arXiv 2211.033362022-11-07math.AP

随机电磁场中Vlasov-Poisson和Vlasov-Poisson-Fokker-Planck系统的局部适定性

The Vlasov-Poisson and Vlasov-Poisson-Fokker-Planck systems in stochastic electromagnetic fields: local well-posedness

Jacob Bedrossian, Stavros Papathanasiou

摘要：

本文在$\mathbb T^d \times \mathbb R^d$中构造了受外部空间规则时变白电磁场作用的Vlasov-Poisson（VP）和Vlasov-Poisson-Fokker-Planck（VPFP）系统的唯一时变局部强解.初始条件采用具有$σ> d/2 + 1$的$H^σ$（除多项式速度权重外）。我们还表明，解决方案的VPFP是即时$C^\infty_{x,v}$由于亚椭圆正则化，如果外部力场是光滑的。动力学方程中的外力作为速度的随机输运而产生，这意味着，连同非线性中$x$和$v$之间的各向异性，局部理论比受到附加随机力或随机输运的可比流体力学方程稍微复杂一些。虽然随机电磁场经常在等离子体物理学文献中讨论，但据我们所知，这是第一次对非线性随机动力学方程的强解进行数学研究。

英文摘要：

In this paper, we construct unique, local-in-time strong solutions to the Vlasov-Poisson (VP) and Vlasov-Poisson-Fokker-Planck (VPFP) systems subjected to external, spatially regular, white-in-time electromagnetic fields in $\mathbb T^d \times \mathbb R^d$. Initial conditions are taken $H^σ$ with $σ> d/2 + 1$ (in addition to polynomial velocity weights). We additionally show that solutions to the VPFP are instantly $C^\infty_{x,v}$ due to hypoelliptic regularization if the external force fields are smooth. The external forcing arises in the kinetic equation as a stochastic transport in velocity, which means, together with the anisotropy between $x$ and $v$ in the nonlinearity, that the local theory is a little more complicated than comparable fluid mechanics equations subjected to either additive stochastic forcing or stochastic transport. Although stochastic electromagnetic fields are often discussed in the plasma physics literature, to our knowledge, this is the first mathematical study of strong solutions to nonlinear stochastic kinetic equations.

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