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, v) = (u · ∇) v 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 θ _{H^s(Ω )} and u _{H^s(Ω )} by ∇θ _{L∞(Ω )} and ∇u _{L∞(Ω )}. Furthermore, ∇θ _{L∞(Ω )} can control ∇u _{L∞(Ω )} by using vorticity transportation equations. At last, ∇θ _{Mφ(Ω )} can control ∇θ _{L∞(Ω )}. Thus, we can ?nd a blowup criterion in the form of lim_t→T* ∫^t₀ ∇θ(·,τ) _{Mφ(Ω )}dτ =∞.