Stability of nonswirl axisymmetric solutions to the Navier–Stokes equations

Abstract

The existence of global regular axisymmetric solutions to the Navier–Stokes equations without swirl and in a finite axisymmetric cylinder is proved. The solutions are such that norms bounded with respect to time are controlled by the same constant for all . Assuming that the initial velocity and the external force are sufficiently close to the initial velocity and the external force of a nonswirl axisymmetric solutions, we prove existence of global regular axisymmetric solutions which remain close to the nonswirl axisymmetric solution for all time. In this sense, we have stability of nonswirl axisymmetric solutions. However, to prove this, we need a smallness condition on the angular component of vorticity of the external force for the nonswirl solution.