On regularity estimates for the axially symmetric Navier-Stokes Cauchy problem and the critical-wedge occurrence problem

We consider the Cauchy problem for the axially symmetric Navier-Stokes equations in R^3.

Our aim is to derive estimates for the scaled vorticity components Phi = omega_r/r and Gamma = omega_phi/r, measured in the energy norm X(t).

The original closure mechanism depends on the relation between the L^s norm and the L-infinity norm of the angular velocity component v_phi.

We identify a critical wedge in the corresponding phase geometry, defined through a smoothly localized velocity profile near the axis of symmetry.

The main result is a conditional a priori estimate in which the possible loss of control is measured by the nonlinear interaction accumulated during the times belonging to the critical wedge.

In particular, if the critical-wedge contribution vanishes, the residual-free a priori estimate depending only on the data is recovered.

Under additional regularity assumptions on the force and the initial velocity, we derive the corresponding higher-order Sobolev estimate on every finite time interval on which the wedge residual remains controlled.

The result does not provide an unconditional global regularity theorem; rather, it isolates the only concentration regime not controlled by the two original closure mechanisms.