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), 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)), 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. © 2020, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd