Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2605.31587 |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914618293616640 |
|---|---|
| author | Avadanei, Ovidiu-Neculai |
| author_facet | Avadanei, Ovidiu-Neculai |
| contents | We consider axisymmetric no-swirl solutions to the three-dimensional incompressible Euler equations, with initial velocity in $C^{1,α}\cap L^2$, where $α\in\left[\frac{1}{3},1\right)$. Motivated by Shkoller's Lagrangian clock-and-driver framework for finite-time blow-up below the $C^{1,\frac{1}{3}}$ threshold, we introduce coherent Lagrangian classes of initial data and conditional solutions for which the same clock mechanism yields a supercritical/critical barrier when $α\geq\frac{1}{3}$, with a genuinely depleted barrier for $α>\frac{1}{3}$ and an exponential bound at the critical endpoint $α=\frac{1}{3}$. 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 $\displaystyle \dot{J}(t)\gtrsim -B(t)J(t)-CJ(t)^{3α}$, which rules out Shkoller-type clock collapse for $α\geq\frac{1}{3}$. 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 $α>\frac{1}{3}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_31587 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A conditional Lagrangian clock barrier at the $C^{1,\frac{1}{3}}$ threshold for axisymmetric Euler without swirl Avadanei, Ovidiu-Neculai Analysis of PDEs Primary: 35Q35, Secondary: 35Q31 We consider axisymmetric no-swirl solutions to the three-dimensional incompressible Euler equations, with initial velocity in $C^{1,α}\cap L^2$, where $α\in\left[\frac{1}{3},1\right)$. Motivated by Shkoller's Lagrangian clock-and-driver framework for finite-time blow-up below the $C^{1,\frac{1}{3}}$ threshold, we introduce coherent Lagrangian classes of initial data and conditional solutions for which the same clock mechanism yields a supercritical/critical barrier when $α\geq\frac{1}{3}$, with a genuinely depleted barrier for $α>\frac{1}{3}$ and an exponential bound at the critical endpoint $α=\frac{1}{3}$. 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 $\displaystyle \dot{J}(t)\gtrsim -B(t)J(t)-CJ(t)^{3α}$, which rules out Shkoller-type clock collapse for $α\geq\frac{1}{3}$. 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 $α>\frac{1}{3}$. |
| title | A conditional Lagrangian clock barrier at the $C^{1,\frac{1}{3}}$ threshold for axisymmetric Euler without swirl |
| topic | Analysis of PDEs Primary: 35Q35, Secondary: 35Q31 |
| url | https://arxiv.org/abs/2605.31587 |