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 $Ω\subset\mathbb{R}^3$ with slip-type boundary conditions. Our aim is to estimate $X_s(t):=\|ω_r/r\|_{V(Ω^t)}+\|ω_φ/r\|_{V(Ω^t)}$. The closure mechanism depends on the relation between the $L^s$ and $L^\infty$ norms of the angular component $v_φ$. For fixed $A,c_0>0$, we identify the critical wedge $W_{A,c_0}:=\{t\in(0,T):\|v_φ(t)\|_{L^s(Ω)}>A,\ \|v_φ(t)\|_{L^s(Ω)}/\|v_φ(t)\|_{L^\infty(Ω)}<c_0\}$, 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 $\int_{Ω^t}(v_φ/r)ΦΓ\,dx\,dt'$, restricted to $W_{A,c_0}$, then $X_s(t)\leqΨ_{s,A,c_0}(\mathrm{data},\int_0^tE_{W,s}(τ)\,dτ)$ for $0<t<T$, where $Ψ_{s,A,c_0}$ 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.