Estimates For Some Fluid Dynamics Equations In Critical Spaces

This thesis contains three parts.The first part is about Meyers' type estimates for Stokes system.We get the gradient estimates for Stokes system by perturbation skill and interpolation method.Furthermore,we eventually get the critical estimates for Navier-Stokes equations by Sobolev imbedding theorem.The second part is about the critical estimates for Navier-Stokes equations in the whole space in two dimension.We obtain some critical estimates in Hardy,B-MO spaces for solutions to the initial value problem of heat equation.In fact the critical estimates of heat equation play an important role for Navier-Stokes equa-tions.To obtain the critical estimates,we get it by heat kernel method.Combined with Coifmann-Lions-Meyer-Semmes' result on compensated compactness in Hardy spaces,we also give some critical estimates for the second order derivatives of global weak solutions to Navier-Stokes equations.The third part is about Euler equations' regularity in bounded Besov spaces.Euler equations are one of the most important basic fluid dynamics system describing the motion of inviscid ideal fluids.For the Besov spaces in Rn and Sobolev spaces with the integer index in bounded domain,the well-posedness of Euler equations has been well-known so far.In this chapter we are concerned with the well-posedness of Euler equations in Besov spaces of bounded smooth domain.We first get a priori estimates in Wps(?)space by energy method.Then we derive a priori estimates for more general Besov spaces by interpolation argument.At last,by utilizing the a priori estimates and constructing suitable iteration sequences,we obtain that 3-D Euler equations have local solutions and 3-D Euler equations admit global solutions in subcritical case.In addition,for the borderline case we also establish the similar result.