Incompressible Euler Blowup at the $C^{1,\frac{1}{3}}$ Threshold
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918484048347136 |
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| author | Shkoller, Steve |
| author_facet | Shkoller, Steve |
| contents | We prove finite-time Type-I blowup for the three-dimensional incompressible Euler equations in the axisymmetric no-swirl class, with initial velocity in $C^{1,α}(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$, odd symmetry in $z$, and $0<α<\tfrac13$, for an explicit class of finite-energy initial data. The singularity forms at a stagnation point on the symmetry axis. The on-axis axial strain and the global vorticity norm blow up at the Type-I rates $-\partial_z u_z(0,0,t)\sim (T^*-t)^{-1}$ and $\|ω(\cdot,t)\|_{L^\infty}\sim (T^*-t)^{-1}$, while the meridional Jacobian collapses according to $J(t)\sim (T^*-t)^{1/(1-3α)}$. The proof introduces a Lagrangian clock-and-strain framework that replaces the Eulerian self-similar ansatz used in prior work with a Lagrangian flow decomposition. The collapse dynamics are governed by a Riccati law for the on-axis axial strain, coupled to a clock ODE for the meridional Jacobian. The decisive step is a non-perturbative strain-pressure comparison showing that the pressure Hessian cannot cancel the quadratic compressive strain responsible for collapse. This gives a dynamical explanation of the threshold $α=\tfrac13$. The blowup mechanism is structurally stable and persists for an open set of admissible angular profiles in a weighted Hölder topology. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_10945 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Incompressible Euler Blowup at the $C^{1,\frac{1}{3}}$ Threshold Shkoller, Steve Analysis of PDEs 35Q31, 76B03, 35B44 We prove finite-time Type-I blowup for the three-dimensional incompressible Euler equations in the axisymmetric no-swirl class, with initial velocity in $C^{1,α}(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$, odd symmetry in $z$, and $0<α<\tfrac13$, for an explicit class of finite-energy initial data. The singularity forms at a stagnation point on the symmetry axis. The on-axis axial strain and the global vorticity norm blow up at the Type-I rates $-\partial_z u_z(0,0,t)\sim (T^*-t)^{-1}$ and $\|ω(\cdot,t)\|_{L^\infty}\sim (T^*-t)^{-1}$, while the meridional Jacobian collapses according to $J(t)\sim (T^*-t)^{1/(1-3α)}$. The proof introduces a Lagrangian clock-and-strain framework that replaces the Eulerian self-similar ansatz used in prior work with a Lagrangian flow decomposition. The collapse dynamics are governed by a Riccati law for the on-axis axial strain, coupled to a clock ODE for the meridional Jacobian. The decisive step is a non-perturbative strain-pressure comparison showing that the pressure Hessian cannot cancel the quadratic compressive strain responsible for collapse. This gives a dynamical explanation of the threshold $α=\tfrac13$. The blowup mechanism is structurally stable and persists for an open set of admissible angular profiles in a weighted Hölder topology. |
| title | Incompressible Euler Blowup at the $C^{1,\frac{1}{3}}$ Threshold |
| topic | Analysis of PDEs 35Q31, 76B03, 35B44 |
| url | https://arxiv.org/abs/2603.10945 |