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

TL;DR

This paper establishes new results on the global regularity and long-term behavior of solutions to the 2D Boussinesq equations without diffusivity, in bounded domains, advancing understanding of fluid dynamics under these conditions.

Contribution

It provides the first proof of global regularity for solutions in certain function spaces and improves previous boundedness results for the 2D Boussinesq equations without diffusivity.

Findings

Global boundedness of solutions in D(A)×H^1

Global regularity in V×H^1 for bounded domain and ℝ²

Global regularity in D(A)×H^2

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.