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Zbl 1213.35051
Ju, Qiangchang; Li, Hailiang; Li, Yong; Jiang, Song
Quasi-neutral limit of the two-fluid Euler-Poisson system. (English)
[J] Commun. Pure Appl. Anal. 9, No. 6, 1577-1590 (2010). ISSN 1534-0392; ISSN 1553-5258/e

Let $\Bbb T^d\subset\Bbb R^d$ be the $d$-dimensional torus ($d=2,3$). This paper deals with the motion of a plasma consisting of electrons and ions with density $n_e,n_i$, charge $q_e=-1$, $q_i=1$, and scaled mean velocity $u_e,u_i,$ respectively. This motion is described by the two-fluid compressible Euler-Poisson system: $$\cases\partial_tn_{\alpha}+\text{div}(n_{\alpha}u_{\alpha})=0,\\ \partial_t(n_{\alpha}u_{\alpha})+\text{div}(n_{\alpha}u_{\alpha}\otimes u_{\alpha})+\nabla p_{\alpha}(n_{\alpha}) =-q_{\alpha}n_{\alpha}\nabla \varphi-\dfrac{n_{\alpha}u_{\alpha}}{\tau_{\alpha}},\\ -\lambda^2\Delta \varphi=n_i-n_e, \endcases\tag1 $$ for $\alpha=e,i,x\in\Bbb T^d$, with initial conditions of the form $$\cases n_{\alpha}(0,x)=\sum\limits_{j=0}^{m}\lambda^{2j}n_{\alpha,j}(x)+\lambda^{2(m+1)}n^{\lambda}_{\alpha,m+1}(x),\\ u_{\alpha}(0,x)=\sum\limits_{j=0}^{m}\lambda^{2j}u_{\alpha,j}(x)+\lambda^{2(m+1)}u^{\lambda}_{\alpha,m+1}(x), \endcases\tag2 $$ for $\alpha=i,e$. Here the small parameter $\lambda$ is the Debye length, the relaxation time constants $\tau_{\alpha}$ will be assumed to be $=1$, and $p_{\alpha}(n_{\alpha})= a^2_{\alpha}n_{\alpha}^{\gamma_{\alpha}}$ with $ \gamma_{\alpha}\geq 1$ and $a_{\alpha}>0.$ The initial data are assumed to be well-prepared i.e., $$n_{e,0}(x)=n_{i,0}(x)=n_o(x),\qquad u_{e,0}(x)=u_{i,0}(x)=u_o(x).$$ The existence of a solution to (1) (2), for sufficiently small $\lambda$, that has the following asymtotic expansion $$(n^{\lambda}_{\alpha},u^{\lambda}_{\alpha},\varphi^{\lambda})(t,x)= \sum_{j\geq 0}\lambda^{2j}(n^j_{\alpha},u^j_{\alpha},\varphi^j)(t,x),\quad\alpha=e,i,\quad (t,x) \in [0,T]\times\Bbb T^d$$ with leading terms $$n_e^o(t,x)=n_i^o(t,x)=n^o(t,x),\qquad u_e^o(t,x)=u_i^o(t,x)=u^o(t,x)$$ is proved, in suitable functional spaces, provided $(n^o(t,x),u^o(t,x))$ is a unique smooth solution of the damped compressible Euler equations with initial data $n^o(0,x)=n_o(x), u^o(0,x)=u_o(x)$, $x\in\Bbb T^d$, in a time interval $[0,T]$, $T>0$.
[Denise Huet (Nancy)]

MSC 2000:: *35B25 Singular perturbations (PDE)
35B40 Asymptotic behavior of solutions of PDE
35C20 Asymptotic expansions of solutions of PDE
35L60 First-order nonlinear hyperbolic equations
35Q35 Other equations arising in fluid mechanics
35Q31
82D10 Plasmas

Keywords: quasi-neutral limit; two-fluid Euler-Poisson system; compressible Euler equations; small Debye length

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