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, 12071192)、河北自然科学基金 (A2018203309, A2022203004) 和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

Li Xin, Hao Wenjuan, Liu Yang

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, 12071192), the Hebei Natural Science Foundation of China (A2018203309, A2022203004) and the 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.

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

参考文献

[1] Fan J, Zhou Y.Global well-posedness of the Navier-Stokes-omega equations. Appl Math Lett, 2011, 24(11): 1915-1918
[2] Koch H, Tataru D.Well-posedness for the Navier-Stokes equations. Adv Math, 2001, 157(1): 22-35
[3] Paicu M, Zhang Z.Global well-posedness for 3D Navier-Stokes equations with ill-prepared initial data. J Inst Math Jussieu, 2014, 13(2): 395-411
[4] Craig W, Huang X, Wang Y.Global wellposedness for the 3D inhomogeneous incompressible Navier-Stokes equations. J Math Fluid Mech, 2013, 15(4): 747-758
[5] Chen Q, Miao C, Zhang Z.Global well-posedness for the 3D rotating Navier-Stokes equations with highly oscillating initial data. Pacific J Math, 2013, 262(2): 263-283
[6] Larios A, Pei Y, Rebholz L.Global well-posedness of the velocity-vorticity-Voigt model of the 3D Navier-Stokes equations. J Differ Equations, 2019, 266(5): 2435-2465
[7] 孙小春, 何港晶. Navier-Stokes-Coriolis 方程解的长时间存在性. 数学物理学报, 2022, 42A(5): 1416-1423
Sun X C, He G J. Long time existence of the solutions for the Navier-Stokes-Coriolis equations. Acta Math Sci, 2022, 42A(5): 1416-1423
[8] Hajduk K W, Robinson J C.Energy equality for the 3D critical convective Brinkman-Forchheimer equations. J Differ Equations, 2017, 263(11): 7141-7161
[9] Kalantarov V, Zelik S.Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities. Commun Pure Appl Anal, 2012, 11(5): 2037-2054
[10] Markowich P A, Titi E S, Trabelsi S. Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model. Nonlinearity, 2016, 29(4): Article 1292
[11] Wang B, Lin S.Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation. Math Method Appl Sci, 2008, 31(12): 1479-1495
[12] You Y, Zhao C, Zhou S.The existence of uniform attractors for 3D Brinkman-Forchheimer equations. Discrete Contin Dyn Syst, 2012, 32(10): 3787-3800
[13] Xueli S, Xi D, Baoming Q.Dimension estimate of the global attractor for a 3D Brinkman-Forchheimer equation. Wuhan University Journal of Natural Sciences, 2023, 28(1): 1-10
[14] Caucao S, Esparza J.An augmented mixed FEM for the convective Brinkman-Forchheimer problem: a priori and a posteriori error analysis. Journal of Computational and Applied Mathematics, 2024, 438: 115517
[15] Hajduk K W, Robinson J C.Energy equality for the 3D critical convective Brinkman-Forchheimer equations. J Differ Equations, 2017, 263(11): 7141-7161
[16] Ghidaglia J M, Temam R.Long time behavior for partly dissipative equations: the slightly compressible 2D-Navier-Stokes equations. Asymptotic Anal, 1988, 1(1): 23-49
[17] Kalantarov V, Zelik S.Asymptotic regularity and attractors for slightly compressible Brinkman-Forchheimer equations. Appl Math Optim, 2021, 84(3): 3137-3171
[18] Córdoba A, Córdoba D.A maximum principle applied to quasi-geostrophic equations. Comm Math Phys, 2004, 249(3): 511-528
[19] 郭柏灵, 蒲学科, 黄凤辉. 分数阶偏微分方程及其数值解. 北京: 科学出版社, 2011
Guo B L, Pu X K, Huang F H. Fractional Partial Differential Equations and their Numerical Solutions. Beijing: Science Press, 2011
[20] Nguyen H Q.Global weak solutions for generalized SQG in bounded domains. Anal PDE, 2018, 11(4): 1029-1047
[21] Constantin P, Ignatova M, Nguyen H Q.Inviscid limit for SQG in bounded domains. SIAM J Math Anal, 2018, 50(6): 6196-6207
[22] Liu Y, Sun C Y.Inviscid limit for the damped generalized incompressible Navier-Stokes equations on ${{T}}^{2}$. Discrete Contin Dyn Syst Ser S, 2021, 14(12): 4383-4408
[23] 孙小春, 吴育联, 徐郜婷. 分数阶不可压缩 Navier-Stokes-Coriolis 方程解的整体适定性. 数学物理学报, 2024, 44A(3): 737-745
Sun X C, Wu Y L, Xu G T. Global well-posedness for the fractional Navier-Stokes equations with the Coriolis force. Acta Math Sci, 2024, 44A(3): 737-745
[24] Pata V.Uniform estimates of Gronwall type. J Math Anal Appl, 2011, 373(1): 264-270
[25] Kalantarov V, Zelik S.Finite-dimensional attractors for the quasi-linear strongly-damped wave equation. J Differ Equations, 2009, 247(4): 1120-1155
[26] Pata V, Zelik S.Smooth attractors for strongly damped wave equations. Nonlinearity, 2006, 19(7): 1495-1506
[27] Zelik S.Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Commun Pure Appl Anal, 2004, 3(4): 921-934
[28] Mei X Y, Savostianov A, Sun C Y, Zelik S.Infinite energy solutions for weakly damped quintic wave equations in ${{R}^{3}}$. Trans Amer Math Soc, 2021, 374(5): 3093-3129
[29] Babin A V, Vishik M I.Attractors of Evolution Equations. Holland: Elsevier, 1992
[30] Vishik M I, Chepyzhov V V.Trajectory attractors of equations of mathematical physics. Russian Math Surveys, 2011, 66(4): 637-731
[31] Ladyzhenskaya O.Attractors for Semi-groups and Evolution Equations (Lezioni Lincee). Cambridge: Cambridge University Press, 1991
[32] Miranville A, Zelik S.Attractors for dissipative partial differential equations in bounded and unbounded domains. Handbook of Differential Equations: Evolutionary Equations, 2008, 4: 103-200
[33] Temam R.Infinite Dimensional Dynamical Systems in Mechanics and Physics. Berlin: Springer-Verlag, 1997
[34] Efendiev M, Miranville A, Zelik S.Exponential attractors for a nonlinear reaction-diffusion system in ${{R}^{3}}$. C R Acad Sci Paris, 2000, 330(8): 713-718
[35] Fabrie P, Galusinski C, Miranville A, Zelik S.Uniform exponential attractors for a singular perturbed damped wave equation. Discrete Contin Dyn Syst, 2004, 10(1/2): 211-238
[36] Adams J F.Stable Homotopy and Generalised Homology. Chicago: University of Chicago Press, 1974