Large time existence of solutions to the Navier-Stokes equations in axially symmetric domains with inflow and outflow

Abstract

We prove a long time existence of special regular solutions to the Navier-Stokes equations in an axially symmetric domain in $\mathbb{R}^3$, with boundary slip conditions and with inflow and outflow. We assume that an initial angular component of velocity and an angular component of the external force and angular derivatives of the cylindrical components of initial velocity and of the external force are sufficiently small in corresponding norms. We assume also that inflow and outflow is sufficiently close to homogeneous. Then there exists a solution such that velocity belongs to $W_{5/2}^{2,1}(\Omega^T)$ and gradient of pressure to $L_{5/2}(\Omega^T)$, and we do not have restrictions on~$T$.