WEAKLY NONLINEAR-DISSIPATIVE APPROXIMATIONS OF HYPERBOLIC-PARABOLIC SYSTEMS WITH ENTROPY

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1 WEAKLY NONLINEAR-DISSIPATIVE APPROXIMATIONS OF HYPERBOLIC-PARABOLIC SYSTEMS WITH ENTROPY NING JIANG AND C. DAVID LEVERMORE Abstract. Hyperbolic-parabolic systems have spatially homogenous stationary states. When the dissipation is weak, one can derive weakly nonlinear-dissipative approximations that govern perturbations of these constant states. These approximations are quadratically nonlinear. When the original system has an entropy, the approximation is formally dissipative in a natural Hilbert space. We show that when the approximation is strictly dissipative it has global weak solutions for all initial data in that Hilbert space. We also prove a weak-strong uniqueness theorem for it. In addition, we give a Kawashima type criterion for this approximation to be strictly dissipative. We apply the theory to the compressible Navier-Stokes system.. Introduction We consider hyperbolic-parabolic systems over the 2π-periodic domain T d that have the form. t U + x F U = x [DU x U ]. These have spatially homogenous stationary solutions. When the dissipation is weak, one can derive a weakly nonlinear-dissipative approximation that governs perturbations Ũ about any constant solution U o. These approximations have the form.2 t Ũ + AŨ + QŨ, Ũ = DŨ, where A = F U U o x is the linearization of the convection operator about U o, while quadratic operator Q and the linear operator D are formally given by.3 QY, Y = lim T 2T DY = lim T 2T e ta x [ 2 F UUU o e ta Y, e ta Y ] dt, e ta DU o : 2 x e ta Y dt. Such weakly nonlinear-dissipative approximations arise when studying incompressible limits of the compressible Navier-Stokes system [8, 32], global regularity of fast rotating Navier-Stokes and Euler equations [, 2, 3], asymptotic limits in equations of geophysical fluid dynamics [, 2], and fast singular limits of hyperbolic and parabolic PDE s [35, 4, 5]. We show that if the original system. has a thrice differentiable convex entropy structure then the approximating system.2 is formally dissipative in the Hilbert space H whose inner product is given by Ũ Ṽ = H H 2π d UU U o Ũx, Ṽ x dx, T d Date: April 6, 29.

2 2 N. JIANG AND C. D. LEVERMORE where H UU U o is the Hessian of the strictly convex entropy density HU at U o. This dissipation property follows because the entropy structure implies that A is skew-adjoint in H, that D is nonpositive definite in H, and that Q formally satisfies the cyclic identity.4 = Ũ Q Ṽ, W H + Ṽ Q W, Ũ H + W Q Ũ, Ṽ H. We show that if D is also strictly dissipative then the approximating system.2 has a Leraytype global weak solution for all initial data in H. We cannot establish the uniqueness of these solutions. Indeed, when. is the Navier-Stokes system of gas dynamics then.2 includes the incompressible Navier-Stokes system as a subsystem. The uniqueness question therefore includes the uniqueness question for the incompressible Navier-Stokes system. We can however use the cyclic identity.4 to prove a so-called weak-strong uniqueness theorem for the approximating system.2. It will be easily seen that D will be nonnegative definite if and only if the linear operators A = F U U o x and D = DU o : x 2 satisfy the Kawashima condition:.5 no nonconstant eigenfunction of A is in the null space of D. In one spatial dimension it is known that the Kawashima condition implies that D is strictly dissipative [25]. We give a stronger Kawashima-type criterion for D to be strictly dissipative in higher dimensions. Our paper is laid out as follows. Section 2 presents the entropy structure we will impose on system.. Section 3 presents the weakly nonlinear-dissipative approximation.2 and its relation to the Kawashima condition.5. Section 4 develops the properties of the averaged operators.3 that we will need later. This includes a proof of the cyclic identity.4. Section 5 presents our Kawashima-type criterion for D to be strictly dissipative. Section 6 contains our existence and weak-strong uniqueness theorems. Finally, section 7 applies the theory to the compressible Navier-Stokes system of gas dynamics. 2. Hyperbolic-Parabolic Systems with Entropy We consider hyperbolic-parabolic systems over T d in the divergence form 2. t U + x F U = x [DU x U ], where Ux, t is a density vector over x, t T d R + that takes values in U c R N. Here T d = R d /2πZ d is the 2π-periodic torus and U c is the closure of a convex domain U R N. We assume that the flux tensor F : U R d N is twice continuously differentiable such that 2.2 t U + x F U = is hyperbolic, while the diffusion tensor D : U R d d N N is continuously differentiable such that 2.3 t U = x [DU x U] is parabolic. Recall that system 2.2 is said to be hyperbolic if for every U U and every ξ R d the N N matrix F U U ξ is diagonalizable within the reals i.e. it has a complete set of real eigenvectors. System 2.2 is said to be strictly hyperbolic if moreover the eigenvalues of F U U ξ are distinct. Recall that system 2.3 is said to be parabolic if for every U U and every ξ R d the N N matrix DU : ξ 2 is diagonalizable within the reals and has nonnegative eigenvalues. System 2.3 is said to be strictly parabolic if moreover the eigenvalues of DU : ξ 2 are positive. Many studies of hyperbolic-parabolic systems assume that system 2.2 is strictly hyperbolic while system 2.3 is strictly parabolic. We will not do that here. Rather, we

3 WEAKLY NONLINEAR-DISSIPATIVE APPROXIMATIONS 3 will assume that system 2. has a strictly convex entropy and satisfies certain nonsingularity conditions. 2.. Entropy Structure. We say that H : U R is a strictly convex entropy for the system 2. when H is twice continuously differentiable over U and for every U U 2.4 i H UU U is positive definite, ii H UU U F U U ξ is symmetric for every ξ R d, iii H UU U DU is symmetric and nonnegative definite. The existence of such a strictly convex entropy implies that system 2. is hyperbolic-parabolic. The compressible Navier-Stokes system is a hyperbolic-parabolic system that is neither strictly hyperbolic nor strictly parabolic, yet has a strictly convex entropy. We will study this example in Section 7. There are many other systems from physics that fit into this framework [5]. Condition ii in 2.4 implies there exists J : U R d that is twice continuously differentiable such that 2.5 H U UF U U = J U U. It follows that if Ux, t is a classical solution of 2. that takes its values in U then it satisfies 2.6 t HU + x JU = x [H U U DU x U ] x H U U DU x U. Condition iii in 2.4 implies that for every differentiable U : T d R N with Ux U 2.7 x H U U DU x U = x U T H UU UDU x U. We thereby see that 2.6 is a local dissipation law for HU. When 2.6 is integrated over T d one obtains the global dissipation law d HU dx = x H U U DU x U dx dt 2.8 = x U T H UU UDU x U dx Nonsingularity Condition. The class of stationary classical solutions of 2. that take values in U is constrained by the entropy structure. It follows from 2.8 that every such solution satisfies x U T H UU UDU x U dx =, which by 2.7 implies that x U T H UU UDU x U =. Conditions i and iii then imply that DU x U =, which when plugged into 2. with t U = yields x F U =. We thereby see that every classical solution of 2. that takes its values in U satisfies 2.9 x F U =, DU x U =. In general this is not enough information to conclude that U is a constant.

4 4 N. JIANG AND C. D. LEVERMORE We consider hyperbolic-parabolic systems over T d that are nonsingular in the sense that for every U U and every continuously differentiable Ũ : Td R N one has that } F U U x Ũ = 2. = x Ũ =. DU x Ũ = With this additional assumption we can prove the following. Lemma 2.. If the hyperbolic-parabolic system 2. has a strictly convex entropy 2.4 and satisfies the nonsingularity condition 2. then its only stationary, spatially periodic classical solutions that take their values in U are constant solutions. In the next section we will strengthen the nonsingularity condition Weakly Nonlinear-Dissipative Approximations We now consider weakly nonlinear-dissipative WND approximations to hyperbolic-parabolic systems of the form 2.. In order to see how these approximations depend on the choice of dependent variables, we express 2. in the form 3. t UW + x F UW = x [DUW x UW ]. Here the mapping U : W U is assumed to be a twice continuously differentiable bijection and have a nonsingular Jacobian. This means that its inverse mapping U : U W will also be a twice continuously differentiable bijection and have a nonsingular Jacobian. 3.. Linearization and the Kawashima Condition. In order to motivate our approximation, let us first consider the linearization of the hyperbolic-parabolic system 3. about some constant state W o W. This is 3.2 t W + Ao x W = Bo : 2 x W, where A o R d N N, and B o R d d N N are defined by 3.3 A o = R o F U U o R o, B o = R o DU o R o, with U o U and R o R N N given by 3.4 U o = UW o, R o = V UW o. If H : U R is a strictly convex entropy for system 2. then system 3.2 is symmetrized by the positive definite matrix 3.5 G o = R T o H UU U o R o. Specifically, one sees from 2.4 and 3.3 that G o A o ξ = R T o H UU U o F U U o ξr o is symmetric for every ξ R d, G o B o = R T o H UU U o DU o R o is symmetric and nonnegative definite. The solutions of 3.2 thereby satisfy the local dissipation law t 2 W T G o W + x 2 W T G o A o W = x W T G o B o x W x W T G o B o x W. When this equation is integrated over T d one obtains the global dissipation law d 2 dt W T G o W dx = x W T G o B o x W dx.

5 WEAKLY NONLINEAR-DISSIPATIVE APPROXIMATIONS 5 The initial-value problem for system 3.2 is therefore naturally well-posed in the Hilbert space H = L 2 dx; R N equipped with the inner product W W 2 H = T W 2π d G o W2 dx, T d where G o is given by 3.5. By arguing as we did in the nonlinear settings, every stationary, spatially periodic classical solution of system 3.2 must satisfy A o x W =, Bo x W =. We can conclude that x W = if system 2. satisfies the nonsingularity condition 2.. In that case the only stationary, spatially periodic classical solutions of system 3.2 are the constant solutions. It is natural to ask if every solution of system 3.2 in H will decay to a constant solution as t. Kawashima [23, 24] gave an elegant characterization of when this is the case in terms of the skew-adjoint operator A and the self-adjoint operator D that are formally given by 3.6 A = A o x, D = B o : 2 x. Because A = A o x is skew-adjoint in H, its spectrum, SpA, is purely imaginary. When the spatial domain is bounded SpA only contains eigenvalues. Moreover, these eigenvalues and their corresponding eigenfunctions are easily computed by Fourier methods when the spatial domain is T d. Kawashima gave the following characterization. Theorem 3.. Every solution of system 3.2 in H will decay to a constant solution as t if and only if 3.7 no nonconstant eigenfunction of A is in the null space of D. Remark. The Kawashima condition 3.7 is clearly necessary for every solution of system 3.2 in H to decay to a constant solution as t. Indeed, if V is nonconstant eigenfunction of A for the eigenvalue iω such that V lies in the null space of D then the real part of e iωt V is a real-valued solution of 3.2 that does not decay to a constant solution as t. Remark. The Kawashima condition 3.7 is stronger than our nonsingular condition 2.. It has been used to obtain similar results regarding the existence, regularity, and decay as t of global solutions to nonlinear systems [4, 8, 9, 25, 26, 34, 37]. Villani has generalized it in his theory of hypercoercivity [38] where he discusses its relation to the older Hörmander hypoellipticity condition [2] Weak Dissipation, Two-Time Asymptotics. It is more interesting to consider regimes in which the dissipation is weak. Introduce the nondimensional small parameter ɛ > so that 3.8 t UW ɛ + x F UW ɛ = ɛ x [DUW ɛ x UW ɛ ], Let W o W and set Then W ɛ = W o + ɛ W + ɛ 2 W 2 + Oɛ 3. UW ɛ = U o + ɛ R o W + ɛ 2 R o W 2 + S o W, W + Oɛ 3, where U o and R o are given by 3.4 while S o is defined by S o W, W = 2 R o W W UW o W, W.

6 6 N. JIANG AND C. D. LEVERMORE To first order in ɛ we see that W satisfies 3.9 t W + Ao x W =. The solution of this equation is given by W t = e ta W in, where A = A o x. Because A is skew-adjoint on H, the solution operator e ta is strongly continuous, one parameter group of unitary operators on H. This approximation cannot be valid uniformly in time because the solutions of 3.9 do not decay as t and 2 there are generally nonconstant stationary solutions of 3.9 when d 2. In order to overcome these problems, one has to introduce a slow time scale τ = ɛ t into the asymptotics and consider ɛ τ UW ɛ + t UW ɛ + x F UW ɛ = ɛ x [DUW ɛ x UW ɛ ]. To first order we see that W still satisfies 3.8. Hence, 3. W t, τ = e ta Ỹ τ, where the τ dependence of Ỹ has yet to be determined. To the second order in ɛ we see that 3. t W 2 + A W 2 = τ W x Q o W, W + B o : 2 x W t + AS o W, W, where 3.2 Q o W, W = 2 R o F UU U o R o W, Ro W. The right-hand side of 3. is an almost periodic function of t. For W 2 to be an almost periodic, we must require lim T 2T e ta[ τ W + x Q o W, W B o : 2 x W ] dt =. Hence, because W is given by 3., we see that Ỹ τ satisfies 3.3 τ Ỹ + QỸ, Ỹ = DỸ, where the operators Q and D are formally defined by 3.4 QY, Y = lim T 2T DY = lim T 2T with the operators Q and D being given by e ta Qe ta Y, e ta Y dt, e ta De ta Y dt, QW, W = x Q o W, W, DW = B o : 2 x W. It is easily checked from formulas 3.4 that 3.5 Qe ta W, e ta W = e ta QW, W, De ta = e ta D. It then follows from 3. and 3.3 that W t, ɛ t satisfies t W + A W + ɛ Q W, W = ɛ D W.

7 WEAKLY NONLINEAR-DISSIPATIVE APPROXIMATIONS 7 Setting ɛ =, we call this the weakly nonlinear-dissipative approximation of the hyperbolicparabolic system Change of Dependent Variables. Suppose we had expressed the system 3.6 in terms of different dependent variables W : 3.6 t U W ɛ + x F U W ɛ = ɛ x [DU W ɛ x U W ɛ ], where U : W U is a twice continuously differentiable bijection with a nonsingular Jacobian. Let W o be the unique constant state such that U W o = U o = UW o. If we approximate solutions of 3.6 that are near W o just as we approximated solutions of 3.8 that are near W o then to leading order we obtain where A o = To A o T o with 3.7 T o = W t W + A o x W =, U U W W = R =W o o R o. In other words, the leading order approximation of the transformed system 3.6 is the leading order approximation of the original system transformed by the associated linear change of variables W = T o W. Remarkably, the same transformation property holds for the weakly nonlinear-dissipative approximation! Specifically, if the weakly nonlinear-dissipative approximation of the primed system 3.6 is t W + A W + ɛ Q W, W = ɛ D W, then it is related to the weakly nonlinear-dissipative approximation of the unprimed system by 3.8 A = T o AT o, Q W, W = T o QT o W, T o W, D = T o DT o, where T o is again given by 3.7. In other words, it is simply the original nonlinear-dissipative approximation transformed by the associated linear change of variables W = T o W. This fact allows us to derive the weakly nonlinear-dissipative approximation of system 3.8 in any set of dependent variables we choose becasue the result is unique up to the transformation Averaged Operators In this section we collect some properties of the averaged operators Q and D. 4.. Spectral Formulas for the Averaged Operators. Because A = A o x is skew-adjoint in H, its spectrum, SpA, is purely imaginary. Define σa = { ω R : iω SpA }. For every ω σa let H ω denote the eigenspace associated with the eigenvalue iω of A and let E ω be the H-orthogonal projection onto H ω. Then for every ω σa one has E ω W W 2 H = W E ω W 2 H for every W, W 2 H, E 2 ω = E ω, H ω = E ω H. For every ω σa let Cω denote the smooth functions in H ω i.e. let Cω = H ω C T d. We will assume that each Cω is dense in H ω and is contained in the domains of D and Q. This assumption holds because we are working over the periodic domain T d. In general settings it would mean that every smooth eigenfunction of A must also satisfy any boundary conditions

8 8 N. JIANG AND C. D. LEVERMORE associated with D, which is typically is not true. Our periodic setting we have explicit formulas for each E ω, however we will put off using them as long as possible. The Spectral Decomposition Theorem implies that e ta = e iωt E ω for every t R. ω σa If W H has nonzero components only in a finite number of the H ω then there are a finite number of nonzero terms in the decomposition e ta W = e iωt E ω W. ω σa If we assume moreover that each of these components is smooth then by using this decomposition in 3.4 we obtain the spectral formulas QW, W = E ω +ω 2 QE ω W, E ω2 W, 4. DW = ω, ω 2 σa ω σa E ω DE ω W, where we understand that E ω +ω 2 = when ω + ω 2 σa Bounds on the Averaged Operators. The spectral formulas 4. only apply to those W H that have nonzero components in only a finite number of the H ω, each of which is smooth. Denote this set by ω Cω. While this set is dense in H, it must be extended to larger classes of W. This is done by continuity once we obtain appropriate bounds on the forms associated with the operators Q and D. We begin by recalling some bounds on the forms associated with the operators A, Q, and D in terms of norms that are invariant under e ta. Specifically, we employ the H s spaces that are the completion of ω Cω in the norms defined for every s R by W H s = + ξ 2 s Ŵ ξ Go ξ Z d 2, Ŵ ξ = e iξ x W x dx. 2π d T d Here Ŵ ξ is the Fourier coefficient associated with the wave vector ξ Zd. Lemma 4.. There there exist positive constants C A and C D such that for every W, W 2 C T d and every s R one has the bounds W AW 2 H CA W H s 4.2 xw 2 H s. W DW 2 CD x W H s xw 2 H s. H For every s > d/2 there exist a positive constant CQ s such that for every W, W 2, W 3 C T d and every s one has the bounds W QW 2, W 3 H C s Q x W H s W 2 H W 3 H, 4.3 W QW 2, W 3 H 2 s CQ W s H s x W 2 W H s 3 H s + W 2 H s x W 3 H s. Proof. We refer the reader to [6] for similar proofs. Because the H s norms are invariant under e ta we can prove the following.

9 WEAKLY NONLINEAR-DISSIPATIVE APPROXIMATIONS 9 Lemma 4.2. For every s > d/2, s, and W, W 2, W 3 ω Cω one has the bounds W DW 2 H CD x W H s xw 2 H s, 4.4 W QW 2, W 3 H C s Q x W H s W 2 H W 3 H, W QW 2, W 3 2 s C s H Q W H s x W 2 W H s 3 H s + W 2 H s x W 3 H s, where C D and C s Q are the constants appearing in 4.2 and 4.3 of Lemma 4.. Proof. Because the H s norms are invariant under e ta, and because e ta and x commute, for every s and every W C T d we have 4.5 e ta W H s = W H s, x e ta W H s = x W H s. We first prove the bound on D in 4.4. From the definition of D given in 3.4 we see that W DW 2 = lim H T 2T = lim T 2T W e ta De ta W 2 e ta W De ta W 2 Hence, the bound on D in 4.2 and the invariances 4.5 imply W DW 2 H lim T 2T C D lim T 2T C D = lim T 2T e ta W De ta W 2 H dt x e ta W H x e ta W 2 H dt H dt H dt. x W H x W 2 H dt = C D x W H x W 2 H. This proves the bound on D in 4.4. We now prove the bounds on Q in 4.4. From the definition of Q given in 3.4 we see that W QW 2, W 3 = lim H T 2T = lim T 2T W e ta Qe ta W 2, e ta W 3 H dt e ta W Qe ta W 2, e ta W 3 H dt. Hence, the first bound on Q in 4.3 and the invariances 4.5 imply W QW 2, W 3 H lim T 2T CQ s lim T 2T CQ s = lim T 2T e ta W Qe ta W 2, e ta W 3 H dt x e ta W H s e ta W H 2 e ta W H 3 dt x W H s W 2 H W 3 H dt = C s Q x W H s W 2 H W 3 H. This proves the first bound on Q in 4.4. The second bound on Q in 4.4 follows similarly from the second bound on Q in 4.3 and the invariances 4.5, thereby proving Lemma 4.2.

10 N. JIANG AND C. D. LEVERMORE 4.3. Averaged Quadratic Convection Operator. Until now the entropy has played no role in our analysis of the weakly nonlinear-dissipative approximation. It will now play a central role. Specifically, the fact system 2. has a strictly convex entropy implies that its flux F U has a special form [7, 6]. We will use this form to show that the averaged quadratic convection operator satisfies a cyclic identity and some new bounds Special Form of the Flux. Henceforth we will assume that the hyperbolic-parabolic system 2. has a strictly convex entropy that is thrice continuously differentiable over the convex domain U. Define the set V = {H U U : U U}. One can show that V R N is a domain and that the mapping H U : U V is a diffeomorphism. For every V V we then define H V R and J V R d by 4.6 H V = V T U HU, J V = V T F U JU, where U U is uniquely determined by H U U = V. Because the entropy flux JU is related to HU and F U by 2.5, one can show that the mappings H : V R and J : V R d are continuously differentiable with 4.7 H V V = U, J V V = F U. It follows that F U is given by 4.8 F U = J V HU U for every U U. Because we have assumed that H is thrice continuously differentiable while F is twice continuously differentiable over U, it follows from 4.7 that both H and J are thrice continuously differentiable over V. This contrasts with J, which is twice continuously differentiable over U Cyclic Identity. The fact that the flux F U of system 2. has the form 4.7 is central to our proof of the following identity. Lemma 4.3. Cyclic Identity. If system 2. considered over the spatial domain T d has a strictly convex entropy H satisfying 2.4 that is thrice continuously differentiable over U then for every W, W 2, W 3 C T d one has the cyclic identity 4.9 W QW 2, W 3 + W H 2 QW 3, W + W H 3 QW, W 2 =, H where Q is defined by 3.4. Proof. We will first establish 4.9 for the particular Q associated with the conserved densities. In that case W = U. The extension of 4.9 to general Q then follows from the change of variable formula 3.8. We will begin by establishing 4.9 when U, U 2, U 3 ω Cω. Once this is done, the extension of 4.9 to U, U 2, U 3 C T d then follows by a density argument that uses the bounds on Q from Lemma 4.2. Let U, U 2, U 3 ω Cω. From the definition of Q given by 3.4 we see that 4. U QU 2, U 3 = lim H T 2T = lim T 2T U e ta Qe ta U 2, e ta U 3 H dt Ũ t QŨ2t, Ũ3t H dt. where Ũt = e ta U, Ũ2t = e ta U 2, and Ũ3t = e ta U 3 are quasiperiodic functions of t.

11 WEAKLY NONLINEAR-DISSIPATIVE APPROXIMATIONS By 3.2 and 3.4 we see that Ũ QŨ2, Ũ3 = Ũ H T G o x Q o Ũ2, Ũ3 dx = x Ũ T G o Q o Ũ2, Ũ3 dx, T d T d where G o = H UU U o and Q o Ũ2, Ũ3 = 2 F UUU o Ũ2 Ũ3. By differentiating 4.8 twice and evaluating at U o we obtain F U U o = J V V V o G o, F UU U o Ũ 2 Ũ3 = J V V V V o G o Ũ 2 G o Ũ 3 + J V V V o H UUU U o Ũ 2 Ũ3. where V o = H U U o. Hence, and A = A o x = J V V V o G o x, x Ũ T G o Q o Ũ2, Ũ3 = 2 xũ T G o JV V V V o G o Ũ 2 G o Ũ xũ T G o JV V V o H UUU U o Ũ 2 Ũ3 = 2 J V V V V o G o x Ũ G o Ũ 2 G o Ũ H UUUU o t Ũ Ũ2 Ũ3. Summing the above relation with its cyclic permutations gives x Ũ T G o Q o Ũ2, Ũ3 + x Ũ T 2 G o Q o Ũ3, Ũ + x Ũ T 3 G o Q o Ũ, Ũ2 = 2 x [J V V V V o G o Ũ G o Ũ 2 G o Ũ 3 ] + 2 t[ HUUU U o Ũ Ũ2 Ũ3]. Integrating this over T d we obtain Ũ QŨ2, Ũ3 + Ũ H 2 QŨ3, Ũ + Ũ H 3 QŨ, Ũ2 H = d H 2 UUU U o Ũ dt Ũ2 Ũ3 dx. T d The time average of this equation yields U QU 2, U 3 + U H 2 QU 3, U + U H 3 QU, U 2 =, H because the time average of the time derivative of a bounded function vanishes, thereby proving the lemma. The cyclic identity 4.9 yields the following bound. Lemma 4.4. If system 2. considered over the spatial domain T d has a strictly convex entropy H satisfying 2.4 that is thrice continuously differentiable over U then for every s > d/2 and every W, W 2 C T d one has the bound 4. W QW, W 2 H 2 Cs Q W H 2 x W 2 H s, where QW, W 2 is defined by 3.4 and CQ s is the constant appearing in 4.3 of Lemma 4.. Proof. The cyclic identity 4.9 and the fact that QW, W 2 = QW 2, W imply 2 W QW, W 2 + W H 2 QW, W =. H It thereby follows from the second bound in 4.4 that W QW, W 2 H W 2 2 QW, W H 2 Cs Q W H 2 x W 2 H s, where CQ s is the constant appearing in 4.3 of Lemma 4..

12 2 N. JIANG AND C. D. LEVERMORE 4.4. Averaged Dissipation Operator. Because the averaged dissipation operator D is nonpositive semidefinite, its Friedrichs extension is a self adjoint nonpositive semidefinite operator with domain DomD. For every V H and every ω σa define V ω to be the component of V in H ω so that V ω = E ω V H ω, V = V ω. ω σa We claim that the nonpositve definite Hermitian form associated with D extends to the domain HermD given by { 4.2 HermD = V H : } V ω DV ω H <. ω σa Recall that V ω DomD for every ω σa and that D is nonpositive over DomD, whereby V ω DV ω H for every ω σa. Then for every V HermD one has by 4. that 4.3 V DV H = V ω DV ω H. ω σa We now characterize when the Hermitian form associated with D is nondegenerate. Lemma V DV if and only if the Kawashima condition 3.7 holds. H < for every nonconstant V HermD, Remark. This kind of characterization was first proved by Kawashima [23, 24]. Proof. First we show that 4.4 implies the Kawashima condition 3.7 holds. Suppose not. Then there exists a nonconstant V H such that V H ω for some ω σa and V NullD. But then V = V ω HermD with V DV H = V ω DV ω H = V DV H =, which contradicts 4.4. Therefore 4.4 implies the Kawashima condition 3.7 holds. Next we show that the Kawashima condition 3.7 implies 4.4. Suppose that 4.4 is false. Then there exists a nonconstant V HermD such that = V DV = V H ω DV ω H. However this is equivalent to which is equivalent to ω σa V ω DV ω H = for every ω σa, V ω NullD for every ω σa, But V implies that V ω for some ω σa. But then for this ω we have V ω is nonconstant, V ω H ω, and V ω NullD, which contradicts the Kawashima condition 3.7. Therefore the Kawashima condition 3.7 implies 4.4, and the proof of Lemma 4.5 is complete.

13 WEAKLY NONLINEAR-DISSIPATIVE APPROXIMATIONS 3 5. Strictly Dissipative Approximations The global existence theory presented in the next section will require that D be strictly elliptic. When this is the case we say that the WND approximation is strictly dissipative. For many physical systems the WND approximation has this property. Indeed, the averaged dissipation operator has played an important role in the study of large time behavior of solutions to hyperbolic-parabolic systems of conservations laws and discrete Boltzmann equations. Kawashima [25, 26] showed that in one space dimension the averaged dissipation operator was strictly dissipative whenever the Kawashima condition 3.7 holds. In that case he showed that solutions of the original system are well approximated by solutions to an effective artificial viscosity system constructed using the averaged dissipation operator. Motivated by this idea, Hoff and Zumbrun [8, 9] studied multi-dimensional diffusion waves for the barotropic Navier- Stokes system through an artificial viscosity system constructed with the averaged dissipation operator. Recently, Bianchini-Hanouzet-Natalini [4] used the same idea to study the large-time behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy. In each case the averaged dissipation operator was shown to be strictly dissipative through a detailed spectral analysis. To our knowledge there is no proof that the Kawashima condition 3.7 implies that the averaged dissipation operator is strictly dissipative in multidimensional settings. Here we give a stronger criterion that does the job in our spatially periodic setting. In the Fourier representation we have ÂV ξ = ia o ξ V ξ, DV ξ = Bo ξ V ξ, where A o ξ and B o ξ are the families of G o -symmetric matrices in R N N defined for every ξ R d by A o ξ = ξ A o, B o ξ = ξ 2 : B o. One then has ê ta V ξ = e itaoξ V ξ, whereby DV ξ = Dξ V ξ, Dξ T = lim e itaoξ B o ξe itaoξ dt. T 2T Our Kawashima-type criterion for strict dissipativity is given by the following. Theorem 5.. Suppose that for some α > there exists β > such that 5. G o B o ξ + α 2 A o ξ T G o B o ξa o ξ β G o for every ξ S d. Then there exists δ > such that 5.2 V DV H δ xv 2 H for every V C 2 T d. Remark. The Kawashima condition is satisfied whenever 5. holds. Our proof of Theorem 5. requires the following lemma. Lemma 5.. Let G C N N be Hermitian positive definite i.e. G = G >. Let A C N N be G-Hermitian i.e. GA = A G and B C N N be G-Hermitian nonnegative definite i.e. GB = B G. Let C A and C B be constants such that A G C A and B G C B. Suppose that for some α > there exists β > such that 5.3 GB + α 2 A GBA β G.

14 4 N. JIANG AND C. D. LEVERMORE Then there exists δ > depending only on α, β, C A, and C B, such that 5.4 lim T 2T Proof. Let v C N be nonzero and define e ita GBe ita dt δ G. ft = v e ita GBe ita v. This is a smooth, nonnegative, quasiperiodic, real-valued function that will vanish whenever e ita v is a null vector of B. In order to prove 5.4 we must obtain a lower bound for its average. The idea of the proof is to show that ft cannot be too small for long. The first two derivatives of ft are 5.5 ft = iv e ita A GB GBAe ita v, ft = v e ita 2A GBA A 2 GB GBA 2 e ita v. Roughly speaking, we will show that ft is dominated by its first term when ft is small. For every η > one has v e ita A 2 GB + GBA 2 e ita v η 2 v e ita GBe ita v + η 2 v e ita A 2 GBA 2 e ita v, The first term on the right-hand side above is just ft/η 2 while the second can be bounded by η 2 CA 4C B v Gv. If we set η 2 = α 2 β/2ca 4C B then we obtain the bound 5.6 v e ita A 2 GB + GBA 2 e ita v 2C 4 A C B α 2 β ft + α2 β 2 v Gv. Let Ω ɛ = {t R : ft < ɛ v Gv}, where ɛ > satisfies α 4 β β 5.7 ɛ α 4 β + CA 4C B 4. For every t Ω ɛ we obtain from 5.3, 5.5, 5.6, and 5.7 the lower bound ft v e ita 2α 2 GB + 2A GBAe ita v 2α 2 + 2C A 4C B α 2 β 5.8 2α 2 β 2α 2 + 2C A 4C B ɛ α2 β v Gv α 2 β 2 α 2 β v Gv. For every t Ω ɛ we obtain from 5.5, 5.6, and 5.7 the upper bound 5.9 ft v e ita 2A GBAe ita v + 2C A 4C B α 2 β ft + α2 β 2 v Gv 2CAC 2 B + 2C A 4C B α 2 β ɛ + α2 β v Gv 2 2C 2 AC B + α 2 β v Gv. ft α2 β 2 v Gv Because f is continuous the set Ω ɛ is open and is therefore a countable union of disjoint open intervals: Ω ɛ = k Na ɛ k, b ɛ k.

15 WEAKLY NONLINEAR-DISSIPATIVE APPROXIMATIONS 5 Let a, b be any one of these intervals. Because ft satisfies the lower bound 5.8 while ft < ɛ v Gv for every t a, b, it is clear that the interval a, b must be bounded. We will begin by bounding b a above and below. Because f is continuous over the bounded interval [a, b], and because a and b are not in Ω ɛ, it follows that fa = fb = ɛ v Gv, and that f takes its minimum at a point t o a, b, at which ft o =. Then 5. ft = ft o + We claim that b a satisfies the bounds 5. t o ft dt = ft o + t o t o ft2 dt 2 dt. ɛ v Gv ft o 2C 2 A C B + α2 β 8 b a2 v Gv ɛ v Gv α 2 β. The upper bound of 5. is obtained from 5. by using the lower bound 5.8 for ft. For every t a, b we have Evaluating this at t = a and t = b yields ft ft o + 2 t t o 2 α 2 β v Gv. ɛ v Gv ft o + 2 t o a 2 α 2 β v Gv, ɛ v Gv ft o + 2 b t o 2 α 2 β v Gv. Because ft o while max{t o a 2, b t o 2 } 4 b a2, we obtain ɛ v Gv 8 b a2 α 2 β v Gv, which yields the upper bound of 5.. The lower bound of 5. is obtained from 5. by using the upper bound 5.9 for ft. For every t a, b we have Evaluating this at t = a and t = b yields ft ft o + 2 t t o 2 2C 2 AC B + α 2 β v Gv, ɛ v Gv ft o + 2 t o a 2 2C 2 AC B + α 2 β v Gv, ɛ v Gv ft o + 2 b t o 2 2C 2 AC B + α 2 β v Gv. Because min{t o a 2, b t o 2 } 4 b a2, we obtain ɛ v Gv ft o + 8 b a2 2C 2 AC B + α 2 β v Gv, which yields the lower bound of 5.. Next, we bound the average of ft over a, b from below. By again using the lower bound 5.8 for ft in 5. we obtain b a b a ft dt ft o + b t o 2 + b t o t o a + t o a 2 6 ft o + b 8 a2 α 2 β v Gv. The lower bound of 5. then implies 5.2 b a b a ft dt ft o + α 2 β ɛ v Gv ft o 2C 2 A C B + α2 β α 2 β v Gv α 2 β ɛ 2C 2 A C B + α2 β v Gv.

16 6 N. JIANG AND C. D. LEVERMORE We are now ready to prove 5.4 with δ given by 5.3 δ = α 2 β ɛ 2C 2 A C B + α2 β. For every T > let KT ɛ = {k N : aɛ k, bɛ k, T } and Ω ɛ T = a ɛ k, b ɛ k. k K ɛ T Then using the fact that ft ɛ over [, T ] Ω ɛ, the lower bound 5.2, and the fact that δ given by 5.3 satisfies δ < ɛ, we find T ft dt [ ft dt + bk ] ft dt 2T 2T [,T ] Ω ɛ k KT ɛ a k [ meas [, T ] Ω ɛ ɛ + ] b ɛ k a ɛ 2T k δ v Gv k KT ɛ [ meas [, T ] Ω ɛ Ω ɛ ] T δ v Gv. 2T The set [, T ] Ω ɛ Ω ɛ T is contained in the union of the at most two disjoint intervals a ɛ k, bɛ k that contain and T. So its measure is bounded above by twice the upper bound for b a given by 5. namely, by meas 2ɛ [, T ] Ω ɛ Ω ɛ 2 T 4. α 2 β Hence, we obtain T [ ft dt 2 ] 2ɛ 2 δ v Gv. 2T T α 2 β Letting T in this inequality yields 5.4. The limit of the left-hand side exists because ft is quasiperiodic. We are now ready to prove Theorem 5. with the aid of Lemma 5.. Proof of Theorem 5.. First observe that for every nonzero ξ R d we set ξ = ξ/ ξ and have T Dξ = lim e itaoξ B o ξe itaoξ dt T 2T = ξ 2 lim T = ξ 2 lim T 2T 2T Then by the fact A o ξ is G o -symmetric we have 5.4 G o Dξ = ξ 2 lim T 2T = ξ 2 lim T 2T e it ξ Ao ξ B o ξe it ξ Ao ξ dt e itao ξ B o ξe itao ξ dt. G o e itao ξ B o ξe itao ξ dt e itao ξ T G o B o ξe itao ξ dt.

17 WEAKLY NONLINEAR-DISSIPATIVE APPROXIMATIONS 7 We now apply Lemma 5. with G = G o, A = A o ξ, B = B o ξ, C A = max { A o ξ Go : ξ S d }, C B = max { B o ξ Go : ξ S d }. We find that for every ξ S d we have the lower bound lim T 2T e itao ξ T G o B o ξe itao ξ dt δ G o, where δ is given by 5.3. Because δ only depends on α, β, C A, and C B, it is independent of ξ. Combining this lower bound with 5.4 yields G o Dξ ξ 2 δ G o for every ξ R d. Then for every V C 2 T d the Plancherel identity implies V DV = V H ξ G o Dξ V ξ δ ξ 2 V ξ G o V ξ = δ x V 2 H. ξ L d ξ L d But this is 5.2, thereby proving the theorem. 6. Global Weak Solutions The weakly nonlinear-dissipative approximation is 6. t W + AW + QW, W = DW. Following the Leray theory for the incompressible Navier-Stokes system, we will show that if D is strictly dissipative then 6. has global weak solutions for all initial data W in H. This result includes the Leray theory, so it cannot be improved easily. The key to obtaining global solutions in the Leray theory for the incompressible Navier- Stokes system is a so-called energy estimate. This designation is a bit misleading because, as we shall see, the estimate is better understood as an entropy estimate. 6.. Notion of Weak Solution. We call W C[, ; w-h L 2 loc dt; V a Leray-type weak solution of weakly nonlinear-dissipative approximation 6., if W satisfies the following weak form of 6.: 6.2 = V W t 2 H V W t H 2 2 t t AV W t T d x V G o QW t, W t dx dt + 2 t H dt for every function V V. These solutions satisfy the entropy inequality W t 2 H W DW H dt 2 W in 2 H. Of course, for every sufficiently nice W one has the identities 6.4 W AW H =, W QW, W =. H T d x V G o B x W t dx dt,

18 8 N. JIANG AND C. D. LEVERMORE 6.2. Existence Theorem. The main theorem of this paper, the global existence of Leray-type weak solution to the weakly nonlinear-dissipative approximation 6. is as follows. Theorem 6.. Let 6. be the weakly nonlinear-dissipative approximation about a constant state U o of the hyperbolic-parabolic system 2. with a strictly convex entropy 2.4. Suppose that the diffusive operator D in 6. is strictly dissipative, i.e. that there exists δ > such that 6.5 V DV H δ xv 2 H for every V C 2 T d. Then, for every W in V, there exists a solution W C[, ; w-h L 2 loc dt; V satisfying 6.2 and 6.3. Remark. If condition 5. is satisfied then D will satisfy 6.5 and the above theorem insures the existence of at least one global weak solution. Remark. In Section 7.2, we apply our theory to the Navier-Stokes system of gas dynamics. The resulting averaged system includes the incompressible Navier-Stokes system as a subsystem. The question of uniqueness for system 6. is thereby at least as hard as that of uniqueness for weak solutions of the incompressible Navier-Stokes system. Proof. The strategy for our proof was introduced by Leray in the context of the incompressible Navier-Stokes system [28], see also [27, 36]. It is now classical compactness argument that has since been used to prove existence of global weak solutions for other equations []. It proceeds in four steps. We begin by constructing a sequence of approximate solutions. We then show that this sequences is relatively compact, first in some weak topologies and then in a strong topology. Finally, we show that limit points of this sequence satisfy 6.2 and are thereby weak solutions of 6.. This strategy strikes a balance between the fact that compactness is easier to establish for weaker topologies and the fact that passing to the limit in nonlinear terms requires convergence in a strong topology Step : Constructing Approximate Solutions. One can construct a sequence of approximation solutions W n by any method that yields a consistent weak formulation and an energy relation. Here we do this with the Galerkin method. Let {H n } n= be a sequence of subspaces of H such that each H n lies within C T d, has dimension n, and satisfies H n H n+. Assume moreover that this sequence is complete. Let P n denote the orthogonal projection from H onto H n. Completeness implies that for every V H one has P n V V as n. The Galerkin approximation of dimension n is the system 6.6 t W n + P n AW n + P n QW n, W n = P n DW n, where W n takes values in H n. This is a system of n ODEs. Its nonlinearities are quadratic, hence locally Lipschitz. The Picard existence theorem insures that system 6.6 has local solutions. Taking inner product with W n, and applying the identities in 6.4, we obtain the energy identity: W nt 2 H W n DW n H dt = 2 W in n 2 H, for every t >. This energy identity immediately implies a global L 2 bound on the approximate solutions W n, which thereby exists for all time.

19 WEAKLY NONLINEAR-DISSIPATIVE APPROXIMATIONS Step 2: Establish Weak Compactness. We claim that the approximate solutions W n are relatively compact in C[, ; w-h w-l 2 loc dt; w-v. First, from the energy identity 6.7, we that W n t is uniformly bounded in H thus relatively compact in w-h for every t >. Next, from the dissipation property of D, see 6.5, we have 6.8 δ x W n 2 H dt Wn DW n H dt C. Thus W n is relatively compact in w-l 2 loc [, ; w-v. We need only to verify that W n is equicontinuous in C[, ; w-h thus by Arzela-Ascoli theorem the weak compactness is established. The equicontinuity can be derived from the weak form of the Galerkin system Ṽ Wn t 2 W n t 2 = Ṽ H AWn t dt H t 2 t Ṽ QWn t, W n t H dt + 2 t Ṽ DWn t H dt, for every function Ṽ V. We first prove the equicontinuity for test function Ṽ V C T d, which is followed from the first and third bounds in the Lemma 4.2. Then we extend the class of test functions to V by standard density argument, thereby finishing the proof of Step Step 3: Establish Strong Compactness. We claim that W n is relatively compact in strong topology of L 2 loc [, ; H. It is a direct consequence of the weak compactness result in Step 2 and the fact that the injection 6. C[, ; w-h w-l 2 locdt; w-v L 2 loc[, ; H is continuous Step 4: Pass to the Limit. Step 2 ensures that there is a subsequence of W n, which we also refer to as W n, converges in C[, ; w-h w-l 2 loc dt; w-v to a limit W C[, ; w-h L 2 loc dt; V. Step 3 ensures the convergence of W n to W in L 2 loc [, ; H. All that remains is to show that the limit W satisfies the weak form 6.2 as well as the energy inequality 6.3. Toward this end we check convergence of each term in the respective regularized versions, 6.9 and 6.7, respectively. Again, we first consider the test function V in the class V C T d then use density argument later. First V Wn t 2 W n t V W t H 2 W t as n, H because of the relative compactness of W n in C[, ; w-h. The convergence of the first term on the righthand side of 6.9 is trivial. Note that 2 V QWn t, W n t 2 dt = H x V G o QW n t, W n t dx dt, T d t t and QW n t, W n t is quadratic in W n. Thus the strong compactness of W n in L 2 loc [, ; H in Step 3 ensures the convergence of above term. We also note that 2 V DWn t 2 dt = H x V G o B x W n t dx dt, t T d t The convergence of above term is straightforward. Thus, we show that the limit W satisfy the weak form 6.2, thus is a weak solution to WND approximation 6..

20 2 N. JIANG AND C. D. LEVERMORE Now, to recover the energy inequality 6.3 from 6.7, first we note that for the initial data term Wn in H W in H. The convergence of W n in C[, ; w-h and L 2 loc [, ; H, together with the fact that the norm of the weak limit is an eventual lower bound to the norms of the sequence, yields W 2 H lim inf n W nt 2 H. Similarly, the convergence of W n in w-l 2 loc [, ; w-v implies W DW H dt lim inf n Thus, we finish the proof of global Leray type weak solutions. W n DW n H dt Uniqueness Theorem. Uniqueness can never be asserted by such a compactness argument, but generally requires the knowledge of additional regularity of the solution. For example, here we will prove the following weak-strong theorem. Theorem 6.2. Let U, U 2 C[, ; w-h L 2 loc dt; V be two weak solutions of the WND system with initial data U in, U2 in H. Let s > max{d/2, }. If U L 2 [, T ]; H s L [, T ]; V s for some T > then U C[, T ]; w-h s and for every t [, T ] one has the energy equality 6. U t 2 U DU 2 H H dt = 2 U in 2 and the stability bound 6.2 U 2 t U t H exp C s Q H, x U t H s dt U2 in U in H. In particular, if U in 2 = U in then U 2 t = U t for every t [, T ]. Remark. Equation 6. is simply the assertion that the energy inequality satisfied by the strong solution U is in fact an equality. The bound 6.2 is a basic weak-strong stability bound, from which the uniqueness assertion follows immediately. The key to the proof of Theorem 6.2 will be provided by the following lemma. Lemma 6.. Let U, U 2 C[, ; w-h L 2 loc dt; V be two weak solutions of the WND system with initial data U in, U2 in H. Let s > max{d/2, }. If U L 2 [, T ]; H s L [, T ]; V s for some T > then U C[, T ]; w-h s and for every t [, T ] one has 6.3 U t U 2 t + QU, U H U 2 H + U QU 2, U 2 H dt DU U 2 H + U DU 2 H dt = U in U2 in. H This lemma will be proved later. Now we will use it to prove Theorem 6.2. Proof of Theorem 6.2. The energy equality 6. follows by setting U 2 = U in equation 6.3 of Lemma 6., using the cyclic identity 4.9 to see that U QU, U =, and H multiplying the result by. 2

21 WEAKLY NONLINEAR-DISSIPATIVE APPROXIMATIONS 2 We now derive the stability bound 6.2. Add the energy inequalities for U and U 2 and subtract equation 6.3 from the result to obtain U 2 t U t 2 QU, U H U 2 H + U QU 2, U 2 H dt Upon letting W = U 2 U so that U 2 = U + W, we see that QU, U U U2 U DU 2 U H dt 2 U2 in U in 2. H H + U QU 2, U 2 H = U QU, U H + W QU, U H + U QU, U H + U QW, U H + U QU, W H + U QW, W H. The cyclic identity 4.9 implies that U QU, U H = and W QU, U H + U QW, U H + U QU, W H =. We thereby see that relation 6.5 reduces to QU, U U 2 H + U QU 2, U 2 H = U QW, W H. When this relation is placed into 6.4, we obtain t W t 2 U QW, W H H dt W DW H dt 2 W in 2, H where W in = U2 in U in. The third bound in 4.4 gives U QW, W H C s x Q U H s W 2 We combine this bound with the fact W DW to see that 6.6 yields the inequality H 2 W t 2 W in 2 + C s H Q x U H s W 2 H dt. H 2 The stability bound 6.2 then follows by the Gronwall Lemma. All that remains is to prove Lemma 6.. Proof of Lemma 6.. We begin by showing that U L 2 [, T ]; H s L [, T ]; V s implies U C[, T ]; w-h s. From equation 6. we see that for every W H s we have W U t 2 U t H = 2 W AU t + QU t, U t DU t dt. H t We have the following estimates: 2 W AU t dt t2 H C A W H s U t H s dt. t t By the first bound in 4.3 of Lemma 4.2 with s = s we have 2 W DU t dt t2 H C D x W H s x U t H s dt t C D W H s t 2 H. t U t V s dt.

22 22 N. JIANG AND C. D. LEVERMORE By the third bound in 4.3 of Lemma 4.2 with s = s we have 2 W QU t, U t dt t2 H 2s C s Q W H U s t H 2 dt. s t t Because U L 2 [, T ]; H s L [, T ]; V s it follows that W U t H is a continuous function of t over [, T ] for every W H s. Hence, U C[, T ]; w-h s. We now prove that 6.3 holds for every t [, T ]. Let α DR d and β DR be mollifiers such that α, β, suppβ, ] and αx dx =, βt dt =. R d R For every ɛ > define Θ ɛ DT d R by Θ ɛ x, t = ɛ d+ l Z d α x + 2πl For each i =, 2 and ɛ > define U iɛ = Θ ɛ U i, so that for every x T d and t we have U iɛ x, t = Θ ɛ U i x, t = Θ ɛ x x, t t U i x, t dx dt. T d Each U iɛ is a smooth function over R d R + that satisfies ɛ β t ɛ. t U iɛ + AU iɛ + Θ ɛ QU i, U i = DU iɛ. Because A is skew-adjoint we thereby see that d Uɛ U 2ɛ dt H = t U ɛ U 2ɛ H + U ɛ t U 2ɛ H = Θ ɛ QU, U U 2ɛ H U ɛ Θ ɛ QU 2, U 2 H + DU ɛ U 2ɛ + U H ɛ DU 2ɛ H. Upon integrating this equation over [, t] we obtain 6.7 We claim that 6.8 Uɛ t U 2ɛ t + Θɛ QU H, U U 2ɛ H + U ɛ Θ ɛ QU 2, U 2 H dt lim ɛ DUɛ U 2ɛ H + U ɛ DU 2ɛ H dt = U ɛ U 2ɛ H. lim Uɛ t U 2ɛ t = U ɛ H t U 2 t, H lim Uɛ U 2ɛ = U in ɛ H U2 in, H t Θɛ QU, U U 2ɛ H dt = QU, U U 2 H dt, Uɛ Θ ɛ QU 2, U 2 H dt = U QU 2, U 2 H dt, lim ɛ lim DUɛ U 2ɛ ɛ H + U ɛ DU 2ɛ H dt = DU U 2 H + U DU 2 H dt.

23 WEAKLY NONLINEAR-DISSIPATIVE APPROXIMATIONS 23 Once these limits are established we can then pass to the limit in 6.7 to obtain 6.3 and thereby complete the proof of Lemma 6.. The limits 6.8 are established by using the bounds from Lemma 4.2 and the convergence and boundedness properties of convolution. Now the first two limits are direct consequence of the convergence property of convolution in C[, T ]; w-h s. Because U and U 2 are continuous in time in w-h s and H respectively, we have, for any t >, U ɛ t U t in w-h s thus U ɛ t U t in H, U 2ɛ t U 2 t in w-h, as ɛ, which imply that for every t >, Uɛ t U 2ɛ t H U t U 2 t H, as ɛ. Thus we prove the first two limits. To prove the third limit, first, we have 6.9 Θɛ QU, U U 2ɛ H QU, U U 2 H dt = Θɛ QU, U QU, U U 2ɛ H dt + QU, U U 2ɛ U 2 H dt. Note that U is a strong solution, i.e. U L [, T ]; H s L 2 [, T ]; V s for s > d/2, and the structure of QU, U is the derivative of U multiplying U, thus QU, U L 2 [, T ] ; H. Then the convergence in L 2 [, T ]; H and the boundedness of mollifier imply that each integral on the right-hand side of 6.9 goes to as ɛ. We leave the proof of the fourth limit to the last step because it is the hardest one. We prove the fifth limit first. Applying the first inequality in Lemma 4.2 with s = and the Hölder inequality, we have DUɛ U 2ɛ H t DU U 2 H dt = DUɛ DU U 2ɛ H + DU U 2ɛ U 2 H dt C D x U ɛ U L 2 [,T ] ;H x U 2ɛ L 2 [,T ] ;H + C D x U L 2 [,T ] ;H x U 2ɛ U 2 L 2 [,T ] ;H. Note that x U iɛ = x U i ɛ, for i =, 2, the boundedness of xu iɛ in L 2 [, T ] ; H, and the convergence x U i ɛ xu i in L 2 [, T ] ; H as ɛ, we finish the proof of the fifth limit. The main difficulty is that U 2 is only a Leray weak solution, so QU 2, U 2 is not in L 2, thus the method to prove the third limit is not applicable here. However, we have the following identity: for any functions U, V so that U T G o V L [, T ] ; L dx and T >, 6.2 Θɛ U V dt = H U Θɛ V dt. H

24 24 N. JIANG AND C. D. LEVERMORE Proof of 6.2: By changing the order of integration, = = = Θɛ U V H dt T d T d t α x x ɛ d+ ɛ β t t ɛ T d α x x ɛ d+ ɛ T d U Θɛ V H dt. Applying 6.2 to U and QU 2, U 2, we have U T x, t dx dt G o V x, t dx dt β t t V T x, t dx dt G o Ux, t dx dt ɛ Uɛ Θ ɛ QU 2, U 2 H dt = Θɛ U ɛ QU 2, U 2 H dt. Thus from the second inequality in Lemma 4.2, Θɛ U ɛ U QU 2, U 2 H dt Note that C s QT 2 We claim that for any W L 2 [, T ]; H s, x Θ ɛ U ɛ U L [,T ];H s dt U 2 2 L dt;h. x Θɛ U ɛ = Θɛ x U ɛ. lim Θ ɛ W ɛ W L 2 [,T ] ;H s as ɛ, ɛ which simply followed from the triangle inequality Θ ɛ W ɛ W L 2 [,T ];H s Θ ɛ W ɛ W ɛ L 2 [,T ];H s + W ɛ W L 2 [,T ];H s, and the convergence property of convolution in L 2 [, T ]; H s. We thereby prove the limit. We have now established all the limits asserted in 6.8 and have thereby completed the proof of Lemma Application to the Compressible Navier-Stokes System 7.. Compressible Navier-Stokes System. The compressible Navier-Stokes system of gas dynamics is an important example of a nonsingular hyperbolic-parabolic system with the strictly convex entropy. It governs the mass density ρx, t, bulk velocity ux, t, and temperature θx, t over Ω R d in the form t ρ + x ρu =, 7. t ρu + x ρu u + pi + S =, t 2 ρ u 2 + ρε + x 2 ρ u 2 u + ρεu + pu + S u + q =, where the specific energy ε and pressure p are given by thermodynamic equations-of-state, ε = ερ, θ and p = pρ, θ, while the stress S and heat flux q are given by the constitutive relations 7.2 S = µ x u + x u T 2 D I x u λ I x u, q = κ x θ.

25 WEAKLY NONLINEAR-DISSIPATIVE APPROXIMATIONS 25 Here the coefficients of shear viscosity µ, bulk viscosity λ and thermal conductivity κ are given by formulas µ = µρ, θ, λ = λρ, θ, and κ = κρ, θ that come either from a nonequilibrium kinetic theory or from fits to experimental data, while D is the dimension of the underlying microscopic world usually D = 3. We require that D max{2, d}. Equations 7. express the local conservation of mass, momentum, and energy. The constitutive relations 7.2 for S and q model viscosity and thermal conductivity, which arise due to deviations of the gas from local thermodynamic equilibrium. Equations 7. reduce to the compressible Euler system when one sets S = and q =. The thermodynamic equations-of-state for the specific energy and pressure, ε = ερ, θ and p = pρ, θ, are assumed to be twice continuously differentiable over ρ, θ R 2 + and to satisfy 7.3 θ ερ, θ >, ρ pρ, θ >, for every ρ > and θ >. In addition, they must satisfy the Maxwell relation p ρ 2 ρ ε + θ 2 θ =. θ This implies the existence of a function σ = σρ, θ that satisfies the differential relation d σ ε = ε d + p θ θ θ d. ρ This is equivalent to 7.4 dσ = θ dε p ρ 2 θ dρ. We can identify σ with the specific entropy. We make use of the convective form of system 7., t ρ + u x ρ + ρ x u =, 7.5 ρ t u + u x u + x p + x S =, ρ t ε + u x ε + p x u + S : x u + x q =, and the differential specific entropy relation 7.4 to see that ρ t σ + u x σ = ρ θ tε + u x ε p ρθ tρ + u x ρ = θ S : xu θ x q. This can be put into the divergence form 7.6 t ρσ + x ρuσ + q = θ θ S : xu θ q xθ. 2 The local form of the second law of thermodynamics and the constitutive relations 7.2 imply that for any values of ρ, θ, x u + x u T 2 D I x u, x u, and x θ one has the inequality θ S : xu θ q xθ = µ x u + 2 x u T 2 D 2 I x u 2 + λ x u 2 + κ x θ 2. Because these values can be independently specified at any point in Ω, the above inequality implies that µρ, θ, λρ, θ, and κρ, θ for every ρ > and θ >. Of course,