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Journal of Applied Mathematics
Volume 2012, Article ID 957185, 8 pages
http://dx.doi.org/10.1155/2012/957185
Research Article

On a Quasi-Neutral Approximation to the Incompressible Euler Equations

College of Mathematics and Information Science, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

Received 10 February 2012; Revised 25 April 2012; Accepted 26 April 2012

Academic Editor: Roberto Natalini

Copyright © 2012 Jianwei Yang and Zhitao Zhuang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We rigorously justify a singular Euler-Poisson approximation of the incompressible Euler equations in the quasi-neutral regime for plasma physics. Using the modulated energy estimates, the rate convergence of Euler-Poisson systems to the incompressible Euler equations is obtained.

1. Introduction

In this paper, we shall consider the following hydrodynamic system: 𝜕𝑡𝑛𝜆𝑛+div𝜆𝐮𝜆=0,𝑥𝒯3𝜕,𝑡>0,𝑡𝐮𝜆+𝐮𝜆𝐮𝜆=𝜙𝜆,𝑥𝒯3,𝑡>0,Δ𝜙𝜆=𝑛𝜆1𝜆,𝑥𝒯3,𝑡>0(1.1) for 𝑥𝒯3 and 𝑡>0, subject to the initial conditions 𝑛𝜆,𝐮𝜆𝑛(𝑡=0)=𝜆0,𝐮𝜆0(1.2) for 𝑥𝒯3. In the above equations, 𝒯3 is 3-dimensional torus and 𝜆>0 is small parameter. Here 𝑛𝜆, 𝐮𝜆, 𝜙𝜆 denote the electron density, electron velocity, and the electrostatic potential, respectively.

System (1.1) is a model of a collisionless plasma where the ions are supposed to be at rest and create a neutralizing background field. Then the motion of the electrons can be described by using either the kinetic formalism or the hydrodynamic equations of conservation of mass and momentum as we do here. The self-induced electric field is the gradient of a potential that depends on the electron’s density 𝑛𝜆 through the linear Poisson equation Δ𝜙𝜆=(𝑛𝜆1)/𝜆.

To solve uniquely the Poisson equation, we add the condition 𝒯3𝑛𝜆𝑑𝑥=1. Passing to the limit when 𝜆 goes to zero, it is easy to see, at least at a very formal level, that (𝑛𝜆,𝐮𝜆,𝜙𝜆) tends to (𝑛𝐼,𝐮𝐼,𝜙𝐼), where 𝑛𝐼=1 and 𝜕𝑡𝐮𝐼+𝐮𝐼𝐮𝐼=𝜙𝐼,div𝐮𝐼=0.(1.3) In other words, 𝐮𝐼 is a solution of the incompressible Euler equations. The aim of this paper is to give a rigorous justification to this formal computation. We shall prove the following result.

Theorem 1.1. Let 𝐮𝐼 be a solution of the incompressible Euler equations (1.3) such that 𝐮𝐼([0,𝑇],𝐻𝑠+3(𝒯3)) and 𝒯3𝐮𝐼𝑑𝑥=0 for 𝑠>5/2. Assume that the initial value (𝑛𝜆0,𝐮𝜆0)𝐻𝑠+1 is such that 𝒯3𝑛𝜆0𝑑𝑥=1,𝒯3𝐮𝜆0𝑀𝑑𝑥=0,𝑠𝐮(𝜆)=𝜆0𝐮𝐼02𝐻𝑠+1+1𝜆𝑛𝜆0𝜆Δ𝜙𝐼0(𝑡)12𝐻𝑠0(when𝜆0),𝐮𝐼0=𝐮𝐼𝑡=0.(1.4) Then, there exist 𝜆0 and 𝐶𝑇 such that for 0<𝜆𝜆0 there is a solution (𝑛𝜆,𝐮𝜆)([0,𝑇],𝐻𝑠+1(𝒯3)) of (1.1) satisfying 𝐮𝜆(𝑡)𝐮𝐼(𝑡)2𝐻𝑠+1+1𝜆𝑛𝜆(𝑡)𝜆Δ𝜙𝐼12𝐻𝑠𝐶𝑇𝜆+𝑀𝑠(𝜆)(1.5) for any 0𝑡𝑇.

Concerning the quasi-neutral limit, there are some results for various specific models. In particular, this limit has been performed for the Vlasov-Poisson system [1, 2], for the drift-diffusion equations and the quantum drift-diffusion equations [3, 4], for the one-dimensional and isothermal Euler-Poisson system [5], for the multidimensional Euler-Poisson equations [6, 7], for the bipolar Euler-Poisson system [8, 9], for the Vlasov-Maxwell system [10], and for Euler-Maxwell equations [11]. We refer to [1215] and references therein for more recent contributions.

The main focus in the present note is on the use of the modulated energy techniques for studying incompressible fluids. We will mostly restrict ourselves to the case of well-prepared initial data. Our result gives a more general rate of convergence in strong 𝐻𝑠 norm of the solution of the singular system towards a smooth solution of the incompressible Euler equation. We noticed that the quasi-neutral limit with pressure is treated in [5, 6]. But the techniques used there do not apply here.

It should be pointed that the model that we considered is a collisionless plasma while the model in [6, 7] includes the pressure. Our proof is based on the modulated energy estimates and the curl-div decomposition of the gradient while the proof in [6, 7] is based on formal asymptotic expansions and iterative methods. Meanwhile, the model that we considered in this paper is a different scaling from that of [16]. Furthermore, our convergence result is different from the convergence result in [16].

2. Proof of Theorem 1.1

First, let us set𝑛(𝑛,𝐮,𝜙)=𝜆1𝜆Δ𝜙𝐼,𝐮𝜆𝐮𝐼,𝜙𝜆𝜙𝐼.(2.1) Then, we know the vector (𝑛,𝐮,𝜙) solves the system 𝜕𝑡𝐮+𝐮+𝐮𝐼𝐮+(𝐮)𝐮𝐼𝜕=𝜙,𝑡𝑛+𝐮+𝐮𝐼𝜕𝑛=(𝑛+1)div𝐮𝜆𝑡Δ𝜙𝐼+divΔ𝜙𝐼𝐮+𝐮𝐼,𝑛Δ𝜙=𝜆,(2.2) where 𝐮𝐯=3𝑖,𝑗=1(𝜕𝑥𝑖𝐮/𝜕𝑥𝑗)(𝜕𝑥𝑗𝐯/𝜕𝑥𝑖). In fact, from (1.3), we get Δ𝜙𝐼=𝐮𝐼𝐮𝐼.

As in [16], we make the following change of unknowns: (𝑑,𝐜)=(div𝐮,curl𝐮).(2.3) By using the last equation in (2.2), we get the following system: 𝜕𝑡𝑑+𝐮+𝐮𝐼𝑛𝑑=𝜆𝐮+2𝐮𝐼𝜕𝐮,𝑡𝐜+𝐮+𝐮𝐼𝐜=𝐜𝐮+𝐮𝐼+curl𝐮𝐼𝐮𝑑𝐜curl(𝐮)𝐮𝐼,𝜕𝑡𝑛+𝐮+𝐮𝐼𝜕𝑛=(𝑛+1)𝑑𝜆𝑡Δ𝜙𝐼+divΔ𝜙𝐼𝐮+𝐮𝐼.(2.4) This last system can be written as a singular perturbation of a symmetrizable hyperbolic system: 𝜕𝑡𝐯+3𝑗=1𝐮+𝐮𝐼𝑗𝜕𝑥𝑗1𝐯=𝜆𝒦𝜆𝐯+(𝐯)+𝒮(𝐯)+𝜆(𝐯),(2.5) where (𝐮+𝐮𝐼)𝑗 denotes the 𝑖th component of (𝐮+𝐮𝐼) and where 𝑑𝐜𝑛𝐯=,𝒦𝜆=0,0𝟎1𝟎𝟎𝟎𝜆𝟎0,(𝐯)=𝑑𝐜𝑑𝑛𝒮(𝐯)=𝐮+2𝐮𝐼𝐮𝐜𝐮+𝐮𝐼+curl𝐮𝐼𝐮curl(𝐮)𝐮𝐼0,0𝟎𝜕(𝐯)=𝑡Δ𝜙𝐼+divΔ𝜙𝐼𝐮+𝐮𝐼.(2.6)

Now, let us set 𝒜𝜆0=1𝟎0𝟎𝐼𝑑𝟎0𝟎1/𝜆 and for |𝛼|𝑠 with 𝑠>𝑑/2, 𝐸𝜆𝛼,𝑠1(𝑡)=2𝒜𝜆0𝜕𝛼𝑥𝐯,𝜕𝛼𝑥𝐯=12𝜕𝛼𝑥𝑑2+𝜕𝛼𝑥𝐜2+1𝜆𝜕𝛼𝑥𝑛2,𝐸𝜆𝑠(𝑡)=|𝛼|𝑠𝐸𝜆𝛼,𝑠(𝑡).(2.7) It is easy to know that system (2.5) is a hyperbolic system. Consequently, for 𝜆>0 fixed, we have a result of local existence and uniqueness of strong solutions in 𝐶([0,𝑇],𝐻𝑠), see [17]. This allows us to define 𝑇𝜆 as the largest time such that 𝐸𝜆𝑠(𝑡)𝑀𝜆,𝑡0,𝑇𝜆,(2.8) where 𝑀𝜆 which is such that 𝑀𝜆0 when 𝜆 goes to zero will be chosen carefully later. To achieve the proof of Theorem 1.1, and in particular inequality (1.5), it is sufficient to establish that 𝑇𝜆𝑇, which will be proved by showing that in (2.8) the equality cannot be reached for 𝑇𝜆<𝑇 thanks to a good choice of 𝑀𝜆.

Before performing the energy estimate, we apply the operator 𝜕𝛼𝑥 for 𝛼3 with |𝛼|𝑠 to (2.5), to obtain𝜕𝑡𝜕𝛼𝑥𝐯+3𝑗=1𝐮+𝐮𝐼𝑗𝜕𝑥𝑗𝜕𝛼𝑥1𝐯=𝜆𝒦𝜆𝜕𝛼𝑥𝐯+𝜕𝛼𝑥(𝐯)+𝜕𝛼𝑥𝒮(𝐯)+𝜆𝜕𝛼𝑥(𝐯)+Σ𝜆𝛼,(2.9) where Σ𝜆𝛼=3𝑗=1𝐮+𝐮𝐼𝑗𝜕𝑥𝑗𝜕𝛼𝑥𝐯𝜕𝛼𝑥𝐮+𝐮𝐼𝑗𝜕𝑥𝑗𝐯.(2.10)

Along the proof, we shall denote by 𝐶 a number independent of 𝜆, which actually may change from line to line, and by 𝐶() a nondecreasing function. Moreover (,) and stand for the usual 𝐿2 scalar product and norm, 𝑠 is the usual 𝐻𝑠 Sobolev norm, and 𝑠, is the usual 𝑊𝑠, norm.

Now, we proceed to perform the energy estimates for (2.9) in a classical way by taking the scalar product of system (2.9) with 𝒜𝜆0𝜕𝛼𝑥𝐯. Then, we have 𝑑𝐸𝑑𝑡𝜆𝛼,𝑠𝒜(𝑡)=𝜆0𝜕𝛼𝑥𝐯,3𝑗=1𝐮+𝐮𝐼𝑗𝜕𝑥𝑗𝜕𝛼𝑥𝐯+1𝜆𝒜𝜆0𝜕𝛼𝑥𝐯,𝒦𝜆𝜕𝛼𝑥𝐯+𝒜𝜆0𝜕𝛼𝑥𝐯,𝜕𝛼𝑥+𝒜(𝐯)𝜆0𝜕𝛼𝑥𝐯,𝜕𝛼𝑥𝒜𝒮(𝐮)+𝜆𝜆0𝜕𝛼𝑥𝐯,𝜕𝛼𝑥+𝒜(𝐮)𝜆0𝜕𝛼𝑥𝐯,Σ𝜆𝛼=6𝑖=1𝑖.(2.11)

Let us start the estimate of each term in the above equation. For 1, since 𝒜𝜆0 is symmetric and div𝐮𝐼=0, by Cauchy-Schwartz’s inequality and Sobolev’s lemma, we have that 1=12div𝐮𝒜𝜆0𝜕𝛼𝑥𝐯,𝜕𝛼𝑥𝐯div𝐮0,𝐸𝜆𝑠𝐸(𝑡)𝐶𝜆𝑠(𝑡)3/2.(2.12) Next, since 𝒜𝜆0𝒦𝜆 is skew-symmetric, we have that 2=0.(2.13) For 3, by a direct calculation, one gets 3𝜕=𝛼𝑥𝐜,𝜕𝛼𝑥1(𝑑𝐜)𝜆𝑛,𝜕𝛼𝑥𝜕(𝑑𝑛)𝛼𝑥𝐜𝜕𝛼𝑥+1(𝑑𝐜)𝜆𝜕𝛼𝑥𝑛2𝜕𝛼𝑥𝑑𝐸𝐶𝜆𝑠(𝑡)3/2.(2.14) Here, we have used the basic Moser-type calculus inequalities [18].

To give the estimate of the term 4, we split it in two terms. Specifically, we can deduce that 4𝜕=𝛼𝑥𝑑,𝜕𝛼𝑥𝐮+2𝐮𝐼+𝜕𝐮𝛼𝑥𝐜,𝜕𝛼𝑥𝐜𝐮+𝐮𝐼+curl𝐮𝐼(𝐮curl𝐮)𝐮𝐼𝑑𝑠𝐮+2𝐮𝐼𝐮𝑠+𝐜𝑠𝐜𝐮+𝐮𝐼𝑠+curl𝐮𝐼𝐮𝑠+curl(𝐮)𝐮𝐼𝑠𝐸𝐶𝜆𝑠+𝐸𝜆𝑠3/2.(2.15) Here, we have used the curl-div decomposition inequality 𝐮𝑠𝐶𝑑𝑠+𝐜𝑠.(2.16)

For 5, we have that 5𝑛𝑠𝜕𝑡Δ𝜙𝐼+divΔ𝜙𝐼𝐮+𝐮𝐼𝑠𝐶𝑛𝑠1+𝑑𝑠+𝐜𝑠𝐶𝜆+𝐸𝜆𝑠.(2.17)

To estimate the last term, that is, 5, by using basic Moser-type calculus inequalities and Sobolev’s lemma, we have 6=𝜕𝛼𝑥𝑑,3𝑗=1𝐮+𝐮𝐼𝑗𝜕𝑥𝑗𝜕𝛼𝑥𝑑𝜕𝛼𝑥𝐮+𝐮𝐼𝑗𝜕𝑥𝑗𝑑+𝜕𝛼𝑥𝐜,3𝑗=1𝐮+𝐮𝐼𝑗𝜕𝑥𝑗𝜕𝛼𝑥𝐜𝜕𝛼𝑥𝐮+𝐮𝐼𝑗𝜕𝑥𝑗𝐜+1𝜆𝜕𝛼𝑥𝑛,3𝑗=1𝐮+𝐮𝐼𝑗𝜕𝑥𝑗𝜕𝛼𝑥𝑛𝜕𝛼𝑥𝐮+𝐮𝐼𝑗𝜕𝑥𝑗𝑛𝐶𝑑𝑠𝐮+𝐮𝐼0,𝜕𝛼𝑥𝑑+𝑑0,𝜕𝛼𝑥𝐮+𝐮𝐼+𝐶𝐜𝑠𝐮+𝐮𝐼0,𝜕𝛼𝑥𝐜+𝐜0,𝜕𝛼𝑥𝐮+𝐮𝐼+𝐶𝜆𝑛𝑠𝐮+𝐮𝐼0,𝜕𝛼𝑥𝑛+𝑛0,𝜕𝛼𝑥𝐮+𝐮𝐼𝐶𝑑2𝑠+𝐜2𝑠+1𝜆𝑛2𝑠𝑑𝑠+𝐜𝑠𝐸+1𝐶𝜆𝑠+𝐸𝜆𝑠3/2.(2.18)

Now, we collect all the previous estimates (2.12)–(2.18) and we sum over 𝛼 to find 𝑑𝐸𝑑𝑡𝜆𝑠𝐶𝜆+𝐶𝐸𝜆𝑠+𝐸𝜆𝑠2.(2.19) By using (2.8), we get with 𝑀𝜆1 that 𝑑𝐸𝑑𝑡𝜆𝑠𝐶𝜆+𝐶𝐸𝜆𝑠,𝑡0,𝑇𝜆.(2.20) Hence, by the Gronwall inequality, we get that 𝐸𝜆𝑠𝑀(𝑡)𝑠𝑒(𝜆)+𝐶𝑡𝜆𝐶𝑡,𝑡0,𝑇𝜆.(2.21)

Consequently, if we choose 𝑀𝜆=(𝑀𝑠(𝜆)+𝐶𝑡𝜆)1/2, we see that we cannot reach equality in (2.8) for 𝑇𝜆<𝑇. This proves that 𝑇𝜆>𝑇 and that (2.21) is valid on [0,𝑇].

Acknowledgment

The authors acknowledge partial support from the Research Initiation Project for High-Level Talents (no. 40118) of North China University of Water Resources and Electric Power.

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