Vorticity growth beyond bounded relative vorticity for axisymmetric Euler flows without swirl

Abstract

We establish upper bounds for vorticity and related quantities in axisymmetric Euler flows without swirl in $\mathbb{R}^3$, assuming that the initial relative vorticity $ω_0/r$ belongs to $L^1\cap L^p$ with $3<p\le \infty$. For each fixed $p \in (3,\infty]$, we obtain polynomial-in-time upper bounds for the radial moments, the radial velocity, and the $L^p$ norm of vorticity. We then consider initial data with unbounded relative vorticity whose growth near the axis is controlled by $Φ(\log \tfrac{1}{r})$ for a subexponential profile $Φ$, so that $ω_0/r$ belongs to $L^p$ for every $p<\infty$ but not to $L^{\infty}$. For such data, we derive profile dependent upper bounds for the vorticity maximum. In particular, for any polynomial function $Φ(s)=s^k$ with $k\ge 0$, we show that $\|ω(\cdot,t)\|_{L^{\infty}}$ is bounded by $(1+t)^{\frac{4}{3}+k} (\log (2+t) )^{\frac{2k}{3}}$. Moreover, for initial data satisfying $ω_0/r \in L^1\cap L^p$ with a fixed $p\in (3,\infty)$, under additional symmetry and vertical confinement assumptions, we obtain a subexponential-in-time upper bound for the vorticity maximum, thereby improving the classical exponential bound for such data.