双流体 Euler-Poisson 方程的长波长极限

数学物理学报 ›› 2023, Vol. 43 ›› Issue (2): 399-420.

双流体 Euler-Poisson 方程的长波长极限

李敏¹(),蒲学科^2,^*()

¹山西财经大学应用数学学院太原030006
²广州大学数学与信息科学学院广州510006

收稿日期:2021-11-11 修回日期:2022-09-13 出版日期:2023-04-26 发布日期:2023-04-17
通讯作者: 蒲学科，E-mail:puxueke@gmail.com
作者简介:李敏，E-mail: minlisxcj@163.com
基金资助:
国家自然科学基金(11871172);国家自然科学基金(12201366);广州市科技计划项目(202201020132);山西省基础研究计划(20210302124380);山西省高等学校科技创新项目(2020L0257)

Long-Wavelength Limit for the Two-Fluid Euler-Poisson Equation

Li Min¹(),Pu Xueke^2,^*()

¹Faculty of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006
²School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006

Received:2021-11-11 Revised:2022-09-13 Online:2023-04-26 Published:2023-04-17
Supported by:
NSFC(11871172);NSFC(12201366);Science and Technology Projects in Guangzhou(202201020132);Applied Basic Research Program of Shanxi(20210302124380);Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi(2020L0257)

摘要/Abstract

摘要：

该文研究双流体Euler-Poisson方程的长波长极限. 首先, 借助长波尺度变换和奇异摄动方法, 建立了双流体 Euler-Poisson 方程到 Korteweg-de Vries(KdV)方程的形式推导. 然后, 当 $m_{i}/m_{e}\neq T_{i}/T_{e}$ 时, 通过深入分析余项方程的结构, 并利用能量方法, 从数学上严格证明此极限过程的合理性. 其结果表明: 在 KdV 方程解存在的时间范围内, 双流体 Euler-Poisson 方程的解可收敛到 KdV 方程的解.

关键词: 双流体 Euler-Poisson 方程, 长波长极限, 奇异摄动方法

Abstract:

In this paper, we justify rigorously the long-wavelength limit for the two-fluid Euler-Poisson matrix. Firstly, under a long wave scaling, we establish the formal derivation of the Korteweg-de Vries (KdV) matrix from the two-fluid Euler-Poisson matrix by using a singular perturbation method. Then, with the aid of deep analysis of the complicated coupling structure of the two-fluid Euler-Poisson system and delicate energy estimates depending on such a structure, we prove the convergence of solutions of the Euler-Poisson system to that of the KdV matrix mathematically rigorously when $m_{i}/m_{e}\neq T_{i}/T_{e}$ in a time interval on which the KdV dynamics can be seen.

Key words: Two-fluid Euler-Poisson matrix, Long-wavelength limit, Singular perturbation method

中图分类号:

O175.29

李敏, 蒲学科. 双流体 Euler-Poisson 方程的长波长极限[J]. 数学物理学报, 2023, 43(2): 399-420.

Li Min, Pu Xueke. Long-Wavelength Limit for the Two-Fluid Euler-Poisson Equation[J]. Acta mathematica scientia,Series A, 2023, 43(2): 399-420.

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