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

Abstract

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.

85 pages. Substantially revised version. The main result is now stated as a conditional a priori estimate involving a critical-wedge residual. The higher-regularity argument in Section 7 has been corrected and expanded. The title, abstract, main results, and references have been updated. No unconditional global regularity claim is made