On regularity estimates for axially symmetric Navier-Stokes equations in a cylinder and the critical-wedge occurrence problem

Abstract

We consider the axisymmetric Navier-Stokes equations in a finite cylinder $\Omega \subset {\mathbb{R}}^3$ with slip-type boundary conditions. Our aim is to estimate $X_s(t):=\Vert {\omega }_r/r{\Vert }_{V({\Omega }^t)}+\Vert {\omega }_{\phi }/r{\Vert }_{V({\Omega }^t)}$. The closure mechanism depends on the relation between the $L^s$ and $L^{\infty }$ norms of the angular component $v_{\phi }$. For fixed $A,c_0>0$, we identify the critical wedge $W_{A,c_0}:=\{t\in (0,T):\Vert v_{\phi }(t){\Vert }_{L^s(\Omega )}>A,\Vert v_{\phi }(t){\Vert }_{L^s(\Omega )}/\Vert v_{\phi }(t){\Vert }_{L^{\infty }(\Omega )}<c_0\}$, with the ratio interpreted as $+\infty$ when the denominator vanishes. The main result is a conditional a priori estimate in which the possible loss of control is measured by a critical-wedge residual. If $E_{W,s}$ denotes the positive, non-closable part of the nonlinear interaction ${\int }_{{\Omega }^t}(v_{\phi }/r)\Phi \Gamma dxd{t}^{\prime }$, restricted to $W_{A,c_0}$, then $X_s(t)\le {\Psi }_{s,A,c_0}(\mathrm{data},{\int }_0^t{E}_{W,s}(\tau )d\tau )$ for $0<t<T$, where ${\Psi }_{s,A,c_0}$ is increasing. If the residual vanishes, in particular when the trajectory does not enter the critical wedge, the original data-dependent a priori estimate is recovered. Under additional regularity assumptions on the force and initial velocity, a corresponding higher Sobolev estimate for $v$ and $\nabla p$ follows with the same conditional dependence.

52 pages. Substantially revised version. The unconditional global regularity claim has been replaced by a conditional estimate involving the critical-wedge residual. Title, abstract, main results, proofs, and conclusions were revised. arXiv admin note: substantial text overlap with arXiv:2507.14964, arXiv:2405.16670, arXiv:2501.18302