VPB 方程的边界层分析——献给陈化教授 70 寿辰

数学物理学报 ›› 2026, Vol. 46 ›› Issue (2): 840-876.

• • 上一篇

VPB 方程的边界层分析——献给陈化教授 70 寿辰

刘慧¹(), 江宁²^,^*(), 罗益龙³(), 唐少君⁴()

¹ 淮南师范学院金融与数学学院安徽淮南 232038
² 武汉大学数学与统计学院, 武汉 430072
³ 湖南大学数学学院长沙 410082
⁴ 武汉理工大学数学与统计学院武汉 430070

收稿日期:2026-04-06 修回日期:2026-04-12 出版日期:2026-04-26 发布日期:2026-04-27
通讯作者: 江宁 E-mail:huiliu@hnnu.edu.cn;njiang@whu.edu.cn;luoylmath@hnu.edu.cn;shaojun.tang@whut.edu.cn
作者简介:刘慧, Email：huiliu@hnnu.edu.cn
罗益龙, Email：luoylmath@hnu.edu.cn
唐少君, Email：shaojun.tang@whut.edu.cn
基金资助:
国家重点研发计划项目(2023YFA1010300);安徽省高校科研项目(2024AH051744);淮南师范学院科研基金(824001);国家自然科学基金(12371224);国家自然科学基金(11971360);国家自然科学基金(11731008);国家自然科学基金(12221001);国家自然科学基金(12201220);国家自然科学基金(12201480);广东省基础与应用基础研究基金(2024A1515012358);湖南大学中央高校基本科研业务费专项资金(531118011008);中央高校基本科研业务费专项资金(223114007);中央高校基本科研业务费专项资金(104972025KFYjc0117)

Boundary Layer Analysis of the Vlasov-Poisson-Boltzmann Equations with Maxwell Reflection Boundary Conditon in Half-Space

Hui Liu¹(), Ning Jiang²^,^*(), Yilong Luo³(), Shaojun Tang⁴()

¹ School of Finance and Mathematics, Huainan Normal University, Anhui Huainan 232038
² School of Mathematics and Statistics, Wuhan University, Wuhan 430072
³ School of Mathematics, Hunan University, Changsha 410082
⁴ Department of Mathematics, Wuhan University of Technology, Wuhan 430070

Received:2026-04-06 Revised:2026-04-12 Online:2026-04-26 Published:2026-04-27
Contact: Ning Jiang E-mail:huiliu@hnnu.edu.cn;njiang@whu.edu.cn;luoylmath@hnu.edu.cn;shaojun.tang@whut.edu.cn
Supported by:
National Key R&D Program of China(2023YFA1010300);Scientiffc Research Project of Universities in Anhui Province(2024AH051744);Huainan Normal University Research Fund Program(824001);NSFC(12371224);NSFC(11971360);NSFC(11731008);NSFC(12221001);NSFC(12201220);NSFC(12201480);Guang Dong Basic and Applied Basic Research Foundation(2024A1515012358);Fundamental Research Funds for the Central Universities of Hunan University(531118011008);Fundamental Research Funds for the Central Universities of Wuhan University of Technology(223114007);Fundamental Research Funds for the Central Universities of Wuhan University of Technology(104972025KFYjc0117)

摘要/Abstract

摘要：

作为动理学方程的典型模型, Vlasov-Poisson-Boltzmann 方程 (以下简称 VPB 方程) 描述了等离子体中带电粒子在粒子碰撞及电场的耦合作用下的运动过程. 对于尺度化的 VPB 方程, 当 Knudsen 数趋于零时, 其流体动力学极限在物理和数学上都是一个十分重要的问题. 该文对半空间上具有 Maxwell 反射边界条件的 VPB 方程, 利用 Hilbert 展开方法, 当 Knudsen 数足够小时, 得到可压缩 Euler-Poisson 方程的流体极限及其边界层的形式分析过程. 通过采用归纳的方法形式推导出各阶内部、粘性层的宏观流体方程, 以及 Knudsen 层满足的方程, 体现出了边界层之间的耦合关系以及形式上的求解过程.

关键词: Vlasov-Poisson-Boltzmann 方程, Maxwell 反射边界条件, 边界层分析, Hilbert 展开

Abstract:

As a typical model of kinetic equations, the Vlasov-Poisson-Boltzmann (VPB) equation describes the motion of charged particles in a plasma under the coupled effects of particle collisions and electric fields. For the scaled VPB equation, its hydrodynamic limit as the Knudsen number tends to zero is a problem of great importance in both physics and mathematics. In this paper, for the VPB equation on a half-space with the Maxwell reflection boundary condition, using the Hilbert expansion method, we formally analyze the fluid limit of the compressible Euler-Poisson system and its boundary layer when the Knudsen number is sufficiently small. By means of an inductive approach, we formally derive the macroscopic fluid equations for the interior and viscous layer at each order, as well as the equations governing the Knudsen layer, thereby revealing the coupling relations among the boundary layers and presenting a formal solution procedure.

Key words: Vlasov-Poisson-Boltzmann equations, Maxwell reflection boundary condition, boundary layer analysis, Hilbert expansion

中图分类号:

O175.2

刘慧, 江宁, 罗益龙, 唐少君. VPB 方程的边界层分析——献给陈化教授 70 寿辰[J]. 数学物理学报, 2026, 46(2): 840-876.

Hui Liu, Ning Jiang, Yilong Luo, Shaojun Tang. Boundary Layer Analysis of the Vlasov-Poisson-Boltzmann Equations with Maxwell Reflection Boundary Conditon in Half-Space[J]. Acta mathematica scientia,Series A, 2026, 46(2): 840-876.

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VPB 方程的边界层分析——献给陈化教授 70 寿辰

Boundary Layer Analysis of the Vlasov-Poisson-Boltzmann Equations with Maxwell Reflection Boundary Conditon in Half-Space

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