Stability and Convergence of Relaxation Schemes to Hyperbolic Balance Laws via a Wave Operator

Transcription

1 Stability and Convergence of elaxation Schemes to Hyperbolic Balance Laws via a Wave Operator In: Journal of Hyperbolic Differential Equations 215, 12-1, Alexey Miroshnikov and Konstantina Trivisa Abstract This article deals with relaxation approximations of nonlinear systems of hyperbolic balance laws. We introduce a class of relaxation schemes and establish their stability and convergence to the solution of hyperbolic balance laws before the formation of shocks, provided that we are within the framework of the compensated compactness method. Our analysis treats systems of hyperbolic balance laws with source terms satisfying a special mechanism which induces weak dissipation in the spirit of Dafermos [11], as well as hyperbolic balance laws with more general source terms. The rate of convergence of the relaxation system to a solution of the balance laws in the smooth regime is established. Our work follows in spirit the analysis presented in [1, 14] for systems of hyperbolic conservation laws without source terms. 1 Introduction elaxation approximations of hyperbolic balance laws is of essence for the investigation of models arising in continuum mechanics and kinetic theory of gases, and serve as a ground stage for the design of numerical schemes for hyperbolic balance laws. In this article we introduce a class of relaxation schemes for the approximation of solutions to the hyperbolic balance law t u + d xj F j u = Gu, u n, x, t d [, 1.1 j=1 and address the issues of stability and convergence. The class of relaxation schemes introduced in this work are of the form d t u + xj v j = j=1 t v i + A i xi u = 1 ε v i F i u + i x, t, i = 1,..., d 1.2 Department of Mathematics, University of Massachusetts, Amherst, MA 13, USA Department of Mathematics, University of Maryland, College Park, MD 2742, USA 1

2 with v i n, A i symmetric, positive definite matrix, and i x, t = 1 d xi Gux1,..., x i 1, z, x i+1,..., x d, t dz. 1.3 Excluding v i for all i = 1,..., d from the equation we obtain d d t u + xj F j u = Gu + ε A j u xj x j u tt j=1 j=1 1.4 that approximates the system of balance laws 1.1. For the relaxation approximations considered in this work the stabilization mechanism is the regularization by the wave operator in 1.4. The convergence properties of relaxation systems and associated relaxation schemes for scalar conservation laws are presently well understood see [2, 8, 14, 17]. When the zero-relaxation limit is a system of conservation laws, the dissipative effect of relaxation is often subtle to capture, yet there is a vast literature on convergence results in that direction see [1, 23, 21, 28, 29] and the references there rein. By contrast, the relaxation approximation of nonlinear systems of hyperbolic balance laws presents major additional challenges and it is the subject of current intense research investigation. It is well known that standard methods that solve correctly systems of conservation laws can fail in solving systems of balance laws, especially when approaching equilibria or near to equilibria solutions. In addition, standard approximating procedures produce often unstable methods when they are applied to coupled systems of conservation or balance laws. Even at the theoretical level, the presence of the production term source term in the equation results to the amplification in time of even small oscillations in the solution. Due to these challenges special mechanisms that induce dissipation are often desirable and have been proven effective in obtaining long-term stability cf. Dafermos [11], [12]. 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: A new class of relaxation schemes 1.2 is introduced. The novelty of the relaxation systems proposed in this work lies in the introduction of the global term { i x, t} in The presence of this term in the relaxation system allows us to relax both the flux and the antiderivative of the source term simultaneously. We refer the reader to the article [19], where a relevant idea was proposed for the numerical treatment of shallow water equations. The present article is the first step towards the construction of fully discrete schemes and the development of numerical methods for the approximation of solutions to complex nonlinear multidimensional systems of hyperbolic balance laws arising in applications. Our analysis provides a rigorous proof of the relaxation limit and a rate of convergence before the formation of shocks. A comparison is presented between the relaxation system introduced here and an alternative relaxation system for which the source term Gu appears in the right-hand side of the first 2

3 equation see 7.1 in Section 6. Note, dealing with 7.1 one faces arduous challenges. More specifically, the time derivative of the source term appears in the energy functional posing enormous challenges in the analysis, an additional hypothesis is required for the establishment of stability see H7, Section 6, the issue of compactness is problematic. The presence of a source term in our system requires us to modify the relative entropy method significantly. The modified relative method presented in this work relies on a relative potential Section 8. The introduction of this concept is required in order to deal with the source term G which typically satisfies no growth conditions. The relative potential assists in tracking the contribution of the source term; it becomes a part of a Lyapunov functional which monitors the evolution of the difference of the solutions and enables us to establish a convergence rate of order Oε 2. In the case of the general source G the terms associated with the source are treated as error cf. Section 6.4. The reader should contrast the relaxation system presented in this work with other relaxation systems proposed in the literature [16, 17], where relaxation approximations to hyperbolic balance laws were proposed and rigorously established. In [16] relaxation approximations are constructed by the introduction of special variables the so-called internal variables. In that setting applications to physical systems in elasticity and combustion theory are presented. In the former case, relaxation is introduced via stress approximation, whereas in the latter case via approximation of pressure. eference [17] treats the Cauchy problem for 2 2 semilinear and quasilinear hyperbolic systems with a singular relaxation term and presents the convergence to equilibrium of the solutions of these problems as the singular perturbation parameter tends to. In the center of our analysis lies the entropy structure of the balance law and the dissipative nature of the source term. The main ingredients of our approach can be formulated as follows: The representation of the global term in the formulation of the relaxation system enables us to obtain the stability estimates in Section 3. These estimates are used subsequently to establish the stability of the relaxation approximations and in fact justify the dissipative character of our systems. The entropy structure of the balance law provides the basis for the use of the compensated compactness method. ecall that a pair of functions η = ηu, q = qu are called the entropy-entropy flux pair if η, q solve the linear hyperbolic system Dq = DηDF. In Section 5, we show that the relaxation approximations satisfy t ηu ε + x qu ε compact set of H 1 loc O for a certain class of entropy-entropy flux pairs η q, which is one of the main ingredients for the establishment of convergence results within the compensated compactness method cf. Serre [24]. The Lyapunov functional construction in Section 8 relies on the relative entropy via a Chapman- Enskog-type expansion and is used to provide a simple and direct convergence framework before formation of shocks as well as suitable error estimates. 3

4 A physically motivated dissipation mechanism associated with the source term in 1.1. The concept of weak dissipation for hyperbolic balance laws was introduced by Dafermos in [11]. The reader may contrast the result of Theorem for weakly dissipative source terms with Theorem which corresponds to the case of a general source. The outline of our article is as follows: In Section 2 we present the basic notation and hypothesis. In Section 3 we present the stability estimates which yield the stability of the relaxation systems. The compactness properties of the approximate solutions are discussed in Section 4. Section 5 is devoted to error estimates for smooth solutions via the relative entropy method as well as proofs of convergence. The multidimensional case is treated in Section 7. Section 8 presents applications in elasticity and combustion. 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,, F i, i = 1,..., d denote the mappings G,, F i : n n. In our presentation, Gu, u, F i u are treated as column vectors. 2. D denotes the differential with respect to the state vectors u n. When used in conjunction with matrix notation, D represents a row operation: 2.1 Entropy Structure D = [ / u 1,..., / u n ]. Some additional assumptions on the system 1.1 read: The system 1.1 is equipped with a globally defined entropy ηu and corresponding fluxes q i u, i = 1,..., d, such that η : n is strictly convex, DηuDF i u = Dq i u β I D 2 ηu 1 2 αi, u n, for α, β >, ηu η =, Dη =. H1 We recall that 2 2 as well as physical systems of hyperbolic conservation laws are always equipped with an entropy-entropy flux pair. The same holds true for symmetric hyperbolic systems [12]. 2.2 Subcharacteristic condition The Whitham relaxation subcharacteristic condition presented below will be essential for the dissipativiness of our system [2, 12, 15, 17]. 4

5 Case d = 1. Systems with a strictly convect entropy. For α > in H1 there exists ν > and symmetric, positive definite matrix A such that 1 2 AD 2 ηu + D 2 ηua αdf u DF u νi, u n. H2 The positivity of this term is required for dissipativiness in Lemma General case d 1. Systems with a strictly convect entropy. For α > in H1 there exists ν > and symmetric, positive definite matrices A j, j = 1,..., d such that 2.3 Dissipation 1 2 ξ j Aj D 2 ηu + D 2 ηua i ξj α d 2 DF u ξ j ν The following hypotheses will be relevant to our subsequent discussion. i=1 d ξ j, i=1 ξ 1,..., ξ d n, u n. The source term Gu is weakly dissipative in the sense of Definition Definition We say that the source Gu is weakly dissipative, if H2* Dηu Dηū Gu Gū, u, ū n. H3-a An alternative condition on the source G, exploited in Theorem 5.4.1, reads: Suppose that for every compact set A there exists L A > such that Gu Gū L A u ū, u n, ū A. H3-b Through out the article we use the assumptions that G = and G C n. 2.4 Source potential For the weakly dissipative source G we employ an additional assumption that it is a gradient: Suppose there exists a potential u : n such that Gu = Du u =, D = Du = Gu C 1 + u, u n. H4 5

6 3 Stability estimates 3.1 Systems with a strictly convex entropy η, d = 1. The balance law 1.1 for d = 1 reads and the relaxation model by t u + x v = t v + A x u = 1 ε t u + x F u = Gu, u n 3.1 x v F u + G ux, t 3.2 dx with A symmetric, positive definite matrix. In that case the second order relaxation system 1.4 reads t u + x F u = Gu + εau xx u tt. 3.3 We now consider the hyperbolic system 3.1 that is equipped with the entropy-entropy flux pair η q, with η strictly convex, and establish stability results for the relaxation model 3.3. Lemma Suppose u u ε x, t is a smooth solution to the equation 3.3 on [, T ], η q is the entropy-entropy flux pair of 3.1 and ᾱ is fixed. Then, the following energy identity holds t [ηu + εu t ε2 ᾱ u t 2 + ε 2 ᾱu x Au x 1 s ] + ε 2 u 1 t 2ᾱI D 2 ηu + ετu t dτds u t + x qu + εᾱ u t + DF uu x 2 + εu t ᾱi D 2 ηu u t + εu x D 2 ηua ᾱdf u DF u u x = x εdηuau x + 2ε 2 ᾱu t Au x + DηuGu + 2εᾱu t Gu. 3.4 Proof. Multiplying 3.3 by u t we obtain 1 t 2 ε u t εu x Au x + u t 2 + u t DF uu x = x εu t Au x + u t Gu. Similarly, multiplying 3.3 by Dηu we get t ηu + εdηuu t + x qu D 2 Aux ηuu x u t D 2 ηuu t + ε = x εdηuau x + DηuGu

7 Now, we multiply 3.5 by 2ᾱε, add 3.6 and use the identity 1 s ηu + εu t = ηu + εdηuu t + ε 2 u t D 2 ηu + ετu t dτ ds u t 3.7 to deduce 3.4. This proves the lemma. We now establish the stability of solutions {u ε }. For that we will require the matrix A to satisfy the subcharacteristic condition H2 and make use of hypotheses H3-a-H4 to control the source G. In the sequel we will use the notation ϕt := u ε 2 + ε 2 u ε x 2 + ε 2 u ε t 2 dx 3.8 with u ε denoting a solution of 3.3. Proposition Weakly dissipative source. Let {u ε x, t} be a family of smooth solutions to the equation 3.3 on [, T ]. Suppose that u u ε decays fast at infinity and that: a1 H1 holds true, and the positive definite, symmetric matrix A is such that the subcharacteristic condition H2 is valid. a2 The conditions H3-a and H4 for the source G hold true. Then for all t [, T ] ϕt + ε u ε x, t dx + t DηuGu dxdt C ϕ + ε u ε x, dx with ϕ defined in 3.8 and C = CA, α, β > independent of both ε and T. Proof. By H1 we have u t 1 2 αi 1 s 3.9 D 2 ηu + ετu t dτds u t 1 2 α u t Then, integrating the identity 3.4, with ᾱ = α, and using hypotheses H3-a, H4 we obtain ηu + εu t ε2 α u t 2 + ε 2 αu x Au x + 2εαu dxdt t + εν u x εα u t 2 dxdt t εα u t + DF uu x 2 + DηuGu dxdt ηu + εu t + cε 2 α u t 2 + ε 2 u x Au x + 2εαu dx with 1 2 c 1. From H1 it follows that c 1 ϕt < ηu + εu t αε2 u t 2 + ε 2 αu x Au x dx < c 2 ϕt 3.12 for some c 1, c 2 that depend on α, β and A. Then, combining 3.11 and 3.12, we obtain

8 Proposition General source. Let {u ε x, t} be a family of smooth solutions to the equation 3.3 on [, T ]. Suppose that u u ε decays fast at infinity and that: a1 H1 holds true, and the positive definite, symmetric matrix A, is such that the subcharacteristic condition H2 is valid. a2 The condition H3-b for the source G holds true. Then, ϕt C ϕ, t [, T ] 3.13 with ϕ defined in 3.8 and C = CA, α, β, T, L > independent of ε. Proof. Integrating the energy identity 3.4, with ᾱ = α, and using H1, H2 together with relations 3.1, 3.12 we obtain for τ [, T ] τ c 1 ϕτ c 2 ϕ + DηuGu + 2εαu t Gu dxdt Since G =, H3-b implies Gu L u and therefore, in view of H1, Then 3.8 and 3.14 imply DηuGu + 2εαu t Gu αl u 2 + αε 2 u t 2 + αl 2 u 2. τ ϕτ c ϕ + ϕt dt with c > depending on c 1, c 2 and L. Then, we conclude 3.13 via the Gronwall lemma. 4 Compactness properties This section focuses on systems of hyperbolic balance laws with a strictly convex entropy, d = Systems with a strictly convex entropy, d = 1. Starting with a family {u ε } smooth solutions of 3.3 on [, the goal in this section is to control the dissipation measure and to show that t ηu ε + x qu ε lies in a compact set of H 1 loc + for a certain class of entropy-entropy flux pairs η, q. In the proof we use Murat s lemma [22]. Lemma Murat s Lemma [22]. Let O be an open subset of m and {φ j } a bounded sequence of W 1,p O for some p > 2. In addition let φ j = χ j + ψ j, where {χ j } belongs in a compact set of H 1 O and {ψ j } belongs in a bounded set of the space of measures MO. Then {φ j } belongs in a compact set of H 1 O. 8

9 In the presence of uniform L -bounds the compensated compactness framework cf. Tartar [27], DiPerna [13] guarantees compactness of approximate solutions and implies that, along a subsequence, u ε u a.e. x, t. In the absence of L -bounds, convergence of viscosity approximations in the literature has been established in the context of elastodynamics by Shearer [25] and Shearer and Serre [26]. The objective in that context is to establish the reduction of the generalized Young measure to a point mass and to show strong convergence. Theorem Weakly dissipative source. Let {u ε } be a family of smooth solutions of 3.3 on [, T ] emanating from smooth initial data. The family {u ε } is assumed to decay fast at infinity. Let the hypotheses of Proposition remain valid so that the stability estimate 3.9 holds true with η q entropy-entropy flux pair satisfying H1, and A a symmetric, positive-definite matrix subject to H2. Then, for entropy pairs η, q satisfying η L, q L, D η L, D 2 η L C 4.1 and D ηvgv C M DηvGv, v n, 4.2 the family { t ηu ε + x qu ε } lies in a compact set of H 1 ε loc [, T ]. 4.3 Proof. Let {u ε } be a family of smooth solutions of 3.3 on [, T ]. The goal is to control the dissipation measure and to establish 4.3 for a class of entropy-entropy flux pairs η, q. It suffices to establish 4.3 for entropy-entropy flux pairs η, q satisfying 4.1. This class of entropy pairs has been used in the literature in a different context in order to establish the reduction of the generalized Young measure to a point mass and to show strong convergence. Starting from 3.3 we obtain t ηu ε + x qu ε = ε x D ηu ε Au ε x ε t D ηu ε u ε t εu ε x D 2 ηu ε Au ε x + εu ε t D 2 ηu ε u ε t + D ηu ε Gu ε := I 1 + I 2 + I 3 + I 4 + I 5. From 3.9 and 4.1, the terms I 1, I 2 lie in compact set of H 1, and the terms I 3, I 4 are bounded in L 1. By 3.9 DηGu ε is bounded in L 1 by initial data due to the weak dissipation assumption and therefore by 4.2 the term D ηgu ε is in a bounded set of L 1 as well. Therefore, by Murat s lemma [22], the sum I i lies in a bounded set of W 1,. Following, similar line of argument an analogous result for systems of hyperbolic balance laws with general source term is established. Theorem General source. Let {u ε } be a family of smooth solutions of 3.3 on [, T ] emanating from smooth initial data. The family {u ε } is assumed to decay fast at infinity. Let the hypothesis of Proposition remain valid so that the stability estimate 3.13 holds true with η q entropy-entropy flux pair satisfying H1, and A a symmetric, positive-definite matrix subject to H2. Then, for entropy pairs η, q satisfying the family { } t ηu ε + x qu ε ε η L, q L, D η L, D 2 η L C 4.4 lies in a compact set of H 1 loc [, T ]

10 Proof. The proof follows similar line of argument as the one in Theorem Starting from 3.3 we obtain t ηu ε + x qu ε = ε x D ηu ε Au ε x ε t D ηu ε u ε t εu ε x D 2 ηu ε Au ε x + εu ε t D 2 ηu ε u ε t + D ηu ε Gu ε := I 1 + I 2 + I 3 + I 4 + I 5. From 3.13 and 4.4, the terms I 1, I 2 lie in compact set of H 1, the terms I 3, I 4 are bounded in L 1, whereas I 5 = D ηu Gu c u 2 is by Lemma bounded in L 1. Therefore, by Murat s lemma [22], the sum I i lies in a bounded set of W 1,. 5 Error estimates via the relative entropy method, d = 1 In this section, we establish the convergence of solutions of 3.3 to solutions of 3.1 before the formation of shocks. In the spirit of [1] we use the modified relative entropy method [12] by introducing a functional H rel ū, u ε, which monitors the difference between the solutions ū to the equilibrium and the solutions u ε to the relaxation systems. The presence of the source G in our work, however, requires us to modify the method significantly. More specifically, in order to treat the weakly dissipative source G, which satifies typically no growth restrictions, we need to introduce the relative potential rel ū, u ε see 5.18 in Section 6.3. This potential becomes part of a Lyapunov functional monitoring the evolution of the difference between the two sources. In the case of the general source G the terms associated with the source are treated as error cf. Section The decay functional and relative entropy identity. Let η q be an entropy-entropy flux pair satisfying H1. We define the corresponding relative entropy-entropy flux pair by H rel ū, u ε = η u ε + εu ε ū t ηū Dηū u ε + εu ε ū t ū and set the functional Q rel ū, u ε = qu ε qū DηūF u ε F ū 5.1 Gū, u ε = H ū, u ε + ε 2 u ε ū t αi D 2 η u ε ū t + ε 2 αu ε ū x Au ε ū x, 5.2 where A is a symmetric, positive definite matrix, α > a fixed constant defined in H1 and D 2 η = 1 s D 2 η u ε + ετu ε ū t dτds

11 Lemma elative entropy identity. Let ū, u ε be smooth solutions to 3.1, 3.3, respectively. Then, the following energy identity holds t Gū, u ε + x Q rel ū, u ε + εα u ε ū t + DF u ε u ε ū x 2 { } + ε u ε ū t αi D 2 ηu ε u ε ū t { } + ε u ε ū x D 2 ηu ε A αdf u ε DF u ε u ε ū x { = x ε Dηu ε Dηū Au ε ū x + 2αε 2 Au ε } u ū ε x ū t D 2 ηūū x F u ε F ū DF ūu ε ū + a 1 + a 2 + b 1 + b 2 + 2εαc 1 + c 2 + d 1 + d 2 + 2εαd 3, 5.4 where a 1 = ε D 2 ηu ε D 2 ηū ū t u ε ū t a 2 = ε Dηu ε Dηū ū tt b 1 = ε D 2 ηu ε D 2 ηū ū x Au ε ū x b 2 = ε Dηu ε Dηū Aū xx 5.5 c 1 = εaū xx ū tt u ε ū t c 2 = DF u ε DF ū ū x u ε ū t are the error terms, and d 1 = Dηu ε Dηū Gu ε Gū d 2 = Gū Dηu ε Dηū D 2 ηūu ε ū 5.6 d 3 = Gu ε Gū u ε ū t. are the terms associated with the source G. Proof. By H1, 3.1, and 3.3 we have Similarly, t ηu ε ηū + x qu ε qū = ε Dηu ε Au ε xx Dηu ε u ε tt + Dηu ε Gu ε DηūGū. t u ε ū + x F u ε F ū = εau ε xx u ε tt + Gu ε Gū and hence, after multiplying the above identity by Dηū, we have t Dηūu ε ū + x Dηū F u ε F ū = D 2 ηūū t u ε ū + x Dηū F u ε F ū + ε DηūAu ε xx Dηūu ε tt + DηūGu ε Gū The existence of the entropy-pair η q is equivalent to the property D 2 ηvdf v = DF v D 2 ηv, v n 11

12 and hence, using 3.1, the first term on the right-hand side of 5.8 can be expressed as D 2 ηūū t u ε ū = D 2 ηūū x DF ūu ε ū + Gū D 2 ηūu ε ū. Combining 5.7, 5.8 and the above identity we obtain t ηu ε ηū Dηūu ε ū + x qu ε qū DηūF u ε F ū = D 2 ηūū x F u ε F ū DF ūu ε ū + ε Dηu ε Dηū Au ε xx ε Dηu ε Dηū u ε tt + Dηu ε Dηū Gu ε Gū + Gū Dηu ε Dηū D 2 ηūu ε ū. 5.9 Next, we express the second and third terms on the right-hand side of 5.9 as Dηu ε Dηū u ε tt = t Dηu ε Dηū u ε ū t D 2 ηu ε u ε ū t u ε ū t D 2 ηu ε D 2 ηū ū t u ε ū t + Dηu ε Dηū ū tt Dηu ε Dηū Au ε xx = x Dηu ε Dηū Au ε ū x D 2 ηu ε u ε ū x Au ε ū x and observe that D 2 ηu ε D 2 ηū ū x Au ε ū x + Dηu ε Dηū Aūxx η u ε + εu ε ū t = ηu ε + εdηu ε u ε ū t + ε 2 u ε ū t D 2 η u ε ū t. Then 5.1, 5.9 and the last three identities imply { } t H rel ū, u ε ε 2 u ε ū t D 2 η u ε ū t + x Q rel ū, u ε { D 2 ηu ε u ε u ū ε x ū x D 2 ηu ε u ε } u ū ε t ū t + ε { } = x εdηu ε DηūAu ε ū x D 2 ηūū x F u ε F ū DF ūu ε ū a 1t + a 2t + b 1x + b 2x + d 1 + d 2 with the last six terms on the right-hand side of the above identity defined in 5.5 1,2,3,4 and 5.6 1,2. The identity 5.1 is supplemented by a correction accounting for the fact that the third term is indefinite. The correcting identity is obtained by multiplying the equation u ε ū t + DF u ε u ε ū x = εau ε ū xx εu ε ū tt + εaū xx ū tt + Gu ε Gū DF u ε DF ūū x 12

13 by u ε ū t and integrating by parts, which leads to t { 1 2 ε uε t ū t εuε ū x Au ε ū x } + u ε t ū t 2 + DF u ε u ε ū x u ε ū t = x { ε Au ε ū x u ε ū t } + c 1t + c 2t + d with the terms on the right-hand side defined in 5.5 5,6 and Finally, multiplying 5.11 by 2αε and adding the resulting identity to 5.1 we obtain Preliminary estimate of G In this section we establish a preliminary estimate for the functional Gū, u ε employed in the proofs of Theorem and Theorem For this purpose we define Ψt := ū u ε 2 + ε 2 ū u ε x 2 + ε 2 ū u ε t 2 dx 5.12 used in our further analysis, where ū, u are smooth solutions to the equilibrium and relaxation system, respectively. Lemma Let ū, u ε be smooth solutions of 3.1, 3.2, respectively and suppose that both ū, u ε decay sufficiently fast at infinity. Suppose that: a1 H1 holds true and the positive definite, symmetric matrix A, is such that the subcharacteristic condition H2 is valid. a2 For some M > D 2 F u M, D 3 ηu M, u n. Then, i There exists c 1, c 2 > independent of ε > such that c 1 Ψt Gū, u ε dx c 2 Ψt ii The functional Gū, u ε is positive definite and satisfies d Gū, u ε dx + ε c u ε x ū x 2 + u ε t ū t 2 dx dt CT, ū Ψt + ε 2 + d 1 + d 2 + 2εαd 3 dx 5.14 with d 1, d 2, d 3 defined in 5.6 and C = CT, ū > independent of ε. 13

14 Proof. From H1 and we have Also, H1 and 5.3 imply β u ε + εu ε ū t ū 2 H rel ū, u ε α u ε + εu ε ū t ū αi αi D 2 η = αi 1 s D 2 η u ε + ετu ε ū t dτds 1 2αI Combining 5.2, 5.15, 5.16 and recalling that A is symmetric, positive definite we get Next, integrating 5.4 we use H2 and 5.16 to conclude d Gū, u ε dx + ε c ū u ε x 2 + ū u ε t 2 dx dt { D 2 ηūū x F u ε F ū DF ūu ε ū + a1 + a 2 + b 1 + b 2 + 2εαc 1 + c 2 + d1 + d 2 + 2εαd 3 } dx By a3 we have D 2 ηūū x F u ε F ū DF ūu ε ū dx C u ε ū 2 L 2 and the error terms in 5.5 are estimated by 5.17 a 1 L 1 εc ū t L u ε ū L 2 u ε t ū t L 2 b 1 L 1 εc ū x L u ε ū L 2 u ε x ū x L 2 a 2 L 1 εc ū tt L 2 u ε ū L 2 b 2 L 1 εc ū xx L 2 u ε ū L 2 and εc 1 L 1 ε 2 C ū tt L 2 + ū xx L 2 u ε t ū t L 2 εc 2 L 1 εc ū x L u ε ū L 2 u ε t ū t L 2, where C is a generic constant that depends on α, M and norms of ū. Then, by 5.13, 5.17 and the above estimates we obtain Error estimates. Weakly dissipative source Gu To establish the convergence result for weakly dissipative source we introduce the relative potential rel ū, u ε := u ε ū Dūu ū 5.18 which is well-defined whenever G C 1 n. As we will see in the next theorem the smoothness of G, which in our work is in general assumed to be C n, will have an impact on the rate of convergence. Theorem Let ūx, t be a smooth solution of the equilibrium system 3.1, defined on d [, T ]. Let {u ε } be a family of smooth solutions of the relaxation system 3.2 on [, T ]. Suppose that both ū and u ε decay sufficiently fast at infinity and that: 14

15 a1 H1 holds true and the positive definite, symmetric matrix A, is such that the subcharacteristic condition H2 is valid. a2 For some M > D 2 F u M, D 3 ηu M, u n. a3 The conditions H3-a, H4 on the source term hold true. Then for all t [, T ] Ψt + ε with C = Cū, α, β, ν, M, T > independent of ε. u ε x, tdx C Ψ + ε u ε x, dx + ε 5.19 If, in addition, G C 2 n then for all t [, T ] [ Ψt + ε rel ū, u ε ] x, t dx C Ψ + ε Proof. By H1, H3-b, and 5.6 we obtain [ rel ū, u ε ] x, dx + ε d 1 Dηu ε DηūGu ε Gū d 2 α Gū L u ε ū 2 εd 3 ε t u ε + εc 1 + u ε ū t + ε Gū u ε t ū t Then, combining 5.14 and 5.21 we obtain d Gū, u ε dx + ε c u ε x ū x 2 + u ε t ū t 2 dx dt C Ψt + ε 2 + 2εαC ū t L 1 + ū t L u ε dx 2εα d u ε dx + 2εα2 Gū 2 dx + ε c u ε t ū t 2 dx. dt c 2 Integrating the above inequality, and using H1, H4, 5.13 and 5.12, we obtain Ψt + ε u ε x, tdx t { } C Ψ + ε u ε x, dx + Ψs + ε u ε x, s dx ds + εt + ε 2 t and conclude 5.19 via the Gronwall lemma. Suppose now that G C 2 n. Then, using H4 and 5.18, we obtain 5.22 t rel ū, u ε = Gu ε u ε t + Gū ū t + Gū u ε ū t D 2 ūū t u ε ū = Gū Gu ε u ε t D 2 ūū t u ε ū

16 Then, using H4, and 5.23, we obtain d 3 = Gu ε Gū u ε ū t = t rel ū, u ε D 2 ūū t u ε ū Gu ε Gū ū t 5.24 = t rel ū, u ε + ū t Du ε Dū D 2 ūu ε ū. Then, combining 5.14 with ,2 and 5.24, we obtain d Gū, u ε dx + ε c u ε x ū x 2 + u ε t ū t 2 dx dt CT, ū Ψt + ε 2 2εα d rel ū, u ε dx dt { + 2εα ū t Du ε Dū D 2 ūu ε ū } dx Now, we estimate the last term on the right-hand side of By assumption on ū there exists a compact set K n such that ūx, t K, x, t [, T ]. Thus, we can take large enough M > such that K B M and ūx, t v > 1 for all v B M C, x, t [, T ], 5.26 where B M denotes a ball of radius M. Then, since G = D C 2 n, we obtain λ M G := sup v + Dv + D 2 v + D 3 v < v B M Fix x, t [, T ]. Suppose ux, t B M. Then by 5.27 Du ε Dū D 2 ūu ε ū λ M G u ε ū Suppose now that u ε x, t B M C. Then, by 5.26 we have u ε x, t ūx, t > 1. Hence u ε x, t = rel ū, u ε + ū + Dūu ū max rel, u ε ū λ M G Then, using 5.27, 5.29 and H4, we obtain Du ε Dū D 2 ūu ε ū Du 2 maxu ε, u ε ū 2 ε u ε λm G 2 maxu ε, u ε ū 2 C + λ M G C max rel, u ε ū

17 for some C > that depends on λ M G and C. Since x, t is arbitrarily chosen, combining 5.28 and 5.3, we conclude Du ε Dū D 2 ūu ε ū C max rel u ε, u ε ū for all x, t [, T ]. Thus 2εα { ū t Du ε Dū D 2 ūu ε ū } dx C ε rel ū, u ε dx + ε ū u ε 2 dx Then, integrating the inequality 5.25 in time, and using the 5.13, 5.12, and 5.32, we obtain [ Ψt + ε rel ū, u ε ] x, t dx [ C Ψ + ε rel ū, u ε ] x, dx 5.33 τ { [ + Ψτ + ε rel ū, u ε ] x, τdx} dτ + ε 2 which along with the Gronwall lemma implies Error estimates. General source Gu We now consider the source term that satisfies the alternative hypothesis H3-b. Theorem Let ūx, t be a smooth solution of the equilibrium system 3.1, defined on [, T ]. Let {u ε } be a family of smooth solutions of the relaxation system 3.2 on [, T ]. Suppose that both ū and u ε decay sufficiently fast at infinity and that: a1 H1 holds true and the positive definite, symmetric matrix A, is such that the subcharacteristic condition H2 is valid. a2 For some M > D 2 F u M, D 3 ηu M, u n. a3 The condition H3-b on the source term holds true. Then, Ψt C Ψ + ε 2, t [, T ] 5.34 with Ψ defined in 5.12 and C = Cū, α, β, ν, M, T > independent of ε. Proof. Let A n denote a set such that ūx, t A for every x, t. Then H1, H3-a, H4 and 5.6 imply d 1 u ε ū Gu ε Gū L A u ε ū 2 d 2 α Gū L u ε ū 2 εd 3 εl A u ε ū u ε ū t L 2 A u ε ū 2 + ε 2 u ε ū t

18 Then, combining 5.12, 5.14 and 5.35, we obtain d Gū, u ε dx + ε c u ε x ū x 2 + u ε t ū t 2 dx C Ψt + ε 2. dt Integrating the above inequality, and using 5.13, we obtain and conclude 5.34 via the Gronwall lemma. 6 Multidimensional case t Ψt C Ψ + Ψs ds + ε 2 In this section we state our results for multidimensional systems. The proofs of these theorems follow similar line of argument as the ones presented in the earlier parts of this article, and therefore are here omitted. We define ϕt := u ε 2 + ε 2 Du ε 2 + ε 2 u ε t 2 dx 6.1 d Ψt := ū u ε 2 + ε 2 Dū Du ε 2 + ε 2 ū u ε t 2 dx 6.2 which are used in the next two subsections. 6.1 Systems with weakly dissipative source G, d 1 The first result is the analog of Proposition on stability. Proposition Let {u ε x, t} be a family of smooth solutions to the system 1.4 on d [, T ]. Suppose that u u ε decays fast at infinity and that: a1 H1 holds true, and the positive definite, symmetric matrices A j, j = 1,..., d are such that the subcharacteristic condition H2* is valid. a2 The conditions H3-a and H4 for the source G hold true. Then for all t [, T ] ϕt + ε u ε x, t dx + d t d DηuGu dxdt C ϕ + ε u ε x, dx d with ϕt defined in 6.1, u defined in H4 and C = CA, α, β > independent of ε, T. The next theorem is the analog of compactness Theorem

19 Theorem Let d 1. Let {u ε } be a family of smooth solutions of 3.3 on d [, T ] emanating from smooth initial data. The family {u ε } is assumed to decay fast at infinity. Let the hypotheses of Proposition remain valid so that the stability estimate 6.3 holds true with η q entropy-entropy flux pair satisfying H1, and A j, j = 1,..., d, symmetric, positive-definite matrices subject to H2*. Then for an entropy pair η q satisfying the growth conditions η L, q L, D η L, D 2 η L C and the family D ηvgv C M DηvGv, v n { t ηu ε + d j=1 } xj q j u ε lies in a compact set of H 1 loc ε d [, T ]. The next theorem is the analog of Theorem on convergence. Theorem Let ūx, t be a smooth solution of the system 3.1, defined on d [, T ]. Let {u ε } be a family of smooth solutions of the relaxation system 3.2 on d [, T ]. Suppose that both ū and u ε decay sufficiently fast at infinity and that: a1 H1 holds true and the positive definite, symmetric matrices A j, j = 1,..., d are such that the subcharacteristic condition H2* is valid. a2 For some M > D 2 F j u M, j = 1,..., d, D 3 ηu M, u n. a3 The conditions H3-a and H4 for the source term G hold true. Then for all t [, T ] Ψt + ε u ε x, tdx C Ψ + ε u ε x, dx + ε d d with Ψ defined in 6.2, u defined in H4 and C = Cū, α, β, ν, M, T > independent of ε. where If, in addition, G C 2 n then for all t [, T ] [ Ψt + ε rel ū, u ε ] [ x, t dx C Ψ + ε rel ū, u ε ] x, dx + ε 2 d d rel ū, u ε := u ε ū Dūu ū. 19

20 6.2 Systems with general source G, d 1 Proposition Let {u ε x, t} be a family of smooth solutions to 1.4 on d [, T ]. Suppose that u u ε decays fast at infinity and that: a1 H1 holds true, and positive definite, symmetric matrices A j, j = 1,..., d are such that the subcharacteristic condition H2* is valid. a2 The condition H3-b on the source G holds true. Then ϕt C ϕ, t [, T ] 6.4 with ϕ defined in 6.1 and C = CA, α, β, T, L > independent of ε and T. Theorem Let d 1. Let {u ε } be a family of smooth solutions of 3.3 on d [, T ] emanating from smooth initial data. The family {u ε } is assumed to decay fast at infinity. Let the hypotheses of Proposition remain valid so that the stability estimate 6.4 holds true with η q entropy-entropy flux pair satisfying H1, and A j, j = 1,..., d, symmetric, positive-definite matrices subject to H2*. Then for an entropy pairs η q satisfying the family { t ηu ε + d j=1 η L, q L, D η L, D 2 η L C } xj q j u ε lies in a compact set of H 1 loc ε d [, T ]. Theorem Let ūx, t be a smooth solution of the system 1.1, defined on [, T ]. Let {u ε } be a family of smooth solutions of the relaxation system 1.2 on d [, T ]. Suppose that both ū and u ε decay sufficiently fast at infinity and that: a1 H1 holds true and the positive definite, symmetric matrix A, is such that the subcharacteristic condition H2* is valid. a2 For some M > D 2 F j u M, j = 1,..., d, D 3 ηu M, u n. a3 The condition H3-b for the source term G holds true. Then Ψt C Ψ + ε 2, t [, T ] with Ψ defined in 6.2 and C = Cū, α, β, ν, M, T > independent of ε. 2

21 7 An alternative relaxation model In this section we consider an alternative relaxation model for the system of hyperbolic balance laws 1.1 given by d t u + xj v j = Gu j=1 7.1 t v i + A i xi u = 1 vi F i u, i = 1,..., d ε with u, v i n and A i symmetric, positive definite matrix. Excluding v i from the equation and assuming G C 1 n, we obtain d d t u + xj F j u = Gu + ε t Gu + ε A j u xj x j u tt j=1 j=1 7.2 that approximates the system of balance laws 1.1. We remark that the treatment of such relaxation systems presents several challenges. More specifically, the time derivative of the source term appear in the energy functional posing enormous difficulties in the analysis, an additional hypothesis is required for the establishment of stability see H7, Section 6, the issue of compactness is problematic. In the next two subsections we will study the stability and compactness properties of solutions to 7.2 in order to point out the advantages of the relaxation model 1.2. We restrict the analysis to weakly dissipative systems of hyperbolic balance laws equipped with strictly convex entropy, d = Weakly dissipative systems with strictly convex entropy, d = 1 The system 7.2 for d = 1 reads t u + x F u = Gu + ε t Gu + εau xx u tt 7.3 where A is symmetric, positive definite matrix. Following the arguments of Lemma we obtain: Lemma Suppose u u ε x, t is a smooth solution to the equation 7.3 on [, T ], η q is the entropy-entropy flux pair of 3.1 and ᾱ is fixed. Then, the following energy identity holds t [ηu + εu t ε2 ᾱ u t 2 + ε 2 ᾱu x Au x 1 s ] + ε 2 u 1 t 2ᾱI D 2 ηu + ετu t dτds u t + x qu + εᾱ u t + DF uu x 2 + εu t ᾱi D 2 ηu u t + εu x D 2 ηua ᾱdf u DF u u x = x εdηuau x + 2ε 2 ᾱu t Au x + Dηu Gu + ε t Gu + 2εᾱu t Gu + ε t Gu

22 In our further analysis we will employ the following elementary lemma. Lemma Weak dissipation of a gradient. Suppose ηu satisfies H1, and Gu C 1 satisfies H3-a, H4. Then DGu = D 2 u for all u n. 7.5 In addition, if ηu : n satisfies D 2 ηu, then Proof. From H3-a, H4 it follows that D 2 ηudgu = D 2 ηud 2 u for all u n. 7.6 z D 2 ηudguz = z D 2 ηud 2 uz for all u, z n. Fix u n. Let λ, v be an eignepair of D 2 u. Then the above inequality implies v D 2 ηud 2 uv = λ v D 2 ηuv. ecalling that D 2 ηu > and v, we conclude that λ. Hence all eigenvalues of D 2 u are nonnegative. Since D 2 u is a symmetric matrix this is equivalent to D 2 u. Next, suppose ηu is a strictly convex function. Then 7.6 follows from the fact that D 2 η, D 2 are symmetric, nonnegative definite matrices. emark Note that the requirement of uniform convexity for ηu in Lemma can be replaced with strict convexity. Thus, when G is a gradient, weak dissipation of G with respect to some strictly convex entropy automatically yields weak dissipation with respect to any convex entropy, a property which is not in general satisfied. To establish the stability of solutions to 7.3 we will employ an additional hypothesis: Suppose there exists S : n such that DηuDGu = DSu, Su S =. u n H5 emark We note that the hypothesis H5 is a severe assumption. It is motivated by the following observation. Suppose that the entropy ηu = 1 2 u 2, and H3-a, H4 hold true. Then, DηuDGu = u D 2 u = DSu with Su = Duu u. Moreover, by Lemma it follows that D 2 and hence Su S =. Proposition Let {u ε x, t} be a family of smooth solutions to the equation 7.3 defined on [, T ]. Suppose that u u ε decays fast at infinity and that: a1 H1 holds true, and the positive definite, symmetric matrix A is such that the subcharacteristic condition H2 is valid. 22

23 a2 The conditions H3-a, H4, and H5 for the source G C 1 n hold true. Then for all t [, T ] ϕt + ε { } t Su ε x, t + u ε x, t dx + DηuGu dxdt { } 7.7 C ϕ + ε Su ε x, + u ε x, dx with ϕ defined in 3.8 and C = CA, α, β > independent of both ε and T. Proof. Using H4, H5 we rewrite the last four terms on the right-hand side of 7.4, with ᾱ = α, as follows DηuGu + εdηu t Gu + 2εu t Gu + 2ε 2 αu t t Gu 7.8 = DηuGu 2ε 2 αu t D 2 uu t ε t Su + 2αu =: I 1 + I 2. Since G C 1, by H1, H3-a, H4 and Lemma αu t D 2 uu t. 7.9 Thus, by H3-a and 7.9 we conclude I 1. Then, integrating the identity 7.4, with ᾱ = α, and employing hypotheses H1, H2, H3-a, H4, H5 along with 3.1, 3.12 and 7.8, we conclude Compactness issues Let {u ε } be a family of smooth solutions of 7.3 on [, T ] emanating from smooth initial data. The family {u ε } is assumed to decay fast at infinity. Let the hypotheses of Lemma remain valid so that the stability estimate 7.7 holds true with η q entropy-entropy flux pair satisfying H1, and A a symmetric, positive-definite matrix subject to H2. Let the entropy pairs η, q satisfy 4.1 and the growth condition holds 4.2. We now show that, in general, one may not expect compactness from the family { t ηu ε + x qu ε }. We follow the arguments of Theorem Starting from 7.3 we obtain t ηu ε + x qu ε = ε x D ηu ε Au ε x ε t D ηu ε u ε t εu ε x D 2 ηu ε Au ε x + εu ε t D 2 ηu ε u ε t + D ηu ε Gu ε + εd ηu ε DGu ε u ε t := I 1 + I 2 + I 3 + I 4 + I 5 + I 6. As before from 4.1, 4.2 and 7.7, it follows that the terms I 1, I 2 lie in compact set of H 1, and the terms I 3, I 4, I 5 are bounded in L 1. However, the term I 6 is, in general, neither in a bounded set of the space of measures nor in a compact set of H 1 : for the former one must have a control of εdg in L 2, and for the latter a control of εg in L 2, which follows from the relation I 6 = t D ηu ε εgu ε D 2 ηu ε u ε t εgu ε. The stability estimate 7.7, however, does not provide such bounds. Thus, Murat s lemma is not applicable and the issue of compactness appears problematic. 23

24 8 Applications 8.1 Elasticity system Consider the relaxation of the isothermal/isentropic elasticity system with a source term: [ ] [ ] [ ] u v = Gu, v =. 8.1 v σu gu, v t x In the context of gas flow, u is specific volume u = 1/ρ, thus constrained by u >. In the context of the thermoelastic bar, u is the strain, likewise constrained by u >. Finally, in the context of shearing motion, u is shearing, which may take both positive and negative values. In the gas case, it is traditional to use the pressure p = σ, instead of σ. In the present context, the stress σu is assumed to satisfy σ = and < γ < σ u < Γ for all u n. 8.2 We assume that gu, v, with g, =, satisfies one of the following: i Either g is independent of u, that is gu, v = gv, and satisfies gv g v v v, v, v 8.3 which corresponds to a frictional damping. ii or for every compact set A 2 there exists L A > such that gu, v gū, v L A u ū + v v 8.4 for all u, v 2, ū, v A. The system 8.1 is equipped with the entropy - entropy flux pair η, q given by ηu, v = 1 2 v2 + Σu, qu, v = σuv with Σu := For the system 8.1 the second order relaxation system 3.3 reads [ ] [ ] [ ] [ ] [ ] u v u u = + ε A v σu gu, v v v t Observe that by 8.2 and 8.5 hypotheses H1-H2 are satisfied with x xx u tt στdτ α = max2γ, 1, β = minγ, 1, A = 2αI. 8.7 Suppose that 8.3 holds true. Then Dηu, v ηū, v Gu, v Gū, v = v vgv g v 24

25 and hence G satisfies H3-a. Furthermore, setting u, v := v gθdθ we get Gu, v = Du, v,, = which gives H4. Thus, one may employ Proposition to obtain the stability estimate 3.9 and Theorem to obtain the error estimate 8.6 that holds before the formation of shocks. Similarly, suppose 8.4 holds true. Then clearly G satisfies H3-b and one may use Proposition and Theorem to obtain the stability estimate 3.13 and the error estimate 8.6, respectively. The stability estimates 3.9 and 3.13 suffice to apply the L p theory of compensated compactness. In the spirit of [21, Theorem 1] we prove the following convergence theorem. Theorem Let σu satisfy 8.2 and suppose also u u g u for u u and g, g L 2 L. 8.8 Let u ε, v ε be a family of smooth solutions of 8.1 defined on [, T emanating from smooth initial data subject to ε-independent bounds ϕ = u ε 2 + v ε2 dx + ε 2 u ε x 2 + u ε t 2 dx C. Let A be a symmetric, positive definite matrix satisfying 8.7 and let gu, v satisfy either 8.3 or 8.4. Then, along a subsequence if necessary, u ε u, v ε v, a.e. x, t and in L p loc, T, for p < 2. Proof. Let u ε, v ε be a family of solutions to 8.6. The proof uses the theory of compensated compactness [27]. Typically, in such proofs, the goal is to control the dissipation measure and to show that { } t ηu ε, v ε + x qu ε, v ε ε lies in a compact set of H 1 loc, T 8.9 for a class of entropy-entropy flux pairs η q for the equations of elasticity. In the presence of uniform L -bounds, the theorem of DiPerna [13] guarantees compactness of approximate solutions and implies that, along a subsequence, u ε u and v ε v a.e. x, t. In the present case L 1 -estimates are only available in the special case that A is a multiple of the identity matrix see [23] and, in view of 3.9 and 3.13, the natural stability framework is in the energy norm. Convergence of viscosity approximations to the equations of elastodynamics in the energy framework is carried out in Shearer [25] for the genuine-nonlinear case and Serre- Shearer [26] for loss of genuine nonlinearity at one point. In [25] two classes of entropies, with growth controlled by the wave-speeds at infinity, are constructed [25, Lemma 2] for which Tartar s commutation relation is justified see [25, Lemma 2] and are used to show that the support of the generalized Young measure is a point mass [25, Lemma 7, Theorem 1-iii]. When σu has one inflection point, the reduction of the Young measure is performed in [26, Lemma 3] and [26, Section 5]. 25

26 To ensure the dissipation estimate, we employ the growth assumption 8.2, the subcharacteristic condition and the assumption on the source 8.3, 8.4. Then it suffices to establish 8.9 for entropy pairs η q satisfying η L, q L, D η L, D 2 η L C. 8.1 This class of entropy pairs contains under the assumption 8.2 the test-pairs that are used in [25, 26] in order to prove the reduction of the generalized Young measure to a point mass and to show strong convergence in L p loc for p < 2. Hypothesis 8.8 reflects the assumptions needed in those works. To complete the proof, we show that 8.9 holds for entropy-entropy flux pair η q satisfying 8.1. First suppose that gu, v satisfies 8.3. Then by 8.5 D ηu, vgu, v = η v u, vgv C M + DηGu, v, M = max v 1 gv The inequality 8.11 is the analog of the condition 4.2 used in Theorem in which case, the condition 4.2 is automatically satisfied for the elasticity system 8.1. Then by 8.1, 8.11, and Theorem we conclude 8.9. Suppose now that gu, v satisfies 8.4. Then G satisfies H3-b and hence from 8.1, 8.11, and Theorem we obtain 8.9. We remark that the alternative relaxation system 7.3 reads [ ] [ ] [ ] [ ] [ ] u v u = + ε + ε A v σu gu, v gu, v v Then setting t x t xx [ ] u v tt Su, v := gvv u, v we get Dηu, vdgu, v = vg v = DSu, v, S, = which gives H5. Thus, the stability of solutions to 8.12 follows from Proposition Isentropic combustion model The governing equations for chemical reaction from unburnt gases to burnt gases in certain physical regimes in Lagrangian coordinates read [7]: 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, ZZ = 8.13 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 26

27 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. In this section we address the problem of relaxation to the isentropic combustion model v u u + P v,z = 8.14 Z KϕΘv, Z t x that arises naturally from the system 8.13 by externally regulating r to ensure s = s [1]; in the sequel we suppress the variable s and use the notation P v, Z := P v, s, Z, Θv, Z := Θv, s, Z. We impose the following requirements on P, Θ see [16] for the motivation: 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, Z [, 1]. P ZZ τ, Zdτ < C, Z P v, Z < C, v, Z [, 1]. 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 8.15 for all v, Z, v, Z [, 1]. Under a1-a3 the system 8.14 admits an entropy-entropy flux pair η, q of the form: ηv, u, Z = 1 v 2 u2 P τ, Zdτ + BZ, qv, u, Z = P v, Zu with BZ an arbitrary function. To ensure the convexity of η we assume, in addition to a1-a3, that B Z > Γ C 2 + C, Z [, 1], in which case < min γ 2, 1 D 2 ηv, u, Z max 1, Γ + C, 2 Γ C C For the system 8.14 the second order relaxation system 3.3 reads v u u u u + P v,z = + ε A v v Z KϕΘv, Z Z Z t x 27 xx tt

28 Clearly by 8.16 hypotheses H1 and H2 are satisfied with α = max 1, Γ + C, 2 Γ C C, β = min γ 2, 1, A = 2αI while, in view of 8.15, the source G satisfies H3-b. Thus, one may use Proposition 3.1.3, Theorem and Theorem to conclude about the stability, compactness and the error estimates before the formation of shocks, respectively. 9 Acknowledgements A.M. thanks. Young and A. Tzavaras for several fruitful conversations during the course of this investigation. This material is based upon work supported by the National Science Foundation under Grant No , while K.T. was in residence at the Mathematical Sciences esearch Institute in Berkeley, California, during the Program on On Optimal Transport in Geometry and Dynamics in Fall 213. K.T. thanks MSI and the organizers of the program for the hospitality, support and for providing an excellent academic environment for scientific research. K.T. also acknowledges the support in part by the National Science foundation under the grant DMS and the support by the Simons Foundation under the grant # K.T. thanks I. Kyza for several discussions in the course of the investigation and for bringing to her attention the work by Katsaounis and Makridakis [19]. eferences [1] Ch. Arvanitis, Ch. Makridakis, A.E. Tzavaras, Stability and Convergence of a Class of Finite Element Schemes for Hyperbolic Conservation Laws, SIAM J. on Numer. Analysis 25, 42, 4, [2] D. Aregba-Driollet,. Natalini, Convergence of relaxation schemes for conservation laws, Appl. Anal. 1996, 61, [3] D. Aregba-Driollet,. Natalini, Discrete kinetic schemes for multi-dimensional conservation laws, SIAM J. Numer. Anal. 2, 37, 6, [4] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Diff. Equations, 25, , 2. [5] Y. Brenier,. Natalini and M. Puel, On the relaxation approximation of the incompressible Navier-Stokes equations, Proc. Amer. Math. Soc., 132, , 24. [6] F. Berthelin and A. Vasseur, From Kinetic Equations to Multidimensional Isentropic Gas Dynamics Before Shocks, SIAM J. Math. Anal. 36, , 25. [7] G.-Q. Chen, D. Hoff and K. Trivisa, Global Solutions to a Model for Exothermically eacting, Compressible Flows with Large Discontinuous Initial Data. Arch. ation. Mech. Anal. 23, 166,

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