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 \(\varphi\).


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