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Blow-up criterion for 2-D Boussinesq equations in bounded domain |
| HU Langhua, JIAN Huaiyu |
| Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China; |
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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 Ut + 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 Hs(Ω ) and u Hs(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 limt→T* ∫t0 ∇θ(·,τ) Mφ(Ω )dτ =∞.
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Issue Date: 05 December 2007
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