Boltzmann方程的适定性和平擦碰撞极限研究

批准号:
12001552
项目类别:
青年科学基金项目
资助金额:
24.0 万元
负责人:
周玉龙
依托单位:
学科分类:
混合型、退化型偏微分方程
结题年份:
2023
批准年份:
2020
项目状态:
已结题
项目参与者:
周玉龙
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中文摘要
朗道(Landau)逼近问题,即如何说明从玻尔兹曼(Boltzmann)方程推导朗道方程的合理性和正当性,是统计物理中的重要数学问题。该问题物理背景清晰、研究历史悠久,吸引了包括菲尔兹奖得主Villani在内的众多优秀数学家。然而,朗道逼近问题的现有理论结果局限于有限时间弱解或者局部经典解,本项目拟从整体经典解层面回答该问题。主要研究内容有:(1)在平擦碰撞极限过程中,刻画线性化玻尔兹曼算子的渐近行为,重点探究其耗散结构和谱隙估计的变化规律;(2)建立平擦碰撞主导的玻尔兹曼方程的整体适定性,并证明解的正则性传播;(3)证明玻尔兹曼方程的解,在平擦碰撞极限过程中,收敛到朗道方程的解,并给出含显式收敛率的渐近展开公式。通过本项目的研究,申请人期望在空间非均匀情形,在低正则函数空间,从整体经典解层面,正面回答朗道逼近问题,旨在完善该问题的现有数学结果,得到朗道方程推导过程的正当性。
英文摘要
Landau approximation problem, i.e., how to justify the process of deriving the Landau equation from the Boltzmann equation, is an important mathematical problem in statistical physics. The problem has a clear physical background and a long research history, and has attracted many excellent mathematicians including Fields medal winner Villani. However, existing theoretical results of the problem are limited within finite-time weak solution framework or local-in-time classical solution. The project is intended to solve the problem at the level of global-in-time classical solution. The main contents of the project include: (1) characterize the asymptotic behavior of the linearized Boltzmann operator in the grazing limit process, in particular, investigate its dissipation structure and spectral gap estimate; (2) establish global well-posedness of Boltzmann equation with grazing collision concentration, and prove propagation of regularity; (3) prove the solution of the Boltzmann equation converges to that of the Landau equation in the grazing limit, additionally, explicit convergence rate will be given in the asymptotic formula. Once the project is successfully finished, the Landau problem is positively answered in a global-in-time classical solution sense. More importantly, the setting is spatially inhomogeneous and the solution belongs to a space with low regularity, which well complements existing literature of the important problem and justifies the derivation of the Landau equation from the Boltzmann equation.
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DOI:10.1137/22m1515963
发表时间:2023-03
期刊:SIAM J. Math. Anal.
影响因子:--
作者:Lingbing He;Yuanjie Lei;Yu-Long Zhou
通讯作者:Lingbing He;Yuanjie Lei;Yu-Long Zhou
Asymptotic analysis of the linearized Boltzmann collision operator from angular cutoff to non-cutoff
DOI:10.4171/aihpc/28
发表时间:2018-05
期刊:Annales de l'Institut Henri Poincaré C, Analyse non linéaire
影响因子:--
作者:Lingbing He;Yu-Long Zhou
通讯作者:Lingbing He;Yu-Long Zhou
DOI:10.4171/AIHPC/72
发表时间:2024
期刊:Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire
影响因子:--
作者:Renjun Duan;Ling-Bing He;Tong Yang;Yu-Long Zhou
通讯作者:Yu-Long Zhou
DOI:https://doi.org/10.1016/j.aim.2023.109234
发表时间:2023
期刊:Advances in Mathematics
影响因子:--
作者:Yu-Long Zhou
通讯作者:Yu-Long Zhou
DOI:https://doi.org/10.1016/j.jfa.2023.110197
发表时间:2024
期刊:Journal of Functional Analysis
影响因子:--
作者:Tong Yang;Yu-Long Zhou
通讯作者:Yu-Long Zhou
角度非截断量子Boltzmann方程的数学理论研究
- 批准号:--
- 项目类别:省市级项目
- 资助金额:15.0万元
- 批准年份:2024
- 负责人:周玉龙
- 依托单位:
国内基金
海外基金
