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

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 $X_s(t):=\|\omega_r/r\|_{V(\Omega^t)}+\|\omega_\varphi/r\|_{V(\Omega^t)}$.

The closure mechanism depends on the relation between the $L^s$ and $L^\infty$ norms of the angular component $v_\varphi$.

For fixed $A,c_0>0$, we identify the critical wedge $W_{A,c_0}:=\{t\in(0,T):\|v_\varphi(t)\|_{L^s(\Omega)}>A,\ \|v_\varphi(t)\|_{L^s(\Omega)}/\|v_\varphi(t)\|_{L^\infty(\Omega)}<c_0\}$, 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 $\int_{\Omega^t}(v_\varphi/r)\Phi\Gamma\,dx\,dt'$, restricted to $W_{A,c_0}$, then $X_s(t)\leq\Psi_{s,A,c_0}(\mathrm{data},\int_0^tE_{W,s}(\tau)\,d\tau)$ 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.