Blow-up criterion for 2-D Boussinesq equations in bounded domain

Abstract

We extend the results for 2-D Boussinesq equations from ℝ² to a bounded domain Ω. First, as for the existence of weak solutions, we transform Boussinesq equations to a nonlinear evolution equation U_t + A(t, U) = 0. In stead of using the methods of fundamental solutions in the case of entire ℝ², we study the qualities of F(u, υ) = (u · ▽)υ to get some useful estimates for A(t, U), which helps us to conclude the local-in-time existence and uniqueness of solutions. Second, as for blow-up criterions, we use energy methods, Sobolev inequalities and Gronwall inequality to control $\left\| \theta \right\|_{H^s (\Omega )} $$ and $\left\| u \right\|_{H^s (\Omega )} $$ by $\left\| {\nabla \theta } \right\|_{L^\infty (\Omega )} $$ and $\left\| {\nabla u} \right\|_{L^\infty (\Omega )} $$. Furthermore, $\left\| {\nabla \theta } \right\|_{L^\infty (\Omega )} $$ can control $\left\| {\nabla u} \right\|_{L^\infty (\Omega )} $$ by using vorticity transportation equations. At last, $\left\| {\nabla \theta } \right\|_{M_\phi (\Omega )} $$ can control $\left\| {\nabla \theta } \right\|_{L^\infty (\Omega )} $$. Thus, we can find a blow-up criterion in the form of $\lim _{t \to T^ * } \int_0^t {\left\| {\nabla \theta ( \cdot ,\tau )} \right\|_{M_\phi (\Omega )} d\tau = \infty } $$.

Keywords

Boussinesq equation / nonlinear evolution equation / existence and uniqueness / M_ϕ space / blow-up criterion

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