On a Quasi-Neutral Approximation of the …downloads.hindawi.com/journals/isrn.applied.mathematics/...1 P. Degond, “Mathematical modelling of microelectronics semiconductor devices,”

International Scholarly Research NetworkISRN Applied MathematicsVolume 2012, Article ID 581710, 7 pagesdoi:10.5402/2012/581710

Research ArticleOn a Quasi-Neutral Approximation ofthe Incompressible Navier-Stokes Equations

Zhiqiang Wei and Jianwei Yang

College of Mathematics and Information Science, North China University ofWater Resources and Electric Power, Zhengzhou 450011, China

Correspondence should be addressed to Jianwei Yang, [email protected]

Received 21 June 2012; Accepted 10 October 2012

Academic Editors: R. Cardoso and X.-S. Yang

Copyright q 2012 Z. Wei and J. Yang. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

This paper considers a pressureless Euler-Poisson system with viscosity in plasma physics in thetorus T3. We give a rigorous justification of its asymptotic limit toward the incompressible NavierStokes equations via quasi-neutral regime using the modulated energy method.

1. Introduction

We will consider the following system:

∂tuε + (uε · ∇)uε = μΔuε +∇Vε,

∂tnε + div (nεuε) = 0,

ΔVε =nε − 1ε2

,

(1.1)

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∫T3 n

εdx = 1. Passing to thelimit 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

∂tuNS +(uNS · ∇

)uNS = μΔuNS +∇VNS,

div uNS = 0.(1.2)

2 ISRN Applied Mathematics

In other words, uNS is a solution of the incompressible Navier-Stokes equations. Theaim of this paper is to give a rigorous justification to this formal computation.

The Euler-Poisson systemwith viscosity (1.1) is a physical model involving dissipationsee [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 viscousfluid in real world.

It should be pointed out that there have been a lot of interesting results about the topicon the quasi-neutral (or called zero-Debye length) limit, for the readers to see [2–5] for isen-tropic 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, andtherein references. We also mention that the above limit has been studied in [14, 15]. But inthis present paper, the convergence result and the method of its proof is different from thatof [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 that nε(x, ·) has total massequal to 1 and the mean values of uε vanish, that is, m(uε) = (1/(2π)3)

∫T3 u

εdx = 0. Wealso 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-typecalculus inequalities, Young inequality, and Gronwall inequality.

The paper is organized as follows. In Section 2 we state our main result. Estimates andproofs are given in Section 3.

2. Main Result

Throughout the paper, we will denote by C a number independent of ε, which actually maychange from line to line. Moreover (·, ·) and ‖ · ‖ stand for the usual L2 scalar product andnorm, ‖ · ‖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, leadsto the statement of our main result.

Theorem 2.1. Let uNS be a solution of the incompressible Euler equations (1.2) such that uNS ∈([0, T],Hs+3(T3)) and

∫T3 uNSdx = 0 for s > (5/2). Assume that (nε

0,uε0) be a sequence of initial

data such that∫T3 n

ε0 dx = 1,

∫T3 uε

0 dx = 0 and∥∥∥uε

0 − uNS0

∥∥∥s+1

≤ Cε,

∥∥nε0 − 1

∥∥s ≤ Cε2

(2.1)

with uNS0 = 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 (nε

0,uε0) belonging to C([0, Tε],Hs ×Hs+1(T3)) with lim infε→ 0Tε ≥ T . Moreover

for any T1 < T and ε small enough,

∥∥∥uε(t) − uNS(t)∥∥∥s≤ Cε,

‖nε(t) − 1‖s ≤ Cε2,

(2.2)

for any 0 ≤ t ≤ T1.

ISRN Applied Mathematics 3

3. Proof of the Theorem

If (uε, nε) is a solution to system (1.1), we introduce

uε = uNS + εu,

nε = 1 + ε2(n + ΔVNS

),

Vε = VNS + V.

(3.1)

Since the pressure VNS in the incompressible Navier-Stokes equation is given by

ΔVNS = ∇uNS : ∇uNS, (3.2)

where, ∇u : ∇v =∑3

i,j=1(∂xiu/∂xj )(∂xjv/∂xi). Then the vector (u1, n1,V1) solves the system

∂tu + uNS · ∇u =∇Vε

− ε(u · ∇)u − (u · ∇)uNS + μΔu,

∂tn + uNS · ∇n = − divuε

− εdiv((

n + ΔVNS)u)− ∂tΔVNS − uNS · ∇ΔVNS,

ΔV = n.

(3.3)

As in [16], we make the following change of unknowns:

d = divu, c = curlu. (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:

∂td + uNS · ∇d =n

ε− 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.

(3.5)

This last system can be written as a singular perturbation of a quasilinear symmetrizablehyperbolic system. Setting W

ε = (d, c, n)T yields

∂tWε +A(t, x, ∂x)Wε =

1εKW

ε − εB(t, x, ∂x)Wε + S(Wε) + μN(Wε) + R, (3.6)


where

A(t, x, ∂x) = diag(uNS · ∇,uNS · ∇I3,uNS · ∇

), B(t, x, ∂x) = diag(u · ∇,u · ∇I3,u · ∇),

K =

⎛

⎝0 0 10 0 0−1 0 0

⎞

⎠, 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

⎞

⎠.

(3.7)

For |α| ≤ s with s > d/2, we set

Eλα,s(t) =

12

(‖∂αxd‖2 + ‖∂αxc‖2 + ‖∂αxn‖2

),

Eλs(t) =

∑

|α|≤sEλα,s(t).

(3.8)

Before performing the energy estimate, we apply the operator ∂αx for α ∈ N3 with |α| ≤ s

to (3.6), to obtain

∂t∂αxW

ε +A(t, x, ∂x)∂αxWε =

1εK∂αxW

ε − εB(t, x, ∂x)∂αxWε + ∂αxS(W

ε) + μ∂αxN(Wε)

+ [∂αx,A(t, x, ∂x)]Wε − ε[∂αx, B(t, x, ∂x)]Wε + ∂αxR.

(3.9)

Now, we proceed to perform the energy estimates for (3.9) in a classical way by taking thescalar product of system (3.9) with ∂αxW

ε.Let us start the estimate of each term. First, sinceA(t, x, ∂x) is symmetric and divuNS =

0, we have that

(A(t, x, ∂x)∂αxWε, ∂αxW

ε) = −∫

T3divuNS

(|∂αxd|2 + |∂αxc|2 + |∂αxn|2

)dx = 0. (3.10)

Next, since K is skew-symmetric, we have that

1ε(K∂αxW

ε, ∂αxWε) = 0. (3.11)

By integration by parts, we have

−ε(B(t, x, ∂x)∂αxWε, ∂αxW

ε) = ε

∫

T3divu|∂αxW

ε|2dx ≤ ‖divu‖0,∞Eεs(t) ≤ (Eε

s(t))3/2. (3.12)

For later estimates in this paper, we recall some results on Moser-type calculus inequalities inSobolev spaces [17, 18].


Lemma 3.1. Let s ≥ 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 (∂αx(uv) − u∂αxv) ∈ L2(T3) and

‖∂αx(uv) − u∂αxv‖ ≤ Cs

(‖∇u‖0,∞

∥∥∥D|α|−1v

∥∥∥ +

∥∥∥D|α|u

∥∥∥‖v‖0,∞

), (3.13)

where

∥∥∥Dhu

∥∥∥ =

∑

|α|=h‖∂αxu‖, ∀h ∈ N. (3.14)

Moreover, if s ≥ 3, then the embedding Hs−1(T3) ↪→ L∞(T3) is continuous and one has

‖uv‖s−1 ≤ Cs‖u‖s−1‖v‖s−1, ‖∂αx(uv) − u∂αxv‖ ≤ Cs‖u‖s‖v‖s−1. (3.15)

By using basic Moser-type calculus inequalities and Sobolev’s lemma, we have

(∂αxS(Wε), ∂αxW

ε) ≤ CEεs(t) + Cε(Eε

s(t))3/2. (3.16)

After a a direct calculation, one gets

μ(∂αxN(Wε), ∂αxWε) = −μ

∫

T3

(|∇∂αxd|2 + |∇∂αxc|2

)dx. (3.17)

To estimate the commutator, we have

([∂αx,A(t, x, ∂x)]Wε, ∂αxWε)

=∫([

∂αx,uNS · ∇

]d∂αxd +

[∂αx,u

NS · ∇]c∂αxc +

[∂αx,u

NS · ∇]n∂αxn

)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).

(3.18)


Also, we have

− ε([∂αx, B(t, x, ∂x)]Wε, ∂αxW

ε)

= −ε∫([∂x,u · ∇]d∂αxd + [∂x,u · ∇]c∂αxc + [∂x,u · ∇]n∂αxn)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(Eε

s(t))3/2.

(3.19)

Here, we have used the inequality

‖u‖s ≤ C‖∇u‖s−1 ≤ C(‖d‖s−1 + ‖c‖s−1). (3.20)

Finally, the Young inequality gives

(∂αxR, ∂αxW

ε) ≤∥∥∥∂tΔVNS + uNS · ∇ΔVNS

∥∥∥s‖n‖s ≤ C(1 + Eε

s(t)). (3.21)

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

dtEεs(t) ≤ C

(1 + Eε

s(t) + ε(Eεs)

3/2(t)). (3.22)

We can conclude using a standard Gronwall’s lemma, that if the solution (uNS,VNS) ofNavier-Stokes equations (1.2) is smooth on the time interval [0, T], for any T1 < T there existsε0 such that the sequence (Wε)ε<ε0 is bounded in C([0, T1],Hs(T3)). Then we have

Wε = (divu, curlu, n),

uε = uNS + εu,

nε = 1 + ε2(n + ΔVNS

).

(3.23)

The assumptions that we havemade 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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On a Quasi-Neutral Approximation of the …downloads.hindawi.com/journals/isrn.applied.mathematics/...1 P. Degond, “Mathematical modelling of microelectronics semiconductor devices,”