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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

##### MSC:
 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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##### References:
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