广义 Brinkman-Forchheimer 方程的渐近性态

数学物理学报 ›› 2025, Vol. 45 ›› Issue (1): 74-91.

广义 Brinkman-Forchheimer 方程的渐近性态

李心(),郝文娟(),刘洋^*()

燕山大学理学院河北秦皇岛 066004

收稿日期:2023-09-04 修回日期:2023-12-25 出版日期:2025-02-26 发布日期:2025-01-08
通讯作者: * 刘洋, E-mail:lliuyang@ysu.edu.cn
作者简介:李心, E-mail:li_xin@ysu.edu.cn|郝文娟, E-mail:18732388934@163.com
基金资助:
国家自然科学基金(11801493);国家自然科学基金(12071192);河北自然科学基金(A2018203309);河北自然科学基金(A2022203004);河北省教育厅高等学校科技计划青年基金(QN2020203)

The Asymptotic Behavior of the Generalized Brinkman-Forchheimer Equation

Xin Li(),Wenjuan Hao(),Yang Liu^*()

School of Science, Yanshan University, Hebei Qinhuangdao 066004

Received:2023-09-04 Revised:2023-12-25 Online:2025-02-26 Published:2025-01-08
Supported by:
NSFC(11801493);NSFC(12071192);Hebei Natural Science Foundation of China(A2018203309);the Hebei Natural Science Foundation of China(A2022203004);Hebei Provincial Department of Education Higher Science and Technology Plan Youth Fund(QN2020203)

摘要/Abstract

摘要：

该文研究了定义在有界域上的三维轻微可压缩广义 Brinkman-Forchheimer 方程解的适定性和长时间性态问题. 该方程模拟了由 Lévy 耗散主导的穿越多孔介质流体的传输过程. 首先, 运用经典紧致性方法和先验估计证明了方程在能量空间上解的适定性. 其次, 引入系统分解思想: 一方面, 用局部化方法证明了方程收缩部分在初始能量空间中的有界性; 另一方面, 通过瞬时光滑化方法得到了方程光滑部分在高阶能量空间中的指数耗散性, 并最终验证了该方程在初始相空间中全局吸引子和指数吸引子的存在性.

关键词: 轻微可压缩 Brinkman-Forchheimer 方程, 适定性, 正则性与部分光滑性, 全局吸引子, 指数吸引子.

Abstract:

This article investigated the well-posedness and long-term behavior problems of solutions to 3D compressible generalized Brinkman-Forchheimer equation defined on a bounded domain. The equation simulates the transport process of fluid through porous medium dominated by Lévy dissipation. Firstly, the classical compactness method and a prior estimation were used to prove the well posedness of the solution of the equation in the energy space. Secondly, introduce the concept of system decomposition: on the one hand, the localization method was used to prove the boundedness of the contraction part of the equation in the initial energy space; on the other hand, the exponential dissipation of the smooth part of the equation in the high-order energy space is obtained by the instantaneous optical smoothing method, and the existence of the global attractor and the exponential attractor of the equation in the initial phase space is finally verified.

Key words: slightly compressible Brinkman-Forchheimer equation, well-posedness, regularity and partial smoothing, global attractor, exponential attractor.

中图分类号:

O175.2

李心, 郝文娟, 刘洋. 广义 Brinkman-Forchheimer 方程的渐近性态[J]. 数学物理学报, 2025, 45(1): 74-91.

Xin Li, Wenjuan Hao, Yang Liu. The Asymptotic Behavior of the Generalized Brinkman-Forchheimer Equation[J]. Acta mathematica scientia,Series A, 2025, 45(1): 74-91.

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