Research On The Blow-up Problem For 2-D Boussinesq Equations In A Bounded Domain

This paper focuses mainly on the 2-D Boussinesq equations,which is a classicPDE model in atmosphere and Ocean fluid mechanics and in mathematical fluid me-chanics．In the paper,we successfully extend the serials of results for 2-D Boussi-nesq equations．such as the local-in-time existence and uniqueness of weak solutions inSobolev space and blow-up criterions,from R² to a bounded domainΩ．First,as for the local-in-time existence and uniqueness．we make use ofHelmholtz-Hodge orthogonal theorem to transfer Boussinesq equations to a nonlinearevolution equation U_t+A(t,U)=0．By virtue of some qualities of the nonlinearoperator F(u,v)=(u·(?))v,we get some useful estimates for the nonlinear oper-ator A(t,U),which helps us to conclude the local-in-time existence and uniquenessof solutions．Then,concerning on the blow-up criterions of weak solutions of Boussinesq equa-tions,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)} cancontrol‖(?)u‖_{L^∞(Ω)} by using the vorticity transportation equation．At last,‖(?)θ‖_{L^∞(Ω)}can be controlled by‖(?)θ‖_M?(Ω)．Thus,we can find a blow-up criterion in form of(?)．...