On the Regularity of Weak Solutions of the Boussinesq Equations in Besov Spaces

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 Ḃ∞,∞−1(ℝ3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\dot {B}_{\infty ,\infty }^{-1}(\mathbb {R}^{3})$\end{document}, that, if the solution of the Boussinesq equation (1) below (starting with an initial data in H2) is such that (∇u,∇𝜃)∈L2(0,T;Ḃ∞,∞−1(ℝ3))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\nabla u,\nabla \theta )\in L^{2}(0,T;\dot {B}_{\infty ,\infty }^{-1}(\mathbb {R}^{3}))$\end{document}, 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 𝜃.