Quasineutral limit of the pressureless Euler–Poisson equation

Accepted Manuscript Quasineutral limit of the pressureless Euler–Poisson equation Xueke Pu PII: DOI: Reference:

S0893-9659(13)00354-6 http://dx.doi.org/10.1016/j.aml.2013.12.008 AML 4480

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Applied Mathematics Letters

Received date: 26 September 2013 Revised date: 12 December 2013 Accepted date: 12 December 2013 Please cite this article as: X. Pu, Quasineutral limit of the pressureless Euler–Poisson equation, Appl. Math. Lett. (2013), http://dx.doi.org/10.1016/j.aml.2013.12.008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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QUASINEUTRAL LIMIT OF THE PRESSURELESS EULER-POISSON EQUATION XUEKE PU Abstract. In this short paper, we consider the quasineutral limit for the pressureless Euler-Poisson system for ions. By applying the modulated energy method, it shows that the weak solutions for the Euler-Poisson system converge weakly to the strong solutions of the compressible Euler equation as the Debye length tends to zero.

1. Introduction The isothermal Euler-Poisson system of plasma reads as follows  ε ε ε   ∂t n + ∇ · (n u ) = 0, ∂t uε + uε · ∇uε + Ti ∇ ln nε = −∇φε , (EPε )  ε  ε∆φε = eφ − nε ,

(1.1a) (1.1b) (1.1c)

where n, u and φ are respectively the density, velocity of ions and the electrical potential at time t ∈ R+ and position x ∈ Td = (R/Z)d , the d-dimensional torus. Physically, Ti is the constant temperature of the ions, ε is a small parameter representing the squared scaled Debye length given by ε = 0 κB Ti /(N0 e2 L2 ), where 0 is the vacuum permittivity, κB is Boltzmann constant, N0 is the density and L is the characteristic length. The case ε < 0 corresponding to attractive forces between particles of the same sign will not be considered in this paper. For typical plasma applications, the Debye length ε ≈ 10−8 is very small compared to the characteristic length of physical interest and it is therefore necessary to consider the limiting problem when ε → 0. This is the so called quasineutral problem for the Euler-Poisson system, which was extensively studied in the past decades. See [1,2,4,6–8] and the references therein for the quasineutral limit for the Euler-Poisson system or the Vlasov-Poisson system. The cold ions approximation means from the physical point of view that Ti  Te . It turns out that approximation is highly relevant for terrestrial plasmas and widely used in plasma physics, especially in the study of tokamak plasmas. In this case, we are lead to the following pressureless Euler-Poisson system (1.3) with Ti = 0 for ions. The quasineutral limit was studied by Cordier and Grenier in [2] when Ti > 0 fixed, and was studied by us when Ti = 0 very recently in [8]. In the following, we only concentrate on the case when Ti = 0 and this assumption is made throughout this paper. Formally, when ε → 0, we obtain the following compressible Euler equation  ∂t n + ∇(nu) = 0, (1.2a) (CE) ∂t u + u · ∇u + ∇ ln n = 0. (1.2b) ε

2000 Mathematics Subject Classification. 35Q53; 35Q35. Key words and phrases. Quasineutral limit; pressureless Euler-Poisson equation; Debye length; modulated energy method. This work is supported by NNSF under #11001285. 1

2

X. PU

This limit was recently studied rigorously by the author in the case of smooth solutions. It was proven that the smooth solutions (as far as they exist) of (1.3) converge to the solution of the compressible Euler system (1.2). But we don’t know what happens when only weak entropy solutions are considered. Note that the existence of weak entropy solution was proved by using Glimm’s method in [3]. We will address this point in the sequel. First, we give the following proposition concerning the local existence of smooth solutions for the compressible Euler system (1.2). The proof is classical and can be found in [2, 5]. The Euler system has to be supplemented by suitable initial conditions. As in [2], we shall assume for simplicity that the initial conditions (n0 , u0 ) satisfy n0 − 1, u0 ∈ H s (Td ) and n0 ≥ σ > 0 for some s > d2 + 1 and some constant σ. Proposition 1.1. Let n0 > 0, u0 be such that n0 ∈ L1 (Td ), ln n0 ∈ H s (Td ) and u0 ∈ H s (Td ) for s > d2 + 1, then there exists a unique local smooth solution n > 0 and u to (1.2) such that \ ln n, u ∈ C([0, T ∗ ); H s (Rd )) C 1 ([0, T ∗ ); H s−1 (Rd )), (1.3) for some T ∗ > 0.

The main result of this short paper is stated in the following Theorem 1.1. Let n0 > 0, u0 satisfy the assumptions of Proposition 1.1, and n, u the corresponding strong solution of system (1.2). Assume that the sequence of initial data nε0 , uε0 satisfy Eε (0) ≤ C uniformly in ε, and R ε ε 2   Rn0 |u0 − u0 | dx → 0 (1.4) ε |∇φε0 |2 dx → 0 
ε ε R φε0 e ln(eφ0 /n0 ) − eφ0 + n0 dx → 0, where φε0 is solution of the nonlinear Poisson equation ε

ε∆φε0 = eφ0 − n0 .

(1.5)

Then nε weakly converges to n and nε uε weakly converges to nu in L1 . Furthermore, we have the following local in time strong convergences Z nε |uε − u|2 dx → 0, in L∞ t , (1.6) √ ε √ ∞ 2 φ e → n, in Lt (Lx ). Here and hereafter, the energy functional Eε is defined by Z Z Z 1 ε ε ε 2 ε φε Eε (t) = n |u | dx + (φ − 1)e dx + |∇φε |2 dx. 2 2

(1.7)

In the following section, we will prove Theorem 1.1 by using the modulated energy method of [1] (also referred to as the relative entropy method). 2. Proof of Theorem 1.1 In this section, we will prove Theorem 1.1 by using the modulated energy method. We begin with the following Proposition 2.1. Let ε > 0 and (nε , uε , φε ) a √ strong solution to (1.1) on [0, T ], then 2 is uniformly bounded with Eε (t) = Eε (0) for any t ∈ [0, T ]. In particular k nε uε kL∞ t Lx respect to ε.

QUASINEUTRAL LIMIT FOR THE PRESSURELESS EULER-POISSON EQUATION

3

Proof. We compute the derivative in time of the first term in Eε , by using the transport equation (1.1a) to obtain Z Z Z 1 1 d nε |uε |2 dx = ∂t nε |uε |2 dx + nε uε · ∂t uε dx 2 dt 2 Z Z 1 (2.1) =− ∇ · (nε uε )|uε |2 dx + nε uε · ∂t uε dx 2 Z Z = nε uε ⊗ uε : ∇uε dx + nε uε · ∂t uε dx Now, thanks to the equation (1.1b) satisfied by the velocity field uε , we obtain Z Z Z nε uε · ∂t uε dx = − nε uε ⊗ uε : ∇uε dx − nε uε · ∇φε dx Z Z = − nε uε ⊗ uε : ∇uε dx − ∂t nε φε dx.

(2.2)

ε

Using (1.1c), we obtain, by setting mε = eφ , that Z Z Z ε − ∂t nε φε dx =ε ∂t ∆φε φε dx − ∂t eφ φε dx Z Z (2.3) d ε d ε 2 ε ε |∇ | dx − m (ln m − 1)dx. =− 2 dt dt Putting the equalities together, we obtain Z Z Z 1 d ε d d (2.4) nε |uε |2 dx = − |∇φε |2 dx − mε (ln mε − 1)dx. 2 dt 2 dt dt This completes the proof by integrating in time. The last statement is easy since the second term has a lower bound independent of ε.  1 Lemma 2.1. Let Jε (t, x) = nε uε , then Jε is uniformly bounded in L∞ t Lx with respect to ε.

Proof. We first observe that for any z ∈ R, (z − 1)ez ≥ −1. Then from the definition of the energy functional Eε (t), we know Z Z ε nε |uε |2 dx ≤2Eε (t) − 2 (φε − 1)eφ dx (2.5) Td Td ≤2Eε (0) + 2|Td |,

where |Td | is the Lebesgue measure of the torus Td . Applying the H¨ older inequality then yields Z Z Z ε ε ε ε 2 |n u |dx ≤ n |u | dx + nε dx. (2.6) This completes the proof.

Td

Td

Td



Next, we prove our main Theorem 1.1. Proof of Theorem 1.1. Let (n, u) be a solution to (1.2) verifying the regularity of (1.3). We modulate energy and consider the following relative entropy Z Z Z ε 1 Hε (t) = nε |uε − u|dx + (mε ln(mε /n) − mε + n) dx + |∇φε |2 dx, (2.7) 2 2 ε

where mε = eφ . By a simple inequality 1 + ln z ≤ z for all z > 0, we easily obtain Z Z √ √ ( a − b)2 dx ≤ (a ln(a/b) − a + b)dx.

(2.8)

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X. PU

In what follows, we will show that this functional is in fact a Lyapunov functional. We shall show that Hε satisfies some stability estimate: Hε (t) ≤ Hε (0) + Gε (t) + C

Z

t

0

k∇ukL∞ Hε (s)ds,

(2.9)

with Gε (t) → 0 uniformly in time. Since the energy Eε (t) is nondecreasing, we know d Hε (t) ≤ dt

    Z 1 2 1 2 ∂t nε |u| − uε dx + nε ∂t |u| − uε dx 2 2 Z Z − ∂t (mε ln n) dx + ∂t ndx

Z

(2.10)

=I1ε (t) + I2ε (t) + I3ε (t) + I4ε (t) = I ε (t). By using (1.1a), we obtain I1ε (t)

+

I2ε (t)

= = =

Z

Z

Z

n (∂t + u · ∇) ε

ε



 1 2 ε |u| − u dx 2

nε (u − uε ) · ((∂t + uε · ∇)u)dx +

Z

nε u · ∇φε dx

(2.11)

n (u − u ) · ((∂t + u · ∇)u)dx Z Z + nε (u − uε ) · ((u − uε ) · ∇u)dx + nε u · ∇φε dx. ε

ε

For the last term in this equality, we obtain Z

Z Z ε nε u · ∇φε dx = eφ ∇φε · udx − ε ∆φε ∇φε · udx Z Z Z ε φε ε ε = ∇(e ) · udx − ε (∇ · (∇φ ⊗ ∇φ )) · udx + (∇|∇φε |2 ) · udx 2 Z Z Z ε ε = − eφ ∇ · udx + ε D(u) : (∇φε ⊗ ∇φε )dx − |∇φε |2 ∇ · udx, 2

(2.12)

where D(u) = (∂xi uj + ∂xj ui )/2 is the symmetric part of ∇u = (∂xi uj )i,j . For the term I3ε and I3ε , we have Z Z Z ε ∂t n ε I3ε (t) + I4ε (t) = − eφ dx − ∂t (eφ ) ln ndx + ∂t ndx n Z Z Z φε e ε = (1 − )∂t ndx − ε ∂t ∆φ ln ndx − ε ∂t nε ln ndx n Z Z Z ε eφ = (1 − )∂t ndx − ε ∂t ∆φε ln ndx + ε ∇ · (nε uε ) ln ndx, n

(2.13)

QUASINEUTRAL LIMIT FOR THE PRESSURELESS EULER-POISSON EQUATION

thanks to (1.1a). Finally, we obtain that Z Z ε φε ∂t n I (t) = − e ( + ∇ · u + u · ∇ ln n)dx + ∂t ndx n Z + nε (u − uε ) · (∂t u + u · ∇u + ∇ ln n)dx Z Z + ε ∆φε u · ∇ ln ndx − ε ∂t ∆φε ln ndx Z Z − nε (u − uε ) · ((u − uε ) · ∇u)dx + ε D(u) : (∇φε ⊗ ∇φε )dx Z ε |∇φε |2 ∇ · udx, − 2 where we have used the fact that Z Z Z nε (u − uε ) · ∇ ln ndx = nε u · ∇ ln ndx − nε uε · ∇ ln ndx Z Z Z φε ε = e u · ∇ ln ndx − ∆φ u · ∇ ln ndx − nε uε · ∇ ln ndx. Let

A(u, n) = Since

we have

Z



∂t ln n + ∇ · u + u · ∇ ln n ∂t u + u · ∇u + ∇ ln n

(u · ∇u + ∇ ln n) ndx = Z

A(u, n) ·



n 0



Z



.

5

(2.14)

(2.15)

(2.16)

∇ · (nu)dx = 0,

(2.17)

Z

(2.18)

dx =

∂t ndx.

It then follows that   Z Z Z ε −eφ + n ε dx + ε ∆φ u · ∇ ln ndx − ε ∂t ∆φε ln ndx I ε (t) = A(u, n) · nε u − nε uε Z Z ε ε ε (2.19) − n (u − u ) · ((u − u ) · ∇u)dx + ε D(u) : (∇φε ⊗ ∇φε )dx Z ε − |∇φε |2 ∇ · udx. 2

The last three integrals are bounded by Z Z ε ε ε 2 B (t) ≤ c n |u − u | k∇ukL∞ dx + c |φε |2 k∇ukL∞ dx. x x

(2.20)

We now consider Z Z Z Z G ε (t) :=ε ∆φε u · ∇ ln ndxds − ε ∂t ∆φε ln ndxds Qt Qt Z Z Z Z √ √ √ √ ε =− ε ε∇φ · ∇ (u · ∇ ln n) dxds − ε ε∇φε · ∂s ∇ ln ndxds (2.21) Qt Qt t Z √ √ ε∇φε · ∇ ln ndx , + ε 0

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√ 2 is uniformly bounded, we obtain where Qt = Rd × (0, t). Since k ε∇φε kL∞ t Lx
√ √ ε ε 2 2 + k ln nkW 1,∞ (0,t;H 1 ) G (t) ≤C εk ε∇φ kL∞ k∇ (u · ∇ ln n) kL∞ t Lx t Lx →0,

as ε → 0,

(2.22)

locally uniformly in t. Therefore, integrating (2.10) in (0, t) yields   Z t Z tZ ε −eφ + n ε A(u, n) · dxds + G (t) + c k∇ukL∞ Hε (s)ds. Hε (t) ≤ Hε (0) + nε u − nε uε 0 0 (2.23) Now we choose (n, u) to be solutions of A(n, u) = 0, with initial conditions (n, u)|t=0 = (n0 , u0 ). In other words, (n, u) is a solution to the Euler system (1.2). Therefore, since Hε (0) → 0 and G ε (t) → 0, we deduce by Gronwall inequality that Hε (t) → 0,

(2.24)

when ε → 0, uniformly with respect to time. √ R ε ε √ 2 This, together with (2.8), implies that eφε → n strongly in L∞ t Lx and Ω n |u − u|2 dx → 0 in L∞ t . On the other hand, from Lemma 2.1 we deduce that there is a vague limit J in the sense of measures on L∞ ([0, T ] × Ω) up to the extraction of a subsequence. R Since (n, J) 7→ |J − nu|2 /ndx is convex and lower semi-continuous with respect to the weak convergence of measures, the weak convergence nε * n and nε uε =: J ε * J in the sense of measures yields Z Z |J ε − nε uε |2 |J − nu|2 dx ≤ lim inf dx. (2.25) ε→0 n nε This implies that J = nu.  Acknowledgment. This work is supported by the Fundamental Research Funds for the Central Universities under Project No. CQDXWL-2012-011. References [1] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25(3&4), (2000)737-754. [2] S. Cordier and E. Grenier, Quasineutral limit of an Euler-Poisson system arising from plasma physics, Comm. Partial Differential Equations, 35(5&6), (2000)1099-1113. [3] S. Cordier, Y. Peng, Syst` eme Euler-Poisson non lin´ eaire. Existence globale de solutions faibles entropiques, Math. Model. Numer. Anal., 32, (1998)1-23. [4] D. Han-Kwan, Quasineutral limit of the Vlasov-Poisson system with massless electrons, Comm. Partial Differential Equations, 36(8), (2011)1385-1425. [5] A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematial Sciences, 53, Springer-Verlag, New York-Berlin, 1984. [6] N. Masmoudi, From Vlasov-Poisson system to the incompressible Euler system, Comm. Partial Differential Equations, 26(9&10), (2001)1913-1928. [7] S. Wang, Quasineutral limit of Euler-Poisson system with and without viscosity. Comm. Partial Differential Equations, 29(2004), no. 3-4, 419-456. [8] X. Pu, B. Guo, Quasineutral limit of the Euler-Poisson equation for a cold, ion-acoustic plasma, arXiv: 1304.0187v1. Xueke Pu Department of Mathematics, Chongqing University, Chongqing 401331, P.R.China E-mail address: [email protected]