Relaxation-time limit in the multi-dimensional bipolar nonisentropic Euler–Poisson systems

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ScienceDirect J. Differential Equations 258 (2015) 3546–3566 www.elsevier.com/locate/jde

Relaxation-time limit in the multi-dimensional bipolar nonisentropic Euler–Poisson systems Yeping Li a,∗ , Zhiming Zhou b a Department of Mathematics, East China University of Science and Technology, Shanghai 200237, PR China b Department of Mathematics, Shanghai Normal University, Shanghai 200234, PR China

Received 3 November 2013; revised 31 December 2014 Available online 3 February 2015

Abstract In this paper, we consider the multi-dimensional bipolar nonisentropic Euler–Poisson systems, which model various physical phenomena in semiconductor devices, plasmas and channel proteins. We mainly study the relaxation-time limit of the initial value problem for the bipolar full Euler–Poisson equations with well-prepared initial data. Inspired by the Maxwell iteration, we construct the different approximation states for the case τ σ = 1 and σ = 1, respectively, and show that periodic initial-value problems of the certain scaled bipolar nonisentropic Euler–Poisson systems in the case τ σ = 1 and σ = 1 have unique smooth solutions in the time interval where the classical energy transport equation and the drift-diffusive equation have smooth solution. Moreover, it is also obtained that the smooth solutions converge to those of energy-transport models at the rate of τ 2 and those of the drift-diffusive models at the rate of τ , respectively. The proof of these results is based on the continuation principle and the error estimates. © 2015 Elsevier Inc. All rights reserved.

MSC: 35B25; 35M20; 35L45 Keywords: Nonisentropic; Bipolar; Euler–Poisson systems; Error estimates; Relaxation-time limit

* Corresponding author.

E-mail address: [email protected] (Y. Li). http://dx.doi.org/10.1016/j.jde.2015.01.020 0022-0396/© 2015 Elsevier Inc. All rights reserved.

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1. Introduction In this paper, we are concerned with the multi-dimensional bipolar hydrodynamic model, which is given by the following scaled multi-dimensional bipolar non-isentropic (full) Euler– Poisson equation (see [2,3,5]) ⎧ ∂s ρ1 + div(ρ1 u1 ) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ∂s (ρ1 u1 ) + div(ρ1 u1 ⊗ u1 + P1 In ) = ρ1 ∇φ − (ρ1 u1 ), ⎪ ⎪ τ ⎪ ⎪  ⎪ ⎪  1 1 3  ⎪ 2 ⎪ ⎪ ⎪ ∂s (ρ1 E1 ) + div(ρ1 u1 E1 + P1 u1 ) = ρ1 u1 ∇φ − σ 2 ρ1 |u1 | + 2 ρ1 θ1 − TL (x) , ⎪ ⎨ ∂s ρ2 + div(ρ2 u2 ) = 0, (1.1) ⎪ ⎪ ⎪ 1 ⎪ ⎪ ∂s (ρ2 u2 ) + div(ρ2 u2 ⊗ u2 + P2 In ) = −ρ2 ∇φ − (ρ2 u2 ), ⎪ ⎪ τ ⎪ ⎪ ⎪ ⎪  1 1 3  ⎪ 2 ⎪ θ (ρ E ) + div(ρ u E + P u ) = −ρ u ∇φ − |u | + − T (x) , ρ ρ ∂ ⎪ s 2 2 2 2 2 2 2 2 2 2 2 2 2 L ⎪ ⎪ σ 2 2 ⎪ ⎪ ⎩ φ = ρ1 − ρ2 − b(x), for (x, s) ∈ Ω × [0, +∞). Here Ω is the n-dimensional torus and In is the unit matrix of order n. The unknown variables ρi , ui , θi (i = 1, 2) and φ are the charge densities, velocities, temperatures and electrostatic potential (noting that ∇φ stands for the electric field). Pi (i = 1, 2) and Ei (i = 1, 2) represent the pressure and the total energy, respectively. The dimensionless parameters τ and σ are the respective momentum relaxation-time and energy relaxation-time. TL(x) > 0 is a given lattice temperature of semiconductor devices, and b(x) > 0 stands for the density of fixed, positively charged background ions (doping profile). Eqs. (1.1)1 and (1.1)3 express the conservation of particle number, (1.1)2 and (1.1)4 express the balance equation of momentum, and (1.1)3 and (1.1)6 express the balance equation of energy, respectively. The system (1.1) is closed with the aid of definition of total energy and the equation of state: Ei = |ui |2 /2 + ei ,

Pi = Pi (ρi , ei ),

i = 1, 2,

where ei (i = 1, 2) is the internal energy. We also note that when we only investigate the dynamics of one particle in semiconductor devices, plasmas and channel proteins, the system (1.1) reduces to the unipolar isentropic and non-isentropic Euler–Poisson equations, when the pressures are only the functions of the densities, (1.1) is the bipolar isentropic Euler–Poisson equations. More details about the hydrodynamic equations from the semiconductor devices and the plasma physics, can be found in [11,23,25], etc. It is well known that when τ goes to zero and τ σ = 1 or σ = 1 in (1.1), we can obtain the bipolar energy transport equations or the bipolar drift-diffusion equations, which are frequently utilized for numerical device simulations, see [4,7,6]. This singular limit is called the relaxation time limit or the relaxation limit. Hence the relaxation limit from the Euler–Poisson equation to the energy transport equation or the drift-diffusion equation is an important problem not only in mathematics but also in engineering. Recently, many efforts have been made for this important limit problem of the bipolar Euler–Poisson equations. Zhou and Li [31], and Li [18] considered the zero-electron-mass limit, relaxation-time limit and quasi-neutral limit of the stationary solutions for the one-dimensional and multi-dimensional bipolar isentropic hydrodynamic model

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with some proper boundary conditions. Marcati and Natalini [22], Hsiao and Zhang [10], and Jüngel and Peng [12,13] studied the relaxation-time limit of the weak solutions for the onedimensional bipolar isentropic hydrodynamic model in real line and over the bound domain, respectively. Some combined limits of the weak solutions for the one-dimensional bipolar isentropic Euler–Poisson equations were also discussed by Gasser and Marcati in [9]. Lattanzio [15] and Li [17] considered the relaxation-time limit for L∞ -solutions and the local smooth solutions of the multi-dimensional bipolar isentropic hydrodynamic equations, respectively. Ali and Jüngel [1], and Li and Zhang [20] studied the relaxation limit for the classical solutions of the initial value problem for the multi-dimensional bipolar Euler–Poisson equations in the Sobolev space and Besov space, respectively. Li [19] also studied the relaxation-time limit of the threedimensional bipolar isentropic hydrodynamic model with boundary effects. Moreover, there are many literatures about the relaxation-time limit of the corresponding unipolar isentropic and nonisentropic Euler–Poisson equations, for example, see [8,14,16,24,26,27,30] and the references therein. As far as we know, no results on this limit problem of the multi-dimensional bipolar full Euler–Poisson system (1.1) can be found. In this paper, we will study the relaxation-time limit of the initial value problem for the multi-dimensional bipolar full Euler–Poisson system (1.1) with well-prepared initial data. For the sake of simplicity, we only consider the perfect fluids so that ei = 32 θi and Pi = ρi θi for i = 1, 2. Using the scaling t = τs , we have ⎧ 1 ⎪ ⎪ ∂t ρi + div(ρi ui ) = 0, i = 1, 2, ⎪ ⎪ τ ⎪ ⎪ ⎪ ⎪ 1 1 1 1 ⎪ ⎪ ∂t (ρi ui ) + div(ρi ui ⊗ ui ) + ∇(ρi θi ) = (−1)i−1 ρi ∇φ − 2 ρi ui , i = 1, 2, ⎪ ⎪ τ τ τ τ ⎪ ⎨  3 5 1 1 1 2 2 (1.2) ∂t ρi |ui | + ρi θi + div ρi |ui | + ρi θi ui ⎪ ⎪ 2 2 τ 2 2 ⎪ ⎪  ⎪ ⎪  1 1 3  ⎪ i−1 1 2 ⎪ = (−1) ρi ui ∇φ − ρi |ui | + ρi θi − TL (x) , i = 1, 2, ⎪ ⎪ ⎪ τ τσ 2 2 ⎪ ⎪ ⎩ φ = ρ1 − ρ2 − b(x). Further, we rewrite the momentum equations in (1.2)2 as ρi ui = (−1)i−1 τρi ∇φ − τ ∇(ρi θi ) − τ div(ρi ui ⊗ ui ) − τ 2 ∂t (ρi ui )   = (−1)i−1 τρi ∇φ − τ ∇(ρi θi ) + O τ 2 , i = 1, 2.

(1.3)

If τ σ = 1, letting us substitute the truncations ρ1 u1 = τρ1 ∇φ − τ ∇(ρ1 θ1 ) and ρ2 u2 = −τρ2 ∇φ − τ ∇(ρ2 θ2 ) into the mass equations and energy equations in (1.2) respectively, we have ∂t ρi = (ρi θi ) + (−1)i div(ρi ∇φ),

i = 1, 2,

and   3 5  ρi θi + div θi (−1)i−1 ρi ∇φ − ∇(ρi θi ) 2 2      3  = (−1)i+1 (−1)i−1 ρi ∇φ − ∇(ρi θi ) ∇φ − ρi θi − TL (x) + O τ 2 , 2

 ∂t

i = 1, 2.

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Then, we immediately obtain the following bipolar energy-transport model: ⎧     ∂t ρ1e =  ρ1e θ1e − div ρ1e ∇φ e , ⎪ ⎪ ⎪   ⎪ ⎪  e e  3 e e 5 e e e ⎪ ⎪ ∂t ρ θ + div θ1 ρ1 ∇φ − ∇ ρ1 θ1 ⎪ ⎪ ⎪ 2 1 1 2 ⎪ ⎪   e e  e 3 e  e  e e ⎪ ⎪ ⎪ = ρ1 ∇φ − ∇ ρ1 θ1 ∇φ − ρ1 θ1 − TL , ⎪ ⎪ ⎨    2  ∂t ρ2e =  ρ2e θ2e + div ρ2e ∇φ e , ⎪   ⎪ ⎪  e e  3 e e 5 e e e ⎪ ⎪ ∂t ρ θ − div θ2 ρ2 ∇φ + ∇ ρ2 θ2 ⎪ ⎪ ⎪ 2 2 2 2 ⎪ ⎪   e e  e 3 e  e  ⎪ ⎪ e e ⎪ ⎪ = ρ2 ∇φ + ∇ ρ2 θ2 ∇φ − ρ2 θ2 − TL , ⎪ 2 ⎪ ⎪ ⎩ φ e = ρ1e − ρ2e − b(x).

(1.4)

Moreover, from (1.2), we also have for i = 1, 2 ρi ui = (−1)i−1 τρi ∇φ − τ ∇(ρi θi ) − τ div(ρi ui ⊗ ui ) − τ 2 ∂t (ρi ui ), and   2 1 1 2 1 2 θ i = TL + τ σ 2 − |ui | − τ ∂t θi + ui ∇θi + θi div ui . 3 2τ σ τ 3τ τ If let σ = 1, the above four equations show that ui = O(τ ) (i = 1, 2) and θi = TL (x) + O(τ ) (i = 1, 2) formally. With those, we iterate the momentum equations in (1.2) once to obtain   ρi ui = (−1)i−1 τρi ∇φ − τ ∇(ρi θi ) + O τ 2 ,

i = 1, 2.

(1.5)

Substituting the truncations into the mass equations in (1.2), we immediately get the other limit equation, which is the classical drift-diffusion equation as follows  d   d  ⎧ d d ⎪ ⎨ ∂t ρ1 = ρ1 TL  − divρ1 ∇φ , ∂t ρ2d =  ρ2d TL + div ρ2d ∇φ d , ⎪ ⎩ φ d = ρ1d − ρ2d − b(x).

(1.6)

The main aim of our paper is to justify the above Maxwell iteration procedure (1.3)–(1.4), and (1.5)–(1.6) vigorously, respectively. The idea is from Yong in [30], where he proved the convergence from the unipolar isentropic hydrodynamic models to the drift-diffusive models at the rate of τ 2 . Now we state our main results as follows Theorem 1.1. Let s > 1 +

n 2

be an integer. Assume that TL = TL (x) and b = b(x) satisfy TL (x), b(x) ∈ H s+1 (Ω),

and the bipolar energy-transport equation (1.4) has a solution

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     ρ1e , θ1e , ρ2e , θ2e ∈ C [0, T∗ ]; H s+3 (Ω) ∩ C 1 [0, T∗ ]; H s+1 (Ω)

with positive lower bounds. Then, for sufficiently small τ and τ σ = 1, the system (1.2) with periodic initial data 

  e e  e e e ρ1τ , uτ1 , θ1τ , ρ2τ , uτ2 , θ2τ (0, x) = ρ1τ (0, x), , u1τ , θ1τ , ρ2τ , ue2τ , θ2τ

(1.7)

has a unique solution (ρ1τ , uτ1 , θ1τ , ρ2τ , uτ2 , θ2τ , ∇φ τ ) ∈ C([0, T∗ ]; H s (Ω)). Furthermore, there exist constants K1 , K2 > 0, independent of τ but dependent on T∗ ∈ (0, +∞), such that

 

e e e e

sup ρ1τ − ρ1τ , uτ1 − ue1τ , θ1τ − θ1τ , ρ2τ − ρ2τ , uτ2 − ue2τ , θ2τ − θ2τ ≤ K1 τ 2 , s

(1.8)



sup ∇φ τ − ∇φτe s ≤ K2 τ 2 ,

(1.9)

t∈(0,T∗ )

and

t∈(0,T∗ )

where   ∇(ρie θie ) e e e ρiτ = ρie , (−1)i−1 ∇φτe − , ueiτ , θiτ , θ i ρie   φτe = −1 ρ1e − ρ2e − b(x) . 

Theorem 1.2. Let s > 1 +

n 2

(i = 1, 2), (1.10)

be an integer. Suppose that TL = TL (x) and b = b(x) satisfy

b(x) ∈ H s+1 (Ω),

TL (x) ∈ H s+3 (Ω),

and the bipolar drift-diffusion equation (1.6) has a solution 

     ρ1d , ρ2d ∈ C [0, T∗ ]; H s+3 (Ω) ∩ C 1 [0, T∗ ]; H s+1 (Ω)

with positive lower bounds. Then, for sufficiently small τ and σ = 1, the system (1.2) with periodic initial data 

  d d d d d d ρ1τ , uτ1 , θ1τ , ρ2τ , uτ2 , θ2τ (0, x) = ρ1τ , u1τ , θ1τ , ρ2τ , u2τ , θ2τ (0, x),

(1.11)

has a unique solution (ρ1τ , uτ1 , θ1τ , ρ2τ , uτ2 , θ2τ , ∇φ τ ) ∈ C([0, T∗ ]; H s (Ω)). Furthermore, there exist constants K3 , K4 > 0, independent of τ but dependent on T∗ ∈ (0, +∞), such that

 

d d d d

sup ρ1τ − ρ1τ , uτ1 − ud1τ , θ1τ − θ1τ , ρ2τ − ρ2τ , uτ2 − ud2τ , θ2τ − θ2τ ≤ K1 τ, s

(1.12)



sup ∇φ τ − ∇φτd s ≤ K4 τ,

(1.13)

t∈(0,T∗ )

and

t∈(0,T∗ )

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where d ρiτ = ρid ,

udiτ = (−1)i−1 τ ∇φ d − τ

ρ1d

,

i = 1, 2,

  φτd = φ d = −1 ρ1d − ρ2d − b ,

   ∇(ρid TL ) 2 ∇(ρid TL ) 2 − τ  i−1 d i−1 d = TL (x) + τ ∇TL (−1) ∇φ −  − (−1) ∇φ − 3  ρid ρid  ∇(ρid TL ) 2 − TL div (−1)i−1 ∇φ d − , i = 1, 2. (1.14) 3 ρid 

d θiτ

∇(ρ1d TL )

Remark 1.3. Theorems 1.1 and 1.2 characterize the relaxation limit of the full bipolar Euler– Poisson system toward the bipolar energy transport equation and the bipolar drift-diffusion equation with the different convergence rates. Meanwhile, no smallness conditions on the initial data are required by Theorems 1.1 and 1.2. Moreover, Theorems 1.1 and 1.2 only deal with the case where the initial data are well-prepared. For more general periodic initial data, the initial layers will occur, and similar results of form (1.8)–(1.9) and (1.12)–(1.13) may still be verified by using the matched expansion methods, e.g., see [17,28]. This will be shown in a forthcoming paper. Finally, we can consider the similar limit problem even some combined limit problems of the smooth solutions for the general quantum Euler–Poisson equation, which is left for the forthcoming future. This paper is organized as follows. In Section 2 we make some necessary preliminaries. First, we rewrite the bipolar isentropic hydrodynamical model as a symmetrizable hyperbolic system and review the convergence-stability result. Then we list some Moser-type calculus inequalities in Sobolev spaces and a nonlinear Gronwall-type inequality. The approximation solutions (1.10) and (1.14) are discussed in Section 3 and Section 4 is devoted to the proofs of Theorems 1.1 and 1.2. Notations. |U | denotes some norm of a vector or matrix U . L2 = L2 (Ω) is the space of square integrable (vector- or matrix-valued) functions on the d-dimensional unit torus Ω = (0, 1]n . For a nonnegative integer s, H s (Ω) is defined as the space of functions whose distribution derivatives of order ≤ s are all in L2 . We use U s to denote the standard norm of U ∈ H s , and U ≡ U 0 . When A is a function of another variable t as well as x, we write A(·, t) s to recall that the norm is taken with respect to x while t is viewed as a parameter. In addition, we denote by C([0, T ], X) (resp., C 1 ([0, T ], X)) the space of continuous (resp., continuously differentiable) functions on [0, T ] with values in a Banach space X. 2. Preliminaries In this section, we give briefly some results, which are used in the proof of Theorems 1.1 and 1.2. To begin with, we recall an elementary fact. Proposition 2.1. (See [30].) ∇−1 is a bounded linear operator on L2 (Ω). This proposition can be easily proved by using the Fourier transformation. It is this proposition that requires the initial data to be periodic.

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Having this proposition, wee see that (1.2) for smooth solution (ρ1 , u1 , θ1 , ρ2 , u2 , θ2 ) with ρi , θi (i = 1, 2) > 0 is equivalent to ⎧ 1 1 ⎪ ⎪ ∂t ρ1 + u1 · ∇ρ1 + ρ1 div u1 = 0, ⎪ ⎪ τ τ ⎪ ⎪ ⎪ ⎪ 1 1 θ1 1 1 1 ⎪ ⎪ ∂t u1 + (u1 · ∇)u1 + ∇ρ1 + ∇θ1 = ∇−1 (ρ1 − ρ2 − b) − 2 u1 , ⎪ ⎪ τ τ ρ1 τ τ τ ⎪ ⎪  ⎪ ⎪   1 2 2 1 1 1 ⎪ ⎪ ⎪ − |u1 |2 − θ1 − TL (x) , ⎨ ∂t θ1 + u1 · ∇θ1 + θ1 div u1 = τ 3τ 3 τ 2 2τ σ τσ 1 1 ⎪ ⎪ ⎪ ∂t ρ2 + u2 · ∇ρ2 + ρ2 div u2 = 0, ⎪ ⎪ τ τ ⎪ ⎪ ⎪   1 1 θ2 1 1 1 ⎪ ⎪ ⎪ ∂t u2 + (u2 · ∇)u2 + ∇ρ2 + ∇θ2 = − ∇−1 ρ1 − ρ2 − b − 2 u2 , ⎪ ⎪ τ τ ρ2 τ τ τ ⎪ ⎪  ⎪ ⎪   1 2 2 1 1 1 ⎪ ⎪ − |u2 |2 − θ2 − TL (x) , ⎩ ∂t θ2 + u2 · ∇θ2 + θ2 div u2 = 2 τ 3τ 3 τ 2τ σ τσ

(2.1)

and φ = −1 (ρ1 − ρ2 − b). Obviously, (2.1) constitutes a symmetrizable hyperbolic system with the symmetrizer  2 θ 3 θ2 3 A0 = diag 12 , θ1 In , , 22 , θ2 In , , 2 ρ2 2 ρ1 since ∇−1 (ρ1 − ρ2 − b) is a zero-order (but nonlocal) term. Then, for fixing τ with τ σ = 1 or σ = 1, according to the local existence theory for IVPs of symmetrizable hyperbolic systems (see Theorem 2.1 in [21]), there is a time interval [0, T ] so that (2.1) with the initial data (ρ1 , u1 , θ1 , ρ2 , u2 , θ2 )(x, 0) = (ρ1 , u1 , θ1 , ρ2 , u2 , θ2 )(x, τ ) ∈ H s (Ω)(s > n2 + 1) and (ρ1 , u1 , θ1 , ρ2 , u2 , θ2 )(x, τ ) ∈ G0 has a unique H s (Ω) solution   (ρ1 , u1 , θ1 , ρ2 , u2 , θ2 )(x, t) ∈ C [0, T ], H s (Ω) ,

(ρ1 , u1 , θ1 , ρ2 , u2 , θ2 )(x, t) ∈ G.

Here G0 = {(ρ1 , u1 , θ1 , ρ2 , u2 , θ2 )(x, τ ) : ρi (x, τ ) > 0 and θi (x, τ ) > 0 for i = 1, 2} and G = {(ρ1 , u1 , θ1 , ρ2 , u2 , θ2 )(x, t) : ρi (x, t) > 0 and θi (x, t) > 0 for i = 1, 2}. Define     Tτ = sup T > 0 : ρ1τ , uτ1 , θ1τ , ρ2τ , uτ2 , θ2τ ∈ C [0, T ], H s (Ω) . Namely, [0, Tτ ) is the maximal time interval of H s (Ω) existence. Note that Tτ depends on G and may tends to zero as τ goes to a certain singular point, say 0. Next, let us recall the convergence stability lemma, which is established in [29,30]. That is, for general singular limit problems of IVPs for quasi-linear first-order symmetrizable hyperbolic systems depending (singularly) on parameters in several space variables: ⎧ d  ⎪ ⎪ ⎨ Ut + Aj (U, )Uxj = Q(U, ) , j =1 ⎪ ⎪ ⎩ U (x, 0) = U¯ (x)

(2.2)

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here represents a parameter in a topological space, Aj (U, ) (j = 1, 2, · · · , n) and Q(U, ) are sufficiently smooth functions of U ∈ G ⊂ Rn , and U¯ (x, ) is periodic in x with period (1, 1, · · · , 1) ∈ Rn . Assume U¯ (x, ) ∈ G0 ⊂⊂ G for all (x, ) and U¯ (x, ) ∈ H s (Ω) with s > n2 + 1 an integer. Fix , according to the local existence theory for IVPs of symmetrizable hyperbolic systems (see Theorem 2.1 in [21]), there is a time interval [0, T ] so that (2.2) has a unique H s solution   U ∈ C [0, T ], H s (Ω) . Define   T = sup T > 0 : U ∈ C [0, T ], H s (Ω) . Namely, [0, T ) is the maximal time interval of H s (Ω) existence. Note that T depends on G and may tends to zero as goes to a certain singular point, say 0. In order to show that lim →0 T > 0, which means the stability (see [29,30]), we make the following assumption. Convergence assumption. There exists T∗ > 0 and U ∈ L∞ ([0, T∗ ], H s (Ω)) for each , satisfying 

U (x, t) ⊂⊂ G, x,t,

such that for t ∈ [0, min{T∗ , T }),   supU (x, t) − U (x, t) = o(1), x,t



sup U (x, t) − U (x, t) s = O(1) t

as tends to the singular point. With such a convergence assumption, we are in a position to state the following fact established in [29,30]. Lemma 2.2. Assume U¯ (x, ) ∈ G0 ⊂⊂ G for all (x, ) and U¯ (x, ) ∈ H s (Ω) with s > n2 + 1, and that the convergence assumption holds. Let [0, T ) be the maximal time interval such that (2.2) has a unique H s (Ω)-solution U ∈ C([0, T ], H s (Ω)). Then T > T∗ for all in a neighborhood of the singular point. a , ua , θ a , ρ a , In terms of Lemma 2.2, our task in the subsequent is reduced to finding a (ρ1τ 1τ 1τ 2τ with a = e or d such that the convergence assumption holds. Below, we will use this lemma with G replaced by its compact subsets. To close this section, we give some Moser-type calculus inequalities in Sobolev spaces and a nonlinear Gronwall-type inequality, which will be used in the subsequent analysis. a ) ua2τ , θ2τ

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Lemma 2.3. (See [21].) (i) Let f, g ∈ H s (Rn ) with s ≥ [d/2] + 1. Then, for any multi-index α with |α| ≤ s, it holds that

α

∂ (f g) ≤ Cs f s g s . (ii) For integer s ≥ [n/2] + 1 and multi-index α with |α| ≤ s, it holds that

 α  α

∂ , f g = ∂ (f g) − f ∂ α g ≤ Cs f s g s−1 . (iii) Let A = A(x, v) be a smooth function satisfying A(x, 0) = 0 and v ∈ H s (Rn ) with s ≥ [n/2] + 1. For any multi-index α with |α| ≤ s, it holds that

α

∂ A(x, v) ≤ Cs |A|



C s+1

 1 + v s−1 v s . s

Here Cs is a generic constant depending only on s. Lemma 2.4. (See [29].) Suppose ψ(t) is a positive C 1 -function of t ∈ [0, T ) with T ≤ ∞, m > 1 and b1 (t), b2 (t) are integrable on [0, T ). If ψ  (t) ≤ b1 (t)ψ(t) + b2 (t)ψ m (t), then there exists δ > 0, depending only on m, C1b and C2b , such that sup ψ(t) ≤ eC1b , t∈[0,T )

whenever ψ(0) ∈ (0, δ]. Here t C1b := sup t∈[0,T )

T b1 (s)ds

and C2b := sup

0

t∈[0,T )



max b1 (t), 0 dt.

0

3. Construction of approximation solution a , ua , θ a , ρ a , ua , θ a ) In this part, we propose a construction of the approximation (ρ1τ 1τ 1τ 2τ 2τ 2τ (a = e, d) in the convergence assumption for the bipolar isentropic hydrodynamic model (2.1). To begin with, when τ σ = 1, let (ρ1e , θ1e , ρ2e , θ2e ) solves the nonlocal bipolar energy-transport equations

Y. Li, Z. Zhou / J. Differential Equations 258 (2015) 3546–3566

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     ⎧ ∂t ρ1e =  ρ1e θ1e − div ρ1e ∇−1 ρ1e − ρ2e − b , ⎪ ⎪  ⎪  ⎪    e e  ⎪ 3 e e 5 e e −1 e e ⎪ ⎪ ∂ ρ ρ θ ∇ − ρ − b − ∇ ρ θ ρ θ + div t ⎪ 1 2 1 1 ⎪ 2 1 1 2 1 1 ⎪ ⎪ ⎪ ⎪       3    ⎪ ⎪ ⎨ = ρ1e ∇−1 ρ1e − ρ2e − b − ∇ ρ1e θ1e ∇−1 ρ1e − ρ2e − b − ρ1e θ1e − TL (x) , 2 (3.1)      ⎪ ∂t ρ2e =  ρ2e θ2e − div ρ2e ∇−1 ρ1e − ρ2e − b , ⎪ ⎪   ⎪ ⎪    e e  ⎪ 3 e e 5 e e ⎪ −1 e e ⎪ ⎪ ⎪ ∂t 2 ρ2 θ2 − div 2 θ2 ρ2 ∇ ρ1 − ρ2 − b + ∇ ρ2 θ2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ = −ρ e ∇−1 ρ e − ρ e − b + ∇ ρ e θ e ∇−1 ρ e − ρ e − b − 3 ρ e θ e − T (x). L 2 1 2 2 2 1 2 2 2 2 Inspired by the Maxwell iteration-described in the Introduction, we construct the formal approximation solution as in (1.10): e ρiτ = ρie ,

ueiτ = (−1)i−1 τ ∇φ e − τ

  φτe = φ e = −1 ρ1e − ρ2e − b .

∇(ρie θie ) , ρie

e θiτ = θie ,

i = 1, 2

Having this formal approximation solution, we define for i = 1, 2, ∂t ueiτ + (ueiτ · ∇)ueiτ /τ τ    ∇(ρie θie ) = ∂t (−1)i−1 ∇−1 ρ1e − ρ2e − b − ρie    ∇(ρie θie ) + (−1)i−1 ∇−1 ρ1e − ρ2e − b − ρie    ∇(ρie θie ) , · ∇ (−1)i−1 ∇−1 ρ1e − ρ2e − b − ρie

e Ri1 =

and e Ri2 =

 e e   e e 2 e |ue |2 ρ1τ uiτ |ui τ e |2 ρiτ ρiτ |uiτ | 1 1 iτ + div + ∂ t 2 τ 2 2 τ2

= ∂t

 ρ e |(−1)i−1 ∇−1 (ρ e − ρ e − b) −

+

i

1

2

∇(ρie θie ) 2 | ρie

2 ρie |(−1)i−1 ∇−1 (ρ1e − ρ2e − b) −

∇(ρie θie ) 2 | ρie

2  ((−1)i−1 ρ e ∇−1 (ρ e − ρ e − b) − ∇(ρ e θ e ))|∇−1 (ρ e − ρ e − b) − i i 2 1 2 1 2 + div 2

Then it is easy to verify that

∇(ρie θie ) 2 | ρie

.

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Y. Li, Z. Zhou / J. Differential Equations 258 (2015) 3546–3566

⎧ 1 1 e e e ⎪ ⎪ + ueiτ · ∇ρiτ + ρiτ div ueiτ = 0, i = 1, 2, ∂t ρiτ ⎪ ⎪ τ τ ⎪ ⎪ e ⎪  e ⎪ 1 1 e 1 θiτ 1 e 1 ⎪ e e e ⎪ ∂ u + · ∇ u + ∇ρiτ + ∇θiτ = (−1)i−1 ∇φτe − 2 ueiτ + τ Ri1 , u t ⎪ iτ iτ iτ e ⎪ τ τ ρ τ τ τ ⎪ iτ ⎪ ⎨  e e  e e 2 ρiτ |uiτ | ρiτ |uiτ |2 3 e e 5 e e 1 e ∂ θ θ + ρ div + ρ + ⎪ iτ iτ iτ iτ uiτ ⎪ t 2 2 τ 2 2 ⎪ ⎪ ⎪  e e 2 ⎪ ⎪  ρiτ |uiτ | 3 e e ⎪ i−1 1 e e e e ⎪ = (−1) , i = 1, 2, ρ u ∇φ − + ρiτ θiτ − TL + τ 2 Ri2 ⎪ ⎪ ⎪ τ iτ iτ τ 2 2 ⎪ ⎪   ⎩ e e e φτ = −1 ρ1τ − ρ2τ −b , or

⎧ 1 1 e ⎪ e e ⎪ + ueiτ ∇ρiτ + ρiτ div ueiτ = 0, i = 1, 2, ∂t ρiτ ⎪ ⎪ τ τ ⎪ ⎪ ⎪ e ⎪  1 1 θiτ 1 e ⎪ e ⎪ ∂t ueiτ + ueiτ · ∇ ueiτ + ⎪ e ∇ρiτ + τ ∇θiτ ⎪ ⎪ τ τ ρ ⎪ iτ ⎪ ⎪ ⎪ ⎨ = (−1)i−1 1 ∇φ e − 1 ue + τ R e , i = 1, 2, τ i1 τ τ 2 iτ ⎪ ⎪ ⎪ ∂t θ e + 1 ue · ∇θ e + 2 θ e div ue ⎪ iτ iτ iτ iτ ⎪ ⎪ 3τ iτ ⎪ τ ⎪ e ⎪    2 e e 2 1 1  e 2  e 2 2 Ri2 ⎪ ⎪ ⎪ − = − − θ − T R + , u τ u τ L ⎪ iτ iτ iτ i1 e ⎪ 3 τ2 2 3 3 ρiτ ⎪ ⎪ ⎪   ⎩ e e e − ρ2τ −b . φτ = −1 ρ1τ

i = 1, 2,

(3.2)

i = 1, 2,

e , R e , R e and R e . From Lemma 2.3, we have the following regularity of R11 12 21 22

Lemma 3.1. Let s > 1 +

n 2

be an integer. Assume that TL = TL (x) and b = b(x) satisfy TL (x), b(x) ∈ H s+1 (Ω),

and (ρ1e , θ1e , ρ2e , θ2e ) ∈ C([0, T∗ ]; H s+3 (Ω)) ∩ C 1 ([0, T∗ ]; H s+1 (Ω)) with positive lower bounds, e , R e , R e , R e ∈ C([0, T ], H s ). then ue1τ , ue2τ ∈ C([0, T∗ ], H s+1 (Ω)) for 0 < τ ≤ 1, and R11 ∗ 12 21 22 Similarly, if σ = 1, taking d ρiτ = ρid ,

udiτ = (−1)i−1 τ ∇φ d − τ

∇(ρid TL )

ρid    ∇(ρid TL ) 2 2  d i−1 d θiτ = TL + τ (−1) ∇φ −  3 ρd

,

i = 1, 2,

i

  ∇(ρid TL ) ∇(ρid TL ) 2 i−1 d − div (−1) ∇φ − T − (−1)i−1 ∇φ d − ∇T , L L 3 ρid ρid   φτd = φ d = −1 ρ1d − ρ2d − b , here (ρ1d , ρ2d , φ d ) solves

i = 1, 2,

Y. Li, Z. Zhou / J. Differential Equations 258 (2015) 3546–3566

3557

  d   d ⎧ d d ⎪ ⎨ ∂t ρ1 = ρ1 TL (x) − divρ1 ∇φ , ∂t ρ2d =  ρ2d TL (x) + div ρ2d ∇φ d , ⎪ ⎩ φ d = ρ1d − ρ2d − b(x),

(3.3)

⎧ 1 1 d d d ⎪ ⎪ ∂t ρiτ + udiτ ∇ρiτ + ρiτ div udiτ = 0, i = 1, 2, ⎪ ⎪ τ τ ⎪ ⎪ ⎪ d ⎪  d 1 d 1 θiτ 1 d ⎪ d d ⎪ ⎪ u ∂ u + · ∇ u + ∇ρiτ + ∇θiτ t iτ ⎪ iτ iτ d ⎪ τ τ τ ρ ⎪ iτ ⎪ ⎪ ⎪ ⎨ = (−1)i−1 1 ∇φ d − 1 ud + τ R d , i = 1, 2, τ i1 τ τ 2 iτ ⎪ ⎪ 1 d 2 d ⎪ d d d ⎪ ⎪ ⎪ ∂t θiτ + τ uiτ · ∇θiτ + 3τ θiτ div uiτ ⎪ ⎪  ⎪ ⎪  2 1 1  d 2 1  d ⎪ d ⎪ uiτ − θiτ − TL + τ Ri2 = − , i = 1, 2, ⎪ ⎪ 2 ⎪ 3 2τ τ τ ⎪ ⎪   ⎩ d d d φτ = −1 ρ1τ − ρ2τ −b ,

(3.4)

then we have

where d ( 2 |ud |2 − ud ∇T − 2 T div ud )) ∂t udiτ + (udiτ · ∇)udiτ /τ 1 ∇(ρiτ L iτ iτ 3 iτ 3 L + d τ τ ρi    ∇(ρid TL ) = ∂t (−1)i−1 ∇−1 ρ1d − ρ2d − b − ρid    ∇(ρid TL ) + (−1)i−1 ∇−1 ρ1d − ρ2d − b − ρi    ∇(ρid TL ) · ∇ (−1)i−1 ∇−1 ρ1d − ρ2d − b − ρid

d Ri1 =

−

+

−

∇( 23 ρid TL div((−1)i−1 ∇−1 (ρ1d − ρ2d − b) −

∇(ρid TL ) )) ρid

ρid ∇(ρid ( 23 τ |(−1)i−1 ∇−1 (ρ1d − ρ2d − b) −

∇(ρid TL ) 2 | )) ρid

ρid ∇(ρid ((−1)i−1 ∇−1 (ρ1d − ρ2d − b) − ρid

∇(ρid TL ) )∇TL ) ρid

,

and Ri2 =

 1 1 2 d 1  2 1 2 d d + udiτ ∇θiτ + θiτ div udiτ + udiτ  − udiτ ∇TL − TL div udiτ ∂t θiτ τ τ 3τ 3τ τ 3τ

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Y. Li, Z. Zhou / J. Differential Equations 258 (2015) 3546–3566

     ∇(ρid TL ) 2 2  i−1 −1 d d  = ∂t (−1) ∇ ρ1 − ρ2 − b −  3 ρid    ∇(ρid TL ) − (−1)i−1 ∇−1 ρ1d − ρ2d − b − ∇TL ρid    ∇(ρid TL ) 2 i−1 −1 d d − TL div (−1) ∇ ρ1 − ρ2 − b − 3 ρid       ∇(ρid TL )  2 i−1 −1 d d ∇ (−1)i−1 ∇−1 ρ1d − ρ2d − b + (−1) ∇ ρ1 − ρ2 − b − d 3 ρi     ∇(ρid TL ) ∇(ρid TL ) 2 i−1 −1 d d − ∇TL  − (−1) ∇ ρ1 − ρ2 − b − ρi ρid    ∇(ρid TL ) 2 i−1 −1 d d − TL div (−1) ∇ ρ1 − ρ2 − b − 3 ρid      ∇(ρid TL ) 2 2 2  i−1 −1 d d  + (−1) ∇ ρ1 − ρ2 − b −  3 3 ρid    ∇(ρid TL ) i−1 −1 d d − (−1) ∇ ρ1 − ρ2 − b − ∇TL ρid    ∇(ρid TL ) 2 − TL div (−1)i−1 ∇−1 ρ1d − ρ2d − b − 3 ρid    ∇(ρid TL ) i−1 −1 d d · div (−1) ∇ ρ1 − ρ2 − b − ρid     ∇(ρid TL ) 2 1   . + τ (−1)i−1 ∇−1 ρ1d − ρ2d − b −  3 ρid d , R d , R d and R d . From Lemma 2.3, we also have the following regularity of R11 12 21 22

Lemma 3.2. Let s > 1 +

n 2

be an integer. Assume that TL = TL (x) and b = b(x) satisfy TL (x) ∈ H s+3 (Ω),

b(x) ∈ H s+1 (Ω),

and (ρ1d , ρ2d ) ∈ C([0, T∗ ]; H s+3 (Ω)) ∩ C 1 ([0, T∗ ]; H s+1 (Ω)) with positive lower bounds, then ud1τ , ud2τ ∈ C([0, T∗ ], H s+2 (Ω)) and θ1d , θ2d ∈ C([0, T∗ ], H s+1 (Ω)) for 0 < τ ≤ 1, and d , R d , R d , R d ∈ C([0, T ], H s ). R11 ∗ 12 21 22 4. Proof of Theorem 1.1 and 1.2 In this section, we are going to prove Theorems 1.1 and 1.2. To begin with, we show Theorem 1.1. That is, we mainly derive the error estimates, which are used to prove Theorem 1.1. First of all, since (ρ1e , θ1e , ρ2e , θ2e ) ∈ C([0, T∗ ], H s+2 (Ω)) ∩ C 1 ([0, T∗ ], H s+1 (Ω)) with a pos-

Y. Li, Z. Zhou / J. Differential Equations 258 (2015) 3546–3566

3559

e , θ e , ρe , θ e ) = itive lower bound, there are two positive numbers a and b such that (ρ1τ 1τ 2τ 2τ e e e e e e (ρ1 , θ1 , ρ2 , θ2 )(0, x) ∈ (2a, b) and |u1τ (0, x)|, |u1τ (0, x)| ≤ b for all x. Denote by [0, T1τ ) the maximal time interval where the system (2.1) with the initial data (1.7) has a unique H s -solution (ρ1τ , uτ1 , θ1τ , ρ2τ , uτ2 , θ2τ ) with (a, 2b) × (−2b, 2b)d × (a, 2b) × (a, 2b) × (−2b, 2b)d × (a, 2b). Thanks to Lemma 2.2, it suffices to prove the error estimate in (1.8) for t ∈ [0, min{T∗ , T1τ }). e − ρ τ , U = ue − uτ , Θ = θ e − θ τ for i = 1, 2. From the To this end, we set Ni = ρiτ i i i i i iτ iτ equations in (2.1) and (3.3), it follows that the error satisfies

⎧   1 1 ⎪ e ⎪ + Ni div ueiτ , i = 1, 2, ∂t Ni + uτi · ∇Ni + ρ1τ div Ui = − Ui ∇ρiτ ⎪ ⎪ ⎪ τ τ ⎪  ⎪ ⎪  θiτ 1 τ 1 ⎪ ⎪ ∇Θi + τ ∇Ni ∂ U + u · ∇ Ui + ⎪ ⎪ ⎪ t i τ i τ ρi ⎪ ⎪  e ⎪ τ ⎪ θ θ 1 1 ⎪ iτ i e e ⎪ ⎪ = − · ∇)u − − (U i ⎪ iτ e τ ∇ρiτ ⎪ τ τ ρ ρ ⎪ i iτ ⎪ ⎨ 1 1 i−1 −1 e + (−1) ∇ (N1 − N2 ) − 2 Ui + τ Ri1 , i = 1, 2, ⎪ τ τ ⎪ ⎪ ⎪ ⎪ 3 1 3 ⎪ ⎪ ∂t Θi + uτi ∇Θi + θiτ div Ui ⎪ ⎪ 2 2 τ ⎪  ⎪ ⎪ ⎪ 3 1 1 1  e 2  τ 2  ⎪ e ⎪ = − Ui · ∇θiτ − Θi div ueiτ + − uiτ − ui ⎪ ⎪ ⎪ 2τ τ τ2 2 ⎪ ⎪ ⎪ ⎪ 1 e 3 ⎪ e ⎪ + τ 2 e Ri2 , i = 1, 2. − Θi − τ ueiτ · Ri1 ⎩ 2 ρ1τ (4.1) Then we differentiate (4.1) with ∂xα for a multi-index α satisfying |α| ≤ s to give ⎧  1 ⎪ ∂t ∂xα N1 + uτ1 · ∇∂xα N1 + ρ1τ div ∂xα U1 = F11α , ⎪ ⎪ ⎪ τ ⎪  ⎪ ⎪  α  τ θ1τ 1 1 1 α ⎪ α α α 1α ⎪ ⎪ ∂ U + · ∇ ∂ U + Θ + ∇∂ N u ∇∂ ∂ t x 1 x 1 x 1 x 1 = − 2 ∂x U1 + F2 , ⎪ 1 τ ⎪ τ τ ρ τ ⎪ 1 ⎪ ⎪ ⎪ 3 τ α 1 τ 3 ⎪3 α α ⎪ ⎨ ∂t ∂x Θ1 + u1 ∇∂x Θ1 + θ1 div ∂x U1 = − ∂xα Θ1 + F31α , 2 2τ τ 2  1 τ ⎪ α α τ α 2α ⎪ ⎪ ∂t ∂x N2 + u2 · ∇∂x N2 + ρ2 div ∂x U2 = F1 , ⎪ ⎪ τ ⎪  ⎪ ⎪  α θ2τ 1 τ 1 1 α ⎪ α α α 2α ⎪ ⎪ ∂ U + · ∇ ∂ U + Θ + ∇∂ N u ∇∂ ∂ ⎪ x 2 x 2 x 2 = − 2 ∂x U2 + F2 , τ ⎪ t x 2 τ 2 τ ρ τ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩ 3 ∂ ∂ α Θ + 3 uτ ∇∂ α Θ + 1 θ τ div ∂ α U = − 3 ∂ α Θ + F 2α . t 2 2 2 x 2 3 2 x 2τ 2 x τ 2 2 x

(4.2)

where   1  α τ     1 e iα iα F1iα = − ∂x α Ui · ∇ρiτ + Ni div ueiτ − + f12 , ∂x , ui ∇Ni + ∂xα , ρiτ div Ui = f11 τ τ  e   1 α θiτ θiτ 1 α iα i−1 −1 2 e e F2 = ∂x (−1) ∇ (N1 − N2 ) + τ Ri1 − ∂x (Ui · ∇)uiτ + e − ρ τ ∇ρiτ τ τ ρiτ i

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Y. Li, Z. Zhou / J. Differential Equations 258 (2015) 3546–3566

   θτ 1  α τ iα iα iα + f22 + f23 , ∂x , ui ∇Ui + ∂xα , iτ ∇Ni = f21 τ ρi  iα α e 2 1 F3 = −∂x τ uiτ · Ri1 − τ e Ri2 ρiτ   1 α 3 1 τ  e 2  τ 2  e e − ∂x Ui · ∇θiτ + Θi div uiτ − − uiτ − ui τ 2 τ 2    1 3 α τ iα iα iα + f32 + f33 . ∂x , ui ∇Θi + ∂xα , θiτ div Ui = f31 − τ 2 −

For the sake of clarity, we divide the following into lemmas. Lemma 4.1. Under the conditions of Theorem 1.1, we have d dt

  Ω

θ1τ ρ1τ

2

 τ 2  α 2       ∂ N1  + θ τ ∂ α U1 2 + 3 ∂ α Θ1 2 + θ2 ∂ α N2 2 x x x 1 x τ 2 ρ2

 2 3  2  2 1  + θ2τ ∂xα U2  + ∂xα Θ2  dx + 2 ∂xα U1 , ∂xα Θ1 , ∂xα U2 , ∂xα Θ2

2 τ  2  2  2  2  2  2   C ≤ M ∂xα N1  + ∂xα U1  + ∂xα Θ1  + ∂xα N2  + ∂xα U2  + ∂xα Θ2  dx τ Ω















+ C1 ∂xα N1

F11α + C2 ∂xα U1

F21α + C3 ∂xα Θ1

F31α















+ C4 ∂xα N2

F12α + C5 ∂xα U2

F22α + C6 ∂xα Θ2

F32α ,

(4.3)

where C, Ci (i = 1, 2, · · · , 6) are all generic constants depending only on the range [a, 2b] of ρiτ , θiτ (i = 1, 2) but independent of τ , and        1     2 2 M = div uτ1  + uτ1 · ∇ρ1τ  + uτ1 · ∇θ1τ  + uτ1  + τ θ1τ − TL  + τ ∇ρ1τ  + τ ∇θ1τ  τ         2 2   1  + div uτ2  + uτ2 · ∇ρ2τ  + uτ2 · ∇θ2τ  + uτ2  + τ θ2τ − TL  + τ ∇ρ2τ  + τ ∇θ2τ  . τ θτ

θτ

1

2

Proof. By multiplying (4.2) by (( ρ1τ )2 ∂xα N1 , θ1τ ∂xα U1 , ∂xα Θ1 , ( ρ2τ )2 ∂xα N2 , θ2τ ∂xα U2 , ∂xα Θ2 ), respectively and integrating them with respect to x over T d , we immediately have (4.3) without extra trouble, for similar details, e.g., see [17,30]. 2 For the right-hand side of the inequality in Lemma 4.1, we have the following claim. Lemma 4.2. Set D = D(t) =

(N1 , U1 , Θ1 , N2 , U2 , Θ2 )(·, t) s . τ

Then, for 0 < τ ≤ 1, the following estimates hold:

Y. Li, Z. Zhou / J. Differential Equations 258 (2015) 3546–3566

M ≤ Cτ (1 + D),

(4.4)

2

α

1α

 

∂ N1

F ≤ ε U1 s + C 1 + D 2 N1 2 , x s 1 2 τ

α

1α 3 ∂xα U1 2  

∂ U1

F ≤ + Cτ 4 + C 1 + D 2 U1 2s x 2 4τ 2    + C 1 + D 2s N1 2s + Θ1 2s + N2 2s ,

α

1α ∂xα Θ1 2   U1 2

∂ Θ1

F ≤ + ε 2 s + Cτ 4 + C 1 + D 2 Θ1 2s , x 3 4 τ 2

α

2α

 

∂ N2

F ≤ ε U2 s + C 1 + D 2 N2 2 , x s 1 τ2

α

2α 3 ∂xα U2 2  

∂ U2

F ≤ + Cτ 4 + C 1 + D 2 U2 2s x 2 2 4τ    + C 1 + D 2s N1 2s + Θ2 2s + N2 2s ,

α

2α

∂ Θ2

F ≤ x

3

∂xα Θ2 2 4

U2 2 +ε 2 s τ

3561

  + Cτ 4 + C 1 + D 2 Θ2 2s .

(4.5)

(4.6) (4.7) (4.8)

(4.9) (4.10)

Proof. Recall that   τ ∇(ρ1e θ1e ) ue1τ = τ ∇−1 ρ1e − ρ2e − b − . ρ1e We deduce from the well-known embedding inequality in Sobolev spaces that  





   

div uτ  < C div uτ ≤ C div ue + div uτ − ue ≤ Cτ (1 + D). 1 1 s 1 1τ s 1τ s Then, the remainders of M are estimated similarly. 1α + f 1α . For f 1α , by using Next we turn to estimate C1 F11α ∂xα N1 , where F11α = f11 12 11 e e Lemma 2.2 and the boundedness of (ρ1τ , u1τ ) s+1 indicated in Lemma 3.1, we have

1α α  

= ∂ U1 · ∇ρ e + N1 div ue

τ f11 x 1τ 1τ



  e

U1 s + div ue N1 s ≤ C ∇ρ1τ 1τ s s   ≤ C U1 s + τ N1 s .

(4.11)

1α , the classical estimates for commutators (see Lemma 2.3(ii)) gives For the second term f12

1α  α τ 

 

= ∂ , u ∇N1 + ∂ α , ρ τ div U1

τ f12 x x 1 1





≤ C uτ1 s ∇N1 s−1 + C ρ1τ s div U1 s−1



    ≤ C U1 s + uτ1 s N1 s + N1 s + ρ1τ s U1 s   ≤ C τ (1 + D) N1 s + (1 + τ D) U1 s .

(4.12)

Thus, combining with inequalities (4.11) and (4.12), from Young’s inequality, we end up with

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Y. Li, Z. Zhou / J. Differential Equations 258 (2015) 3546–3566





U1 2 C1 F11α

∂xα N1 ≤ ε 2 s + C(1 + D) N1 2s , τ where ε > 0 is a generic constant (independent of τ ) to be determined below. This is just the inequality (4.5). Recall that 1α 1α 1α F21α = f21 + f22 + f23

can be estimated as follows:

1α

α C2 α 





∂ U1 =

∂ ∇−1 (N1 − N2 ) + τ 2 R e

∂ α U1

C2 f21 x x x 11 τ





 ∂ α U1 

∂ α N1 + ∂ α N2 + τ 2 ∂ α R e

≤C x x x x 11 τ α  ∂ U1  2 ≤C x τ + N1 s + N2 2s τ α   ∂ U1 2 ≤ x 2 + C τ 4 + N1 2s + N2 2s , 4τ

(4.13)

where we have used Lemma 3.1 and Young’s inequality. Moreover, it follows from Lemma 2.3 that

  e

τ

1α

α C2 α





∂ U1 =

∂ (U1 · ∇)ue + θ1τ − θ1 ∇ρ e

∂ α U1

C2 f22 x x 1τ 1τ x e τ

τ ρ1τ ρ1



 e

θ1τ





θ1τ

∂ α U1

∇ρ e

− U1 s ∇ue1τ s +

≤C x 1τ s

ρe τ ρ1τ s 1τ     ∂ α U1 ≤C x τ U1 s + 1 + D s−1 N1 s + Θ1 s τ ≤

   ∂xα U1 2 2(s−1) 2 2 2 N , + C N + C 1 + D + Θ 1 s 1 s 1 s 4τ 2

(4.14)

where the estimate for composed functions enables us to derive A(x, N1 , Θ1 ) :=

θτ θ τ (x) − Θ1 θ1τ TL (x) − 1τ = − 1τ ρ1τ ρ1 ρ1τ (x) ρ1 (x) − N1

as



  s−1  

A(·, N1 , Θ1 ) ≤ C|A| 1 + N1 s + Θ1 s N1 s + Θ1 s s C s+1    ≤ C 1 + D s−1 N1 s + Θ1 s . With this and the estimate for commutators (Lemma 2.3), one has

(4.15)

Y. Li, Z. Zhou / J. Differential Equations 258 (2015) 3546–3566

3563



 τ

1α

α

 α τ ∂xα U1

α θ1







C2 f23 ∂x U1 = C2 ∂x , u1 ∇U1 + ∂x , τ ∇N1



τ ρ1





θ1τ

∂xα U1

τ





u1 s ∇U1 s−1 + τ ∇N1 s−1 ≤C τ ρ1 s  τ

 α

θ1

∂x U1



≤C τ (1 + D) U1 s + τ + A s N1 s τ ρ1 s    ∂xα U1  (4.16) τ (1 + D) U1 s + D 1 + D s−1 N1 s + Θ1 s τ      ∂ α U1 2 ≤ x 2 + C 1 + D 2 U1 2s + C 1 + D 2s N1 2s + Θ1 2s . (4.17) 4τ

≤C

Summing up (4.13)–(4.17) immediately gives the inequality (4.6). Finally, in a similar spirit, C3 F31α ∂xα Θ1 is estimated as

 e

1α

α







∂ Θ1 = C3 ∂ α τ ue R e − τ 2 R12

∂ α Θ1

C3 f31 x x 1τ 11 e

x ρ1τ

e

e  α



∂ Θ1

≤ C τ u1τ s−1 R11 s + τ 2 R12 x s ∂xα U1 2 + Cτ 4 , (4.18) 4τ 2





1α α 3 1 τ  e 2  1τ 2 

e e

= ∂ · ∇θ − Θ div u − − u U − u τ f32 1 1 1 1τ 1τ 1τ

x 2 τ 2 





e

 1

e

e





≤ C ∇θ1τ s U1 s + div u1τ s Θ1 s + 2 u1τ s + U1 s U1 s τ ≤

≤ C(1 + D) U1 s + Cτ Θ1 s ,

(4.19)

and



1α 3  α 1τ 

 

= ∂ , u ∇Θ1 + ∂ α , θ 1τ div U1

τ f33 x 1

2 x 1

1τ

  1τ

≤ u1 s ∇Θ1 s−1 + θ1 s div U1 s−1





≤ Cτ (1 + D) E T s + C(1 + D) E u s .

(4.20)

From (4.18)–(4.20), by using Young’s inequality, we can achieve the inequality (4.7). In the completely same way, we can show (4.8)–(4.10). The proof of Lemma 4.2 is complete. 2 Proof of Theorem 1.1. Substituting these estimates in Lemma 4.2 into the inequality (4.3) in Lemma 4.1, one has d dt

  Ω

θ1τ ρ1τ

2

 τ 2  α 2         ∂ N1  + θ τ ∂ α U1 2 + 3 ∂ α Θ1 2 + θ2 ∂ α N2 2 + θ τ ∂ α U2 2 x x x 1 x 2 x τ 2 ρ2

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Y. Li, Z. Zhou / J. Differential Equations 258 (2015) 3546–3566

2  2 3 1  + ∂xα Θ2  dx + 2 ∂xα U1 , ∂xα Θ1 , ∂xα U2 , ∂xα Θ2

2 τ ≤ Cτ 4 + 2ε

2 

 U1 2s + C 1 + D 2s (N1 , U1 , Θ1 , N2 , U2 , Θ2 ) s . 2 τ

(4.21)

We integrate (4.21) from 0 to t ≤ min{Tτ , T∗ } to reach   Ω

θ1τ ρ1τ

2

 +

 α 2     ∂ N1  + θ τ ∂ α U1 2 + 3 ∂ α Θ1 2 x x 1 x 2

θ2τ ρ2τ

2

 t  α 2   

α   

∂ N2  + θ τ ∂ α U2 2 + 3 ∂ α Θ2 2 dx + 1

∂ (U1 , U2 ) t  , . 2 dt  x x x 2 x 2 2 4τ 0

≤ CT∗ τ 4 +

2ε τ2

t

 

(U1 , U2 ) t  , . 2 dt  + C

t

s

0

2 

1 + D 2s (N1 , U1 , Θ1 , N2 , U2 , Θ2 ) s dt  ,



0

(4.22) where we have used the fact that the initial data are in equilibrium. It is easy to show that

2 C −1 ∂xα (N1 , U1 , Θ1 , N2 , U2 , Θ2 )

  τ 2  2 θ1  α 2 ∂x N1 + θ1τ ∂xα U1  + ≤ τ ρ1 Ω



3  α 2 ∂ Θ1 2 x

 α 2     ∂ N2  + θ τ ∂ α U2 2 + 3 ∂ α Θ2 2 dx x 2 x 2 x

2 ≤ C ∂xα (N1 , U1 , Θ1 , N2 , U2 , Θ2 ) , +

θ2τ ρ2τ

2

(4.23)

since nτi (i = 1,  2) and Tiτ (i = 1, 2) are bounded from below and from above. Now, we chose ε so small that 8ε α:|α|≤s 1 ≤ 1 and sum up (4.23) over all α satisfying |α| ≤ s to get



(N1 , U1 , Θ1 , N2 , U2 , Θ2 )(t, .) 2 s

t ≤ CT∗ τ 4 + C





  2 1 + D 2s (N1 , U1 , Θ1 , N2 , U2 , Θ2 ) t  , . s dt  .

(4.24)

0

Then we apply Gronwall’s lemma to (4.24) and obtain  t 



   2s

(N1 , U1 , Θ1 , N2 , U2 , Θ2 )(t, .) 2 ≤ CT∗ τ 4 exp C 1 + D dt . s

0

In view of (N1 , U1 , Θ1 , N2 , U2 , Θ2 ) s = τ D, it follows from (4.25) that

(4.25)

Y. Li, Z. Zhou / J. Differential Equations 258 (2015) 3546–3566

  t    1 + D 2s dt  ≡ Φ(t), D(t)2 ≤ CT∗ τ 2 exp C

3565

(4.26)

0

thus,   Φ  (t) = C 1 + D 2s Φ(t) ≤ CΦ(t) + CΦ s+1 (t). Applying the nonlinear Gronwall-type inequality in Lemma 2.4 to the last inequality yields Φ(t) ≤ eCT∗

(4.27)

for t ∈ [0, min{T∗ , T1τ }) if we choose so small that Φ(0) = CT∗ 2 ≤ e−CT∗ . Because of (4.26), there exists a constant c, independent of , such that D(T ) ≤ c,

(4.28)

for T ∈ [0, min{T∗ , T1τ }). Finally, Theorem 1.1 is concluded from (4.25) and (4.28). This completes the proof. 2 Since we can obtain the proof of Theorem 1.2 in a similar fashion, we can omit the details here. Acknowledgments The authors would like to thank the referees for the valuable comments and suggestions which greatly improved the presentation of the manuscript. The research of Li is partially supported by the National Science Foundation of China (Grant No. 11171223), the Ph.D. Program Foundation of Ministry of Education of China (Grant No. 20133127110007) and the Innovation Program of Shanghai Municipal Education Commission (Grant No. 13ZZ109). References [1] G. Ali, A. Jüngel, Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasma, J. Differential Equations 190 (2003) 663–685. [2] A.M. Anile, S. Pennisi, Extended thermodynamics of the Blotekjaer hydrodynamical model for semiconductors, Contin. Mech. Thermodyn. 41 (1992) 187–197. [3] A.M. Anile, O. Muscato, Improved hydrodynamical model from carrier transport in semiconductors, Phys. Rev. B 51 (1995) 16728–16740. [4] H.C. Chainais, J.-G. Liu, Y.-J. Peng, Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis, Math. Model. Numer. Anal. 37 (2003) 319–338. [5] D.P. Chen, R.S. Eisenberg, J.W. Jerome, C.W. Shu, A hydrodynamic model of temperature change in open ionic channels, Biophys. J. 69 (1995) 2304–2322. [6] F. Fuchs, F. Poupaud, Asymptotical and numerical analysis of degeneracy effects on the drift-diffusion equations for semiconductors, Math. Models Methods Appl. Sci. 5 (1995) 1093–1111. [7] S. Gadau, A. Jüngel, A three-dimensional mixed finite-element approximation of the semiconductor energytransport equations, SIAM J. Sci. Comput. 31 (2009) 1120–1140.

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