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Weak solutions of the Vlasov–Poisson initial boundary value problem. (English) Zbl 0786.35014
The author considers the Vlasov-Poisson system in the form \[ \partial_ t f+\xi \cdot \nabla_ xf+\nabla_ xu(t,x)\cdot \nabla_ \xi f=0,\qquad -\Delta_ xu=\int f d \xi, \] where \(t \geq 0\), \((x,\xi)\in \Omega \times \mathbb{R}^ N\) with \(\Omega \subset \mathbb{R}^ N\) regular, open, and bounded. Assuming absorbing boundary conditions for the phase space density \(f\) and Dirichlet boundary conditions for the electrostatic potential \(u\) the author proves global existence of weak solutions to the corresponding initial value problem and uses these solutions to investigate homogenization effects.
The inclusion of boundary effects is an important extension of previous existence results for the Vlasov-Poisson system. The only result in this direction that the reviewer knows of is due to Y. Guo [Commun. Math. Phys. 154, No. 2, 245-263 (1993)] who treats the Vlasov-Maxwell system with quite general boundary conditions. The author claims that his results remain true for other boundary conditions, for example for specular reflection, a condition which seems more natural in connection with homogenization. However, specular reflection is in general more difficult to handle than absorbing boundary conditions. For example, in the former case the linear Vlasov equation not necessarily has smooth solutions, and Sect. 2.3 would have to be changed.
The extension to the higher dimensional cases \(N>3\) occupies a large part of the paper and is justified by the author by interpreting \(f\) as the joint position/velocity probability density of a system of \(m\) electrons, \(N=3m\). This is inconsistent with the coupling with the Poisson equation. As explained in reference [22] of the paper, pp. 17-19, the Vlasov- Poisson system arises only after passing to the electron number density on phase space \(\Omega \times \mathbb{R}^ 3\), \(\Omega \subset \mathbb{R}^ 3\), via the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy.
Reviewer: G.Rein (München)

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35D05 Existence of generalized solutions of PDE (MSC2000)
35D10 Regularity of generalized solutions of PDE (MSC2000)
82B40 Kinetic theory of gases in equilibrium statistical mechanics
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