带inflow边界条件的Landau方程解的性态研究

数学物理学报 ›› 2020, Vol. 40 ›› Issue (4): 904-917.

带inflow边界条件的Landau方程解的性态研究

刘莉萍¹(),杨航¹(),马璇^2,*()

¹ 暨南大学信息科学技术学院广州 510632
² 武汉纺织大学数学与计算机学院武汉 430073

收稿日期:2019-10-15 出版日期:2020-08-26 发布日期:2020-08-20
通讯作者: 马璇 E-mail:llp@stu2018.jnu.edu.cn;yhang918@stu2018.jnu.edu.cn;xma@wtu.edu.cn
作者简介:刘莉萍, E-mail:llp@stu2018.jnu.edu.cn|杨航, E-mail:yhang918@stu2018.jnu.edu.cn
基金资助:
国家自然科学基金(11601092);国家自然科学基金(11971201)

The Landau Equation with Inflow Boundary Condition in a Finite Channel

Liping Liu¹(),Hang Yang¹(),Xuan Ma^2,*()

¹ College of Information Science and Technology, Jinan University, Guangzhou 510632
² College of Mathematics and Computer Science, Wuhan Textile University, Wuhan 430073

Received:2019-10-15 Online:2020-08-26 Published:2020-08-20
Contact: Xuan Ma E-mail:llp@stu2018.jnu.edu.cn;yhang918@stu2018.jnu.edu.cn;xma@wtu.edu.cn
Supported by:
the NSFC(11601092);the NSFC(11971201)

摘要/Abstract

摘要：

该文研究有限管道上Landau方程的inflow边值问题.在该文中引入了一个新的函数空间，它的主要特征是在空间变量法向方向具有较低的正则性.该文的主要目的是用能量估计的方法，证明Landau方程在一个新的具有一定正则性的函数空间中存在全局唯一解，此外还得到了解的大时间性态以及解沿切向方向的正则性传播.

关键词: Landau方程, 能量估计, inflow边界, 低正则性

Abstract:

This paper is concerned with the inflow boundary value problem of the Landau equation in a finite channel. Based on an elementary energy method, a global strong solution is established for the corresponding problem in a new function space which has mild regularity in normal direction. Moreover, the large time behaviors and the regularity propagation of the solution are also obtained.

Key words: Landau equation, Energy estimation, Inflow boundary, Mild regularity

中图分类号:

O175.29

刘莉萍,杨航,马璇. 带inflow边界条件的Landau方程解的性态研究[J]. 数学物理学报, 2020, 40(4): 904-917.

Liping Liu,Hang Yang,Xuan Ma. The Landau Equation with Inflow Boundary Condition in a Finite Channel[J]. Acta mathematica scientia,Series A, 2020, 40(4): 904-917.

参考文献 11

1	Villani C. A Review of Mathematical Topics in Collisional Kinetic Theory. Amsterdam: North-Holland, 2002
2	Guo Y . The Landau equation in a periodic box. Comm Math Phys, 2002, 231 (3): 391- 434 doi: 10.1007/s00220-002-0729-9
3	Carrapatoso K , Mischler S . Landau equation for very soft and Coulomb potentials near Maxwellians. Ann PDE, 2017, 3 (1): Artcile 1 doi: 10.1007/s40818-017-0021-0
4	Caflisch R E . The Boltzmann equation with a soft potential. II. Nonlinear, spatially-periodic. Comm Math Phys, 1980, 74 (2): 97- 109 doi: 10.1007/BF01197752
5	Strain R M , Guo Y . Exponential decay for soft potentials near Maxwellian. Arch Ration Mech Anal, 2008, 187 (2): 287- 339 doi: 10.1007/s00205-007-0067-3
6	Strain R M , Guo Y . Almost exponential decay near Maxwellian. Comm Partial Differential Equations, 2006, 31 (1/3): 417- 429
7	Strain R M . Optimal time decay of the non cut-off Boltzmann equation in the whole space. Kinet Relat Models, 2012, 5 (3): 583- 613 doi: 10.3934/krm.2012.5.583
8	Sohinger V , Strain R M . The Boltzmann equation, Besov spaces, and optimal time decay rates in Rn x. Adv Math, 2014, 261, 274- 332 doi: 10.1016/j.aim.2014.04.012
9	Kim J, Guo Y, Hwang H J. A L² to L^∞ approach for the Landau equation. 2016, arXiv: 1610.05346
10	Guo Y, Hwang H J, Jang J W, Ouyang Z. The Landau equation with the specular reflection boundary condition. 2019, arXiv: 1905.00173
11	Duan R J, Liu S Q, Sakamoto S, Strain R M. Global mild solutions of the Landau equation and non-cutoff Boltzmann equations. 2019, arXiv: 1904.12086

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[6]	张晓轶. 复Ginzburg Landau方程的自相似解及其极限行为[J]. 数学物理学报, 2005, 25(5): 652-662.
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