On the Existence and Structures of Almost Axisymmetric Solutions to 3-D Navier–Stokes Equations

Abstract

In this paper, we consider 3-D Navier–Stokes equations with almost axisymmetric initial data, which means that by writing in the cylindrical coordinates, then and are small in some sense (recall axisymmetric means these three quantities vanish). Then with additional smallness assumption on , we prove the global existence of a unique strong solution , and this solution keeps close to some axisymmetric vector field. We also establish some refined estimates for the integral average in variable for . Moreover, as and here depend on , it is natural to expand them into Fourier series in variable. And we shall consider one special form of , with some small parameter to measure its swirl part and oscillating part. We study the asymptotic expansion of the corresponding solution, and the influences between different profiles in the asymptotic expansion. In particular, we give some special symmetric structures that will persist for all time. These phenomena reflect some features of the nonlinear terms in Navier–Stokes equations.