The APDE seminar on Monday, 9/14, will be given by Ovidiu-Neculai Avadanei (Caltech) in-person in Evans 736, and will also be broadcasted online via Zoom from 4:10pm to 5:00pm PST. To participate, please email Adam Black (adamblack@berkeley.edu).
Title: 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. Motivated by this, we define Lagrangian classes of coherent initial data and conditional solutions for which the same mechanism yields a supercritical-critical barrier to blow-up when $\alpha\geq\frac{1}{3}$. 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 cusp-tail, Dini coherence, near-field compatibility, and bounded transverse-distortion hypotheses, this singular value cannot collapse in finite time. In the on-axis case, the criterion reduces to the scalar clock inequality $\displaystyle \dot{J}(t)\gtrsim -B(t)J(t)-CJ(t)^{3\alpha}$, which rules out Shkoller-type clock collapse for $\alpha\geq\frac{1}{3}$. 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 $\alpha>\frac{1}{3}$.