Some Kinds Of Global Solutions For Incompressible Navier - Stokes Equations

In this Ph.D thesis, we study global wellposedness for 3D incompressible Navier-Stokes equations. Under some conditions, we obtain global, large solutions to Navier-Stokes equations.The structure of this thesis is as follows:A brief survey of the background knowledge and the history on the study of Navier-Stokes equations are given in Chapter 1. Also we introduce our results in this thesis.In Chapter 2, we consider a special class of large initial data to the 3D incom-pressible Navier-Stokes equations with gravity. We show that, under such conditions, the incompressible Navier-Stokes equations with gravity are globally wellposed. The important features of the initial data are that the velocity fields minus gravity term are almost parallel to the corresponding vorticity fields in a very large space domain.In Chapter 3, we study 3D axisymmetric incompressible Navier-Stokes equa-tions with one slow variable. We get the global wellposedness under the weighted energy.In Chapter 4, we study 3D incompressible Navier-Stokes equations with damp-ing and with two slow variables. The main feature is that we work in the analytical functions to compensate the loss of the derivatives. We give a very short proof.