Global smooth solutions of the damped Boussinesq equations with a class of large initial data

TL;DR

This paper demonstrates that damping in the Boussinesq equations can ensure global smooth solutions even for large initial data, advancing understanding of regularity in fluid dynamics models.

Contribution

It establishes the global existence of smooth solutions for the damped Boussinesq equations with large initial data, using a novel approach that separates linear and nonlinear components.

Findings

Damping induces exponential decay in solutions.

Large initial data in certain Besov spaces still lead to global regularity.

The method separates linear decay from nonlinear dynamics.

Abstract

The global regularity problem concerning the inviscid Boussinesq equations remains an open problem. In an attempt to understand this problem, we examine the damped Boussinesq equations and study how damping affects the regularity of solutions. In this paper, we consider the global existence to the damped Boussinesq equations with a class of large initial data, whose $B^{s}_{p,r}$ or $\dot{B}^{s}_{p,r}$ norms can be arbitrarily large. The idea is splitting the linear Boussinesq equations from the damped Boussinesq equations, the exponentially decaying solution of the former equations together with the structure of the Boussinesq equations help us to obtain the global smooth solutions.