A regularity criterion for the weak solutions to the Navier-Stokes-Fourier system

Transcription

1 A regularity criterion for the weak solutions to the Navier-Stokes-Fourier system Eduard Feireisl Antonín Novotný Yongzhong Sun Charles University in Prague, Faculty of Mathematics and Physics, Mathematical Institute Sokolovská 83, Prague 8, Czech Republic IMATH Université du Sud Toulon-Var BP 132, La Garde, France Department of Mathematics, Nanjing University, Nanjing, Jiangsu 2193, China Abstract We show that any weak solution to the full Navier-Stokes-Fourier system emanating from the data belonging to the Sobolev space W 3,2 remains regular as long as the velocity gradient is bounded. The proof is based on the weak-strong uniqueness property and parabolic a priori estimates for the local strong solutions. Key words: Navier-Stokes-Fourier system, weak solution, regularity Contents 1 Introduction Weak and dissipative solutions Relative entropy and dissipative solutions Regularity criteria Eduard Feireisl acknowledges the support of the project LL122 in the programme ERC-CZ funded by the Ministry of Education, Youth and Sports of the Czech Republic. Yongzhong Sun is supported by NSF of ChinaGrants No and and the PAPD of Jiangsu Higher Education Institutions. 1

2 2 Preliminaries, main results Weak and dissipative solutions Main result Local existence and weak-strong uniqueness First a priori bounds Local existence of strong solutions Weak strong uniqueness Conditional regularity Energy bounds, temperature Energy bounds, velocity Elliptic estimates Energy estimates for the time derivatives L p L q estimates for the temperature Full regularity Conclusion 22 1 Introduction In continuum mechanics, the motion of a general compressible, viscous, and heat conducting fluid is described by the thermostatic variables - the mass density ϱ = ϱt, x and the absolute temperature ϑt, x, and the velocity field u = ut, x evaluated at the time t and the spatial position x belonging to the reference physical domain R 3. The time evolution of the fluid, emanating from the initial data ϱ, = ϱ, ϑ, = ϑ, u, = u in, 1.1 is governed by the Navier-Stokes-Fourier system of partial differential equations: 2

3 Equation of continuity: t ϱ + div x ϱu = ; 1.2 Momentum equation: t ϱu + div x ϱu u + x pϱ, ϑ = div x Sϑ, x u; 1.3 Total energy balance: [ ] 1 1 t 2 ϱ u 2 + ϱeϱ, ϑ + div x 2 ϱ u 2 + ϱeϱ, ϑ u + div x pϱ, ϑu 1.4 = div x Sϑ, x uu div x qϑ, x ϑ. The symbols p = pϱ, ϑ and e = eϱ, ϑ stand for the pressure and the specific internal energy, respectively. Furthermore, S = Sϑ, x u denotes the viscous stress given by Newton s rheological law: Sϑ, x u = µϑ x u + t xu 2 3 div xui, 1.5 while q = qϑ, x ϑ is the heat flux determined by Fourier s law: qϑ, x u = κϑ x ϑ

4 The reader will have noticed that the effect of bulk viscosity is neglected in 1.5. The reason is to avoid certain technical difficulties in the proof of the main result. Soome indications how to handle the general case are given in Section 5. We restrict ourselves to bounded regular domains R 3, with energetically insulated boundary. More specifically, we consider the standard No slip boundary condition: u =, 1.7 together with No-flux condition: q n =, 1.8 where n is the outer normal vector to. Under these circumstance, the total mass as well as the total energy of the system are constants of motion, specifically, integrating equations 1.2, 1.4 over we obtain ϱt, dx = ϱ dx = M, ϱ u 2 + ϱeϱ, ϑ t, dx = 1 2 ϱ u 2 + ϱ eϱ, ϑ dx = E. 1.1 From the mathematical viewpoint, the Navier-Stokes-Fourier system suffers the same deficiency as most of its counterparts in continuum mechanics - the lack of sufficiently strong a priori bounds. It is known that the problem is well posed, locally in time, in the framework of classical solutions, see Tani [26], or in slightly more general energy spaces of Sobolev type, see Valli [27], [28], Valli and Zajaczkowski, [29], among others. Besides these local existence results, Matsumura and Nishida established in a series of papers [2], [21], [22] global-in-time existence of strong/classical solutions provided the initial data are sufficiently close to an equilibrium solution. Similarly to the well known incompressible Navier-Stokes system, the existence of global-in-time smooth solutions for any physically admissible and large initial data remains an outstanding open problem. 4

5 The system has been also studied in the framework of weak solutions. Hoff and Jenssen [14] established global existence for radially symmetric data in R 3. They also identified one of the main stumbling blocks in the analysis of the Navier-Stokes- Fourier system, namely the hypothetical appearance of vacuum zones, where the density vanishes and the classical understanding of the equations breaks down. More recently, Bresch and Desjardins [4], [5] discovered a new a priori bound on the density gradient leading to global-in-time existence in the truly 3D setting conditioned, unfortunately, by a very specific relation satisfied by the density dependent viscosity coefficients and a rather unrealistic formula for the pressure that has to be infinite negative for ϱ. 1.1 Weak and dissipative solutions We adopt the approach to the Navier-Stokes-Fourier system originated in [9] and further developed and detailed in [11]. Suppose that p and e are interrelated through Gibbs equation: ϑdsϱ, ϑ = Deϱ, ϑ + pϱ, ϑd 1, 1.11 ϱ where s = sϱ, ϑ is a new thermodynamic function called specific entropy. If ϱ, ϑ are smooth and bounded below away from zero and if u is smooth, then the total energy balance 1.4 can be replaced by Thermal energy balance: eϱ, ϑ pϱ, ϑ ϱ t ϑ + u x ϑ div x κϑ x ϑ = Sϑ, x u : x u ϑ div x u ϑ ϑ with Furthermore, dividing 1.12 on ϑ we arrive at Entropy production equation: qϑ, x ϑ t ϱsϱ, ϑ + div x ϱsϱ, ϑu + div x = σ, 1.13 ϑ σ = 1 ϑ Entropy production rate: Sϑ, x u : x u q xϑ ϱ Let us remark that the systems 1.2, 1.3, 1.4, 1.2, 1.3, 1.12, and 1.2, 1.3, 1.14 are perfectly equivalent in the class of classical solutions. 5

6 Another crucial observation, which is the platform of the approach developed in [11], is that we can relax 1.14 to σ 1 Sϑ, x u : x u q xϑ 1.15 ϑ ϱ provided the system is supplemented by the integrated total energy balance 1.1, meaning the relations 1.2, 1.3, 1.1, 1.13, with 1.15 are still equivalent to the original system 1.2, 1.3, 1.4, at least for smooth solutions. The resulting problem is mathematically tractable, more specifically, we report the following results: Existence. The problem { , , 1.13, 1.15 } admits globalin-time weak solutions for any finite energy initial data under certain structural restrictions imposed on p, e, s, and µ, κ presented in Section 2 below, see [11, Theorem 3.1]. Compatibility. Any weak solution of { , , 1.13, 1.15 } that is regular solves the original problem , see [11, Chapter 2]. Weak-strong uniqueness. Any weak solution coincides with a classical solution emanating from the same initial data provided the latter exists, see [12] Relative entropy and dissipative solutions The property of weak-strong uniqueness mentioned above is closely related to the relative entropy inequality introduced in [12]. We start by introducing another thermodynamic function: Ballistic free energy: see [7], together with H Θ ϱ, ϑ = ϱeϱ, ϑ Θϱsϱ, ϑ, 1.16 Relative entropy functional: E ϱ, ϑ, u [ 1 r, Θ, U = 2 ϱ u U 2 + H Θ ϱ, ϑ H ] Θr, Θ ϱ r H Θ r, Θ dx, ϱ 1.17 where {ϱ, ϑ, u} is a weak solution of the Navier-Stokes-Fourier system and {r, Θ, U} is an arbitrary trio of smooth functions satisfying r >, Θ >, U =

7 As observed in [12], the weak solutions of the Navier-Stokes-Fourier system satisfy Relative entropy inequality: [ E ϱ, ϑ, u r, Θ, U ] t=τ t= + τ Θ Sϑ, x u : x u qϑ, xϑ x ϑ ϑ ϑ τ ϱu u t U + ϱu u u : x U pϱ, ϑdiv x U dx dt τ + Sϑ, x u : x U dx dt τ ϱ sϱ, ϑ sr, Θ t Θ + ϱ sϱ, ϑ sr, Θ u x Θ dx dt τ τ + 1 ϱ r qϱ, x ϑ ϑ x Θ dx dt t pr, Θ ϱ r u xpr, Θ dx dt dx dt 1.19 for any trio {r, Θ, U} satisfying Taking {r, Θ, U} the strong solution of the same system emanating from the same initial data gives rise to a Gronwall type inequality for E yielding ϱ = r, ϑ = Θ, u = U, see [12]. As the proof of the weak-strong uniqueness principle uses only the integral inequality 1.19 without any reference to the original system of equations, we may introduce a new class of dissipative solutions satisfying solely the relative entropy inequality 1.19 for any trio of admissible test functions {r, Θ, U}. Note that a similar concept was introduced by DiPerna and Lions [19] in the context of the Euler system. 1.2 Regularity criteria A conditional regularity criterion is a condition which, if satisfied by a weak solution, implies that the latter is regular. Similarly, such a condition may be applied to guarantee that a local strong solution can be extended to a given time interval. The most celebrated conditional regularity criteria are due to Prodi and Serrin [23], [24] and Beal, Kato and Majda [3], Constantin and Fefferman [6] for solutions to the incompressible Navier-Stokes and Euler systems. Recently, similar conditions were obtained also in the context of compressible barotropic fluids and the full Navier-Stokes-Fourier system, see Fan, Jiang and Ou [8], Sun et al. [25], and the references cited therein. In view of the results of Hoff et al. [13], [15], certain discontinuities imposed through the initial data in the compressible Navier-Stokes system propagate in time. In other 7

8 words, unlike its incompressible counterpart, the hyperbolic-parabolic compressible Navier- Stokes system does not enjoy the smoothing property typical for purely parabolic equations. Analogously, a solution of the full Navier-Stokes-Fourier system can be regular only if regularity is enforced by a proper choice of the initial data. Our approach to conditional regularity is based on the concept of weak dissipative solutions satisfying the relative entropy inequality 1.19: In Section 2, we introduce the structural restrictions imposed on the thermodynamic functions p, e, s and the transport coefficients µ, κ so that the Navier-Stokes-Fourier system may possess a global in time weak dissipative solution for any finite energy initial data. Now, suppose that the initial data are more regular, specifically, < ϱ ϱ, < ϑ ϑ, ϱ, ϑ W 3,2, u W 3,2 ; R 3, 1.2 satisfying, in addition, the necessary compatibility conditions u =, x ϑ n =, x pϱ, ϑ = div x Sϑ, u Under these circumstance, the Navier-Stokes-Fouries system admits a local-in-time strong solution { ϱ, ϑ, ũ} constructed by Valli [28] such that ϱ, ϑ C[, T ]; W 3,2, ϑ L 2, T ; W 4,2, t ϑ L 2, T ; W 2,2, 1.22 ũ C[, T ]; W 3,2 ; R 3 L 2, T ; W 4,2 ; R 3, t ũ L 2, T ; W 2,2 ; R Modifying slightly the arguments of [12] to accommodate the strong solutions belonging to the regularity class 1.22 we show in Section 3 that the associated globalin-time weak solution {ϱ, ϑ, u} emanating from the initial data coincides with the strong solution { ϱ, ϑ, ũ} on its existence interval [, T ]. We show that if lim sup ϱt, W 3,2 t T + ϑt, W 3,2 + ut, W 3,2 ;R 3 =, 1.24 then, necessarily, see Section 4. lim sup x ut, L t T ;R 3 3 =,

9 Finally, in Section 5, we observe that the solutions constructed by Valli are classical all necessary derivatives are continuous for any t >. We conclude by the following conditional regularity result for the weak solutions of the Navier-Stokes-Fourier system: Suppose that {ϱ, ϑ, u} is a weak dissipative solution of the Navier-Stokes-Fourier system in, T, emanating from the initial data belonging to the regularity class specified through 1.2, 1.21, and satisfying ess sup x ut, L ;R 3 3 <. t,t Then u is a classical solution of the Navier-Stokes-Fourier system, unique in the class of weak dissipative solutions, see Theorem 2.1 below. 2 Preliminaries, main results We start with a list of structural restrictions imposed on the functions p, e, s, µ, and κ. The interested reader may consult [11, Chapter 1] for the physical background and possible relaxations. We suppose that the pressure is given by the formula pϱ, ϑ = ϑ 5/2 P with P C 1 [, C 3, satisfying P =, P Z >, < ϱ + a ϑ 3/2 3 ϑ4, a >, P Z P ZZ 3 Z < c for all Z >, 2.2 P Z lim >. 2.3 Z Z5/3 Accordingly, in agreement with Gibbs equation 1.11, eϱ, ϑ = 3 ϑ 5/2 ϱ 2 ϱ P + a ϑ 3/2 ϱ ϑ4, 2.4 9

10 and sϱ, ϑ = S ϱ + 4a ϑ 3/2 3 ϑ 3 ϱ, 2.5 where S Z = 3 5 P Z P ZZ 3, lim SZ = Z 2 Z Finally, the transport coefficients are continuously differentiable functions of the temperature satisfying µ1 + ϑ Λ µϑ µ1 + ϑ Λ, µ ϑ < c for all ϑ [,, 2 5 < Λ 1, 2.7 κ1 + ϑ 3 κϑ κ1 + ϑ 3 for all ϑ [, Weak and dissipative solutions It was shown in [11, Theorem 3.1] that under the hypotheses , the problem { , 1.1, } admits a global-in-time weak solution for any initial data satisfying ϱ, ϑ L, ϱ, ϑ > a.a. in, u L 2 ; R Moreover, the weak solution {ϱ, ϑ, u} satisfies the relative entropy inequality 1.19, see [12]. Accordingly, a trio {ϱ, ϑ, u} will be called dissipative solution of the Navier-Stokes- Fourier system provided it obeys the relative entropy 1.19 for any choice of smooth functions {r, Θ, U} satisfying Summarizing the results of [11, Chapter 3] and [12] we obtain: Proposition 2.1 Let R 3 be a bounded domain of class C 2+ν. Suppose that the thermodynamic functions p, e, s and the transport coefficients µ, κ obey the structural hypotheses Finally, let the initial data belong to the class specified in 2.9. Then the Navier-Stokes-Fourier system possesses a dissipative solution {ϱ, ϑ, u} on an arbitrary time interval, T that enjoys the following regularity properties: ϱ a.a. in, T, ϱ C[, T ]; L 1 L, T ; L 5/3 L β, T 2.1 1

11 for a certain β > 5 3 ; ϑ > a.a. in, T, ϑ L, T ; L 4 L 2, T ; W 1,2, 2.11 ϑ 3, logϑ L 2, T ; W 1,2 ; 2.12 u L 2, T ; W α ; R 3, α = 8 5 Λ, ϱu C weak, T ; L 5/4 ; R Main result In accordance with the programme delineated in the introductory part, our main goal is to show that for regular data the weak solution remains regular as long as we can control the gradient of the velocity field. More specifically, we will show the following result. Theorem 2.1 Under the hypotheses of Proposition 2.1, let {ϱ, ϑ, u} be a dissipative solution of the Navier-Stokes-Fourier system on the time interval, T belonging to the regularity class , with the regular initial data satisfying 1.2, together with the compatibility conditions Suppose, in addition, that ess sup x ut, L ;R 3 3 < t,t Then {ϱ, ϑ, u} is a classical solution of the Navier-Stokes-Fourier system satisfying in, T. Remark: For technical reasons, we have omitted the effect of the bulk viscosity in the viscous stress Sϑ, x u. The possibility to extend the result to more general forms of the viscous stress is discussed in Section 5. Here, classical solution means that all functions and all derivatives appearing in the equations are continuous in, T, the functions ϱ, ϑ, u, together with their first order derivatives, are continuous in [, T ] and satisfy the initial conditions 1.1 as well as the boundary conditions 1.7,

12 The condition 2.14 is probably not optimal but difficult to relax in the situation when the transport coefficients depend effectively on the temperature. We would like to point out that, unlike the result of Fan et al. [8], which is a blow-up criterion for regular solutions, Theorem 2.1 is a regularity criterion for the weak dissipative solutions of the Navier-Stokes-Fourier system. Moreover, under the hypotheses of Theorem 2.1, the weak dissipative solutions are known to exist globally in time, while a similar result is not available for the system studied in [8]. Further discussion is postponed to Section 5. Although the rest of the paper is essentially devoted to the proof of Theorem 2.1, we will obtain a few other results that may be of independent interest. 3 Local existence and weak-strong uniqueness To begin, it is convenient to introduce a scaled temperature Ξ = Kϑ, where Kϑ = ϑ Accordingly, the thermal energy balance 1.12 reads [ ϱ κϑ [ ] 1 = Kϑ Sϑ, xu : x u Ξ 3.1 First a priori bounds κz dz. ] eϱ, ϑ t Ξ + u x Ξ Ξ 3.1 ϑ [ ϑ Kϑ ] pϱ, ϑ div x u Ξ. ϑ Our goal is to derive a lower and upper bounds on the temperature. We first rewrite 3.1 in the form t Ξ + u x Ξ D Ξ = DAΞ Bdiv x u Ξ, where D = [ ϱ κϑ ] 1 eϱ, ϑ, A = 1 ϑ Kϑ Sϑ, xu : x u, B = ϑκϑ Kϑ pϱ, ϑ ϑ ϱ 1 eϱ, ϑ. ϑ Now, in view of the hypotheses 2.1, 2.4, and 2.8, it is easy to check that the exist constants B, B such that < B Bt, x B for all t, x. 12

13 Applying the standard comparison argument, we therefore deduce that τ Ξτ, inf Ξ, x exp B div x ut, L x dt, τ. 3.2 In order to obtain an upper bound, we need a similar estimate for the density, namely τ inf ϱ x exp div x ut, L x dt 3.3 τ ϱτ, sup ϱ x exp div x ut, L dt x that follows easily from the equation of continuity 1.2. Now, we may use hypotheses to observe that At, A x ut, 2 L ;R 3 3 ; whence Ξτ, 3.4 [ τ τ ] sup Ξ, x exp D A x ut, 2 L x ;R 3 3 dt + exp B div x ut, L dt for τ [, T ], where D depends only on T inf ϱ x, inf Ξ, x, and div x u L x x dt. 3.2 Local existence of strong solutions Rewriting the system of equations 1.2, 1.3, 1.12 in terms of the new unknowns {ϱ, Ξ, u} and taking the a priori bounds 3.2, 3.4 into account, we can apply the local existence result of Valli [28, Theorem A and Remark 3.3]. Going back to the original variables, we obtain: Proposition 3.1 Under the hypotheses of Proposition 2.1, suppose that the initial data {ϱ, ϑ, u } satisfy 1.2, and the compatibility conditions Then there exists a positive time T such that the Navier-Stokes-Fourier system admits a unique strong solution { ϱ, ϑ, ũ} in the class ϱ, ϑ C[, T ]; W 3,2, ũ C[, T ]; W 3,2 ; R 3, 3.5 ϑ L 2, T ; W 4,2, t ϑ L 2, T ; W 2,2,

14 ũ L 2, T ; W 4,2 ; R 3, t ũ L 2, T ; W 2,2 ; R Moreover, there exist scalar functions ϱ, ϱ, ϑ, ϑ, depending solely on inf ϱ, sup ϱ, inf ϑ, sup ϑ, and on T x ũ 2 L ;R 3 3 dt, such that < ϱτ ϱτ, ϱτ, < ϑτ ϑτ, ϑτ 3.8 for any τ [, T ]. 3.3 Weak strong uniqueness We claim that in view of the result [12] and its generalizations obtained in [1], the dissipative solution obtained in Proposition 2.1 coincides with the strong solution of Proposition 3.1 on the time interval [, T ] provided they start from the same initial data 1.2, This deserves some comments since both [12] and [1] deal with classical solutions having all relevant derivatives continuous and bounded in, T. Given the integrability properties of the dissipative solutions stated in and the regularity of the strong solutions , we can easily check that the trio {r = ϱ, Θ = ϑ, U = ũ} can be taken as test functions in the relative entropy inequality Now, following step by step the arguments of [1, Section 6, formula 79] we deduce from 1.19 the estimate [ E ϱ, ϑ, u ϱ, ϑ, ũ ] t=τ t= + c τ τ τ {[ ] + χ 2 t 1 + ϱ + ϱ sϱ, ϑ where, similarly to [1], we have denoted [ u ũ 2 W 1,α ;R 3 + ϑ ϑ 2 χ 1 te ϱ, ϑ, u ϱ, ϑ, ũ dt res [ 1 + u ũ ] h = [h] ess + [h] res = h [h] ess, res W 1,2 + [ ϑ ϑ] [h] ess = Φϱ, ϑh, Φ C c, 2, Φ 1, Φ = 1 in an open neighborhood of a compact K, 2 ] dt 3.9 } dxdt, res 14

15 where K is chosen to contain the range of [ ϱ, ϑ], specifically, The functions χ i, i = 1, 2 are of the form [ ϱt, x, ϑt, x] K for all x, t [, T ]. χ i t = b i t 1 + t ϑ L, T + 2 ϑ L, T + tũ L, T + 2 ũ L, T, where b i are bounded positive functions determined in terms of the amplitude of ϱ, ϑ, ũ and their spatial gradients in [, T ]. Focusing on the most difficult term, we have χ 2 t ϱ sϱ, ϑ [u ũ] res dx χ 2 t [ϱsϱ, ϑ] res L 4/3 u ũ L 4 ;R 3 ε u ũ 2 W 1,α ;R 3 + cεχ2 2t [ϱsϱ, ϑ] res 2 L 4/3 ε u ũ 2 W 1,α ;R 3 + cεχ2 2tE 3/2 ϱ, ϑ, u ϱ, ϑ, ũ for any ε >, where, exactly as in [1, Section 6.1], we have used the structural properties of s and the embedding W 1,α L 4, α = 8 5 Λ, 2 5 < Λ 1. Going back to 3.9 we may infer that [ E ϱ, ϑ, u ϱ, ϑ, ũ ] t=τ t= + c τ where, by virtue of 3.6, 3.7, τ [ u ũ 2 W 1,α ;R 3 + ϑ ϑ 2 χte ϱ, ϑ, u ϱ, ϑ, ũ dt, χ L 1, T. Applying Gronwall s lemma we obtain the following conclusion: W 1,2 ] dt 3.1 Proposition 3.2 Under the hypotheses of Proposition 3.1, let { ϱ, ϑ, ũ} be the local strong solution specified in Proposition 3.1 and {ϱ, ϑ, u} a dissipative solution obtained in Proposition 2.1, emanating from the same initial data. Then ϱ = ϱ, ϑ = ϑ, u = ũ in [, T ]. 15

16 4 Conditional regularity Our goal is to show that the energy norm ϱt, W 3,2 + ϑt, W 3,2 + ũt, W 3,2 ;R 3 associated to a strong solution of the Navier-Stokes-Fourier system remains bounded in [, T ] as long as sup t, T x ũt, L ;R 3 3 G <. 4.1 We remark that we already know that < ϱτ ϱτ, ϱτ, < ϑ ϑτ, ϑ 4.2 for any τ [, T ], see 3.8, where the bounds depend only on G, the initial data, and the length of the time interval. 4.1 Energy bounds, temperature Multiplying equation 3.1 on [ ϱ κ ϑ e ϱ, ϑ ] 1 Ξ, Ξ = K ϑ, ϑ we obtain, after a routine manipulation, ess sup Ξt, W 1,2 t, T + t Ξ 2 + L 2,T ;L 2 Ξ 2 L 2,T ;W 2,2 cb, data. 4.3 Since ϑ is already known to be bounded, the estimates 4.3 transfer to ϑ. Indeed note that D 2 x ϑ = [ K 1 Ξ ] 1 D 2 x Ξ + [ K 1 Ξ ] D x Ξ 2, where, since Ξ satisfies the homogeneous Neumann boundary condition, the term x Ξ may be estimates in terms of the Gagliardo-Nirenberg inequality Consequently, relation 4.3 implies x Ξ 2 L 4 ;R 3 Ξ L Ξ L ess sup ϑt, W 1,2 t, T + 2 t ϑ + 2 ϑ cb, data. 4.5 L 2,T ;L 2 L 2,T ;W 2,2 16

17 4.2 Energy bounds, velocity Our next goal is to deduce similar bounds for the velocity field. To this end, we write the momentum equation in the form t ũ 1 ϱ div xs ϑ, x ũ = ũ x ũ 1 ϱ = h ϱ where, in accordance with the previous estimates p ϱ, ϑ ϑ p ϱ, ϑ x ϱ, ϱ x ϑ + 1 ϱ sup h 1 t, L 2 t [, T ;R 3 cb, data. ] Taking the scalar product of 4.6 with div x S ϑ, x ũ we obtain where, furthermore, where S ϑ, ũ : t x ũ dx + = h ϱ 1 div x S ϱ ϑ, x ũ 2 p ϱ, ϑ x ϱ 4.6 ϱ dx p ϱ, ϑ x ϱ div x S ϱ ϑ, x ũ dx, S ϑ, ũ : t x ũ dx = d µ ϑ dt 4 xũ + t xũ 2 3 div xũi µ ϑ 4 ϑ t xũ + t xũ 2 3 div xũi In view of the estimate 4.5 we conclude that d µ ϑ dt 4 xũ + t xũ 2 3 div 2 xũi dx + = h ϱ 2 2 dx dx. 1 div x S ϱ ϑ, x ũ 2 dx 4.7 p ϱ, ϑ x ϱ div x S ϱ ϑ, x ũ dx + h 2 dx, h 2 L 2, T ;L 2 cb, data. 17

18 4.3 Elliptic estimates In order to exploit 4.7, we have to show that the L 2 -norm of div x S ϑ, x ũ is equivalent to the L 2 -norm of 2 ũ. We have H div x S ϑ, x ũ = µ ϑ [ ũ + 1 ] 3 xdiv x ũ + h 3, where, in accordance with 4.5, h 3 L, T ;L 2 ;R 3 cb, data. Consequently, we may integrate 4.7 with respect to t and use the standard elliptic estimates to obtain τ τ ũ 2 W 2,2 ;R 3 dt cb, data 1 + x ϱ 2 L 2 ;R 3, τ [, T ]. 4.8 Now, we differentiate the equation of continuity with respect to x to obtain t xi ϱ + ũ x xi ϱ = xi ũ x ϱ xi ϱ div x ũ ϱ xi div x ũ; 4.9 whence d x ϱ 2 dx cb, data x ϱ 2 + x ϱ x div x ũ dx. dt 4.1 Combining 4.8, 4.1 with the standard Gronwall type argument we get the following estimates: sup t [, T ] x ϱt, L 2 ;R 3 cb, data Next, due to 4.6, t ũ L 2, T ;L 2 ;R 3 + ũ L 2, T ;W 2,2 ;R 3 cb, data, 4.12 and, finally, by means of the equation of continuity, t ϱ L 2, T ;L 2 cb, data Energy estimates for the time derivatives Differentiating the momentum equation 1.3 with respect to t and setting Ṽ = tũ we obtain ϱ t Ṽ + ũ x Ṽ div x S ϑ, x Ṽ = t ϱṽ + ϱṽ xũ + h 1,

19 with [ µ h 1 = t ϱũ x ũ + div x ϑ t ϑ x ũ + t xũ 2 ] 3 div xũi x t p ϱ, ϑ, where, in accordance with the previous estimates, Seeing that h 1 L 2, T ;W 1,2 ;R 3 cg, data t ϱṽ + ϱṽ xũ V = Ṽ 2 ϱdiv x ũ we may take the scalar product of 4.14 with Ṽ and, integrating over, we deduce the energy estimates for Ṽ = tũ: sup t ũ L 2 ;R 3 + t ũ L 2, T ;W 1,2 ;R 3 cg, data t [, T ] and, according to Sobolev s embedding theorem, Going back to 1.3 we compute t ũ L 2, T ;L 6 ;R 3 cg, data µ ϑ ũ xdiv x ũ = ϱ t ũ + ũ x ũ µ ϑ x ũ + x ũ t 2 3 div xũi x ϑ + x p ϱ, ϑ; which, combined with 4.16 and the previous estimates, gives rise to 4.18 sup ũt, W 2,2 ;R 3 cg, data t [, T ] Next, bootstraping 4.18 via the elliptic regularity, yields τ τ ũt, 2 W 2,6 ;R 3 dt c 1 + x ϱt, 2 L 6 ;R 3 dt. 4.2 Furthermore, multiplying 4.9 on x ϱ 4 x ϱ yields d dt x ϱ 6 L 6 ;R 3 c x ϱ 6 L 6 ;R 3 + ũ W 2,6 ;R 3 x ϱ 5 L 6 ;R Thus, combining 4.2, 4.21, we may infer that [ ] x ϱt, L 6 ;R 3 + t ϱt, L 6 ;R 3 + ũ L 2, T ;W 2,6 ;R 3 cg, data sup t [, T ] 19

20 4.5 L p L q estimates for the temperature Our next goal is to apply the technique of L p L q estimates to the parabolic equation For this purpose, we first need Hölder continuity for ϑ. To this end, let us write equation 1.12 in terms of ẽ ϱ, Ξ = e ϱ, K 1 Ξ. Note that We obtain ϱ t ẽ ϱ, Ξ + ϱũ x ẽ ϱ, Ξ x Ξ = S ϑ, x ũ : x ũ p ϱ, ϑdiv x ũ := h. x Ξ = divx x e e Θ ϱ, Ξ ϱ t ẽ ϱ, Ξ + ϱũ x ẽ ϱ, Ξ div x ϱ ẽ ϱ, div Ξ x Θ ẽ ϱ, Ξ x ϱ. x ẽ Θ ẽ ϱ, Ξ ϱ ẽ ϱ, = h + div Ξ x Θ ẽ ϱ, Ξ x ϱ, which, after dividing on both sides by ϱ, yields t ẽ ϱ, Ξ x ϱ + ũ ϱ 2 Θ ẽ ϱ, Ξ x ẽ ϱ, Ξ x e div x ϱ Θ ẽ ϱ, Ξ = h ϱ + ϱẽ ϱ, Ξ ϱ 2 Θ ẽ ϱ, Ξ x ϱ 2 ϱ ẽ ϱ, + div Ξ x ϱ Θ ẽ ϱ, Ξ x ϱ Note that x ϱ L, T ; L 6, R 3 according to 4.22, we can apply the standard theory of parabolic equations with bounded measurable coefficients to deduce that ẽ ϱ, Ξ is Hölder continuous in [, T ]. See Ladyzhenskaya et al. [18]. Since ϱ is already Hölder continuous in the set [, T ] cf. the estimates 4.22, we find Now we write 1.12 in the following form: Ξhence ϑ is Hölder continuous in [, T ] t Ξ + ũ x Ξ D x Ξ = Dh e ϱ, ϑ with the diffusion coefficient D = ϱ ϑ being Hölder continuous. The L p L q theory for parabolic equations is now applicable yielding ϑ L p, T ; W 2,6, t ϑ L p, T ; L 6 for any 1 < p <

21 see Amann [1], [2], Krylov [16]. Note that the integrability in the spatial variable is limited by the integrability of the initial data. Remark: Applying similar arguments to the momentum equation 1.3, we could obtain analogous estimates for the velocity field: ũ L p, T ; W 2,6 ; R 3, t ũ L p, T ; L 6 ; R 3 for any 1 < p <, 4.27 however, we do not need this refinement in the future analysis. 4.6 Full regularity Differentiating 4.25 with respect to time yields t Ξt D x Ξt = Dh t + D t x Ξ ũ x Ξt := h According to 4.16, 4.22 and 4.27, it is easy to check that h L 2, T ;L 2 Standard energy method and elliptic estimates yield as well as cg, data Ξ t L, T ;L 2 + Ξ t L 2, T ;W 1,2 cg, data, 4.3 Ξ L, T ;W 2,2 + Ξ L 2, T ;W 3,2 cg, data, 4.31 Ξ t L, T ;W 1,2 + Ξ tt L 2, T ;L 2 cg, data, 4.32 Ξ L, T ;W 3,2 + Ξ t L 2, T ;W 2,2 cg, data, 4.33 Ξ L 2, T ;W 4,2 cg, data Similar estimates for ϑ hold. With these estimates in hand, we can go back to equation 4.14 for Ṽ = tũ and in a similar way to conclude As for the final estimates ũ t L, T ;W 1,2,R 3 + ũ tt L 2, T ;L 2,R 3 cg, data, 4.35 ũ L, T ;W 3,2,R 3 + ũ t L 2, T ;W 2,2,R 3 cg, data ũ L 2, T ;W 4,2,R 3 + ϱ L, T ;W 3,2 cg, data, 4.37 one needs to combine with the transport equation. The treatment is similar to Section 4.3 and we omit the details. Summarizing the previous considerations, we are allowed to state the following result. 21

22 Proposition 4.1 Let T > be given. Suppose that { ϱ, ϑ, ũ} is a strong solution of the Navier-Stokes-Fourier on the time interval [, T ], T T, the existence of which is claimed in Proposition 3.1. Assume, in addition, that Then sup t [, T ] sup x ũt, L ;R 3 G t [, T ] [ ϱt, W 3,2 + ϑt, W 3,2 + ũt, W 3,2 ;R 3 In particular, if T is the maximal existence time, then T = T. 5 Conclusion ] cg, data, T To begin, the standard parabolic regularity theory implies that the solutions belonging to the class specified in Proposition 3.1 are, in fact, classical solutions, meaning all relevant derivatives appearing in the system are continuous in the open set, T, the fields ϑ, ϱ are continuously differentiable up to the boundary, and the boundary conditions 1.7, 1.8 hold, see Matsumura and Nishida [21], [22]. Consequently, combining Proposition 3.1 with Proposition 4.1 we obtain the conclusion of Theorem 2.1. We conclude the paper by comments concerning the hypotheses of Theorem 2.1. In the light of the results obtained by Fan et al. [8], the optimal regularity criterion is expected to be the bound T x u L ;R 3 3 dt <. Note, however, that, unlike [8], we have to handle the transport coefficients that depend effectively on the temperature. As a matter of fact, the hypothesis 2.14 is only use to deduce the estimate It is also plausible to expect that 2.14 could be relaxed to T x u p L ;R 3 3 dt < for a suitable exponent p. Still the fact that we have to handle the more complex constitutive relations than in [8] makes the problem rather difficult. The regularity of the strong solutions seems optimal, at least in view of the result [12]. Indeed the time derivative t ũ of the strong solutions should be at least of class L 2, T ; L ; R 3 for the technique of [12] to be applicable. 22

23 The next remark concerns the possibility to include the more general class of Newtonian stress, namely Sϑ, x u = µϑ x u + t xu 2 3 div xui + ηϑdiv x ui, where we have added the bulk viscosity, with the coefficient η = ηϑ. Clearly, we can show the same result if ηϑ µϑ = A a constant. Moreover, our method still applies if ηϑ µϑ small, where small is determined in terms of the optimal constant λ in the elliptic estimate v div xvi λ v W 2,2 ;R 3, v =. L 2 ;R 3 3 The general case may be handled in the following way: 1. After establishing the uniform bounds in Section 3.1, apply the theory of Krylov and Safonov [17] for parabolic equation in non-divergence form to equation 3.1 to obtain uniform bounds on ϑ in the Hölderian norm. 2. Use the regularity theory for elliptic systems with Hölder coefficients in Section 4.3. Finally, we remark that the methods of the present paper could be extended to other types of boundary conditions of the Navier type for the velocity and to more general classes of domains including certain unbounded domains like exterior domains or a half-space. References [1] H. Amann. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In Function spaces, differential operators and nonlinear analysis Friedrichroda, 1992, volume 133 of Teubner-Texte Math., pages Teubner, Stuttgart,

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