INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES

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1 INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES Weak solutions to the full Navier-Stokes-Fourier system with slip boundary conditions in time dependent domains Ondřej Kreml Václav Mácha Šárka Nečasová Aneta Wróblewska-Kamińska Preprint No PRAHA 215

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3 Weak solutions to the full Navier-Stokes-Fourier system with slip boundary conditions in time dependent domains Ondřej Kreml 1 Václav Mácha 1 Šárka Nečasová 1 Aneta Wróblewska-Kamińska 2 1 Institute of Mathematics of the Academy of Sciences of the Czech Republic Žitná 25, Praha 1, Czech Republic 2 Institute of Mathematics, Polish Academy of Sciences Śniadeckich 8, -656 Warszawa, Poland Abstract We consider the compressible Navier-Stokes-Fourier system on time-dependent domains with prescribed motion of the boundary, supplemented with slip boundary conditions for the velocity. Assuming that the pressure can be decomposed into an elastic part and a thermal part, we prove global-in-time existence of weak solutions. Our approach is based on the penalization of the boundary behavior, viscosity, and the pressure in the weak formulation. Moreover, the thermal energy equation is in the weak formulation replaced by the thermal energy inequality complemented with the global total energy inequality. In the approximation scheme the thermal energy inequality is consider to be satisfied in the renormalized sense. Keywords: compressible Navier-Stokes-Fourier equations, time-varying domain, slip boundary conditions 1 Introduction The flow of a compressible viscous heat conducting fluid is in the absence of external forces described by the following system of partial differential equations t ϱ + div x ϱu =, 1.1 t ϱu + div x ϱu u + x pϱ, ϑ = div x S x u, 1.2 t ϱe + div x ϱe + pu + div x q = div x S x uu. 1.3 These equations are mathematical formulations of the balance of mass, linear momentum and total energy respectively. Here ϱ is the density of the fluid, u denotes the velocity and E = 1 2 u 2 + eϱ, ϑ the specific total energy given as sum of the kinetic energy and the internal energy e being a function of density and temperature ϑ. The work of O.K., V.M. and Š.N. was supported by Grant of GA ČR GA13-522S and by RVO The work of A.W.-K. was supported by Grant of National Science Center Sonata, No 213/9/D/ST1/

4 The stress tensor S is determined by the standard Newton rheological law S x u = µ x u + t xu 2 3 div xui + ηdiv x ui, µ >, η. 1.4 For simplicity, throughout the rest of this paper we assume the viscosity coefficients µ and η to be constant, however the case of µ, η being dependent on temperature in a suitable way can be also treated, see Section 5. The Fourier law for the heat flux q has the following form: Motivated by [4] we assume the following state equation for the pressure q = κϑ x ϑ, κ >. 1.5 pϱ, ϑ = p e ϱ + ϑp ϑ ϱ, 1.6 with the additional assumption that p ϑ is a non-decreasing function of the density vanishing for ϱ =. Consequently the Maxwell relation yields the form of the specific internal energy as with the elastic potential eϱ, ϑ = P e ϱ + Qϑ 1.7 P e ϱ = and the thermal energy being related to the specific heat at constant volume c v by Qϑ = ϑ ϱ 1 p e z z 2 dz 1.8 c v zdz, c v z c v > for all z, 1.9 Assuming smoothness of the flow the system can be rewritten using the equation for the thermal energy instead of the balance of total energy t ϱ + div x ϱu =, 1.1 t ϱu + div x ϱu u + x pϱ, ϑ = div x S x u, 1.11 t ϱqϑ + div x ϱqϑu + div x q = S x u : x u ϑp ϑ ϱdiv x u We study the system of equations on a moving domain Ω = with the prescribed movement of the boundary and on time interval [, T ] with T <. More precisely, the boundary of the domain occupied by the fluid is described by means of a given velocity field Vt, x, where t and x R 3. Assuming V is regular, we solve the associated system of differential equations and set d dt Xt, x = V t, Xt, x, t >, X, x = x, 1.13 Ω τ = X τ, Ω, where Ω R 3 is a given domain, Γ τ = Ω τ, and Q τ = {t, x t, τ, x Ω τ }. The impermeability of the boundary of the physical domain is described by the condition u V n Γτ = for any τ,

5 where nt, x denotes the unit outer normal vector to the boundary Γ t. Moreover, we assume the Navier type boundary conditions in the form [Sn] tan + ζ [u V] tan Γτ =, ζ, 1.15 where ζ represents a friction coefficient. If ζ =, we obtain the complete slip while the asymptotic limit ζ gives rise to the standard no-slip boundary conditions. Concerning the heat flux we consider the conservative boundary conditions q n = for all t [, T ], x Γ t For physical motivation of correct description of the fluid boundary behavior, see ulíček, Málek and Rajagopal [1], Priezjev and Troian [16] and the references therein. Finally, the problem is supplemented by the initial conditions ϱ, = ϱ, ϱu, = ϱu, ϑ, = ϑ, ϱqϑ, = ϱq = ϱ Qϑ in Ω The main goal of this paper is to show the existence of global-in-time weak solutions to problem for any finite energy initial data. The existence theory for the barotropic Navier-Stokes system on fixed spatial domains in the framework of weak solutions was developed in the seminal work by Lions [12], and later extended in [9] to a class of physically relevant pressure-density state equations. Then it was extended by Feireisl to the full Navier-Stokes-Fourier system [4, 5]. The investigation of incompressible fluids in time dependent domains started with a seminal paper of Ladyzhenskaya [11], see also [13, 14, 15] for more recent results in this direction. Compressible fluid flows in time dependent domains in barotropic case were examined in [6] for the no-slip boundary conditions and in [7] for the slip boundary conditions. The aim of this paper is to extend this result to the full system. We proceed in the following way. 1. In order to deal with the slip boundary condition, we introduce to the weak formulation of the momentum equation a term 1 T u V n ϕ n ds x dt, ε > small, 1.18 ε Γ t which was originally proposed by Stokes and Carey in [17]. This extra term allows to consider the system in time independent domain which is divided by impermeable boundary Γ t. In order to handle the behaviour of fluid in the solid domain Q c T we use the following three in fact four level penalization scheme 2. In addition to 1.18, we introduce a variable shear viscosity coefficient µ = µ ω, where µ ω remains strictly positive in the fluid domain Q T but vanishes in the solid domain Q c T as ω. 3. We introduce the heat conductivity coefficient κ ν t, x, ϑ which remains strictly positive in the fluid domain Q T but vanishes in the solid domain Q c T as ν. For technical reasons this procedure needs to be done in two steps, more precisely we consider a step function κ ν = κ in the fluid part and κ ν = νκ in the solid part. This function is then mollified to get smooth κ ν,ξ, ξ > with κ ν,ξ κ ν as ξ. 4. Similarly to the existence theory developed in [9], we introduce the artificial pressure in the momentum equation p ϱ, ϑ = pϱ, ϑ + ϱ β, β 4, >, 3

6 5. Keeping ε,, ν, ξ and ω > fixed, we solve the modified problem in a bounded reference domain R 3 chosen in such a way that Ω τ for any τ [, T ]. To this end, we adapt the existence theory for the compressible Navier-Stokes system with variable viscosity coefficients developed in [5]. 6. We take the initial density ϱ vanishing outside Ω and letting ε for fixed, ω > we obtain a two-fluid system where the density vanishes in the solid part, T \ Q T of the reference domain together with the thermal pressure p ϑ. It follows that the choice of the pressure is fundamental. 7. Passing with ξ we recover the system with jump in the heat conductivity coefficient and justify the choice of the boundary condition of the test function in the weak formulation of the thermal energy balance. 8. Letting the viscosity vanish in the solid part, we perform the limit ω, where the extra stresses disappear in the limit system. The desired conclusion results from the final limit process, where we set ν = ν and let. The paper is organized as follows. In Section 2, we introduce all necessary preliminary material including a weak formulation of the problem and state the main result. Section 3 is devoted to the penalized problem and to uniform bounds and existence of solutions at the starting level of approximations. In Section 4, the limits for ε, ξ, ω, and are preformed successively. Section 5 discusses possible extensions and applications of the method. 2 Preliminaries 2.1 Hypotheses Hypotheses imposed on constitutive relations and transport coefficients are motivated by the general existence theory for the Navier-Stokes-Fourier system developed in [4, 5]. Hypotheses on the pressure: p e C[, C 1,, p e =, p eϱ a 1 ϱ γ 1 b for all ϱ >, 2.1 for certain constants γ > 3 2, a 1, a 2, b >, p e ϱ a 2 ϱ γ + b for all ϱ, p ϑ C[, C 1,, p ϑ =, p ϑ is a non-decreasing function of ϱ [,, p ϑ ϱ cϱ γ 3 for all ϱ

7 Hypotheses on the heat conductivity coefficient: κ = κϑ belongs to the class C 2 [,, k 1 ϑ α + 1 κϑ k 2 ϑ α + 1 for all ϑ, with constants k 1, k 2 >, α 4 and α 12γ 1 γ. 2.3 Hypothesis on the thermal energy: Q = Qϑ = ϑ c vz dz, where c v C 1 [, is such that there exist < c < c <, c1 + ϑ α 2 1 c v ϑ c1 + ϑ α We would like to emphasise that the lower bound on c v is restrictive compare with [4, Section 3]. However, this assumption is essential since it allows us to derive estimates on the velocity see Weak formulation, main result In the weak formulation, it is convenient to consider the continuity equation 1.1 in the whole physical space R 3 provided the density ϱ is extended to be equal to zero outside the fluid domain, specifically τ ϱϕτ, dx ϱ ϕ, dx = ϱ t ϕ + ϱu x ϕ dxdt 2.5 Ω τ Ω for any τ [, T ] and any test function ϕ Cc [, T ] R 3. Moreover, equation 1.1 is also satisfied in the sense of renormalized solutions introduced by DiPerna and Lions [2]: τ bϱϕτ, dx bϱ ϕ, dx = bϱ t ϕ + bϱu x ϕ + bϱ b ϱϱ div x uϕ dxdt 2.6 Ω τ Ω for any τ [, T ], any ϕ Cc [, T ] R 3, and any b C 1 [,, b =, b r = for large r. Of course, we suppose that ϱ a.e. in, T R 3. Similarly, the momentum equation 1.11 is replaced by a family of integral identities ϱu ϕτ, dx Ω τ ϱu ϕ, dx Ω 2.7 = τ for any τ [, T ] and any test function ϕ C c ϱu t ϕ + ϱ[u u] : x ϕ + pϱ, ϑdiv x ϕ S x u : x ϕ dxdt [, T ] R 3 ; R 3 satisfying ϕ n Γτ = for any τ [, T ]

8 where Then the impermeability condition 1.14 is satisfied in the sense of traces, specifically, u, x u L 2 Q T ; R 3 and u V nτ, Γτ = for a.a. τ [, T ]. 2.9 According to [4, 5] the equation 1.12 is replaced by two inequalities, the thermal energy inequality t ϱqϑ + div x ϱqϑu Kϑ S : x u ϑp ϑ div x u 2.1 Kϑ = ϑ κz dz and the global total energy inequality. In the case of the problem on a fixed domain Ω, this inequality states simply ϱ 1 ϱu 2 2 u 2 + P e ϱ + Qϑτ, dx + ϱ P e ϱ + ϱ Qϑ dx ϱ Ω Ω for all τ, the elastic potential P e is defined in 1.8. However, in the problem on the moving domain this inequality is no longer such simple and additional terms appear. We have ϱ 1 ϱu 2 Ω τ 2 u 2 + P e ϱ + Qϑτ, dx + ϱ P e ϱ + ϱ Qϑ dx Ω 2ϱ τ ϱu Vτ, dx Ω τ ϱu V, dx Ω + S x u : x V ϱu t V ϱu u : x V pϱdiv x V dxdt for all τ. We also emphasise that the boundary condition 1.16 is replaced by the inequality T ϑ n on Γ t for all t [, T ] Following these considerations, in the weak formulation of the problem the temperature ϑ satisfies T ϱqϑ t ϕ + ϱqϑu ϕ + Kϑ ϕ dxdt ϑp ϑ div x u S : x uϕ dxdt + ϱ Qϑ ϕ dx 2.14 Ω for any ϕ C [, T ] R 3, ϕ, ϕt =, ϕ n Γτ = for any τ [, T ]. Definition 2.1 We say that the trio ϱ, u, ϑ is a variational solution of problem with boundary conditions and initial conditions 1.17 if ϱ L, T ; L γ R 3, u, x u L 2 Q T ; R 3, ϱu L, T ; L m R 3 ; R 3 for some m > 6 5, ϱqϑ L, T ; L 1 R 3 L 2, T ; L q R 3 for some q > 6 5, log ϑ, ϑp ϑϱ L 2 Q T, Kϑ L 1 Q T, relations , 2.12 and 2.14 are satisfied. 6

9 At this stage, we are ready to state the main result of the present paper: Theorem 2.1 Let Ω R 3 be a bounded domain of class C 2+ν, and let V C 1 [, T ]; Cc 3 R 3 ; R 3 be given. Assume that the pressure pϱ, ϑ takes the form 1.6 with p e, p ϑ C[, C 1, complying with the hypothesis 2.1, 2.2 with a certain γ > 3/2. Moreover let q be given by 1.5 with κϑ complying with the hypothesis 2.3 and let hypothesis 2.4 be satisfied with α 12γ 1 γ. Furthermore let the initial data fulfill and ϱ L γ R 3, ϱ, ϱ, ϱ R 3 \Ω =, ϱu = a.a. on the set {ϱ = }, ϑ L Ω, ϑ ϑ > on Ω. Ω 1 ϱ ϱu 2 dx < Then the problem with boundary conditions and initial conditions 1.17 admits a variational solution on any time interval, T in the sense specified through Definition 2.1. The rest of the paper is devoted to the proof of Theorem Penalization For the sake of simplicity, we restrict ourselves to the case ζ = in 1.15 and η = in 1.4. As we shall see in Section 5, the main ideas of the proof presented below require only straightforward modifications to accommodate the general case. 3.1 Penalized problem - weak formulation Choosing R > such that V [,T ] { x >R} =, Ω { x < R} we take the reference domain = { x < 2R}. Next, the shear viscosity coefficient µ ω is taken such that and µ ω C c [, T ] R 3, < µ ω µ ω t, x µ in [, T ], µ ω τ, Ωτ = µ for any τ [, T ] 3.1 µ ω a.e. in, T \ Q T as ω. 3.2 Similarly, we introduce variable heat conductivity coefficient as follows: κ ν t, x, ϑ = χ ν t, xκϑ, where and its mollification κ ν,ξ t, x, ϑ = χ ν,ξ t, xκϑ, where χ ν = 1 in Q T and χ ν = ν in, T \ Q T, χ ν,ξ = χ ν Φ ξ. 3.3 Here denotes the convolution in time-space i.e. R 4, Φt, x is a standard mollifier such that Φ ξ t, x t,x as ξ in the sense of distributions. 7

10 Finally, let ϱ, ϱu and ϑ be initial conditions as specified in Theorem 2.1. We define modified initial data ϱ,, ϱu, and ϑ, so that ϱ,, ϱ,, ϱ, R3 \Ω =, ϱ γ, + ϱβ, dx c, ϱ, ϱ in L γ, {ϱ, < ϱ }, 3.4 ϱu, = { ϱu if ϱ, ϱ, otherwise 3.5 and for ϑ, C 2+ν R 3 it holds ϑ, n Γ =, < ϑ ϑ, ϑ on R 3 and ϑ, Ω ϑ in L 1 Ω. 3.6 Now we are ready to state the weak formulation of the penalized problem. Let β > max{4, γ}. Again, we consider ϱ, u to be prolonged by zero outside of, T. The weak formulation of the continuity equations reads as τ ϱϕτ, dx ϱ, ϕ, dx = ϱ t ϕ + ϱu x ϕ dxdt 3.7 for any τ [, T ] and any test function ϕ Cc [, T ] R 3. The momentum equation is represented by the family of integral identities ϱu ϕτ, dx, ϕ, dx ϱu 1 τ V u n ϕ n ds x dt 3.8 ε Γ t τ = ϱu t ϕ + ϱ[u u] : x ϕ + pϱ, ϑdiv x ϕ + ϱ β div x ϕ S ω : x ϕ dxdt, with S ω = µ ω x u + t xu 2 3 div xui for any τ [, T ] and any test function ϕ C c [, T ] ; R 3. The thermal energy inequality in the penalized problem is satisfied in the renormalized sense: T T ϱ + Q h ϑ t ϕ + ϱq h ϑu ϕ + K ν,ξ,h t, x, ϑ ϕ + K h ϑ χ ν,ξ x ϕ hϑϑ α+1 ϕ dxdt 3.9 hϑϑp ϑ div x u hϑ1 S ω : x u + h ϑκ ν,ξ t, x, ϑ x ϑ 2 ϕ dxdt + ϱ, + Q h ϑ, ϕ dx for any ϕ C [, T ] R 3, ϕ, ϕt =, ϕ n = for any τ [, T ] and any h C [, such that h >, h nonincreasing on [,, lim z hz =, h zhz 2h z 2 for all z, 3.1 where Q h, K ν,ξ,h and K h are defined by Q h = ϑ Q zhz dz, ϑ K ν,ξ,h = χ ν,ξ κzhz dz, K h = ϑ κzhz dz. 8

11 Finally, the global total energy inequality reads as T t ψ ϱ 1 2 u 2 + ϱp e ϱ + T β 1 ϱβ + ϱ + Qϑ dxdt + ψ S ω : x u + ϑ α+1 dxdt ϱu, 2 + ϱ, P e ϱ, + 2 ϱ, β 1 ϱβ, + ϱ, + Qϑ, dx + 1 T ψ V u n u n ds x dt ε Γ t for every ψ C [, T ] satisfying ψ = 1, ψt =, t ψ. Definition 3.1 Let ε,, ν, ξ and ω be positive parameters and let β > max{4, γ}. We say that a trio ϱ, u, ϑ is a renormalized solution to the penalized problem with initial data if ϱ L, T ; L γ R 3 L, T ; L β R 3, u L 2, T ; W 1,2 ; R 3, ϱu L, T ; L m R 3 ; R 3 for some m > 6 5, ϱqϑ L, T ; L 1 R 3 L 2, T ; L q R 3 for some q > 6 5, log ϑ, ϑp ϑϱ L 2, T, K h ϑ L 1, T for all h as in 3.1, relations and 3.11 are satisfied. The choice of the no-slip boundary condition u = is not essential here. We have the following existence theorem concerning weak solutions to the penalized problem. Theorem 3.1 Let V C 1 [, T ]; C 3 c R 3 ; R 3 be given. Assume that the pressure satisfies the constitutive equation 1.6 with p e, p ϑ C[, C 1, complying with hypothesis 2.1, 2.2 with certain γ > 3/2. Moreover let q be given by 1.5 with assumption 2.3, let hypothesis 2.4 be satisfied and let the initial data fulfill 3.4, 3.5 and 3.6. Finally, let β > max{4, γ}, and ε, ω, ν, ξ, >. Then the penalized problem admits a renormalized solution on any time interval, T in the sense specified by Definition 3.1. The existence of global-in-time solutions to the penalized problem can be shown by means of the method developed in [5] to handle the nonconstant viscosity coefficients, more precisely see Proposition 4.3 in [5]. Indeed, for ε > fixed, the extra penalty term in 3.8 can be treated as a compact perturbation. The presence of terms involving χ ν,ξ x, t in the thermal energy inequality 3.9 does not cause any additional difficulties since this function is smooth and bounded away from zero. In addition, since β > 4, the density is square integrable and we may use the regularization technique of DiPerna and Lions [2] to deduce the renormalized version of 3.7, namely τ bϱϕτ, dx bϱ, ϕ, dx = bϱ t ϕ + bϱu x ϕ + bϱ b ϱϱ div x uϕ dxdt 3.12 for any ϕ and b as in

12 3.2 Modified energy inequality and uniform bounds Since the vector field V vanishes on the boundary it may be used as a test function in 3.8. Combining the resulting expression with the energy inequality 3.11, we obtain 1 2 ϱ u 2 + ϱp e ϱ + β 1 ϱβ + ϱ + Qϑ τ, dx 3.13 τ + τ + µ ω x u + t xu 2 3 div xui S ω : x u + ϑ α+1 dxdt + 1 ε 1 ϱu, 2 + ϱ, P e ϱ, + 2ϱ, + τ ϱu Vτ, ϱu, V, Γ t u V n 2 ds x dt β 1 ϱβ, + ϱ, + Qϑ, dx dx : x V ϱu t V ϱu u : x V pϱ, ϑdiv x V β 1 ϱβ div x V dxdt. Since the vector field V is regular suitable manipulations with the Hölder, Young and Poincare inequalities and thermodynamical hypothesis yield 1 2 ϱ u 2 + ϱp e ϱ + β 1 ϱβ + ϱ + Qϑ τ, dx + τ S ω : x u + ϑ α+1 dxdt τ u V n 2 ds x dt ε Γ t 1 ϱu, 2 + ϱ, P e ϱ, + 2ϱ, β 1 ϱβ, + ϱ, + Qϑ, dx ϱu, V, dx + CV,. As p e ϱ c1 + ϱp e ϱ for all ϱ, relation 3.14 gives rise to the following bounds independent of parameters ε and ξ: ess sup t,t ess sup t,t ϱut, L 2 ;R 3 c, 3.15 ϱp e ϱt, dx c yielding ess sup ϱt, Lγ c, 3.16 t,t Since we obtain that µ ω 2 ess sup t,t xu + t xu 2 3 div xui T 2 = ϱt, β c L β [ µ ω x u + t xu 2µ ] ω 3 div xui : x u, µ ω x u + t xu 2 3 div xui 2 dxdt c

13 Moreover, T Γ t u V n 2 ds x dt εc We note also that the total mass is conserved, meaning ϱτ, dx = ϱ, dx = ϱ, dx c for any τ [, T ]. 3.2 Ω Thus, relations 3.15, 3.18, 3.2, combined with the generalized version of Korn s inequality see [8, Theorem 1.17], imply that T Moreover directly from the energy estimates we get that ess ut, 2 dt cω W 1,2 ;R 3 sup t,t ϱ + Qϑ L 1 c, 3.22 ϑ L α+1,t c y the proper choice of renormalization function h the same methods as in Section 5.1 [5] gives and moreover We emphasise that all constants in depend on. 3.3 Pressure estimates log ϑ L 2,T c, 3.24 ϑ α/2 L 2,T c, 3.25 ϑ α+1 λ 2 L 2,T c for any < λ < We use the technique based on the ogovskii operator in order to derive the following estimate p e ϱ + ϑp ϑ ϱ ϱ λ + ϱ β+λ dxdt ck for a certain λ > 3.27 for any compact K [, T ] such that For details we refer reader to [5, Section 4.2] or [1]. K 4 Singular limits K τ [,T ] {τ} Γ τ =. In this section, we perform successively the singular limits ε, ξ ω, and with ν = ν. 11

14 4.1 Penalization limit Firstly, we proceed with ε in 3.7, 3.8, 3.9 and 3.13 while other parameters ξ, ν, ω and remain fixed. Let {ϱ ε, u ε, ϑ ε } ε> be the corresponding sequence of renormalized solutions of the penalized problem given by Theorem 3.1. The estimates 3.16, 3.21 together with the equation of continuity 3.7, imply that and, up to a subsequence, ϱ ε ϱ in C weak [, T ]; L γ, u ε u weakly in L 2, T ; W 1,2, R 3. Directly from 3.19 we derive in the limit as ε that u V nτ, Γτ = for a.a. τ [, T ]. 4.1 Consequently, in accordance with 3.15, 3.16 and the compact embedding L γ W 1,2, we obtain and, due to the embedding W 1,2 L 6, ϱ ε u ε ϱu weakly-* in L, T ; L 2γ/γ+1 ; R 3, 4.2 ϱ ε u ε u ε ϱu u weakly in L 2, T ; L 6γ/4γ+3 ; R 3, where the bar denotes a weak limit of a composed function. Finally we deduce from 3.8 that for any space-time cylinder ϱ ε u ε ϱu in C weak [T 1, T 2 ]; L 2γ/γ+1 O; R 3 T 1, T 2 O [, T ], [T 1, T 2 ] O τ [,T ] {τ} Γ τ =. Since L 2γ/γ+1 W 1,2, we conclude that Strong convergence of temperature ϱu u = ϱu u a.a. in, T. y assumption 2.4 Qϑ ε 2 L 2,T C ϑ ε 2 dxdt +,T C {ϑ ε<1} ϑ ε ϑ ε 2,T ϑ α 2 ε 2 dxdt dxdt + ϑ ε 2 ϑ ε α 2 dxdt + {ϑ ε 1},T ϑ α 2 ε 2 dxdt C log ϑ ε 2 L 2,T + ϑ α 2 ε 2 L 2,T 4.3 and, with help of 3.24 and 3.25, we get Qϑ ε L2,T ;W 1,2 c,

15 consequently, Next, according to Lemma 6.3 in [4] applied to + ϱ ε Qϑ ε we obtain Similarly as in Subsection in [4] one gets Pointwise convergence of density Qϑ ε Qϑ weakly in L 2, T ; W 1, ϱ ε Qϑ ε + ϱqϑ strongly in L 2, T ; W 1, Next, we show pointwise convergence of the sequence {ϱ ε } ε>. We define ϑ ε ϑ strongly in L 2, T Ω. 4.7 p ϱ, ϑ = pϱ, ϑ + ϱ β, T k ϱ = min{ϱ, k}. Similarly as in [5] we establish the effective viscouse pressure identity: p ϱ, ϑt k ϱ p ϱ, ϑ T k ϱ = 4 3 µ ω which holds only on compact sets K [, T ] satisfying K τ [,T ] {τ} Γ τ =. Following [5], we introduce the oscillations defect measure osc q [ϱ ε ϱ]k = sup lim sup k ε and use 4.8 to conclude that where the constant c is independent of K. Thus T k ϱdiv x u T k ϱdiv x u, 4.8 K T k ϱ ε T k ϱ q dxdt, osc γ+1 [ϱ ε ϱ]k cω <, 4.9 osc γ+1 [ϱ ε ϱ][, T ] cω, 4.1 which implies, by virtue of the procedure developed in [4], the desired conclusion Passing to the limit with ε ϱ ε ϱ a.e. in, T Passing to the limit in 3.7 we obtain τ ϱϕτ, dx ϱ, ϕ, dx = ϱ t ϕ + ϱu x ϕ dxdt 4.12 for any τ [, T ] and any ϕ C c [, T ] R 3. 13

16 The limit in the momentum equation 3.8 is more delicate. Since we have at hand only the local estimates 3.27 on the pressure, we have to restrict ourselves to the class of test functions ϕ C 1 [, T ]; W 1, ; R 3, supp[div x ϕτ, ] Γ τ =, ϕ n Γτ = for all τ [, T ] In accordance with 4.11 and 3.27, the momentum equation reads ϱu ϕτ, dx ϱu, ϕ, dx 4.14 = τ ϱu t ϕ + ϱ[u u] : x ϕ + pϱ, ϑdiv x ϕ + ϱ β div x ϕ µ ω x u + t xu 2 3 div xui : x ϕ dxdt for any test function ϕ as in In addition, as already observed, the limit solution {ϱ, u} satisfies also the renormalized equation The strong convergence of the temperature 4.7 is sufficient for the non-linear terms in thermal energy equation to pass to their limits counterparts. In particular: and ϱ ε Qϑ ε ϱqϑ weakly in L 2, T ; L q1 with q 1 = ϱ ε Qϑ ε u ε ϱqϑu weakly in L 2, T ; L q2 with q 2 = Moreover, if p ϑ satisfies hypothesis 2.2 we have 6γ 6 + γ ϑ ε p ϑ ϱ ε div x u ε ϑp ϑ ϱdiv x u weakly in L 1, T 6γ 3 + 4γ. in order to provide it one may use 3.25 and additional information from artificial pressure and since h satisfies 3.1, hϑ ε ϑ ε p ϑ ϱ ε div x u ε hϑϑp ϑ ϱdiv x u weakly in L 1, T. Due to the convexity of the function { hϑ µ [M, ϑ] 2 M : M + λtr[m]2 if ϑ, M R 3 3 if ϑ < and [4, Lemma 4.8], we get T T hϑs ω x u : x uϕ dxdt lim inf hϑ ε S ω x u ε : x u ε ϕ dxdt 4.15 ε where ϕ is as in 3.9. In order to treat the most-right term in 3.9 we apply [4, Corollary 2.2]. Let us set T χ ν,ξ κϑ ε h ϑ ε ϑ ε 2 dxdt = T χ ν,ξ G ϑ ε ϑ ε 2 dxdt with G z = κz h z The sequence χ ν,ξ Gϑ ε is uniformly bounded in L 1+λ, T for certain λ > and, consequently, χ ν,ξ Gϑ ε χν,ξ Gϑ weakly in L 1, T. Due to a.e. convergence of ϑ ε ϑ we have χ ν,ξ κϑε h ϑ ε 14

17 χν,ξ κϑ h ϑ strongly in L 2. Thus, as ϑ ε ϑ weakly in L 2, we get χ ν,ξ Gϑ ε χ ν,ξ Gϑ weakly in L 1, T. Then convexity of Φ = 2 and [4, Corollary 2.2] provide T T χ ν,ξ κϑ h ϑ ϑ 2 ϕ dxdt lim inf χ ν,ξ κϑ ε h ϑ ε ϑ ε 2 ϕ dxdt, 4.17 ε where ϕ is as in 3.9. Since ϑ ε is uniformly bounded in L α+1 and due to properties of h, the sequence hϑ ε ϑ α+1 ε uniformly integrable in L 1 see [4, Proposition 2.1]. Then a.e. convergence of ϑ ε provides y the same token and also T T T T hϑ ε ϑ α+1 ε ϕ dxdt hϑϑ α+1 ϕ dxdt as ε. K ν,ξ,h ϑ ε ϕ dxdt K h ϑ ε χ ν,ξ ϕ dxdt T T K ν,ξ,h ϑ ϕ dxdt as ε, K h ϑ χ ν,ξ ϕ dxdt as ε, where ϕ is as in 3.9. Thus, the renormalized thermal energy inequality 3.9 takes in the limit the same form. Finally, we obtain the following energy inequality when passing ε ϱ 1 2 u 2 + ϱp e ϱ + τ + τ β 1 ϱβ + ϱ + Qϑ τ dx + S ω : x u + ϑ α+1 dxdt 1 ϱu, 2 + ϱ, P e ϱ, + 2 ϱ, β 1 ϱβ, + ϱ, + Qϑ, dx + µ ω x u + t xu 2 3 div xui ϱu Vτ, ϱu, V, dx : x V ϱu t V ϱu u : x V pϱ, ϑdiv x V β 1 ϱβ div x V Fundamental lemma and extending the class of test functions dxdt. Our next goal is to use the specific choice of the initial data ϱ, to get rid of the density-dependent terms in 4.14 supported by the solid part, T \ Q T. To this end, we proceed the same way as in the barotropic case, more precisely we use [7, Lemma 4.1]. Lemma 4.1 Let ϱ L, T ; L 2, ϱ, u L 2, T ; W 1,2 ; R 3 be a weak solution of the equation of continuity, specifically, τ ϱτ, ϕτ, ϱ ϕ, dx = ϱ t ϕ + ϱu x ϕ dxdt 4.19 for any τ [, T ] and any test function ϕ C 1 c [, T ] R 3. In addition, assume that u Vτ, n Γτ = for a.a. τ, T, 4.2 and that ϱ L 2 R 3, ϱ, ϱ \Ω =. is

18 Then ϱτ, \Ωτ = for any τ [, T ]. y virtue of Lemma 4.1, the momentum equation 4.14 reduces to ϱu ϕτ, dx Ω τ ϱu, ϕ, dx Ω 4.21 τ = ϱu t ϕ + ϱ[u u] : x ϕ + pϱ, ϑdiv x ϕ + ϱ β div x ϕ µ ω x u + t xu 2 3 div xui : x ϕ dxdt τ µ ω x u + t xu 2 \ 3 div xui : x ϕ dxdt for any test function ϕ as in We remark that in this step we crucially need the extra pressure term ϱ β ensuring the density ϱ to be square integrable. Next, we argue the same way as in [7, Section 4.3.1] to conclude, that the momentum equation 4.21 holds in fact for any test function ϕ such that ϕ C c [, T ] R 3 ; R 3, ϕτ, n Γτ = for any τ [, T ] The discontinuous heat conductivity coefficient The aim of this section is to proceed to a limit with ξ. In other words we pass from a smooth function χ ν,ξ presented in 3.3 to a jump function χ ν. We denote by u ξ, ϑ ξ and ϱ ξ the sequences of functions satisfying relations 4.12, 4.21, 3.9 and 4.18 which were constructed in a previous section. To pass to the limit ξ we use the a priori estimates from Section 3.2 to conclude that there exist ϱ, u and ϑ such that ϱ ξ ϱ in C weak [, T ]; L γ C weak [, T ]; L β, u ξ u weakly in L 2, T ; W 1,2, R 3, ϑ ξ ϑ strongly in L 2, T. Using the same procedures as in Section 4.1 we pass to the limit in most of the terms in the equations and inequalities 4.12, 4.21, 3.9 and We treat in detail only the terms directly involving χ ν,ξ. Firstly, we consider the term T I ξ := χ ν,ξ K h ϑ ξ ϕ dxdt + K h ϑ ξ χ ν,ξ ϕ dxdt, with ϕ as for 3.9. y integration by parts we derive T I ξ = χ ν,ξ K h ϑ ξ ϕ dxdt = As a corollary of 3.22, 3.23, 3.26 and we get up to a subsequence T ϑ ξ ϑ weakly in L 2, T ; W 1,2, χ ν,ξ κϑ ξ hϑ ξ ϑ ξ ϕ dxdt. ϑ ξ ϑ strongly in L p, T ; L p for all p [1, α + 1, ϑ α ξ ϑ α weakly in L 2, T ; W 1,2,

19 and also χ ν,ξ t, x κϑ ξhϑ ξ 1 + ϑ α ξ Indeed, if we denote f ξ = χ ν,ξ t, x κϑ ξhϑ ξ 1+ϑ α ξ χ ν t, x κϑhϑ 1 + ϑ α Further, using interpolation between 3.22 and 3.26 we get strongly in L 2, T and f = χ ν t, x κϑhϑ, it follows that f 1+ϑ α ξ f almost everywhere. ϑ ξ L α+ 3 2,T c and f ξ L α+ 7 4, T uniformly. Consequently, f ξ 2 is uniformly integrable and from the Vitali convergence theorem we get Finally, we have χ ν,ξ t, xκϑ ξ hϑ ξ ϑ ξ = χ ν,ξt, xκϑ ξ hϑ ξ Since a ξ tends strongly to χνt,xκϑhϑ 1+ϑ α , we get 1 + ϑ α ξ ϑ ξ + in L 2, T and b ξ tends to 1 α ϑ α ξ =: a ξ b ξ. χ ν,ξ t, xκϑ ξ hϑ ξ ϑ ξ χ ν t, xκϑhϑ ϑ in D, T. We conclude, that for ϕ such that ϕ n Γt = we get ϑ + 1 α ϑ α weakly in L 2, T 12 T T T I ξ χ ν K h ϑ ϕ dxdt = χ ν K h ϑ ϕ dxdt χ ν K h ϑ ϕ dxdt \ T T T = χ ν K h ϑ ϕ dxdt + χ ν K h ϑ ϕ dxdt = χ ν K h ϑ ϕ dxdt \ T = K ν,h ϑ ϕ dxdt. It remains to handle the term T χ ν,ξ κϑ ξ h ϑ ξ x ϑ ξ 2 dxdt. However, since χ ν,ξ χ ν a.e in as ξ, we may proceed similarly as in 4.16 and Hence, after passing with ξ the thermal energy inequality has the form T ϱ + Q h ϑ t ϕ + ϱq h ϑu ϕ + K ν,h t, x, ϑ ϕ hϑϑ α+1 ϕ dxdt 4.25 T hϑϑp ϑ div x u hϑ1 S ω : x u + h ϑκ ν t, x, ϑ x ϑ 2 ϕ dxdt + ϱ, + Qϑ, ϕ dx for any ϕ C [, T ] R 3, ϕ, ϕt =, ϕ n =, ϕ n Γτ = for any τ [, T ] and any h C [, such that 3.1 holds. The equation of continuity 4.12, momentum equation 4.21 and the global total energy inequality 4.18 remain in the same forms. 17

20 4.3 Vanishing viscosity limit In this section we let ω in order to get rid of the last integral in Let {ϱ ω, u ω, ϑ ω } ω> be the solution constructed in the previous section. Let us recall that the viscosity coefficient has the following form see 3.1, 3.2 µ = const > in Q T, µ ω = µ ω a.e. in, T \ Q T. From 3.18 we deduce that and The last estimate yields τ τ T T \ µ ω \ µ ω \ µω µω xu ω + t xu ω 2 3 div xu ω I xu ω + t xu ω 2 3 div xu ω I 2 dxdt c dxdt c. x u ω + t xu ω 2 3 div xu ω I : x ϕ dxdt = x u ω + t xu ω 2 3 div xu ω I : x ϕ dxdt as ω for any fixed ϕ. As we know from Lemma 4.1, the density ϱ ω is supported by the fluid region Q T. We use 3.15, 4.26 and Korn s inequality to infer T x u ω 2 dxdt c. y the arguments of Section 4.1, we let ω and obtain the same form of the continuity equation The momentum equation takes the form ϱu ϕτ, dx Ω τ ϱu, ϕ, dx Ω 4.27 = τ ϱu t ϕ + ϱ[u u] : x ϕ + pϱ, ϑdiv x ϕ + ϱ β div x ϕ S x u : x ϕ dxdt for any test function ϕ as in We would like to emphasise that the compactness of the density is necessary only in the fluid part Q T so a possible loss of regularity of u ω outside Q T is irrelevant. 18

21 The thermal energy equation 4.25 can be written in a form T ϱ ω + Q h ϑ ω t ϕ + K ν,h ϑ ω ϕ hϑ ω ϑ α+1 ω ϕ + ϱ ω Q h ϑ ω u ω ϕ dxdt T + Q h ϑ ω t ϕ + K ν,h ϑ ω ϕ hϑ ω ϑ α+1 ω ϕ dxdt \ T hϑω ϑ ω p ϑ div x u ω + hϑ ω 1S ω : x u ω + h ϑ ω κ ν ϑ ω x ϑ ω 2 ϕ dxdt T + hϑω 1S ω : x u ω + h ϑ ω κ ν ϑ ω x ϑ ω 2 ϕ dxdt + ϱ, + Qϑ, ϕ dx 4.28 \ for any ϕ C [, T ] R 3, ϕ, ϕt =, ϕ n =, ϕ n Γτ = for any τ [, T ] and any h C [, such that 3.1 holds. Note, that the last integral on the right hand side is negative and thus can be omitted. y the same arguments as in Section 4.1 we get after passing with ω to zero T T ϱ + Q h ϑ t ϕ + K ν,h ϑ ϕ hϑϑ α+1 ϕ + ϱq h ϑu ϕ dxdt T + Q h ϑ t ϕ + K ν,h ϑ ϕ hϑϑ α+1 ϕ dxdt \ hϑϑpϑ div x u + hϑ 1S : x u + h ϑκ ν ϑ x ϑ 2 ϕ dxdt + ϱ, + Qϑ, ϕ dx The total energy inequality 4.18 can be written in the following way Ω τ Ω 1 ϱ ω 1 2 u ω 2 + ϱ ω P e ϱ ω + + β 1 ϱβ ω + ϱ ω + Qϑ ω τ, dx + Qϑ ω τ, dx \Ω τ τ τ + S ω : x u ω + ϑ α+1 ω dxdt + S ω : x u ω + ϑ α+1 ω dxdt \ + ϱ, P e ϱ, + ϱ, β 1 ϱβ, + ϱ, + Qϑ, dx + ϱ ω u ω Vτ, dx ϱu, V, dx Ω τ Ω µ ω x u ω + t xu ω 2 3 div xu ω I : x V ϱ ω u ω t V ϱ ω u ω u ω : x V pϱ ω, ϑ ω div x V ϱu, 2 2 τ β 1 ϱβ ωdiv x V τ dxdt + Qϑ, dx + µ ω x u ω + t xu ω 2 \Ω \ 3 div xu ω I : x V dxdt. 4.3 On the right hand side we pass to the limit in the same way as in the momentum equation. The term τ \ S ω : x u ω on the left hand side can be neglected since it is positive and the rest of the terms converge due to the same 19

22 arguments as in Section 4.1. This way as ω we obtain Ω τ + 1 Ω τ ϱ 1 2 u 2 + ϱp e ϱ + ϱu, 2 2 β 1 ϱβ + ϱ + Qϑ τ dx + Qϑτ, dx \Ω τ τ τ + S : x u + ϑ α+1 dxdt + ϑ α+1 dxdt \ + ϱ, P e ϱ, + ϱ, β 1 ϱβ, + ϱ, + Qϑ, dx + ϱu Vτ, dx ϱu, V, dx Ω τ Ω x u + t xu 2 3 div xui : x V ϱu t V ϱu u : x V pϱ, ϑdiv x V β 1 ϱβ div x V dxdt + Qϑ, dx \Ω µ 4.4 Vanishing artificial pressure In this section we make the final limit procedure while setting ν = ν in a suitable way. In order to derive estimates independent of, which we need further, we proceed in the following way. Starting from 4.31 we get Ω τ 1 2 ϱ u 2 + ϱp e ϱ + β 1 ϱβ + ϱ + Qϑ τ, dx cv 1 + x u L2 Q T + τ 1 2 ϱ u 2 + ϱp e ϱ + ϱqϑ dxdt, 4.32 where cv may depend also on, T and initial conditions but is independent of. Using the Gronwall inequality, we get ess sup τ,t 1 2 ϱ u 2 + ϱp e ϱ + β 1 ϱβ + ϱ + Qϑ τ, dx + Qϑτ, dx \Ω τ In the same way as in [5] one may derive from equation 4.29 that T x u 2 L 2 Q T S : x u dxdt T T + ϑ α+1 dxdt c 1 + x u L2 Q T ϑp ϑ ϱ div x u dxdt + c1 + x u L2 Q T T 1 2 ϑ 2 p 2 ϑϱ dxdt x u L2 Q T + c1 + x u L2 Q T

23 Further, T T ϑ 2 ϱ 2γ 3 dxdt 4 ϱϑ α α 2 dx ϱ 2αγ 12 3α 12 Ωt c ess sup ϱqϑ dx t,t α 4 α dx 2 α T dt ϱ γ dx 2γα 12 3γα dt c1 + x u L2,T 4 α + 2γα 12 3γα 4.35 provided α 12γ 1 γ. Note that 4 α + 2γα 12 3γα < 2 for any α 4 and γ 3 2. Combining this and 4.34 we obtain with constant independent of. Going back to 4.33 we get ess sup t,t ess ess x u L 2 Q T c 4.36 sup t,t sup t,t T T ϱut, L 2 Q T c, 4.37 ϱt, Lγ c, 4.38 ϱt, β L β c, 4.39 ϑ α+1 dxdt c, 4.4 Qϑ dxdt c, 4.41 with c independent of. Let {ϱ, u, ϑ } > be a solution constructed in the previous section as a consequence of letting ω. Now we pass to the limit with in the weak formulations of the equations and inequalities 4.12, 4.27, 4.29, The key point here is again the strong convergence of densities which is obtained using the results on propagation of oscillations mentioned already in Section 4.1.2, for more details see [4]. Having this, we pass to the limit in the continuity equation 4.12 which together with Lemma 4.1 yields 2.5. We also easily pass to the limit in the momentum equation 4.27 with test function ϕ as in Concerning the total energy inequality 4.31 we proceed in the same steps as in [4, Section 7.5] to obtain It remains to pass to the limit in the thermal energy inequality Thermal energy inequality In order to get a proper limit in thermal energy equation, we set ν = 2, we take hϑ = 1 1+ϑ z, z, 1 in 4.29 and we let to. In order to prove that the terms integrated over the solid part \ vanish in the limit we proceed in the following way. We have, due to 4.4 and the Hölder inequality, T T hϑ ϑ α+1 ϕ dxdt c T ϑ α+1 z dxdt = c c z α+1 α+1 z α+1 ϑ α+1 z T ϑ α+1 dxdt z α+1 dxdt α+1 z α+1 c z α+1 as. 21

24 Similarly, T T Q h ϑ t ϕ dxdt c c α α+1 Finally, by the choice ν = 2 we have T K ν,h ϑ ϕ, dxdt \ T ϑ dxdt + ϑ α 2 dxdt T 1 α+1 T ϑ α+1 dxdt c T ϑ 2 \ T κz dz ϕ dxdt ϑ α+1 dxdt \ 2 ϑ + ϑ α+1 ϕ dxdt as. 1 2 as. We omit the details concerning the limit in the terms integrated on the fluid part, since it is done in the same way as in [4, Section 7.5.5] by first proceeding with keeping z fixed and then passing with z. In the limit, 2.14 is obtained, which completes the proof of Theorem Discussion The assumption on monotonicity of the pressure is not necessary, the same result can be obtained for a non-monotone pressure adopting the method developed in [3]. The case of nonconstant viscosities µ = µϑ and η = ηϑ for bounded, continuously differentiable, globally Lipschitz functions µ, η with µ bounded away from zero, can be also treated by adopting the technique of [5]. As pointed out in the introduction, the general Navier slip conditions 1.15 are obtained introducing another boundary integral in the weak formulation, namely T Γ t ζu V ϕ ds x dt Taking ζ = ζx as a singular parameter, we can deduce results for mixed type no-slip - partial slip boundary conditions prescribed on various components of Γ t. References [1] M. ulíček, J. Málek, and K.R. Rajagopal. Navier s slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity. Indiana Univ. Math. J., 56:51 86, 27. [2] R.J. DiPerna and P.-L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98: , [3] E. Feireisl. Compressible Navier-Stokes equations with a non - monotone pressure law. J. Differential Equations, 184:97 18,

25 Powered by TCPDF [4] E. Feireisl. Dynamics of viscous compressible fluids. Oxford University Press, Oxford, 24. [5] E. Feireisl. On the motion of a viscous, compressible, and heat conducting fluid. Indiana Univ. Math. J., 53: , 24. [6] E. Feireisl, J. Neustupa, and J. Stebel. Convergence of a rinkman-type penalization for compressible fluid flows. J. Differential Equations, 251:596 66, 211. [7] E. Feireisl, O. Kreml, Š. Nečasová, J. Neustupa, J. Stebel. Weak solutions to the barotropic Navier-Stokes system with slip boundary conditions in time dependent domains. J. Differential Equations , no. 1, [8] E. Feireisl and A. Novotný. Singular limits in thermodynamics of viscous fluids. irkhäuser-verlag, asel, 29. [9] E. Feireisl, A. Novotný, and H. Petzeltová. On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids. J. Math. Fluid Mech., 3: , 21. [1] E. Feireisl and H. Petzeltová. On integrability up to the boundary of the weak solutions of the Navier-Stokes equations of compressible flow. Commun. Partial Differential Equations, 253-4: , 2. [11] O. A. Ladyzhenskaja. An initial-boundary value problem for the Navier-Stokes equations in domains with boundary changing in time. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. LOMI, 11:97 128, [12] P.-L. Lions. Mathematical topics in fluid dynamics, Vol.2, Compressible models. Oxford Science Publication, Oxford, [13] J. Neustupa. Existence of a weak solution to the Navier-Stokes equation in a general time-varying domain by the Rothe method. Math. Methods Appl. Sci., 326: , 29. [14] J. Neustupa and P. Penel. A weak solvability of the Navier-Stokes equation with Navier s boundary condition around a ball striking the wall. In Advances in mathematical fluid mechanics, pages Springer, erlin, 21. [15] J. Neustupa and P. Penel. A weak solvability of the Navier-Stokes system with Navier s boundary condition around moving and striking bodies. Recent Developments of Mathematical Fluid Mechanics, Editors: Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M. To appear in irkhauser, series: Advances in Mathematical Fluid Mechanics. [16] N. V. Priezjev and S.M. Troian. Influence of periodic wall roughness on the slip behaviour at liquid/solid interfaces: molecular versus continuum predictions. J. Fluid Mech., 554:25 46, 26. [17] Y. Stokes and G. Carrey. On generalised penalty approaches for slip, free surface and related boundary conditions in viscous flow simulation. Inter. J. Numer. Meth. Heat Fluid Flow, 21:668 72,