On Asymptotic Properties of the Boussinesq Equations

We address the long time behavior of the Boussinesq system coupling the Navier-Stokes equations driven by density with the non-diffusive equation for the density. We construct solutions of the system justifying previously obtained a priori bounds.

[1]  A. Kiselev,et al.  Small scale formation for the 2D Boussinesq equation , 2022, 2211.05070.

[2]  Gautam Iyer,et al.  Growth of Sobolev norms and loss of regularity in transport equations , 2021, Philosophical Transactions of the Royal Society A.

[3]  Igor Kukavica,et al.  Asymptotic properties of the Boussinesq equations with Dirichlet boundary conditions , 2021, Discrete and Continuous Dynamical Systems.

[4]  T. Elgindi,et al.  Finite-Time Singularity Formation for Strong Solutions to the Boussinesq System , 2017, Annals of PDE.

[5]  Weiwei Hu,et al.  An approximating approach for boundary control of optimal mixing via Navier-Stokes flows , 2019, Journal of Differential Equations.

[6]  I. Kukavica,et al.  Global Sobolev persistence for the fractional Boussinesq equations with zero diffusivity , 2019, 1910.11307.

[7]  I. Kukavica,et al.  Long Time Behavior of Solutions to the 2D Boussinesq Equations with Zero Diffusivity , 2019, Journal of Dynamics and Differential Equations.

[8]  Jiahong Wu,et al.  A global regularity result for the 2D Boussinesq equations with critical dissipation , 2014, Journal d'Analyse Mathématique.

[9]  C. Doering,et al.  Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusion , 2018, Physica D: Nonlinear Phenomena.

[10]  Weiwei Hu,et al.  Partially dissipative 2D Boussinesq equations with Navier type boundary conditions , 2017, Physica D: Nonlinear Phenomena.

[11]  A. Stefanov,et al.  ON THE GLOBAL REGULARITY OF THE 2D CRITICAL BOUSSINESQ SYSTEM WITH α>2/3∗ , 2017 .

[12]  I. Kukavica,et al.  Persistence of regularity for solutions of the Boussinesq equations in Sobolev spaces , 2016, Advances in Differential Equations.

[13]  N. Ju Global Regularity and Long-time Behavior of the Solutions to the 2D Boussinesq Equations without Diffusivity in a Bounded Domain , 2015, 1508.06176.

[14]  C. Foias,et al.  On the Attractor for the Semi-Dissipative Boussinesq Equations , 2015, 1507.00080.

[15]  Igor Kukavica,et al.  Persistence of regularity for the viscous Boussinesq equations with zero diffusivity , 2015, Asymptot. Anal..

[16]  Zhifei Zhang,et al.  The Two-Dimensional Incompressible Boussinesq Equations with General Critical Dissipation , 2014, SIAM J. Math. Anal..

[17]  E. Titi,et al.  Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion , 2013 .

[18]  I. Kukavica,et al.  On the regularity for the Boussinesq equations in a bounded domain , 2013 .

[19]  C. Cao,et al.  Global Regularity for the Two-Dimensional Anisotropic Boussinesq Equations with Vertical Dissipation , 2013 .

[20]  E. Titi,et al.  Title Global well-posedness for the 2 D Boussinesq system with anisotropic viscosity and without heat diffusion , 2013 .

[21]  C. Cao,et al.  Global regularity results for the 2D Boussinesq equations with vertical dissipation , 2011 .

[22]  R. Temam,et al.  Boundary layer associated with the Darcy–Brinkman–Boussinesq model for convection in porous media , 2011 .

[23]  Ming-Jun Lai,et al.  Initial Boundary Value Problem for Two-Dimensional Viscous Boussinesq Equations , 2011 .

[24]  L. Berselli,et al.  On the Boussinesq system: regularity criteria and singular limits , 2011 .

[25]  F. Rousset,et al.  Global Well-Posedness for Euler–Boussinesq System with Critical Dissipation , 2010 .

[26]  Lorenzo Brandolese,et al.  Large time decay and growth for solutions of a viscous Boussinesq system , 2010, 1003.4921.

[27]  Min Chen,et al.  Long-time asymptotic behavior of two-dimensional dissipativeBoussinesq systems , 2009 .

[28]  R. Danchin,et al.  Les theoremes de Leray et de Fujita-Kato pour le systeme de Boussinesq partiellement visqueux. The Leray and Fujita-Kato theorems for the Boussinesq system with partial viscosity , 2008, 0806.4083.

[29]  Taoufik Hmidi,et al.  On the global well-posedness of the Boussinesq system with zero viscosity , 2007, 0711.3198.

[30]  T. Hmidi,et al.  On the global well-posedness of the two-dimensional Boussinesq system with a zero diffusivity , 2007, Advances in Differential Equations.

[31]  Dongho Chae,et al.  Global regularity for the 2D Boussinesq equations with partial viscosity terms , 2006 .

[32]  Thomas Y. Hou,et al.  GLOBAL WELL-POSEDNESS OF THE VISCOUS BOUSSINESQ EQUATIONS , 2004 .

[33]  Dongho Chae,et al.  Local existence and blow-up criterion for the Boussinesq equations , 1997, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[34]  Hantaek Bae Navier-Stokes equations , 1992 .

[35]  Yoshikazu Giga,et al.  Abstract LP estimates for the Cauchy problem with applications to the Navier‐Stokes equations in exterior domains , 1991 .

[36]  P. Lions,et al.  Ordinary differential equations, transport theory and Sobolev spaces , 1989 .

[37]  R. Temam Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

[38]  R. Rautmann Approximation Methods for Navier-Stokes Problems , 1980 .

[39]  J. Cooper SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .