Global Regularity and Long-time Behavior of the Solutions to the 2D Boussinesq Equations without Diffusivity in a Bounded Domain

Abstract

New results are obtained for global regularity and long-time behavior of the solutions to the 2D Boussinesq equations for the flow of an incompressible fluid with positive viscosity and zero diffusivity in a smooth bounded domain. Our first result for global boundedness of the solution $(u,\theta )\in D(A)\times H^1$ improves considerably the main result of the recent article [7]. Our second result on global regularity of the solution $(u,\theta )\in V\times H^1$ for both bounded domain and the whole space ${\mathbb{R}}^2$ is a new one. It has been open and also seems much more challenging than the first result. Global regularity of the solution $(u,\theta )\in D(A)\times H^2$ is also proved.

Cited by in corpus (7)

• On asymptotic stability of the 3D Boussinesq equations without thermal conduction
• Global well-posedness for the 2D Boussinesq Equations with Zero Viscosity
• Small scale formation for the 2D Boussinesq equation
• Global Regularity for the Two-dimensional Boussinesq Equations Without Diffusivity in Bounded Domains
• Fractional regularity, global persistence, and asymptotic properties of the Boussinesq equations on bounded domains
• Asymptotic stability of the 2D Boussinesq equations without thermal conduction
• Long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity