A bound from below for the temperature in compressible Navier-Stokes equations

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1 A bound from below for the temperature in compressible Navier-Stokes equations Antoine Mellet, Alexis Vasseur September 5, 7 Abstract: We consider the full system of compressible Navier-Stokes equations for heat conducting fluid. We show that the temperature is uniformly positive for t t (for any t > ) for any solutions with finite initial entropy. The assumptions on the viscosity and conductivity coefficients are minimal (for instance, the solutions constructed by E. Feireisl in ] verify all the requirements). Introduction and main result In this article, we consider ρ(t, x), θ(t, x) and u(t, x) R 3 solutions of the following system of equations: t ρ div(ρ u) =, () c v t (ρθ) c v div(ρu θ) Rρ θdivu = µ D(u) λ divu div(κ θ) () for (t, x) R. Those equations are the usual continuity and temperature equations of the full system of compressible Navier-Stokes equations (for heat conducting fluids) when the specific heat at constant volume c v is assumed to be constant and for a pressure law of the form P (ρ, θ) = p e (ρ) Rρθ. We recall that ρ denotes the density of the fuid, u its velocity field and θ its temperature. The strain tensor in ()-(4) is given by D(u) = u t u], University of British Columbia, Department of Mathematics, 984 Mathematics Road, Vancouver, BC V6T Z University of Texas at Austin, department of Mathematics, University Station C, Austin, TX 787

2 and µ and λ are the two Lamé viscosity coefficients (possibly depending on the temperature θ or on the density ρ) satisfying µ > µ 3λ (3) (µ is sometime called the shear viscosity of the fluid, while λ is usually referred to as the second viscosity coefficient). The coefficient κ in () is the heat conductivity coefficient. Without loss of generality, we can take c v = (by rescaling the other coefficients). Those equations are usually supplemented by the momentum equation: t (ρu) div(ρu u) P (ρ, θ) div(µd(u)) (λdivu) =. (4) Altogether, Equations (), () and (4) describe the motion of a general viscous, heat conducting and compressible fluid. We want to emphasis however that for the purpose of this paper we will not need equation (4). The equations () and () are set in a smooth bounded subset of R 3, and we consider the following boundary conditions: u(t, x) =, θ(t, x) n(x) = on (5) where n denotes the exterior normal vector to. The existence of solutions for the full system of Navier-Stokes equations for compressible fluids (), () and (4) is a very delicate problem which has been addressed in particular by E. Feireisl in, 3]. We will not discuss this issue in this paper. Instead, we are interested in the properties of the temperature θ(t, x) (assuming that it exists). More precisely, we will show that under reasonable assumptions on the initial data (which imply in particular that the initial temperature may not vanish on an open set), the temperature cannot vanish for positive time and is uniformly bounded away from zero on any time interval t, T ] with T > t >. The proof is inspired by De Giorgi s proof of Hölder regularity for the solutions of elliptic equation with discontinuous coefficients. Similar arguments was used in the context of Reaction-Diffusion systems by Alikakos ]. In fluid mechanics, the method is used in 5] to obtain partial regularity results for incompressible Navier- Stokes system of equations, and in 4] to obtain L bounds on the velocity for the isentropic compressible Navier-Stokes system of equations. The method relies mainly on the energy (or entropy) inequality for such equations. We now have to make precise the notion of solutions we are going to use. As it turns out, the proof only makes use of the fact that solutions of ()-() satisfy (at least formally) the following inequality: d dt ρφ(θ) dx µφ (θ) D(u) dx λφ (θ) divu dx κφ (θ) θ dx R ρθφ (θ)divu dx (6)

3 for some appropriate functions φ. We thus give the following definition: Definition We say that (ρ, u, θ) is an admissible solution of ()-() if the following inequality holds t t ρφ(θ)(t, x) dx µφ (θ) D(u) dx dτ λφ (θ) divu dx dτ t s t s κφ (θ) θ dx dτ R ρθφ (θ)divu dx dτ ρφ(θ)(s, x) dx, s for any function φ of the form ( )] C φ(θ) = ln θ ε where C and ε are two positive constants. s = ln C ln(θ ε)], We stress out the fact that (ρ, u, θ) do not need to satisfy ()-() even in some very weak sense to be admissible in the sense of this definition. In section, we will prove that smooth solutions of ()-() satisfies (7). More importantly, we will check that the method can be applied to the variational solutions constructed by E. Feireisl in ]. Strictly speaking, those variational solutions are not known to satisfy inequalities (7). But in ] (Equation (7.97) page 85), it is shown that there exists an approximated family of solution (ρ δ, u δ, θ δ ) satisfying: t (ρ δ δ)φ(θ δ )(t, x) dx ( δ)µφ (θ δ ) D(u δ ) dx dτ t s t λ( δ)φ (θ δ ) divu δ dx dτ κφ (θ δ ) θ δ dx dτ s t s (8) R ρθ δ φ (θ δ )divu δ dx dτ (δ ρ δ )φ(θ δ )(s, x) s t δ θ α δ φ (θ δ ) dx dτ, s for a given α and any function φ of the form ( )] C φ(θ) = ln = ln C ln(θ ε)], θ ε where C and ε are two positive constants independent on δ. This inequality is even better than (7) (since there is no problem close to the vacuum ρ = ), except for the damping term of the temperature of the form δθ α. We will see in the last section of this paper that this damping term can be dealt with and thus that our result holds for Feireisl s solutions. The important point will be that we obtain an estimate for θ δ which is independent of δ, and is thus satisfied by lim δ θ δ. We can now state our main result: (7) 3

4 Theorem Assume that the coefficients µ, λ and κ (which may depend on ρ or θ) satisfies µ 3λ ν(θ) > for all θ, (9) ν(θ) Cθ for small θ, () κ(θ) κ > for small θ, () for some function ν(θ) and constant κ. Let (ρ, u, θ) be an admissible solution of ()-() such that and ρ L (, T ; L p ()) for some p > 3, u L (, T ; H ()). Assume moreover that (ρ, θ ) satisfy ρ ln(/θ )] dx <. () Then, for every T > t >, there exists a constant δ t,t > such that: θ(s, x) δ t,t for all s (t, T ) and almost all x. Furthermore, if there exists a constant δ > such that θ (x) δ for almost every x, then the constant δ t,t does not depend on t. We will see in the next section that the assumptions on (ρ, u, θ) are satisfied by the variational solutions constructed by Feireisl 3, ] when the pressure law is given by: P (ρ, θ) = Cρ γ Rρ θ, with γ > 3. (3) This leads to the following corollary: Corollary Let (ρ, u, θ) be a solution a la Feireisl of the Navier-Stokes system of equations ()-(4). Assume that µ, λ and κ satisfy (9)-() and that the pressure law is given by (3). If the initial data (ρ, u, θ ) has finite energy and finite physical entropy, then for every T > t >, there exists a constant δ t,t > such that: θ(s, x) δ t,t, for every s (t, T ) and almost every x. The paper is organized as follows: In section, we check that smooth integrable solutions of ()-(), as well as Feireisl s solutions are admissible in the sense of Definition. Section 3 is devoted to the proof of Theorem. In the last section, we complete the proof of Corollary. 4

5 Admissible solutions At least formally, the notion of admissible solutions introduced by Definition is consistent with equations (), (): As a matter of fact, a simple computation shows that if (ρ, θ) is a smooth solution of ()-() then for any function φ we have: t (ρφ(θ)) div(ρuφ(θ)) = µφ (θ) D(u) λφ (θ) divu Rρθφ (θ)divu div(κ φ(θ)) φ (θ)κ θ. Integrating with respect to x (assuming that all integrations are valid) and using the boundary conditions (5), we immediately deduce that (7) holds (with equality) for any smooth function φ. However, the existence of smooth solutions satisfying the necessary integrability conditions is not known. It is therefore important to check that admissible solutions actually exist. The system of equations ()-() has been studied in great details by E. Feireisl in, 3]. We recall the following result (see ]): For all δ > and for some α >, there exists (ρ, u, θ) solution of t ρ div(ρ u) =, (4) c v t (ρθ) c v div(ρu θ)rρ θdivu δθ α = µ D(u) λ divu div(κ θ)(5) such that the following inequality holds (see ], Equation (7.97) p. 85): t (ρ δ)q h (θ)(t, x) dx ( δ) µq h(θ) D(u) dx dτ s t ( δ) λq h(θ) divu dx dτ s t t κq h(θ) θ dx dτ δq h(θ)θ α dx dτ s s t R ρθq h(θ)divu dx dτ (ρ δ)q h (θ)(s, x) dx, s (6) where Q h (θ) = θ h(z) dz with h non-increasing function in C (, )) such that h() <, lim z h(z) = and h (z) h(z) (h (z)) for all z. (7) In particular, the function h(z) = zε zε C satisfies all the conditions (including (7)), so we deduce that (6) holds with { ln(θ ε) ln(ε) if θ ε C Q h (θ) = ln(c) ln(ε) if θ ε C 5

6 Taking φ(θ) = Q h (θ) ln(c) ln(ε) and using the fact that d dt ρ(t, x) dx =, we easily see that (ρ, u, θ) satisfies t (ρ δ)φ(θ)(t, x) dx ( δ)µφ (θ) D(u) dx dτ s t ( δ)λφ (θ) divu dx dτ s t t (8) κφ (θ) θ dx dτ δφ (θ)θ α dx dτ s s t R ρθφ (θ)divu dx dτ (ρ δ)φ(θ)(s, x) dx, for any function φ of the form ( )] C φ(θ) = ln θ ε s = ln C ln(θ ε)]. where C and ε are two positive constants. We stress out the fact that (5) and (8) contain an additional damping term δθ α and thus that the solution of Feireisl is not exactly admissible in the sense of Definition. For the sake of clarity, however, we prove our main result without the damping term, and we show in the last section that this term can be handled without additional difficulties. 3 Proof of Theorem 3. Setting of the problem and technical lemmas The proof of Theorem is inspired by De Giorgi s method for the regularity of the solutions of elliptic equations and relies crucially on inequality (7). We begin by introducing the sequence of real numbers = e M k], k N, where M is a positive number to be chosen later (note that lim k = e M ). We then define the sequences of functions (φ ) k N by ( )] Ck φ (θ) = ln θ ε. The starting point of the proof is the following Lemma: Lemma 3 Let (T k ) be a sequence nonnegative numbers and assume that (ρ, u, θ) is an admissible solution of ()-(). Define ( ) T ν(θ) U := sup ρφ (θ) dx T k t T θ ε {θε } D(u) dx dt T T k T k κ(θ) (θ ε) {θε } θ dx dt. 6

7 Then the following inequalities holds for all k N: If T k = for all k N, then: T U R ρ θ θ ε {θε } divu dx dt If (T k ) k N is an increasing sequence of positive numbers, then: T U R ρ θ θ ε {θε } divu dx dt T k Tk Note that the right hand side of (9) is bounded by T R ρ divu dx dt ρ ln(/θ )] dx, ρ φ (θ ) dx. (9) ρφ (θ) dx dt. () and so the assumptions on ρ, u, ρ and θ in Theorem give the existence of a constant C independent of ε and k such that U C. () Theorem will be proved by showing that for suitable choices of constant M (independent on ε) and sequence (T k ) k N, the sequence (U ) k N goes to zero as k goes to infinity. Using the definition of U, this will imply that θ(t, x) ε e M and the main theorem will follow since this inequality holds for any ε >. In order to establish the convergence to zero of U, we will prove in the next sections that both terms in the right hand side of () can be controlled by U γ,ε for some γ >. Proof of Lemma 3. The lemma follows directly from (7) (with C = ). We note that: φ (θ) = θ ε {θε } φ (θ) (θ ε) {θε }. Integrating (7) with respect to time, we deduce that for all σ, t such that σ T k t T, we have t ν(θ) ρφ (θ)(t, x) dx σ θ ε {θε } D(u) dx ds t κ σ (θ ε) {θε } θ dx ds t R ρ θ θ ε {θε } divu dx ds ρφ (θ)(σ, x) dx. σ 7

8 and thus ρφ (θ)(t, x) dx t T k t T k ν(θ) θ ε {θε } D(u) dx ds κ (θ ε) {θε } θ dx ds T R ρ θ θ ε {θε } divu dx ds ρφ (θ)(σ, x) dx. If T k = for all k, then taking σ = and the supremum over t T k, T ], we deduce (9). If (T k ) k N is an increasing sequence of positive numbers, then taking the mean value with respect to σ in, T k ], we get: ρφ (θ)(t, x) dx t T k t T k ν(θ) θ ε {θε } D(u) dx ds κ (θ ε) {θε } θ dx ds T R ρ θ θ ε {θε } divu dx ds Tk ρφ (θ)(s, x) dx ds. T k Finally, taking the supremum over t T k, T ], we deduce (). In the next sections, we will attempt to control the two terms in the right hand side of (9) and () by some power of U,ε. For that purpose, we need the following technical lemma: Lemma 4 There exists a constant C (depending only on, T, ρ (x) dx and ρ L (,T,L 3 ())) such that, for every function F with ρf L (, T ; L ()) and F L (, T ; L ()), we have: F L (,T ;L 6 ()) C( ρf L (,T ;L ()) F L (,T ;L ())). Proof. First note that there exists two constants ε, C > (independent of t) such that (t) = {x ρ(t, x) > ε} C. Indeed, we have: ρ dx = ρ(t, x) dx ε ρ L (L 3 ) (t) /3, and so, for ε small enough, we can take: ( C = ρ ) 3/ dx ε. ρ L (L 3 ) 8

9 Now, for any fixed time t, we write: F (t) L 6 () F (t) F (t, y) dy L 6 F (t) F (t, y) dy L 6 5/6 F (t, x) dx (t) (t) 5/6 (t) (t) /6 F (t, y) dy F (t, x) dx F (t, y) dy The first term in the right hand side is bounded by F L () (which is in L (, T ) by definition) and by definition of (t), the second term is bounded by: /6 εc ρf (t, x) dx, which lies in L (, T ), and therefore in L (, T ). Finally, the last term can be written in the following way: F (t, x) dx F (t, y) dy (t) (t) F (t, x) F (t, y)] dx dy C (t) F (t, x) F (t, y) dx dy, C which is classically bounded by F L (). We can now prove the technical lemma which will be essential in the next sections: Lemma 5 If p, q, α and β satisfy q = 6 5α β > and p = >, () β α then there exist a constant C depending on the dimension, T, ρ (x) dx > and ρ L (,T,L 3 ()) such that: ρ α φ (θ) β L p L q CU βα CU β, where L p L q denotes the space L p (T k, T ; L q ()) 9

10 Proof. Before starting the proof, we observe that () yields: κ (θ ε) {θε } θ dx κ φ (θ) dx and thus Now, we write φ (θ) L ((T k,t ) ) CU. (3) ρ α φ (θ) β L p L q ρ α/β φ (θ) β L p β L q β (ρφ (θ)) α/β φ (θ) α/β β L p β L q β ( ) β (ρφ (θ)) α/β L L β/α φ (θ) α/β L α/β L α/β 6 (4) whenever the coefficients α, β, p and q are such that: q β = α β α/β, 6 p β = α/β. Note that those conditions are equivalent to (). So in order to conclude, we only need to control the two norms in the right hand side of (4). For the first one, using the definition of U, it is readily seen that Next, we note that (ρφ (θ)) α/β L L β/α U α/β. (5) φ (θ) α/β L α/β L 6 α/β and Lemma 4 yields: = T ( T k /6 φ (θ) dx) 6 dt = φ (θ) α/β L (T k,t,l 6 ()) ] α/β φ (θ) L (T k,t,l 6 ()) C ( ρφ (θ) L (T k,t ;L ()) φ (θ) L ((T k,t ) )). and so φ (θ) L (T k,t,l 6 ()) C φ (θ) L ((T k,t ) ) C ρ φ (θ) dx. Finally (3) and the definition of U leads to φ (θ) L (T k,t,l 6 ()) CU CU. (6)

11 We deduce φ (θ) α/β L α/β L 6 α/β CU α/β CU α/β. (7) We can now conclude, since (4), together with (5) and (7), yields: ρ α φ (θ) β L p L q ( ( C U α β U α/β )) β U α/β CU βα CU β. 3. Control of the pressure term The next lemma shows that the first term in the right hand side of (9) and () can be controlled by U,ε in a nonlinear fashion: Lemma 6 If ρ L (, T ; L p ()) for some p > 3, then there exists γ > and β (, ) such that: T R ρ θ θ ε {θε } divu dx dt C T ν(θ) θ ε {θε } D(u) dx dt ln C ] β U γ,ε C. k Proof. Cauchy-Schwarz and Young s inequality yield: T R ρ θ θ ε {θε } divu dx dt T T C C T ν(θ) θ ε {θε } D(u) dx dt θ ν(θ) {θε }ρ dx dt T ν(θ) θ ε {θε } D(u) dx dt {θε Ck }ρ dx dt, where we have used Hypothesis () in the last inequality. Next, we note that when θ ε, we have φ,ε (θ) ln C. It follows that {θε Ck } ln C ] β φ,ε(θ) β

12 for any β >. Using Lemma 5, we deduce: T {θε Ck }ρ dx dt ln C ] β T ρ α ρ α φ,ε (θ) β dx dt ln C ] β ρ α C L p L q ρα φ,ε (θ) β L p (L q ) k ln C ] β ( ) ρ α C L p U αβ L q,ε U β,ε, k with α, β, p and q satisfying () and p, q the conjugate exponents. Lemma 6 follows if we can show that we can choose α, β, p and q satisfying () such that γ = min( α β, β) > and ρ α L p C. L q First, we note that if (α β)/ >, then () implies: p < α, 3 q < α (which requires α < and β > ). The corresponding conditions for the conjugate exponents read: Therefore we need to have ρ α L p = L q T p > α q 3 > ( α). ( ) ( α) ρ 3( α) 3 ( α) dx α α dt < C, for some α (, ). This is satisfied for some α close enough to since ρ L (, T ; L p ()) for some p > Proof of Theorem We are now ready to complete the proof of Theorem. Lemmas 3 and 6 yield: U C ln C ] β U γ,ε C ρ φ (θ ) dx, k (8) k

13 if T k = for all k, and U C ln C ] β U γ C Tk ρφ (θ) dx dt k (9) k T k if (T k ) k N is an increasing sequence of positive numbers. To complete the proof, we will use (8) in the case when the initial datum is bounded away from zero and (9) in the general case Case without initial layer In this subsection we assume that the initial datum verifies: θ (x) δ >. In that case, we take T k = for every k and we choose M such that e M/ < δ. This implies in particular that φ (θ ) = k N, for any ε >. Inequality (8) thus becomes U C ln C ] β U γ,ε C, k for some γ >. Moreover our choice of constants yield: C = e M k, and thus ln C ] β = kβ M β. We deduce: U β M β U γ,ε k N. Since γ >, it is a classical result that for M large enough (depending only on U and thus independent on ε), we have: In particular, this yields T lim U =. k ] κ(θ) ln e M θ ε dx dt =, 3

14 and ] ρ ln e M θ ε The first equality gives (using ()) that ln e M θε dx =. ] is constant in for all t, and the second equality implies that this constant is (unless ρ(t, ) = which contradicts the conservation of mass). We deduce that for almost every t, T ] and x R 3 we have θ(t, x) ε e M. Since this inequality holds for any ε >, the theorem follows Case with the time layer We now remove the assumption that the initial temperature is bounded away from zero and only assume that the initial entropy is finite, i.e. R 3 ρln(/θ )] dx <, which is enough to guarantee (see ()) that U,ε C < with C independent on ε. Let t be a fixed positive number and set T k = t ( k ). The only thing left to do is to show that we can control the second term in the right hand side of (9). For that, we note that we have (using (6)): ] /6 Tk ρφ (θ) dx dt T φ (θ) 6 dx 5/6 ρ 6/5 {θε Ck }] dt ] / T 5/3 φ (θ) L L 6 ρ 6/5 {θε Ck }] dt ( ) T 5/3 / U / U ρ 6/5 {θε Ck }] dt]. Moreover, proceeding as in the proof of Lemma 6, we check that when θ ε, we have φ,ε (θ) ln C and so {θε Ck } ln C ] α φ,ε(θ) α, for any α >. We deduce Tk ρφ (θ) dx dt ( U / U ) ln C ] 5 6 α T 4 5/3 / ρ 6/5 φ,ε (θ) dx] dt] α.

15 The last term can be bounded as follows: ( 5/3 ( ) 5 ρ 6/5 φ,ε (θ) dx) α ρ 6/5 α α dx 3 ( α) ( ρφ,ε (θ) dx) 5α/3. If we take α > 3/5, we deduce that there exists positive numbers β, β, and γ > such that: Tk ρφ (θ) dx dt ln C ] β ρ β L T k T k C L puγ,ε. k (3) with p = 6/5 α α. Note that for α close to 3/5, we have p close to 3/ and so ρ L L p < C Finally, recalling that = e M k] and T k = t ( k ), we get: ln C ] = k M and so (9) and (3) yield and U C βk M β U γ,ε C β k t M U γ β T k = k t,ε. The proof can now be completed as in the case without time initial layer. 4 Proof of Corollary In this last section, we show how to deal with the temperature damping term which is necessary in Feireisl s result to prove the existence of admissible solutions. The important point is to get estimates that are independent of δ, so that the result holds when passing to the limit δ. We only treat the case without initial layer (the general case is left to the reader). We thus have φ (θ ) = and we can take T k = for all k N. Let (ρ δ, u δ, θ δ ) be the solution of (4), (5) satisfying (6). We define: U,δ := sup t T ( (ρ δ δ)φ (θ δ ) dx T T k ) T ν(θ δ ) θ δ ε {θ δ ε } D(u δ ) dx dt κ(θ δ ) (θ δ ε) {θ δ ε } θ δ dx dt. Using (8), it is readily seen that Lemma 3 with the additional damping term becomes: T θ T δ U,δ R ρ δ θ δ ε θ α δ {θ δ ε } divu δ dx dt δ θ δ ε {θ δ ε } dx dt. 5

16 Together with Lemma 6, this implies: U,δ β M β U γ,ε,δ δ T θ α δ θ δ ε {θ δ ε } dx dt. (3) In order to control the last term in the right hand side, we note that when θ δ ε, we have (provided that M is large enough, so that C ): θ α δ θ δ ε for all ε >. Next, we note that when θ δ ε, we have φ,ε (θ δ ) ln C It follows that {θδ ε } ln C ] β φ,ε(θ δ ) β for any β >. Using Lemma 5 with ρ replaced by ρ δ, we deduce: T δ {θδ ε } dx dt δ α ln C ] β T (ρ δ) α φ,ε (θ δ ) β dx dt δ α ln C Cδ α ln C ] β T /p /q ] β (. (ρ δ) α φ,ε (θ δ ) β L p L q ) U αβ,ε,δ U β,ε,δ, with α, β, p and q satisfying () and p, q the conjugate exponents, with p > and q > close to. As in Section 3., we can choose α < and β > such that γ = min( α β, β) > if we take p and q close enough to. Since δ α, we deduce U,δ β M β U γ,ε,δ where all the constants are independent of δ (and ε). As in the previous section, this implies that there exists a constant η T > (independent on δ) such that: θ δ (t, x) η T, t T, x. Passing to the limit with respect to δ, this shows that Feireisl s solution to (), (), (4) satisfies θ η T > in, T ]. References ] N. D. Alikakos. L p bounds of solutions of reaction-diffusion equations. Comm. Partial Differential Equations, 4(8):87 868,

17 ] Eduard Feireisl. Dynamics of viscous compressible fluids, volume 6 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 4. 3] Eduard Feireisl. On the motion of a viscous, compressible, and heat conducting fluid. Indiana Univ. Math. J., 53(6):75 738, 4. 4] A. Mellet and A. Vasseur. L p estimates for quantities advected by a compressible flow. Preprint, 6. 5] Alexis Vasseur. A new proof of partial regularity of solutions to Navier-Stokes equations. To appear in NoDEA. 7