Step 5Coming back with (3.39) to the momentum equation (3.28), we obtain

Z

%0u0 '(0; ) dx (3.40)

=

Z T0

Z

%u @t'+ %u u : rx'+ p(%; #)divx' S(#;rxu) : rx'

dx dt

for any ' 2 C1c ([0; T ) ;R3), 'j@ = 0.Moreover, estimate (2.8) yields boundedness of the sequencess

(#n)

#n

rxun + (rxun)T 2

3divun

;

s(#n)

#ndivun;

p(#n)

#nrx#n (3.41)

in L2((0; T ) )); whence by the lower weak continuity combined with (3.39) and (3.24) one getsZ T0

Z

'

#

S(#;rxu) : rxu q(#;rx#) rx#

#

dx dt (3.42)

lim infn!1

Z T0

Z

'

#

S(#n;rxun) : rxun q(#n;rx#n) rx#n

#n

dx dt;

for any 0 ' 2 Cc([0; T ] ).Thus eectuating the limit n!1 in (2.6) (with %n; #n;un on place of %; #;u), we getZ

%0s(%0; #0)'(0; ) dx+Z T0

Z

'

#

S(#;rxu) : rxu q(#;rx#) rx#

#

dx dt (3.43)

Z T0

Z

%s(%; #)@t'+ %s(%; #)u rx'+ q(#;rx#)

# rx'

dx dt

for any ' 2 C1c ([0; T ) ), ' 0.

3.3 Strong convergence of densities

3.3.1 Eective viscous ux

The main result of this section is the following lemma.

Lemma 3.8

Tk(%)divxu Tk(%)divxu = 143(#) + (#)

p(%; #)Tk(%) p(%; #) Tk(%)

: (3.44)

Lemma 3.8 will be proved in several steps.

21

Step 1First we introduce operators A = rx1 and R = rx rx1,

(r1)j(v) = F1h ijjj2F(v)()

i; (rr1)ij(v) = F1

hijjj2 F(v)()

i;

where F denotes the Fourier transform

[F(v)]() = 123

ZR3

v(x)exp(i x)dx:

We recall that these operators have the following properties:

(i) A is a continuous linear operator from L1 \ L2(R3) to L2 + L1(R3;R3) and from Lp(R3) toL3p=(3p)(R3;R3) for any 1 < p < 3.

(ii) R is a continuous linear operator from from Lp(R3) to Lp(R3;R33) for any 1 < p

function ' = r1[Tk(%)1]. This procedure yields

limn!1

Z T0

Z

(t; x)p(%n; #n)Tk(%n) S(#n;un) : R[1Tk(%n)]

dxdt

=

Z T0

Z

(t; x)(p(%; #) Tk(%) S(#;u) : R[1Tk(%)]

dxdt

limn!1

Z T0

Z

Tk(%n)un R[%nun] %n(un un) : R[1Tk(%n)]

dxdt

Z T0

Z

Tk(%)u R[%u] %(u u) : R[1Tk(%)]

dxdt;

(3.45)

where (R[z])j =P3

k=1Rjk[zk]. In fact, other terms that should eventually appear at the right hand sideare zero by virtue of standard compactness arguments.

Step 3Next, we shall need the following lemma in the spirit of Meyer [37], see [17, Theorem 10.27]

Lemma 3.9 (Commutators I) Let Un * U in Lp(R3;R3), vn * v in L

q(R3), where

1

p+

1

q=

1

r< 1:

ThenvnR[Un]R[vn]Un * vR[U]R[v]U

in Lr(R3;R3).

Lemma 3.9 is a consequence of the Dic-curl lemma 3.5. Indeed, realizing that

U R(V)V R(U) = X R(V) +Y R(U);

whereX = UR(U); Y = R(V)V:

Seeing that divX = divY = 0 and that curlR(U) = curlR(V) = 0, we obtain employing Lemma 3.5.

Vn R[Un]Un R[Vn]* V R[U]U R[V]

provided Vn * V in Lq(R3), Un * U in L

p(R3;R3). Taking in the last formula subsequently Vn =(vn; 0; 0), (0; vn; 0), and (0; 0; vn) we get Lemma 3.9.

Step 4Combining this lemma with the convergence established in the rst two lines of (3.26), we get

Tk(%n)R[%nun] %nun R[1Tk(%n)](t)!

Tk(%)R[%u] %u R[1Tk(%)]

(t)

23

weakly in Lr(;R3) with some r > 6=5 for all t 2 [0; T ]; whence by the compact imbedding Lr() ,!,!W1;2() and the Lebsegue dominated convergence theorem used over (0; T ) we concludeR T

0

R

un

Tk(%n)R[1%nun] %nun R[1Tk(%n)]

dx

! R T0

R

u Tk(%)R[1%u] %u R[1Tk(%)]

dx

Thus (3.45) yields Z T0

Z

(t; x)p(%; #)Tk(%) p(%; #) Tk(%)

dxdt

=

Z T0

Z

(t; x)S(#;u) : R[1Tk(%)] S(#;u) : R[1Tk(%)]

dxdt:

(3.46)

Step 5Hereafter we will need another another commutator lemma in the spirit of Meyer [37], see [17, Theorem10.28]

Lemma 3.10 (Commutators II) Let w 2 W 1;r(R3), z 2 Lp(R3;R3), 1 < r < 3, 1r + 1p 13 < 1s < 1.Then for all such s we have

kR[wz] wR[z]kW;s(R3;R3) CkwkW 1;r(R3)kzkLp(R3;R3);where 3 =

1s 16 1r . Here, k kW;s(R3) denotes the norm in the Sobolev{Slobodetskii space W ;s(R3).

We can writeZ T0

Z

(t; x)S(#;u) : R[1Tk(%)] dxdt =Z T0

Z

(t; x) 2

3(#) + (#)

Tk(%) divxu dxdt

+

Z T0

Z

Tk(%)nR :

h(#)

rxu+ (rxu)T

i (#)R :

hrxu+ (rxu)T

iodxdt

+

Z T0

Z

Tk(%)(#)R :hrxu+ (rxu)T

idxdt;

where R : (Z) =P3i;j=1Rij(Zij). WhenceZ T0

Z

(t; x)S(#;u) : R[1Tk(%)] dxdt = limn!1

Z T0

Z

(t; x)43(#n) + (#n)

Tk(%n) divxun dxdt

+ limn!1

Z T0

Z

Tk(%n)!(#n;un) dxdt

(3.47)and Z T

0

Z

(t; x)S(#;u) : R[1Tk(%)] dxdt =Z T0

Z

(t; x)43(#) + (#)

Tk(%) divxu dxdt

+R T0

R

Tk(%)!(#;u) dxdt;

(3.48)

24

where

!(#n;un) =Rh(t; x)(#n)

run + (run)T i (t; x)(#n)R : run + (run)T :Thanks to Lemma 3.10, the sequence !(#n;un) is bounded in L

1(0; T ;W ;q(;R3)) with some 2 (0; 1),q > 1; whence curlUn is compact in W

1;r((0; T ) ;R33) (and of course divVn is compact inW1;r((0; T ) ;R33), cf. (2.9)) with some r > 1, where

Vn [Tk(%n); Tk(%n)un]; Un [!(#n;un); 0; 0; 0]:We may thus apply a convenient version of Div-curl lemma (see e.g. [17, Theorem 10.21]) to these4-dimensional vector elds to get

!(#n;un)Tk(%n)* !(#;u) Tk(%);

where, due to (3.39),!(#;u) = !(#;u):

This result in combination with (3.46) and (3.47) yields the eective viscous ux identity (3.44).

3.3.2 Oscillations defect measure and renormalized continuity equation

Going back to (3.2), we deduce employing the hypotheses(1.18{1.22) that

p(%; #) = d% + pm(%; #); for some d > 0; (3.49)

where @%pm(%; #) 0.We calculate

d lim supn!0

Z T0

Z

1 + #jTk(%n) Tk(%)j+1dxdt (3.50)

d lim supn!1

Z T0

Z

1 + #

(Tk(%n) Tk(%)

(%n %)dxdt Z T

0

Z

1 + #

% Tk(%) % Tk(%)

dxdt

Z T0

Z

1 + #

p(%; #)Tk(%) p(%; #) Tk(%)

dxdt;

where 2 C1c ((0; T ) ), 0. To get the rst inequality we have used (b a) b a andTk(b) Tk(a) b a, with any b a 0. To derive the second one, we have used the inequalities% % and Tk(%) Tk%) that hold due to the convexity of functions % ! % and the concavity offunctions %! Tk(%). Finally, to derive the third one, we have employed (3.49), the fact that p(%; #)g(%) =p(; #)g() w limn!1 p(%n; #)g(%n) for any bounded function g. This result holds since the sequence#s converges almost everywhere to #, see (3.39)), and due to the well known relation between the weaklimits of monotone functions

pm(; #)Tk() pm(; #) Tk(%) 0; (3.51)cf. Lemma 3.7.

Next, we verify thatZ T0

Z

jTk(%n) Tk(%)jqdx =Z T0

Z

1

(1 + #)jTk(%n) Tk(%)jq

1 + #

dxdt

25

ch Z T

0

Z

1

1 + #jTk(%n) Tk(%)j+1dxdt

iq=(+1);

where q > 2, provided ( + 1) = q and ( + 1)=( + 1 q) 17=3, cf. (3.17).The expression Z T

0

Z

1

1 + #jTk(%n) Tk(%)j+1dxdt

that stays at the right hand side of the last inequality can be calculated from the eective viscous uxidentity (3.44); using (3.13) we nd thatZ T

0

Z

1

1 + #jTk(%n) Tk(%)j+1dxdt c(n)kTk(%n) Tk(%)kL2((0;T ))

kTk(%n) Tk(%)k(q2)=(2q2)L1((0;T )) kTk(%n) Tk(%)kq=(2q2)Lq((0;T )):Consequently, one concludes that

oscq[%n ! %]((0; T ) ) supk>0

lim supn!0

Z T0

Z

jTk(%n) Tk(%)jqdxdt with some q > 2; (3.52)

where the expression at the left hand side is called oscillations defect measure, see [17, Chapter 3, Section3.7.5].

On the other hand, relation (3.52) implies that the limit quantities %, u satisfy the renormalizedequation of continuity (2.9), see [17, Lemma 3.8], that reads

Lemma 3.11 (Renormalized continuity equation) Let Q R3 be open set and let

%n * % in L1((0; T )Q);

un * u in Lr((0; T )Q;R3);

run * ru in Lr((0; T )Q;R33); r > 1:Let

oscq[%n ! %]((0; T )Q)

for a.a. 2 (0; T ), where the second term at the right hand side tends to 0 as k !1. Reasoning as in(3.51) we verify that

p(%; #)Tk(%) p(%; #) Tk(%) 0:Consequently, there holds Z

% log % % log %

() dx 0:

Finally, since the function z ! z log z is strictly convex on (0;1), we necessarily have% log % % log % = 0 a. e. in (0; T ) :

%n ! % a.e. in (0; T ) : (3.55)With (3.39) and (3.55) at hand, momentum equation (3.28) turns into (2.5) and entropy inequality

(3.43) turns into (2.6). Moreover, clearlyZ T0

0(t)Z

12%nu

2n + %ne(%n; #n)

dxdt!

Z T0

0(t)Z

12%u2 + %e(%; #)

dxdt;

cf. last line in (3.26), (3.23) and (3.39), (3.55). The latter limit yields (2.7). Theorem 2.3 is proved.

4 Proof of Theorem 2.3. Relative entropy inequality.

We deduce a relative entropy inequality satised by any weak solution to the Navier-Stokes-Fouriersystem. To this end, consider a trio (r;;U) of smooth functions, r and bounded below away fromzero in [0; T ] , and Uj@ = 0.

Taking ' = 12 jUj2 as a test function in (2.4), we getZ

1

2%jUj2(; ) dx =

Z

1

2%0jU(0; )j2 dx+

Z 0

Z

%U @tU+ %u rxU U

dx dt: (4.1)

Similarly, the choice ' = U in (2.5) gives rise toZ

%u U(; ) dxZ

%0u0 U(0; ) dx (4.2)

=

Z 0

Z

%u @tU+ %u u : rxU+ p(%; #)divxU S(#;rxu) : rxU

dx dt:

Combining relations (4.1), (4.2) with the total energy balance (2.7) we may infer thatZ

1

2%juUj2 + %e(%; #)

(; ) dx =

Z

1

2%0ju0 U(0; )j2 + %0e(%0; #0)

dx (4.3)

+

Z 0

Z

%@tU+ %u rxU

(U u) p(%; #)divxU+ S(#;rxu) : rxU

dx dt:

Now, take ' = > 0 as a test function in the entropy inequality (2.6) to obtainZ

%0s(%0; #0)(0; ) dxZ

%s(%; #)(; ) dx (4.4)

27

+Z 0

Z

#

S(#;rxu) : rxu q(#;rx#) rx#

#

dx dt

Z 0

Z

%s(%; #)@t+ %s(%; #)u rx+ q(#;rx#)

# rx

dx dt:

Thus, the sum of (4.3), (4.4) readsZ

1

2%juUj2 + %e(%; #)%s(%; #)

(; ) dx (4.5)

+

Z 0

Z

#

S(#;rxu) : rxu q(#;rx#) rx#

#

dx dt

=

Z

1

2%0ju0 U(0; )j2 + %0e(%0; #0)(0; )%0s(%0; #0)

dx

+

Z 0

Z

%@tU+ %u rxU

(U u) p(%; #)divxU+ S(#;rxu) : rxU

dx dt

Z 0

Z

%s(%; #)@t+ %s(%; #)u rx+ q(#;rx#)

# rx

dx dt:

Next, we take ' = @%H(r;) as a test function in (2.4) to deduce thatZ

%@%H(r;)(; ) dx =Z

%0@%H(0;)(r(0; );(0; ) dx

+

Z 0

Z

%@t

@%H(r;)

+ %u rx

@%H(r;)

dx dt;

which, combined with (4.5), gives rise toZ

1

2%juUj2 +H(%; #) @%(H)(r;)(% r)H(r;)

(; ) dx+ (4.6)

Z 0

Z

#

S(#;rxu) : rxu q(#;rx#) rx#

#

dx dt

Z

1

2%0ju0 U(0; )j2 dx

+

Z

H(0;)(%0; #0) @%(H(0;))(r(0; );(0; ))(%0 r(0; ))H(0;)(r(0; );(0; ))

dx

+

Z 0

Z

%@tU+ %u rxU

(U u) p(%; #)divxU+ S(#;rxu) : rxU

dx dt

Z 0

Z

%s(%; #)@t+ %s(%; #)u rx+ q(#;rx#)

# rx

dx dt:

Z 0

Z

%@t

@%H(r;)

+ %u rx

@%H(r;)

dx dt

28

+Z 0

Z

@t

r@%(H)(r;)H(r;)

dx dt:

Furthermore, seeing that

@y (@%H(r;)) = s(r;)@y r@%s(r;)@y+ @2%;%H(r;)@y%+ @2%;#H(r;)@y

for y = t; x, we may rewrite (4.6) in the formZ

1

2%juUj2 +H(%; #) @%(H)(r;)(% r)H(r;)

(; ) dx+ (4.7)

Z 0

Z

#

S(#;rxu) : rxu q(#;rx#) rx#

#

dx dt

Z

1

2%0ju0 U(0; )j2 dx

+

Z

H(0;)(%0; #0) @%(H(0;))(r(0; );(0; ))(%0 r(0; ))H(0;)(r(0; );(0; ))

dx

+

Z 0

Z

%@tU+ %u rxU

(U u) p(%; #)divxU+ S(#;rxu) : rxU

dx dt

Z 0

Z

%s(%; #) s(r;)

@t+ %

s(%; #) s(r;)

u rx+ q(#;rx#)

# rx

dx dt:

+

Z 0

Z

%r@%s(r;)@t+ r@%s(r;)u rx

dx dt

Z 0

Z

%@2%;%(H)(r;)@tr + @

2%;#(H)(r;)@t

dx dt

Z 0

Z

%u @2%;%(H)(r;)rxr + @2%;#(H)(r;)rx

dx dt

+

Z 0

Z

@t

r@%(H)(r;)H(r;)

dx dt:

In order to simplify (4.7), we recall several useful identities that follow directly from Gibbs' relation(1.13):

@2%;%(H)(r;) =1

r@%p(r;);

r@%s(r;) = 1r@#p(r;); (4.8)

@2%;#(H)(r;) = @% (%(#)@#s) (r;) = (#)@%%@#s(%; #)

(r;) = 0;

andr@%(H)(r;)H(r;) = p(r;):

29

Thus relation (4.7) can be nally written in a more concise formZ

1

2%juUj2 + E(%; #jr;)

(; ) dx+

Z 0

Z

#

S(#;rxu) : rxu q(#;rx#) rx#

#

dx dt (4.9)

Z

1

2%0ju0 U(0; )j2 + E(%0; #0jr(0; );(0; ))

dxZ

0

Z

%(uU) rxU (U u) dx dt+Z 0

Z

%s(%; #) s(r;)

U u

rx dx dt

+

Z 0

Z

%@tU+U rxU

(U u) p(%; #)divxU+ S(#;rxu) : rxU

dx dt

Z 0

Z

%s(%; #) s(r;)

@t+ %

s(%; #) s(r;)

U rx+ q(#;rx#)

# rx

dx dt

+

Z 0

Z

1 %

r

@tp(r;) %

ru rxp(r;)

dx dt;

where E was introduced in (2.12).On the other hand, we observe that

Z

%u rp(r;)r

dx =

Z

%(U u) rp(r;)

r %rU rp(r;)

dx

Z

%(U u) rp(r;)

r+1 %

r

U rp(r;) + p(r;)divU

dx

Using the latter identity in the formula (4.9), we get the relative entropy inequality in the form (2.10).Theorem 2.3 is proved.

5 Weak-strong uniqueness

In this section we prove Theorem 2.4 by applying the relative entropy inequality (2.10) to r = ~%, = ~#,and U = ~u, where (~%; ~#; ~u) is a classical (smooth) solution of the Navier-Stokes-Fourier system such that

~%(0; ) = %0; ~u(0; ) = u0; ~#(0; ) = #0:

Accordingly, the integrals depending on the initial values on the right-hand side of (2.10) vanish, and weapply a Gronwall type argument to deduce the desired result, namely,

% ~%; # ~#; and u ~u:

Here, the hypothesis of thermodynamic stability formulated in (1.14) will play a crucial role.

30

5.1 Preliminaries, notation

Following [17, Chapters 4,5] we introduce essential and residual component of each quantity appearingin (2.10). To begin, we choose positive constants %, %, #, # in such a way that

0 < % 12

min(t;x)2[0;T ]

~%(t; x) 2 max(t;x)2[0;T ]

~%(t; x) %;

0 < # 12

min(t;x)2[0;T ]

~#(t; x) 2 max(t;x)2[0;T ]

~#(t; x) #:

In can be shown, as a consequence of the hypothesis of thermodynamic stability (1.14), or, morespecically, of (1.16), (1.17), that

E(%; #j~%; ~#) c8

Z 0

Z

%s(%; #) s(~%; ~#)

@t ~#+ %

s(%; #) s(~%; ~#)

~u rx ~#+ q(#;rx#)

# rx ~#

dx dt

+

Z 0

Z

1 %

~%

@tp(~%; ~#) %

~%u rxp(~%; ~#)

dx dt:

In order to handle the integrals on the right-hand side of (5.4), we proceed by several steps:

Step 1:We have Z

%ju ~uj2jrx~uj dx dt 2krx~ukL1(;R33)Z

1

2%ju ~uj2 dx: (5.5)

Step 2: Z

%s(%; #) s(~%; ~#)

~u u

rx ~# dx

krx ~#kL1(;R3)

2%

Z

hs(%; #) s(~%; ~#)iess

ju ~uj dx+ Z

h%s(%; #) s(~%; ~#)ires

ju ~uj dx ;where, by virtue of (5.2),Z

hs(%; #) s(~%; ~#)iess

ju ~uj dx ku ~uk2L2(;R3) + c() Z

E(%; #j~%; ~#) dx (5.6)

for any > 0.Similarly, by interpolation inequality,Z

h%s(%; #) s(~%; ~#)ires

ju ~uj dx ku ~uk2L6(;R3) + c()

h%s(%; #) s(~%; ~#)ires

2L6=5()

for any > 0, where, again by (5.2)

h%s(%; #) s(~%; ~#)ires

2L6=5()

cZ

E(%; #j~%; ~#) dx5=3

: (5.7)

Combining (5.6), (5.7) and (5.3), we conclude thatZ

%s(%; #) s(~%; ~#)

~u u

rx ~# dx

(5.8) krx ~#kL1(;R3)

ku ~uk2

W 1;20 (;R3)+ c()

Z

E(%; #j~%; ~#) dx

ku ~uk2W 1;20 (;R

3)+K(; )

Z

E(%; #j~%; ~#) dx

for any > 0. Here and hereafter, K(; ) is a generic constant depending on , ~%, ~u, ~# through thenorms induced by (2.13), and %, #, while K() is independent of but depends on ~%, ~u, ~#, %, # throughthe norms (2.13).

32

Step 3:WritingZ

%@t~u+ ~u rx~u

(~u u) dx =

Z

%

~%(~u u)

divxS(~#;rx~u)rxp(~%; ~#)

dx

=

Z

1

~%(% ~%)(~u u)

divxS(~#;rx~u)rxp(~%; ~#)

dx+

Z

(~u u) divxS(~#;rx~u)rxp(~%; ~#)

dx;

we observe that the rst integral on the right-hand side can be handled in the same way as in Step 2,namely,Z

1

~%(% ~%)(~u u)

divxS(~#;rx~u)rxp(~%; ~#)

ess

dx K(; ) k[% ~%]essk2L2()+ku ~uk2L2(;R3);Z

1

~%(% ~%)(~u u)

divxS(~#;rx~u)rxp(~%; ~#)

res

dx

K(; )k[%]resk2L6=5() + k[1]resk2L6=5()

+ ku ~uk2L6(;R3);

while, integrating by parts, Z

(~u u) divxS(~#;rx~u)rxp(~%; ~#)

dx

=

Z

S(~#;rx~u) : rx(u ~u) + p(~%; ~#)divx(~u u)

dx

Thus using again (5.1), (5.3) and continuous imbedding W 1;2() ,! L6() we arrive atZ

%@t~u+ ~u rx~u

(~u u) dx

Z

S(~#;rx~u) : rx(u ~u) + p(~%; ~#)divx(~u u)

dx (5.9)

+cjrx~%j; jrx ~#j; jr2x~uj

ku ~uk2

W 1;20 (;R3)+ c()

Z

E(%; #j~%; ~#) dx

Z

S(~#;rx~u) : rx(u ~u) + p(~%; ~#)divx(~u u)

dx

+ku ~uk2W 1;2(;R3) +K(; )Z

E(%; #j~%; ~#) dx

for any > 0.

Step 4:Next, we get Z

%s(%; #) s(~%; ~#)

@t ~# dx =Z

%hs(%; #) s(~%; ~#)

iess@t ~# dx+

Z

%hs(%; #) s(~%; ~#)

ires@t ~# dx;

33

where Z

%hs(%; #) s(~%; ~#)

ires@t ~# dx

(5.10) k@t ~#kL1()

Z

[%s(%; #)]res dx+ ks(~%; ~#)kL1()Z

[%]resdx K()

Z

E(%; #j~%; ~#) dx;

while Z

%hs(%; #) s(~%; ~#)

iess@t ~# dx

=

Z

(% ~%)hs(%; #) s(~%; ~#)

iess@t ~# dx+

Z

~%hs(%; #) s(~%; ~#)

iess@t ~# dx;

where, with help of Taylor-Lagrange formula,Z

(% ~%)hs(%; #) s(~%; ~#)

iess@t ~# dx

(5.11)

sup(%;#)2[%;%][#;#]

j@%s(%; #)j+ sup(%;#)2[%;%][#;#]

j@#s(%; #)jk@t ~#kL1()

Z

h[% ~%]ess[% ~%]ess+ [# ~#]essidx K() Z

E(%; #j~%; ~#) dx:

Finally, we writeZ

~%hs(%; #) s(~%; ~#)

iess@t ~# dx =

Z

~%hs(%; #) @%s(~%; ~#)(% ~%) @#s(~%; ~#)(# ~#) s(~%; ~#)

iess@t ~# dx

Z

~%h@%s(~%; ~#)(% ~%) + @#s(~%; ~#)(# ~#)

ires@t ~# dx+

Z

~%h@%s(~%; ~#)(% ~%) + @#s(~%; ~#)(# ~#)

i@t ~# dx;

where the rst two integrals on the right-hand side can be estimated exactly as in (5.10), (5.11). Thuswe conclude that

Z

%s(%; #) s(~%; ~#)

@t ~# dx K()

Z

E(%; #j~%; ~#) dx (5.12)

Z

~%h@%s(~%; ~#)(% ~%) + @#s(~%; ~#)(# ~#)

i@t ~# dx:

Step 5:Similarly to Step 4, we get

Z

%s(%; #) s(~%; ~#)

~u rx ~# dx K()

Z

E(%; #j~%; ~#) dx (5.13)

Z

~%h@%s(~%; ~#)(% ~%) + @#s(~%; ~#)(# ~#)

i~u rx ~# dx:

Step 6:

34

Finally, we have Z

1 %

~%

@tp(~%; ~#) %

~%u rxp(~%; ~#)

dx (5.14)

=

Z

(~% %) 1~%

@tp(~%; ~#) + ~u rxp(~%; ~#)

dx+

Z

p(~%; ~#)divxu dx

+

Z

(% ~%)1~%rxp(~%; ~#) (u ~u) dx;

where, by means of the same arguments as in Step 2,Z

(% ~%)1~%rxp(~%; ~#) (u ~u) dx

cjrx~%j; jrx ~#j

ku ~uk2W 1;2(;R3) +

Z

E(%; #j~%; ~#) dx

for any > 0. Resuming this step, we haveZ

1 %

~%

@tp(~%; ~#) %

~%u rxp(~%; ~#)

dx (5.15)

Z

(~% %) 1~%

@tp(~%; ~#) + ~u rxp(~%; ~#)

dx+

Z

p(~%; ~#)divxu dx+

ku ~uk2W 1;2(;R3) +K(; )Z

E(%; #j~%; ~#) dx:

Step 7:Summing up the estimates (5.5), (5.8), (5.9), (5.12 - 5.15), we can rewrite the relative entropy in-

equality (5.4) in the form Z

1

2%ju ~uj2 + E(%; #j~%; ~#)

(; ) dx (5.16)

+

Z 0

Z

~#

#S(#;rxu) : rxu S(~#;rx~u) : (rxurx~u) S(#;rxu) : rx~u

!dx dt

+

Z 0

Z

q(#;rx#) rx ~#

#

~#

#

q(#;rx#) rx##

!dx dt

Z 0

ku ~uk2W 1;2(;R3) +K(; )

Z

1

2%ju ~uj2 + E(%; #j~%; ~#)

dx

dt

+

Z 0

Z

p(~%; ~#) p(%; #)

divx~u dx dt

+

Z 0

Z

(~% %)1~%

h@tp(~%; ~#) + ~u rxp(~%; ~#)

idx dt

35

Z 0

Z

~%@%s(~%; ~#)(% ~%) + @#s(~%; ~#)(# ~#)

h@t ~#+ ~u rx ~#

idx dt

for any > 0.

Step 8:Our next goal is to control the last three integrals on the right-hand side of (5.16). To this end, we

use (4.8) to obtain Z

(~% %)1~%

h@tp(~%; ~#) + ~u rxp(~%; ~#)

idx

Z

~%@%s(~%; ~#)(% ~%) + @#s(~%; ~#)(# ~#)

h@t ~#+ ~u rx ~#

idx

=

Z

~%(~# #)@#s(~%; ~#)h@t ~#+ ~u rx ~#

idx+

Z

(~% %)1~%@%p(~%; ~#) [@t~%+ ~u rx~%] dx;

where, as ~%, ~u satisfy the equation of continuity (1.1),Z

(~% %)1~%@%p(~%; ~#) [@t~%+ ~u rx~%] dx =

Z

(~% %)@%p(~%; ~#)divx~u dx: (5.17)

Finally, using (4.8) once more, we deduce thatZ

~%(~# #)@#s(~%; ~#)h@t ~#+ ~u rx ~#

idx (5.18)

Z

~%(~# #)h@ts(~%; ~#) + ~u rxs(~%; ~#)

idx

Z

(~# #)@#p(~%; ~#)divx~u dx

=

Z

(~# #)"1~#

S(~#;rx~u) : rx~u q(

~#;rx ~#) rx ~#~#

! divx

q(~#;rx ~#)

~#

!#dx

Z

(~# #)@#p(~%; ~#)divx~u dx

Seeing that Z

p(~%; ~#) @%p(~%; ~#)(~% %) @#p(~%; ~#)(~# #) p(%; #)

divx~u dx

ckdivx~ukL1()

Z

E(%; #j~%; ~#) dx

we may use the previous relations to rewrite (5.16) in the formZ

1

2%ju ~uj2 + E(%; #j~%; ~#)

(; ) dx (5.19)

+

Z 0

Z

~#

#S(#;rxu) : rxu S(~#;rx~u) : (rxurx~u)

36

S(#;rxu) : rx~u~# #~#

S(~#;rx~u) : rx~u!dxdt

+

Z 0

Z

q(#;rx#) rx ~#

#

~#

#

q(#;rx#) rx##

+(~# #)q(~#;rx ~#) rx ~#

~#2+q(~#;rx ~#)

~# rx(# ~#)

!dxdt

Z 0

ku ~uk2W 1;2(;R3) +K(; )

Z

1

2%ju ~uj2 + E(%; #j~%; ~#)

dx

dt

for any > 0.

5.3 Viscous terms and terms related to the heat conductivity

Step 1Concerning the viscous term, we shall investigate separately the cases 0 < # < ~# and # ~#.

In the rst case, we have

1f0

ku ~uk2L2(0;T ;W 1;2(;R3)) cZ 0

Z

%(u ~u)2 + E(%; # j ~%; ~#)

dxdt;

where and c are convenient positive constants.

Step 2For the heat conducting term, we have similarly,

~#

#

q(#;r#) r##

+q(#;r#) r~#

# q(

~#;r~#)~#

r(~# #) #~#

~#

q(~#;r~#) r~#~#

= ~#(#)jr(log # log ~#))j2 + ~#q(~#;r log ~#) q(#;r log ~#)

r(log # log ~#)

+(# ~#)q(~#;r log ~#) r(log # log ~#):Whence,

Z 0

Z

~#

#

q(#;r#)#

r# q(#;r#)#

r~#+ q(~#;r~#)~#

r(~# #) + #~#

~#

q(~#;r~#)~#

: r~#!

dxdt

(5.21)

kp(#)rx(log # log ~#)k2L2((0;);R3) c

Z 0

Z

E(%; # j ~%; ~#)dxdt:

where and c are convenient positive constants.Finally, putting together (5.19), (5.20{5.21) we getZ

1

2%ju ~uj2 + E(%; #j~; ~#)

(; ) dx (5.22)

+

p(#)rx log #rx log ~#

2

L2((0;))+ ku ~uk2L2(0;T ;W 1;2(;R3))

c

Z 0

Z

12%(u ~u)2 + E(%; # j ~%; ~#)

dxdt

for a. a. 2 (0; T ). We conclude by applying the Gronwall lemma to this inequality. Theorem 2.4 isproved.

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