Smoothing Effect And Well-posedness Of Fractional Dissipation Boussinesq Equations In Critical Sobolev Space

In this paper,we consider the fractional dissipation 2D Boussinesq equations with initial data in the critical Sobolev space.And we investigate local well-posedness,local smoothing effect,global well-posedness and blow-up criterion of the equations based on Littlewood-Paley theory,which improve some of the previous results(suncritical,critical or supercritical)for Boussinesq equations.The main conclusions as follows:(1)We first establish a priori estimates of the solutions by some optimized commutator estimatesin the space Lp(0,T;H2-(p-1)/p2α(R2))× Lp(0,T;H2-(p-1)/p2β(R2))with some suitable p,then we prove the equations exist unique local solutions.(2)We get the local smoothing effect of solutions by adopt iterative technique and commutator estimates of higher regularity.(3)We prove the blow-up criterion of temperature of equations in the critical Sobolev space,and combine the conclusion of local smoothing effect to get global well-posedness of solutions.(4)We obtain a generalized blow-up by improve the nonlinar lower bounds estimates of fractional Laplacian in[P.Constantin,V.Vicol,Geom.Funct.Anal,2012,22:12891321].