The global solutions of axisymmetric Navier–Stokes equations with anisotropic initial data

Abstract

Consider the Cauchy problem of incompressible Navier–Stokes equations in three dimension with axisymmetric initial data. If the swirl component satisfies some certain regularity criteria in weighted spaces, the existence of the time-global solution has been proved in our previous work. However, it is more or less evident that the solution is anisotropic in vertical direction (i.e., the direction parallel to the axis of symmetry) and the horizontal directions (in the plane perpendicular to the axis of symmetry). In this paper, we are interested in constructing regularity criteria and time-global solution in anisotropic Lebesgue spaces.