On a Quasi-neutral Approximation to the Incompressible Euler Equations

Copyright q 2012 J. Yang and Z. 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. 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.


Introduction
In this paper, we shall consider the following hydrodynamic system: for x ∈ T 3 and t > 0, subject to the initial conditions n λ , u λ t 0 n λ 0 , u λ 0 1.2 for x ∈ T 3 .In the above equations, T 3 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 T 3 n λ dx 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 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 T 3 and T 3 u I dx 0 for s > 5/2.Assume that the initial value n λ 0 , u λ 0 ∈ H s 1 is such that

1.4
Then, there exist λ 0 and C T such that for 0 < λ ≤ λ 0 there is a solution n λ , u λ ∈ 0, T , H s 1 T 3 of 1.1 satisfying 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 driftdiffusion 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 wellprepared 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 .

Proof of Theorem 1.1
First, let us set Then, we know the vector n, u, φ solves the system where ∇u : In fact, from 1.3 , we get Δφ I ∇u I : ∇u I .As in 16 , we make the following change of unknowns: By using the last equation in 2.2 , we get the following system: 2.4 This last system can be written as a singular perturbation of a symmetrizable hyperbolic system: where u u I j denotes the ith component of u u I and where  E λ α,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 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 ∂ α x for α ∈ N 3 with |α| ≤ s to 2.5 , to obtain where 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 L 2 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 λ 0 ∂ α x v.Then, we have

2.11
Let us start the estimate of each term in the above equation.For I 1 , since A λ 0 is symmetric and div u I 0, by Cauchy-Schwartz's inequality and Sobolev's lemma, we have that

2.12
Next, since A λ 0 K λ is skew-symmetric, we have that

2.13
For I 3 , by a direct calculation, one gets

2.14
Here, we have used the basic Moser-type calculus inequalities 18 .
To give the estimate of the term I 4 , we split it in two terms.Specifically, we can deduce that

2.15
Here, we have used the curl-div decomposition inequality

2.16
For I 5 , we have that

2.17
To estimate the last term, that is, I 5 , by using basic Moser-type calculus inequalities and Sobolev's lemma, we have 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 .