Asymptotic properties of the Boussinesq equations with Dirichlet boundary conditions

We address the asymptotic properties for the Boussinesq equations with vanishing thermal diffusivity in a bounded domain with no-slip boundary conditions. We show the dissipation of the L norm of the velocity and its gradient, convergence of the L norm of Au, and an o(1)-type exponential growth for ‖Au‖L2 . We also obtain that in the interior of the domain the gradient of the vorticity is bounded by a polynomial function of time.

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

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

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

[4]  R. Temam.  Behaviour at Time t=0 of the Solutions of Semi-Linear Evolution Equations. , 1982 .

[5]  C. Cao,et al.  Global Regularity for the Two-Dimensional Anisotropic Boussinesq Equations with Vertical Dissipation , 2013, Archive for Rational Mechanics and Analysis.

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

[7]  I. Kukavica.  On the dissipative scale for the Navier-Stokes equation , 1999 .

[8]  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.

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

[10]  Wolf von Wahl,et al.  On the regularity of the pressure of weak solutions of Navier-Stokes equations , 1986 .

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

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

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

[14]  A global regularity result for the 2D Boussinesq equations with critical dissipation , 2014, Journal d'Analyse Mathématique.

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

[16]  James C. Robinson.  Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors , 2001 .

[17]  C. Doering,et al.  Applied analysis of the Navier-Stokes equations: Index , 1995 .

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

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

[20]  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.

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

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

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

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

[25]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

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

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

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

[29]  Y. Giga,et al.  On the Stokes operator in exterior domains , 1988 .

[30]  T. Hou,et al.  Finite Time Blowup of 2D Boussinesq and 3D Euler Equations with $$C^{1,\alpha }$$ Velocity and Boundary , 2019, Communications in Mathematical Physics.

[31]  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.

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

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

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

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

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

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