带有阻尼项的Boussinesq方程解的大时间性态

数学物理学报 ›› 2021, Vol. 41 ›› Issue (5): 1415-1427.

带有阻尼项的Boussinesq方程解的大时间性态

蔡晓静*(),周艳杰

^{北京工商大学数学与统计学院北京 100048}

收稿日期:2020-07-21 出版日期:2021-10-26 发布日期:2021-10-08
通讯作者: 蔡晓静 E-mail:caixj@th.btbu.edu.cn
基金资助:
国家重点研发计划课题(2017YFC1500301);国家自然基金联合基金重点项目合作项目(U1839206);北京工商大学科研启动基金(19008020161)

Asymptotic Behavior for the Damped Boussinesq

Xiaojing Cai*(),Yanjie Zhou

^{School of Mathematics and Statistics, Beijing Technology and Business University, Beijing 100048}

Received:2020-07-21 Online:2021-10-26 Published:2021-10-08
Contact: Xiaojing Cai E-mail:caixj@th.btbu.edu.cn
Supported by:
the National Key Research and Development Project(2017YFC1500301);the Joint Key Project of the NSFC and the China Earthquake Administration(U1839206);the Research Foundation for Advanced Talents of Beijing Technology and Business University(19008020161)

摘要/Abstract

摘要：

利用Fourier区域分解研究带有阻尼项的三维Boussinesq方程解的渐近行为，得到了具体的衰减率.此外，并和热方程作比较，得到了衰减的上界.

关键词: Boussinesq方程, 阻尼项, Fourier分解, 衰减

Abstract:

This paper mainly focus on the asymptotic behavior of the solutions for the 3D Boussinesq equations with damping term. We obtain the explicit time decay rate of the solutions by means of the Fourier splitting method. In addition, we get the upper-bound of the time decay results compared with the heat equation.

Key words: Boussinesq Equations, Damping term, Fourier Splitting Method, Decay

中图分类号:

O175.29

蔡晓静,周艳杰. 带有阻尼项的Boussinesq方程解的大时间性态[J]. 数学物理学报, 2021, 41(5): 1415-1427.

Xiaojing Cai,Yanjie Zhou. Asymptotic Behavior for the Damped Boussinesq[J]. Acta mathematica scientia,Series A, 2021, 41(5): 1415-1427.

参考文献 18

1	Bresch D , Desjardins B . Existence of global weak solution for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Comm Mech Phys, 2003, 238 (1/2): 211- 223
2	Bresch D , Desjardins B , Lin Chi-Kun . On some compressible fluid models: Korteweg, lubricatior and shallow water systems. Comm Partial Diff Equa, 2003, 28 (3/4): 843- 868
3	Cai X , Jiu Q . Weak and strong solutions for the incompressible Navier-Stokes equations with damping. J Math Anal Appl, 2008, 343 (2): 799- 809 doi: 10.1016/j.jmaa.2008.01.041
4	Ye Z . Regularity and decay of 3D incompressible MHD equations with nonlinear damping terms. Colloq Math, 2015, 139, 185- 203 doi: 10.4064/cm139-2-3
5	Cai X , Lei L . L² decay of the incompressible Navier-Stokes equations with damping. Acta Math Sci, 2010, 30B (4): 1235- 1248
6	Schonbek M E . L² decay for weak solutions of the Navier-Stokes equations. Arch Ration Mech Anal, 1985, 88, 209- 222 doi: 10.1007/BF00752111
7	Schonbek M E . Lower bounds of rates of decay for solutions to the Navier-Stokes equations. J Amer Math Soc, 1991, 4 (3): 423- 449 doi: 10.1090/S0894-0347-1991-1103459-2
8	Zhang L H . Sharp rate of decay of solutions to 2-dimensional Navierr-Stokes equations. Comm Partial Diff Equa, 1995, 20 (1/2): 119- 127
9	Zhou Y . A remark on the decay of solutions to the 3-D Navier-Stokes equations. Math Methods Appl Sci, 2007, 30 (10): 1223- 1229 doi: 10.1002/mma.841
10	Jia Y , Zhang X , Dong B . The asymptotic behavior of solutions to three-dimensional Navier-Stokes equations with nonlinear damping. Nonlinear Anal: RWA, 2011, 12, 1736- 1747 doi: 10.1016/j.nonrwa.2010.11.006
11	Zhou Y . Regularity and uniqueness for the 3D incompressible Navier-Stokes equations with damping. Appl Math Lett, 2012, 25, 1822- 1825 doi: 10.1016/j.aml.2012.02.029
12	Zhang Z , Wu X , Lu M . On the uniqueness of strong solutions to the incompressible Navier-Stokes equations with damping. J Math Anal Appl, 2011, 377, 414- 419 doi: 10.1016/j.jmaa.2010.11.019
13	Jiang Z , Zhu M . The large time behavior of solutions to 3D Navier-Stokes equations with nonlinear damping. Math Mech Appl Sci, 2012, 35, 97- 102 doi: 10.1002/mma.1540
14	Caffarelli L , Kohn R , Nirenberg L . Partial regularity of suitably weak solutions of the Navier-Stokes equations. Comm on Pure and Applied Math, 1982, 25, 771- 831
15	Leray J . Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math, 1934, 63, 193- 248 doi: 10.1007/BF02547354
16	Schonbek M E , Schonbek T P , Süli E . Large time behavier of solution to the Magneto-Hydrodynamics equations. Math Ann, 1996, 304, 717- 756 doi: 10.1007/BF01446316
17	Morimoto H . On non-stationary Boussinesq equations. Proc Japan Acad Ser A, 1991, 67, 159- 161 doi: 10.2183/pjab.67.159
18	Morimoto H . On the existence and uniqueness of the stationary solution to the equation of natural convection. Tokyo J Math, 1991, 14, 217- 226

[1]	章春国, 付煜之, 刘宇标. 二维Mindlin-Timoshenko板系统的稳定性与最优性[J]. 数学物理学报, 2021, 41(5): 1465-1491.
[2]	刘志卿, 方钟波. 具有不一定递减核的线性粘弹性波动方程振动传递问题的一般衰减估计[J]. 数学物理学报, 2021, 41(5): 1428-1444.
[3]	李远飞,石金诚,朱慧珊,黄诗淇. 热弹性方程的快速增长或衰减估计[J]. 数学物理学报, 2021, 41(4): 1042-1052.
[4]	谢倩倩,翟小平,董柏青. N维不可压无阻尼Oldroyd-B模型的最优衰减[J]. 数学物理学报, 2021, 41(3): 762-769.
[5]	陈卿. 可压Navier-Stokes方程解的最高阶导的最佳衰减估计[J]. 数学物理学报, 2021, 41(2): 345-356.
[6]	郭连红. 关于Boussinesq方程组无粘极限的研究[J]. 数学物理学报, 2021, 41(1): 91-99.
[7]	李远飞,李志青. 具有非线性边界条件的瞬态热传导方程的二择一结果[J]. 数学物理学报, 2020, 40(5): 1248-1258.
[8]	仉志余,宋菲菲,李同兴,俞元洪. 带阻尼项的二阶非线性中立型Emden-Fowler微分方程的振动准则[J]. 数学物理学报, 2020, 40(4): 934-946.
[9]	廖梦兰,郭斌. 具变指数源项和强阻尼项的波动方程解的渐近稳定性[J]. 数学物理学报, 2020, 40(1): 146-155.
[10]	李凯,杨晗,王凡. 三维带有衰减项的不可压缩磁流体力学方程组弱解与强解的研究[J]. 数学物理学报, 2019, 39(3): 518-528.
[11]	邱华,谢常平,房少梅. 三维广义MHD方程和Boussinesq方程正则性准则的注记[J]. 数学物理学报, 2019, 39(2): 316-328.
[12]	贺艺军,高怀红,王华,李顺勇. 一类具对数非线性项的伪p-拉普拉斯方程的整体解和爆破的注记[J]. 数学物理学报, 2019, 39(1): 125-132.
[13]	姚绍文, 程志波. 一类带阻尼项的奇性Duffing方程周期解的存在性[J]. 数学物理学报, 2018, 38(3): 543-548.
[14]	冯保伟, 苏克勤. Rⁿ空间中带有密度函数的四阶粘弹方程的一致衰减性[J]. 数学物理学报, 2018, 38(1): 122-133.
[15]	叶耀军, 胡月. 一类高阶波动方程的整体解及指数衰减估计[J]. 数学物理学报, 2018, 38(1): 110-121.