The mathematical description of the interaction between a plasma and a solid surface is a major issue that still remains challenging. In this paper, we model this interaction as a stationary and bi-kinetic Vlasov-Poisson-Ampere boundary value problem with boundary conditions that are consistent with the physics. In particular, we show that the wall potential can be determined from the ampibolarity of the particle flows as the unique solution of a non linear equation. Based on variational techniques, our analysis establishes the well-posedness of the model, provided that the incoming ion distribution satisfies a moment condition that generalizes the historical Bohm criterion of plasma physics. Quantitative estimates are also given, together with numerical illustrations that validate the robustness of our approach.

中文翻译：

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双动力鞘的最小化公式

等离子体和固体表面之间相互作用的数学描述是一个仍然具有挑战性的主要问题。在本文中，我们将这种相互作用建模为一个稳态和双动力学 Vlasov-Poisson-Ampere 边界值问题，其边界条件与物理学一致。特别是，我们表明壁电位可以从作为非线性方程的唯一解的粒子流的波峰性确定。基于变分技术，我们的分析建立了模型的适定性，前提是进入的离子分布满足概括了等离子体物理学的历史玻姆标准的矩条件。还给出了定量估计，以及验证我们方法稳健性的数值说明。