We investigates the globally stabilizing effects of the geometry of the domain and the solution in studying the regularity issue on the three-dimensional incompressible Navier-Stokes and Euler system. We establish the global existence and uniqueness of the smooth solution to the Cauchy problem for the three-dimensional incompressible Navier-Stokes and Euler system, and, also, to the initial boundary value problem for the 3D Navier-Stokes equations, in spherical coordinates for a class of the smooth large initial data. This is the first result on the global existence and uniqueness of the smooth solution to the 3D incompressible Navier-Stokes and Euler equations in spherical coordinates. The related problems the axisymmetric Navier-Stokes equations are surveyed and some results on the singularity formation and global regularity of an axisymmetric model for the 3D incompressible Euler and Navier-Stokes equations will also be reviewed.
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