Global Regularity Criterion For The 2D Boussinesq Equations

As the mode of the Ocean and Mathematical hydrodynamics,Boussinesq System is proposed to understand the vortex stretching effect for 3D flows.This paper aims at the global regularity criterion for the 2D Boussinesq equations with either vertical or horizontal viscosity and diffusivity in Hm(m ≥ 1).That is,we improve the regularity results of[2]from H2 to Hm(m ≥1),and by the same method,we get the results easily to the symmetry of Boussinesq equation in[2];Through the same regularity criterion in[2],we obtain the global regularity straightly without the extensive proof from local solutions to global solutions in time.Moreover,the method used here is suitable for all the cases when some of the parameters are zero.