发表机构
School of Mathematics, Guangxi University; Center for Applied Mathematical of Guangxi (Guangxi University)（广西大学数学学院; 广西应用数学中心（广西大学））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究三维环面均匀剪切流下的Vlasov-Poisson-Boltzmann系统，结合多种分析方法证明自相似分布的全局稳定性，并推导零频系统得到剪切诱导能量增长的精确长时间描述。
AI 中文摘要
我们在三维环面上研究均匀剪切流条件下的Vlasov-Poisson-Boltzmann系统。对于带有Grad角截断的麦克斯韦分子及足够小的剪切率，我们考虑剪切玻尔兹曼方程空间均匀自相似分布附近的扰动。在自相似变量下，我们证明该分布的全局稳定性，包括全局存在性与唯一性、非空间导数在加权L∞空间中的指数衰减，以及重整化扰动的一致有界性。分析结合了Caflisch分解、Guo的L∞-L²估计、宏-微分析以及零频模式的谱研究。我们进一步推导了重整化总能量和合适二阶矩的闭合零频系统，得到剪切诱导能量增长的精确长时间描述。
英文摘要
We study the Vlasov--Poisson--Boltzmann system on the three-dimensional torus under uniform shear flow. For Maxwell molecules with Grad's angular cutoff and sufficiently small shear rate, we consider perturbations around the spatially homogeneous self-similar profile of the sheared Boltzmann equation. In self-similar variables, we prove the global stability of this profile, including global existence and uniqueness, the exponential decay of nonzero-spatial derivatives in weighted $L^\infty$ spaces, and the uniform boundedness of the renormalized perturbation. The analysis combines a Caflisch's decomposition, Guo's $L^\infty$--$L^2$ estimates, macro--micro analysis, and a spectral study of the zero-frequency mode. We further derive a closed zero-frequency system for a renormalized total energy and suitable second-order moments, which yields a precise large-time description of the shear-induced energy growth.
