On the existence and structures of almost axisymmetric solutions to 3-D Navier-Stokes equations

In this paper, we consider 3-D Navier-Stokes equations with almost axisymmetric initial data, which means that by writing $u_0 =u^r_0 e_r+u^\theta_0 e_\theta+u^z_0 e_z$ in the cylindrical coordinates, then $\partial_\theta u^r_0,\,\partial_\theta u^\theta_0$ and $\partial_\theta u^z_0$ are small in some sense (recall axisymmetric means these three quantities vanish). Then with additional smallness assumption on $u^\theta_0$, we prove the global existence of a unique strong solution $u$, and this solution keeps close to some axisymmetric vector field. We also establish some refined estimates for the integral average in $\theta$ variable for $u$. Moreover, as $u^r_0,\,u^\theta_0$ and $u^z_0$ here depend on $\theta$, it is natural to expand them into Fourier series in $\theta$ variable. And we shall consider one special form of $u_0$, with some small parameter $\varepsilon$ 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.