The Well-posedness To The 3D Incompressible Navier-Stokes System And Related Models

As a famous system of equations describing hydrodynamics,the three-dimensional incompressible Navier-Stokes system has been widely used in the field of aerodynamics,aerospace and other research fields.The well-posedness problem of the three-dimensional incompressible Navier-Stokes equations has been one of the focuses of mathematicians for a long time.It is well known that there exist at least one global weak solution and an unique local strong solution for the three-dimensional incompressible Navier-Stokes equations with the finite initial energy.But the regularity and uniqueness of the global weak solution are still a challenging open problem in the field of hydrodynamics.The three-dimensional incompressible Boussinesq system,which is closely related to the three-dimensional incompress-ible Navier-Stokes system,is widely concerned in the fields of atmospheric ocean and geophysics.Since the three-dimensional incompressible Boussinesq equations have similar nonlinear structures to the three-dimensional incompressible Navier-Stokes equations,the study on them is helpful for us to understand the rotation and contraction structures better of the three-dimensional incompressible Navier Stokes equationsThis paper investigates the globally dynamical stabilizing effects of the geom-etry of the domain at which the flow locates and of the geometry structure of the solutions with the finite energy to the three-dimensional incompressible fluid.We establish the global well-posedness to the three-dimensional incompressible Navier-Stokes system for a class of large initial data in the generalized cylindrical coor-dinate system and the generalized spherical coordinate system.We also establish the global well-posedness to the three-dimensional incompressible Boussinesq sys-tem for a class of large initial data in the generalized spherical coordinate system The main research methods include the classical energy estimation,embedding theorem,Sobolev space and the basic theory of partial differential equation The first chapter is the introduction,which introduces the background and current situation to the three-dimensional incompressible Navier-Stokes equations and three-dimensional incompressible Boussinesq equations and some basic knowl-edge.In Chapter 2,the initial boundary value problem of the three-dimensional incompressible Navier-Stokes equations is studied in the generalized cylindrical coordinate system by the special structure of the fluid functions in the generalized cylindrical coordinate transformation.We get the existence and uniqueness of the global strong solutions for the initial boundary value problem of three-dimensional incompressible Navier-Stokes equations.Moreover,the global strong solution is also smooth when the initial date is sufficiently smoothIn Chapter 3,we study the global well-posedness of the initial boundary value problem of the three-dimensional incompressible Navier-Stokes equations in the generalized spherical coordinate system.We obtain the existence and uniqueness of the global strong solution of the initial boundary value problems of the three-dimensional incompressible Navier-Stokes equations in a bounded domain by using the special structure of the fluid functions in the generalized spherical coordinate transformation.In addition,we obtain the existence and uniqueness of the global smooth solution of the Cauchy problem to the Navier-Stokes equations without swirl when the initial date is sufficiently smoothIn Chapter 4,the initial boundary value problem of the three-dimensional incompressible Boussinesq equations is studied in the generalized spherical coor-dinate system.By using the special structure of the fluid functions in the general-ized spherical coordinate transformation,we obtain the existence and uniqueness of the global strong solution for the initial boundary value problem of the three-dimensional incompressible Boussinesq equations.In addition,we obtain the exis-tence and uniqueness of the global smooth solution of the Cauchy problem to the Boussinesq equations without swirl when the initial date is sufficiently smoothIn Chapter 5,we establish some new regularity criteria only involve the norms of velocity or vorticity in a thin cylinder including the symmetry with infinite height.It should be remarked that the regularity criteria are independent of ?,and therefore our results can be thought as a generation of the Navier-Stokes equations by assuming ??const.