On a Quasi-Neutral Approximation to the Incompressible Euler Equations

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: 𝜕𝜕_𝑡𝑛^𝜆+div𝑛^𝜆𝐮^𝜆=0,𝑥∈𝒯³,𝑡>0,_𝑡𝐮^𝜆+Δ𝜙𝐮^𝜆_𝜆⋅∇𝐮^𝜆=∇𝜙^𝜆,𝑥∈𝒯³,𝑡>0,=𝑛^𝜆−1𝜆,𝑥∈𝒯³,𝑡>0(1.1) for 𝑥∈𝒯³ and 𝑡>0, subject to the initial conditions 𝑛^𝜆,𝐮^𝜆(𝑡=0)=𝑛^𝜆0,𝐮^𝜆0(1.2) for 𝑥∈𝒯³. In the above equations, 𝒯³ is 3-dimensional torus and 𝜆>0 is small parameter. Here 𝑛^𝜆, 𝐮^𝜆, 𝜙^𝜆 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 𝑛^𝜆 through the linear Poisson equation Δ𝜙^𝜆=(𝑛^𝜆−1)/𝜆.

To solve uniquely the Poisson equation, we add the condition ∫_𝒯3𝑛^𝜆𝑑𝑥=1. Passing to the limit when 𝜆 goes to zero, it is easy to see, at least at a very formal level, that (𝑛^𝜆,𝐮^𝜆,𝜙^𝜆) tends to (𝑛^𝐼,𝐮^𝐼,𝜙^𝐼), where 𝑛^𝐼=1 and 𝜕_𝑡𝐮^𝐼+d𝐮i^𝐼v⋅𝐮∇_𝐼𝐮^𝐼=∇𝜙^𝐼,=0.(1.3) In other words, 𝐮^𝐼 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 𝐮^𝐼 be a solution of the incompressible Euler equations (1.3) such that 𝐮^𝐼∈([0,𝑇],𝐻^𝑠+3(𝒯³)) and ∫_𝒯3𝐮^𝐼𝑑𝑥=0 for 𝑠>5/2. Assume that the initial value (𝑛^𝜆0,𝐮^𝜆0)∈𝐻^𝑠+1 is such that _𝒯3𝑛^𝜆0𝑑𝑥=1,_𝒯3𝑀𝐮^𝜆0𝑑𝑥=0,_𝑠(𝜆)∶=‖‖𝐮^𝜆0−𝐮^𝐼0‖‖^2𝐻_𝑠+1+1𝜆‖‖𝑛^𝜆0−𝜆Δ𝜙^𝐼0(𝑡)−1‖‖^2𝐻_𝑠⟶0(when𝜆⟶0),𝐮^𝐼0=𝐮^𝐼∣_𝑡=0.(1.4) Then, there exist 𝜆₀ and 𝐶_𝑇 such that for 0<𝜆≤𝜆₀ there is a solution (𝑛^𝜆,𝐮^𝜆)∈([0,𝑇],𝐻^𝑠+1(𝒯³)) of (1.1) satisfying ‖‖𝐮^𝜆(𝑡)−𝐮^𝐼(𝑡)‖‖^2𝐻_𝑠+1+1𝜆‖‖𝑛^𝜆(𝑡)−𝜆Δ𝜙^𝐼−1‖‖^2𝐻_𝑠≤𝐶_𝑇𝜆+𝑀_𝑠(𝜆)(1.5) for any 0≤𝑡≤𝑇.

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 [12–15] 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 𝐻^𝑠 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 set(𝑛,𝐮,𝜙)=𝑛^𝜆−1−𝜆Δ𝜙^𝐼,𝐮^𝜆−𝐮^𝐼,𝜙^𝜆−𝜙^𝐼.(2.1) Then, we know the vector (𝑛,𝐮,𝜙) solves the system 𝜕𝜕_𝑡𝐮+𝐮+𝐮^𝐼⋅∇𝐮+(𝐮⋅∇)𝐮^𝐼=∇𝜙,_𝑡𝑛+𝐮+𝐮^𝐼⋅∇𝑛=−(𝑛+1)dΔiv𝜙𝐮=−𝑛𝜆𝜕_𝑡Δ𝜙^𝐼+divΔ𝜙^𝐼𝐮+𝐮^𝐼,𝜆,(2.2) where ∇𝐮∶∇𝐯=∑^3𝑖,𝑗=1(𝜕_𝑥𝑖𝐮/𝜕_𝑥𝑗)(𝜕_𝑥𝑗𝐯/𝜕_𝑥𝑖). In fact, from (1.3), we get Δ𝜙^𝐼=∇𝐮^𝐼∶∇𝐮^𝐼.

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+𝐮^𝐼⋅∇𝑑=𝑛𝜕𝜆−∇𝐮+2𝐮^𝐼∶∇𝐮,_𝑡𝐜+𝜕_𝑡𝐮+𝐮^𝐼⋅∇𝐜=𝐜⋅∇𝐮+𝐮^𝐼+curl∇𝐮^𝐼⋅𝐮−𝑑𝐜−curl(𝐮⋅∇)𝐮^𝐼,𝑛+𝐮+𝐮^𝐼⋅∇𝑛=−(𝑛+1)𝑑−𝜆𝜕_𝑡Δ𝜙^𝐼+divΔ𝜙^𝐼𝐮+𝐮^𝐼.(2.4) This last system can be written as a singular perturbation of a symmetrizable hyperbolic system: 𝜕_𝑡𝐯+³_𝑗=1𝐮+𝐮^𝐼^𝑗𝜕_𝑥𝑗𝐯=1𝜆𝒦^𝜆𝐯+ℒ(𝐯)+𝒮(𝐯)+𝜆ℛ(𝐯),(2.5) where (𝐮+𝐮^𝐼)^𝑗 denotes the 𝑖th component of (𝐮+𝐮^𝐼) and where 𝒮(𝐯)𝐯==⎛⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎝𝑑𝑛𝐜⎞⎟⎟⎟⎟⎠,𝒦^𝜆=−⎛⎜⎜⎜⎜⎝−∇0𝟎𝜆𝐮𝟎𝟎𝟎+01𝟎2⎞⎟⎟⎟⎟⎠𝐮,_𝐼ℒ(𝐯)=⎛⎜⎜⎜⎜⎝−−0𝑑𝑑𝑛𝐜⎞⎟⎟⎟⎟⎠,𝐜⋅∇𝐮+𝐮_𝐼∶∇𝐮ℛ(𝐯)=⎛⎜⎜⎜⎜⎝−𝜕_𝑡+curl∇𝐮00𝟎^𝐼⋅𝐮−curl(𝐮⋅∇)𝐮^𝐼⎞⎟⎟⎟⎟⎠,Δ𝜙^𝐼+divΔ𝜙^𝐼𝐮+𝐮^𝐼⎞⎟⎟⎟⎟⎠.(2.6)

Now, let us set 𝒜^𝜆0=^{1𝟎0𝟎𝐼}_𝑑^{𝟎0𝟎1/𝜆} and for |𝛼|≤𝑠 with 𝑠>𝑑/2, 𝐸^{𝜆𝛼,𝑠}(𝑡)=12𝒜^𝜆0𝜕^𝛼𝑥𝐯,𝜕^𝛼𝑥𝐯=12‖‖𝜕^𝛼𝑥𝑑‖‖²+‖‖𝜕^𝛼𝑥𝐜‖‖²+1𝜆‖‖𝜕^𝛼𝑥𝐸_𝜆𝑠𝑛‖‖²,(𝑡)=_{|𝛼|≤𝑠}𝐸^{𝜆𝛼,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],𝐻^𝑠), see [17]. This allows us to define 𝑇^𝜆 as the largest time such that 𝐸^𝜆𝑠(𝑡)≤𝑀_𝜆,∀𝑡∈0,𝑇^𝜆,(2.8) where 𝑀_𝜆 which is such that 𝑀_𝜆→0 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 𝑇^𝜆≥𝑇, which will be proved by showing that in (2.8) the equality cannot be reached for 𝑇^𝜆<𝑇 thanks to a good choice of 𝑀_𝜆.

Before performing the energy estimate, we apply the operator 𝜕^𝛼𝑥 for 𝛼∈ℕ³ with |𝛼|≤𝑠 to (2.5), to obtain𝜕_𝑡𝜕^𝛼𝑥𝐯+³_𝑗=1𝐮+𝐮^𝐼^𝑗𝜕_𝑥𝑗𝜕^𝛼𝑥𝐯=1𝜆𝒦^𝜆𝜕^𝛼𝑥𝐯+𝜕^𝛼𝑥ℒ(𝐯)+𝜕^𝛼𝑥𝒮(𝐯)+𝜆𝜕^𝛼𝑥ℛ(𝐯)+Σ^𝜆𝛼,(2.9) where Σ^𝜆𝛼=³_𝑗=1𝐮+𝐮^𝐼^𝑗𝜕_𝑥𝑗𝜕^𝛼𝑥𝐯−𝜕^𝛼𝑥𝐮+𝐮^𝐼^𝑗𝜕_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 𝐻^𝑠 Sobolev norm, and ‖⋅‖_𝑠,∞ is the usual 𝑊^𝑠,∞ 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 𝒜^𝜆0𝜕^𝛼𝑥𝐯. Then, we have 𝑑𝑑𝑡𝐸^{𝜆𝛼,𝑠}(𝑡)=−𝒜^𝜆0𝜕^𝛼𝑥𝐯,³_𝑗=1𝐮+𝐮^𝐼^𝑗+1𝜕_𝑥𝑗𝜕^𝛼𝑥𝐯𝜆𝒜^𝜆0+𝒜𝜕^𝛼𝑥𝐯,𝒦^𝜆𝜕^𝛼𝑥𝐯+𝒜^𝜆0𝜕^𝛼𝑥𝐯,𝜕^𝛼𝑥ℒ(𝐯)_𝜆0=₆𝜕^𝛼𝑥𝐯,𝜕^𝛼𝑥𝒮(𝐮)+𝜆𝒜^𝜆0𝜕^𝛼𝑥𝐯,𝜕^𝛼𝑥ℛ(𝐮)+𝒜^𝜆0𝜕^𝛼𝑥𝐯,Σ^𝜆𝛼_𝑖=1ℐ_𝑖.(2.11)

Let us start the estimate of each term in the above equation. For ℐ₁, since 𝒜^𝜆0 is symmetric and div𝐮^𝐼=0, by Cauchy-Schwartz’s inequality and Sobolev’s lemma, we have that ℐ₁=12div𝐮𝒜^𝜆0𝜕^𝛼𝑥𝐯,𝜕^𝛼𝑥𝐯≤‖div𝐮‖_0,∞𝐸^𝜆𝑠(𝑡)≤𝐶𝐸^𝜆𝑠(𝑡)^3/2.(2.12) Next, since 𝒜^𝜆0𝒦^𝜆 is skew-symmetric, we have that ℐ₂=0.(2.13) For ℐ₃, by a direct calculation, one gets ℐ₃=−𝜕^𝛼𝑥𝐜,𝜕^𝛼𝑥(𝑑𝐜)−1𝜆𝑛,𝜕^𝛼𝑥≤‖‖𝜕(𝑑𝑛)_𝛼𝑥𝐜‖‖‖‖𝜕^𝛼𝑥(𝑑𝐜)‖‖+1𝜆‖‖𝜕^𝛼𝑥≤𝐶𝐸_𝜆𝑠𝑛‖‖²‖‖𝜕^𝛼𝑥𝑑‖‖(𝑡)^3/2.(2.14) Here, we have used the basic Moser-type calculus inequalities [18].

To give the estimate of the term ℐ₄, we split it in two terms. Specifically, we can deduce that ℐ₄=+−𝜕𝜕^𝛼𝑥𝑑,𝜕^𝛼𝑥∇𝐮+2𝐮^𝐼∶∇𝐮_𝛼𝑥≤‖𝑑‖_𝑠𝐜,𝜕^𝛼𝑥𝐜⋅∇𝐮+𝐮^𝐼+curl∇𝐮^𝐼⋅𝐮−curl(𝐮⋅∇)𝐮^𝐼‖‖∇𝐮+2𝐮^𝐼∶∇𝐮‖‖_𝑠≤+𝐶‖𝐜𝐸‖_𝑠𝜆𝑠‖‖𝐜⋅∇𝐮+𝐮^𝐼‖‖_𝑠+‖‖curl∇𝐮^𝐼⋅𝐮‖‖_𝑠+‖‖curl(𝐮⋅∇)𝐮^𝐼‖‖_𝑠+𝐸^𝜆𝑠^3/2.(2.15) Here, we have used the curl-div decomposition inequality ‖∇𝐮‖_𝑠≤𝐶‖𝑑‖_𝑠+‖𝐜‖_𝑠.(2.16)

For ℐ₅, we have that ℐ₅≤‖𝑛‖_𝑠‖‖𝜕_𝑡Δ𝜙^𝐼+divΔ𝜙^𝐼𝐮+𝐮^𝐼‖‖_𝑠≤≤𝐶𝐶𝜆‖𝑛+‖_𝑠𝐸_𝜆𝑠1+‖𝑑‖_𝑠+‖𝐜‖_𝑠.(2.17)

To estimate the last term, that is, ℐ₅, by using basic Moser-type calculus inequalities and Sobolev’s lemma, we have ℐ₆=𝜕^𝛼𝑥𝑑,³_𝑗=1𝐮+𝐮^𝐼^𝑗+𝜕𝜕_𝑥𝑗𝜕^𝛼𝑥𝑑−𝜕^𝛼𝑥𝐮+𝐮^𝐼^𝑗𝜕_𝑥𝑗𝑑_𝛼𝑥𝐜,³_𝑗=1𝐮+𝐮^𝐼^𝑗+1𝜕_𝑥𝑗𝜕^𝛼𝑥𝐜−𝜕^𝛼𝑥𝐮+𝐮^𝐼^𝑗𝜕_𝑥𝑗𝐜𝜆𝜕^𝛼𝑥𝑛,³_𝑗=1𝐮+𝐮^𝐼^𝑗≤𝐶‖𝑑‖_𝑠𝜕_𝑥𝑗𝜕^𝛼𝑥𝑛−𝜕^𝛼𝑥𝐮+𝐮^𝐼^𝑗𝜕_𝑥𝑗𝑛‖‖∇𝐮+𝐮^𝐼‖‖_0,∞‖‖𝜕^𝛼𝑥𝑑‖‖+‖∇𝑑‖_0,∞+𝐶‖𝐜‖_𝑠‖‖𝜕^𝛼𝑥𝐮+𝐮^𝐼‖‖‖‖∇𝐮+𝐮^𝐼‖‖_0,∞‖‖𝜕^𝛼𝑥𝐜‖‖+‖∇𝐜‖_0,∞+𝐶‖‖𝜕^𝛼𝑥𝐮+𝐮^𝐼‖‖𝜆‖𝑛‖^𝑠‖‖∇𝐮+𝐮^𝐼‖‖_0,∞≤𝐶‖𝑑‖_2𝑠‖‖𝜕^𝛼𝑥𝑛‖‖+‖∇𝑛‖_0,∞‖‖𝜕^𝛼𝑥𝐮+𝐮^𝐼‖‖+‖𝐜‖^2𝑠+1𝜆‖𝑛‖^2𝑠‖𝑑‖^𝑠≤𝐶𝐸_𝜆𝑠+‖𝐜‖_𝑠+1+𝐸^𝜆𝑠^3/2.(2.18)

Now, we collect all the previous estimates (2.12)–(2.18) and we sum over 𝛼 to find 𝑑𝑑𝑡𝐸^𝜆𝑠≤𝐶𝜆+𝐶𝐸^𝜆𝑠+𝐸^𝜆𝑠².(2.19) By using (2.8), we get with 𝑀_𝜆≪1 that 𝑑𝑑𝑡𝐸^𝜆𝑠≤𝐶𝜆+𝐶𝐸^𝜆𝑠,∀𝑡∈0,𝑇^𝜆.(2.20) Hence, by the Gronwall inequality, we get that 𝐸^𝜆𝑠(𝑡)≤𝑀_𝑠(𝜆)+𝐶𝑡𝜆𝑒^𝐶𝑡,∀𝑡∈0,𝑇^𝜆.(2.21)

Consequently, if we choose 𝑀_𝜆=(𝑀_𝑠(𝜆)+𝐶𝑡𝜆)^1/2, we see that we cannot reach equality in (2.8) for 𝑇^𝜆<𝑇. This proves that 𝑇^𝜆>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.