On the regularity of axially-symmetric solutions to the incompressible Navier-Stokes equations in a cylinder

Abstract

We consider the axisymmetric Navier-Stokes equations in a finite cylinder $\Omega \subset R^3$. We assume that $v_r$, $v_{\phi }$, ${\omega }_{\phi }$ vanish on the lateral boundary ∂Ω of the cylinder, and that $v_z$, ${\omega }_{\phi }$, ${\partial }_z{v}_{\phi }$ vanish on the top and bottom parts of the boundary ∂Ω, where we used standard cylindrical coordinates, and we denoted by $\omega =\mathrm{curl}v$ the vorticity field. We use weighted estimates and $H^3$ Sobolev estimate on the modified stream function to derive three order-reduction estimates. These enable one to reduce the order of the nonlinear estimates of the equations, and help observe that the solutions to the equations are “almost regular”. We use the order-reduction estimates to show that the solution to the equations remains regular as long as, for any $p\in (6,\infty )$, ${\Vert v_{\phi }\Vert }_{L_t^{\infty }{L}_x^p}/{\Vert v_{\phi }\Vert }_{L_t^{\infty }{L}_x^{\infty }}$ remains bounded below by a positive number.