A conditional Lagrangian clock barrier at the $C^{1,\frac{1}{3}}$ threshold for axisymmetric Euler without swirl

Abstract

We consider axisymmetric no-swirl solutions to the three-dimensional incompressible Euler equations, with initial velocity in $C^{1,\alpha }\cap L^2$, where $\alpha \in \left[\frac{1}{3},1\right)$. In a major breakthrough, Shkoller introduced a clock-and-driver framework that he used in order to prove finite-time type I blow-up below the $C^{1,\frac{1}{3}}$ threshold in this setting. Motivated by this, we define Lagrangian classes of coherent conditional solutions for which the same mechanism yields a supercritical-critical barrier to blow-up when $\alpha \ge \frac{1}{3}$. When $\alpha >\frac{1}{3}$, the aforementioned barrier is genuinely depleted, whereas at the critical endpoint $\alpha =\frac{1}{3}$, we obtain an exponential bound preventing blow-up. In the general case, we formulate a matrix-clock criterion in terms of the smallest singular value of the deformation gradient and show that, under transverse cusp-tail, longitudinal, off-clock, Dini, and suitable geometric coherence hypotheses, this singular value cannot collapse in finite time. In particular, we also show that the class of such coherent solutions includes the smooth ones locally in time. In the on-axis case, the criterion reduces to the scalar clock inequality $\dot{J}(t)\gtrsim -B(t)J(t)-CJ(t{)}^{3\alpha }$, which rules out Shkoller-type clock collapse for $\alpha \ge \frac{1}{3}$. These results do not enlarge the known Lorentz-space global regularity classes. Rather, they in particular identify the supercritical Lagrangian obstruction dual to Shkoller's subcritical blow-up mechanism in the case $\alpha >\frac{1}{3}$.

28 pages. This version shows that the coherent solution classes contain the classical $C^{1,\alpha }\cap L^2$ solutions locally in time, and further clarifies their relation to the existent Lorentz space-based regularity classes. The main results, driver estimates, as well as the matrix-clock mechanism and analogous on-axis mechanism remain all unchanged