A Regularity Criterion for the Angular Component of Velocity in the norm Lq(0,T;Lₚ(Ω )),\;3/p +2/q<1,\;q< ∞, in Axisymmetric Navier-Stokes Equations in a Cylinder

We consider the axisymmetric Navier-Stokes equations in a finite cylinder Ω ⊂ R³ . We assume that vᵣ , v_φ , ω _φ vanish on the lateral part of boundary ∂ Ω of the cylinder, and that vz , ω _φ , ∂ zv_φ vanish on the top and bottom parts of the boundary ∂ Ω , where we used standard cylindrical coordinates, and we denoted by ω =curlv the vorticity field. We use H³ Sobolev estimates for the modified stream function (stream function divided by radius) and energy type estimates for gradient of swirl to derive two order reduction estimates. Finally, using the estimate v_φ Lq(0,T;Lₚ(Ω )) ≤ A, where A is a given number and 3/p + 2/q < 1 , q<∞ we prove the existence of global regular axially-symmetric solutions.