A new regularity criterion for the 3D-Boussinesq equations in homogeneous Besov and Triebel-Lizorkin spaces

This paper aims to establish a new regularity criterion for weak solutions to the three-dimensional (3D) Boussinesq equations in the homogeneous Besov spaces Ḃsp,q (ℝ3) and the homogeneous Triebel–Lizorkin spaces Ḟsp,q (ℝ3). This regularity is proven if the velocity ω satisfies the following condition ∫I ||ω||2Ḃp,q (ℝ3)2 dt<∞ and ∫I ||ω||F ̇p,qs (ℝ3)2 dt<∞; I := ]0,T[ with some conditions on exponents s, p and q.