RELATIVE ENTROPY IN HYPERBOLIC RELAXATION FOR BALANCE LAWS. Alexey Miroshnikov and Konstantina Trivisa. Abstract

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1 RELATIVE ENTROPY IN HYPERBOLIC RELAXATION FOR BALANCE LAWS In: Communications in Mathematical Sciences, 12-6, 214)) Alexey Miroshnikov and Konstantina Trivisa Abstract We present a general framework for the approximation of systems of hyperbolic balance laws. The novelty of the analysis lies in the construction of suitable relaxation systems and the derivation of a delicate estimate on the relative entropy. We provide a direct proof of convergence in the smooth regime for a wide class of physical systems. We present results for systems arising in materials science, where the presence of source terms presents a number of additional challenges and requires delicate treatment. Our analysis is in the spirit of the framework introduced by Tzavaras [23] for systems of hyperbolic conservation laws. 1 Introduction We present a general framework for the approximation of systems of hyperbolic balance laws, t u + α f α u) = gu) ux, t) R n, x R d, 1.1) by relaxation systems presented in the form of the extended system t U + α F α U) = 1 ε RU) + GU), Ux, t) RN, x R d 1.2) in the regime where the solution of the limiting system as ε ) is smooth. Motivated by the structure of physical models and the analysis in [6], we deal with relaxation systems of type 1.2) which are equipped with a globally defined, convex entropy HU) satisfying t HU) + α Q α U) = 1 DHU)RU) + DHU)GU). 1.3) ε Department of Mathematics and Statistics, University of Massachusetts Amherst, MA 13, USA, amiroshn@math.umass.edu. Department of Mathematics, University of Maryland, College Park, MD 2742, USA, trivisa@math.umd.edu. 1

2 The problem of numerical approximation of nonlinear hyperbolic balance laws is extremely challenging. In the present article we identify a class of relaxation schemes suitable for the approximation of solutions to certain systems of hyperbolic balance laws arising in continuum physics. The relaxation schemes proposed in our work provide a very effective mechanism for the approximation of the solutions of these systems with a very high degree of accuracy. The main contribution of the present article to the existing theory can be characterized as follows: This work provides a general framework describing how, given a physical system governed by a hyperbolic balance law, one can construct an extended system endowed with a globally defined, convex entropy HU) and the resulting relaxation system for its approximation. It has the potential of being of use for the construction of suitable approximating schemes for a variety of hyperbolic balance laws. Our analysis treats a large class of physical systems such as the system of elasticity 6.1) c.f. Section 6), two phase flow models 7.2) c.f. Section 7), and general symmetric hyperbolic systems 8.1) c.f. Section 8). In the latter application the relaxation of the hyperbolic system is obtained by relaxing n-vector flux components. Our framework is applicable in the multidimensional setting and provides a rigorous proof of the relaxation limit and a rate of convergence for a large class of physically relevant hyperbolic balance laws. As it is well known, results for multidimensional systems of hyperbolic balance laws are limited in the literature. In addition, our analysis treats a large class of source terms: those satisfying a special mechanism that induces dissipation as well as more general source term. We establish convergence of weak solutions of 1.2) to solutions of the equilibrium system 1.1) via a relative entropy argument which relies on 1.3). The proof provides a rate of convergence. The relative entropy method relies on the weak-strong uniqueness principle established by Dafermos for systems of conservation laws admitting convex entropy functional [9], see also DiPerna [13]. In addition to the pioneer work of Dafermos and DiPerna, the relative entropy method has been successfully used to study hydrodynamic limits of particle systems [4, 14, 22, 21, 25], hydrodynamic limits from kinetic equations to multidimensional macroscopic models [1, 3, 17], as well as the convergence of numerical schemes in the context of three-dimensional polyconvex elasticity [18, 2]. The main ingredients of our approach can be formulated as follows: A relative entropy inequality which provides a simple and direct convergence framework before formation of shocks. The reader may contrast the present framework to the classic convergence framework for relaxation limits, which proceeds through analysis of the linearized collision or relaxation) operator [26]. 2

3 Physically grounded structural hypotheses imposed on the relaxation system. These structural hypotheses will be of use for the derivation of the relative entropy inequality and for the proof of the desired convergence. The relative entropy computation hinges on entropy consistency [23], that is, the restriction of the entropy pair H Q α on the manifold of Maxwellians M := {U R N : RU) = } = {U R N : U = Mu), u R n } induces an entropy pair for the equilibrium system 1.1) in the form ηu) = HMu)), q α = Q α Mu)). A physically motivated dissipation mechanism in the sense of H8)) associated with the source term in 1.2) with respect to the manifold of Maxwellians on which relaxation takes place. The dissipation mechanism on 1.2) induces weak dissipation on the equilibrium balance law 1.1) due to source consistency requirement c.f. Section 2.4). The concept of weak dissipation for hyperbolic balance laws was introduced by Dafermos in [11]. To realize the role of dissipation in the present context, the reader may contrast the result of Theorem 3.1 for weakly dissipative source terms with Theorem 3.2 which corresponds to the case of a general source. The paper is organized as follows: In Section 2 we present the structural hypotheses on 1.2) which are of use in the derivation of the relative entropy inequality and the proof of the desired convergence. In Section 3 we present the main theorems of this article for two different classes of source terms. In Section 4 we define the concept of relative entropy H r U ε, Mū)) and entropy fluxes Q r αu ε, Mū)). Section 5 contains the proof of the main result, which is based on error estimates for the approximation of the conserved quantities by the solution of the relaxation system. Applications to nonlinear elasticity and two phase flow models combustion) are presented in Section 6 and Section 7, respectively. Finally, Section 8 provides a general framework describing how, given a physical system governed by a symmetric hyperbolic balance law, one can construct an extended system and the resulting relaxation system for its approximation. 2 Notation and Hypotheses For the convenience of the reader we collect in this section all the relevant notation and hypotheses. Here and in what follows: 1. G, R, F α, α = 1,..., d denote the mappings G, R, F α : R N R N, whereas g, f denote the maps g, f : R n R n. In our presentation, GU), RU), F α U), gu), fu) are treated as column vectors. 3

4 2. D, D u denote the differentials with respect to the state vectors U R N and u R n respectively. When used in conjunction with matrix notation, D and D u represent a row operation: D = [ / U 1,..., / U N ], D u = / u 1,..., / u n ). 3. The symbol α denotes the derivative with respect to x α, α = 1,..., d. The summation convention over repeated indices is employed throughout the article: repeated indices are summed over the range 1,..., d. Motivated by theoretical studies [15, 18, 23] as well as computations devoted to the approximation of the hyperbolic systems of conservation laws and kinetic equations [5] by relaxation schemes, our analysis is based on the following assumptions: The manifold M of Maxwellians the equilibrium solutions U eq to the equation RU) = ) can be parameterized by n conserved quantities U eq = Mu), u R n. H1) RU) satisfies the nondegeneracy condition { dimn RMu))) = n dimr RMu))) = N n H2) There exists a projection matrix P : R N R n with rankp = n corresponding to Maxwellians that determines the conserved quantity u = PU and satisfies PMu) = u and PRU) = for all u R n, U R N. H3) In this case, the corresponding system of balance laws for conserved quantities is given by ) ) t u + α PF α Mu) = PG Mu) 2.1) which can be rewritten in the form with f, g defined by t u + α f α u) = gu) f α u) := PF α Mu)), gu) := PG Mu) ). 2.2) The system of balance laws 2.1) is resulting by applying P to 1.2), letting ε, and then using the fact that at the equilibrium U eq = Mu), u = PU eq. Our analysis exploits the entropy structure of the relaxation systems under consideration. Below are stated the main structural assumptions on 1.2). 4

5 2.1 Entropy Structure Some additional assumptions on the system 1.2) read: The system 1.2) is equipped with a globally defined entropy HU) and corresponding fluxes Q α U), α = 1,..., d, such that H : R N R is convex, DHU)DF α U) = DQ α U). H4) The entropy HU) is such that DU) := DHU)RU), U R N. H5) The entropy equation for the relaxation system 1.2) in that case is given by t HU) + α Q α U) = 1 DU) + DHU)GU). 2.3) ε Entropy consistency. The restriction of the entropy pair H, Q α, ηu) := H Mu) ), q α u) := Q α Mu) ), H6) on the equilibrium manifold M is an entropy pair η q α for the system of conserved quantities 2.1), that is, D u ηu)d u f α u) = D u q α u), u R n. In that case smooth solutions to 2.1) satisfy the additional balance law t HMu)) + α Q α Mu)) = D u ηu)gu). 2.4) In the sequel, we present some implications on the geometry of the manifold M obtained as a consequence of the entropy structure of the relaxation systems. We refer the reader to [23] for the details of the derivation in a relevant setting. 2.2 Properties of H, Q α on the manifold M The geometric implications of the assumptions are the following [23]: DHU)RU), RMu)) = rankp = n, PMu)) = u, u R n, U R N RDRMu))) = N P) DHMu)) [ DRMu))A ] =, u R n, A R N DHMu))V =, V R N with PV =. 2.5) 5

6 Thus, the entropy consistency hypothesis H6) along with the property 2.5) 3 imply that the gradients of entropies η, H are related by Then, in view of 2.2) 2, we have D u ηu)pa = DHMu))A, A R N. 2.6) D u ηu)gu) = DHMu))GMu)), u R n and thus the entropy equation 2.4) for conserved quantities may be written as 2.3 Dissipation t HMu)) + α Q α Mu)) = DHMu))GMu)). 2.7) Making use of the dissipation incorporated in the term DU) = DHU)RU) we introduce an additional hypothesis, which plays the role of relative dissipation, a measure of the distance between a relaxation state vector U R N and its equilibrium version Mu) R N with u = PU) on the manifold of Maxwellians M. More precisely, We assume that for some ν > [ DHU) DHMu)) ][ RU) RMu)) ] ν U Mu) 2 H7) for arbitrary U R N with u = PU. Note that H7) is stronger then the following assumption: For every ball B r R N there exists ν r > such that [ DHU) DHMu)) ][ RU) RMu)) ] ν r U Mu) 2 H7 ) for U, Mu) B r, where u = PU, which will be of use in Theorem 3.3. Our analysis handles a large class of source terms. The following hypothesis will be relevant to our subsequent discussion. The source term GU) is weakly dissipative with respect to the manifold M in the sense of Definition 2.1. Definition 2.1. We say that the source GU) is weakly dissipative with respect to the manifold M, if for all arbitrary U, Mū) R N [ DHU) DH Mū) )][ GU) GMū)) ]. H8) An alternative condition on the source GU), exploited in Theorem 3.2, reads: For every compact set A R N there exists L A > such that GU) GŪ) L A U Ū for all U RN, Ū A. H9) 6

7 2.4 Source consistency We first note that the hypothesis H8) is less restrictive than the requirement for G to be weakly dissipative, hence a special name for it: M-weakly dissipativity. We next point out that H8) requires certain consistency between the source terms GU) and gu) which are related by 2.2). Namely, take an arbitrary u R n and set U = Mu) in H8). Then, recalling 2.6) one concludes that H8) implies that the source gu) in the system 2.1) is weakly dissipative, that is ) ) D u ηu) D u ηū) gu) gū), u, ū R n. 2.8) Thus, H8) makes sense only when the source gu) in the equilibrium system 2.1) is weakly dissipative in the sense of 2.8). 2.5 Weak solutions and entropy admissibility We introduce the notions of weak solutions and entropy admissibility following the discussion in [12, Sec. 4.3, 4.5]. Definition 2.2. A locally bounded measurable function Ux, t), defined on R d [, T ) and taking values in an open set O R N, is a weak solution to t U + α F α U) = 1 ε RU) + GU), Ux, ) = U x), 2.9) with F, R, G : O R N Lipschitz, if T R d { } t Φ U + α Φ Fα U) dxdt + R d Φx, ) U x)dx T [ 1 ] + Φx, t) R ε RU) + GU) dxdt = d 2.1) for every Lipschitz test function Φx, t), with compact support in R d [, T ) and values in M 1 N. Definition 2.3. Assume that H, Q α is an entropy-entropy flux pair of 2.9). Then, a weak solution Ux, t) of 2.9), in the sense of Definition 2.2, defined in R d [, T ), is entropy admissible, relative to H, if T R d { } t φ HU) + α φ Q α U) dxdt + T + φx, t) DHU) R d φx, ) HU x))dx R d [ 1 ] ε RU) + GU) dxdt 2.11) for every nonnegative Lipschitz test function φx, t), with compact support in R d [, T ). 7

8 Remark 2.1. Note, a smooth solution U ε of 1.2) satisfies 2.11) identically as an equality and therefore it is admissible. It is worth pointing out that relaxation systems of type 1.2) are often designed to produce global smooth solutions. We refer the reader to [16, 26] as well as [12, Section 5.2] for further remarks. A more detailed discussion about the existence of smooth solutions follows in the sequel. 3 Main Results In this section we present the main results of this article. 3.1 M-weakly dissipative source GU) Theorem 3.1. Let ūx, t) be a smooth solution of the equilibrium system 2.1), defined on R d [, T ], with initial data ū x). Let {U ε x, t)} be a family of admissible weak solutions of the relaxation system 1.2) on R d [, T ), with initial data U ε x), and let u ε x, t) = PU ε x, t) denote the conserved quantity associated to U ε. Assume H1)-H8) hold and suppose that: i) HU), F U) in the relaxation system 1.2) satisfy for some M, µ, µ > µi D 2 HU) µ I, DF α U) < M, U R N. 3.1) ii) ηu) = HMu)), fu) = PF Mu)) satisfy for some K > D 3 uηu) K, D 2 ufu) K, u R n. 3.2) Then, for R > there exist constants C = CR, T, ū, M, K) > and s > independent of ε such that ) H r x, t)dx C H r x, ) dx + ε, a.e. t [, T ). ER) x <R x <R+st Moreover, if the initial data satisfy H r x, ) dx as ε, CD) x <R+sT then ess sup U ε Mū) 2 x, t) dx as ε. CS) t [,T ) x <R 8

9 3.2 General source GU) We now drop the assumption H8) which leads to the following theorem. Theorem 3.2. Let ū be a smooth solution of 2.1), defined on R d [, T ], with initial data ū, and {U ε } a family of admissible weak solutions of 1.2) on R d [, T ), with initial data U ε. Assume H1)-H7), H9) hold. Suppose that HU), F U), ηu), and fu) satisfy i)-ii) of Theorem 3.1. Then, for R > there holds the estimate ER) for some constants C = CR, T, ū, M, K, L) > and s = sm, µ ) > independent of ε. Moreover, if the initial data satisfy CD), then CS) holds. 3.3 Uniformly bounded ū, {U ε } If a priori bounds on the family of solutions {U ε } are available, then it is possible to weaken the requirements i) ii) of Theorems 3.1, 3.2. For example, one may weaken the assumption for H to be uniformly convex and DF α, D 2 uf α, and D 3 uη to be uniformly bounded. Theorem 3.3. Let ū be a smooth solution of 2.1), defined on R d [, T ], with initial data ū, and {U ε } a family of admissible weak solutions of 1.2) on R d [, T ), with initial data U ε. Assume H1)-H6), H7 ) hold. Suppose that: i) {U ε }, {Mu ε )} and Mū) take values in a ball B r R N. ii) HU) C 2 R N ) is strictly convex. F U), ηu), fu) are smooth. iii) The source GU) either satisfies H8) or is locally Lipschitz. Then, for R > there holds the estimate ER) for some constant C = C R, T, B r, ū W 1, C T,R) )) > and the constant 1 s = µ r sup D 2 HU)DF α V ), U,V B r α where C T,R) denotes a cone C T,R) = { x, t) : < t < T, x < R + st t) }, and µ r > is a constant such that µ r I < D 2 HU), U B r. Moreover, if the initial data satisfy CD), then CS) holds. 9

10 4 Relative Entropy To compare the solution U ε of the relaxation system 1.2) and the solution ū of the equilibrium system 2.1), we employ the notion of the relative entropy [9]. We define the relative entropy and entropy-fluxes [23] among the two solutions by H r U ε, Mū)) := HU ε ) HMū)) DHMū)) [ U ε Mū) ] Q r αu ε, Mū)) := Q α U ε ) Q α Mū)) DHMū)) [ F α U ε ) F α Mū)) ]. 4.1) By H5) we have DU ε ) = DHU ε )RU ε ), 4.2) that expresses the entropy dissipation of the relaxation system 1.2). In view of 2.5) 3 and the fact that RMu)) = for all u R n, DU ε ) may be written in an alternative form DU ε ) = [ DHU ε ) DHMu ε )) ][ RU ε ) RMu ε )) ] 4.3) where u ε = PU ε. Finally, we denote by SU ε, Mū)) := [ DHU ε ) DHMū)) ][ GU ε ) GMū)) ] 4.4) the term not necessarily dissipative) associated with the source GU). Let U U ε x, t) be a smooth solution of the relaxation system 1.2), ux, t) = PUx, t) be the conserved quantity associated to U and ūx, t) be a smooth solution of the equilibrium system 2.1). Then the relative entropy H r U, Mū)) satisfies Lemma 4.1 Relative entropy identity ). Suppose ūx, t) is a smooth solution of the equilibrium system 2.1), defined on R d [, T ], with initial data ū x). Let U U ε x, t) be any admissible weak solution of the relaxation system 1.2) on R d [, T ), with initial data U x), and let ux, t) = PUx, t) denote the conserved quantity associated to U. Then the relative entropy H r U, Mū)) satisfies T { } t φ H r U, ū) α φ Q r αu, ū) dxdt R d φx, ) H r U, Mū ))dx 4.5) R d T { 1 } ε D S + J 1 + J 2 + J 3 + J 4 dxdt R d φ for every nonnegative Lipschitz test function φx, t), with compact support in 1

11 R d [, T ), where J 1 := D 2 uηū) α ū ) fα u) f α ū) D u f α ū)u ū) ) J 2 := D 2 uηū) α ū ) [ P Fα U) F α Mu)) ] J 3 := gū) ) D u ηu) D u ηū) D 2 uηū) u ū) 4.6) J 4 := [ DHU) DHMu)) ] GMū)). If, in addition, {U ε } are smooth solutions, then they identically satisfy 2.11) as equality. As a consequence, the inequality 4.5) for the relative entropy H r becomes the identity t H r + α Q r α + 1 ε D + S = J 1 + J 2 + J 3 + J 4, x, t) R d [, T ). 4.7) Proof. Let us fix any nonnegative, Lipschitz continuous test function φx, t), compactly supported in R d [, T ). Since ū is smooth, from 2.7) it follows that ηū) = HMū)) satisfies the entropy identity t HMū)) + α Q α Mū)) = DHMū))GMū)) which in its the weak form reads T ) t φ HMū)) + α φ Q α Mū)) dxdt R d + φx, ) HMū )))dx + R d T R d φ DHMū))GMū))dxdt =. 4.8) Recall that U, an admissible weak solution of 1.2), with initial data U, must satisfy the inequality 2.11). Thus, upon subtracting 4.8) from 2.11), we obtain T { } t φ HU) HMū))) + α φ Q α U) Q α Mū))) dxdt R d T { [ 1 ] + φ DHU) R ε RU) + GU) d + φx, ) HU ) HMū )))dx. R d } DHMū))GMū)) dxdt Next, recalling that ū is a smooth solution of t ū + α PF α Mū) ) = PG Mū) ) and that PMū) = ū we obtain the identity T { } t Φ PMū) + α Φ PF α Mū)) dxdt R d + Φx, ) PMū x))dx + R d T 11 R d Φx, t) PGMū))dxdt = 4.9) 4.1) 4.11)

12 where Φx, t) is a Lipschitz continuous vector field with compact support in R d [, T ) and values in M 1 n. Also, since U is a weak solution of 1.2), it must satisfy 2.1) which, with Φ = Φ P M 1 N, reads T R d { } t Φ PU + α Φ PF α U) dxdt + Φx, ) PU x)dx + R d in view of the property PRU) =. T R d Φx, t) PGU)dxdt = 4.12) Now, we subtract 4.12) from 4.11), set the Lipschitz continuous vector field Φ = φ D u ηū), and recall the geometric relation 2.6), to get T { t φ DHMū)) [ U Mū) ] R d + α φ DHMū)) [ F α U) F α Mū)) ]} dxdt T { D 2 + φ u ηū) t ū ) [ ] P U Mū) R d + D 2 uηū) α ū ) [ P Fα U) F α Mū)) ] + DHMū)) [ GU) GMū) ]} dxdt + φx, ) DHMū )) [ U x) Mū ) ] dx = R d 4.13) The existence of an entropy pair η q α is equivalent to the property D 2 uηv)d u f α v) = D u f α v) D 2 uηv), v R n and therefore, in view of 4.1), we have ) D 2 uηū) t ū = D 2 uηū) D u f α ū) α ū + gū) = D u f α ū) D 2 uηū) α ū + D 2 uηū)gū). Hence we must have D 2 u ηū) t ū ) P [ U Mū) ] + D 2 u ηū) α ū ) P [ Fα U) F α Mū)) ] = D 2 uηū) α ū ) ) f α u) f α ū) D u f α ū)u ū) 4.14) + D 2 uηū) α ū ) P [ Fα U) F α Mu)) ] + D 2 u ηū)gū) ) u ū) where we used 2.2), the fact that u = PU, and PMū) = ū. 12

13 Combining 4.9) with 4.13) ) and recalling 4.2), 4.3) we obtain T { } t φ H r U, ū) + α φ Q r αu, ū) dxdt R d T + R d φ { 1 ε D S + J 1 + J 2 + J 4 D 2 uηū)gū) ) u ū) + [ DHMu)) DHMū)) ] } GMū)) dxdt + φx, ) H r U, Mū ))dx. R d Observe that, in view of 2.2), 2.6), we have [ DHMu)) DHMū)) ] GMū)) = Du ηu) D u ηū) ) gū) 4.15) and hence [ ] DHMu)) DHMū)) GMū)) D 2 u ηū)gū) ) u ū) = gū) ) D u ηu) D u ηū) D 2 uηū) u ū) = J ) Then from 4.15), 4.16) we get the desired inequality 4.5). 5 Proof of Theorems via Error Estimates To investigate the convergence of solutions {U ε } of the relaxation system 1.2) to Mū) in the smooth regime, one employs the inequality 4.5) derived in the previous section. The preliminary analysis of the inequality indicates that the evolution of H r, t) depends heavily on the properties of the entropy HU), flux F U), dissipative source RU) and, especially, the source GU). 5.1 Proof of Theorem 3.1 Proof. The argument follows along the lines of [1, Theorem 5.2.1]. Fix ε >. Since U ε is an admissible weak solution of 1.2) it must satisfy 2.11). Then [12, Lemma 1.3.3] implies that the map t HU ε, t)) is continuous on [, T )\F in L A) weak, for any compact subset A R d, where F is at most countable. We now fix R > and any point t [, T ) of L weak continuity of HU ε, t)) and let C t,r) denote the cone { } C t,r) = x, τ) : < τ < t, x < R + st τ) where s is a constant selected later. To prove the statement of the theorem we need to monitor the evolution of the quantity Ψτ) = Ψτ ; t, R) := H r x, τ) dx, τ t. x <R+st τ) 13

14 Clearly U ε, ū satisfy the assumptions of Lemma 4.1 and hence there holds the relative entropy inequality T R d { H r x, τ) t φ Q r αx, τ) α φ + 1 ε φd } dxdτ H r x, )φx, )dx R d T R d φ { S + J 1 + J 2 + J 3 + J 4 } dxdτ 5.1) where D, S, J k, k = 1,..., 4 defined by 4.4) and φ is nonnegative Lipschitz continuous function compactly supported in R d [, T ). Since the family {U ε } together with ū are not necessarily uniformly bounded, to handle the flux term Q r we need to exploit the uniform convexity of the entropy HU). From 4.1) and the assumption i) it follows that there exists c 1 > independent of ε such that H r U ε, Mū) ) c 1 U ε Mū) ) Now, by 4.1) 2 the relative entropy flux Q r α maybe written as Q r αu ε, Mū)) = 1 DQ α Ûβ))[ U ε Mū) ] dβ 1 [ DHMū)) DF α Ûβ))[ U ε Mū) ]] dβ 5.3) where Ûβ) := βu ε + 1 β)mū). Recalling H4) we have and hence 5.3) becomes Q r α = 1 DQ α Û)[ U ε Mū) ] = DHÛ)DF αû)[ U ε Mū) ] [DHÛβ)) DHMū)) ][ DF α Ûβ))[ U ε Mū) ]] dβ = [ U ε Mū) ] 1 1 β D 2 HŨ)DF αû) dγ dβ ) [U ε Mū) ] 5.4) where Ũβ, γ) := βγ U ε + 1 βγ)mū). Then, from 5.4) and i) we conclude that Q r α c 2 U ε Mū) 2 5.5) α for some c 2 = c 2 M, µ ) > independent of ε. Hence, in view of 5.2) and 5.5), we can choose s > such that sh r x, τ) + α x α x Qr αx, τ) >, x, τ) R d [, T ). 5.6) 14

15 Next, take δ > such that t + δ < T and select the test function φ = φx, τ) as follows cf. [12, Theorem 5.3.1]) φx, τ) = θτ)γx, τ) where 1, τ < t θτ) = 1 1 τ t), t τ t + δ δ, t + δ τ, 1, τ >, x R st τ) < γx, τ) = 1 1 ) x R st τ), τ >, < x st τ) R < δ δ, τ >, δ < x R st τ) and use it in 5.1). This gives 1 δ = t+δ t + 1 δ + 1 ε t x <R t t H r x, τ) dxdτ H r x, ) dx x <R+st x <R+st τ) < x R st τ)<δ x <R+st τ) sh r x, τ) + α D dxdτ + Oδ) S + J1 + J 2 + J 3 + J 4 ) dxdτ. x ) α x Qr αx, τ) dxdτ 5.7) We next let δ + in 5.7). The second integrals in 5.7) is nonnegative in view of 5.6). Recalling H7) and using the fact that H r U ε, τ), ū, τ)) is weak continuous in L at τ = t we conclude H r x, t)dx + ν U ε Mu ε ) 2 dxdτ + S dxdτ x <R ε C t,r) C t,r) ) 5.8) H r x, )dx + J 1 + J 2 + J 3 + J 4 dxdτ. x <R+st C t,r) We next estimate the terms on the right-hand side of 5.8). Recalling 4.6) and using i), ii), and the Young s inequality we obtain J 1 + J 3 dxdτ C U ε Mū) 2 dxdτ C t,r) C t,r) J 2 + J 4 dxdτ ν U ε Mu) 5.9) 2 dx + Cε, C t,r) ε C t,r) 15

16 where the constant C = Ct, R, u, ū, M, K) > depends on the norms ū W 1, C t,r) ), ū W 1,2 C t,r) ), 5.1) and constants M, K are introduced in i), ii). Finally, by H8) C t,r) SU ε, Mū)) dxdτ. 5.11) Then, combining 5.8)-5.11) and recalling i) 1 we conclude Ψt; t, R) Ψ; t, R) + C ε + t ) Ψτ ; t, R) dτ. Since R > and t [, T ] in the above inequality are arbitrary, we conclude via the Gronwall lemma. Remark 5.1. The terms J 1, J 3 in the proof of Theorem 3.1) are bounded by CH r U ε, Mū)), in view of 3.1) 1, 5.9) 1. This is one of the key features of the calculations that eventually leads to the use of the Gronwall lemma. The term SU ε, Mū)) has a quadratic structure similar to that J 1, J 3 and thus, one may think that there is no need in requiring H8). To this end, we point out that if H8) does not hold, then one has to make sure that SU ε, Mū)) dx c C t,r) H r U ε, Mū)) dx C t,r) 5.12) with c = ct, R) > independent of ε in order to exploit Gronwall lemma), and this is not true in general. In this case, to ensure 5.12), one has to impose certain regularity conditions on the source function GU). 5.2 Proof of Theorem 3.2 In this section we drop the assumption H8) and following Remark 5.1 require the source GU) to satisfy H9). This will ensure 5.12) and thus following the analysis in the proof of Theorem 3.1 we obtain the result. 5.3 Proof of Theorem 3.3 In the previous two sections we established convergence of weak solutions of the relaxation system 1.2) to the equilibrium system via the error estimate on the cone C R,t). Observe, however, that the bounds imposed on D 2 H, D 3 uη and D u f α, DF α in Theorem 3.1 are global. In particular, the requirement that H is uniformly convex on R N which is used to handle the flux Q r α on the boundary of the cone, see 5.6)) is a very stringent condition that narrows significantly the class of systems to which our error analysis may be applied. Let us note at this point that if a priori local) bounds on the family of solutions {U ε } are available, then it is possible to weaken the requirements 16

17 i) ii) of Theorems 3.1, 3.2. For example, one may weaken the assumption for H to be uniformly convex and DF α, D 2 uf α, and D 3 uη to be uniformly bounded. This is indeed the case and the proof of Theorem 3.3 follows using the line of argument presented in the proof of Theorem Application to Elasticity Consider the relaxation of the isothermal/isentropic) elasticity system: ) ) ) u v = gu, v) = v σu) g 2 u, v) with the stress σu) such that t x 6.1) σ) = and < γ < σ u) < Γ for all u R n. 6.2) We assume that the source gu, v) satisfies one of the following: i) Either g is independent of u, that is gu, v) = gv), and satisfies g2 v) g 2 v) ) v v ), v, v R, 6.3) ii) or for every compact set A R 2 there exists L A > such that g 2 u, v) g 2 ū, v) L A u ū + v v ) 6.4) for all u, v) R 2, ū, v) A. The system 6.1) is equipped with the entropy - entropy flux pair η, q given by ηu, v) = 1 2 v2 + Σu), qu, v) = σu)v with Σu) := u στ)dτ. 6.5) Relaxation via stress approximation. Consider the following extended system which approximates the stress σu): u v v α + Eu = 1 Ru, v, α) + Gu, v, α) 6.6) ε α with t Ru, v, α) =,, hu) α ), and the function hu) defined by x Gu, v, α) =, g2 u, v), ) hu) = σu) Eu with E > Γ. 6.7) 17

18 Observe that as ε, the variable α tends to its equilibrium state α eq = hu). Thus, the corresponding equilibrium states u eq, v eq satisfy 6.1). This motivates the parameterization of the manifold of Maxwellians by Mu, v, α) = u, v, hu) ) which implies H1). Next, we easily check that dim N RMu, v))) = 2, dim R RMu, v))) = 1 which verifies H2) for n = 2, N = 3. Also, the structure of 6.1), 6.6) suggests the choice of the projection matrix [ ] 1 P = 1 for which PMu, v) = u, v), PRu, v, α) = and hence H3) is satisfied. At this point, we identify the corresponding entropy-entropy flux pair of the system 6.6) and verify the remaining hypotheses that allow one to apply the theory developed in the preceding sections. In view of the requirement 6.2) for the stress σu), hu) : R R is strictly decreasing, onto and satisfies h) =. Hence h 1 : R R is well-defined. Then, we set Hu, v, α) := 1 2 v Eu2 + αu Qu, v, α) := α + Eu)v. α h 1 ξ)dξ It is easy to check that H, Q is the entropy-entropy flux pair for the system 6.6). Next, we observe that the entropy H maybe written as where Hu, v, α) = v2 2 + γu2 α + Êu)2 + ψα) + 4 2Ê ψα) := From 6.2) we have α h 1 ξ) ξ ) dξ, Ê := E γ Ê 2 >. 6.8) ψ α) = h 1 α) 1 Ê 1 = E σ h 1 α)) ) 1 E γ 2 ) Then 6.8), 6.9) imply that there exist µ, µ > such that γ 6.9) 2E γ)e γ 2 ) >. µi D 2 Hv, u, α) µ I, u, v, α) R 3 6.1) 18

19 and hence the pair H, Q satisfies H4). Next, we compute and observe that by 6.2) DHu, v, α) = Eu + α, v, u h 1 α)) DHu, v, α)ru, v, α) = u h 1 α) ) α hu) ) 1 E α hu) ) ) which implies H5). We next check the entropy consistency between the systems 6.1), 6.6). First we observe that Also, we have where qu, v) := QMu, v)) = hu) + Eu)v = σu)v. 6.12) ηu, v) := HMu, v)) = 1 2 v2 + Σu) + ku), hu) ku) := 1 2 Eu2 + hu)u h 1 ξ)dξ u σξ) dξ. From 6.7) it follows that k) =, and k u) = for all u R and hence ηu, v) = 1 2 v2 + Σu). 6.13) Then, 6.5), 6.13), 6.12) imply H6). Next, notice that u, v, α) Mu, v) = α hu) and hence 4.2), 4.3), and 6.11) imply H7) with ν = 1 E. Finally, we observe that [ DHu, v, α) DHMū, v)) ][ Gu, v, α) GMū, v)) ] = v v ) g 2 u, v) g 2 ū, v) ) = [ Dηu, v) Dηū, v) ][ gu, v) gū, v) ] 6.14) for each u, v, α), Mū, v) R 3. Then, if the source gu, v) satisfies 6.3), then 6.14) implies H8). If, on the other hand, gu, v) satisfies 6.4), then 6.14) implies H9). Thus, if { u ε, v ε, α ε ) } is a uniformly bounded family of weak solutions, one may apply Theorem 3.3 to establish convergence before formation of shocks. If such a priori information is not available, then, in addition to 6.2)-6.4), require that σ u) K, u R. 6.15) In that case, from 6.1), 6.15) it follows that 3.1), 3.2) hold and therefore one may apply Theorems 3.1, 3.2 depending on the type of source term). 19

20 Remark 6.1. Replacing 6.3) with the weakly dissipative condition g2 v) g 2 v) ) v v ) c v v 2, v, v R, the relaxation system falls into the framework of [16, 26], which provides global smooth solutions for small initial data. The case of Lipschitz source terms can also be handled following similar line of argument as in [16, 26]. Note that the same follows for the combustion model presented below, which has a Lipschitz source term. 7 Application to Combustion The governing equations for chemical reaction from unburnt gases to burnt gases in certain physical regimes read [8]: t v x u = t u + x P v, s, Z)) = t Ev, s, Z) u2 + qz ) t + xup v, s, Z)) = r t Z + KφΘv, s, Z))Z =. 7.1) The state of the gas is characterized by the macroscopic variables: the specific volume vx, t), the velocity field ux, t), the entropy sx, t) and the mass fraction of the reactant Zx, t), whereas the physical properties of the material are reflected through appropriate constitutive relations which relate the pressure P v, s, Z), internal energy Ev, s, Z) with the macroscopic variables. Here, and in what follows, q represents the difference in the heats between the reactant and the product, K denotes the rate of the reactant, whereas φθ) is the reaction function. The function rx, t) represents a source term additional radiating heat density). Isentropic combustion. In this section we address the problem of relaxation to the isentropic combustion model: v u u + P v,z) Z t x = KφΘv, Z)) 7.2) that arises naturally from 7.1) by externally regulating r to ensure s = s [9], in which case we suppress variable s and use the notation P v, Z) := P v, s, Z), Θv, Z) := Θv, s, Z). 7.3) Motivation for assumptions. Our main objective is to find a proper extended system associated with the system 7.2) that models isentropic processes with specific volume v away from both zero and vacuum, that is, when v has upper and lower bounds, v v V for some v, V, ). 7.4) 2

21 For the rest of the paper we assume that the a priori bound 7.4) holds. The physics of isentropic) thermodynamical processes determined by the equations 7.1) and compatible with the Clausius-Duhem inequality require the choice of the pressure P v, Z) and temperature Θv, Z) which are compatible with the following properties: for v [v, V ], s = s, and Z [, 1] P v, z) = v Ev, s, Z) >, Θv, Z) = s Ev, s, Z) > 7.5) for some appropriate) energy function Ev, s, Z) > with E Z v, s, Z) >. 7.6) We remark that such a function Ev, s, Z) is known to exist for the system 7.1) as long as v, s have lower and upper bounds [8]. For technical convenience, outside of the interval 7.4), we redefine the constitutive law Ev, s, Z) ensuring that the functions P v, Z), Θv, Z) are defined for all v R, Z [, 1] with bounded derivatives as indicated below. Conditions on P, Θ. a1) Motivated by the physical property v P < we assume that a2) There exists C > such that v < γ < v P v, Z) < Γ, v R, Z [, 1]. 7.7) P ZZ τ, Z)dτ < C, Z P v, Z) < C, v R, Z [, 1]. 7.8) a3) The composition φ Θ of the rate and constitutive temperature functions satisfies for some L > φθv, Z)) φθ v, Z)) L v, Z) v, Z) 7.9) for all v, Z), v, Z) R [, 1]. Under a1)-a3) the system 7.2) admits an entropy-entropy flux pair η, q of the form: ηv, u, Z) = 1 v ) 2 u2 P τ, Z)dτ + BZ) 7.1) qv, u, Z) = P v, Z)u, where BZ) is an arbitrary function. Relaxation via approximation of pressure. In the spirit of the example for the elasticity system 6.1) we define hv, Z) := P v, Z) Ev with E > Γ. 7.11) 21

22 We now approximate the pressure P v, Z) by the linear combination α+ev). This leads to the extended system, where v u u α + Ev Z α t x = 1 Rv, u, Z, α) + Gv, u, Z, α), 7.12) ε Rv, u, Z, α) = [,,, hv, Z) α ] Gv, u, Z, α) = [,, KφΘ)Z, ]. 7.13) Note that as ε, α tends to its equilibrium state α eq = hv eq, Z eq ). Then α eq + Ev eq = P v eq, Z eq ) and hence v eq, u eq, Z eq ) solves 7.2). This motivates the parameterization of the manifold of Maxwellians M by Mv, u, Z) = [ v, u, Z, hv, Z) ] which yields H1). Next, we compute DRv, u, Z, α) = h v v, Z) h Z v, Z) 1 from which we conclude dim N DRMv, u, Z)) ) = 3, dim R DRMv, u, Z)) ) = 1 which verifies H2) for n = 3, N = 4. We choose the projection matrix P = for which PMv, u, Z) = v, u, Z), PRv, u, Z, α) = and hence H3) holds. Entropy of the extended system. We next specify the entropy-entropy flux pair of the relaxation system 7.12). By 7.7) < E Γ < h v v, Z) < E γ. 7.14) Hence there exists jα, Z) : R [, 1] R such that jhv, Z),Z) = v, hjα, Z),Z) = α 7.15) 22

23 for all v, α R, Z [, 1]. Thus, we define α Hv, u, Z, α) := u2 2 jξ, Z)dξ + αv + Ev2 h,z) 2 + BZ) Qu, v, α) := α + Ev ) u with BZ) an arbitrary function such that where the constant m > is to be specified. B Z) > m >, Z [, 1] 7.16) It is easy to check that H, Q is the entropy-entropy flux pair for 7.12). To show that HU) is strictly convex, however, is less trivial and therefore, for the convenience of a reader, we provide detailed calculations. Recalling that E > Γ > γ we rewrite Hv, u, Z, α) as follows: with u 2 Hu, v, Z, α) = 2 + γv2 4 ) α + Êv ) 2 + ψα, Z) + 2Ê α ψα, Z) := jξ, Z)dξ α2 h,z) 2Ê + BZ), Ê := E γ 2. We now show that there exists Λ > such that 7.17) Λ 1 I D 2 ψα, Z) ΛI 7.18) by establishing the bounds on the eigenvalues of D 2 ψα, Z) = j αα, Z) Ê 1 j Z α, Z) ) j Z α, Z) B α Z) ZZ h,z) jξ, Z)dξ. 7.19) Differentiating 7.15) 2 and recalling 7.11) we get j α α, Z) = 1 ), j Z α, Z) = P Zjα, Z), Z) h v jα, Z), Z h v jα, Z), Z) 7.2) and hence by 7.8), 7.14) 1 E γ j αα, Z) 1 E Γ, j Z α, Z) C E Γ. 7.21) Then by 7.21) 1 γ [ 2E γ)ê D 2 ψα, Z) ]11 Γ 1 2 γ E Γ)Ê. 7.22) 23

24 Next, using 7.15), 7.2) 1,2 we compute α ) Z jξ, Z)dξ = and hence = = h,z) α h,z) α h,z) ZZ α h,z) ) j Z ξ, Z)dξ jh, Z),Z)h Z, Z) P Z jξ, Z), Z)j α ξ, Z)dξ = ) jξ, Z)dξ = P Z jα, Z), Z)j Z α, Z) + jα,z) jα,z) P Z τ, Z)dτ P ZZ τ, Z) dτ. Then, by 7.8), 7.21) 2 we conclude α ZZ jξ, Z) dξ) C 1 + C ). 7.23) E Γ The analysis of the above inequalities motivates to choose m := m + C 1 + C ) E Γ with m := [ ) ] C2 2 2E γ) Ê + 1 E Γ γ in which case by 7.16), 7.19) and 7.23) we obtain [ ] < m D 2 ψα, Z) 2m. 7.24) Combining 7.21) 2, 7.22), and 7.24) we get [ ] 1 det D 2 ψα, Z) = λ 1 λ 2, 7.25) where λ 1, λ 2 R denote the largest and smallest eigenvalues of D 2 ψ, respectively. Observe that 7.19), 7.22), and 7.24) imply < λ 1 Λ := 2m + Γ γ 2 E Γ)Ê + C 2 ). 7.26) E Γ Then, from 7.25), 7.26) we obtain the estimate 7.18). Combining 7.17), 7.18) we conclude that for some µ, µ > and this yields H4). 22 µi D 2 Hv, u, Z, α) µ I 7.27) 24

25 Now, recalling 7.13), 7.15), and 7.21) 1 we obtain DHv, u, Z, α)rv, u, Z, α) = jhv, Z), Z) jα, Z) ) hv, Z) α ) [ 1 ) ] hv, ) 2 = j α shv, Z) + 1 s)α, Z ds Z) α 1 E γ ) 2 1 hv, Z) α = Mv, u, Z) v, u, Z, α) 2 E γ which implies that the entropy H satisfies hypotheses H5), H7). Next, we observe that 7.11), 7.15), and 7.2) 1 imply hv,z) h,z) jξ, Z)dξ = v h v τ, Z)τ dτ = hv, Z)v + Ev2 2 + v P v, Z)dτ. Thus, the entropy pair H, Q restricted to the equilibrium manifold satisfies ηv, u, Z) := HMv, u, Z)) = u2 2 v P v, Z)dτ + BZ) qv, u, Z) := QMv, u, Z)) = hv, Z) + Ev ) u = P v, Z)u. Then, 7.28) together with 7.1) yields H6). 7.28) Consider an arbitrary compact set A R R [, 1] R. Then, by 7.9) for all v, u, Z, α) R R [, 1] R, v, ū, Z, ᾱ) A, we have Gv, u, Z, α) G v, ū, Z, ᾱ) = KφΘv, Z))Z KφΘ v, Z)) Z K Z φθv, Z)) φθ v, Z)) + φθ v, Z)) Z Z ) L + L A ) K v, u, Z, α) v, ū, Z, ᾱ), where L A > denotes a constant for which, in view of 7.9), there holds φθ v, Z)) L A, v, ū, Z, ᾱ) A. 7.29) The estimate 7.29) implies that the source G satisfies the hypothesis H9) on R R [, 1] R, the state space of 7.12) with initial data such that Z, ) 1. Thus, if the family { v ε, u ε, Z ε, α ε ) } is uniformly bounded, one may apply Theorem 3.3 to establish convergence before the formation of shocks. If such a priori information is not available, then, in addition to a1)- a3), require that D 2 P v, Z) K, B Z) < K, v R, Z [, 1] R. 7.3) In that case, from 7.27), 7.3) it follows that 3.1), 3.2) hold and therefore one may apply Theorem

26 8 General framework for symmetric hyperbolic systems, d = 1 In this section we present a general strategy indicating how starting from a symmetric hyperbolic system one can construct an extended relaxation system. Consider the hyperbolic balance law such that: t u + x fu) = gu), u, fu), gu) R n 8.1) The flux fu) has symmetric D u fu). Thus, fu) = DΦ u) for some Φu) : R n R. h1) Φ is convex and for some Γ, γ > such that < γ < D 2 Φu) < Γ, u R n. h2) For each compact A R n there exists L A > such that gu) gū) L A u ū, u R n, ū A. h3) By h1) the system 8.1) admits entropy-entropy flux pair ηu) = Φu), qu) = 1 2 DΦu) 2. Relaxation via flux approximation. Next, we approximate the flux fu) by the combination α + DE u), where α R n is a new vector variable and Eu) : R n R is a convex function such that for some E, δ > there holds E + δ)i D 2 ueu) E I, E > Γ > γ > δ >. h4) This leads to the relaxation system for variables u, α R n ) ) u α + DE u) + = 1 Ru, α) + Gu, α) 8.2) α ε with We now define t x Ru, α) = [, hu) α ], Gu, α) = [ gu), ]. 8.3) Σu) := Eu) Φu), hu) := D u Σ u) = fu) D u E u). 8.4) Then, by h2), h4) we have The mapping D 2 ue E I > E + δ γ)i > D 2 uσ > E Γ)I. 8.5) as implied by the following lemma c.f. Zeidler [27]). D u Σ : R n R n is onto 8.6) 26

27 Lemma 8.1. Suppose V u) : R n R n is a C 1 -mapping such that DV u) : R n R n is invertible for all u R n and the map V u) is coercive, that is, V u) u > c u 2, u R n for some fixed constant c >. Then, V must be surjective and covers all of R n. Observe that as ε, α tends to its equilibrium state α eq = hu eq ) in which case the corresponding equilibrium state u eq satisfies 8.1). This suggests the parameterization of the manifold of Maxwellians by which yields H1). Next, observe that Mu) = [ u, hu) ] dim N DRMu))) = n, dim RDRMu))) = n which verifies H2) with N = 2n. The structure of 8.1), 8.2) suggests the choice of the projection matrix P = [ I, ] : R 2n R n for which PMu) = u, PRu, α) = which implies H3). To construct the entropy-entropy flux pair for the relaxation system 8.2) we exploit the ideas of the analysis of A. Tzavaras [24]. By 8.5), 8.6) the map DΣ is bijective. This motivates the definition of jα) : R n R n by jα) = D u Σ) 1 α), α R n. 8.7) Then, by the inverse mapping theorem, D α jα) is symmetric and hence there exists Jα) : R n R such that D α Jα) = jα) = DΣ) 1 α) D 2 αjα) = [ D 2 uσ D α J α)) ] 1 = [ D 2 u Σ jα)) ] ) Furthermore, by 8.5) we obtain that Jα) is uniformly convex with E + Γ) 1 I > D 2 αjα) > E + δ γ) 1 I. 8.9) We next define Hu, α) = Eu) + α u + Jα) Qu, α) = 1 α + DE u) ) It is easy to verify that H, Q is the entropy-entropy flux pair for the system 8.2). To show that Hu, α) is strictly convex, we compute the Hessian [ ] D 2 D 2 u Eu) I Hu, α) = I D 2 αjα) 27

28 and write u, α) [ D 2 Hu, α) ] u, α) = u [ D 2 Eu) ] u + 2α u + α [ D 2 Jα) ] α. Then, recalling h4), 8.5), and 8.9) we get the estimates and u, α) [ D 2 Hu, α) ] u, α) > 1 2 γ δ) u γ δ) α 2 E δ γ)) E + δ γ ) u, α) [ D 2 Hu, α) ] u, α) E + δ + 1) u 2 + E Γ) ) α 2. The above inequalities and the fact that γ > δ imply that there exist µ, µ > such that µ I D 2 Hu, α) µi, u, α) R n+n 8.11) and hence we conclude that the pair H, Q satisfies H4). Next, we compute and observe that by 8.4) 1, 8.7), DHu, α) = [ D u Eu) + α, u + D α Jα) ] jhu)) = DΣ) 1 hu)) = DΣ) 1 DΣ u)) = u. 8.12) Hence recalling 8.3) 1, 8.4), 8.8) 1, and 8.9) we obtain DHu, α)ru, α) = = u + D α Jα) ) α hu) ) = jα) jhu)) ) ) α hu) = α hu) ) [ 1 D 2 J sα + 1 s)hu) ) ] α ) ds hu) 1 α hu) 2 1 = u, α) Mu) 2. E + δ γ) E + δ γ) The last inequality implies that H satisfies hypotheses H5), H7). Next, observe that by 8.4), 8.1) ηu) := HMu)) = Hu, hu)) = Eu) + hu) u + Jhu)) = Φu) + Σu) [D u Σu)]u + Jhu)). 8.13) Then, by 8.4), 8.8) 1, and 8.12) D u ηu) = D u Φu) [ D 2 uσu) ] u + jhu)) [ D 2 uσu) ] = D u Φu) 28

29 and we conclude ηu) = Φu) + C for some C R. Similarly, by 8.4), 8.1) qu) := QMu)) = Qu, hu)) = 1 2 hu) + DE u) 2 = 1 2 DΦu) ) By the discussion in the beginning of the section we conclude that η, q defined in 8.13), 8.14) is an entropy-entropy flux pair of 8.1) which implies H6). Now, take an arbitrary compact set C R n R n and define A = { ū R n : for some ᾱ R n ū, ᾱ) C } which is compact as well. Then, by h3) for all u, α) R n+n, ū, ᾱ) C Gu, α) Gū, ᾱ) = gu) gū) L A u, α) ū, ᾱ). The above estimate shows that Gu, α) satisfies H9). Thus, the relaxation system 8.2) satisfies H1)-H7), H9). Thus, if { u ε} is uniformly bounded family of weak solutions, one may apply Theorem 3.3 to establish convergence. If such a priori information is not available, then, in addition to h1)-h4), require that D 3 Φu) K, u R n. 8.15) In that case, from 8.11), 8.15) it follows that 3.1), 3.2) hold and therefore one may apply Theorem 3.2 to establish convergence in the smooth regime. 9 Acknowledgements The authors thank Thanos Tzavaras for several fruitful conversations and suggestions during the course of this investigation. A.M. thanks Robin Young for advice and helpful comments regarding the combustion model. K.T. acknowledges the support by the National Science foundation under the grant DMS and the support by the Simons Foundation under the grant # References [1] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Diff. Equations, 25, , 2. [2] Y. Brenier, R. Natalini and M. Puel, On the relaxation approximation of the incompressible Navier-Stokes equations, Proc. Amer. Math. Soc., 132, , 24. [3] F. Berthelin and A. Vasseur, From Kinetic Equations to Multidimensional Isentropic Gas Dynamics Before Shocks, SIAM J. Math. Anal., 36, ,

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