A Regularity Criterion of Weak Solutions to the 3D Boussinesq Equations

The paper deals with the regularity criteria for the weak solutions to the 3D Boussinesq equations in terms of the partial derivatives in Besov spaces. It is proved that the weak solution (u,theta) becomes regular provided that ( backward difference hu, backward difference h theta)is an element of L1(0,T & x37e;B center dot infinity,infinity 0(R3))\,\nabla _{h}\theta )\in L<^>{1}(0,T;\overset{\cdot }{B }_{\infty ,\infty }<^>{0}({\mathbb {R}}<^>{3})) \end{aligned}$$\end{document}Our results improve and extend the well-known result of Dong and Zhang (Nonlinear Anal 11:2415-2421, 2010) for the Navier-Stokes equations.