Weak solutions for steady compressible Navier-Stokes-Fourier system in two space dimensions

Transcription

1 Nečas Center for Mathematical Modeling Weak solutions for steady compressible Navier-Stokes-Fourier system in two space dimensions Antonín Novotný and Milan Pokorný Preprint no Research Team 1 Mathematical Institute of the Charles University Sokolovská 83, Praha 8

2 Weak solutions for steady compressible Navier Stokes Fourier system in two space dimensions Antonín Novotný 1 and Milan Pokorný 2 1. Université du Sud Toulon-Var, BP 132, La Garde, France novotny@univ-tln.fr 2. Mathematical Institute of Charles University, Sokolovská 83, Praha 8, Czech Republic pokorny@karlin.mff.cuni.cz Abstract We consider steady compressible Navier Stokes Fourier system in a bounded twodimensional domain. We show the existence of a weak solution for arbitrarily large data for the pressure law p, ϑ γ + ϑ if γ > 1 and p, ϑ ln α ϑ if γ = 1, α > 0, depending on the model for the heat flux. 1 Introduction, main result We study the following system of partial differential equations which models the flow of a compressible heat conducting fluid 1.1 div u = 0, 1.2 div u u div S + p = f, 1.3 div Eu = f u divpu + divsu divq. The first equation expresses the balance of mass, the second the balance of momentum and the last one the balance of total energy. Here and in the sequel, the scalar quantity is the density of the fluid, the vector u is the velocity field, the tensor S is the viscous part of the stress tensor, p is the pressure, f the given external force, E the specific total energy and q the heat flux. Mathematics Subject Classification N10, 35Q30 Keywords. steady compressible Navier Stokes Fourier system; weak solution; entropy inequality; Orlicz spaces; compensated compactness; renormalized solution 1

3 We consider system in a bounded domain R 2 together with the following boundary conditions at 1.4 u = 0, 1.5 q n + Lϑϑ Θ 0 = 0, where n is the outer normal to, Θ 0 and the constant L are given. Note that 1.5 expresses the fact that the heat flux through the boundary is proportional to the difference of the temperature ϑ inside and the known temperature Θ 0 outside. Finally, the total mass of the fluid M is given, i.e. 1.6 dx = M > 0. In order to close system we have to specify the constitutive laws for the quantities S, p, E and q. First, the fluid will be assumed newtonian, i.e. [ ] 1.7 S = Sϑ,u = µϑ u + u T divui + ξϑ divui, where µϑ and ξϑ are the viscosity coefficients. We assume the functions to be globally Lipschitz such that 1.8 c ϑ µϑ, 0 ξϑ. Due to the global Lipschitz continuity we also have 1.9 µϑ,ξϑ c ϑ. The heat flux obeys the Fourier law 1.10 q = qϑ, ϑ = κϑ ϑ, with κ C[0, such that 1.11 c ϑ m kϑ c ϑ m, m > 0. The specific total energy E has the form 1.12 E = E,ϑ,u = 1 2 u 2 + e,ϑ with e, the specific internal energy. Note at this moment that for sufficiently smooth solutions, is equivalent with 1.2 and the balance of internal energy 1.13 div eu + divq = S : u p divu. Finally, we have to specify the pressure and the internal energy. We will assume two cases. For simplicity, in the case γ > 1 we consider 1.14 p = p,ϑ = γ + ϑ. 2

4 Recall that, in agreement with the second law of thermodynamics, there exists a function of and ϑ, called entropy, such that 1.15 which due to 1.13 obeys the equation 1 1 De,ϑ + p,ϑd = Ds,ϑ ϑ q 1.16 div su + div = ϑ S : u q ϑ. ϑ ϑ 2 Gibbs relation 1.15 immediately implies that the internal energy fulfills the Maxwell relation 1.17 e, ϑ = 1 2 Assuming the pressure law 1.14 we have 1.18 e = e,ϑ = γ 1 For simplicity, we assume gϑ = c v ϑ and thus p,ϑ ϑ p,ϑ ϑ γ 1 + gϑ s,ϑ = ln ϑcv + s 0. Note that instead of 1.14 we can also treat, similarly as in [9] or [10], more general pressure law 1.20 p,ϑ = γ 1 e,ϑ, γ > 1, where due to 1.15 the pressure has the form 1.21 p,ϑ = ϑ γ γ 1 P ρ P C 1 0,. Assuming additionally ϑ 1 γ 1, 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 K <, Z > 0, γ 1 Z we can get the same result as below. The main ideas are the same, however, many additional technicalities occur, similar to those presented in [9]. Therefore we will not consider this general model and restrict ourselves to

5 Next case concerns γ = 1. We are not able to set simply γ = 1 as the resulting model does not have a good physical meaning. Moreover, due to the lack of a priori estimates, we will consider slightly more regular model, i.e p,ϑ = lnα ϑ, α > 0. Note that α = 0 corresponds to a possible generalization of the case γ = 1 in Unfortunately, we will have to require α > 0. The pressure law 1.23 implies due to e,ϑ = 1 α + 1 lnα c v ϑ, where we did the same choice of the unknown function of temperature as above. Note that the entropy remains unchanged cf Again, a more general model with asymptotic behaviour as in could be considered, however, in order to avoid additional technicalities, we restrict ourselves to the model presented above. We consider weak solutions to our problem as follows: Definition 1 The triple, u, ϑ is called a weak solution to system , or 1.15, 1.23 if > 0 a.e. in, L sγ ; R, s > 1, dx = M, u W 1,2 0 ; R 2, ϑ > 0 a.e. in, ϑ W 1,r ; R r < 2, and 1.25 u ψ dx = 0 ψ C ; R, 1.26 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 2, f uψ + p,ϑu ψ dx Lϑϑ Θ 0 ψ dσ ψ C ; R. We will also need the notion of the renormalized solution to the continuity equation Definition 2 Let u W 1,2 loc R2 ; R 2 and L q loc R2 ; R solve div u = 0 in D R 2. Then the pair,u is called a renormalized solution to the continuity equation, if divb u + b b div u = 0 in D R 2 for all b C 1 [0, W 1, 0, with zb z L 0,. We aim to prove the following two results 4

6 Theorem 1 Let C 2 be a bounded domain in R 2, f L ; R 2, Θ 0 K 0 > 0 a.e. at, Θ 0 L 1. i Let γ > 1, m > 0. Then there exists a weak solution to , in the sense of Definition 1. ii Let α max{2, 2 }, m > 0. Then there exists a weak solution to , 1.15, m 1.23 in the sense of Definition 1. Moreover,,u, extended by zero outside of, is a renormalized solution to the continuity equation in the sense of Definition 2. The first existence result for arbitrarily large data in the theory of steady compressible Navier Stokes equations goes back to P.L. Lions see [6], where the existence was shown for γ > 5 N = 3 and γ > 1 N = 2. These results have been recently improved by 3 Frehse, Steinhauer and Weigant see [3], [4] up to γ > 4 N = 3 and γ 1 N = 2; 3 see also related papers by Plotnikov and Sokolowski [13], [14], [15]. These results concern the barotropic case, i.e. only with ϑ = const. Note that the reason why it is possibly even relatively easily to get the existence of a weak solution for γ = 1 in two space dimensions is connected with the fact that in order to pass to the limit from the approximation to the original system, for p = a, the weak convergence of the density is sufficient, which simplifies the problem considerably. Note also that similar problem with γ = 1 as in this paper, for barotropic flow, was considered in the evolutionary case by Erban [1]. Concerning the heat conducting fluid, the first result for large data in the steady case goes back to Mucha and Pokorný [7], where the existence for shown for γ > 3, m > 3γ 1 3γ 7 for Navier boundary conditions for the velocity. Note that the solution is more regular in this case, i.e. the density is bounded and the velocity and temperature belong to W 1,q for any q <. In [8] the same authors got the existence up to γ > 7 for both the Dirichlet 3 and Navier boundary condition for the velocity. Similar result in two space dimensions is due to Pecharová and Pokorný; here γ > 2 and m > γ 1, see [12]. All these results are γ 2 proved for constant viscosities. In [9], [10] the authors of this paper studied the problem for temperature dependent viscosity, similarly as here. They proved the existence of a variational entropy solution i.e. the balance of total energy is replaced by entropy inequality and global total energy balance for γ > and m > max{ 2, 2, 2 γ4γ 1 }. These solutions are weak 8 3 3γ 1 9 4γ 2 3γ 2 solutions for γ > 4, m > max{1, 2γ } or γ > 5, m > 1. The aim of this paper is to 3 33γ 4 3 extend these results to the twodimensional problem. The plan of the paper is following. First, we recall several useful results, mostly covering properties of Orlicz and Sobolev Orlicz spaces needed in the case of Next we introduce the approximative system based on the approach from [9] and briefly recall the main steps in the existence proof and the first three limit passages. The last limit passage requires new a priori estimates which will be shown in Section 4. Section 5 is devoted to the proof of strong convergence of density, for both 1.14 and

7 2 Preliminaries 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. We will use notation L p ; R, u L p ; R 2, and S L p ; R 2 2. The generic constants are denoted by C and their values may vary even in the same formula or in the same line. We also use summation convention over twice repeated indeces, from 1 to 2; e.g. u i v i means 2 i=1 u iv i. We will need the following version of the Korn inequality Lemma 1 We have for u W 1,2 0 ; R 2, ϑ > 0 and Sϑ,u satisfying Sϑ,u : u 2.1 dx C u 2 1,2 and Sϑ,u : u dx C u 2 ϑ 1,2. Proof. As Sϑ,u : u dx ϑ c 1 u + u T divui : u dx = c 1 u 2 dx, similarly in the second case, 2.1 follows from standard Friedrichs inequality. Next we recall basic properties of a certain class of Orlicz spaces. Let Φ be the Young function. We denote by E Φ the set of all measurable functions u such that Φ ux dx < +. Next we introduce the Luxemburg norm u Φ, i.e. { 1 } 2.2 u Φ = inf k > 0; Φ k ux dx 1. The Orlicz space L Φ is the set of all measurable functions u such that u Φ is finite. Note that for any u E Φ we have see e.g. [5] 2.3 u Φ Φ ux dx + 1. Recall that a Young function is said to satisfy the 2 -condition if there exist k > 0 and c 0 such that Φ2t kφt t c. If c = 0, we speak about the global 2 -condition. We will work with the following classes of Young functions. For α 0 and β 1 we denote by L z β ln α 1+z the class of Orlicz spaces generated by Φz = z β ln α 1 + z. Note that for our choice of α and β this Young function fulfills the global 2 -condition and we have E z β ln α 1+z = L z β ln α 1+z. 6

8 Note further that the definition of the Luxemburg norm yields for β 1 and α > u z β ln α 1+z 1 + ux β ln α β 1 + ux dx. Next, recall that the complementary function to z ln α 1 + z behaves as e z1/α for z large. We denote by E e1/α and L e1/α the corresponding sets of functions. Note that this Young function does not satisfy the 2 -condition. Due to the generalized Hölder inequality we have 2.5 uv 1 u z ln α 1+z v e1/α. As we work in two space dimensions, recall see e.g. [5] i.e. for u W 1,2 0 W 1,2 0 L e z 2 1, 2.6 u e2 C u 1, We also need to estimate u Φ. The definition of the Luxemburg norm immediately yields for any > u Φz = u Φz ; especially 2.8 u eα C u eα + 1, and for u z ln α 1+z C u z ln α 1+z + 1. Next we consider the following problem 2.10 div ϕ = f in, ϕ = 0 at. If f L p, 1 < p < and f dx = 0, then there exists a solution to 2.10 such that 2.11 ϕ 1,p C f p, see e.g. [11]. A similar result holds also in a certain class of Orlicz spaces. We have for the Young function Φ satisfying the global 2 -condition such that for certain γ 0, 1 the function Φ γ is quasiconvex that for f L Φ satisfying the same compatibility condition as above 2.12 ϕ Φ C f Φ, see [16]. Especially for α 0 and β > ϕ z β ln α 1+z C f z β ln α 1+z. 7

9 3 Approximative system For the sake of simplicity, set 3.1 p γ,ϑ = γ + ϑ, γ > 1, lnα ϑ, γ = 1. Similarly, e γ,ϑ denotes the corresponding specific internal energy cf or 1.18, s γ,ϑ the corresponding specific entropy We take η,ε, > 0 and N N. We denote X N = span{w 1, w N } W 1,2 0 ; R 2 with {w i } i=1 a complete orthogonal system in W 1,2 0 ; R 2 such that w i W 2,q ; R 2 for all i N and all q <. We look for a triple N,η,ε,,u N,η,ε,,ϑ N,η,ε, we skip the indeces in what follows in this section such that W 2,q ; R, u X N and ϑ W 2,q ; R, 1 q < arbitrary, where 1 2 u u wi 1 2 u u : wi + S η ϑ,u : w i dx 3.2 pγ,ϑ + β + 2 divw i dx = f w i dx for all i = 1, 2,...,N, 3.3 ε ε + div u = εh a.e. in, and 3.4 div κ η ϑ + ϑ B + ϑ 1 ε + ϑ ϑ ϑ + div e γ,ϑu = S η ϑ,u : u + ϑ 1 p γ,ϑ divu + ε 2 β β a.e. in, with β and B sufficiently large, 3.5 S η ϑ,u = µ [ ] ηϑ u + u T divui + ξ ηϑ 1 + ηϑ 1 + ηϑ divui, h = M/, µ η, ξ η, κ η standard regularizations of functions µ, ξ, κ extended by constants µ0, ξ0, κ0 to the negative real line together with the boundary conditions at 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. Similarly as in [9] we can proof the following 8

10 Theorem 2 Let ε,, η and N be as above, β and B be sufficiently large and all assumption made in Section 1 and Section 3 be satisfied. Let 1 γ < and ε be sufficiently small with respect to. Then 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. We will not go into details of the proof of Theorem 2 as it contains tedious and long computations. The proof is based on a priori estimates coming from the entropy inequality and on the application of the Schaeffer fixed point theorem which is a version of the Schauder fixed point theorem. The details in the case of 3D problem can be found in [9] and the proof in 2D follows precisely the same line and is only slightly easier. We also omit the details of the limit passages N, η 0 + and ε 0 +. Let us only mention that the approximative energy balance 3.4 is in fact only balance of internal energy. From it and the balance of momentum we can deduce entropy equality on the level N < and due to sufficient a priori estimates needed also for the existence result we can pass with N. At this step we immediately lose the entropy equality and get only entropy inequality. In order to pass with η 0 + we also lose the balance of the internal energy; we must switch before the limit passage to the balance of the total energy which is similarly as on the level of the original system just a consequence of balance of internal energy and momentum. At this moment we have sufficient regularity to verify this fact rigorously, which is not the case for weak solution to the original system. The next limit passage, ε 0 +, is more delicate due to the loss of compactness of density. However, exactly as in the 3D case, using technique based on compensated compactness, we can prove that also the density sequence is strongly convergent in certain L p ; R which is enough to pass to the limit. The proof is exactly the same as in the case γ > 1 for 0 +, only somewhat easier; hence there is no need to repeat this procedure twice here. See also [9] for more details. The above mentioned limit passages give us for any > 0 the existence of L sβ ; R, s < 2 arbitrary, 0 a.e. in, 1,2 dx = M, u W 0 ; R, ϑ W 1,2, ϑ > 0 a.e. in such that 3.8 u ψ dx = 0 for all ψ W 1,r ; R, r > 2β, 2β u u : ϕ+sϑ,u : ϕ p γ,ϑ+ β + 2 div ϕ dx = for all ϕ W 1,r 0 ; R 3, r > 2, 1 2 u 2 e γ,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 1 dx u ψ dx β 1 β 9 f ϕ dx

11 for all ψ C 1 ; R. Finally, we also get 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 γ,ϑu ψ dx ϑ L + ϑ B 1 + ϑ Θ 0 ψ dσ ϑ for all nonnegative ψ C 1 ; R. In the following section we will investigate estimates of,u,ϑ independent of which will allow us to pass with A priori estimates 4.1 First, in 3.11, we set ψ 1. It yields ϑ κϑ + ϑ B + ϑ ϑ 2 dx + L + ϑ B 1 Θ 0 dσ ϑ Similarly, using the same test function in Lϑ + ϑ B dσ = u f dx + 1 Sϑ,u : u + ϑ 2 dx ϑ L + ϑ B 1 dσ. L + ϑ B 1 Θ0 dσ + ϑ 1 dx. We plug in estimate 4.2 into 4.1. Thus we need to control the density, multiplied by. To this aim, we use as test function in 3.9 the solution to cf. [9, Section 5] with div ϕ = M in, ϕ = 0 at ϕ 1,q C q, 1 < q <. Hence, after relatively standard computations, β 3 2 β+1 const. Due to this additional a priori bound we have 4.3 u 1,2 + ϑ m ln ϑ 2 + ϑ 1 1, + ϑ B 2 r <, arbitrary, and from 4.2 also 4.4 ϑ m 2 2 m 1,2 C 2 2+ ϑ ϑ 1, + ϑ m 2 2 m 2 C ϑ B 2 r + ϑ 2 1 C f u dx. At this moment, we need to show estimates of the density which are independent of. We proceed separately for γ > 1 and γ = 1. 10

12 4.1 Estimates of the density for γ > 1 The aim is to use as test function in 3.9 a suitable function which produces the estimates of the density. Similarly as above, we take ϕ, solution to div ϕ = s 1γ 1 s 1γ dx a.e. in, 4.5 ϕ = 0 ϕ 1, s s 1 at, s 1γ C sγ, for s chosen suitably below. We have sγ + s 1γ+1 ϑ dx + β + 2 s 1γ dx = 1 γ ϑ + β + 2 dx s 1γ dx f ϕ dx + Sϑ,u : ϕ dx u u : ϕ dx = I 1 + I 2 + I 3 + I 4. Due to the fact that 4.4 implies 4.7 ϑ q Cq,ε1 + ε sγ for arbitrary small ε > 0 and arbitrary large q <, there is no problem to estimate I 1, using interpolation between L 1 and L sγ for. Further, assuming sγ > 2, we have I 2 C f 2 sγ. Next, for q sufficiently close to I 3 C 1 + ϑ u ϕ dx C1 + ϑ q u 2 ϕ s, s 1 i.e. we get the restriction s < 2. Then I 3 C1 + ϑ q u 2 s 1γ sγ C s 1γ+ε sγ. Finally, taking s so close to 2 that sγ > 2 for any γ > 1, we have Altogether, estimates above read I 4 sγ u 2 q ϕ s s sγ C. C s 1γ+1 sγ. Note that we can take any s < 2 thus for arbitrary γ > 1 we can ensure sγ > 2. Hence 4.3, 4.4 and 4.8 imply 4.9 u 1,2 + ϑ r + ln ϑ 2 + ϑ q + ln ϑ q + ϑ 1 1, + sγ C with any r < 2, q <, and any s < 2. Furthermore 4.10 ϑ B ϑ ϑ B q + ϑ B q, + ϑ β+1 C for arbitrary q <. 11

13 4.2 Estimates of the density for γ = 1 Next we consider the case γ = 1. We replace problem 4.5 by 4.11 div ϕ = M a.e. in, ϕ = 0 at, ϕ 1,q C q, and we use this ϕ as test function in 3.9. It reads 3 ln α β ϑ dx dx = M lnα ϑ + β + 2 dx f ϕ dx + Sϑ,u : ϕ dx u u : ϕ dx = J 1 + J 2 + J 3 + J 4. Estimates of J 1 and J 2 are easy; thus we concentrate ourselves only on the last two terms. We have 1 J 3 C + ϑ u ϕ dx C u ϑ 1 2 ϕ 2 2 dx. We now use 2.5 to get 1 1 J 3 C 1 + ϑ 2 2 e m 2 ϕ z ln, m 1+z and thus 2.12, 2.8, 2.9, 2.6, 4.4 and 4.10 yield J 3 C1 + ε 2 hence 2.4 implies for ε > 0, arbitrarily small, Thus z 2 ln m 2 ; 1+z J 3 C ln ε m 1 + dx. For J 4 we proceed similarly: 2 J 4 u 2 ϕ dx ϕ ϕ 2 u e u 4 2 dx C z ln 2 1+z u 2 e21 + z 2 ln 2 1+z 1 C1 + u 2 1, ln dx. J 4 C ln dx 12

14 consider separately K and > K for K sufficiently large and the estimates above yield for α max{2, 2 } m ln α 1 + dx C. We get with s = 2 or, more precisely, together with We may now pass to the limit in the weak formulation. Using and 4.13 we get a subsequence denoted again by index such that 4.14 in L sγ ; R, s < 2 γ > 1 or s = 2 γ = 1, 4.15 u u in W 1,2 0 ; R 2, u u in L q ; R 2, q <, 4.16 ϑ ϑ in W 1,r ; R, r > 2, ϑ ϑ in L q ; R, q <. Passing to the limit in the weak formulation we have 4.17 u ψ dx = 0 for all ψ W 1,r ; R, r > 2, 4.18 u u : ϕ + Sϑ,u : ϕ p γ,ϑ div ϕ dx = for all ϕ W 1,r 0 ; R 2, r > 2, 1 2 u 2 e γ,ϑ u ψ + κϑ ϑ : ψ dx Lϑ Θ 0 ψ dσ = f uψ dx + Sϑ,uu + pγ,ϑu ψ dx for all ψ C 1 ; R. Note that we have also used the fact that β+1 we could also pass to the limit in the entropy inequality 3.11 to get ϑ 1 Sϑ,u : u + κϑ ϑ 2 ψ dx ϑ : ψ κϑ ϑ ϑ 2 s γ,ϑu ψ dx + L ϑ ϑ Θ 0ψ dσ f ϕ dx dx C. Finally, for all ψ C 1 ; R, nonnegative. However, we will not use this fact in the sequel. To finish the proof of Theorem 1, we have to verify that strongly at least in L 1 ; R. We will show this in the last section, separately for γ > 1 and γ = 1. 13

15 5 Strong convergence of the density Before starting, we recall several lemmas which will be useful throughout this section. The proof of them can be found in [2, Appendix]. Lemma 2 Renormalized continuity equation Assume that 5.1 b C[0, C 1 0,, lim s 0 +sb s bs R, b s Cs λ, s 1,, λ a 2 1. Let L a ; R, a 2, 0 a.e. in, u W 1,2 0 ; R 2 be such that R 2 u ψ dx = 0 for all ψ C0 R 2 ; R with, u extended by zero outside of. Then the pair,u is a renormalized solution to the continuity equation, i.e. we have for all b as specified in b u ψ + b b divuψ dx = 0 R 2 for all ψ C 0 R 2 ; R. We will work with the following operators v F 1 [ iξ ξ 2 Fvξ ], R[v] ij 1 ij v = F 1 [ ξi ξ j ξ 2 Fvξ ] with F the Fourier transform. Denote also [ R[v] i = F 1 ξi ξ ] j ξ Fv jξ. 2 We have Lemma 3 Continuity properties of 1 and 1 Operator R is a continuous operator from L p R 2 ; R to L p R 2 ; R 2 2 for any 1 < p <. Operator 1 is a continuous linear operator from the space L 1 R 2 ; R L 2 R 2 ; R to L 2 R 2 ; R 2 + L R 2 ; R 2 and from L p R 2 ; R to L 2p 2 p R 2 ; R 2 for any 1 < p < 2. Lemma 4 Commutators I Let U U in L p R 2 ; R 2, v v in L q R 2 ; R, where Then in L r R 2 ; R 2. 1 p + 1 q = 1 r < 1. v R[U ] R[v ]U vr[u] R[v]U 14

16 Lemma 5 Commutators II Let w W 1,r R 2 ; R, z L p R 2 ; R 2, 1 < r < 2, 1 < p <, < 1 < 1. Then for all such s we have r p 2 s where a 2 = 1 s p 1 r. R[wz] wr[z] a,s,r 2 C w 1,r,R 2 z p,r 2, 5.1 Strong convergence for γ > 1 Using as test function in 3.9 the function ϕ = ζx 1 1 and in 4.18 the function ϕ = ζx 1 1, ζ C 0 ; R, 1 being the characteristic function of the set, we get applying also 4.17 or 3.8, respectively lim 0 + lim 0 + = ζx ζx ζx p γ,ϑ Sϑ,u : R[1 ] dx ζx p γ,ϑ Sϑ,u : R[1 ] dx u R[1 u ] u u : R[1 ] u R[1 u] u u : R[1 ] dx. dx Note that is bounded in L s ; R for a certain s > 2 and thus in C; R 2. Equality 5.4 together with lemmas above implies the following effective viscous flux identity Lemma 6 Under assumption above, for γ > 1, 5.5 p γ,ϑ µϑ + ξϑ divu = p γ,ϑ µϑ + ξϑ divu a.e. in. Proof. We apply Lemma 4 with v = in L s R 2 ; R, s > 2, U = u u in L 2 R 2 ; R 2, with all functions extended by zero outside to the whole R 2. Then R[1 u ] R[1 ] u R[1 u] R[1 ] u in L r R 2 ; R 2, for a certain r > 1. Consequently ζxu R[1 u ] R[1 ]u dx ζxu R[1 u] R[1 ]u dx. 15

17 Hence equation 5.4 reduces to lim ζx p γ,ϑ Sϑ,u : R[1 ] dx = ζx p γ,ϑ Sϑ,u : R[1 ] dx. and Next ζxµϑ u + u T : R[1 ] dx = +R : R : [ ζxµϑ u + u T] dx, [ R : ζxµϑ u + u T] = ζx2µϑ divu [ ζxµϑ u + u T] ζxµϑ R : [ u + u T]. Similar formulas hold also for the limit term. Then, using Lemma 5 with w = ζxµϑ 1 + ϑ, r < 2 and z i = j u i + i u j, j = 1, 2, 3, p = 2 and recalling that µ is globally Lipschitz, R : [ ζxµϑ u + u T] ζxµϑ R : [ u + u T] a,s,r 2 C with 1 < s < 2, a < 2 s. As W a,s L q for any q < 2 and the density s is bounded in L s ; R, s > 2, we get [ R : ζxµϑ u + u T] ζxµϑ R : [ u + u T] [ ζxµϑ u + u T] ζxµϑr : [ u + u T] R : in L 1 ; R. Lemma 6 is thus proved. Recall that L s ; R for s > 2 and thus it fulfills the renormalized continuity equation. We use also see e.g. [2, Appendix] Lemma 7 Weak convergence, monotone functions Let P, G CR CR be a couple of nondecreasing functions. Assume that n L 1 ; R is a sequence such that P n P, G n G, P ng n P G in L1 ; R. i Then a.e. in. P G P G ii If, in addition, Gz = z, P CR,P non-decreasing 16

18 and P = P where we have denoted by = G then P = P. Using Lemma 2 with the renormalization function b = ln implies 5.7 divu dx = 0 as well as i.e. 5.8 Thus formula 5.5 implies i.e. in particular divu dx = 0, divu dx = 0. p γ,ϑ = p γ,ϑ, γ+1 = γ which, due to Lemma 7 and properties of the L p -spaces, yields immediately the strong convergence of the density. Theorem 1 for γ > 1 is proved. 5.2 Strong convergence for γ = 1 This case is slightly more delicate as we are not able to get 5.4. However, we can replace the test function ϕ = ζx 1 1 by ϕ = ζx 1 1 Θ for 0 < Θ < 1; similarly for the limit problem. Thus, exactly as in Lemma 6 we can get Lemma 8 Under assumption above, for γ = 1, 0 < Θ < 1 arbitrary 5.9 p 1,ϑ Θ µϑ + ξϑ Θ divu = p 1,ϑ Θ µϑ + ξϑ Θ divu a.e. in. Proof. Using as test function in 3.9 ϕ = ζx 1 1 Θ and in 4.18 ϕ = ζx 1 1 Θ we get as above note that 1 1 Θ 1 1 Θ in C; R 2 lim ζx p 1,ϑ Θ Sϑ,u : R[1 Θ ] dx 0 + = ζx p 1,ϑ Θ Sϑ,u : R[1 Θ] dx lim ζx Θ u R[1 u ] u u : R[1 Θ ] dx 0 + ζx Θu R[1 u] u u : R[1 Θ] dx. 17

19 Now we apply Lemma 4 with v = Θ Θ in L 2 Θ R 2 ; R, U = u u in L q R 2 ; R 2, q < 2, arbitrarily close to 2. It yields similarly as above ] [ ] 5.11 u [ Θ R[1 u ] R[1 Θ ] u u ΘR[1 u] R[1 Θ] u in L 1 R 2 ; R. Next we write ζxµϑ u + u T : R[1 Θ ] dx [ = R : ζxµϑ u + u T] Θ dx [ = ζx2µϑ divu Θ dx + R : ζxµϑ u + u T] Θ dx ζxµϑ R : [ u + u T] Θ dx and apply exactly as above Lemma 5 with w = ζxµϑ 1 + ϑ, r < 2 and z i = j u i + i u j, j = 1, 2, 3, p = 2 to get 5.9. Note that due to Lemma 2 we have the continuity equation satisfied also in the renormalized sense. Thus for 0 < Θ < div Θ u + Θ 1 Θ divu = 0 in D R 2 and passing to the limit 5.13 div Θ u + Θ 1 Θ divu = 0 in D R 2. Repeating the computations from [11, Lemma 4.39] we arrive at here we need that L 2 ; R! 5.14 div Θ 1 Θ u = 1 Θ Θ 1 Θ 1 Θ Θ divu Θ divu. Therefore, using 5.9 and 4.17, 0 < Θ < 1, 5.15 div Θ 1 Θ u = 1 Θ Θ 1 Θ 1 p 1,ϑ Θ p 1,ϑ Θ Θ µϑ + ξϑ in D R 2, where all functions are extended by 0 outside of. Hence, testing 5.15 by ψ 1 reads p 1,ϑ Θ p 1,ϑ Θ 5.16 Θ 1 Θ 1 dx = 0. µϑ + ξϑ It is not difficult to see cf. [11, Subsection 4.9.4] that 0 in L 1 { Θ = 0}. Thus, due to the strong convergence of the temperature and monotonicity of the mapping t t2 1 + t lnα 1 + t 18

20 we arrive at lnα 1 + Θ + ϑ 1+Θ = lnα 1 + Θ + ϑ Θ. It immediately implies due to Lemma 7 recall that ϑ > 0 a.e. in due to a priori estimate Θ = Θ a.e. in ; thus also Θ = 1+Θ a.e. in, giving the strong convergence of the density. The proof of Theorem 1 is complete. Acknowledgment. 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. 201/08/0315 and by the project LC06052 Jindřich Nečas Center for Mathematical Modeling. References [1] Erban, R.: On the existence of solutions to the Navier-Stokes equations of a two-dimensional compressible flow, Math. Methods Appl. Sci , No. 6, [2] Feireisl, E., Novotný, A.: Singular Limits in Thermodynamics of Viscous Fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser, Basel, [3] Frehse, J., Steinhauer, M., Weigant, W.: The Dirichlet Problem for Steady Viscous Compressible Flow in 3-D, preprint University of Bonn, SFB 611, No , [4] 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 , [5] Kufner, A., John, O., Fučík S.: Function spaces, Academia, Praha [6] Lions, P.-L.: Mathematical Topics in Fluid Dynamics, Vol.2: Compressible Models, Oxford Science Publication, Oxford, [7] Mucha, P.B., Pokorný, M.: On the steady compressible Navier Stokes Fourier system, Comm. Math. Phys , [8] Mucha, P.B., Pokorný, M.: Weak solutions to equations of steady compressible heat conducting fluids, Math. Models Methods Appl. Sci No. 5, [9] Novotný, A., Pokorný M.: Steady compressible Navier Stokes Fourier system for monoatomic gas and its generalizations, submitted. [10] Novotný, A., Pokorný M.: Weak and variational solutions to steady equations for compressible heat conducting fluids, submitted. [11] Novotný, A., Straškraba, I,.: Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, Oxford, [12] Pecharová, P., Pokorný M.: Steady compressible Navier Stokes Fourier system in two space dimensions, submitted. 19

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