分数阶 Burgers 方程时间周期弱解的唯一性与渐近稳定性

数学物理学报 ›› 2023, Vol. 43 ›› Issue (6): 1710-1722.

分数阶 Burgers 方程时间周期弱解的唯一性与渐近稳定性

徐斐¹(),张勇^2,^*()

¹江苏大学数学科学学院江苏镇江 212013
²江苏大学应用系统分析研究院江苏镇江 212013

收稿日期:2022-10-08 修回日期:2023-04-10 出版日期:2023-12-26 发布日期:2023-11-16
通讯作者: *张勇，E-mail: 18842629891@163.com
作者简介:徐斐，E-mail: xufeiujs@126.com
基金资助:
国家自然科学基金(11971202)

Uniqueness and Asymptotic Stability of Time-Periodic Solutions for the Fractional Burgers Equation

Xu Fei¹(),Zhang Yong^2,^*()

¹School of Mathematical Sciences, Jiangsu University, Jiangsu Zhenjiang 212013
²Institute of Applied System Analysis, Jiangsu University, Jiangsu Zhenjiang 212013

Received:2022-10-08 Revised:2023-04-10 Online:2023-12-26 Published:2023-11-16
Supported by:
NSFC(11971202)

摘要/Abstract

摘要：

该文主要讨论实轴上分数阶 Burgers 方程时间周期弱解的性质. 首先利用伽辽金逼近和傅里叶展开的方法, 得到了分数阶 Burgers 方程对应的线性化问题时间周期弱解的存在性. 然后通过构造合适的压缩映射, 得到了非线性方程时间周期弱解的存在唯一性, 并进一步证明了该时间周期弱解是渐近稳定的. 此外, 该文中的方法也可以用来处理有界区间上的问题, 并进一步提升了文献 [5] 中已有的结果.

关键词: 唯一性, 分数阶, 压缩映射, 渐近稳定性

Abstract:

The paper is concerned with the time-periodic (T-periodic) problem of fractional Burgers equation on the real line. Based on the Galerkin approximates and Fourier expansion, we first prove the existence of T-periodic solution to a linearized version. Then, the existence and uniqueness of T-periodic solution for the nonlinear equation are established by constructing a suitable contraction mapping. Furthermore, we show that the unique T-periodic solution is asymptotically stable. In addition, our method can be extended to the classical forced Burgers equation in a bounded region, which improves the known result.

Key words: Uniqueness, Fractional order, Contraction mapping, Asymptotic stability

中图分类号:

O175

徐斐, 张勇. 分数阶 Burgers 方程时间周期弱解的唯一性与渐近稳定性[J]. 数学物理学报, 2023, 43(6): 1710-1722.

Xu Fei, Zhang Yong. Uniqueness and Asymptotic Stability of Time-Periodic Solutions for the Fractional Burgers Equation[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1710-1722.

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