On the regularity of weak solutions of the Boussinesq equations in Besov spaces Dedicated to Enrique Zuazua on the occasion of his sixtieth birthday

TL;DR

This paper extends a regularity result for weak solutions of the Boussinesq equations in Besov spaces, showing that under certain conditions on the velocity, solutions remain smooth indefinitely, without requiring conditions on temperature.

Contribution

It proves that weak solutions of the Boussinesq equations remain smooth if the velocity satisfies a specific Besov space condition, removing the need for temperature conditions.

Findings

Weak solutions remain smooth under velocity conditions in Besov spaces.

The result generalizes previous work by relaxing temperature regularity assumptions.

The proof applies to solutions starting from initial data in H^2.

Abstract

The main issue addressed in this paper concerns an extension of a result by Z. Zhang who proved, in the context of the homogeneous Besov space $\dot{B}_{\infty ,\infty }^{-1}(\mathbb{R}% ^{3})$, that, if the solution of the Boussinesq equation (\ref% {eq1.1}) below (starting with an initial data in $H^{2}$) is such that $% (\nabla u,\nabla \theta )\in L^{2}\left( 0,T;\dot{B}_{\infty ,\infty }^{-1}(% \mathbb{R}^{3})\right)$, then the solution remains smooth forever after $T$. In this contribution, we prove the same result for weak solutions just by assuming the condition on the velocity $u$ and not on the temperature $\theta$.