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

. 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 2 -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 2 -decay rate of global smooth solutions to the multi-dimensional non-isentropic bipolar hydrodynamic model.


Introduction
In this paper, we consider the following three-dimensional non-isentropic bipolar hydrodynamic model: Y. Li ( Here (t, x) ∈ R + × R 3 , and the unknown variables ρ i , m i , T i (i = 1, 2), and φ are the charge densities, current densities, temperatures, and electrostatic potential.The coefficients τ 1 , τ 2 , 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 L2 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 [2,3].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 [11,16,20], etc.Recently, many efforts were made for the isentropic bipolar hydrodynamic equations from semiconductors or plasmas.Zhou and Li [25] and Tsuge [22] 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 [26] 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/2) and in the Besov space, respectively.Ju [10] 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 2 -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 http://www.mii.lt/NAEuler-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 2 = κ = 1, τ 1 = 2τ 2 and T L1 = T L2 = T L , then we can rewrite (1) as ( We also prescribe the initial data as The main result in this paper is stated in the following theorem. Theorem 1.Let (ρ, m, T L ) be constant state with ρ > 0 and T L > 0. Assume that and there is some positive constant C > 0 such that, for i = 1, 2, and there is some positive constant C > 0 such that, for i = 1, 2 and |α| l, |β| l + 1, Remark 2. Compared with the Euler equations with damping in [23], 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 [12].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 2, we reformulate the original problem in terms of the perturbed variable, and present the L 2 decay rate of the linearized equations.The global existence and L 2 -convergence rates of smooth solutions will be shown in Section 3.
Moreover, from ( 9) and ( 21), the Fourier transform for the electric field is From the above equality, we can define Here L, L, L are the Fourier transform of function L, L, L, respectively.From Lemma 1, the estimates of L,L, L is given as following.
), then for m = 1, 2, we have 3 Global existence and L 2 -decay rate In this section, we are going to establish the global existence and show the L 2 -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 [12].The key point is the electric field ∇Φ can be expressed by the Riesz potential as a nonlocal term which together with the L p estimates of Riesz potential leads to 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 [12].The details are omitted.
Theorem 2. Assume that (n 10 , m 10 , θ 10 , n 20 , m 20 , θ 20 )(x) ∈ H 4 (R 3 ).Then, there is a time T > 0 such that the IVP (7)-( 8) has a unique global smooth solution 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 where Due to the property of Riesz potential (see [21]), we have It seems that the estimates of the high order derivatives of ∇Φ come from the bounds of n 1 and n 2 .That is, if (n 1 , m 1 , n 2 , m 2 , ∇Φ) ∈ S, it is obviously that for all 0 s t, So we should obtain the estimate of ∇Φ itself.
Proof.By Duhamel principle, it is easy to verify that the solution U = (n 1 , m 1 , θ 1 , n 2 , m 2 , θ 2 , ∇Φ) of the IVP problem ( 7)-( 8) can be expressed as and Y. Li First, from (26) and Lemma 1, we have For the estimates of the nonlinear terms, we first have for i = 1, 2, with the aid of which appeared in [21].Further, from ( 25) and ( 33)-(34), we can get that, for i = 1, 2, and Thus, one have Next, we have the following estimate of the nonlinear terms f i (U ) (i = 1, 2) and g i (U ) (i = 1, 2): and which together with (26) yield Nonlinear Anal.Model.Control, 20(3):305-330 with the help of ( 35)-( 38).Finally, we can show which together with (35)-(38), and (42) leads to Moreover, in the completely same way, from (27), we can obtain and from (28), we can show The estimates of n 2 , m 2 and θ 2 can be obtained by the completely similar way.We omit the details here.For the time decay rate for ∇Φ, from Lemma 2 and ( 35)-(38), it is easy to get This completes the proofs.
Next, we are going to derive the estimates of higher order derivatives of (n 1 , m 1 , θ 1 , n 2 , m 2 , θ 2 ).For simplicity, we denote u i = m i /(n i + 1), i = 1, 2. From ( 7)-( 8), we derive the system for (n 1 , u 1 , n 2 , u 2 , Φ) as Further, we have http://www.mii.lt/NA for i = 1, 2, here In the following, we define 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 2 , m 2 , ∇Φ) ∈ S is the solution of IVP ( 7)-( 8) on [0, T ] for any T > 0, which satisfies (25).Then, for any t ∈ [0, T ], there exists C such that d dt E(t) + ∇Φ(t) Proof.From ( 22)- (25) and Sobolev inequality, we know that From equalities (43)-(45) and assumption (25), we also have Moreover, it is easy to see that 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.
Estimate A. Multiplying (47) with i = 1 by n 1t + λn 1 (0 < λ 1) and integrating it by parts over R 3 yields First, we have By integrating by parts, we can obtain So, (51)-(53) give For the estimate of I 2 , we will use the following equality http://www.mii.lt/NA which comes from (43).And it implies that Since the last term above is Then we have Now, we rewrite I 3 as follows, where I 3,j represents every term in the above equality respectively.First, By using (55) for ∇ • u 1t , we have The estimation of other terms, {I 3,j } j 4 is similar, so we omit the details.Combining above inequalities gives Thus, if Λ(T ) δ 0 is sufficiently small, we obtain for all t ∈ [0, T ] Similarly, from (47) with i = 2, we can show Moreover, noting that which together with (56)-(57) yields http://www.mii.lt/NAEstimate B. Next, multiplying (48) by u i (i = 1, 2) and integrating it over R 3 gives By (43), (46), we have From (53), we have It is easy to see that By combining the above estimates and the assumption (25), we get for i = 1, 2 Moreover, we can deal with the coupled term as follows: Estimate C. By differentiating (48) with respect to x l and integrating its product with u ix l (i = 1, 2) over R 3 , respectively, we have Similar to the proof of (59), by (43), we have By symmetry, such as we have after some tedious but straightforward calculation that From ( 23) and ( 24), we also have the following estimate: