Some problems on the regularity of solutions for the 3D Navier-Stokes equations

Abstract

In this paper, we review some recent work on the Cauchy problem of the three-dimensional incompressible Navier-Stokes equations in recent years. It is well known that the three-dimensional incompressible Navier-Stokes system has the global Leray-Hopf weak solutions. When the weak solution satisfies the Prodi-Serrin condition, the solution is regular. We obtain some new results in the regularity conditions. Especially, for axisymmetric system, when the rotation speed is zero, it is well known that the system is globally well-posed. We find a new conserved quantity, and obtain some new advances in the regularity conditions of axisymmetric solutions with nonzero swirl, also get the global well-posedness results with the small initial rotating speed. Furthermore, we also get the similar result for the inhomogeneous system. In the end, we consider the global well-posedness of a class of generalized Navier-Stokes systems with the higher order horizontal viscosity $D_h^{2\alpha}$, $\alpha\geq\frac43$.