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Régularité de la solution des équations cinétiques en physiques de plasmas. (French) Zbl 0613.35070
Sémin., Équations Dériv. Partielles 1985-86, Exposé No. 18, 11 p. (1986).
The author considers the Vlasov-Poisson system: \[ \partial_ tf+v.\nabla_ xf-\nabla_ x\phi.\nabla_ vf=0;\quad \Delta \phi =\pm \int f(x,v,t)dv;\quad \phi \to 0\quad for\quad | x| \to +\infty \] where the signs \(+\) or - correspond respectively to attractive or repulsive forces. The aim of the paper is to analyse the phenomenon of regularity loss for this system, and of their generalizations. Many theorems are demonstrated on the upper bounds of the magnitudes related to the problem; however the author points out that in particular in the case of dimension \(d=3\), the physically most interesting, no complete answer is known. Concluding the existence of steady state solutions, their stability, the asymptotic behaviour, the calculation of explicit solutions are indicated as interesting open problems.
Reviewer: S.Nocilla

35Q99 Partial differential equations of mathematical physics and other areas of application
82D10 Statistical mechanical studies of plasmas
35B65 Smoothness and regularity of solutions to PDEs
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