American Institute of Mathematical Sciences

Advanced
January  2005, 12(1): 1-12. doi: 10.3934/dcds.2005.12.1

Global well-posedness of the viscous Boussinesq equations

Thomas Y. Hou 1, and  Congming Li 2,

1. 

Department of Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA 91125, United States
2. 

Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO 80309-0524

Received  November 2004 Revised  November 2004 Published  December 2004

We prove the global well-posedness of the viscous incompressible Boussinesq equations in two spatial dimensions for general initial data in $H^m$ with $m\ge 3$. It is known that when both the velocity and the density equations have finite positive viscosity, the Boussinesq system does not develop finite time singularities. We consider here the challenging case when viscosity enters only in the velocity equation, but there is no viscosity in the density equation. Using sharp and delicate energy estimates, we prove global existence and strong regularity of this viscous Boussinesq system for general initial data in $H^m$ with $m \ge 3$.
Keywords: Boussinesq equations, vortex stretching, fluid mechannics., global existence.
Mathematics Subject Classification: 76B03, 76D03, 76D05, 35M1.
Citation: Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1
[1]

Šárka Nečasová, Joerg Wolf. On the existence of global strong solutions to the equations modeling a motion of a rigid body around a viscous fluid. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1539-1562. doi: 10.3934/dcds.2016.36.1539
[2]

Yuming Qin, Yang Wang, Xing Su, Jianlin Zhang. Global existence of solutions for the three-dimensional Boussinesq system with anisotropic data. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1563-1581. doi: 10.3934/dcds.2016.36.1563
[3]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045
[4]

Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira, Élder J. Villamizar-Roa. Global existence for an attraction-repulsion chemotaxis fluid model with logistic source. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 423-447. doi: 10.3934/dcdsb.2018180
[5]

Miao Liu, Weike Wang. Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1203-1222. doi: 10.3934/cpaa.2014.13.1203
[6]

Igor Chueshov, Irena Lasiecka. Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 777-809. doi: 10.3934/dcds.2006.15.777
[7]

Jimmy Garnier, FranÇois Hamel, Lionel Roques. Transition fronts and stretching phenomena for a general class of reaction-dispersion equations. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 743-756. doi: 10.3934/dcds.2017031
[8]

D. L. Denny. Existence of solutions to equations for the flow of an incompressible fluid with capillary effects. Communications on Pure and Applied Analysis, 2004, 3 (2) : 197-216. doi: 10.3934/cpaa.2004.3.197
[9]

Myeongju Chae, Kyungkeun Kang, Jihoon Lee. Existence of smooth solutions to coupled chemotaxis-fluid equations. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2271-2297. doi: 10.3934/dcds.2013.33.2271
[10]

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 and Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115
[11]

Xiaojing Xu. Local existence and blow-up criterion of the 2-D compressible Boussinesq equations without dissipation terms. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1333-1347. doi: 10.3934/dcds.2009.25.1333
[12]

Wenru Huo, Aimin Huang. The global attractor of the 2d Boussinesq equations with fractional Laplacian in subcritical case. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2531-2550. doi: 10.3934/dcdsb.2016059
[13]

Renhui Wan. Global well-posedness for the 2D Boussinesq equations with a velocity damping term. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2709-2730. doi: 10.3934/dcds.2019113
[14]

Xiaoqiang Dai, Shaohua Chen. Global well-posedness for the Cauchy problem of generalized Boussinesq equations in the control problem regarding initial data. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4201-4211. doi: 10.3934/dcdss.2021114
[15]

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
[16]

Feng Li, Yuxiang Li. Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5409-5436. doi: 10.3934/dcdsb.2019064
[17]

Seung-Yeal Ha, Bingkang Huang, Qinghua Xiao, Xiongtao Zhang. A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data. Communications on Pure and Applied Analysis, 2020, 19 (2) : 835-882. doi: 10.3934/cpaa.2020039
[18]

Marco Di Francesco, Alexander Lorz, Peter A. Markowich. Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1437-1453. doi: 10.3934/dcds.2010.28.1437
[19]

Daniela Giachetti, Maria Michaela Porzio. Global existence for nonlinear parabolic equations with a damping term. Communications on Pure and Applied Analysis, 2009, 8 (3) : 923-953. doi: 10.3934/cpaa.2009.8.923
[20]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1473-1493. doi: 10.3934/dcdss.2020083

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (305)
  • HTML views (0)
  • Cited by (209)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy