数学物理学报 ›› 2021, Vol. 41 ›› Issue (6): 1853-1863.

• 论文 • 上一篇    下一篇

空间非均匀线性Boltzmann方程在硬势情形下解的最优指数收敛速率

孙宝燕*()   

  1. 烟台大学数学与信息科学学院 山东烟台 264005
  • 收稿日期:2021-07-22 出版日期:2021-12-26 发布日期:2021-12-02
  • 通讯作者: 孙宝燕 E-mail:bysun@ytu.edu.cn
  • 基金资助:
    烟台大学博士科研启动基金(2219008)

Optimal Exponential Decay for the Linear Inhomogeneous Boltzmann Equation with Hard Potentials

Baoyan Sun*()   

  1. School of Mathematics and Information Sciences, Yantai University, Shandong Yantai 264005
  • Received:2021-07-22 Online:2021-12-26 Published:2021-12-02
  • Contact: Baoyan Sun E-mail:bysun@ytu.edu.cn
  • Supported by:
    the Scientific Research Foundation of Yantai University(2219008)

摘要:

该文研究空间非均匀线性Boltzmann方程在周期区域上非角截断硬势情形下解的渐近行为.在带有多项式权的$L_{v}^{1} L_{x}^{2}\left(\langle v\rangle^{k}\right)$空间中得到了解的最优指数收敛速率.将从谱理论和半群理论的角度来对该方程进行分析,主要方法是借助于在Hilbert空间中的强制性估计、算子分解以及空间拉大理论等.

关键词: 线性Boltzmann方程, 硬势, 多项式权, 谱隙, 指数衰减

Abstract:

In this paper, we consider the asymptotic behavior of solutions to the linear spatially inhomogeneous Boltzmann equation for hard potentials in the torus. We obtain an optimal rate of exponential convergence towards equilibrium in a Lebesgue space with polynomial weight $L_{v}^{1} L_{x}^{2}\left(\langle v\rangle^{k}\right)$. This model is analyzed from a spectral point of view and from the point of view of semigroups. Our strategy is taking advantage of the spectral gap estimate in the Hilbert space with inverse Gaussian weight, the factorization argument and the enlargement method.

Key words: Linear Boltzmann equation, Hard potentials, Polynomial weight, Spectral gap, Exponential decay

中图分类号: 

  • O175