Global Regularity Problems For Some Navier-Stokes Systems

In this thesis, we discuss well-posedness, regularity problems and blowup criteria related to several kinds of Navier-Stokes systems.After introducing some necessary notations, preliminary and known results in the first two chapters, in Chapter3we begin to consider a variant of Kazhikhov-Smagulov’s model for incompressible fluids with mass diffusion. In the case of two dimensional space, we prove the global well-posedness for the initial(-boundary) value problems of this system, without any size restriction on the initial data. In Chapter4we concern Brenner’s Navier-Stokes system with mass diffusion, which is a compressible counterpart of the Kazhikhov-Smagulov’s model. Similar results as in Chapter3in2D case are obtained, as well as blowup criterion and weak-strong uniqueness results in3D case. Finally in Chapter5we consider the classical compressible Navier-Stokes(-Fourier) system and give some blowup criterias for its local strong solution. It is shown that the Lipschitz norm of the velocity controls regularity of the local strong solution under more general setting in the sense that the fluid is not necessary ideal and the viscosities(and heat conductivity) may depend on the density(and temperature).Comparing to known results on the well-posedness and global regularity related to incompressible Navier-Stokes equations, in our case the main difficulty lies on the fact that either the viscosity coefficients may depend on the density or the fluid is com-pressible. A general approach has been developed to overcome these difficulties. The main idea is first to obtain Holder estimates(incompressible case) or upper and lower bounds of the density(compressible case), via the methods of De Giorgi-Nash-Moser’s iteration. Then energy method, combined with elliptic estimates, yields global-in-time a priori estimates. Together with the local existence result, we finally obtain the ex-istence of global regular solution or blowup criteria of local strong solution to the corresponding initial-boundary value problem.We believe the methods developed in this thesis can be applied to the study of other similar models in fluid dynamics.