Some Topics On The Global Well-posedness Of 3-D Navier-Stokes Equations

The incompressible Navier-Stokes equations describe the motion of an incom-pressible fluid with positive viscosity.One fundamental and very important open prob-lem is:for given regular initial data,whether the incompressible 3-D Navier-Stokes equations generate a unique global solution in the corresponding functional spaces?Many experts in analysis and PDE has seriously taken this problem into consideration,but so far we can only prove global well-posedenss for the initial data having some spe-cial structure or being sufficiently small,and for general large initial data,we only have local well-posedness.In this work we consider the existence and uniqueness of global solution to the 3-D Navier-Stokes equations.It is composed of four parts.In Chapter 2,we first prove the global well-posedness of 3-D anisotropic Navier-Stokes system provided that the vertical viscous coefficient of the system is sufficiently large compared to some critical norm of the initial data.Then we shall adapt the proof to show the global well-posedness of the classical 3-D Navier-Stokes equations with the initial data varying fast enough in the vertical direction and the third component of the initial velocity being sufficiently small.Chapter 3 is devoted to the regularity criteria concerning only one conpoment of the velocity.Let us consider an initial vorticity belonging to 3I L2 for the classical 3-D Navier-Stokes equation.We prove that if the Fujita-Kato solution associated with this initial data blows up at a finite time T*,then for any p ?]4??[,q1 ?[1,2[,?>0,q2?[2,(1/p + ?)-1[,k?]1,?[,and any unit vector e,the LP estimate in time of(?)blows up at T*.The order of spacial derivatives required in this norm must be positive but can be arbitrarily close to 0,yet we can not succeed in reducing the derivative estimate to be zero-th ordef.The next two chapters are concerning the axi-symmetric Navier-Stokes equaitons.We prove the local well-posedness of 3-D axi-symmetric Navier-Stokes system with initial data in the critical Lebesgue spaces.We also obtain the global well-posedness result with small initial data.Furthermore,with the initial swirl component of the veloc-ity being sufficiently small,we can still prove the global well-posedness of the system.These will be done in Chapter 4.In Chapter 5,as a natural generalization of the initial data in the critical Lebesgue spaces considered in the previous chapter,we shall consider finite measure as initial data.In particular,we consider the case W?(t)drdz?(?)?0,where n can be any positive integer,(?)i is some positive constant,and ?xi is the Dirac mass at point xi=(ri,zi)?? with ri>0.Then we prove that under some natural restrictions the axi-symmetric solution is unique,and give an accurate quantitative short time estimate for this solution.