Global well-posedness for the 2D Boussinesq equations with a velocity damping term

TL;DR

This paper proves the global existence and uniqueness of smooth solutions for the 2D Boussinesq equations with velocity damping, when initial data is near a specific equilibrium, addressing an open question in the field.

Contribution

It establishes global well-posedness for the 2D Boussinesq equations with velocity damping near a nontrivial equilibrium, providing a positive answer to an open problem.

Findings

Global well-posedness for the equations with damping

Solution stability near equilibrium state

Resolution of an open question in the literature

Abstract

In this paper, we prove global well-posedness of smooth solutions to the two-dimensional incompressible Boussinesq equations with only a velocity damping term when the initial data is close to an nontrivial equilibrium state $(0,x_2)$. As a by-product, under this equilibrium state, our result gives a positive answer to the question proposed by [ACWX] (see P.3597).