Blowup Criterion of Strong Solutions to the Three-Dimensional Double-Diffusive Convection System

Abstract

In this paper, we investigate some sufficient conditions for the breakdown of local strong solutions to the three-dimensional double-diffusive convection system. We obtain the classical blowup criterion for strong solutions $$(u, \theta , s)$$, i.e., $$u \in L^{q}\left( 0, T ; L^{p}\left( {\mathbb {R}}^{3}\right) \right) $$, for $$\frac{2}{q}+\frac{3}{p}=1$$, $$3<p \le \infty $$ or $$\nabla u \in L^{q}\left( 0, T ; L^{p}\left( {\mathbb {R}}^{3}\right) \right) $$, for $$\frac{2}{q}+\frac{3}{p}=2$$, $$\frac{3}{2}<p \le \infty $$. Moreover, we get a further blowup criterion, which involves partial derivatives of the partial components of the velocity field (i.e., $$\nabla _{h}{\tilde{u}}$$ ) belonging to the Besov spaces, where $$\nabla _{h}=(\partial _{1}, \partial _{2})$$ and $$ {\tilde{u}}=(u_{1}, u_{2})$$.