On a Quasi-Neutral Approximation to the Incompressible Euler Equations

Journal of Applied Mathematics, Jun 2012

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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On a Quasi-Neutral Approximation to the Incompressible Euler Equations

Journal of Applied Mathematics Hindawi Publishing Corporation On a Quasi-Neutral Approximation to the Incompressible Euler Equations Jianwei Yang 0 Zhitao Zhuang 0 Roberto Natalini 0 College of Mathematics and Information Science, North China University of Water Resources and Electric Power , Zhengzhou 450011 , China 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. - 1. Introduction In this paper, we shall consider the following hydrodynamic system: for x ? T3 and t > 0, subject to the initial conditions ?tn? ?tu? div n?u? u? ? ? u? 0, ???, 3 x ? T , t > 0, 3 x ? T , t > 0, ??? n? ? 1 ? 3 x ? T , t > 0 n?, u? t 0 n0?, u0? for x ? T3. In the above equations, T3 is 3-dimensional torus and ? > 0 is small parameter. Here n?, u?, ?? denote the electron density, electron velocity, and the electrostatic potential, respectively. 1.1 1.2 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 n? through the linear Poisson equation ??? n? ? 1 /?. the limTitowsohlevne ?ungioqeusetloy zthereoP, oitisissoenaseyq utoatsieoen,,awt eleaadstdatthae vcoernydfitoiromn alT3lenv?edl,xtha1t. Pna?s,sui?n,g?t?o tends to nI , uI , ?I , where nI 1 and ?tuI uI ? ? uI div uI 0. ??I , 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. iTsh0se,uTochre,tHmhast13.1T.3Let aunId beT3ausIodlxution0 offorthse >inc5o/m2p.rAessssiubmleeEthualetrtheqeuiantiitoianlsva1lu.3e snu0?c,hu0?tha?t uHI s ?1 1.3 1.4 1.5 Ms ? : ? u0 ? u0 Hs 1 I 2 1 ? T T 3 n0?dx 3 u0?dx n0 ? ???0I t ? 1 ? 1, 0, 2 Hs ?? 0 when ? ?? 0 , uI0 uI |t 0. Then, there exist ?0 and CT such that for 0 < ? ? ?0 there is a solution n?, u? ? of 1.1 satisfying 0, T , Hs 1 T3 u? t ? uI t 2 Hs 1 1 ? n? t ? ???I ? 1 2 Hs ? CT ? Ms ? 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 driftdiffusion 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 wellprepared initial data. Our result gives a more general rate of convergence in strong Hs 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 . 2.1 2.2 2.3 2.4 2.5 2. Proof of Theorem 1.1 First, let us set Then, we know the vector n, u, ? solves the system n, u, ? n? ? 1 ? ???I , u? ? uI , ?? ? ?I . ?tu u uI ? ?u u ? ? uI ??, ? n 1 div u ? ? ?t??I ?? n , ? ?tn u uI ? ?n div ??I u uI where ?u : ?v i3,j 1 ?xi u/?xj ?xj v/?xi . In fact, from 1.3 , we get ??I As in 16 , we make the following change of unknowns: ?uI : ?uI . d, c div u, curl u . By using the last equation in 2.2 , we get the following system: ?td u uI ? ?d n ? ? ? u 2uI : ?u, ?tc u uI ? ?c c ? ? u uI curl ?uI ? u ? dc ? curl u ? ? uI , ?tn u uI ? ?n ? n 1 d ? ? ?t??I div ??I u uI . This last system can be written as a singular perturbation of a symmetrizable hyperbolic system: ?tv 3 1 ? K?v where u uI j denotes the ith component of u uI and where ? d? v ? c? , n and for |?| ? s with s > d/2, E??,s t 1 2 A0???xv, ??xv Es? t Along the proof, we shall denote by C a number independent of ?, which actually may change from line to line, and by C ? a nondecreasing function. Moreover ?, ? and ? stand It is easy to know that system 2.5 is a hyperbolic system. Consequently, for ? > 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 T ? as the largest time such that Es? t ? M?, ?t ? 0, T ? , where M? which is such that M? ? 0 when ? 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 T ? ? T , which will be proved by showing that in 2.8 the equality cannot be reached for T ? < T thanks to a good choice of M?. Before performing the energy estimate, we apply the operator ??x for ? ? N3 with |?| ? s to 2.5 , to obtain where 3 ? ?xS v ???xR v ??, ? ?? ? 3 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 A0???xv. Then, we have ddt E??,s t ? ?? A0???xv, 3 Let us start the estimate of each term in the above equation. For I1, since A0? is symmetric and div uI 0, by Cauchy-Schwartz?s inequality and Sobolev?s lemma, we have that I1 1 2 div uA0???xv, ??xv ? div u 0,?Es? t ? C Es? t 3/2 . Next, since A0?K? is skew-symmetric, we have that For I3, by a direct calculation, one gets I2 0. I3 1 ? ??xc, ??x dc ? ? n, ??x dn ? ??xc ??x dc ? C Es? t 3/2 . 1 ? ??xn 2 ??xd 2.11 2.12 2.13 2.14 that Here, we have used the basic Moser-type calculus inequalities 18 . To give the estimate of the term I4, we split it in two terms. Specifically, we can deduce I4 ? ??xd, ??x ? u 2uI : ?u ??xc, ??x c ? ? u uI curl ?uI ? u ? curl u ? ? uI Here, we have used the curl-div decomposition inequality ?u s ? C d s c s . For I5, we have that I5 ? n s ?t??I div ??I u u I s ? C n s 1 ? ? C? Es . d s c s and Sobolev?s lemma, we have To estimate the last term, that is, I5, by using basic Moser-type calculus inequalities ? d s ? u 2uI : ?u c s c ? ? u u I ? ? C Es ? Es 3/2 . s s curl ?uI ? u s curl u ? ? u I s C c s u u ? I n s 2 d s ? C ? ? C Es u u ? I c s2 3/2 ? Es 1 ? By using 2.8 , we get with M? 1 that Hence, by the Gronwall inequality, we get that ddt Es? ? C? CEs?, ?t ? Acknowledgment The authors acknowledge partial support from the Research Initiation Project for High-Level Talents no. 40118 of North China University of Water Resources and Electric Power. 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Jianwei Yang, Zhitao Zhuang. On a Quasi-Neutral Approximation to the Incompressible Euler Equations, Journal of Applied Mathematics, 2012, DOI: 10.1155/2012/957185