Regularity Of Solutions For Two Kinds Of Fluid Equations

This thesis studies the regularity of solutions for two kinds of fluid equations.The contents are organized as followsIn chapter 1,we introduce the background of magnetic Benard equations and Leray-a Navier Stokes equations and the related academic results.At the same time,it introduces the conclusions of this thesisIn chapter 2,we obtain a new rugularity criterion for the smooth solution to three dimensional magnetic Benard equations without thermal diffusion in terms of pressure.We prove that if ? ?L2(0,T;L3/r(R3)),with 0<r ?1,then the solution(u,b,?)to the magnetic Benard equations can be extended beyond time t=T.Meanwhile we also show that provided that ??L9-2r/2r(0,T;L3/r(R3))with 0<r ?1,the solution(u,b,?)can also be extended smoothly beyond t=T.Finally,we also obtain the regularity criteria on Morrey space,Multiplier space,BMO space and Besov space by imposing some growth conditions only on the pressure.In chapter 3,we establish the global existence and uniqueness of the three dimensional Leray-?Navier-Stokes equations with mixed partial dissipations less than 5/2 in H1-functional setting.This result can be achieved heavily based on the potential equation u=(Id-??)u.