On the domain dependence of solutions to the compressible Navier-Stokes equations of an isothermal fluid

Transcription

1 Nečas Center for Mathematical Modeling On the domain dependence of solutions to the compressible Navier-Stokes equations of an isothermal fluid Nikola Hlaváčová Preprint no Research Team 1 Mathematical Institute of the Charles University Sokolovská 83, Praha 8

2 On the domain dependence of solutions to the compressible Navier-Stokes equations of an isothermal fluid Nikola Hlaváčová January 1, Introduction The aim of this paper is to study the behaviour of the variational solutions to equations describing viscous compressible isothermal fluids with nonlinear stress tensors in a sequence of domains { n } n=1, which converges to a domain. This convergence of domains is defined by the Sobolev-Orlicz capacity. We prove that the solutions converge to a solution of the respective equations in. This problem was first studied in [1] for barotropic fluids. The result obtained in the paper can be applied to generalization of the existence result proved in [6], [4] and [9], where C 2+µ -regularity of the boundary of the domain was required. After the convergence of the sequence n, the existence result covers all having C,1 -regularity of its boundary. Moreover, the results provide mathematical apparatus for shape optimization. In the paper is supposed the knowledge of theory of Orlicz and Sobolev-Orlicz spaces in the range of [2]. In the following text we will work with Orlicz spaces associated with Young functions Φ 1 z := z ln1 + z, Φ γ z := 1 + z ln γ 1 + z for γ > 1, Mz := e z z 1 and their complementary functions Ψ 1, Ψ γ and M. Next we denote Φ 1 z the α Young functions with the asymptotic growth z ln 1 α z for z z >, α 1,, and Ψ 1 z their complementary functions. We know that Young functions Φ γ, α The author thanks the Jindřich Nečas Center for Mathematical Modeling the project LC652 financed by MŠMT for its support. 1

3 γ 1, satisfy the global 2 -condition see [9, Lemma 1.33, page 14]. Relationship between Young functions Φ γ1 and Φ γ2 is Φ γ1 Φ γ2 for < γ 1 < γ 2. For their complementary functions it then holds Ψ γ2 Ψ γ1. It follows from [9, Lemma 1.32, page 13], that Ψ 1 and M are equivalent Young functions. The same holds for Φ 1 and M. We define the spaces X := {v : R N ; v =, Dv L M }, Y := {v :, T R N ; vt = for a.a. t, T, Dv L M, T } and their norms v X := Dv M,, v Y := Dv M,,T. We consider a system of equations composed of the continuity equation and the momentum equation t ρ + divρu = in 1.1 t ρu + divρu u + ρ div SDu = ρf in, 1.2 where is a bounded domain in R N. The system is completed by the boundary condition ut, x =, t, T, T >, x 1.3 and the initial conditions ρ, x = ρ x, x, 1.4 ρu, x = q, x. 1.5 In addition, we assume that the stress tensor S satisfies these conditions: 1. S is coercive, i.e. for any function v X, SDv : Dv dx M Dv dx S is monotone, i.e. SDv SDw : Dv Dwϕ dx 1.7 for any v, w X and ϕ C, ϕ, 2

4 3. S is bounded in the following sense: M SDv dx c 1 + M Dv dx 1.8 for any functions v X and let Sv εw M Sv for ε and any function v Y such that Dv L M, T and any w C, T, 4. if {Du n } n=1 L M, T is a sequence such that Du n M Du in L M, T and then lim inf n t T SDu n : Du n dxdt c for all n N, SDu n : Du n dx dt t SDu : Du dx dt 1.9 for all t [, T ] and any. Note that such a tensor really exist. As an example we can take { M Du Du for Du, SDu = Du 2 for Du =. For the proof that it satisfies the above mentioned conditions see [9, page 43]. Definition 1.1. Couple ρ, u is called the variational solution of system if the density ρ is a nonnegative function, continuity equation 1.1 is satisfied in the sense of distribution in R N and in the sense of renormalized solution, i.e. t bρ + divbρu + b ρρ bρ div u = in D, T R N 1.1 for any b C 1 [, such that b and b are bounded provided ρ and u were extended to be zero outside, u satisfies 1.3 in the sense of traces and equation 1.2 holds in space D, T, 3

5 the energy inequality τ Eτ + Su : Du dx dt E + τ holds for a.a. t [, T ], where Et = 1 ρt ut 2 + ρt ln ρt dx, 2 E = 1 q 2 + ρ ln ρ dx, 2 ρ the initial conditions are satisfied in the sense ρtη dx = ρ η dx, lim t + lim t + ρtut η dx = q η dx, ρf u dx dt 1.11 η D, η D, Definition 1.2. Assume that Φ is a Young function. Let { n } n=1 be a sequence of open sets in R N. We say that n converges to an open set R N with respect to Φ, denoted by Φ n, if for any compact set K there exists m N such that the sets n \ are bounded and K n n m, 1.12 cap Φ n \ pro n Note that { } cap Φ K = inf Φ v dx; v DR N, v 1 in K R N for each compact K R N. See [7, page 134]. The main result states: Theorem 1.3. Let { n } n=1 be a sequence of open sets in R N Ψ such that 2 n, where is a nonempty open set. Assume that tensor S satisfies conditions Let ρ n, u n be a variational solution of the problem in, T n with the driving force f n E Ψ βr, T n, r 2, β > 2, and initial data ρ n 4

6 L Φβ n, and q n 2 L 1 n such that qn 2 ρ n for ρ >. Let T f n dx dt be uniformly bounded, n Ψ β r := if ρ n = and qn 2 ρ n L 1 n f n f in L Ψ βr, T ; L Ψ βr, 1.14 ρ n q n ρ in L Φβ R N, 1.15 q in L Φ β2 R N 1.16 and q n 2 q 2 in L 1 R N ρ n ρ Then passing to subsequences if necessary we have ρ n ρ in C[, T ]; L weak Φ β R N, and u n u n in L M, T ; W 1 L Ψ2 R N, Du n Du in L M, T ; L M R N ρ n u n ρu in C[, T ]; L weak Φ β2 R N, where ρ, u is a variational solution of the problem in, T driven by the force f with initial data ρ and q. 2 Auxiliary assertions Let us denote R N an open ball such that n for all n m. Ψ Lemma 2.1. Let 2 n and Φ w 2 n w in W 1 L Ψ2, where w n W 1 L Ψ2 n. Then w W 1 L Ψ2. Proof: According to 1.13 there exist functions ϕ n D such that ϕ n 1, ϕ 1 in V n n \, ϕ n in W 1 L Ψ2, where V n n \ is an open neighbourhood of n \. Put where T k are the cut-off functions, v n = 1 ϕ n T k w n, T k z = kt z k, k 1, 2.1 5

7 with T C R such that T z = T z for every z R, T being concave in, and { z for z 1, T z = 2 for z 3. Functions T k w n are obviously bounded in W 1 L Ψ2 thus passing to subsequence if necessary T k w n Φ 2 T k w in W 1 L Ψ2. Furthermore, it is known from [2, page 358] that W 1 L Ψ2 L p for arbitrary p N. Hence T k w n T k w in L p. In the same way boundedness of w n in W 1 L Ψ2 implies w n w in L p and thus it follows from the uniqueness of the limit that T k w = T k w and Φ we do not have to pass to the subsequence. Now we show that v 2 n Tk w in W 1 L Ψ2. For ψ E Φ2 one has 1 ϕ n T k w n T k wψ dx ϕ n T k w n ψ dx + T k w n T k wψ dx, where and ϕ n T k w n ψ dx 2k ϕ n Ψ2 ψ Φ2 T k w n T k wψ dx for n for n by virtue of the E Φ2 -weak convergence of T k w n and 1 ϕ n T k w n T k wψ dx ϕ n T k w n ψ dx + 1 ϕ n T k w n T k wψ dx ϕ n T k w n ψ dx + ϕ n T k w n ψ dx + + T k w n T k wψ dx, 6

8 where ϕ n T k w n ψ dx 2k ϕ n Ψ2 ψ Φ2 for n and ϕ n T k w n ψ dx ϕ n T k w n T k wψ dx + ϕ n T kw wψ dx ϕ n T k w n T k wψ dx + ϕ n w Ψ2 ψ Φ2 for n as a consequence of the E Φ2 -weak convergence of T k w n and the strong convergence of ϕ to zero in L, and T k w n T k wψ dx for n, which follows from the E Φ2 -weak convergence of T k w n. Consequently T k w W 1 L Ψ2 because obviously v n W 1 L Ψ2 for every n N. ut this also means that w W 1 L Ψ2. In the following text we will often use these two lemmas: Lemma 2.2 Korn s inequality. Let u W 1,p for all p > 1. Then u 1,p cp2 p 1 Du p. Lemma 2.3. Let u L Ψ2, v L Ψ1 and u p cp v p, p 2, where the constant c does not depend on p. Then u Ψ2 c v M. For the proofs see [6] Lemma 2.2 and [9, page 17] Lemma 2.3. Ψ Remark 2.4. If 2 n and Dw n M Dw in L M, where Dw n L M n, then it follows from Korn s inequality that passing to subsequence if necessary Φ w 2 n w in W 1 L Ψ2 and according to the previous lemma w W 1 L Ψ2. Thus Dw L M. 7

9 Lemma 2.5. If Du L M, T and u = at, then u L q, T ; L for q [1,. Proof: First we show that Du L q, T ; L p for arbitrary p [1,. Indeed, p Du p dx c M 1 q Du dx + 1 c M 1 q Du dx + 1, because z p e z q T + c and a a p + 1, and thus we can write q Du p p T q dx dt c M 1 q Du dx dt + 1 T c q 1 M Du dx dt + 1 where we have used Jensen s inequality. From the fact that Du L q, T ; L p it follows from Korn s inequality that u L q, T ; W 1,p for arbitrary p [1,. Since W 1,p L for p > N, we get u L q, T ; L. Lemma 2.6. Let Du L M, T and ρ L, T ; L Φβ, β > 2, be a solution of 1.1 in the sense of distributions. Then ρ dx = ρ dx for a.e. t, T. Proof: Let us recall the fact that if we extend the functions ρ and u to be zero outside, then the equation 1.1 is satisfied in the space D, T R N see [9, page 45, Lemma 4.1]. In a similar way like in [1, page 5] we take a sequence ϕ j DR N, ϕ j, sup x R N ϕ j x < 1/j, ϕ j 1 for j and the test functions of the form ϕ j xψt with ψ D, T to deduce τ ρτϕ j dx = ρϕ j dx + ρu ϕ j dx dt R N R N R N for a.a. τ, T. Since ρ L, T ; L Φβ R N and u L 2, T ; L R N, then ρu L 2, T ; L 1 R N. Indeed, T 2 T ρu dt ρt 1 ut 2 dt R N ρ 2 L,T ;L 1 R N T ut 2 dt = ρ 2 L,T ;L 1 R N u 2 L 2,T ;L R N. 8

10 Recall that ρt L 1 R N because it is extended to be zero outside, which is a bounded domain. Thus for j we infer τ ρτ dx = ρτ dx = lim ρϕ j dx + ρu ϕ j dx dt = R N j R N R N = lim ρϕ j dx = ρ dx = ρ dx. j R N R N R N Recall [9, Lemma 1.47, page 22]: Lemma 2.7. Let u L M, T. Then u L M, T ; L M. Let further v L Φ1, T. Then u L Φ 1α, T ; L Φ 1β for α, β 1, such that = 1. α β We proceed in a similar way as in the proof of Lemma 2.7 to prove following lemma: Lemma 2.8. Let u L Ψγ, T, γ 1. Then u L Ψγ, T ; L Ψγ. Proof: For functions ϕ L Φγ, Φ γ ϕ dx 1, and ψ L Φγ, T, T Φ γ ψ dt 1 we have T T sup ψ sup uϕ dx dt Ψ γ u dx dt+ ψ ϕ T + sup sup ϕψ ln γ 1 + ϕψ dx dt + c ψ cu sup cu cu + sup ψ ψ sup ψ sup ϕ T T T ψ T = cu sup + sup ψ sup ϕ sup ϕ sup ϕ sup ϕ T ψ T ϕψ ln 1 γ + ϕ 1 + ψ dx dt + 1 γ ϕψ ln1 + ϕ + ln1 + ψ dx dt + 1 ϕψ ln γ 1 + ϕ dx dt+ ϕψ ln γ 1 + ψ dx dt + 1 = ψ dt sup ϕ ψ ln γ 1 + ψ dt sup ϕ ϕ ln γ 1 + ϕ dx+ ϕ dx + 1 <, where we have used Young inequality and Jensen s inequality. 9

11 3 Apriori estimates Let us denote θ k z := T k Φ β z, where T k z are the cut-off functions defined by 2.1. Consider the equation 1.1 and put bρ n = θ k ρ n in the renormalized continuity equation to infer d θ k ρ n t dx θ k ρ n t θ dt kρ n t div u n t dx =. n n Now letting k we obtain d Φ β ρ n t dx Φ β ρ n t ρ n tφ dt βρ n t div u n t dx =. n n In the next step we use the fact that εm z c and εm2cz are for arbitrary ε and c complementary Young functions, M is equivalent with Φ 1 and Φ 1 satisfies the 2 -condition. Employing the inequality Φ 1 zφ γz Φ γ z Φ γ z + c, z, γ > 1, we have d Φ β ρ n t dx = Φ β ρ n t ρ n tφ dt βρ n t div u n t dx n n div cε 1 Φ 1 Φ β ρ n t ρ n tφ un t βρ n t dx + ε 1 M dx n n c div un t cε 1 Φ β ρ n t dx ε 1 M dx n n c cε 1 Φ β ρ n t dx ε 1 M Du n t dx, n n where the constant c is taken from the inequality div u n c Du n. We add the obtained inequality to 1.11 and estimate the remaining term on 1

12 the right-hand side as follows τ τ ρ n f n u n dx dt = r ρn f n r ρn u n dx dt n n 1 r ε 1 τ 1 r 2 ρ n f n r dx dt + 1 τ n r ε 2 ρ n u n r dx dt n 1 r ε 1 τ 1 r 2 Φ β ρ n dx dt + 1 n r ε 1 τ 1 r 2 Ψ β f n r dx dt + n + 1 τ r ε 2 ρ n dx u n r L n dt 3.1 n τ Φ β ρ n dx dt + 1 n r ε 1 τ 1 r 2 Ψ β f n dx dt + r n + 1 τ r ε 2 ρ n L 1 ncn n r 1 M Du n dx dt + 1 n τ τ cε 2 Φ β ρ n dx dt + ε 2 cr, N M Du n dx dt + cε 2, r, N, n n 1 r ε 1 1 r 2 Using 1.6 we arrive to 1 ρn τ u n τ 2 + ρ n τ ln ρ n τ dx + Φ β ρ n τ dx+ 2 n n τ + 1 ε 1 ε 2 cr, N M Du n dx dt n τ cε 1, ε 2 Φ β ρ n dx dt + cε 1, ε 2, r, N. n From the integral Gronwall inequality we have Altogether 1 2 Consequently n Φ β ρ n τ dx cε 1, ε 1, r, N, T. ρn τ u n τ 2 + ρ n τ ln ρ n τ dx + Φ β ρ n τ dx+ n n τ + 1 ε 1 ε 2 cr, N M Du n dx dt n cε 1, ε 1, r, N, T. n ρ n t u n t 2 dx cε 1, ε 1, r, N, T, for a.a. t [, T ],

13 ρ n t ln ρ n t dx cε 1, ε 1, r, N, T, for a.a. t [, T ], 3.3 n Φ β ρ n t dx cε 1, ε 1, r, N, T, for a.a. t [, T ], 3.4 n T MDu n dx dt cε 1, ε 1, r, N, T. 3.5 n 4 Limit passages 4.1 Continuity equation Consider a sequence {ρ n, u n } n=1 of variational solutions of the problem on corresponding sets n. We extend this functions to be zero in R N \ n and take arbitrary open ball R N such that n for all n m. We can see from 3.4 that passing to subsequence if necessary ρ n ρ in L, T ; L Φβ. 4.1 In the same way we get from 3.5 and Lemma 2.7 that passing to a subsequence if necessary Du n Du in L M, T ; L M. 4.2 Moreover, we know from Korn s inequality that u n u in L M, T ; W 1 L Ψ2, 4.3 where u L M, T ; W 1 L Ψ2 from Lemma 2.1. Recall that u n are extended to be zero outside n, thus u n L 2,T ;L = u n L 2,T ;L n c Du n LM,T ;L M n < cε 1, ε 1, r, N, T, which follows from 3.5 and the proof of Lemma 2.8. From continuity equation 1.1 we obtain for ϕ W 1 L Ψβ and ψ L 2, T the estimate T T ψt t ρ n, ϕ dt = ψt ρ n u n ϕx dx dt T ψt ρ n t Φβ u n t ϕ Ψβ dt ψ 2 ρ n L,T ;L Φβ u n L 2,T ;L ϕ Ψβ, where we have used the conclusion of Lemma 2.5. Hence t ρ n are uniformly bounded in L 2, T ; W 1 L Φβ. Since ρ n are uniformly bouded in the space L, T ; L Φβ and W 1 L Ψβ W 1,p C E Ψβ, p > N, 12

14 i.e. L Φβ W 1 L Φβ, one has from [8, page 85] that passing to a subsequence if necessary ρ n ρ in C[, T ]; W 1 L Φβ and i.e. ρ n ρ in C[, T ]; L weak Φ β, 4.4 ρ n ρψ dx C[,T ] for n for every ψ E Ψβ. Since ρ n are extended to be zero outside n the same holds in C[, T ]; L weak Φ β R N. Now we are going to show the weak- convergence of functions ρ n u n to ρu. At first we deduce for ϕ L Ψ β2 the estimate ρ n tu n t ϕ dx = ρn tu n t ρ n tϕ dx ρ n t u n t 2 dx + ρ n t ϕ 2 dx c + Φ β ρ n t dx + Ψ β ϕ 2 dx, for a.a. t [, T ], where we have used estimate 3.2. Therefore ρ n u n ρu in L, T ; L Φ β2. It remains to show that ρu = ρu. Take arbitrary open ball 1 1 this implies 1 n for every n m. For ϕ W 1 L Ψβ 1 and ψ L Φ1, T it holds T ψt T ψt ρ n u n ρu ϕx dx dt 1 T ρ n ρu n ϕx dx dt + ψt ρu n u ϕx dx dt. 1 1 For the first integral we have T ψt ρ n ρu n ϕx dx dt 1 T ψt ρ n t ρt W 1 L Φβ 1 u n t ϕ W 1 L Ψβ 1 dt, 13

15 where u n t ϕ W 1 L Ψβ 1 = u n t ϕ Ψβ, 1 + u n t ϕ Ψβ, 1 u n t,n ϕ Ψβ, 1 + u n tϕ Ψβ, 1 + u n t ϕ Ψβ, 1 Du n t M,n ϕ Ψβ, 1 + u n t Ψβ, n ϕ,1 + u n t,n ϕ Ψβ, 1 Du n t M,n ϕ Ψβ, 1 + ϕ,1 + ϕ Ψβ, 1 c Du n t M,n ϕ W 1 L Ψβ 1, where we have used the assumption that β > 2 and Korn s inequality, thus T ψt ρ n ρu n ϕx dx dt 1 c ψt Φ1 ρ n ρ C[,T ];W 1 L Φβ 1 Du n LM,T ;L M n ϕ W 1 L Ψβ 1. In the case of the second integral we use weak- convergence 4.3. It only remains to check that ρtϕ E Φ2 1 = L Φ2 1 Φ 2 satisfies the 2 -condition. ut this is easy because for σ L Ψ2 1 it holds ρtϕσ dx c ρ L,T ;L Φβ 1 ϕ σ Ψ2. Altogether ρ n u n ρu in L M, T ; W 1 L Φβ 1 for arbitrary 1 1 and consequently from the uniquenes of the limit, the definition of open sets and the prolongation of ρ n, u n, ρ and u we have ρ n u n ρu in L, T ; L Φ β Since the ball was arbitrary, we have deduced that couple ρ, u satisfies the equation t ρ + divρu = in D, T R N 4.6 and also in the sense of renormalized solution see [9, Lemma 4.2, page 46]. 4.2 Momentum equation In the foregoing part we assumed that the open ball R N satisfies n for n m. ut here we will consider the case. Now we are going to show the uniform boundedness of ρ n u n u n in the space L q, T ; L Φβ, q [1,. Let ϕ L Ψβ and ψ L q, T, = 1. q q Then according to Lemma 2.5 and 3.5 for u = u n and = n it follows T T ψt ρ n u n 2 ϕx dx dt ψt ρ n t Φβ u n t 2 ϕ Ψβ dt ψ q ρ n L,T ;L Φβ u n 2 L 2q,T ;L ϕ Ψ β 4.7 ψ q ρ n L,T ;L Φβ u n 2 L 2q,T ;L n ϕ Ψ β. 14

16 In the next step we obtain from the momentum equation for ϕ W 1 L Ψ 12 and ψ L Ψ 12, T the estimate T T ψt t ρ n u n, ϕx dt ψt ρ n u n u n : Dϕx dx dt + }{{} T + ψt ρ n div ϕx dx dt + }{{} + I 2 T ψt ρ n f n ϕx dx dt. }{{} I 4 I 1 T ψt SDu n : Dϕx dx dt + }{{} I 3 We now estimate integrals I i, i = 1, 2, 3, 4, one by one: T I 1 = ψt ρ n u n u n : Dϕx dx dt T ψt ρ n tu n t u n t Φβ Dϕ Ψβ dt ψ q ρ n u n u n L q,t ;L Φβ Dϕ Ψβ because we have already proved that ρ n u n u n are uniformly bounded in the space L q, T ; L Φβ, T T I 2 = ψt ρ n div ϕx dx dt ψt ρ n t Φβ div ϕ Ψβ dt ψ 1 ρ n L,T ;L Φβ div ϕ Ψβ, where we have used the fact that β > 2, and therefore L Ψ 12 L Ψβ, I 3 = T T ψt SDu n : Dϕx dx dt ψt SDu n t Φ 12 Dϕ Ψ 12 dt ψ Ψ 12 SDu n LΦ 12,T ;L Φ 12 Dϕ Ψ 12 T ψ Ψ 12 Dϕ Ψ 12 c 1 M SDu n dx dt + 1 T ψ Ψ 12 Dϕ Ψ 12 c 2 M Du n dx dt + 1 T ψ Ψ 12 Dϕ Ψ 12 c 2 M Du n dx dt + 1 n 15

17 from [9, Lemma 1.47, page 22] and the third assumption on the stress tensor S, T T I 4 = ψt ρ n f n ϕx dx dt ψt ρ n t Φβ f n t Ψβ ϕ dt c ψ Φ βr ρ n L,T ;L Φβ f n LΨ βr,t ;L Ψ βr ϕ in view of Lemma 2.8 and the fact that L Ψ βr L Ψβ, which gives the result that t ρ n u n are uniformly bouded in L Φ 12, T ; W 1 L Φ 12. Since ρ n u n are moreover uniformly bounded in L, T ; L Φ β2 and i.e. W 1 L Ψ 12 W 1 L Ψβ W 1,p C E Ψ 12, we can write see [8, page 85] and L Φ 12 W 1 L Φβ W 1 L Φ 12, ρ n u n ρu in C[, T ]; W 1 L Φβ 4.8 ρ n u n ρu in C[, T ]; L weak Φ β Now we are going to show the convergence of ρ n u n u n. For ϕ W 1 L Ψβ and ψ L Φ1, T we have T ψt ρ n u n u n ρu u : ϕx dx dt T ψt ρ n u n ρu u n : ϕx dx dt + T + ψt ρu u n u : ϕx dx dt. For the first integral whereas T T ψt ρ n u n ρu u n : ϕx dx dt ψt ρ n tu n t ρtut W 1 L Φβ u n tϕ W 1 L Ψβ dt, u n tϕ W 1 L Ψβ = u n tϕ Ψβ, + u n tϕ Ψβ, u n t,n ϕ Ψβ, + u n tϕ Ψβ, + u n t div ϕ Ψβ, Du n t M,n ϕ Ψβ, + u n t Ψβ, n ϕ, + u n t,n div ϕ Ψβ, c Du n t M,n ϕ W 1 L Ψβ, 16

18 where we have used Korn s inequality and the fact β > 2, thus T ψt ρ n u n ρu u n : ϕx dx dt c ψ Φ1 ρ n u n ρu C[,T ];W 1 L Φβ Du LM,T ;L M m ϕ W 1 L Ψβ, which converges to zero according to 4.8. For the convergence of the second integral we use weak- convergence 4.3. It is possible because T 2 T ρ n u n ϕx dx dt ρ n t 2 Φ β u n t 2 ϕ 2 Ψ β dt ρ n t 2 L,T ;L Φβ u n t 2 L 2,T ;L ϕ 2 Ψ β, and thus ρ n u n are bounded in L 2, T ; L Φβ. Altogether ρ n u n u n ρu u in L M, T ; W 1 L Φβ, i.e. from uniqueness of the limit function ρ n u n u n ρu u in L q, T ; L Φβ for q [1,. In the case of ρ n f n we have for ψ E Φβ, T T ψt ρ n f n ρf dx 4.1 T T ψt ρ n ρf n dx + ψt ρf n f dx and we can apply weak- convergence 4.1 because it can be easily checked that ψtf n L 1, T ; E Ψβ and weak- convergence 1.14 because ψtρ L Φβ, T ; L Φβ E Φ βr, T ; E Φ βr, really for σ 1 L Ψβ and σ 2 L Ψβ, T we have T T ψtσ 2 t ρσ 1 x dx dt ψtσ 2 t ρt Φβ σ 1 Ψβ dt c ρ L,T ;L Φβ σ 1 Ψβ ψ Φβ σ 2 Ψβ Next obviously SDu n M SDu in L M, T from 3.4 and the properties of S. From definition of the open set it follows that ρ, u satisfy the equation t ρu + divρu u + ρ div SDu = ρf in D, T

19 Similarly to [9, page 62] we use Steklovov s functions to prove that couple ρ, u satisfies in the sense of distributions the identity d 1 dt 2 ρ u 2 dx + SDu : Du dx ρ div u dx = ρu f dx Nevertheless, for ρ n, u n we have d 1 dt n 2 ρ n u n 2 dx+ SDu n : Du n dx ρ n div u n dx = ρ n u n f n dx. n n n 4.13 After subtraction of the identities we obtain τ ϕ h t SDu n : Du n dx SDu : Du dx dt = n τ = ϕ 1 ht n 2 ρ n u n 2 1 dx 2 ρ u 2 dx dt + τ + ϕ h t ρ n div u n dx ρ div u dx dt + n τ + ϕ h t ρ n f n u n dx ρf u dx dt, n where ϕ h C, T, ϕ h 1, ϕ h 1 for h a.e. in [, T ]. Let Q, Q, be a suitable set generated by a finite unification of balls. We treat the above integrals one by one τ ϕ ht 2 ρ n u n 2 1 dx 2 ρ u 2 dx dt = where = τ τ + τ n 1 ϕ ht ϕ ht n\q Q ρ n u n 2 dx \Q ρn u n 2 ρ u 2 dx dt, 1 2 ρ u 2 dx dt + ϕ 1 ht ρn u n 2 ρ u 2 dx dt for n Q 2 from the weak- convergence of ρ n u n 2 in L q, T ; L Φβ Q, see 4.7, and τ ϕ 1 ht n\q 2 ρ n u n 2 1 dx \Q 2 ρ u 2 dx dt ch χ n\q Ψβ + χ \Q Ψβ < ε, 18

20 as a consequence of see [7, page 136], the boundedness ρ n u n 2 in L q, T ; L Φβ Q and the suitable choice of Q depending on h. We argue similarly for τ ϕ h t ρ n f n u n dx ρf u dx dt. n The convergence of ρ n f n u n can be shown in the same way as convergence 4.1 as a consequence of 1.14 and 4.5. For the remaining term we use the same method as at the beginning of Section 3 to derive from the continuity equations the identities and n ρ n τ lnρ n t + δ ρ n lnρ n + δ dx = ρτ lnρt + δ ρ lnρ + δ dx = τ τ n ρ 2 n δ + ρ n div u n dx dt, ρ 2 div u dx dt, δ + ρ where δ, 1. Now we can take a ball, n. From zero prolongation of ρ and ρ n and the convexity of functional ρ lnρ + δ dx we have ρτ lnρτ + δ dx ρ n τ lnρ n τ + δ dx = n = ρτ lnρτ + δ ρ n τ lnρ n τ + δ dx lnρ + δ + ρ ρ ρ n dx for n ρ + δ as a consequence of 4.4 and similarly ρ n lnρ n + δ dx ρ lnρ + δ dx for n n as a consequence of It follows from the foregoing estimates τ ϕ h t ρ n div u n dx ρ div u dx dt = n τ τ = ϕ h t 1 ρ n div u n dx dt ϕ h t 1 ρ div u dx dt + n τ δρ τ n δρ + div u n dx dt div u dx dt + n δ + ρ n δ + ρ τ ρ 2 τ n ρ 2 + div u n dx dt div u dx dt n δ + ρ n δ + ρ ϕ h 1 M ρ n L,T ;L Φβ n Du n LM,T ;L M n + + ϕ h 1 M ρ L,T ;L Φβ Du LM,T ;L M + + δ Du n LM,T ;L M n + δ Du LM,T ;L M + cn Kc 1 h + c 2 n + δ 19

21 where c 1 h for h, c 2 n for n and δ can be arbitrarily small. Altogether τ τ lim inf ϕ h t SDu n : Du n dx dt ϕ h t SDu : Du dx dt n n c 1 h + c 2 h 4.14 Now we take a sequence of open sets Q k defined similarly as Q such that Q k, \ Q k, and functions ψ k C Q k, ψ k 1, ψ k 1 a.e. in. From the monotonicity of the stress tensor S we infer τ ϕ h t SDu n SDv : Du n Dvψ k dx dt, n i.e. τ ϕ h t τ SDu n : Du n dx dt n ϕ h t n τ ϕ h t SDu n : Du n ψ k dx dt n ψ k dx dt, SDu n : Dv + SDv : Du n SDv : Dv and thus letting n and using 4.14 τ ϕ h t SDu : Du dx dt τ ϕ h t SDu : Dv + SDv : Du SDv : Dv ψ k dx dt. In view of the Lebesgue theorem for k τ ϕ h t SDu SDv : Du Dv dx dt cδ, h. Since cδ, h is arbitrarily small we get using the Lebesgue theorem τ SDu SDv : Du Dv dx dt. Then put v = u λψ, where ψ C, T and λ R. As a consequence to the monotonicity and the third property of the stress tensor S we infer div SDu = div SDu in L Φ 1α, T ; W 1 L Φ 12, α > 2. Moreover, SDu L M, T in view of 1.6, 1.8 and

22 4.3 Energy inequality In the same way as in [9] we deduce d 1 dt 2 ρ u 2 + ρ ln ρ dx + SDu : Du dx = ρf u dx. We know, that the energy inequality 1.11 holds for ρ n, u n, i.e. 1 ρn τ u n τ 2 + ρ n τ ln ρ n τ τ dx + Su n : Du n dx dt 2 n n 1 q n 2 τ + ρ n ln ρ n dx + ρ n f 2 n u n dx dt. n n ρ n If we multiply this inequality by ψ D, T, pass to the limit for n and use convergences proved in forgoing sections and the fourth condition on stress tensor S, we deduce 1 T ψτ ρ u 2 + ρ ln ρ T dx dτ + ψτ q 2 T + ρ ln ρ dx ψτ dτ + ρ τ Su : Du dx dt dτ τ ψτ ρf u dx dt dτ. T Since ψ was arbitrary, we deduce the energy inequality 1.11 for a.a. τ, T and Theorem 1.3 is proved because the convergence of boundary conditions is obvious. 5 Application Assume that the stress tensor S satisfies another condition: 5. S satisfies the estimate T SDv 1 SDv 2 dxdt ct, κ for v i M κ, i = 1, 2, where T Dv 1 t Dv 2 t dt M κ := { v C[, T ]; W 1,2 L, T ; W 1, ; vt + vt κ for a.a. t [, T ]}. Then we have following existence result proved in [9, page 73]: 21

23 Theorem 5.1. Assume that C 2+µ and tensor S satisfies conditions Let f L Ψ βr, T, β > 2, r >. For given initial data ρ L Φβ and q 2 L 1 such that q 2 ρ := if ρ = and q 2 ρ L 1 for ρ > there exist functions ρ L, T ; L Φβ, u Y, such that couple ρ, u is the variational solution of In addition d 1 dt 2 ρ u 2 + ρ ln ρ dx + SDu : Du dx = ρf u dx in D, T. From Theorem 1.3 it follows: Corollary 5.2. For r 2 the conclusion of Theorem 5.1 keeps valid even if C,1. References [1] Feireisl, E., Novotný, A., Petzeltová, H., On the domain dependence of solutions to the compressible Navier-Stokes equations of a barotropic fluid, Math. Methods Appl. Sci. 25, No.12, [2] Kufner, A., John, O., Fučík, S., Function Spaces, Prague: Publishing House of the Czechoslovak Academy of Sciences. XV, 454 p. Dfl [3] Lukeš, J., Zápisky z funkcionální analýzy, Prague: Karolinum Press 23 [4] Mamontov, A. E., Global solvability of the multidimensional Navier-Stokes equations of a compressible fluid with nonlinear viscosity I., Sib. Math. J. 4, No.2, [5] Mamontov, A. E., Global solvability of the multidimensional Navier-Stokes equations of a compressible fluid with nonlinear viscosity II., Sib. Math. J. 4, No.3, [6] Mamontov, A. E., Orlicz spaces in the existence problem of global solutions of viscous compressible nonlinear fluid equations, Sb. Math. 19, No.8, [7] Rudd, M., A direct approach to Orlicz Sobolev capacity, Nonlinear Anal., Theory Methods Appl. 6, No. 1 A, [8] Simon, J., Compact Sets in the Space L p, T ;, Ann. Mat. Pura Appl., IV. Ser. 146,

24 [9] Vodák, R., Existence řešení Navierových-Stokesových rovnic pro proudění izotermálních stlačitelných tekutin a jeho kvalitativní vlastnosti, Ph.D. Thesis, Olomouc 23 23