Linear Growth of the Vorticity Maximum for Axisymmetric Euler Flows Without Swirl

Abstract

We study long-time vortex stretching for three-dimensional axisymmetric Euler flows without swirl in the anti-parallel class associated with the head-on collision of two coaxial vortex rings. This geometry motivated Childress's \(t^{4/3}\) conjecture for the vorticity maximum in the full axisymmetric no-swirl class [S.~Childress, \emph{Physica D} \textbf{237} (2008), 1921--1925]. For unit-strength relative-vorticity patches in this class, we prove that the outer radius, which is exactly the vorticity maximum, reaches the linear scale on an arbitrarily large fixed proportion of every sufficiently large dyadic interval: for every \(0<\eta<1\), there exist \(c_\eta>0\) and \(T_\eta>1\) such that \[ \left| \left\{t\in[T,2T]: \mathcal R_\omega(t) =\|\boldsymbol\Omega(t)\|_{L^\infty(\mathbb R^3)} \ge c_\eta t \right\} \right|\ge(1-\eta)T \qquad(T\ge T_\eta). \] The same estimate for \(\mathcal R_\omega(t)\) holds for all data considered below. For every nontrivial compactly supported initial datum in this class that is odd in \(z\) and non-positive for \(z>0\), we also prove \[ \lim_{t\to\infty} \frac{P(t)[\log(2+t)]^{5/2}}{(1+t)^{3/2}} =+\infty. \] To the best of our knowledge, this is the first radial-moment lower bound with exponent greater than one. For unit-strength patches, the same moment bound also yields the full-time estimate \[ \lim_{t\to\infty} \frac{\|\boldsymbol\Omega(t)\|_{L^\infty(\mathbb R^3)} [\log(2+t)]^{5/4}}{(1+t)^{3/4}} =+\infty. \] For general data, we further obtain a quantitative Eulerian form of simultaneous radial escape and collision. The proof uses two monotone mixed moments, a compactly supported multiplier, and an exterior \(L^2\) estimate for the velocity generated by interior vorticity.