Weak and variational solutions to steady equations for compressible heat conducting fluids

Transcription

1 Nečas Center for Mathematical Modeling Weak and variational solutions to steady equations for compressible heat conducting fluids Antonín Novotný and Milan Pokorný Preprint no Research Team 1 Mathematical Institute of the Charles University Sokolovská 8, Praha 8

2 Weak and variational solutions to steady equations for compressible heat conducting fluids Antonín Novotný 1 and Milan Pokorný 2 1. Université du Sud Toulon-Var, BP 2012, La Garde, France novotny@univ-tln.fr 2. Mathematical Institute of Charles University, Sokolovská 8, Praha 8, Czech Republic pokorny@karlin.mff.cuni.cz Abstract We study steady compressible Navier Stokes Fourier system in a bounded three dimensional domain. Considering a general pressure law of the form p = γ 1 e, we show existence of a variational entropy solution i.e. solution satisfying balance of mass, momentum, entropy inequality and global balance of total energy for γ > which is a weak solution i.e. also the weak formulation of total energy balance is satisfied provided γ > 4. These results cover at least two physically reasonable cases, namely γ = 5 monoatomic gas and γ = 4 relativistic gas. 1 Introduction, main result We consider a system of partial differential equations in R 1.1 div u = 0, 1.2 div u u div S + p = f, 1. div Eu = ρf u divpu + divsu divq which describes the steady flow of a compressible, heat conducting fluid in a bounded domain. Here, 0 is the density of the fluid, u the velocity field. The third unknown quantity is the temperature ϑ > 0. It appears in implicitly, as the stress tensor S, the pressure p, the total energy E and the heat flux q depend on the temperature. More precisely, we assume the fluid to be newtonian with 1.4 S = Sϑ,u = µϑ [ u + u T 2 ] divui + ξϑ divui, Mathematics Subject Classification N10, 5Q0 Keywords. Steady Navier Stokes Fourier system; variational entropy solution; weak solution; large data; renormalized solution 1

3 where µϑ > 0, ξϑ 0 are globally Lipschitz functions, 1.5 c ϑ µϑ in R +. Note that due to the global Lipschitz continuity, 1.6 µϑ,ξϑ c ϑ in R +. Next, we assume generalization of the pressure law for monoatomic gas 1.7 p,ϑ = γ 1 e,ϑ with e, being the specific internal energy, and the heat capacity ratio γ > 1. The quantity E, the specific total energy, is defined as the sum of specific internal and kinetic energies, 1.8 E,ϑ,u = 1 2 u 2 + e,ϑ. Notice that state equation 1.7 is physically relevant at least in two situations: if γ = 5, it describes a monoatomic gas and if γ = 4, it describes a relativistic gas. In general, whatever the situation is, the physical values of γ stay below or are equal to 5, cf. Eliezer, Ghatak, Hora [2]. Finally, the heat flux q fulfills the Fourier law 1.9 qϑ = κϑ ϑ, with the heat conductivity 1.10 c 1 + ϑ m κϑ c ϑ m, m > 0. We assume that there exists a quantity called specific entropy, defined by the Gibbs relation 1.11 Ds,ϑ = 1 1 De,ϑ + p,ϑd ϑ up to an additive constant. The Gibbs relation is equivalent with the so-called Maxwell relation e, ϑ 1.12 = 1 2 p,ϑ ϑ p,ϑ, ϑ assuming p, e C 2 0, 2. The Maxwell relation implies, in view of 1.7 and smoothness of p and e, that the pressure 1.1 p,ϑ = ϑ γ γ 1 ρ P, ϑ 1 γ 1 2

4 where P C 1 0, is a nonnegative function. Note that 1.7 can be then understood as definition of the specific internal energy e,ϑ = γ 1. p,ϑ We will further assume that 1.14 P C 1 [0, C 2 0,, P0 = 0, P 0 = p 0 > 0, P Z > 0, Z > 0, PZ lim = p Z Z γ > 0, 0 < 1 γpz ZP Z c 5 <, Z > 0. γ 1 Z Hypotheses and translate the conditions of the thermodynamic stability p > 0, ϑ e > 0. Condition 1.14 is physically relevant if one of the component of the gas in the extremal regime, where is large is a classical γ = 5 or relativistic ϑ γ 1 γ γ = 4 gas. In the whole paper, the state equations 1.7, 1.1 and conditions 1.14 can be replaced by 1.15 p,ϑ = γ + ϑ, e,ϑ = 1 γ 1 γ 1 + c v ϑ, c v > 0, in which case, in agreement with 1.11, 1.16 s,ϑ = log + c v log ϑ. System will be considered with the boundary conditions at 1.17 u = 0, 1.18 qϑ n + Lϑϑ Θ 0 = 0, where L is a given function of the temperature, 1.19 L C0,, c ϑ l Lϑ c ϑ l, l R, and Θ 0 > 0 is a given function at the boundary. Note that 1.18 expresses the fact that the heat flux through the boundary is proportional to the difference of the temperature ϑ inside and the temperature Θ 0 outside. Finally, we prescribe the total mass 1.20 dx = M > 0. The first results concerning the weak solvability of the Navier-Stokes equations concern a more simply looking Navier-Stokes system for isentropic flows which is formally system with ϑ being taken constant and where p γ. Introducing the notion of the effective viscous flux and employing the concept of renormalized solutions to the continuity equation, P.L. Lions proved in his seminal work [12] existence of weak renormalized solutions to the isentropic Navier-Stokes equations with γ > 5. It is to be

5 noticed, however. that all physically reasonable values of the coefficient γ belong to the range [1, 5] Later on, the application of Feireisl s concept of oscillation defect measures [7] to the steady case in [16] made possible to show existence for small values of γ provided better estimates for the density than classical were known. In fact, in contrast with the previous methods, this method allows to prove strong convergence of the density also in situations when the density is uniformly bounded only in L p with p < 2. New ideas, providing the improved estimates of density have been suggested independently by Plotnikov, Sokolowski [17] [18], [19] and by Frehse, Goj, Steinhauer [8]. The paper [19] of Plotnikov and Sokolowski contains also an existence result, where, unfortunately, the integral form of the conservation of mass is violated. A first rigorous proof of existence of weak solutions for a certain γ < 5 appeared in [1]. More precisely, the authors considered γ > 1+ 1, however, for space periodic boundary conditions to avoid problems near boundary. Finally, a new method, allowing to treat γ > 4 in three space dimensions and γ 1 in two space dimensions for Dirichlet boundary conditions, up to the boundary, is proposed in Frehse, Steinhauer, Weigant [9] and [10]. Compactness properties of the full system have been investigated in P.L. Lions [12] under a posteriori assumption about the boundedness of density in L p spaces with p large. The question whether the system provides these bounds was however left open. Existence questions for system with the total energy balance replaced in the weak formulation by the internal energy balance have been re-considered in [1] and the existence of a weak solution was shown for γ > and m > γ 1, l = m 1. Instead γ 7 of 1.17, the Navier boundary conditions with friction for the velocity were imposed, the pressure law was taken p,ϑ = γ + ϑ and the viscosity was assumed to be a constant. Note that in this case L ; R and u W 1,q ; R, ϑ W 1,q ; R for all q <. The method from [1] has been extended to γ > 7 for both no-slip and slip boundary conditions in [14]. The condition for m = l 1 remains the same. However, as the integrability of the density and velocity gradient is lower than for the case γ >, the balance of internal energy had to be replaced by the balance of total energy. In all above mentioned papers, the value of the heat capacity ratio γ is larger than 5 and thus far beyond the physically reasonable cases. Our goal is to develop an existence theory of weak solutions for arbitrary large external data for compressible heat conducting gases including physically reasonable parameters γ which are close to 1. The first step to this goal has been done [15]; the viscosity was taken temperature dependent, as in 1.5, and using only standard tools Bogovskii type estimates of the density we were able to get existence of a solution for γ >. These solutions are 2 however solely variational entropy solutions see Definition 2 in the case γ, 5] or 2 m 1. In the present paper we will use [15] as a starting point for proving existence for values of γ closer to 1. For the sake of completeness, we will anyway in the next section recall the entire approximation process. This result will be achieved by combination of the concept of variational entropy solutions see e.g. [5, Chapter ] with the Feireisl s oscillation defect measure technique and with the application of the Frehse, Steinhauer, Weigant weighted estimates of density 4

6 [9] to the complete Navier-Stokes-Fourier system. The result itself is quite surprising: It gives an example of the situation where for the complete Navier-Stokes-Fourier system one may get better results than for the mathematically simpler looking but physically less reasonable barotropic approximation. We would like to present our main result now. As mentioned above, we will work with two notions of a solution, weak see Definition 1 and variational entropy solution see Definition 2. More precisely, Definition 1 The triple,u,ϑ is called a weak solution to system , if 0 a.e. in, L 6γ 5 ; R, 1,2 dx = M, u W 0 ; R, ϑ > 0 a.e. in, ϑ W 1,r ; R L m ; R L l+1 ; R with u 2 L5; 6 R, uϑ L 1 ; R, Sϑ,uu L 1 ; R, ϑ m ϑ L 1 ; R and 1.21 u ψ dx = 0 ψ C ; R, 1.22 u u : ϕ p,ϑ div ϕ + Sϑ,u : ϕ dx = u 2 + e,ϑ u ψ dx = Sϑ,uu ψ + κϑ ϑ ψ dx f ϕ dx ϕ C 0 ; R, f uψ + p,ϑu ψ dx Lϑϑ Θ 0 ψ dσ ψ C ; R. As mentioned above, in certain cases we are not able to pass to the limit in the total energy balance and the less we can expect it for the internal energy balance; thus we introduce Definition 2 The triple, u, ϑ is called a variational entropy solution to system , if 0 a.e. in, L γ ; R, 1,2 dx = M, u W 0 ; R, ϑ > 0 a.e. in, ϑ L m ; R with u 2 L 1 ; R, ϑ L 1 ; R, ϑ 1 Sϑ,uu L 1 ; R, 1 + ϑ l, 1 ϑ L1 ; R, ϑ m ϑ 2 L 1 ; R and ϑ m ϑ L 1 ; R, equalities 1.21 ϑ 2 ϑ and 1.22 are satisfied in the same sense as in Definition 1, and we have the entropy inequality Sϑ,u : u + κϑ ϑ 2 Lϑ ψ dx ϑ ϑ 2 ϑ Θ 0ψ dσ ϑ : ψ Lϑψ dσ + κϑ s,ϑu ψ dx ϑ for all nonnegative ψ C ; R, together with the global total energy balance 1.25 Lϑϑ Θ 0 dσ = f u dx. 5

7 Remark 1.1 Note that any sufficiently smooth solution in the sense of Definition 2 is actually a classical solution to It can be shown similarly as in the case of the evolutionary system and we refer thus to [4] or [5]. We will also work with renormalized solutions to the continuity equation 1.1. Recall that for, u sufficiently smooth, equality 1.26 below is a direct consequence of multiplication of 1.1 by b. We have Definition Let u W 1,2 loc R ; R and L 6 5 loc R ; R solve div u = 0 in D R. Then the pair,u is called a renormalized solution to the continuity equation, if 1.26 divb u + b b div u = 0 in D R for all b C 1 [0, W 1, 0, with zb z L 0,. We aim to prove Theorem 1 Let C 2 be a bounded domain in R, f L ; R, Θ 0 K 0 > 0 a.e. at, Θ 0 L 1. Let γ > + 41, m > max { 2, 2, 2 γ4γ 1 8 γ 1 9 4γ γ 2}. Let l = 0. Then 2 there exists a variational entropy solution to in the sense of Definition 2. Moreover,,u is a renormalized solution to the continuity equation. Additionally, if m > max{1, 2 γ } and γ > 4, then the solution is a weak solution in γ 4 the sense of Definition 1. Note that in [15] we proved the existence of weak solutions for γ > 5, but assuming only m > 1. For γ = 5, 2 γ = 10 > 1 and thus for γ γ 4 9 5, 12 the result from [15] is in 7 this sense stronger, as larger interval for m is allowed. On the other hand, concerning the variational solution, we extend considerably the interval for γ, and concerning the weak solution, we are able to treat γ > 4, however with additional restriction on m. Before starting with the proof of Theorem 1, we recall several properties of functions p, ϑ, e, ϑ and s, ϑ, resulting from 1.7, The calculations are similar to those performed in [5] in the case γ = 5 and thus we leave them as an easy exercise for the reader. We have for K 0 a fixed constant 1.27 c 8 ϑ p,ϑ c 9 ϑ, for K 0 ϑ 1 γ 1 {, ϑ γ γ 1 for K c 10 γ p,ϑ c 0 ϑ 1 γ 1, 11 ρ γ for > K 0 ϑ 1 γ 1. Further 1.28 p, ϑ p = d γ + p m,ϑ, d > 0, with > 0 in 0, 2, 6 p m,ϑ 0 in 0, 2.

8 For the internal energy defined by γ 1 p γ 1 e,ϑ c 12 γ 1 + ϑ, e, ϑ c γ ϑ in 0, 2. Moreover, the specific entropy s,ϑ defined by the Gibbs relation 1.11 satisfies 1.0 We also have s, ϑ ϑ s, ϑ = 1 ϑ = 1 e, ϑ ϑ ϑ p,ϑ 2 = 1 γ 1 + e,ϑ = 1 2 γp ϑ 1 γ 1 ϑ 1 γ 1 ϑ 1 γ 1 p, ϑ, ϑ P ϑ 1 γ 1 > s,ϑ c ln + ln ϑ in 0, 2, s,ϑ c ln in 0, 1,, s,ϑ c 15 > 0 in 0, 1 1,, s,ϑ c lnϑ in 0, 1 0, 1. Next section will contain several preliminary results, together with basic idea of the approximative scheme to solve our problem, and first three limit passages. In Section we prove new a priori estimates inspired by the method introduced in [9] combined with ideas from [15]. Last section will be devoted to the limit passage closing the proof of Theorem 1. In what follows, we use standard notation for the Lebesgue space L p endowed with the norm p, and Sobolev spaces W k,p endowed with the norm k,p,. If no confusion may arise, we skip the domain in the norm. The vector-valued functions will be printed in boldface, the tensor-valued functions with a special font. Moreover, we will use notation L p ; R, u L p ; R, and S L p ; R. The generic constants are denoted by C and their value may change even in the same formula or in the same line. 2 Preliminaries, approximative system The approximations introduced below in this section involve four parameters: N, η 0 +, ε 0 + and 0 +. The estimates and the limit passages at levels N, η 0 + and ε 0 + can be taken over from [15] modulo several minor modifications. The purpose of this section is to recall concisely this procedure. More details are available in [15]. We start with one preliminary result 7

9 2.1 Korn s inequality Lemma 1 Let u W 1,2 0 ; R, ϑ > 0 and Sϑ,u satisfy Then Sϑ,u : u dx C u ϑ 1,2, Sϑ,u : u dx C u 2 1,2. Proof. We have Sϑ,u : u µϑ dx u + u T 2 ϑ ϑ divui : u dx C u 2 + u T u 2 divu2 dx = C u 2 dx + C divu 2 dx C 1 u 2 1,2, similarly also the other inequality. 2.2 Existence of a solution to the approximative system Let us now introduce the approximating procedure. We fix N N and ε, and η > 0 we pass subsequently with N, η 0 +, ε 0 + and finally with 0 +, thus the assumption ε sufficiently small with respect to does not cause any problems and denote by X N = span {w 1,...,w N } W 1,2 0 ; R with {w i } i=1 an orthonormal basis in W 1,2 0 ; R. Note that due to the smoothness of we may additionally assume that w i W 2,q ; R for all 1 q < e.g. the eigenfunctions of the Laplace operator subject to the homogeneous Dirichlet boundary conditions. Note that we set immediately l = 0 and assume L to be constant. We look for the triple N,η,ε,,u N,η,ε,,ϑ N,η,ε, denoted briefly,u,ϑ such that W 2,q ; R, u X N and ϑ W 2,q ; R, 1 q < arbitrary, where 1 2 u u wi 2 u 1 u : wi + S η ϑ,u : w i 2.2 for all i = 1, 2,...,N, p,ϑ + β + 2 divw i dx = f w i dx 2. ε ε + div u = εh a.e. in, dx and 2.4 div κ η ϑ + ϑ B + ϑ 1 ε + ϑ ϑ ϑ + div e,ϑu = S η ϑ,u : u + ϑ 1 p,ϑ divu + ε 2 β β a.e. in, 8

10 with β max{8, γ}, B 2m + 2, S η ϑ,u = µ ηϑ [ u + u T 2 ] 1 + ηϑ divui + ξ ηϑ 1 + ηϑ divui, where h = M, µ η, ξ η and κ η are suitable regularizations of µ, ξ and κ, respectively, that conserve 1.5, 1.6 and 1.10 and that converge uniformly on compact subsets of [0, to µ, ξ and κ, respectively. We consider system together with boundary conditions at the no-slip boundary condition for the approximative velocity is included in the choice of X N κη ϑ + ϑ B + ϑ 1 ε + ϑ ϑ n = 0, ϑ n + L + ϑ B 1 ϑ Θ η 0 + ε ln ϑ = 0, with Θ η 0 a smooth approximation of Θ 0 such that Θ η 0 is strictly positive at. We have Theorem 2 Let ε,, η and N be as above, β max{8, 2γ} and B 2m + 2. Let ε be sufficiently small with respect to. Under the assumptions of Theorem 1 and assumptions made above in this section, there exists a solution to system such that W 2,q ; R q <, 0 in, dx = M, u X N, and ϑ W 2,q ; R q <, ϑ CN > 0. The detailed proof of Theorem 2 can be found in [15]. Let us only recall the main steps here. We consider mapping with where 2.7 T : X N W 2,q ; R X N W 2,q ; R Tv,z = u,r, S η e z,u : w i dx = 1 2 v v : wi 2 v 1 v wi + p, e z + β + 2 divw i + f w i i = 1, 2,...,N, div κ 2.8 η e z + e zb + e z ε + e z r = div e, e z v + S η e z,v : v +e z p, e z divv + ε 2 β β a.e. in, with, unique solution to see Lemma dx 2.9 ε ε + div v = εh in, n = 0 at, 9

11 together with the boundary conditions at 2.10 κη e z + e zb + e z ε + e z r n + L + e B 1z e z Θ n 0 + εr = 0. Note that the fixed point of T provided it exists corresponds to r = ln ϑ in We apply see e.g. [] Lemma 2 Let T : X X be a continuous, compact mapping, X a Banach space. Let for any t [0, 1] the fixed points tt u = u be bounded. Then T possesses at least one fixed point in X. Concerning problem 2.9 we have see e.g. [16] Lemma Let ε > 0, h = M. Let v X N. Then there exists unique solution to 2.9 such that W 2,p ; R for all p <, dx = M and 0 in. Moreover, the mapping S : v ρ is continuous and compact from X N to W 2,p ; R. For the operator T it holds see [1] for a similar result Lemma 4 Under the assumptions of Theorem 2, for p >, the operator T is a continuous and compact operator from X N W 2,p ; R into itself. In order to finish the proof of Theorem 2, we have to show see [15] Lemma 5 Let assumptions of Theorem 2 be satisfied. Let p >. Then there exists C > 0 such that all solutions to ttu,r = u,r fulfill u 1,2 + r 2,p + ϑ 2,p C, where ϑ = e r and C is independent of t [0, 1]. The proof is based on: i Testing 2.2 by u i.e. linear combinations of w i ii Integrating 2.4 over, together with 2.6 iii Integrating over the entropy version of the approximative energy balance 2.4 i.e. 2.4 divided by ϑ iv Combing identities from steps i iii we get u 1,2 + ϑ B + ϑ 1,2 + β C with C independent of t and also of η and N v Properties of X N and standard regularity results for elliptic equations imply u 2,q + 2,q CN 10

12 vi Standard tools as Kirchoff transform and regularity results for elliptic problems finally yield r 2,p + ϑ 2,q CN 2. Limit passage N As a byproduct of the proof of Theorem 2 we can get the following uniform estimates 2.11 u N 1,2 + N β + ϑ N B + ϑ N 1,2 + ϑ 2 N 1 + ϑ 1 N 1, + ϑ 4 N ϑ N N 2,2 Cε,. Thus, extracting suitable subsequences if necessary, we can get a triple,u,ϑ, limit of Nk,u Nk,ϑ Nk in spaces given by estimates 2.11, a solution to u u ϕ 2 u 1 u : ϕ + S η ϑ,u : ϕ dx p,ϑ + β + 2 div ϕ dx = f ϕ dx ϕ W 1,2 0 ; R, 2.1 ε ε + div u = εh a.e. in, with 2.14 n = 0 a.e. at 2.15 κη ϑ + ϑ B + ϑ 1 ε + ϑ ϑ ϑ : ψ e,ϑu ψ dx L + + ϑ B 1 ϑ Θ η 0 + ε ln ϑ ψ dσ = S η ϑ,u : u + ϑ 1 p,ϑ divu + ε 2 β β ψ dx, for all ψ C 1 ; R. Note that in order to get 2.15 we need S η ϑ N,u N : u N S η ϑ,u : u in L 1 ; R. This fact does not follow from 2.11 but we may get it realizing that we can use as test function in 2.12 the limit function u, together with the limit passage in 2.2 with w i replaced by u N. Last but not least, we can also get the entropy inequality 2.16 ϑ 1 S η ϑ,u : u + ϑ 2 + κ η ϑ + ϑ B + ϑ 1 ε + ϑ ϑ 2 ψ dx ϑ ϑ 2 κη ϑ + ϑ B + ϑ 1 ε + ϑ ϑ : ψ s,ϑu ψ dx ϑ ϑ L + ϑ B 1 + ϑ Θ η ϑ 0 + ε ln ϑ ψ dσ + F ε, for all ψ C 1 ; R, nonnegative, with F ε = oε as ε

13 2.4 Limit passage η 0 + We can use again 2.11 to pass to the limit in the approximative continuity equation, momentum equation and entropy inequality switching to subsequences if necessary 1 2 u u ϕ 1 2 u u : ϕ + Sϑ,u : ϕ dx p,ϑ + β + 2 div ϕ + f ϕ dx ϕ W 1, 6B B 2 0 ; R, 2.18 ε ψ + ψ dx u ψ dx = εh ψ dx ψ W 1,6 5 ; R, ϑ 1 Sϑ,u : u + ϑ 2 + κϑ + ϑ B + ϑ 1 ε + ϑ ϑ 2 ψ dx ϑ ϑ 2 κϑ ϑ B + ϑ 1 ε + ϑ ϑ : ψ s,ϑu ψ dx ϑ ϑ L + ϑ B 1 + ϑ Θ 0 + ε ln ϑ ψ dσ + F ε, ϑ for all ψ C 1 ; R, nonnegative, with F ε as above. The main difficulty in this step appears in the limit passage for the energy balance. We are not anymore able to guarantee the strong convergence of u η in L 2 ; R and thus we are not able to recover in the limit the balance of internal energy. However, we may consider instead of it the balance of total energy which we get summing the approximative balance of the internal energy 2.15 and the approximative momentum equation 2.12 tested by u η ψ i.e. the balance of kinetic energy. Doing so we may now pass with η 0 + as the most difficult term S ηu η,ϑ η : u η ψ dx is replaced by S ηu η,ϑ η u η ψ dx, and here the information from 2.11 is sufficient; we get 1 2 u 2 e,ϑ u ψ + κϑ + ϑ B + ϑ 1 ε + ϑ ϑ ϑ : ψ dx L + + ϑ B 1 ϑ Θ 0 + ε ln ϑ ψ dσ = f uψ dx Sϑ,uu + p,ϑu + β + 2u ψ + ϑ ψ εβh β 1 ψ + βu ψ εβ βψ dx 1 β 1 2εh ψ + 2u ψ 2ε 2ψ dx ψ C 1 ; R. Note that, due to the bounds, the temperature is positive a.e. in and a.e. at. 2.5 Limit passage ε 0 + From the entropy inequality, together with our version of Korn s inequality see Lemma 1 we can deduce the following estimates independent of ε: 2.21 u ε 2 1,2 + ϑ ε B B + ϑ ε 2 1,2 + ϑ 1 2 ε ϑ 2 ε 1 + ϑ ε B B, + ϑ 1 ε 1, C1 + ε 2 6, 12 dx 5

14 with C = C, but independent of ε. There is no information about density which is independent of ε; thus we have to prove it. We can use standard tools as Bogovskiioperator type estimates, i.e. we can use as test function in 2.17 solution to 2.22 with div Φ = s 1β ε 1 Φ = 0 Φ s s 1 s 1, s 1 After standard estimates we can get s 1β ε dx in, at C ε sβ sβ, 1 < s <. 2.2 ε 5 β C and thus, using also the approximative continuity equation with test function ψ = ε, 2.24 u ε 1,2 + ϑ ε B + ϑ ε 1,2 + ϑ 1 2 ε 1,2 + ln ϑ ε 1,2 + ϑ 1 ε 1, + ε 5 β + ε ε 2 C with C independent of ε. Note that we miss information giving us the compactness of density. Passing to the limit ε 0 + we get switching to subsequences, if necessary 2.25 u ψ dx = 0 ψ W 1, 0β 25β 18 ; R, 2.26 u u : ϕ+sϑ,u : ϕ p,ϑ + β + 2 div ϕ dx = f ϕ dx for all ϕ W 1,5 2 0 ; R. Here and in the sequel, g,u,ϑ denotes the weak limit of a sequence g ε,u ε,ϑ ε. Further 1 2 u 2 e,ϑ u ψ + κϑ + ϑ B + ϑ 1 ϑ : ψ dx + L + ϑ B 1 ϑ Θ 0 ψ dσ = f uψ dx Sϑ,uu + p,ϑ + β + 2 u ψ + ϑ ψ u ψ dx β 1 β ψ C 1 ; R, and the entropy inequality ϑ 1 Sϑ,u : u + ϑ 2 + κϑ + ϑ B + ϑ 1 ϑ 2 ψ dx ϑ 2 κϑ ϑ B + ϑ 1 ϑ : ψ s,ϑu ψ dx ϑ L + ϑ B 1 + ϑ Θ 0 ψ dσ ϑ 1 dx

15 for all nonnegative ψ C 1 ; R. In order to show the strong convergence of density which is sufficient to remove the bars in we can combine technique introduced by P.L. Lions in [5] with some of technique from [16, Chapter ]. Using, roughly speaking, as test function in 2.17 ϕ := 1 ε and in 2.26 ϕ := 1, passing to the limit ε 0 +, together with several deep results from harmonic analysis, we end up with 4 p,ϑ + β µϑ + ξϑ divu = p,ϑ + β µϑ + ξϑ divu a.e. in. This fact, together with the theory of renormalized solutions to continuity equation and standard properties of weakly convergent sequences, leads to 2.0 p,ϑ + β + 2 = p,ϑ + β + 2 a.e. in, i.e. β+1 = β, which implies the strong convergence of the density. The reasoning above is somewhat similar and simpler as that one needed for the passage 0 +. The latter is described in all details in Section 4. Thus we have 2.1 u ψ dx = 0 for all ψ W 1, 0β 25β 18 ; R, 2.2 u u : ϕ+sϑ,u : ϕ p,ϑ+ β + 2 div ϕ dx = for all ϕ W 1,5 2 0 ; R, 1 2 u 2 e,ϑ u ψ + κϑ + ϑ B + ϑ 1 ϑ : ψ dx + L + ϑ B 1 ϑ Θ 0 ψ dσ = f uψ dx 2. + Sϑ,uu + p,ϑ + β + 2 u ψ + ϑ ψ 1 dx u ψ dx β 1 β for all ψ C 1 ; R, and ϑ 1 Sϑ,u : u + ϑ 2 + κϑ + ϑ B + ϑ 1 ϑ 2 ψ dx ϑ 2 κϑ ϑ B + ϑ 1 ϑ : ψ s,ϑu ψ dx ϑ L + ϑ B 1 + ϑ Θ 0 ψ dσ, ϑ f ϕ dx for all nonnegative ψ C 1 ; R. More details concerning all estimates and limit passages above are contained in [15]. 14

16 A priori estimates independent of This section will contain new a priori estimates which will finally allow, passing with 0 +, to prove Theorem 1. We will combine the approach from [15] with the ideas from [9] dealing with the barotropic case p = γ, where the existence of a weak solution has been shown for γ > 4. Here, due to the heat conductivity of the fluid, we will be able to get even stronger results. First, we will prove a priori estimates independent of. We have Lemma 6 There exists a constant C, independent of, such that.1 u 1,2 C.2 ϑ m C 1 + u f dx. Proof. Using as test function in the entropy inequality 2.4 ψ 1 reads ϑ κϑ + ϑ B + ϑ dx + Sϑ ϑ. 2,u : u + ϑ 2 dx ϑ L + ϑ B 1 + Θ 0 dσ L + ϑ B 1 dσ. ϑ We have to estimate the second term at the r.h.s. To this aim, we use as test function in 2. ψ 1. It yields.4 Lϑ + ϑ B dσ = u f dx + L + ϑ B 1 Θ0 dσ + ϑ 1 dx. Thus.5 ϑ 1, + ϑ B B, C 1 + u 1,2 6 + ϑ 1 5 dx. Then. and.5 imply by virtue of Hölder and Young inequalities.6 and.7 u 2 1,2 + ϑ m ln ϑ ϑ 1 1, + ϑ B ϑ ϑ B 2 B + ϑ 2 1 C ϑ m C ϑ 1, + ϑ m 2 2 m 2 C B+1 2B B B 1 2 mb+1 mb B+1. 2B 1 B+1 6 5, We need estimates of the density which may depend on, however, in a specific way. We use as test function in 2.2 a solution to 2.22 with s 1 = 1. These relatively β 15

17 easy computations, where we employ.6 and.7, are performed in [15]. We skip them here and only recall that we are able to get.8 β 2 β+1 C provided m > 2 m+2 and β >. Thus, assuming additionally B 6β 8, we can control m 2 the r.h.s. of.7 for bounded which implies.9 u 1,2 + ϑ m ln ϑ 2 + ϑ 1 1, + ϑ B 2 Estimate.9 with.4 gives additionally.10 ϑ m C 1 + Following the idea from [9], we denote for b 1.11 A = b u 2 dx and prove the following lemma Lemma 7 Under the assumptions above, for b 1, Proof. Easily, u f dx 2 2+ ϑ 1 2 u f dx. u 1,2 C, ϑ m C A 1 6b CA 1 b u 2 1 6b 4 5b 4 6b 4 6b 4 5b 4 6b 4 1 u 6b 1 6b 4 1,2, 2 2+ ϑ B 2 B + ϑ 2 1 C. b 1 u 6 6b 4 dx which due to.10 and.9 implies the result. Next we need bounds of the density via standard Bogovskii-operator type estimates, i.e. estimates resulting from testing the momentum equation by a solution to Lemma 8 We have for 1 < s b, s 6m, m > 2, s γ+1 and b 1 b+2 2+m γ.12 u 2 s dx + sγ dx + s 1γ p,ϑ dx + β+s 1γ dx C 1 + A b 2 4s. Proof. Using as test function in 2.2 solution to 2.22 with s 1γ instead of s 1β yields s 1γ + p,ϑ dx + β + 2 dx s 1γ s 1γ β + 2 dx = 1 dx u u : Φ dx + f Φ dx = I 1 + I 2 + I + I 4 + I p,ϑ dx s 1γ dx Su,ϑ : Φ dx

18 First we have, according to 1.27 p,ϑ dx C sγ 1+s 1γ 1 s 1γ 1+s 1γ 1+s 1γ +C 1 ϑ ϑ γ 1 { <K.1 0 ϑ } + 1+s 1γ 1 1+s 1γ 1 ϑ dx γ 1 { <K 0 ϑ } ε s 1γ p,ϑ dx + Cε But 1 s dx dx C 1 γ 1 { <K 0 ϑ } ϑ dx. ϑ dx ϑ dx C ϑ m C1 + A 1 6b 4 C1 + A 4s 6b 4 sγ 1 s dx s 1γ 1+s 1γ as s > 1. The second term can be estimated easily due to interpolation inequalities. Next I u 2 1 s s dx Φ s s 1 s s 1 dx ε sγ dx + Cε u 2 s dx Cε s 1γ p,ϑ dx + Cε u 2 s dx. For b 1, u 2 s dx = b u 24s b 2 u 61 s2 b 61 s2 b b 2 CA 4s b 2 u 1,2 b 2 b sb+2 b 2 1, b sb+2 b 2 provided s < b and s < 1. However, 1 > b for b > 1. b+2 2 b 2 b b+2 Further I 4 C 1 + ϑ u Φ dx C Φ s u s ϑ m, provided s 6m i.e. s > 1 implies m > 2. Then m+2 I 4 ε s 1γ p,ϑ dx + Cε u s 21 + ϑ s m. Hence I 4 ε s 1γ p,ϑ dx + CεA s 6b 4 and for s > 1 we have s < 4s. Finally The lemma is proved. I 5 C Φ s s 1 s ε sγ sγ + Cε. Remark.1 The computations above correspond to assumption s 1γ 1, i.e. s γ+1. It gives additional restriction for the existence of weak solution, γ < 5. We can avoid γ this restriction by more careful and technically more complicated estimates of the term I 1. We will not do it as for γ 5 we have another method giving the existence of a weak solution, see [15]. dx 17

19 Next two lemmas form the core of the method they provide new a priori estimates of the pressure; similar estimates have been used in the papers by Plotnikov and Sokolowski [17], by Březina and Novotný [1] and by Frehse, Steinhauer and Weigant [9]. Our approach follows closely the last mentioned paper. Lemma 9 Let x 0, R 0 < 1dist x 0,. Then.14 p,ϑ x x 0 dx C 1 + p,ϑ α 1 + u 1,2 1 + ϑ m + u 2 1, provided B R0 x 0 { m 2 }.15 α < min 2m, 1. Proof. We use as test function in 2.2 ϕ i x = x x 0 i x x 0 ατ2 with τ 1 in B R0 x 0, R 0 as above, τ 0 outside B 2R0 x 0, τ C R 0. Note that i ϕ j = div ϕ = α + g x x 0 ατ2 1 x, ij x x 0 αx x 0 i x x 0 j τ 2 + g α x x 0 α+2 2 x with g 1, g 2 in L. Then.16 p,ϑ + β + 2 u ατ 2 2 dx + x x 0 α x x 0 u x x 0 2 α α τ 2 dx x x 0 α+2 = p,u + β + x 2 x 0 τ 2 x x0 dx + Su x x 0 α,ϑ : τ 2 dx x x 0 α x x 0 x x 0 + Su,ϑ : x x 0 α τ2 dx f dx x x 0 ατ2 x x 0 u u : x x 0 α τ2 dx. As we have provided x x 0 x x 0 α 1 x x 0 α, Su,ϑ : x x 0 dx C1 + ϑ x x 0 ατ2 m u 2 1 q = m > α, 18

20 implying α < m 2 for m > 2. An important observation is that the second integral on 2m the l.h.s. of.16 is nonnegative. As the other terms on the r.h.s. of.16 contain the part of the gradient of the test function which is bounded and α 1, the proof is finished. Next, we would like to get analogous estimates also near the boundary. Following [9] we consider the test function of the form.17 ϕx = dx dxdx + x x 0 a α with a = 2, x 2 α 0 and dx the distance function to the boundary. Due to the fact that C 2, the distance function d C 2. Moreover, dx = x ξx, where dx ξx is the closest point to x, cf. [20, Exercise 1.15] or [11, Section 14.6]. We thus deduce that there exist c 1, c 2 positive such that: i dx C 2, dx > 0 in, dx = 0 at ii dx c 1 > 0, x with dist x, c 2 iii dx c 1 > 0, x with dist x, c 2 The main properties of ϕ are given in the following lemma Lemma 10 The function ϕ, defined by.17, belongs to W 1,q 0 for 1 q < α α. Moreover,.18 dx ijdx 2 j ϕ i x = dx + x x 0 a + 1 αdx + x x 0 a α 2dx + x x 0 a idx 1+α j dx + 1 αdx + x x 0 a 2dx + x x 0 a idx µ i x 1+α j dx µ j x + αdx[ jdx i x x 0 a i dx j x x 0 a ] 2dx + x x 0 a 1+α α 2 d 2 x i x x 0 a j x x 0 a 2dx + x x 0 a 1+α 1 αdx + x x 0 a, where i = 1, 2,. µ i x = αdx 1 αdx + x x 0 a 1 i x x 0 a, Proof. First, as ϕ = 0 at dx = 0 at and ϕ is continuous in, the trace of ϕ is zero. Next we compute the derivative of ϕ, however, in a slightly different way than in.18:.19 j ϕ i x = αdx idx j x x 0 a dx + x x 0 a 1+α + dx 2 ijdx dx + x x 0 a α 1 αdx + x x0 a i dx j dx. dx + x x 0 a 1+α 19

21 As a = 2 2 α > 1, we have j ϕ i x C dx + x x 0 a α. Thus we need to find for which q [1, the integral 1 dx < +. dx + x x 0 a αq In the general situation with curved boundary we can flatten it and we are left with η 2 dx x + x x 0 a αq R + with η a suitable cut-off function with bounded support. Using cylindrical coordinates we can transform the integral into the form rdr dx r2 + x 2 a 2 + x αq which is finite provided q < α. Hint: use e.g. the change of variable u = α r2 + x 2 a 2, v = x. Finally, formula.18 is a direct consequence of.17. Using function ϕ from.17 as a test function in 2.2 yields Lemma 11 Under assumptions above, we have for α < 9m 6 and x 9m 2 0 p,ϑ.20 x x 0 αdx C 1 + p,ϑ ϑ m u 1,2 + u 2 1. B R0 x 0 Proof. We have p,ϑ + β + 2 div ϕ dx + u u : ϕ dx.21 = Su,ϑ : ϕ dx f ϕ dx. From.18 we see that dx dx div ϕ = dx + x x 0 a + 1 αdx + x x 0 a α 2dx + x x 0 a dx 2 α αdx + x x 0 a dx 2dx + x x 0 a µx 2 α+1 α 2 d 2 x x x 0 a 2 2dx + x x 0 a α+1 1 αdx + x x 0 a C 1 dx 2 dx + x x 0 a C 2. α Thus recall that ϕx + x x 0 a C x x 0 as ϕ is continuously differentiable near the boundary and ϕx 0 = 0, together with a > 1 p,ϑ p,ϑ div ϕ dx C 1 x x 0 dx C α 2 p,ϑ dx. B R0 x 0 20

22 Next u u : ϕ dx C u 2 dx, as the skew symmetric part of ϕ has zero contribution. Note that the positive part of u u : ϕ does not provide any useful information. The term coming from the r.h.s. f can be estimated directly by f 1 ϕ, and Su,ϑ : ϕ dx C u ϑ m provided is such that q < α 9m 6, i.e. α <. α 9m 2 Note that 9m 6 < m 2 for m > 2 9m 2 2m restrictive. Lemma 12 Let 1 b < γ, α < 9m 6 9m 2.22 A = b u 2 dx C u 2 1,2 1 q = m and thus the bound from Lemma 11 is more b 2γ and α >. Then b 1+ p,ϑ 1 + u 1,2 1+ ϑ m + u 2 1 b γ. Proof. Recall that from Lemmas 9 and 11 we have for any x 0 see.14 and.20 p,ϑ.2 x x 0 dx C 1 + p,ϑ α 1 + u 1,2 1 + ϑ m + u 2 1, provided α < 9m 6. 9m 2 Take b < γ and ν = γ αb γ b.24 C b γ x x 0 dx γ α b x x 0 dx = b 2γ < i.e. α >. As γ b Cp,ϑ, we have C Let h be the unique solution to b γ γ 1 1 b γ dx x x 0 α x x 0 1 ν b + p,ϑ 1 + u 1,2 1 + ϑ m + u 2 γ h = b > 0 in, h = 0 at. Then hx = Gx,y by dy with G, the Green function to problem.25. As Gx,y x y, we get from.2.24 that h L with b.26 h C sup x x 0 x x 0 dx. 21 C for all x,y, x y

23 Therefore and A = hx u x 2 dx = 2 u : u h dx, 1.27 A C u 2 u 2 h 2 2 dx. Now Thus B = u 2 h 2 dx = h h u 2 dx u 2 h h dx C h A + u 2 B 1 2. B CA h + u 2 2 h 2. Returning to.27, Young s inequality together with the Friedrichs inequality imply A C u 2 2 h. The last estimate together with.26 and.24 finishes the proof. We will now combine estimates.12 and.22. As in.1, we can show that 1 p,ϑ 1 C sγ s dx + s 1γ 1 s 1γ s 1γ+1 s 1γ+1 p,ϑ dx ϑ dx. Moreover, as we get, combining Lemmas 8 and 12 i.e. A C.28 A C u 2 1 C u 2 s, 1 + A 4s b 2 s + A 6b 4 + A 6b 4 s 1γ s 1γ s b γ s 1γ+1, 1 + A 4s 1 1 b 2 s + A In order to get the desired estimates, we need to fulfill.29 4s s 1 b b 2 γ < 1 and 1 6b 4 6b s 7 b γ s 1γ s 7 b s 1γ + 1 γ < 1 for a certain s > 1 and 1 b < γ. If we collect all assumptions from this section, we get, additionally to.29,.0 s > 1, s b b + 2, s 6m 2 + m, m > 2, s γ + 1 γ, 1 b < γ, b 2γ b γ < 9m 6 9m 2. Note that, for m > 2 and m > 2, for any γ > 1, we can always choose 1 < b < γ and 9 γ 1 s > 1 such that.29 and.0 are fulfilled. Therefore we have 22

24 Lemma 1 Let γ > 1, m > 2 and m > 2. Then there exists s > 1 such that 9 γ 1 is bounded in L sγ ; R and p,ϑ, u and u 2 are bounded in L s ; R. Proof. From Lemma 8 we get the statement of Lemma 1, except for u. But u s 1 2 s u s. γ In order to pass to the limit in the total energy balance, we have to show that.1 lim 0 + β 6 5 β = 0. However, we only have see the proof of Lemma 8.2 β+s 1γ β+s 1γ C. Exactly as in [15], we can repeat the estimates of Lemma 8 with s 1γ > 1 β to get 5. β 6 C, β+η 5 η > 0, with C independent of. Thus. implies.1, due to interpolation of the L 6 5 β -norm between L 1 and L 6 5 β+η. Note that for this we need that ϑ is bounded in L q ; R, q > and u 2 is bounded in L q ; R for q > 6. The former is fulfilled if 5 m > 1, the latter will be considered in the following section. 4 Limit passage 0 + From the estimates in the previous sections we may deduce that there exist subsequences denoted again u,,ϑ such that 4.1 u u in W 1,2 0 ; R, u u in L q ; R, q < 6, in L sγ ; R, γ > 1, m > 2,m > 2 γ 9 γ 1, { ϑ ϑ in W 1,r m } ; R, r = min 2,, m + 1 ϑ ϑ in L q ; R, q < m, ϑ ϑ in L q ; R, q < 2m. As mentioned at the end of the last section, we need to clarify when s from the previous section can be chosen larger that 6. To this aim, we return to From.29.0 and the requirement s > 6 we get b b + 2 > 6 5, b 1 γ b 4 γ γ + 5 < 1, 6m 2 + m > 6 5, 1 b 2 b 2 γ < 1, γ + 1 > 6 γ 5 2

25 together with 4. b < γ m Thus imply 4.4 m > 1, 5 > γ > 4, m > 2γ γ 4. Hence s > 6 5 provided 4.5 m > m > 1 for 5 > γ > 12 7, 2γ for γ γ 4 4, 12 7 ]. Remark 4.1 The presented method implies, due to the condition s γ+1, the restriction γ γ < 5. To avoid it, we can use the method proposed in Remark.1. As large values of γ are physically less interesting, we skip details and rather recommend paper [15], where another method giving existence of a weak solution has been shown. In this case we can pass to the limit in the continuity equation, momentum equation and the total energy balance as well as entropy inequality to get 4.6 u ψ dx = 0 for all ψ C 1 ; R, 4.7 u u : ϕ + Sϑ,u : ϕ p,ϑ div ϕ dx = f ϕ dx for all ϕ C 1 ; R, ϕ = 0 at for the notation, cf Note that we used in the limit passage for the momentum equation that in L sγ ; R, u u in L q ; R, q < 6 and u 2 s C <, s > 1 imply that u u u u in L s ; R. Thus at this moment, there is no restriction on γ. Further ϑ 1 Sϑ,u : u + κϑ ϑ 2 ψ dx 4.8 ϑ : ψ κϑ ϑ ϑ 2 s,ϑu ψ dx + L ϑ ϑ Θ 0ψ dσ, for all nonnegative ψ C 1 ; R. Assuming 4.5, then also 1 2 u 2 e,ϑ u ψ + κϑ ϑ : ψ dx Lϑ Θ0 ψ dσ = f uψ dx + Sϑ,uu + p,ϑu ψ dx 24

26 for all ψ C 1 ; R. However, in the case when 4.5 is not fulfilled, we get only 4.6, 4.7 and 4.8. Instead of 4.9, we can only show taking ψ 1 and passing to the limit 4.10 Lϑ Θ 0 dσ = f u dx. To finish the proof of Theorem 1 we have to verify that strongly in L 1. As the proof is exactly the same as in [15], we will only briefly recall the main steps and in more detail we point out the moment, when the restriction on γ appears. Step i Effective viscous flux Using as test function ζx 1 1 T k, k N with T C [0,, T k z = kt z, Tz = k z for 0 z 1, concave on 0,, 2 for z in the approximative balance of momentum 2.2, and ζx 1 1 T k in its limit version 4.7 with ζx C0 ; R, we get the identity lim ζx p,ϑ T k Sϑ,u : R[1 T k ] dx 0 + = ζx p,ϑ T k Sϑ,u : R[1 T k ] dx lim ζx T k u R[1 u ] u u : R[1 T k ] dx 0 + ζx T k u R[1 u] u u : R[1 T k ] dx; [ ] here R denotes the Riesz operator, R[v] ij = 1 ij v = F 1 ξi ξ j Fvξ with ξ 2 F the Fourier transform. We recall for the proof see e.g. [5, Appendix] Lemma 14 Commutators I Let U U in L p R ; R, v v in L q R ; R, where Then in L s R ; R. Lemma 14 applied to 1 p + 1 q = 1 s < 1. v R[U ] R[v ]U vr[u] R[v]U v = T k T k in L q R ; R, q < arbitrary U = u u in L p R ; R, for certain p > 1, 25

27 yields that ζxu T k R[1 u ] R[1 T k ]u dx ζxu T k R[1 u] R[1 T k ]u dx provided p > 6. Indeed, we need to verify that u 5 p C and u u in L 1 ; R. As u 6 5 u s 1 2 s 5s, we need s 5s < sγ which together with.29.0 implies 4γ 2 γ 2 > 0, i.e. γ > Moreover, and.29 7 yield additionally m > 2 γ4γ 1. The other conditions do not give 9 4γ 2 γ 2 any further restrictions with respect to conditions deduced above and below. Summarizing, 4.11 yields ζx p,ϑt k p,ϑ T k dx = ζx Sϑ,u : R[1 T k ] Sϑ,u : R[1 T k ] dx. In the sequel we will need another specific result from harmonic analysis and one result concerning properties of weakly convergent sequences, see again [5, Appendix] Lemma 15 Commutators II Let w W 1,r R ; R, z L p R ; R, 1 < r <, 1 < p <, < 1 < 1. Then for all such s we have r p s R[wz] wr[z] a,s,r C w 1,r,R z p,r, where a = Here, s p r a,s,r denotes the norm in the Sobolev Slobodetskii space W a,s R. Lemma 16 Weak convergence and monotone operators Let P,G CR CR be a couple of nondecreasing functions. Assume that n L 1 ; R is a sequence such that Then a.e. in. P n P, G n G, P ng n P G P G P G in L1 ; R. 26

28 We can write 4 ζxsϑ,u : R[1 T k ] dx = lim ζx 0 + µϑ + ξϑ divu T k dx + lim T k ϑ R [ζxµϑ u + u T] 0 + ζxµϑ R : [ u + u T] dx, similarly for the limit term. Using Lemma 15 with w = ζxµϑ bounded in W 1,r ; R with r = min{2, m } and z m+1 i = i u j + j u i, j = 1, 2,, bounded in L 2 ; R, we get that R [ζxµϑ u + u T] ζxµϑ R : [ u + u T] is bounded in W a,s ; R with s < 6r, a = r r s 6, provided m+1 [ < 5, i.e. m > 2. m 6 Thus the latter expression converges strongly to R ζxµϑ u+ u T] ζxµϑr : [ ] u + u T in certain L q ; R with q > 1, where we used the strong convergence of ϑ to ϑ. As T k T k in all L p ; R, p <, we get finally the effective viscous flux identity p,ϑt k µϑ + ξϑ T k divu 4 = p,ϑ T k µϑ + ξϑ T k divu. Step ii Renormalized continuity equation We introduce the quantity 4.1 osc q [ ]Q = sup lim sup T k T k q dx k>1 0 + Q called oscillations defect measure. This quantity can be used to verify the validity of the renormalized continuity equation, see [5, Lemma.8] Lemma 17 Let R be open and let Let in L 1 ; R, u u in L r ; R, u u in L r ; R, r > osc q [ ] < for 1 q < 1 1 r, where,u solve the renormalized continuity equation 4.6. Then the limit functions solve 4.6 for all b C 1 [0, W 1, 0,. We have 27

29 2 Lemma 18 Let,u,ϑ be as above and let m > max{, 2 }. Then there exists γ 1 q > 2 such that 4.14 holds true. Proof. We follow the ideas from [5, Section.7]. Recall that p,ϑ = d γ + p m,ϑ, p m,ϑ see We get using Lipschitz continuity of T k and trivial inequality a b γ a γ b γ, a b 0, d lim sup T k T k γ+1 dx d lim sup γ γ T k T k dx = d γt k γt k dx + d γ γ T k T k dx. Using convexity of γ, concavity of T k, strong convergence of the temperature, 1.28 and Lemma 16, we get 4.15 d lim sup 0 + T k T k γ+1 dx p,ϑt k p,ϑ T k dx. Let G k t,x,z = d T k z T k t,x γ+1. Thus and using 4.12, G k,, G k,, p,ϑt k p,ϑ T k 0, 4 µϑ + ξϑ T k divu T k divu for all k 1. Then 1 + ϑ 1 G k t,x, dx C sup divu 2 lim sup T k T k >0 0 + C lim sup T k T k On the other hand, for q > 2 T k T k q dx T k T k q 1 + ϑ q q γ ϑ C 1 + ϑ 1 T k T k γ+1 q dx + C 1 + ϑ γ+1 q dx, γ+1 dx which, using 4.16 with interpolation inequality, yields under assumptions of Lemma 18 the desired result. Step iii Strong convergence of density As,u and,u verify the renormalized continuity equation, we have T k divu dx = 0, 28

30 and Hence, using 4.12, µϑ + ξϑ T k divu dx = 0, i.e. T k divu dx = 0. p,ϑt k p,ϑ T k dx = Tk T k divu dx. As lim k T k 1 = lim k T k 1 = 0, 4.1 and 4.17 together with interpolation inequality imply p,ϑt k p,ϑ T k dx = 0. lim k Returning back to 4.15 we have Therefore, as we have which implies lim k lim sup 0 + T k T k γ+1 dx = 0. 1 T k 1 + T k T k 1 + T k 1, in L 1 ; R in L p ; R 1 p < sγ. The proof of Theorem 1 is finished. We end with several concluding remarks. a As mentioned in [15], assuming also the radiation, i.e. p,ϑ = p M,ϑ + a ϑ4, e,ϑ = e M,ϑ + a ϑ4, κϑ = κ M ϑ + cϑ with p M,, e M, κ M as in this paper, the proof becomes easier as we get immediately m. Therefore we do not consider this situation here. b The slip or Navier boundary conditions u n = 0, Sn τ + αu τ = 0, α 0 can be considered as well. The only changes are the following. For Lemma 1 Korn s inequality we need ξϑ c ϑ and, if α = 0, additionally, cannot be axially symmetric. Definition of weak solution to the momentum equation must be modified the solution and the test function have only normal traces zero. Finally, in Section in Lemma 12 we have to replace.25 by h = b 1 h n = 0 b dx, at. Then we get the same result, following mutatis mutandis the steps above. 29

31 c Looking at conditions in Theorem 1, we get the existence of variational solution for larger interval of γ s than in [15] γ > 4 instead of γ > 5, however, the conditions on m in the interval γ 5, 12 are more restrictive than in [15], where m > 1 was 7 sufficient. Thus the weak solution exists also for γ 5, 12 if m > 1 only, however, 7 the technique from [15] based on Bogovskii-type estimates of the density must be used. This is in full correspondence with the observation in the barotropic case that the technique of Frehse, Steinhauer and Weigant gives for γ > 5 worse result than the standard technique based on the Bogovskii-type estimates. Acknowledgment. This research was initiated during the stay of M.P. at the Université du Sud Toulon Var which is kindly acknowledged for the financial support. The work of M.P. is a part of the research project MSM financed by MSMT and partly supported by the grant of the Czech Science Foundation No. 201/08/015 and by the project LC06052 Jindřich Nečas Center for Mathematical Modeling. References [1] Březina, J., Novotný, A.: On Weak Solutions of Steady Navier-Stokes Equations for Monatomic Gas, Comment. Math. Univ. Carolin , [2] Elizier, S., Ghatak, A., Hora H.: An introduction to equations of states, theory and applications, Cambridge University Press, Cambridge, [] Evans, L.C.: Partial Differential Equations, Graduate Studies in Math. 19, Amer. Math. Soc., Providence, [4] Feireisl, E.: Dynamics of viscous compressible fluids, Oxford University Press, Oxford [5] Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser, Basel, [6] Feireisl, E., Novotný, A.: On a simple model of reacting compressible flows arising in astrophysics, Proc. Royal Soc. Edinbourgh, 15A 2005, [7] Feireisl, E., Novotný, A., Petzeltová, H.: On the existence of globally defined weak solutions to the Navier Stokes equations of compressible isentropic fluids, J.Math. Fluid Mech. 2001, [8] Frehse, J., Goj, S., Steinhauer, M.: L p -estimates for the Navier-Stokes equations for steady compressible flow, Manuscripta Mathematica , [9] Frehse, J., Steinhauer, M., Weigant, W.: The Dirichlet Problem for Steady Viscous Compressible Flow in -D, preprint University of Bonn, SFB 611, No , [10] Frehse, J., Steinhauer, M., Weigant, W.: The Dirichlet Problem for Viscous Compressible Isothermal Navier Stokes Equations in Two-Dimensions, preprint University of Bonn, SFB 611, No , 0

32 [11] Gilbarg, D., Trudinger, N.-S.: Elliptic Partial Differential Equations of Second Order, Springer Verlag, Berlin Heidelberg New York, 198. [12] Lions, P.-L.: Mathematical Topics in Fluid Dynamics, Vol.2: Compressible Models, Oxford Science Publication, Oxford, [1] Mucha, P.B., Pokorný, M.: On the steady compressible Navier Stokes Fourier system, Comm. Math. Phys , [14] Mucha, P.B., Pokorný, M.: Weak solutions to equations of steady compressible heat conducting fluids, Mathematical Models and Methods in Applied Sciences No. 5, [15] Novotný, A., Pokorný M.: Steady compressible Navier Stokes Fourier system for monoatomic gas and its generalizations, submitted. See also Preprint Series of Necas Center for Mathematical Modeling, , Preprint No [16] Novotný, A., Straškraba, I,.: Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, [17] Plotnikov P.I., Sokolowski, J.: On compactness, domain dependence and existence of steady state solutions to compressible isothermal Navier-Stokes equations, J. Math. Fluid Mech , [18] Plotnikov, P.I., Sokolowski, J.: Concentrations of stationary solutions to compressible Navier-Stokes equations, Comm. Math. Phys , [19] Plotnikov, P.I., Sokolowski, J.: Stationary solutions of Navier-Stokes equations for diatomic gases, Russian Math. Surv , [20] Ziemer W.P.: Weakly Differential Functions, Springer Verlag, New York