Global well-posedness of the 3D incompressible Navier-Stokes and Euler equations in orthogonal curvilinear coordinate systems

This paper investigates the globally dynamic stabilizing effects of the geometry of the domain at which the flow locates and of the geometrical structure of the finite-energy solutions to three-dimensional (3D) incompressible Navier-Stokes and Euler systems. We have established the global existence and uniqueness of the smooth solution to the Cauchy problem for 3D incompressible Navier-Stokes and Euler equations for a class of smooth large initial data in orthogonal curvilinear coordinate systems, which has no smallness assumption on initial data for the Cartesian coordinate system. Moreover, we have also established the existence, uniqueness, and exponential decay rate of the global strong solution to the initial-boundary-value problem for 3D Navier-Stokes equations for a class of smooth large initial data and a class of special bounded domain in orthogonal curvilinear coordinate systems. Regarding application, some new classes of large-amplitude global smooth solutions with very complex geometric structures, and that are either non-axisymmetric or non-helical in $ \mathbb{R}^3 $, have been established for 3D incompressible Navier-Stokes and Euler equations. This is the first result on the global existence and uniqueness of the large smooth solution to 3D incompressible Navier-Stokes and Euler equations in general orthogonal curvilinear coordinate systems.