2025/11/05 by Daomin Cao, Cao, Daomin, Junhong Fan +3
#math.AP
paper · pdf · doi:10.48550/arxiv.2511.03171
We study long-time vortex stretching for three-dimensional axisymmetric Euler flows without swirl in the head-on collision geometry of anti-parallel vortex rings that motivated Childress's \(t4/3\) conjecture [S.~Childress, Physica D 237 (2008), 1921--1925] for optimal vorticity growth in the full axisymmetric no-swirl class. For nontrivial compactly supported data that are odd in \(z\), nonpositive for \(z>0\), and satisfy the hypotheses of the main theorem, we prove limt→∞ \fracP(t)[log(2+t)]5/2(1+t)3/2=+∞, P(t):=\iintΠ+r2[-ω(r,z,t)] \dd r\dd z . This is the first lower bound for the radial moment that crosses the linear power barrier. For unit-strength relative-vorticity patches, it yields the full-time pointwise estimate limt→∞ \frac‖\boldsymbolΩ(t)‖L^∞(\mathbb R3) [log(2+t)]5/4(1+t)3/4 =+∞ . Moreover, for every \(A>0\), limT→∞(1)/(T) | \t∈[T,2T]: ‖\boldsymbolΩ(t)‖L^∞(\mathbb R3) ≥ A t(log t)-2 \ |=1. Thus the vorticity maximum reaches a nearly linear lower bound on an almost density-one fraction of every large dyadic time interval. The result is not restricted to patches. For general data in this class, all vorticity \(Lp\)-norms, uniformly over \(1≤ p≤∞\), exceed (1+t)1/4[log(2+t)]-25/12 by a factor tending to infinity. On a density-one fraction of \([T,2T]\), they simultaneously exceed every fixed multiple of \(T1/2(log T)-3/2\). The key new mechanism combines two monotone mixed moments with an exterior harmonic localization of the conserved kinetic energy to force quantitative radial escape.