Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data. (English) Zbl 0593.35076

The authors consider the equations \[ \partial u_{\alpha}/\partial t+(v\cdot \nabla_ x)u_{\alpha}+(q_{\alpha}/m_{\alpha})(E\cdot \nabla_ v)u_{\alpha}=\quad 0,\quad 1\leq \alpha \leq N \] with \(v\cdot \nabla_ x=\sum^{3}_{i=1}v_ i(\partial /\partial x_ i)\), \(E\cdot \nabla_ v=\sum^{3}_{i=1}E_ i(\partial /\partial v_ i)\), where E is the electric field related to \(u_{\alpha}\) by the Poisson equation \[ E=-\nabla_ x\phi =4\pi \sum_{\alpha}q_{\alpha}\int_{R^ 3}((x-y)/| x-y|^ 3)(\int_{R^ 3}u\quad_{\alpha}(y,v,t)dv)dy. \] The existence, the uniqueness of a smooth solution \(u_{\alpha}(x,v,t)\) satisfying the initial conditions \(u_{\alpha}(x,v,0)=u_{\alpha,0}(x,v)\) is proved.
Reviewer: I.Bock


35Q99 Partial differential equations of mathematical physics and other areas of application
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B65 Smoothness and regularity of solutions to PDEs
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