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

We consider the axisymmetric Navier-Stokes equations in a finite cylinder Ω³. We assume that vᵣ, v_φ, ω_φ vanish on the lateral boundary ∂ Ω of the cylinder, and that vz, ω_φ, ∂z v_φ vanish on the top and bottom parts of the boundary ∂ Ω, where we used standard cylindrical coordinates, and we denoted by ω =curl\, v the vorticity field. We use estimates and H³ 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 is ``almost regular''. We use the order-reduction estimates to show that the solution to the equations remains regular as long as, for any p∈ (6,∞), \| v_φ \|L^∞ₜ Lᵖₓ/\| v_φ \|L^∞ₜ L^∞ₓ remains bounded below by a positive number.