Small global solutions to the damped two-dimensional Boussinesq equations

TL;DR

This paper investigates how damping influences the regularity of solutions to the 2D Boussinesq equations, demonstrating the existence of unique global small solutions under minimal smallness conditions in a specific functional space.

Contribution

It establishes the existence of unique global small solutions to the damped 2D Boussinesq equations with minimal smallness assumptions in a homogeneous Besov space.

Findings

Global small solutions exist under minimal smallness assumptions

Damping alone is insufficient to prevent vortex stretching

Solutions are unique in the specified functional setting

Abstract

The two-dimensional (2D) incompressible Euler equations have been thoroughly investigated and the resolution of the global (in time) existence and uniqueness issue is currently in a satisfactory status. In contrast, the global regularity problem concerning the 2D inviscid Boussinesq equations remains widely open. In an attempt to understand this problem, we examine the damped 2D Boussinesq equations and study how damping affects the regularity of solutions. Since the damping effect is insufficient in overcoming the difficulty due to the "vortex stretching", we seek unique global small solutions and the efforts have been mainly devoted to minimizing the smallness assumption. By positioning the solutions in a suitable functional setting (more precisely the homogeneous Besov space $\mathring{B}^1_{\infty,1}$), we are able to obtain a unique global solution under a minimal smallness assumption.