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Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data. (English) Zbl 0810.35089
From the introduction: We study the following system of partial differential equations, which is called the Vlasov-Poisson system ${\partial\over {\partial t}} f(t,x,v)+ v{\partial \over {\partial x}} f(t,x,v)+ E(t,x) {\partial\over {\partial v}} f(t,x,v)=0,\;E(t,x)= \gamma\int_{\mathbb{R}^ n} {{x-y} \over {| x-y|^ n}} \rho(t,y) dy,\tag{1}$ where $$\rho(t,x):= \int_{\mathbb{R}^ n} f(t,x,v)dv$$, $$t\geq 0$$, $$x,v\in \mathbb{R}^ n$$, $$\gamma=\pm 1$$, $$f|_{t=0}= \varphi$$.
We denote this system by (VP). The function $$f$$ describes the development in time of a particle-distribution in the phase-space $$\mathbb{R}^ n \times\mathbb{R}^ n$$. The motion of the particles is, via Vlasov’s equation (1) determined by $$E$$, which is a Newtonian force (attracting) for $$\gamma=-1$$ and a Coulomb force (repulsing) for $$\gamma=1$$. $$E$$ at the time $$t$$ is, via Poisson’s equation, determined by the particle- distribution at $$t$$, so that the problem is nonlinear. The distribution at time 0 is given by $$\varphi$$.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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##### References:
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