GLOBAL CLASSICAL SOLUTIONS AND THE CLASSICAL LIMIT OF THE NON-RELATIVISTIC VLASOV-DARWIN SYSTEM WITH SMALL INITIAL DATA*

doi:10.1007/s10473-023-0507-7

数学物理学报 ›› 2023, Vol. 43 ›› Issue (5): 2043-2060.doi: 10.1007/s10473-023-0507-7

GLOBAL CLASSICAL SOLUTIONS AND THE CLASSICAL LIMIT OF THE NON-RELATIVISTIC VLASOV-DARWIN SYSTEM WITH SMALL INITIAL DATA^*

Yaxian Ma^1,†, Xianwen Zhang²

1. Department of Mathematics, North University of China, Taiyuan 030051, China;
2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

收稿日期:2022-03-02 修回日期:2023-05-02 出版日期:2023-10-26 发布日期:2023-10-25

GLOBAL CLASSICAL SOLUTIONS AND THE CLASSICAL LIMIT OF THE NON-RELATIVISTIC VLASOV-DARWIN SYSTEM WITH SMALL INITIAL DATA^*

Yaxian Ma^1,†, Xianwen Zhang²

1. Department of Mathematics, North University of China, Taiyuan 030051, China;
2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received:2022-03-02 Revised:2023-05-02 Online:2023-10-26 Published:2023-10-25
Contact: †Yaxian Ma, E-mail: 20210059@nuc.edu.cn
About author:Xianwen Zhang, E-mail: xwzhang@hust.edu.cn
Supported by:
National Natural Science Foundation of China (11871024) and the Fundamental Research Program of Shanxi Province (202103021223182).

摘要/Abstract

摘要： We investigate the global classical solutions of the non-relativistic Vlasov-Darwin system with generalized variables (VDG) in three dimensions. We first prove the global existence and uniqueness for small initial data and derive the decay estimates of the Darwin potentials. Then, we show in this framework that the solutions converge in a pointwise sense to solutions of the classical Vlasov-Poisson system (VP) at the asymptotic rate of $\frac{1}{c^2}$ as the speed of light $c$ tends to infinity for all time. Moreover, we obtain rigorously an asymptotic estimate of the difference between the two systems.

关键词: Vlasov-Darwin system, generalized variables, global classical solution, classical limit

Abstract: We investigate the global classical solutions of the non-relativistic Vlasov-Darwin system with generalized variables (VDG) in three dimensions. We first prove the global existence and uniqueness for small initial data and derive the decay estimates of the Darwin potentials. Then, we show in this framework that the solutions converge in a pointwise sense to solutions of the classical Vlasov-Poisson system (VP) at the asymptotic rate of $\frac{1}{c^2}$ as the speed of light $c$ tends to infinity for all time. Moreover, we obtain rigorously an asymptotic estimate of the difference between the two systems.

Key words: Vlasov-Darwin system, generalized variables, global classical solution, classical limit

中图分类号:

35Q83

Yaxian Ma, Xianwen Zhang. GLOBAL CLASSICAL SOLUTIONS AND THE CLASSICAL LIMIT OF THE NON-RELATIVISTIC VLASOV-DARWIN SYSTEM WITH SMALL INITIAL DATA^*[J]. 数学物理学报, 2023, 43(5): 2043-2060.

Yaxian Ma, Xianwen Zhang. GLOBAL CLASSICAL SOLUTIONS AND THE CLASSICAL LIMIT OF THE NON-RELATIVISTIC VLASOV-DARWIN SYSTEM WITH SMALL INITIAL DATA^*[J]. Acta mathematica scientia,Series A, 2023, 43(5): 2043-2060.

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GLOBAL CLASSICAL SOLUTIONS AND THE CLASSICAL LIMIT OF THE NON-RELATIVISTIC VLASOV-DARWIN SYSTEM WITH SMALL INITIAL DATA^*

GLOBAL CLASSICAL SOLUTIONS AND THE CLASSICAL LIMIT OF THE NON-RELATIVISTIC VLASOV-DARWIN SYSTEM WITH SMALL INITIAL DATA^*

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