标题： 有界区域中Boltzmann方程的$L^{p}_{v}L^{\infty}_{x}$中的整体解 显示英文标题

标题： Global solutions in $L^{p}_{v}L^{\infty}_{x}$ for the Boltzmann equation in bounded domains

摘要： 对于有界域中Boltzmann方程解的存在性理论，主要是在一致有界函数类如$L^{\infty}_{x,v}$中发展起来的，如[Duan-Huang-Wang-Yang,2017]，[Duan-Wang,2019]，[Guo,2010]。在本文中，我们研究有界域中Boltzmann方程初边值问题的松弛函数空间$L^{p}_{v}L^\infty_{x}$中的解。我们考虑扩散反射边界条件下硬势能的情况，并假设截断模型。对于加权$L^{p}_{v}L^\infty_{x}$空间中相对熵较小的大初始数据，我们构造了唯一全局时间温和解，该解指数收敛到全局麦克斯韦分布。对增益项的点态估计，在$L^p_v$和$L^2_v$范数下有界，是证明我们主要结果的关键。与[Gualdani-Mischler-Mouhot,2017]相比，我们的工作提供了在存在边界条件情况下趋于平衡的另一种视角。 显示英文摘要

摘要： The existence theory for solutions to the Boltzmann equation in bounded domains has primarily been developed within uniformly bounded function classes, such as $L^{\infty}_{x,v}$, as in [Duan-Huang-Wang-Yang,2017], [Duan-Wang,2019], [Guo,2010]. In this paper, we investigate solutions in relaxed function spaces $L^{p}_{v}L^\infty_{x}$ for the initial-boundary value problem of the Boltzmann equation in bounded domains. We consider the case of hard potential under diffuse reflection boundary conditions and assume cutoff model. For large initial data in a weighted $L^{p}_{v}L^\infty_{x}$ space with small relative entropy, we construct unique global-in-time mild solution that converge exponentially to the global Maxwellian. A pointwise estimate for the gain term, bounded in terms of $L^p_v$ and $L^2_v$ norms, is essential to prove our main results. Relative to [Gualdani-Mischler-Mouhot,2017], our work provides an alternative perspective on convergence to equilibrium in the presence of boundary conditions.