# American Institute of Mathematical Sciences

doi: 10.3934/krm.2020052

## Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem

 Université de Nantes, CNRS UMR 6629, Laboratoire de Mathématiques Jean Leray, 2, rue de la Houssinière, 44332 Nantes

Received  April 2020 Revised  July 2020 Published  November 2020

The mathematical description of the interaction between a collisional plasma and an absorbing wall is a challenging issue. In this paper, we propose to model this interaction by considering a stationary bi-species Vlasov-Poisson-Boltzmann boundary value problem with boundary conditions that are consistent with the physics. In particular, we show that the wall potential can be uniquely determined from the ambipolarity of the particles flows as the unique solution of a nonlinear equation. We also prove that it is an increasing function of the electrons re-emission coefficient at the wall. Based on the Schauder fixed point theorem, our analysis establishes the existence of a solution provided, on the one hand that the incoming ions density satisfies a moment condition that generalizes the Historical Bohm criterion, and on the other hand that the collision frequency does not exceed a critical value whose definition is subordinated to the strict validity of our generalized Bohm criterion.

Citation: Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, doi: 10.3934/krm.2020052
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Schematic characteristic ions trajectories associated with a decreasing potential $\phi$. The solid lines corresponds to characteristic curves originating from $x = 0$ with positive velocities, and they span $D_{i,0}$ the lighter gray region. The dashed lines correspond to characteristic curves originating from the wall with negative velocities, and they span the darker gray region $D_{i,1}$
Schematic characteristic electrons trajectories associated with a decreasing potential $\phi$. The solid lines corresponds to characteristic curves originating from $x = 0$ with positive velocities, and they span $D_{e,0}$ the lighter gray region. The dashed lines correspond to characteristic curves originating from the wall with negative velocities, and they span the darker gray region $D_{e,1}$
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