具有非等温边界条件的退化线性动力学方程的渐近行为

IF 2.4 2区 数学 Q1 MATHEMATICS
Armand Bernou
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引用次数: 0

摘要

研究了变温度下边界处具有广义漫反射的有界区域内的退化线性玻尔兹曼方程,包括具有壁面反射核或重尾反射核的麦克斯韦边界条件和cercignini - lampis边界条件。我们的抽象碰撞设置适用于线性BGK模型,向空间相关稳态的松弛,以及具有胖尾的碰撞核。我们首次证明了稳态的存在性和收敛速度,而没有对温度变化的假设。我们对于Cercignani-Lampis边界条件的结果也没有对调节系数作任何假设。当满足碰撞算子简并性的控制条件时,证明的速率为指数速率,当不满足该假设时,证明的速率仅为多项式速率,这与我们之前关于自由输运方程的结果一致。我们还提供了当稳态有界时不同收敛速率的精确描述,包括下界。我们的方法得到构造常数。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Asymptotic behavior of degenerate linear kinetic equations with non-isothermal boundary conditions
We study the degenerate linear Boltzmann equation inside a bounded domain with a generalized diffuse reflection at the boundary and variable temperature, including the Maxwell boundary conditions with the wall Maxwellian or heavy-tailed reflection kernel and the Cercignani-Lampis boundary condition. Our abstract collisional setting applies to the linear BGK model, the relaxation towards a space-dependent steady state, and collision kernels with fat tails. We prove for the first time the existence of a steady state and a rate of convergence towards it without assumptions on the temperature variations. Our results for the Cercignani-Lampis boundary condition make also no hypotheses on the accommodation coefficients. The proven rate is exponential when a control condition on the degeneracy of the collision operator is satisfied, and only polynomial when this assumption is not met, in line with our previous results regarding the free-transport equation. We also provide a precise description of the different convergence rates, including lower bounds, when the steady state is bounded. Our method yields constructive constants.
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来源期刊
CiteScore
4.40
自引率
8.30%
发文量
543
审稿时长
9 months
期刊介绍: The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools. Research Areas Include: • Mathematical control theory • Ordinary differential equations • Partial differential equations • Stochastic differential equations • Topological dynamics • Related topics
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