Global existence and asymptotic behavior of the solutions to the 3D bipolar non-isentropic Euler Poisson equation

Transcription

1 Nonlinear Analysis: Modelling and Control, Vol., No. 3, ISSN Global existence and asymptotic behavior of the solutions to the 3D bipolar non-isentropic Euler Poisson equation Yeping Li Department of Mathematics East China University of Science and Technology Shanghai 37, China Received: October 1, 13 / Revised: March 9, 14 / Published online: March, 15 Abstract. In this paper, the global existence of smooth solutions for the three-dimensional 3D non-isentropic bipolar hydrodynamic model is showed when the initial data are close to a constant state. This system takes the form of non-isentropic Euler Poisson with electric field and frictional damping added to the momentum equations. Moreover, the L -decay rate of the solutions is also obtained. Our approach is based on detailed analysis of the Green function of the linearized system and elaborate energy estimates. To our knowledge, it is the first result about the existence and L -decay rate of global smooth solutions to the multi-dimensional non-isentropic bipolar hydrodynamic model. Keywords: non-isentropic bipolar Euler Poisson system, asymptotic behavior, Green function, smooth solution, energy estimates. 1 Introduction In this paper, we consider the following three-dimensional non-isentropic bipolar hydrodynamic model: t ρ 1 + m 1 =, m1 m 1 t m ρ 1 T 1 = ρ 1 φ m 1, ρ 1 τ 1 t T 1 + m 1 T 1 + ρ 1 3 T m1 1 κ T 1 3ρ 1 = τ τ 1 3τ 1 τ m 1 ρ 1 ρ 1 T 1 T L1 τ, The research is partially supported by the National Science Foundation of China grant No , the PhD Program Foundation of Ministry of Education of China grant No and the Innovation Program of Shanghai Municipal Education Commission grant No. 13ZZ c Vilnius University, 15

2 36 Y. Li t ρ + m =, m m t m + + ρ T = ρ φ m, ρ τ 1 t T + m T + ρ 3 T m κ T ρ 3ρ = τ τ 1 m 3τ 1 τ ρ T T L, τ λ φ = ρ 1 ρ. 1 Here t, x R +, and the unknown variables ρ i, m i, T i i = 1,, and φ are the charge densities, current densities, temperatures, and electrostatic potential. The coefficients τ 1, τ, κ and λ are the momentum relaxation time, the energy relaxation limit, the heat conduction and the Debye length, respectively. The constants T L1 and T L stand for the lattice temperature. The non-isentropic bipolar hydrodynamic model plays an important role in simulating the behavior of charge carries in submicron semiconductor devices, see the pioneering work by Blotekjaer in [4], and also see [,3]. The model takes the nonisentropic Euler Poisson form, and consists of a set of nonlinear conservation laws for particle number, momentum, and energy, plus Poisson s equation for the electric potential. Moreover, it is worth mentioning that there are a lot of simplified models in the fields of applied and computational mathematics, i.e., we can refer to [11, 16, ], etc. Recently, many efforts were made for the isentropic bipolar hydrodynamic equations from semiconductors or plasmas. Zhou and Li [5] and Tsuge [] discussed the unique existence of the stationary solutions for the one-dimensional bipolar hydrodynamic model with proper boundary conditions. Natalini [18] and Hsiao and Zhang [6] established the global entropy weak solutions in the framework of compensated compactness on the whole real line and bounded domain respectively. Hattori and Zhu [6] proved the stability of steady-state solutions for a recombined one-dimensional bipolar hydrodynamical model. Gasser, Hsiao and Li [5] investigated the large time behavior of smooth small solutions for the one-dimensional bipolar hydrodynamic model, and they found that the frictional damping is the key to the nonlinear diffusive phenomena of hyperbolic waves. Huang and Li [7] also studied the large-time behavior and quasi-neutral limit of L solution for large initial data with vacuum. Huang, Mei and Wang [8] discussed the large time behavior of solution to n-dimensional bipolar hydrodynamic model for semiconductors in switch-on case. Ali and Jüngel [1] and Li and Zhang [15] studied the global smooth solutions of the Cauchy problem for multidimensional bipolar hydrodynamic models in the Sobolev space H l R d l > 1 + d/ and in the Besov space, respectively. Ju [1] discussed the global existence of smooth solutions to the IBVP for the 3D bipolar Euler Poisson system 1. Li and Yang [14] discussed the global existence and L -decay rates of smooth solutions for the three-dimensional isentropic bipolar hydrodynamic model. To our knowledge, there are very few results about the non-isentropic bipolar hydrodynamic model 1. Li [13] investigated the global existence and nonlinear diffusive waves of smooth solutions for the initial value problem of the one-dimensional non-isentropic bipolar hydrodynamic model. Jiang, etc. [9] discussed the quasi-neutral limit of the full bipolar

3 Global existence and asymptotic behavior of 3D BEPS 37 Euler Poisson system and obtained the local existence of smooth solutions for the initial value problem. In this paper, we will discuss the global existence and asymptotic behavior of smooth solutions of the initial value problem for the three-dimensional hydrodynamic model 1 here. For the sake of simplicity, we assume τ = κ = 1, τ 1 = τ and T L1 = T L = T L, then we can rewrite 1 as t ρ 1 + m 1 =, m1 m 1 t m ρ 1 T 1 = ρ 1 φ m 1, ρ 1 t T 1 + m 1 ρ 1 T T 1 m1 ρ 1 3ρ 1 T 1 + T 1 T L =, t ρ + m =, m m t m + + ρ T = ρ φ m, ρ t T + m ρ T + 3 T φ = ρ 1 ρ. m ρ 3ρ T + T T L =, We also prescribe the initial data as ρ 1 t =, x = ρ 1 x >, ρ t =, x = ρ x >, x, m 1, T 1, m, T t =, x = m 1, T 1, m, T x, φ as x. 3 The main result in this paper is stated in the following theorem. Theorem 1. Let ρ, m, T L be constant state with ρ > and T L >. Assume that Θ =: ρ 1 ρ, m 1 m, T 1 T L, ρ ρ, m m, T T L H 4 L 1 is small enough. Then, there is a unique global classical solution ρ 1, m 1, T 1, ρ, m, T, φ of the IVP 3 satisfying ρ 1 ρ, ρ ρ C R +, H 4 C 1 R +, H 3, m 1 m, m m C R +, H 4 C 1 R +, H 3, T 1 T L, T T L C R +, H 4 C 1 R +, H, φ C R +, L 6, φ C R +, H 5, and there is some positive constant C > such that, for i = 1,, x ρ i ρt CΘ 1 + t 3/4 /,, 4 x m i mt CΘ 1 + t 1/4 /,, 5 x T i T L t CΘ 1 + t 3/4 /,, 6 β x φt CΘ 1 + t 1/4 β /, β 3. Nonlinear Anal. Model. Control, 3:35 33

4 38 Y. Li Remark 1. Furthermore, if we assume that Θ 1 =: ρ 1 ρ, m 1 m, ρ ρ, m m H s+l L s =, l is small, then there is a unique global classical 1 solution ρ 1, m 1, T 1, ρ 1, m, T, φ of the IVP 3 satisfying ρ 1 ρ, ρ ρ C R +, H s+l C 1 R +, H s+l 1, m 1 m, m m C R +, H s+l C 1 R +, H s+l 1, T 1 T L, T T L C R +, H s+l C 1 R +, H s+l, φ C R +, L 6, φ C R +, H s+l+1, and there is some positive constant C > such that, for i = 1, and l, β l + 1, x ρ i ρt CΘ1 1 + t 3/4 /, x m i mt CΘ t 1/4 /, x T i T L t CΘ t 3/4 /, β x φt CΘ1 1 + t 1/4 β /. Remark. Compared with the Euler equations with damping in [3], we find that the interaction of the two particles and the additional electric field reduce the decay rate of the momentums, which are seen in the isentropic bipolar case in [14]. Moreover it is interesting studying the existence and stability of the planar diffusion waves for the multidimensional full bipolar Euler Poisson system in switch-on case as in [8], which is left for the forthcoming future. The idea of the proof is outlined as follows. First, we present local-in-time existence of the initial value problem 7 8 by the standard argument of contracting map theorem as in [1]. Next, combining the local existence and global a-priori estimates, we apply the continuity argument to establish global existence of smooth solutions for the nonlinear problem. The key point is to derive the a-priori estimates, in which we show the estimates of the lower order derivatives of solutions by the spectral analysis of the corresponding linearized equations, and obtain the estimates of the higher order derivatives of solutions by elaborate energy estimates. The rest of this paper is outlined as follows. In Section, we reformulate the original problem in terms of the perturbed variable, and present the L decay rate of the linearized equations. The global existence and L -convergence rates of smooth solutions will be shown in Section 3. Notations. Throughout this paper, C > denotes a generic positive constant independent of time. L p 1 p < denotes the space of measurable functions whose p-powers are integrable on, with the norm L p = p dx 1/p, and L is the space of bounded measurable functions on, with the norm L = esssup x, and also simply denote L by. H k k stands for usual Sobolev space with the norm s. Moreover, we denote s + L 1 by Hs L 1. Finally, for f L, the Fourier transform of f is ˆfξ = F[f]ξ = π 3/ fxe ix ξ dx.

5 Global existence and asymptotic behavior of 3D BEPS 39 Solutions of the linearized equation Firstly, we reformulate the nonlinear system for ρ 1, m 1, T 1, ρ, m, T around the equilibrium state ρ, m, T L, ρ, m, T L. Without loss of generality, we can take ρ, m, T L, ρ, m, T L = 1,, 1, 1,, 1. Denote n i = ρ i 1, m i = m i, θ i = T i 1, i = 1,, Φ = φ, then the IVP problem for n 1, m 1, θ 1, n, m, θ, Φ is given by with the initial data t n i + m i =, i = 1,, t m i + n i + θ i 1 i Φ + m i = f i, i = 1,, t θ i + 3 m i 3 θ i + θ i = g i, i = 1,, Φ = n 1 n, lim Φx, t =, x 7 n 1, m 1, θ 1, n, m, θ x, = n 1, m 1, θ 1, n, m, θ x = ρ 1 1, m 1, T 1 1, ρ 1, m, T 1x. 8 Here the nonlinear terms f i, g i i = 1, are defined by f i = 1 i 1 mi m i n i Φ n i θ i, n i + 1 g i = m i n i + 1 θ i 3 θ m i i n i mi n i + 1 m i n i θ i. For simplicity, we replace Φ with the following formulation: Φ = 1 n 1 n. 9 Inserting 9 into 7 and neglecting the nonlinear terms, we have the following linearized nonisentropic bipolar Euler Poisson system: t n i + m i =, i = 1,, t m i + n i + θ i 1 i 1 n 1 n + m i =, i = 1,, t θi + 3 m i 3 θ i + θ i =, i = 1,, 1 with n 1, n, θ 1, m 1, m, θ x, = n 1, m 1, θ 1, n, m, θ x. 11 By setting Ū = n 1, m 1, θ 1, n, m, θ t, IVP 1 11 can be expressed as Ū t = BŪ, Ū = U, t. 1 Nonlinear Anal. Model. Control, 3:35 33

6 31 Y. Li In particular, there is a solution of the following IVP: U t = BU, U = δxi 1 1, which was always called Green function and denoted as Gx, t. And the solution of 1 can be expressed as Ūx, t = G, t U, 13 where is the convolution in x. In the following, we focus on analyzing the properties of G, t U. For this aim, from 1, we have t n 1 + n + m 1 + m =, t m 1 + m + n 1 + n + θ 1 + θ + m 1 + m =, t θ 1 + θ + 3 m 1 + m 3 θ 1 + θ + θ 1 + θ = 14 and t n 1 n + m 1 m =, t m 1 m + n 1 n + θ 1 θ + m 1 m 1 n 1 n =, 15 t θ 1 θ + 3 m 1 m 3 θ 1 θ + θ 1 θ =, By setting Ū1 = n 1 + n, m 1 + m, θ 1 + θ t, and Ū = n 1 n, m 1 m, θ 1 θ t, IVP 14 and 15 can be expressed as and Ū 1t = B 1 Ū 1, Ū 1 = U 1 = n 1 + n, m 1 + m, θ 1 + θ t, t, 16 Ū t = B Ū, Ū = U = n 1 n, m 1 m, θ 1 θ t, t, 17 Let us denote the solution of the following IVP: U it = B i U i, U i = δxi 5 5 by G i x, t. Then the solution of 14 and 15 can be expressed as U i x, t = G i, t n 1 ± n, m 1 ± m, θ 1 ± θ, 18 where is the convolution in x. Applying the Fourier transform to system 16 with respect to x, we have t Û 1 ξ, t = A 1 ξû1ξ, t ξ = ξ1, ξ, ξ 3, Û 1 = I 5 5, where iξ A 1 ξ = iξ I 3 iξ T iξ/3 ξ /

7 Global existence and asymptotic behavior of 3D BEPS 311 The eigenvalues of the matrix A 1 ξ can be computed from the determinant det λi 5 A 1 ξ 7 = λ + 1 λ ξ + λ + 3 ξ + 1 λ + ξ 3 ξ + 1 =, which implies λ 1 = 1 double, and there exist positive constants r 1 and r satisfying r 1 < r such that when ξ > r, λ = b 3 Y 1 3 Y, λ 3,4 = 1 3a 6a b + 3 Y Y ± 6 3 Y 1 3 Y, and when ξ < r 1, λ = b A cos ϑ 3 3a, λ 3,4 = 1 b + A cos ϑ 3a 3 ± 3 sin ϑ. 3 Here a = 1, b = 3 ξ +, c = 7 3 ξ + 1, d = ξ 3 ξ + 1, A = b 3ac, B = bc 9ad, C = c 3bd, Y 1, = Ab + 3a B ± B 4AC, ϑ = arccos Ab 3aB A 3. The semigroup S 1 t = e ta1 is expressed as 4 e ta1ξ = e λit P i ξ = Ĝ 1 11 Ĝ 1 1 Ĝ 1 31 Ĝ 1 1 Ĝ 1 Ĝ 1 3 Ĝ 1 13 Ĝ 1 3 Ĝ 1 33, 19 where the project operators P i ξ i = 1,, 3, 4 can be computed as P i ξ = j i A 1 ξ λ j I λ i λ j. Similarly, for iξ A ξ = iξ 1 + ξ I 3 iξ T, iξ/3 ξ /3 + 1 we have 4 e taξ = e λ it Pi ξ = Ĝ 11 Ĝ 1 Ĝ 31 Ĝ 1 Ĝ Ĝ 3 Ĝ 13 Ĝ 3 Ĝ 33, Nonlinear Anal. Model. Control, 3:35 33

8 31 Y. Li where λ 1 = and λ,3,4 satisfy λ ξ + λ ξ + 1 λ + ξ ξ + 1 =, and the project operators P i ξ i = 1,, 3, 4 can be computed as P i ξ = j i A ξ λ j I λ i λ j. Noting in 19 and, G i 1 i = 1, are 1 3-matrix, G i i = 1, are 3 3-matrix, and G i 1, G i 3 i = 1, are 3 1-matrix, the other are 1 1-matrix. Then, from 19 and, we have G 11 G 1 G 13 G 14 G 15 G 16 G 1 G G 3 G 4 G 5 G 6 Gξ, t = G 31 G 3 G 33 G 34 G 35 G 36 G 41 G 4 G 43 G 44 G 45 G 46, 1 G 51 G 5 G 53 G 54 G 55 G 56 G 61 G 6 G 63 G 64 G 65 G 66 where G ij = 1 G 1 ij + G ij, i = 1,, 3, j = 1,, 3, G ij = 1 G 1 il + G il, i = 1,, 3, j = 4, 5, 6, l = j 3, G ij = 1 G 1 lj G lj, i = 4, 5, 6, j = 1,, 3, l = i 3, G ij = 1 G 1 kl G kl, i = 4, 5, 6, j = 4, 5, 6, k = i 3, l = j 3. Using the idea and argument of [14, 17, 19, 4], we have L -estimate of solution for the IVP 1 11 as follows. Lemma 1. If U L 1 H 4, then for i = 1, 3, 4, 6, j =, 5, we have x G i1 U, G i4 U C1 + t 3/4 / U L 1 + x U, x G i U, G i5 U C1 + t 3/1/m 1/ 1/ / U L m + x U, x G i3 U, G i6 U C1 + t 3/1/m 1/ 1/ / U L m + x U, x G i3 U, G i6 U C1 + t 1/ / U + β x U, 1, x G j1 U, G j4 U C1 + t 1/4 / U L 1 + x U, x G j U, G j5 U C1 + t 3/1/m 1/ / U L m + x U, x G j3 U, G j6 U C1 + t 3/1/m 1/ 1/ / U L m + x U, x G j3 U, G j6 U C1 + t 1/ / U + β x U, 1, where 4, β = 1, and m = 1,.

9 Global existence and asymptotic behavior of 3D BEPS 313 Indeed, with the express of Ĝξ, t, and applying Hausdorff Young inequality, we can show Lemma 1, and the details can be found in [14, 17, 19, 4]. Moreover, from 9 and 1, the Fourier transform for the electric field is ˆĒ = iξ ξ ˆ n 1 ˆ n = iξ ξ G 11, G 1, G 13, G 14, G 15, G 16 Û + iξ ξ G 41, G 4, G 43, G 44, G 45, G 46 Û. From the above equality, we can define where ˆĒ = ˆLÛ = ˆL + ˆLÛ, ˆL = iξ ξ G 11 G 41,,, G 14 G 44,,, ˆL = iξ ξ, G 1 G 4, G 13 G 43,, G 15 G 45, G 16 G 46. Here ˆL, ˆL, ˆL are the Fourier transform of function L, L, L, respectively. From Lemma 1, the estimates of L,L, L is given as following. Lemma. If U L 1 H 4, then for m = 1,, we have x L U C1 + t 1/4 / U L 1 + x U, x L U C1 + t 3/4 / U L 1 + x U. 3 Global existence and L -decay rate In this section, we are going to establish the global existence and show the L -decay rate of the solution of nonlinear problem 7 8. First of all, we give the local existence theory, which can be established in the framework as in [1]. The key point is the electric field Φ can be expressed by the Riesz potential as a nonlocal term Φ = Φt =, x + 1 t m 1 m ds, which together with the L p estimates of Riesz potential leads to t t 1 m 1 m ds C m 1 m ds. k k Then, we can prove the following local-in-time existence of the initial value problem 7 8 by the standard argument of contracting map theorem as in [1]. The details are omitted. Nonlinear Anal. Model. Control, 3:35 33

10 314 Y. Li Theorem. Assume that n 1, m 1, θ 1, n, m, θ x H 4. Then, there is a time T > such that the IVP 7 8 has a unique global smooth solution n 1, m 1, θ 1, n, m, θ, Φ: n 1, m 1, n, m C [, T ], H 4 C 1 [, T ], H 3, θ 1, θ C [, T ], H 4 C 1 [, T ], H, Φ C [, T ], L 6, Φ C [, T ], H 5, satisfying inf t,x [,T ] n i t, x >, and n 1, m 1, θ 1, n, m, θ, t 4 + Φ, t 5 + Φ, t L 6 C. To extend the local existence of solution to be a global solution in time, we need to establish some uniform a priori estimates. For this aim, we will look for the solution in the following space S = { n 1, m 1, θ 1, n, m, θ, Φ H 4 6 H 5 Λt < + }, where { Λt = sup 1+s 1/4 [ Φs + 1+s 3/4+ / D n 1, θ 1, n, θ s s t + 1+s 1/4+ / D m 1, m s ] +1+s 3/4 D 3 n 1, m 1, θ 1, n, m, θ + D 4 n 1, m 1, θ 1, n, m, θ }. Due to the property of Riesz potential see [1], we have D k Φ C D k 1 n 1 n, k 1. 3 It seems that the estimates of the high order derivatives of Φ come from the bounds of n 1 and n. That is, if n 1, m 1, n, m, Φ S, it is obviously that for all s t, D k Φs 1 + s 1/4 k/ Λt k = 1,, 3, D 4 Φs 1 + s 3/4 Λt. So we should obtain the estimate of Φ itself. Lemma 3 The a priori estimate of lower order derivatives of solution. Under the assumption of Theorem 1, suppose that n 1, m 1, θ 1, n, m, θ, Φ S is the solution of IVP 7 8 on [, T ] for any T >, which satisfies the following assumption: 4 n 1, m 1, θ 1, n, m, θ, Φ S, Λt δ for δ

11 Global existence and asymptotic behavior of 3D BEPS 315 Then, for any t [, T ], there exists some constant C such that k xn 1, n, θ 1, θ t CΘ 1 + t 3/4 k/ + C Λt 1 + t 3/4 k/, 3 x n 1, n, θ 1, θ t CΘ 1 + t 3/4 + C Λt 1 + t 3/4, k x m 1, m t CΘ 1 + t 1/4 k/ + C Λt 1 + t 1/4 k/, Here k =, 1,. 3 xm 1, m t CΘ 1 + t 3/4 + C Λt 1 + t 3/4, Φt CΘ 1 + t 1/4 + C Λt 1 + t 1/4. Proof. By Duhamel principle, it is easy to verify that the solution U = n 1, m 1, θ 1, n, m, θ, Φ of the IVP problem 7 8 can be expressed as n 1 = G 11, G 1, G 13, G 14, G 15, G 16 U + t G 1 f 1 + G 13 g 1 + G 15 f + G 16 g s ds, 6 m 1 = G 1, G, G 3, G 4, G 5, G 6 U + t G f 1 + G 3 g 1 + G 5 f + G 6 g s ds, 7 θ 1 = G 31, G 3, G 33, G 34, G 35, G 36 U + t G 3 f 1 + G 33 g 1 + G 35 f + G 36 g s ds, 8 n = G 41, G 4, G 43, G 44, G 45, G 46 U + t G 4 f 1 + G 43 g 1 + G 45 f + G 46 g s ds, 9 m = G 51, G 5, G 53, G 54, G 55, G 56 U + t G 5 f 1 + G 53 g 1 + G 55 f + G 56 g s ds, 3 and θ = G 61, G 6, G 63, G 64, G 65, G 66 U t + G 6 f 1 + G 63 g 1 + G 65 f + G 66 g s ds, 31 t Φ = L U + Lt s, f 1, g 1,, f, g s ds. 3 Nonlinear Anal. Model. Control, 3:35 33

12 316 Y. Li First, from 6 and Lemma 1, we have D k x G 11, G 1, G 13, G 14, G 15, G 16 U C1 + t 3/4 k / U L 1 + U k. For the estimates of the nonlinear terms, we first have for i = 1,, n i, θ i C1 + t 3/ Λt, Dn i, Dθ i C1 + t 1 Λt, with the aid of m i, Φ C1 + t 1 Λt, 33 Dm i, Φ C1 + t 3/4 Λt 34 u L C Du 1/ L D u 1/ for u H, which appeared in [1]. Further, from 5 and 33 34, we can get that, for i = 1,, f i U C n i Φ L + Dm i m i L + Dn i m i L + Dn i n i L + Dn i θ i L + n i L Dθ i C1 + t 7/4 Λt, 35 fi U L 1 C n i Φ + Dm i m i + Dn i m i m i L + Dn i n i + Dn i θ i + n i Dθ i C1 + t 1 Λt, 36 gi U C mi L θ i + Dm i θ i L + n i L + Dn i m i L θ i L + θ i n i L + Dn i m i L + Dn i m i L n i L C Λt 1 + t 9/4, 37 and g i U L 1 Thus, one have C m i θ i + Dm i θ i + Dn i m i L θ i + θ i n i + Dn i m i + n i Dm i + Dn i m i n i L C Λt 1 + t 3/. 38 n 1 G 11, G 1, G 13, G 14, G 15, G 16 U t + G1 f 1 + G 13 g 1 + G 15 f + G 16 g dτ C U L 1 L1 + t 3/4 + C t 1 + t τ 5/4 fi Uτ L + gi Uτ 1 L L 1 L dτ

13 Global existence and asymptotic behavior of 3D BEPS 317 CΘ 1 + t 3/4 + C Λt t 1+t τ 5/4[ 1+τ 7/4 + 1+τ τ 9/4 + 1+τ 3/] dτ CΘ 1 + t 3/4 + C Λt 1 + t 1. Next, we have the following estimate of the nonlinear terms f i U i = 1, and g i U i = 1, : Df i U C Λt 1 + t 3/, 39 Dg i U C Λt 1 + t, 4 D f i U C Λt 1 + t 7/4, 41 and which together with 6 yield D g i U C Λt 1 + t 1, 4 Dn 1 t DG 11, G 1, G 13, G 14, G 15, G 16 U t + D k G 1 f 1 + G 13 g 1 + G 15 f + G 16 g τ dτ C1 + t 5/4 U L 1 H 1 t/ + C 1 + t τ 7/4 fi U L 1 + Dfi Uτ dτ + C t t/ t 1 + t τ 1 f i Uτ + Df i Uτ dτ t/ + C 1 + t τ 7/4 g i U L 1 + Dg i Uτ dτ + C t/ 1 + t τ 1 gi U + Dgi Uτ dτ C1 + t 5/4 U L 1 H 1 + C Λt t/ 1 + t τ 7/4[ 1 + τ τ 3/] dτ Nonlinear Anal. Model. Control, 3:35 33

14 318 Y. Li t + C Λt 1 + t τ 1[ 1 + τ 7/ τ 3/] dτ t/ + C Λt t/ 1 + t τ 7/4[ 1 + τ τ 3/] dτ t + C Λt 1 + t τ 1[ 1 + τ 9/ τ ] dτ t/ CΘ 1 + t 5/4 + C Λt 1 + t 3/ and D n 1 t D G 11, G 1, G 13, G 14, G 15, G 16 U t + D G 1 f 1 + G 13 g 1 + G 15 f + G 16 g τ dτ C1 + t 7/4 U L1 H t/ + C 1 + t τ 9/4 f i U L 1 + D f i Uτ dτ + C t t t 1 + t τ 3/ fi Uτ + D f i Uτ dτ t/ + C 1 + t τ 9/4 gi U L 1 + D g i U dτ + C t/ 1 + t τ 3/ gi U + Dgi U dτ C1 + t 7/4 U L1 H t + C Λt t/ 1 + t τ 9/4[ 1 + τ τ 7/4] dτ + C Λt t/ 1 + t τ 3/[ 1 + τ 7/ τ 7/4] dτ

15 Global existence and asymptotic behavior of 3D BEPS C Λt t/ 1 + t τ 9/4[ 1 + τ τ 3/] dτ + C Λt t t/ 1 + t τ 3/[ 1 + τ 7/ τ 9/4] dτ CΘ 1 + t 7/4 + C Λt 1 + t 7/4 with the help of Finally, we can show D 3 f i U C Λt 1 + t 1, which together with 35 38, and 4 leads to D 3 n 1 t D 3 G 11, G 1, G 13, G 14, G 15, G 16 U t + D k G 1 f 1 + G 13 g 1 + G 15 f + G 16 g τ dτ C1 + t 11/4 U L1 H 3 t/ + C 1 + t τ 11/4 f i U L 1 + D 3 f i Uτ dτ + C + C t t/ t 1 + t τ 3/ fi Uτ + D 3 f i Uτ dτ 1 + t τ g i Uτ + D g i Uτ dτ C1 + t 11/4 U L1 H 3 + C Λt t/ 1 + t τ 11/4[ 1 + τ τ 1] dτ + C Λt + C Λt t t/ t 1+t τ [ 1+τ 7/4 + 1+τ 7/4 + 1+τ 1] dτ 1+t τ [ 1+τ 7/4 + 1+τ 7/4 + 1+τ 1] dτ CΘ 1+t 3/4 + C Λt 1+t 3/4. Nonlinear Anal. Model. Control, 3:35 33

16 3 Y. Li Moreover, in the completely same way, from 7, we can obtain m 1 t CΘ 1 + t 1/4 + C Λ 1 t 1 + t 1, D m 1 t CΘ 1 + t 1/4 k/ + C Λ 1 t 1 + t 1/4 k/, = k = 1,, D 3 m 1 t L CΘ 1 + t 7/4 + C Λ 1 t 1 + t 1, and from 8, we can show θ 1 t CΘ 1 + t 3/4 + C Λ 1 t 1 + t 1, D θ 1 t CΘ 1 + t 3/4 k/ + C Λ 1 t 1 + t 3/4 k/, = k = 1,, D 3 θ 1 t L CΘ 1 + t 9/4 + C Λ 1 t 1 + t 1, The estimates of n, m and θ can be obtained by the completely similar way. We omit the details here. For the time decay rate for Φ, from Lemma and 35 38, it is easy to get Φt t L U + Lt τ, f 1, g 1,, f, g τ dτ C1 + t 1/4 U L1 L + C t This completes the proofs. 1 + t τ 3/4 fi τ L + gi τ 1 L L 1 L dτ CΘ 1 + t 1/4 + C Λt 1 + t 1/4. Next, we are going to derive the estimates of higher order derivatives of n 1, m 1, θ 1, n, m, θ. For simplicity, we denote u i = m i /n i + 1, i = 1,. From 7 8, we derive the system for n 1, u 1, n, u, Φ as t n i + u i = n i u i, i = 1,, 43 t u i + u i u i + n i + 1θ i i Φ = u i, n i + 1 i = 1,, 44 t θ i + u i θ i + 3 θ i + 1 u i 3n i + 1 θ i = 1 3 u i θ i, i = 1,, 45 Φ = n 1 n, Φ, x. 46 Further, we have n itt n i + n it + 1 i 1 n 1 n θ i = R i, 47 u it + u i + n i + θ i 1 i Φ = Q i, 48 t θ i + 3 u i 3 θ i + θ i = S i, 49

17 Global existence and asymptotic behavior of 3D BEPS 31 for i = 1,, here θi + 1 R i = n i + 1 n i n i n i u i + n i u i t + u k ix l u l ix k + u k i u i xk, l,k k Q i = 1 θ i + 1 n i u i u i, n i + 1 S i = u i θ i + 3n i + 1 θ i 3 3 θ i u i. In the following, we define Et = n 1t, n t 3 + n 1, u 1, θ 1, n, u, θ 4, Dt = n1t, n t, n 1, n, θ 1t, θ t 3 + u1, u 4 + θ1, θ 5. Then, we have the following estimate of the solution by basic energy estimate. Lemma 4 The a priori estimate of the high order derivatives of solution. Under the assumption of Theorem 1, suppose n 1, m 1, n, m, Φ S is the solution of IVP 7 8 on [, T ] for any T >, which satisfies 5. Then, for any t [, T ], there exists C such that d Et + Φt dt 4 + Dt CΛtDt + C1 + t 3/u 1, u, Φ. 5 L Proof. From 5 and Sobolev inequality, we know that x n 1, u 1, θ 1, n, u, θ L CΛt. 51 From equalities and assumption 5, we also have 1 Moreover, it is easy to see that x n 1t, u 1t, θ 1t, n t, u t, θ t L CΛt. 5 θ i n i C n i + θ i CΛt, i = 1,. 53 In the following, we will obtain five elementary estimates, denoted by estimates A, B, C, D and E. Then the estimates of the higher derivatives will be considered. Nonlinear Anal. Model. Control, 3:35 33

18 3 Y. Li Estimate A. Multiplying 47 with i = 1 by n 1t + λn 1 < λ 1 and integrating it by parts over yields 1 d n dt 1t + n 1 + λn 1 dx + n 1t + λ n 1 dx R 3 + n 1 n n 1t + λn 1 dx θ 1 n 1t + λn 1 dx R 3 = n 1t + λn 1 R 1 dx = n 1t + λn 1 1 θ n 1 dx n R 3 + n 1t + λn 1 u k 1x l u l 1x k + u k 1 u 1 xk dx l,k k n 1t + λn 1 n1 u 1 + n 1 u 1 t dx =: I 1 + I + I First, we have θ 1 n 1t + λn 1 dx ɛ n 1t + n 1 dx + Cɛ D θ 1 + Dθ1 dx. By integrating by parts, we can obtain I 1 = n 1t + λn 1 1 θ 1+1 n 1 dx 1+n 1 = 1 d 1 θ [ 1+1 n 1 dx + λ 1 θ ] θ1 +1 n 1 dx. dt 1+n 1 1+n 1 1+n 1 t So, give I 1 1 d dt 1 θ n 1 dx + CΛt n 1 t. 1 + n 1 For the estimate of I, we will use the following equality u 1 = n 1 n 1t + u 1 n 1, 55

19 Global existence and asymptotic behavior of 3D BEPS 33 which comes from 43. And it implies that u k 1 u 1 xk dx n 1t = k k + k [ n u k 1 1t 1 + n 1 u k 1 Since the last term above is u k 1n 1t u 1 n 1xk dx 1 + n 1 and k = k,l = k,l + ]xk n 1x n k 1tu k n 1 dx n1t n 1xk u 1 n n 1 n 1tn 1xk u 1 + n 1t u 1 n 1xk dx. 1 + n 1 u k 1u l 1n 1t n 1xl x k 1 + n 1 dx u k 1u l 1n 1xk tn 1xl dx 1 + n 1 k,l u k 1 u l n 1 x k n 1t n 1xl dx, u k 1u l 1n 1xk tn 1xl = 1 u k 1 n + 1xk u k t 1u l 1n 1xk n 1xl t. k>l k,l Then we have I 1 d dt k=l k=l u k 1n 1xk dx + d 1 + n 1 dt k>l + CΛt n 1 + n 1t + u 1. u k 1u l 1n 1xk n 1xl 1 + n 1 dx Now, we rewrite I 3 as follows, I 3 = n 1t + λn 1 n1 u 1 + n 1 u 1 t dx R 3 = n1t u 1 + n 1t u 1 + u 1t n 1 + n 1 u 1t + n 1 u 1 dx n 1t λ n 1 n1 u 1 + n 1 u 1 t dx 7 = I 3,j, j=1 Nonlinear Anal. Model. Control, 3:35 33

20 34 Y. Li where I 3,j represents every term in the above equality respectively. First, 1 I 3,1 + I 3, = n 1t u 1 dx CΛt n 1t. By using 55 for u 1t, we have I 3,3 1 d n 1t n 1t dx + CΛt n 1 + n 1t. dt 1 + n 1 The estimation of other terms, {I 3,j } j 4 is similar, so we omit the details. Combining above inequalities gives I 3 1 d n 1t n 1t dx + CΛt n 1 + n 1t + u 1 + u 1. dt 1 + n 1 Thus, if ΛT δ is sufficiently small, we obtain for all t [, T ] d n dt 1t + n 1 +λn 1 dx + n 1t +λ n 1 dx + n1 n n 1t +λn 1 dx CΛt u Cɛ D θ 1, θ Similarly, from 47 with i =, we can show d n dt t + n +λn dx + n t +λ n dx n 1 n n t+λn dx CΛt u 1 + Cɛ D θ, θ. 57 Moreover, noting that 1 d n 1 n n 1t + λn 1 n t λn dx = n 1 n dx + λ n 1 n dx, dt which together with yields [ d n dt it + n i + λn n 1 n ] i + dx, [ + n it + λ n i ] + λn 1 n dx, CΛt u u 1 + Cɛ D θ 1, θ 1, D θ, θ. 58

21 Global existence and asymptotic behavior of 3D BEPS 35 Estimate B. Next, multiplying 48 by u i i = 1, and integrating it over gives 1 d u i dx + u i dx n i + θ i u i dx + 1 i dt R 3 = u i Q i dx. Φu i dx By 43, 46, we have 1 d dt = u i + n i dx + u i u i dx θ i u i dx + 1 i θ i n i dx u i u i u i dx 1 + n i =: H 1 + H + H 3. Φu i dx n i n i u i dx From 53, we have It is easy to see that H 1 CΛt n i + u i. H + H 3 CΛt n i + u i. By combining the above estimates and the assumption 5, we get for i = 1, 1 d dt u i + n i dx + u i dx θ i u i dx + 1 i Φu i dx CΛt n i + u i. 59 Moreover, we can deal with the coupled term as follows: Φu 1 u = Φ u 1 u = Φ Φ t + Φu 1 n 1 u n 1 d Φ { dx max n1 L, n L }[ Φ + u 1 L dt + u ] 1 d Φ dx CΛt1 + t 3/ u1, u, Φ, dt Nonlinear Anal. Model. Control, 3:35 33

22 36 Y. Li which together with 59 gives 1 d u dt 1 + n 1 + u + n + Φ dx + R 3 θ1 u 1 + θ u dx u 1 + u dx CΛt n 1, n, u 1, u + C1 + t 3/ u 1, u, Φ. 6 Estimate C. By differentiating 48 with respect to x l and integrating its product with u ixl i = 1, over, respectively, we have 1 d uixl + n ixl dx + u ix dt l dx θ ixl u ixl dx + n ixl u ixl dx + 1 i Φ xl u ixl dx = u ixl Q ixl dx. Similar to the proof of 59, by 43, we have 1 d ui + n i dx+ u i dx+ 1 i Φ xl u 1x dt l dx θ ixl u ixl dx C [ u ixl Q ixl dx n i n ix l n it dx l R 3 ] + 1 n ixl n i + 1 u i n i xl dx. By symmetry, such as u i n i + 1 n ix l n ixl dx = 1 ui n ixl dx, n i + 1 we have after some tedious but straightforward calculation that 1 d ui + n i dx+ u i dx θ ix dt l u ixl dx+ 1 i Φ xl u 1xl dx CΛt n i.

23 Global existence and asymptotic behavior of 3D BEPS 37 From 3 and 4, we also have the following estimate: Φ xl u 1xl u xl = Φ xl u1xl u xl = Φ xl Φ xl t + Φ xl u 1 n 1 u n xl 1 d Φ xl dx CΛt n1, n, u 1, u, u 1, u. dt L Therefore, d ui + n i + Φ dx + u i dx dt,, θ ixl u ixl dx, CΛt n 1 + n + u u Estimate D. Multiplying 49 by 3/θ i i = 1,, and taking the derivatives of 49 with respect to x l and multiplying the resultant equation by 3/θ ixl i = 1,, then integrating it over, after a length and straightforward computation, we have d 3 3 dt 4 θ i dx + θ i + θ i dx + θ i u i dx = 3 S i θ i dx CΛt θ i + θ i + n i + u i + u i dx 6 and d dt d dt = θ ix l dx + R 3 3 θ ix l + θ ixl dx + θ ixl u ixl dx S ixl θ ixl dx CΛt θixl + θ ixl + n i + u i dx. 63 Combining 58, 6 and 61 63, we obtain [ nit + n i, u i, θ i 1 + Φ ] + [ n it, n i, + u i 1 + θ i ],, CΛt [ nit, n i + u i 1 + θ i ], + CΛt1 + t 3/ u 1, u, Φ. 64 Nonlinear Anal. Model. Control, 3:35 33

24 38 Y. Li Estimate E. Similarly, taking the th derivatives of 47, the + 1th derivatives of 48, and the + 1th derivatives of 49 for 1 3, with respect to the space variable, respectively, furthermore, multiplying the resultant equations by D n it + λd n i, D +1 u i and D +1 θ i with i = 1,, respectively, then integrating them over, we have [ d D n it + D n i, θ i, u i dt 1 + D Φ ] 1, + [ D n it + D n i, n i 1 + D θ i ], CΛt, [ n i, n it 3 + u i 4 + θ i ] Summing up together for all 3, estimate 5 follows immediately. Proof of Theorem 1. Suppose that n 1, m 1, θ 1, n, m, θ, Φ H 4 6 H 5 correspond respectively to the smooth solutions of the bipolar non-isentropic Euler Poisson system 7 for t [, T ], subject to initial data 8. First, if Λt is rather small, we have d Et + Φt dt 4 + CDt C1 + t 3/ Λt Φt. 66 By the Gronwall s inequality, we get Et + t Φt + C 4 Dτ dτ E + Φ 4 + CΛ3 t, 67 where Φ is explained as 3 as t = and E + Φ 4 CΘ. Hence, from 67 and Lemma 3, we have Λt CΘ + C Λt, t [, T ]. 68 By standard continuous argument, we know that there exist constant C such that Λt CΘ, t [, T ] 69 when the initial data Θ > is sufficiently small. And estimate 68 implies that assumption 5 is valid for all the time t [, T ]. Furthermore, it is easy to get estimates 4 6. The proof of Theorem 1 is completed. Acknowledgments. The author would like to thank the referees for the valuable comments and suggestions, which greatly improved the presentation of the paper.

25 Global existence and asymptotic behavior of 3D BEPS 39 References 1. G. Ali, A. Jüngel, Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasma, J. Differ. Equations, 19: , 3.. A.M. Anile, O. Muscato, Improved hydrodynamical model from carrier transport in semiconductors, Phys. Rev. B, 51: , A.M. Anile, S. Pennisi, Extended thermodynamics of the Blotekjaer hydrodynamical model for semiconductors, Contin. Mech. Thermodyn., 41: , K. Blotekjaer, Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electron Devices, 17:38 47, I. Gasser, L. Hsiao, H.-L. Li, Asmptotic behavior of solutions of the bipolar hydrodynamic fluids, J. Differ. Equations, 19:36-359, L. Hsiao, K.-J. Zhang, The global weak solution and relaxation limits of the initial boundary value problem to the bipolar hydrodynamic model for semiconductors, Math. Models Methods Appl. Sci., 1: ,. 7. F.-M. Huang, Y.-P. Li, Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum, Discrete Contin. Dyn. Sys., A4:455 47, F.-M. Huang, M. Mei, Y. Wang, Large time behavior of solution to n-dimensional bipolar hydrodynamic model for semiconductors, SIAM J. Math. Anal., 43: , S. Jiang, Q.-C, Ju, H.-L. Li, Y. Li, Quasi-neutral limit of the full bipolar Euler Poisson system, Sci. China, Math, 53: , Q.-C. Ju, Global smooth solutions to the multidimensional hydrodynamic model for plasmas with insulating boundary conditions, J. Math. Anal. Appl., 336:888 94, A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations, Progress in Nonlinear Differential Equations, Vol. 39, Birkhäuser, Basel, S. Kawashima, Systems of a Hyperbolic-Parabolic Composite Type, with Applications to the Equations of Magnetohydrodynamics, PhD thesis, Kyoto University, Y.-P. Li, Global existence and asymptotic behavior of solutions to the nonisentropic bipolar hydrodynamic models, J. Differ. Equations, 5: , Y.-P. Li, X.-F. Yang, Global existence and asymptotic behavior of the solutions to the threedimensional bipolar Euler Poisson systems, J. Differ. Equations, 5: , Y.-P. Li, T. Zhang, Relaxation-time limit of the multidimensional bipolar hydrodynamic model in Besov space, J. Differ. Equations, 51: , P.A. Markowich, C.A. Ringhofev, C. Schmeiser, Semiconductor Equations, Springer, Wien, New York, A. Matsumura, T. Nishida, The initial value problem for the equation of motion of compressible viscous and heat-conductive fluids, Proc. Japan. Acad., 55:337 34, Nonlinear Anal. Model. Control, 3:35 33

26 33 Y. Li 18. R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equation, J. Math. Anal. Appl., 198:6 81, G. Ponce, Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Anal., Theory Methods Appl., 9: , A. Sitnko, V. Malnev, Plasma Physics Theory, Chapman & Hall, London, M.E. Taylor, Partial Differential Equation III: Nonlinear Equations, Springer, New York, N. Tsuge, Existence and uniqueness of stationary solutions to a one-dimensional bipolar hydrodynamic models of semiconductors, Nonlinear Anal., Theory Methods Appl., 73: , W.-K. Wang, T. Yang, The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, J. Differ. Equations, 173:41 45, G.-J. Zhang, H.-L. Li, C.-J. Zhu, Optimal decay rate of the non-isentropic compressible Navier Stokes Poisson system in, J. Differ. Equations, 5: , F. Zhou, Y.-P. Li, Existence and some limits of stationary solutions to a one-dimensional bipolar Euler Poisson system, J. Math. Anal. Appl., 351:48 49, C. Zhu, H. Hattori, Stability of steady state solutions for an isentropic hydrodynamic model of semiconductors of two species, J. Differ. Equations, 166:1 3,.