On a Quasi-Neutral Approximation of the Incompressible Navier-Stokes Equations

This paper considers a pressureless Euler-Poisson system with viscosity in plasma physics in the torus T 3 . We give a rigorous justiﬁcation of its asymptotic limit toward the incompressible Navier Stokes equations via quasi-neutral regime using the modulated energy method.


Introduction
We will consider the following system: for x ∈ T 3 and t > 0, n ∈ R, u ∈ R 2 . is small parameter and μ > 0 is a constant viscosity coefficient. To solve uniquely the Poisson equation, we add the T 3 n dx 1. Passing to the limit when → 0, it is easy to see, at least at a very formal level, that n , u tends to n NS , u NS , where n NS 1 and ∂ t u NS u NS · ∇ u NS μΔu NS ∇V NS , div u NS 0.

2 ISRN Applied Mathematics
In other words, u NS is a solution of the incompressible Navier-Stokes equations. The aim of this paper is to give a rigorous justification to this formal computation.
The Euler-Poisson system with viscosity 1.1 is a physical model involving dissipation see 1 , which here could be regarded as a viscous approximation of Euler-Poisson. Formally, it is a kind of new approximation of the incompressible Navier-Stokes equations of viscous fluid in real world.
It should be pointed out that there have been a lot of interesting results about the topic on the quasi-neutral or called zero-Debye length limit, for the readers to see 2-5 for isentropic Euler-Poisson system, 6, 7 for nonisentropic Euler-Poisson system, 8-10 for Vlasov-Poisson system, 11, 12 for drift-diffusion system, 13 for Euler-Maxwell equations, and therein references. We also mention that the above limit has been studied in 14,15 . But in this present paper, the convergence result and the method of its proof is different from that of 14, 15 . The main focus in this paper is on the use of modulated energy techniques and divcurl for studying incompressible fluids. And for that, we assume that n x, · has total mass equal to 1 and the mean values of u vanish, that is, m u 1/ 2π 3 T 3 u dx 0. We also restrict ourselves to the case of well-prepared initial data and the case of periodic torus. Indeed, the quasi-neutral limit is much more difficult without these assumptions.
In this note, we will use some inequalities in Sobolev spaces, such as basic Moser-type calculus inequalities, Young inequality, and Gronwall inequality.
The paper is organized as follows. In Section 2 we state our main result. Estimates and proofs are given in Section 3.

Main Result
Throughout the paper, we will denote by C a number independent of , which actually may change from line to line. 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.
The study of the asymptotic behavior of the sequence u , n , as goes to zero, leads to the statement of our main result. Theorem 2.1. Let u NS be a solution of the incompressible Euler equations 1.2 such that u NS ∈ 0, T , H s 3 T 3 and T 3 u NS dx 0 for s > 5/2 . Assume that n 0 , u 0 be a sequence of initial data such that T 3 n 0 dx 1, T 3 u ε 0 dx 0 and

Proof of the Theorem
If u , n is a solution to system 1.1 , we introduce u u NS u,

3.1
Since the pressure V NS in the incompressible Navier-Stokes equation is given by ΔV n.

3.3
As in 16 , we make the following change of unknowns: d div u, c curl u.

3.4
By using the last equation and taking the curl and the divergence of the first equation in 3.5 , we get the following system: 3.5 This last system can be written as a singular perturbation of a quasilinear symmetrizable hyperbolic system. Setting W d, c, n T yields

3.8
Before performing the energy estimate, we apply the operator ∂ α x for α ∈ N 3 with |α| ≤ s to 3.6 , to obtain

3.9
Now, we proceed to perform the energy estimates for 3.9 in a classical way by taking the scalar product of system 3.9 with ∂ α x W . Let us start the estimate of each term. First, since A t, x, ∂ x is symmetric and div u NS 0, we have that Next, since K is skew-symmetric, we have that By integration by parts, we have For later estimates in this paper, we recall some results on Moser-type calculus inequalities in Sobolev spaces 17, 18 .
Lemma 3.1. Let s ≥ 1 be an integer. Suppose u ∈ H s T 3 , ∇u ∈ L ∞ T 3 , and v ∈ H s−1 T 3 ∩ L ∞ T 3 . Then for all multi-indexes |α| ≤ s, one has ∂ α x uv − u∂ α x v ∈ L 2 T 3 and 3.14 Moreover, if s ≥ 3, then the embedding H s−1 T 3 → L ∞ T 3 is continuous and one has By using basic Moser-type calculus inequalities and Sobolev's lemma, we have After a a direct calculation, one gets To estimate the commutator, we have

3.19
Here, we have used the inequality Finally, the Young inequality gives Notice that, to get the last line, we have used 3.2 . Now, we collect all the previous estimates 3.10 -3.21 and we sum over α to find d dt E s t ≤ C 1 E s t ε E s 3/2 t .

3.22
We can conclude using a standard Gronwall's lemma, that if the solution u NS , V NS of Navier-Stokes equations 1.2 is smooth on the time interval 0, T , for any T 1 < T there exists 0 such that the sequence W < 0 is bounded in C 0, T 1 , H s T 3 . Then we have W div u, curl u, n , u u NS u, n 1 2 n ΔV NS .

3.23
The assumptions that we have made on the initial data imply that 1/ u −u NS , 1/ 2 n−1 is bounded. This proves Theorem 2.1.