Quasi-neutral limit of the non-isentropic Euler Poisson system

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1 Proceedings of the Royal Society of Edinburgh, 136A, , 2006 Quasi-neutral limit of the non-isentropic Euler Poisson system Yue-Jun Peng Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal (Clermont-Ferrand 2), Aubière cedex, France Ya-Guang Wang Department of Mathematics, Shanghai Jiao Tong University, Shanghai , People s Republic of China (ygwang@sjtu.edu.cn) Wen-An Yong Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing , People s Republic of China (MS received 12 September 2005; accepted 18 October 2005) This paper is concerned with multi-dimensional non-isentropic Euler Poisson equations for plasmas or semiconductors. By using the method of formal asymptotic expansions, we analyse the quasi-neutral limit for Cauchy problems with prepared initial data. It is shown that the small-parameter problems have unique solutions existing in the finite time interval where the corresponding limit problems have smooth solutions. Moreover, the formal limit is justified. 1. Introduction In mathematical modelling of plasmas and semiconductor devices, the Euler Poisson system is an active player [1, 3, 6]. For large-scale structures relative to the Debye length, the modelling is usually based on a simplified system. This can be simply derived from the Euler Poisson system by setting the Debye length to zero, i.e. the plasmas are assumed to be electrically neutral. The zero-debye-length (or so-called quasi-neutral) limit of various plasma models has attracted considerable attention in recent years. For a one-dimensional steady Euler Poisson system, the limit was found by Slemrod and Sternberg [10] for prepared boundary data. The steady problem for a potential flow in several spatial variables without the formation of boundary layers was studied in [7]. The case with boundary layers was investigated recently in [8]. By using pseudo-differential techniques, Cordier and Grenier [2] studied the same limit for local smooth solutions of a one-dimensional and isothermal model for plasmas in which the electron density is described by the Maxwell Boltzmann relation with the electrostatic potential. The quasi-neutral limit for local smooth solutions to the isentropic Euler Poisson equations in several spatial variables was studied by Wang [11], with the asymptotic 1013 c 2006 The Royal Society of Edinburgh

2 1014 Y.-J. Peng, Y.-G. Wang and W.-A. Yong expansions being developed only for the leading profiles, and by Peng and Wang [9] with complete asymptotic expansions. The purpose of this paper is to investigate the quasi-neutral limit for Cauchy problems of multi-dimensional non-isentropic Euler Poisson equations with given ion density. By an asymptotic expansion, we formally derive an incompressible type of non-isentropic Euler system for the electron velocity, entropy and the electrostatic potential. When the ion density is constant, the limits of the electron velocity, entropy and electrostatic potential satisfy the classical incompressible nonisentropic Euler equations. Furthermore, under the assumption that the ion density and velocity satisfy certain compatibility conditions which prevent the formation of initial layers, we prove the existence of the asymptotic expansion and rigorously justify the formal limit for periodic initial data by adapting the approach developed in [12,13]. This approach is different from those used in [4,9,11]. In the proof of the existence of the asymptotic expansion, new variables are introduced (which are not necessary in the isentropic Euler Poisson equations) to separate the hyperbolic and elliptic parts of the problem. Furthermore, we conclude that the Cauchy problem of the non-isentropic Euler Poisson equations has a unique classical solution in the time interval when the limit problem for the incompressible type of non-isentropic Euler system admits a classical solution. The paper is arranged as follows. In 2, we derive, by formal asymptotic analysis, an incompressible type of non-isentropic Euler system for the leading terms of the expansion and corresponding linearized equations for the other terms. The expansion is determined in 3 by solving the incompressible Euler system and linearized equations. In 4, we rigorously justify the formal asymptotic expansion and obtain the existence of solutions to the multi-dimensional non-isentropic Euler Poisson system in the time interval where the leading terms exist and are smooth. 2. Asymptotic analysis Denote by n, u, e and φ the respective density, velocity (vector), specific internal energy and electric potential of the electrons in a plasma. The Euler Poisson system consists of the following conservation (or balance) laws: n t + div(nu) =0, (nu) t + div(nu u)+ p = n φ nu, τ p E t + div(eu + pu) =nu φ E e (2.1) Ln, τ w λ 2 φ = n b(t, x), for (t, x) R t R d x. Here p = p(n, e) and E = n(e u 2 ) are the pressure and total energy, respectively; τ p, τ w and e L are positive constants and their physical meanings can be found in [3]; b(t, x) is the given ion density; λ>0 denotes the Debye length. Throughout this paper, we assume that there is a constant b 0 > 0 such that b(t, x) b 0 (t, x) R t R d x.

3 With Non-isentropic Euler Poisson system 1015 p = 2 3 ne, e = 3k B 2m T, e L = 3k B 2m T L and λ 2 = ε sm q 2, the system of equations in [3] is easily recovered from (2.1). We introduce an entropy variable, S, according to the Gibbs relation: ( ) 1 T ds =de + p d. n For the above specific model from [3], S can be taken to be S = k B m ( 3 2 ln T ln n). In the variables (n, u, S), system (2.1) can be rewritten as n t + div(nu) =0, u t + u u + 1 n p = φ u, τ p S t + u S = u 2 E e Ln, Tτ p nt τ w λ 2 φ = n b(t, x), (2.2) for smooth solutions with n>0. Moreover, it is easy to see that, with (p, u, S) as unknowns, (2.2) is equivalent to n p n (p t + u p) + div u = n ( S u 2 E e ) Ln, n Tτ p nt τ w u t + u u + 1 n p = φ u τ p, S t + u S = u 2 E e Ln, Tτ p nt τ w λ 2 φ = n b(t, x). (2.3) It is remarkable that the first three equations in (2.3) constitute a symmetrizable hyperbolic system with A 0 = diag(n p /n, ni d, 1) as its symmetrizer, provided that φ is given. We remark on the entropy variable, S, as follows. Remark 2.1. As usual, ns plays the role of an entropy density function for the physical system under consideration. From (2.2) 1,3 we can easily obtain ( u 2 (ns) t + div(nus) =n E e ) Ln. Tτ p nt τ w Here the right-hand side is the corresponding entropy production density and should be non-negative by the second law of thermodynamics. Thus, we have identified a

4 1016 Y.-J. Peng, Y.-G. Wang and W.-A. Yong possibly new constraint on the parameters: ( u 2 n E e ) ( Ln τw 0 e e L + 1 ) u 2, Tτ p nt τ w τ p 2 under which the mathematical description (2.1) of the physical system is valid. This inequality can be easily guaranteed by requiring that it holds for the initial data in our problem, because we shall study the smooth solutions only locally in time. For simplicity, we shall not emphasize this fact in the following discussion. We specify initial data for (2.1) at t =0as n t=0 = n λ (x), u t=0 =ū λ (x), S t=0 = S λ (x). (2.4) These data are assumed to have the following expansions: m n λ (x) = λ 2j n j (x)+λ 2(m+1) n λ m+1(x), j=0 m ū λ (x) = λ 2j ū j (x)+λ 2(m+1) u λ m+1(x), j=0 m S λ (x) = λ 2j Sj (x)+λ 2(m+1) Sm+1(x), λ j=0 (2.5) where ( n j ) 0 j m will be determined by (ū j, S j ) 0 j m and b(t, x). For example, n 0 and ū 0 are assumed to satisfy the zero-order compatibility conditions in (3.1), which guarantee that no initial layer appears when λ 0. Denote by (n λ,u λ,s λ,φ λ ) a solution to the Euler Poisson system (2.2) with (2.4). We will study its formal expansions with respect to λ (0,λ 0 ] for a certain λ 0 small. Plugging the ansatz n λ (t, x) = λ 2j n j (t, x), j 0 u λ (t, x) = λ 2j u j (t, x), j 0 S λ (t, x) = j 0 λ 2j S j (t, x), (2.6) φ λ (t, x) = λ 2j φ j (t, x) j 0 into (2.2), we obtain the following two results. (i) The leading terms (n 0,u 0,S 0,φ 0 ) satisfy the following equations: n 0 = b(t, x), (2.7)

5 Non-isentropic Euler Poisson system 1017 u 0 t + u 0 u b p0 = φ 0 u0, τ p St 0 + u 0 S 0 = u0 2 T 0 E 0 e L b τ p bt 0, τ (2.8) w div(bu 0 )= b t, u 0 t=0 =ū 0 (x), S 0 t=0 = S 0 (x), where p 0 = p(b, S 0 ), T 0 = T (b, S 0 ) and E 0 = E(b, u 0,S 0 ). Remark that, if the ion density b(t, x) is constant, say b(t, x) = 1 for simplicity, then from (2.8) we see that (u 0,S 0 ) satisfies the non-isentropic incompressible Euler equations. (ii) For any j 1, we find that n j = φ j 1, (2.9) and (u j,s j,φ j ) satisfy the linear system u j t + j u k u j k + 1 b (p S(b, S 0 )S j ) φ j + uj τ p = nj b 2 p0 1 b (p n(b, S 0 )n j )+f j 1 ({n k, n k, S k } k j 1 ), j S j t + u k S j k = g j 1 ({n k,u k,s k } k j ), div(bu j )= n j t u j t=0 =ū j (x), j div(n k u j k ), k=1 S j t=0 = S j (x). (2.10) Here p n and p S denote the partial derivatives of p = p(n, S) with respect to n and S, f j 1, g j 1 are smooth with respect to their arguments, and g j 1 depends linearly on (n j,u j,s j ), and we do not need their explicit formulae. 3. Determination of formal expansions In this section, we determine the formal expansion (2.6) by solving equations (2.8) (2.10). To this end, we give the following hypotheses: (H1) the Cauchy problem (2.1) with (2.4) is defined for x on the torus T d = (R/2π) d ; (H2) S 0 H s (T d ) and s> 1 2d + 1 is an integer; (H3) b 2 Ck ([0,T 0 ]; H s+2 k (T d )) for some T 0 > 0 and b b 0 on [0,T 0 ] T d for some constant b 0 > 0;

6 1018 Y.-J. Peng, Y.-G. Wang and W.-A. Yong (H4) ū 0 H s (T d ) and the zero-order compatibility conditions n 0 (x) =b(0,x), b t (0,x) + div(b(0,x)ū 0 (x)) = 0 (3.1) hold. Once (n 0,u 0,S 0,φ 0 ) has been obtained from (2.7) and (2.8), (n j,u j,s j,φ j ) j 1 can easily be determined by solving the linear problem (2.9), (2.10). Thus, the key is to solve the nonlinear problem (2.8). Dropping the superscript 0, the equations in (2.7) and (2.8) can be rewritten as n = b(t, x), (3.2) b t + div(bu) =0, u t +(u )u + u = 1 p + φ, (3.3) τ p b S t + u S = u 2 E e Lb, (3.4) Tτ p bt τ w where p = p(b, S), T = T (b, S) and E = E(b, u, S), with initial conditions u t=0 =ū 0 (x), S t=0 = S 0 (x). (3.5) At this point, we recall the following elementary fact. Lemma is a linear bounded operator from L 2 (T d ) into H 1 (T d ). Set v = bu + A with A = 1 b t. Equation (3.3) 1 is then equivalent to div v = 0, and (3.3) 2 becomes v t +(u )v + v = b P +(u )A + ub t p b +(u b)u + B, (3.6) τ p b where P = p/b φ and B = A t + A/τ p. Moreover, since div((u )v) = d i,k=1 u i x k v k x i +(u )(div v), (3.7) we apply the divergence operator on (3.6), and use div v = 0 to obtain ( div(b P ) = div (u )A + ub t p b ) +(u b)u + B b d i,j=1 u i x j v j x i, (3.8) where u =(u 1,...,u d ) and v =(v 1,...,v d ). The initial conditions for (v, S) are v t=0 = v 0 (x) def = b(0,x)ū 0 (x)+a(0,x), S t=0 = S 0 (x). (3.9) In order to see that the equations in (3.3) and (3.4) are equivalent to (3.4) together with (3.6) and (3.8), we prove the following lemma.

7 Non-isentropic Euler Poisson system 1019 Lemma 3.2. Let (v, S, P) be a smooth solution to (3.4) together with (3.6) and (3.8) and let its initial values satisfy the compatibility condition div v 0 =0. Then (u, S, φ) with u = v A p(b, S), φ = P (3.10) b b is a smooth solution to (3.3) and (3.4). Proof. It suffices to show that div v = 0. To this end, we apply the divergence operator on (3.6) and use (3.7), (3.8) to deduce that (div v) t +(u )(div v)+ div v =0. τ p Since div v 0 =0,wehavedivv = 0. This completes the proof. Now the new variables v, S and P can be obtained through the following iteration: v 0 (t, x) = v 0 (x), S 0 (t, x) = S 0 (x), (v l+1 ) t +(u l )v l+1 + v l+1 τ p (S l+1 ) t +(u l )S l+1 = f S l = f v l def = with u l =(v l A)/b and initial conditions def = b P l+1 +(u l )A + u l b t p(b, S l ) b ( 1 ul 2 E(b, u l,s l ) e L b T (b, S l ) τ p bτ w v l t=0 = v 0 (x), S l t=0 = S 0 (x), l 1. i,j=1 b +(u l b)u l + B, ), l 0, (3.11) Here P l+1 is determined by the elliptic equation on T d : ( div(b P l+1 ) = div (u l )A + u l b t p(b, S l ) b ) b +(u l b)u l + B d (u l ) i (v l ) j, l 0, (3.12) x j x i and m(p l+1 ) def = 1 (2π) d P l+1 (x, t)dx =0. T d Note that div v l = 0 is not guaranteed in the above iteration. However, if the sequence (v l,s l,p l ) converges in a certain strong topology, then lemma 3.2 shows that the limit v of (v l ) l 1 should satisfy div v = 0. In the following, we denote by s the norm of H s (T d ). For (3.12), we have the following lemma. Lemma 3.3. For (u l,s l ) C([0,T]; H s (T d )) with s> 1 2d +1, (3.12) has a unique solution P l+1 C([0,T]; H s+1 (T d )) with m(p l+1 )=0, and there is a constant C 1 > 0, depending only on b and s, such that P l+1 (t) s C 1 ( p(b, S l (t)) s + v l (t) 2 s +1). (3.13)

8 1020 Y.-J. Peng, Y.-G. Wang and W.-A. Yong This lemma indicates that P l+1 in (3.11), although non-local, can be viewed as a zero-order term like (u l b)u l. Thus, for given smooth (v l,s l ) the linear problem (3.11), (3.12) has a unique smooth solution (v l+1,s l+1,p l+1 ) in a certain time interval possibly depending on l. Moreover, with the proof in [5], we see the convergence of iteration in (3.11). Consequently, we have the following theorem. Theorem 3.4. Under the assumptions (H1) (H4), there is a positive number T depending only on s, b, ū 0 and S 0 such that the problem (3.3) (3.5) admits a unique solution (u, S) (C([0,T ]; H s (T d )) C 1 ([0,T ]; H s 1 (T d ))) 2, φ C([0,T ]; H s (T d )), (3.14) p(b, S) bφ C([0,T ]; H s+1 (T d )) in the class m(φ) =m(p(n, S)/b). Remark 3.5. In the isentropic problem, the solution for the electric potential φ has the regularity φ C([0,T ]; H s+1 (T d )) (see [9]). In the non-isentropic case, this regularity is replaced by p(b, S) bφ C([0,T ]; H s+1 (T d )) and we do not know how to recover this regularity for φ. From (3.6), (3.8), lemmas 3.2 and 3.3 and theorem 3.4, it is easy to deduce the following regularity of solutions. Corollary 3.6. Assume j 1 and (H1) (H4) hold. Assume furthermore that S 0 H s+3j (T d ), b 3j+2 C k ([0,T 0 ]; H s+3j k+2 (T d )) and ū 0 H s+3j (T d ). The solution (u, S, φ) to the problem (3.3) (3.5) then satisfies (u, S) φ p(b, S) bφ ( 3j+1 3j 3j C k ([0,T ]; H s+3j k (T d ))) 2, C k ([0,T ]; H s+3j k (T d )), C k ([0,T ]; H s+3j k+1 (T d )). (3.15) Now let us consider the solvability of (n j,u j,s j,φ j ) j 1. By induction, suppose that (n k,u k,s k,φ k ) 0 k j 1 are already known. Then n j is given by (2.9) and (u j,s j,φ j ) satisfies (2.10) for a linearized Euler system. Let P j = p S (b, S 0 )S j /b φ j. Then, from (2.9) and the relation 1 b (p S(b, S 0 )S j )+ φ j = P j p S (b, S 0 )S j b b 2,

9 Non-isentropic Euler Poisson system 1021 we see that the problem (2.10) can be written as u j t +(u 0 )u j +(u j )u 0 + uj = P j p S (b, S 0 )S j b τ p b 2 + φj 1 (u k )u j k j 1 b 2 p(b, S 0 ) 1 b (p n(b, S 0 ) φ j 1 ) + f j 1 ({n k, n k, S k } k j 1 )S j t +(u 0 )S j +(u j )S 0 j 1 = g j 1 ({n k,u k,s k } k j ) (u k )S j k, k=1 u j t=0 =ū j (x), S j t=0 = S j (x), (3.16) where P j satisfies the following elliptic equation on T d : [ div(b P j ) = div p S (b, S 0 )S j b b + b tu j + u j (u 0 )b b(u j )u 0 1 ] (bu j ) τ p d u 0 k (bu j i ) x i x k i,k=1 [ j 1 + div b (u k )u j k + φj 1 p(b, S 0 ) b k=1 ] (p n (b, S 0 ) φ j 1 )+bf j 1 ({n k, n k, S k } k j 1 ) ( +( t + u 0 ) φ j 1 t + div( φ j 1 u 0 )+ j 1 k=1 k=1 ) div(n k u j k ), (3.17) with m(p j ) = 0. Here ū j should satisfy the following jth-order compatibility condition: ( j ) div( n 0 (x)ū j (x)) = n j t(0,x)+ div( n k (x)ū j k (x)). (3.18) Using a method similar to lemma 3.2, we can prove that the problems (2.10) and (3.16), (3.17) are equivalent under condition (3.18). Indeed, if (u j,s j,p j )isa smooth solution of the problem (3.16), (3.17), by applying the divergence operator on (3.16) and using (2.9), (3.17), we obtain ( ( t + u 0 ) n j t + j k=1 ) div(n k u j k ) =0. Together with the compatibility condition (3.18), this implies that j n j t + div(n k u j k )=0. Therefore, (u j,s j,p j ) is a smooth solution of the problem (2.10).

10 1022 Y.-J. Peng, Y.-G. Wang and W.-A. Yong Analogously to lemma 3.3, for given (u j,s j ) (C([0,T]; H s (T d ))) 2, the problem (3.17) admits a unique solution P j C([0,T]; H s+1 (T d )), which satisfies the estimate P j (t) s+1 C( u j (t) s + S j (t) s +1), t [0,T], (3.19) provided that φ j 1 C([0,T],H s+3 (T d )) C 1 ([0,T],H s+2 (T d )) C 2 ([0,T],H s+1 (T d )) (3.20) and (n k,u k,s k ) (C([0,T],H s+1 (T d )) C 1 ([0,T],H s (T d ))) 3 (3.21) for all 0 k j 1. Using (3.19), we deduce that, under the conditions (ū j, S j ) (H s (T d )) 2 and (3.20), (3.21), the linear problem (3.16) has a unique solution (u j,s j,φ j ) satisfying } (u j,s j ) (C([0,T]; H s (T d )) C 1 ([0,T]; H s 1 (T d ))) 2, (3.22) φ j C([0,T]; H s (T d )), p S (b, S 0 )S j bφ j C([0,T]; H s+1 (T d )). Thus, by employing corollary 3.6 and by induction, we have proved the following theorem. Theorem 3.7. Let T be as determined in theorem 3.4, and let the assumptions of corollary 3.6 hold. For any fixed j 1, given (ū l, S l ) (H s+3(j l) (T d )) 2, let the compatibility conditions (3.18), with j being replaced by l, hold for any 1 l j. Then, for all 1 l j, the linear problem (2.10) with j being replaced by l has a unique solution (u l,s l,φ l ), which satisfies (u l,s l ) (C([0,T ]; H s+3(j l) (T d )) C 1 ([0,T ]; H s+3(j l) 1 (T d ))) 2, φ l C([0,T ]; H s+3(j l) (T d )), p S (b, S 0 )S l bφ l C([0,T ]; H s+3(j l)+1 (T d )). 4. Justification For a fixed positive integer m, weset (n λ,m,u λ,m,s λ,m )= m λ 2j (n j,u j,s j ), j=0 U λ,m =(p(n λ,m,s λ,m ),u λ,m,s λ,m ), where the (n j,u j,s j ) are those constructed in the previous sections. We then have, for λ small, U λ,m (t, ) C([0,T ],H s+1 (T d )) with s> 1 2 d +1, sup U λ,m (t, ) s+1 <, t,λ G 0 := {U λ,m (t, x) :(t, x) [0,T ] T d } (0, ) R d+1 (state space), (4.1)

11 Non-isentropic Euler Poisson system 1023 under the assumptions of theorem 3.7 being held with (j, s) being replaced by (m, s + 1). Here (4.1) 3 uses n 0 def = b b 0 > 0 and λ 1. Furthermore, U λ = U λ,m solves (2.2) with a remainder R λ = R λ (t, x): t U λ + d A j (U λ ) xj U λ = λ 2 F 1 (U λ )+F 2 (U λ )+R λ (4.2) j=1 in (t, x) [0,T ] T d, with R λ satisfying max 0 t T R λ (t, ) s = O(λ 2m ). (4.3) Here A j (U) is such that A 0 (U)A j (U) is a symmetric matrix, F 1 (U) =(0, 1 (n b), 0), ( u 2 F 2 (U) = E e )( Ln n )( S, 0, 1 0, u ), 0. Tτ p nt τ w n p τ p Theorem 4.1. Let s> 1 2 d +1 be an integer. Suppose that Ū(,λ) H s (T d ) for all λ sufficiently small and Ū(,λ) U λ,m(0, ) s = O(λ 2m ) with m>s. There is then λ 0 > 0 such that the Euler Poisson system (2.3) with periodic initial data Ū(,λ) has a unique classical solution (p λ,u λ,s λ ) C([0,T ],H s (T d )) for any 0 < λ λ 0, and there exists a constant K > 0, independent of λ but dependent on T <, such that (p λ p λ,u λ u λ,s λ S λ )(t, ) s Kλ 2m 2s (4.4) for all t [0,T ], where T > 0 is given in theorem 3.4. Remark 4.2. The solution of the Euler Poisson system has another component, φ λ, which cannot be prescribed initially and is unique merely up to a constant. For this component, one can easily deduce from (4.4) the following estimate: for all t [0,T ]. φ λ (t, ) λ 2 1 (n λ b) s+1 Kλ 2m 2s 2 We now turn to the proof of theorem 4.1. Proof. Due to lemma 3.1, the local-in-time existence theory (see [5, theorem 2.1]) for periodic initial-value problems of first-order symmetrizable hyperbolic systems applies to (2.3). Because of (4.1) 3, Ū(,λ) U λ,m(0, ) s = O(λ 2m ) and the embedding theorem, Ū takes values in a compact subset of the state space for 0 <λ λ 0. Thus, there is a convex and open subset G of the state space such that {Ū(x, λ) : (x, λ) Td (0,λ 0 ]} G 0 G (0, ) R d+1. (4.5) Since Ū(,λ) Hs (T d ), by the local-in-time existence theory there is a time interval [0,T] such that (2.3) has a unique classical solution: U λ C([0,T],H s (T d )) and U λ (t, x) G, (t, x) [0,T] T d.

12 1024 Y.-J. Peng, Y.-G. Wang and W.-A. Yong Define T λ = sup{t >0:U λ C([0,T],H s (T d )) and U λ (t, x) G, (t, x) [0,T] T d }. Namely, [0,T λ ) is the maximal time interval for the existence of the H s -solution U λ with values in the precompact set G. Note that T λ depends on G and may tend to zero as λ 0. Thus, we only need to show T λ >T and the error estimate (4.4). Moreover, it suffices to prove (4.4) for t [0, min{t,t λ }), thanks to the convergence-stability principle [13, lemma 9.1], which takes G 0 G in (4.5) as a condition. Now we turn to derive the error estimate (4.4) for t [0, min{t,t λ }). Note that, in this time interval, both U λ and U λ take values in the convex compact set Ḡ (see (4.1) and the definition of T λ ). We see from (2.3) and (4.2) that E = U λ U λ satisfies E t + j A j (U λ )E xj = λ 2 [F 1 (U λ ) F 1 (U λ )] + F 2 (U λ ) F 2 (U λ ) + R λ + j [A j (U λ ) A j (U λ )]U λxj. Differentiating this equation with α (in x) for a multi-index α satisfying α s leads to d E αt + A j (U λ )E αxj = λ 2 G α 1 + G α 2 + G α 3, (4.6) j=1 where E α = α E, G α 1 =(0, 1 E n α, 0) T with E n α = α (n λ n λ ), G α 2 =[F 2 (U λ ) F 2 (U λ )] α + R λ,α and G α 3 = j ([A j (U λ ) A j (U λ )]U λxj ) α + j [A j (U λ ), α ]E xj. Note that (4.6) is a symmetrizable hyperbolic system with ( n A 0 (U λ ) = diag (n λ,s λ ) λ p,n λ I d, 1) n as its symmetrizer. Applying the standard argument (see, for example, [5]) to (4.6) yields d Eα T A 0 (U λ )E α dx Eα T div A(U λ )E α dx + C E α λ 2 G α 1 + G α 2 + G α 3 dt T d T d (4.7) with div A(U λ )= t A 0 (U λ )+ j x j (A 0 A j )(U λ ) and = L 2 (T ). d Since U λ and U λ take values in the convex compact set Ḡ, wehave C 1 E α (t) 2 Eα T A 0 (U λ )E α dx C E α (t) 2. (4.8) T d

13 Non-isentropic Euler Poisson system 1025 From the equations satisfied by n λ and S λ we deduce that div A(U λ )= t A 0 (U λ )+ j xj (u λ j A 0 (U λ )) Thus, we have =( t + u λ + div u λ )A 0 (U λ ) = div u λ (A 0 (U λ ) n λ A 0n (U λ )) + ( u λ 2 T λ E λ e L n λ ) τ p n λ T λ A 0S (U λ ). τ w div A(U λ ) C div u λ + C C E s + C. (4.9) For G α 1 we see from proposition 3.1 that G α 1 C E α 1. (4.10) Moreover, we use the classical Moser-type calculus inequalities in Sobolev spaces to obtain G α 2 C R λ,α + C E α (1 + E s s), G α 3 C A j (U λ ) s E α + C U λxj s E α (1 + E s s) C E α (1 + E s s). (4.11) Here U λ s+1 = O(1) in (4.1) and U λ s U λ U λ s + U λ s E s + C have been used. Now, substituting the inequalities in (4.8) (4.11) into (4.7) gives E α (t) 2 C E α (0) 2 + C t 0 R λ,α (τ) 2 dτ t t + Cλ 4 E(τ) 2 α 1 dτ + C E(τ) 2 α (1 + E(τ) s s)dτ. (4.12) 0 As in the isentropic case [9], we define the following weighted norm: E(t) s,λ = 0 s λ 2 α E α (t). α =0 Recall (4.3) and the assumption of the theorem that sup R λ (t) s,λ = O(λ 2m )= E(0) s,λ. 0 t T It follows from (4.12) that E(t) 2 s,λ Cλ 4m + C t 0 E(τ) 2 s,λ(1 + E(τ) s s)dτ. Thus, we deduce from the Gronwall inequality that { t } E(t) 2 s,λ Cλ 4m exp C (1 + E(τ) s s)dτ λ 4s Φ(t). (4.13) 0

14 1026 Y.-J. Peng, Y.-G. Wang and W.-A. Yong Since E(t) s λ 2s E(t) s,λ Φ(t) for λ<1, it follows that Φ (t) =C(1 + E(t) s s)φ(t) CΦ(t)+CΦ 1+s/2 (t). Applying the nonlinear Gronwall-type inequality in [12] to the last inequality above yields E(t) 2 s Φ(t) e CT for t [0, min{t λ,t })ifwechooseλ so small that Φ(0) = Cλ 4m 4s < e CT. Hence, the theorem is concluded from (4.13). This completes the proof. Remark 4.3. In case U λ,m is defined globally in time, we actually proved the following existence result for the hydrodynamical model: For any T <, there exists a λ 0 = λ 0 (T ) such that, for each λ<λ 0, the Euler Poisson system (2.3) with periodic initial data Ū(,λ) has a unique classical solution (pλ,u λ,s λ ) C([0,T ],H s (T d )). Acknowledgments This work was supported by the European IHP project Hyperbolic and kinetic equations under Contract no. HPRN-CT (Y.-J.P. and W.-A.Y.), and by a key project from the NSFC, the Educational Ministry of China and the Shanghai Science and Technology Committee under Contract no. 03QMH1407 (Y.-G.W.). References 1 F. Chen. Introduction to plasma physics and controlled fusion, vol. 1 (New York: Plenum, 1984). 2 S. Cordier and E. Grenier. Quasineutral limit of an Euler Poisson system arising from plasma physics. Commun. PDEs 25 (2000), A. Jüngel. Quasi-hydrodynamic semiconductor equations (Birkhäuser, 2001). 4 S. Klainerman and A. Majda. Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34 (1981), A. Majda. Compressible fluid flow and systems of conservation laws in several space variables (Springer, 1984). 6 P. Markowich, C. A. Ringhofer and C. Schmeiser. Semiconductor equations (Springer, 1990). 7 Y.-J. Peng. Some asymptotic analysis in steady-state Euler Poisson equations for potential flow. Asymp. Analysis 36 (2003), Y.-J. Peng and Y.-G. Wang. Boundary layers and quasi-neutral limit in steady state Euler Poisson equations for potential flows. Nonlinearity 17 (2004), Y.-J. Peng and Y.-G. Wang. Convergence of compressible Euler Poisson equations to incompressible type Euler equations. Asymp. Analysis 41 (2005), M. Slemrod and N. Sternberg. Quasi-neutral limit for the Euler Poisson system. J. Nonlin. Sci. 11 (2001), S. Wang. Quasineutral limit of Euler Poisson system with and without viscosity. Commun. PDEs 29 (2004), W.-A. Yong. Singular perturbations of first-order hyperbolic systems with stiff source terms. J. Diff. Eqns 155 (1999), W.-A. Yong. Basic aspects of hyperbolic relaxation systems. In Advances in the theory of shock waves (ed. H. Freistühler and A. Szepessy). Progress in Nonlinear Differential Equations and Their Applications, vol. 47, pp (Birkhäuser, 2001). (Issued 6 October 2006 )