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

Authors: Zhiqiang Wei; Jianwei Yang
Journal: ISRN Applied Mathematics
 (2012)

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## Abstract

This paper considers a pressureless Euler-Poisson system with viscosity in plasma physics in the torusT3. We give a rigorous justification of its asymptotic limit toward the incompressible Navier Stokes equations via quasi-neutral regime using the modulated energy method.


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## Body

## 1. Introduction

We will consider the following system:(1.1)∂tuϵ+(uϵ·∇)uϵ=μΔuϵ+∇𝒱ϵ,∂tnϵ+div(nϵuϵ)=0,Δ𝒱ϵ=nϵ-1ϵ2,
for x∈T3 and  t>0, nϵ∈ℝ,  uϵ∈ℝ2. ϵ is small parameter and μ>0 is a constant viscosity coefficient. To solve uniquely the Poisson equation, we add the ∫T3‍nϵdx=1. Passing to the limit when  ϵ→0,  it is easy to see, at least at a very formal level, that (nϵ,uϵ) tends to (nNS,uNS),  where  nNS=1 and
(1.2)∂tuNS+(uNS·∇)uNS=μΔuNS+∇𝒱NS,divuNS=0.In other words,uNS 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 div-curl for studying incompressible fluids. And for that, we assume thatnϵ(x,·) has total mass equal to 1 and the mean values of uϵ vanish, that is, m(uϵ)=(1/(2π)3)∫T3uϵdx=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 Section2 we state our main result. Estimates and proofs are given in Section 3.


## 2. Main Result

Throughout the paper, we will denote byC a number independent of ϵ, which actually may change from line to line. 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.The study of the asymptotic behavior of the sequence(uϵ,nϵ), as ϵ goes to zero, leads to the statement of our main result.Theorem 2.1.
LetuNS be a solution of the incompressible Euler equations (1.2) such that uNS∈([0,T],Hs+3(T3)) and ∫T3uNSdx=0 for  s>(5/2). Assume that (n0ϵ,u0ϵ) be a sequence of initial data such that ∫T3n0ϵdx=1,∫T3u0εdx=0 and
(2.1)∥u0ϵ-u0NS∥s+1≤Cϵ,∥n0ϵ-1∥s≤Cϵ2
with  u0NS=uNS∣t=0. Then there is a sequence (nϵ,uϵ)∈C([0,T],Hs×Hs+1(T3)) of solutions to (1.1) with initial data  (n0ϵ,u0ϵ)  belonging to C([0,Tϵ],Hs×Hs+1(T3)) with liminfϵ→0Tϵ≥T. Moreover for any T1<T and ϵ small enough,
(2.2)∥uϵ(t)-uNS(t)∥s≤Cϵ,∥nϵ(t)-1∥s≤Cϵ2,
for any  0≤t≤T1.


## 3. Proof of the Theorem

If(uϵ,nϵ) is a solution to system (1.1), we introduce
(3.1)uϵ=uNS+ϵu,nϵ=1+ϵ2(n+Δ𝒱NS),𝒱ϵ=𝒱NS+𝒱.
Since the pressure 𝒱NS in the incompressible Navier-Stokes equation is given by
(3.2)Δ𝒱NS=∇uNS:∇uNS,
where,  ∇u:∇v=∑i,j=13(∂xiu/∂xj)(∂xjv/∂xi). Then the vector (u1,n1,𝒱1) solves the system
(3.3)∂tu+uNS·∇u=∇𝒱ϵ-ϵ(u·∇)u-(u·∇)uNS+μΔu,∂tn+uNS·∇n=-divuϵ-ϵdiv((n+Δ𝒱NS)u)-∂tΔ𝒱NS-uNS·∇Δ𝒱NS,Δ𝒱=n.
As in [16], we make the following change of unknowns:
(3.4)d=divu,c=curlu.
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)∂td+uNS·∇d=nϵ-ϵ(u·∇)d-ϵ∇u:∇u-∇u:∇uNS+μΔd,∂tc+uNS·∇c=-ϵ(u·∇)c-ϵ(c·∇)u-(c·∇)uNS+curl(∇uNS·u-u·∇uNS)-ϵdc+μΔd,∂tn+uNS·∇n=-dϵ-ϵ(u·∇)n-ϵ(n+Δ𝒱NS)d-u·∇Δ𝒱NS-(∂t+uNS·∇)Δ𝒱NS.
This last system can be written as a singular perturbation of a quasilinear symmetrizable hyperbolic system. Setting  𝕎ϵ=(d,c,n)T  yields
(3.6)∂t𝕎ϵ+A(t,x,∂x)𝕎ϵ=1ϵK𝕎ϵ-ϵB(t,x,∂x)𝕎ϵ+S(𝕎ϵ)+μN(𝕎ϵ)+R,
where
(3.7)A(t,x,∂x)=diag(uNS·∇,uNS·∇I3,uNS·∇),B(t,x,∂x)=diag(u·∇,u·∇I3,u·∇),K=(001000-100),N(𝕎ϵ)=(ΔdΔc0),R=(00-(∂t+uNS·∇)Δ𝒱NS),S(𝕎ϵ)=(-ϵ∇u:∇u-∇u:∇uNS-ϵ(c·∇)u-(c·∇)uNS+curl(∇uNS·u-u·∇uNS)-ϵdc-ϵ(n+Δ𝒱NS)d-u·∇Δ𝒱NS).
For |α|≤s with   s>d/2, we set
(3.8)ℰα,sλ(t)=12(∥∂xαd∥2+∥∂xαc∥2+∥∂xαn∥2),ℰsλ(t)=∑|α|≤sℰα,sλ(t).Before performing the energy estimate, we apply the operator∂xα for α∈ℕ3 with |α|≤s to (3.6), to obtain
(3.9)∂t∂xα𝕎ϵ+A(t,x,∂x)∂xα𝕎ϵ=1ϵK∂xα𝕎ϵ-ϵB(t,x,∂x)∂xα𝕎ϵ+∂xαS(𝕎ϵ)+μ∂xαN(𝕎ϵ)+[∂xα,A(t,x,∂x)]𝕎ϵ-ϵ[∂xα,B(t,x,∂x)]𝕎ϵ+∂xαR.
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  ∂xα𝕎ϵ.Let us start the estimate of each term. First, sinceA(t,x,∂x) is symmetric and divuNS=0, we have that
(3.10)(A(t,x,∂x)∂xα𝕎ϵ,∂xα𝕎ϵ)=-∫T3divuNS(|∂xαd|2+|∂xαc|2+|∂xαn|2)dx=0.
Next, since K is skew-symmetric, we have that
(3.11)1ϵ(K∂xα𝕎ϵ,∂xα𝕎ϵ)=0.
By integration by parts, we have
(3.12)-ϵ(B(t,x,∂x)∂xα𝕎ϵ,∂xα𝕎ϵ)=ϵ∫T3divu|∂xα𝕎ϵ|2dx≤∥divu∥0,∞ℰsϵ(t)≤(ℰsϵ(t))3/2.
For later estimates in this paper, we recall some results on Moser-type calculus inequalities in Sobolev spaces [17, 18].Lemma 3.1.
Lets≥1 be an integer. Suppose u∈Hs(T3),∇u∈L∞(T3), and v∈Hs-1(T3)∩L∞(T3). Then for all multi-indexes |α|≤s, one has (∂xα(uv)-u∂xαv)∈L2(T3) and
(3.13)∥∂xα(uv)-u∂xαv∥≤Cs(∥∇u∥0,∞∥D|α|-1v∥+∥D|α|u∥∥v∥0,∞),
where
(3.14)∥Dhu∥=∑|α|=h∥∂xαu∥,∀h∈ℕ.
Moreover, if  s≥3, then the embedding  Hs-1(T3)↪L∞(T3) is continuous and one has
(3.15)∥uv∥s-1≤Cs∥u∥s-1∥v∥s-1,∥∂xα(uv)-u∂xαv∥≤Cs∥u∥s∥v∥s-1.By using basic Moser-type calculus inequalities and Sobolev’s lemma, we have(3.16)(∂xαS(𝕎ϵ),∂xα𝕎ϵ)≤Cℰsϵ(t)+Cϵ(ℰsϵ(t))3/2.
After a a direct calculation, one gets
(3.17)μ(∂xαN(𝕎ϵ),∂xα𝕎ϵ)=-μ∫T3(|∇∂xαd|2+|∇∂xαc|2)dx.
To estimate the commutator, we have
(3.18)([∂xα,A(t,x,∂x)]𝕎ϵ,∂xα𝕎ϵ)=∫([∂xα,uNS·∇]d∂xαd+[∂xα,uNS·∇]c∂xαc+[∂xα,uNS·∇]n∂xαn)dx≤C(∥uNS∥s∥∇d∥0,∞+∥uNS∥0,∞∥∇d∥s-1)∥d∥s+C(∥uNS∥s∥∇c∥0,∞+∥uNS∥0,∞∥∇c∥s-1)∥c∥s+C(∥uNS∥s∥∇n∥0,∞+∥uNS∥0,∞∥∇n∥s-1)∥n∥s≤Cℰs(t).
Also, we have
(3.19)-ϵ([∂xα,B(t,x,∂x)]𝕎ϵ,∂xα𝕎ϵ)=-ϵ∫([∂x,u·∇]d∂xαd+[∂x,u·∇]c∂xαc+[∂x,u·∇]n∂xαn)dx≤Cϵ(∥u∥s∥∇d∥0,∞+∥u∥0,∞∥∇d∥s-1)∥d∥s+Cϵ(∥u∥s∥∇c∥0,∞+∥u∥0,∞∥∇c∥s-1)∥c∥s+Cϵ(∥u∥s∥∇n∥0,∞+∥u∥0,∞∥∇n∥s-1)∥n∥s≤Cϵ(ℰsϵ(t))3/2.
Here, we have used the inequality
(3.20)∥u∥s≤C∥∇u∥s-1≤C(∥d∥s-1+∥c∥s-1).
Finally, the Young inequality gives
(3.21)(∂xαR,∂xα𝕎ϵ)≤∥∂tΔ𝒱NS+uNS·∇Δ𝒱NS∥s∥n∥s≤C(1+ℰsϵ(t)).
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
(3.22)ddtℰsϵ(t)≤C(1+ℰsϵ(t)+ε(Esϵ)3/2(t)).
We can conclude using a standard Gronwall's lemma, that if the solution (uNS,𝒱NS) of Navier-Stokes equations (1.2) is smooth on the time interval [0,T], for any  T1<T there exists ϵ0 such that the sequence  (𝕎ϵ)ϵ<ϵ0 is bounded in  C([0,T1],Hs(T3)). Then we have
(3.23)𝕎ϵ=(divu,curlu,n),uϵ=uNS+ϵu,nϵ=1+ϵ2(n+Δ𝒱NS).
The assumptions that we have made on the initial data imply that  (1/ϵ)(uϵ-uNS),(1/ϵ2)(n-1) is bounded. This proves Theorem 2.1.



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