带阻尼Boussinesq方程的一类大初值整体光滑解

数学物理学报 ›› 2025, Vol. 45 ›› Issue (4): 1077-1085.

带阻尼Boussinesq方程的一类大初值整体光滑解

朱伟鹏^1,^*(),李金禄²(),吴星³()

¹佛山大学数学学院广东佛山 528000
²赣南师范大学数学与计算机科学学院江西赣州 341000
³河南农业大学信息与管理科学学院郑州 450002

收稿日期:2024-02-26 修回日期:2024-10-15 出版日期:2025-08-26 发布日期:2025-08-01
通讯作者: *E-mail: mathzwp2010@163.com
作者简介:E-mail: lijinlu@gnnu.edu.cn;|ny2008wx@163.com
基金资助:
国家自然科学基金(12201118);国家自然科学基金(12161004);赣鄱俊才支持计划——主要学科学术和技术带头人培养项目(20232BCJ23009)

Global Smooth Solutions of the Damped Boussinesq Equations with a Class of Large Initial Data

Zhu Weipeng^1,^*(),Li Jinlu²(),Wu Xing³()

¹School of Mathematics, Foshan University, Guangdong Foshan 528000
²School of Mathematics and Computer Sciences, Gannan Normal University, Jiangxi Ganzhou 341000
³College of Information and Management Science, Henan Agricultural University, Zhengzhou, 450002

Received:2024-02-26 Revised:2024-10-15 Online:2025-08-26 Published:2025-08-01
Supported by:
NSFC(12201118);NSFC(12161004);Training Program for Academic and Technical Leaders of Major Disciplines in Ganpo Jun-cai Support Program(20232BCJ23009)

摘要/Abstract

摘要：

无粘性 Boussinesq 方程的全局正则性问题是数学领域的一个公开问题. 为了探究此问题, 该文研究了带阻尼的 Boussinesq 方程, 并分析了阻尼效应对解的正则性的影响. 特别地, 考虑了带阻尼 Boussinesq 方程在一类任意大 $L^\infty$ 范数的初值条件下的全局存在性. 通过带阻尼的 Boussinesq 方程的结构特性和其线性化方程解的指数衰减性质, 得到了全局光滑解的存在唯一性.

关键词: Boussinesq 方程, 整体存在性, 大初值

Abstract:

The global regularity problem concerning the inviscid Boussinesq equations remains an open problem. In an attempt to understand this problem, we examine the damped Boussinesq equations and study how damping affects the regularity of solutions. In this paper, we consider the global existence to the damped Boussinesq equations with a class of large initial data, whose $L^\infty$ norm can be arbitrarily large. The idea is splitting the linear Boussinesq equations from the damped Boussinesq equations, the exponentially decaying solution of the former equations together with the structure of the Boussinesq equations help us to obtain the global smooth solutions.

Key words: Boussinesq equations, global existence, large initial data

中图分类号:

O175.29

朱伟鹏, 李金禄, 吴星. 带阻尼Boussinesq方程的一类大初值整体光滑解[J]. 数学物理学报, 2025, 45(4): 1077-1085.

Zhu Weipeng, Li Jinlu, Wu Xing. Global Smooth Solutions of the Damped Boussinesq Equations with a Class of Large Initial Data[J]. Acta mathematica scientia,Series A, 2025, 45(4): 1077-1085.

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