# On a Quasi-Neutral Approximation to the Incompressible Euler Equations

Authors: Jianwei Yang; Zhitao Zhuang
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 1. Introduction

In this paper, we shall consider the following hydrodynamic system:(1.1)∂tnλ+div⁡(nλuλ)=0,x∈T3,t>0,∂tuλ+(uλ⋅∇)uλ=∇ϕλ,x∈T3,t>0,Δϕλ=nλ-1λ,x∈T3,t>0
for x∈𝒯3 and t>0, subject to the initial conditions
(1.2)(nλ,uλ)(t=0)=(n0λ,u0λ)
for x∈𝒯3. 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=(nλ-1)/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
(1.3)∂tuI+(uI⋅∇)uI=∇ϕI,div⁡uI=0.
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.
LetuI 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 (n0λ,u0λ)∈Hs+1 is such that
(1.4)∫T3n0λdx=1,∫T3u0λdx=0,Ms(λ)∶=‖u0λ-u0I‖Hs+12+1λ‖n0λ-λΔϕ0I(t)-1‖Hs2⟶0(whenλ⟶0),u0I=uI∣t=0.
Then, there exist λ0 and CT such that for 0<λ≤λ0 there is a solution (nλ,uλ)∈([0,T],Hs+1(𝒯3)) of (1.1) satisfying
(1.5)‖uλ(t)-uI(t)‖Hs+12+1λ‖nλ(t)-λΔϕI-1‖Hs2≤CT(λ+Ms(λ))
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 strongHs 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 Theorem1.1

First, let us set(2.1)(n,u,ϕ)=(nλ-1-λΔϕI,uλ-uI,ϕλ-ϕI).
Then, we know the vector (n,u,ϕ) solves the system
(2.2)∂tu+(u+uI)⋅∇u+(u⋅∇)uI=∇ϕ,∂tn+(u+uI)⋅∇n=-(n+1)div⁡u-λ(∂tΔϕI+div⁡(ΔϕI(u+uI))),Δϕ=nλ,
where ∇u:∇v=∑i,j=13(∂xiu/∂xj)(∂xjv/∂xi). In fact, from (1.3), we get ΔϕI=∇uI:∇uI.As in [16], we make the following change of unknowns:
(2.3)(d,c)=(div⁡u,curl⁡u).
By using the last equation in (2.2), we get the following system:
(2.4)∂td+(u+uI)⋅∇d=nλ-∇(u+2uI):∇u,∂tc+(u+uI)⋅∇c=c⋅∇(u+uI)+curl⁡(∇uI⋅u)-dc-curl⁡((u⋅∇)uI),∂tn+(u+uI)⋅∇n=-(n+1)d-λ(∂tΔϕI+div⁡(ΔϕI(u+uI))).
This last system can be written as a singular perturbation of a symmetrizable hyperbolic system:
(2.5)∂tv+∑j=13(u+uI)j∂xjv=1λKλv+L(v)+S(v)+λR(v),
where (u+uI)j denotes the ith component of (u+uI) and where
(2.6)v=(dcn),Kλ=(001000-λ00),L(v)=(0-dc-dn),S(v)=(-∇(u+2uI):∇uc⋅∇(u+uI)+curl⁡(∇uI⋅u)-curl⁡((u⋅∇)uI)0),R(v)=(00-(∂tΔϕI+div⁡(ΔϕI(u+uI)))).Now, let us set𝒜0λ=(1000Id0001/λ) and for |α|≤s with s>d/2,
(2.7)Eα,sλ(t)=12(A0λ∂xαv,∂xαv)=12(‖∂xαd‖2+‖∂xαc‖2+1λ‖∂xαn‖2),Esλ(t)=∑|α|≤sEα,sλ(t).
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
(2.8)Esλ(t)≤Mλ,∀t∈[0,Tλ],
where Mλ which is such that Mλ→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 Tλ≥T, 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∂xα for α∈ℕ3 with |α|≤s to (2.5), to obtain(2.9)∂t∂xαv+∑j=13(u+uI)j∂xj∂xαv=1λKλ∂xαv+∂xαL(v)+∂xαS(v)+λ∂xαR(v)+Σαλ,
where
(2.10)Σαλ=∑j=13[(u+uI)j∂xj∂xαv-∂xα((u+uI)j∂xjv)].Along the proof, we shall denote byC 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 𝒜0λ∂xαv. Then, we have
(2.11)ddtEα,sλ(t)=-(A0λ∂xαv,∑j=13(u+uI)j∂xj∂xαv)+1λ(A0λ∂xαv,Kλ∂xαv)+(A0λ∂xαv,∂xαL(v))+(A0λ∂xαv,∂xαS(u))+λ(A0λ∂xαv,∂xαR(u))+(A0λ∂xαv,Σαλ)=∑i=16Ii.Let us start the estimate of each term in the above equation. Forℐ1, since 𝒜0λ is symmetric and divuI=0, by Cauchy-Schwartz’s inequality and Sobolev’s lemma, we have that
(2.12)I1=12(div⁡uA0λ∂xαv,∂xαv)≤‖div⁡u‖0,∞Esλ(t)≤C(Esλ(t))3/2.
Next, since 𝒜0λ𝒦λ is skew-symmetric, we have that
(2.13)I2=0.
For ℐ3, by a direct calculation, one gets
(2.14)I3=-(∂xαc,∂xα(dc))-1λ(n,∂xα(dn))≤‖∂xαc‖‖∂xα(dc)‖+1λ‖∂xαn‖2‖∂xαd‖≤C(Esλ(t))3/2.
Here, we have used the basic Moser-type calculus inequalities [18].To give the estimate of the termℐ4, we split it in two terms. Specifically, we can deduce that
(2.15)I4=-(∂xαd,∂xα(∇(u+2uI):∇u))+(∂xαc,∂xα(c⋅∇(u+uI)+curl⁡(∇uI⋅u)-curl⁡((u⋅∇)uI)))≤‖d‖s‖∇(u+2uI):∇u‖s+‖c‖s(‖c⋅∇(u+uI)‖s+‖curl⁡(∇uI⋅u)‖s+‖curl⁡((u⋅∇)uI)‖s)≤C(Esλ+(Esλ)3/2).
Here, we have used the curl-div decomposition inequality
(2.16)‖∇u‖s≤C(‖d‖s+‖c‖s).Forℐ5, we have that
(2.17)I5≤‖n‖s‖∂tΔϕI+div⁡(ΔϕI(u+uI))‖s≤C‖n‖s(1+‖d‖s+‖c‖s)≤Cλ+Esλ.To estimate the last term, that is,ℐ5, by using basic Moser-type calculus inequalities and Sobolev’s lemma, we have
(2.18)I6=(∂xαd,∑j=13[(u+uI)j∂xj∂xαd-∂xα((u+uI)j∂xjd)])+(∂xαc,∑j=13[(u+uI)j∂xj∂xαc-∂xα((u+uI)j∂xjc)])+1λ(∂xαn,∑j=13[(u+uI)j∂xj∂xαn-∂xα((u+uI)j∂xjn)])≤C‖d‖s(‖∇(u+uI)‖0,∞‖∂xαd‖+‖∇d‖0,∞‖∂xα(u+uI)‖)+C‖c‖s(‖∇(u+uI)‖0,∞‖∂xαc‖+‖∇c‖0,∞‖∂xα(u+uI)‖)+Cλ‖n‖s(‖∇(u+uI)‖0,∞‖∂xαn‖+‖∇n‖0,∞‖∂xα(u+uI)‖)≤C(‖d‖s2+‖c‖s2+1λ‖n‖s2)(‖d‖s+‖c‖s+1)≤C(Esλ+(Esλ)3/2).Now, we collect all the previous estimates (2.12)–(2.18) and we sum over α to find
(2.19)ddtEsλ≤Cλ+CEsλ+(Esλ)2.
By using (2.8), we get with Mλ≪1 that
(2.20)ddtEsλ≤Cλ+CEsλ,∀t∈[0,Tλ].
Hence, by the Gronwall inequality, we get that
(2.21)Esλ(t)≤(Ms(λ)+Ctλ)eCt,∀t∈[0,Tλ].Consequently, if we chooseMλ=(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].



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*Source: 957185-2012-06-06.xml*