American Institute of Mathematical Sciences

Advanced
May  2010, 9(3): 667-684. doi: 10.3934/cpaa.2010.9.667

On the stability problem for the Boussinesq equations in weak-$L^p$ spaces

Lucas C. F. Ferreira 1, and  Elder J. Villamizar-Roa 2,

1. 

Universidade Estadual de Campinas, Campinas, CEP 13083-970, Brazil
2. 

Universidad Nacional de Colombia, Sede Medellín, Medellín, A.A. 3840, Colombia

Received  April 2009 Revised  September 2009 Published  January 2010

We consider the Boussinesq equations in either an exterior domain in $\mathbb{R}^{n}$, the whole space $\mathbb{R}^{n}$, the half space $\mathbb{R}_{+}^{n}$ or a bounded domain in $\mathbb{R}^{n}$, where the dimension $n$ satisfies $n \geq 3$. We give a class of stable steady solutions, which improves and complements the previous stability results. Our results give a complete answer to the stability problem for the Boussinesq equations in weak-$L^{p}$ spaces, in the sense that we only assume that the stable steady solution belongs to scaling invariant class $L_{\sigma }^{(n,\infty)}\times L^{(n,\infty)}$. Moreover, some considerations about the exponential decay (in bounded domains) and the uniqueness of the disturbance are done.
Keywords: Boussinesq equations, strong solutions., stability.
Mathematics Subject Classification: Primary: 35Q35, 76D03; Secondary: 76M0.
Citation: Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the stability problem for the Boussinesq equations in weak-$L^p$ spaces. Communications on Pure & Applied Analysis, 2010, 9 (3) : 667-684. doi: 10.3934/cpaa.2010.9.667
[1]

Elder J. Villamizar-Roa, Elva E. Ortega-Torres. On a generalized Boussinesq model around a rotating obstacle: Existence of strong solutions. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 825-847. doi: 10.3934/dcdsb.2011.15.825
[2]

Chun-Hsiung Hsia, Tian Ma, Shouhong Wang. Rotating Boussinesq equations: Dynamic stability and transitions. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 99-130. doi: 10.3934/dcds.2010.28.99
[3]

Zhijian Yang, Pengyan Ding, Xiaobin Liu. Attractors and their stability on Boussinesq type equations with gentle dissipation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 911-930. doi: 10.3934/cpaa.2019044
[4]

Yonggeun Cho, Tohru Ozawa. On small amplitude solutions to the generalized Boussinesq equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 691-711. doi: 10.3934/dcds.2007.17.691
[5]

Tong Tang, Hongjun Gao. Local strong solutions to the compressible viscous magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1617-1633. doi: 10.3934/dcdsb.2016014
[6]

Agissilaos G. Athanassoulis, Gerassimos A. Athanassoulis, Mariya Ptashnyk, Themistoklis Sapsis. Strong solutions for the Alber equation and stability of unidirectional wave spectra. Kinetic & Related Models, 2020, 13 (4) : 703-737. doi: 10.3934/krm.2020024
[7]

Claudia Valls. Stability of some waves in the Boussinesq system. Communications on Pure & Applied Analysis, 2006, 5 (4) : 929-939. doi: 10.3934/cpaa.2006.5.929
[8]

Marta Lewicka, Mohammadreza Raoofi. A stability result for the Stokes-Boussinesq equations in infinite 3d channels. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2615-2625. doi: 10.3934/cpaa.2013.12.2615
[9]

Siran Li, Jiahong Wu, Kun Zhao. On the degenerate boussinesq equations on surfaces. Journal of Geometric Mechanics, 2020, 12 (1) : 107-140. doi: 10.3934/jgm.2020006
[10]

T. Tachim Medjo. Existence and uniqueness of strong periodic solutions of the primitive equations of the ocean. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1491-1508. doi: 10.3934/dcds.2010.26.1491
[11]

Youcef Amirat, Kamel Hamdache. Strong solutions to the equations of flow and heat transfer in magnetic fluids with internal rotations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3289-3320. doi: 10.3934/dcds.2013.33.3289
[12]

Jochen Merker. Strong solutions of doubly nonlinear Navier-Stokes equations. Conference Publications, 2011, 2011 (Special) : 1052-1060. doi: 10.3934/proc.2011.2011.1052
[13]

Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395
[14]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115
[15]

Yan Jia, Xingwei Zhang, Bo-Qing Dong. Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion. Communications on Pure & Applied Analysis, 2013, 12 (2) : 923-937. doi: 10.3934/cpaa.2013.12.923
[16]

Shengfu Deng. Generalized multi-hump wave solutions of Kdv-Kdv system of Boussinesq equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3671-3716. doi: 10.3934/dcds.2019150
[17]

Tianwen Luo, Tao Tao, Liqun Zhang. Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3737-3765. doi: 10.3934/dcds.2019230
[18]

Jianhua Huang, Tianlong Shen, Yuhong Li. Dynamics of stochastic fractional Boussinesq equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2051-2067. doi: 10.3934/dcdsb.2015.20.2051
[19]

Mirosław Lachowicz, Andrea Quartarone, Tatiana V. Ryabukha. Stability of solutions of kinetic equations corresponding to the replicator dynamics. Kinetic & Related Models, 2014, 7 (1) : 109-119. doi: 10.3934/krm.2014.7.109
[20]

Zhong Tan, Leilei Tong. Asymptotic stability of stationary solutions for magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3435-3465. doi: 10.3934/dcds.2017146

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences