The Vlasov–Poisson–Fokker–Planck equation in an interval with kinetic absorbing boundary conditions

Hyung Ju Hwang and Jinoh Kim

Stochastic Processes and their Applications, 2019, vol. 129, issue 1, 240-282

Abstract: We study the initial–boundary value problem for the Vlasov–Poisson–Fokker–Planck equations in an interval with absorbing boundary conditions. We first prove the existence of weak solutions of the linearized equation in an interval with absorbing boundary conditions. Moreover, the weak solution converges to zero exponentially in time. Then we extend the above results to the fully nonlinear Vlasov–Poisson–Fokker–Planck equations in an interval with absorbing boundary conditions; the existence and the longtime behavior of weak solutions. Finally, we prove that the weak solution is actually a classical solution by showing the hypoellipticity of the solution away from the grazing set and the Hölder continuity of the solution up to the grazing set.

Keywords: Partial differential equation; Kinetic theory; Vlasov–Poisson–Fokker–Planck equation; Absorbing boundary value problem; Non-linear equation; Feynman–Kac formula (search for similar items in EconPapers)
Date: 2019
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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:129:y:2019:i:1:p:240-282

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DOI: 10.1016/j.spa.2018.02.016

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