The Well-Posedness Of The 3D Incompressible Navier-Stokes System And Related Models In Orthogonal Curvilinear Coordinate Systems

The motion of viscous incompressible fluid can be characterized by Navier-Stokes equations and related models.As we all know,the existence and unique-ness of global solutions of 3D(three-dimensional)incompressible Navier-Stokes e-quations with regular initial values is a important open problem.The existence and uniqueness of solutions of the 3D incompressible Navier-Stokes equations is known for some initial values or small initial values with special structures.In this paper,we mainly study the influence of the geometry of the fluid region and the geometry of velocity field of incompressible flow on the well-posedness of the 3D incompressible Navier-Stokes equations and related models.Here,we also study the regularity of 3D incompressible Navier-Stokes equations in orthogonal curvi-linear coordinate systems and the global well-posedness of smooth large solutions of 3D incompressible Boussinesq equations in orthogonal curvilinear coordinate systems.The main results are as follows:1.The expressions of gradient operator,divergence operator and twist opera-tor are derived by coordinate transformation in orthogonal curvilinear coordinate systems,and the general expressions of the component forms of the 3D incompress-ible Navier-Stokes fundamental equations and the 3D incompressible Boussinesq fundamental equations are derived by using the general forms of equation solutions and basic calculus operations in orthogonal curvilinear coordinate systems,which provides a basis for deducing similar forms in the future.2.The regularity of 3D incompressible Navier-Stokes equation is studied in orthogonal curvilinear coordinate systems.The regularity criterion of the weak so-lution of the incompressible 3D Navier-Stokes equation involving only the vorticity component?³is established in orthogonal curvilinear coordinate systems.When1?p??,in this part,the a priori estimate related to ?H₃u³?L^?(0,T;L^p((?)³)))for the solutions of the 3D incompressible Navier-Stokes equation is also established in or-thogonal curvilinear coordinate systems.These greatly extend the corresponding results of axisymmetric cylindrical flow.3.The global well-posedness of smooth large solution of 3D incompressible Boussinesq is studied in orthogonal curvilinear coordinate systems.Here,a class of nonempty bounded domains in R³is constructed,in which there is a unique global strong solution or smooth solution for the initial boundary value problem of 3D Boussinesq system in orthogonal curvilinear coordinate systems,and the solution has exponential decay rate in time.These results can cover the most known results of large initial value smooth solutions of 3D Boussinesq equation in orthogonal curvilinear coordinate systems.