VPB 方程的边界层分析——献给陈化教授 70 寿辰

Acta mathematica scientia,Series A ›› 2026, Vol. 46 ›› Issue (2): 840-876.

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)

Abstract

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

CLC Number:

O175.2

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

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