Probe Into The Global Well-posedness And Regularity Results For Related Boussinesq Equations

In certain ranges of scales in the atmosphere and in the ocean,the fluid dynamics is controlled by the interaction of gravity and the earth's rotation.The equations at work on incompressible effects,called the rotating Boussinesq equations.The important point is that the effect of rotation and stratification will be produced in the flow.In this paper,we mainly study two important equations:the 2D Regularized-Boussinesq equations and the 3D axisymmetric Boussinesq equations.Considering the two dimensional critical Regularized-Boussinesq equations with ? + ? = 1-?,it is a supercritical problem when ?>0.In particular,when(?,?)=(1-?,0)and the initial data have low regularity,we prove the global well-posedness result.Here we need to overcome one point,namely no estimate for one derivative of p.We introduce G = ?)??-R? to avoid dealing the term(?)1? directly.Then,after the estimates for ?G?L?(0,T;L2?Lm),?G?L1(0,T;B?,1-?)and ?R???L?(0,T;L2?L?),we obtain the upper bound for the term ??v?L1(0,T;L?).For general cases(?,?>0,? + ? ?1-?),we also obtain the global well-posedness result.The difficulties are no estimate for one derivative of p and the diffusion term ??p.We also denote G = ?-R?.Although there is only one term(?)1? appeared in the equation of ?,we have two terms[R?,v · ?]?and(?)1??-?? in the equation of G.Since the commutator[R?,v ·?]?has good properties and(?)1??-??(?<?)has lower regularity compared with(?)1?,after some esti-mates for ?G?L?,(0,T;L2?Lm),?G?L?(0,T;Br,? s)and so on,we obtain the estimate for the blow up term ???L1(0,T;L?).Simultaneously,when ? = 0,using Fourier splitting method,we obtain the decay estimates for the global solution.As for the two dimensional subcritical Regularized-Boussinesq equations(a,?>0,? + ?>1-?),using similar methods to overcome similar difficulties,we obtain the global well-posedness result for ? ? 0 and the decay estimates for the case ? = 0.At last,we consider the three dimensional axisymmetric Boussinesq equations.Based on the difficult point for nonswirl and the term-(?)33(v?)2/e = 2u?wr/r appeared in the equation of w?,we introduce an important quantity wr/r.Considering the system wr/r and G = w?/r-1/2(k>0)(or H = w?/r-(?)?,k = 0),using classical energy methods and the properties of the operator(?),we obtain some Prodi-Serrin type regularity criteria based on rdu3(k ? 0),rdu?(k ? 0)and the regularity criterion based on ru?(k>0).When the initial data u?0 satisfies some small conditions,we obtain the global well-posedness result,some decay estimates for p(t)and upper bound for u(t).Moreover,if ?0 satisfies some small conditions,we obtain more better decay estimates for the global solution(?,u)(t).