Constructs C^α self-similar blowup profiles for 3D Euler vorticity without swirl and proves asymptotically self-similar blowup from C_c^α data, with limiting factorization as α→(1/3)^-.
Incompressible Euler Blowup at the $C^{1,\frac{1}{3}}$ Threshold
4 Pith papers cite this work. Polarity classification is still indexing.
4
Pith papers citing it
abstract
We prove finite-time Type-I blowup for the three-dimensional incompressible Euler equations in the axisymmetric no-swirl class, with initial velocity in $C^{1,\alpha}(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$, odd symmetry in $z$, and $0<\alpha<\tfrac13$, 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 $\|\omega(\cdot,t)\|_{L^\infty}\sim (T^*-t)^{-1}$, while the meridional Jacobian collapses according to $J(t)\sim (T^*-t)^{1/(1-3\alpha)}$. 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 $\alpha=\tfrac13$. The blowup mechanism is structurally stable and persists for an open set of admissible angular profiles in a weighted H\"older topology.
fields
math.AP 4years
2026 4representative citing papers
Constructs self-similar blow-up solutions to axisymmetric Euler equations in d greater than or equal to 3 with initial data in C to the 1,alpha intersect smooth away from origin for alpha less than 1-2/d.
Constructs C^∞ self-similar blowup profiles for 1D models of 3D Euler at α=1/3 using fixed-point around a numerical approximation, plus nearby exact profiles for α slightly below 1/3.
For first-time blowup solutions of the 3D incompressible Euler equations in a bounded domain, the L^infty norms of vorticity derivatives satisfy explicit pointwise-in-time lower bounds, and the associated Gronwall inequality admits wildly oscillating solutions.
citing papers explorer
-
Asymptotically Self-Similar Blowup for 3D Incompressible Euler with $C^{1, 1/3-}$ Velocity II: 3D Profiles, Blowup, and Limiting behavior
Constructs C^α self-similar blowup profiles for 3D Euler vorticity without swirl and proves asymptotically self-similar blowup from C_c^α data, with limiting factorization as α→(1/3)^-.
-
Self-similar blow-up solutions of incompressible Euler equations in $\mathbb R^d, d\geq 3$ with $C^{1,1-2/d-}$ velocity
Constructs self-similar blow-up solutions to axisymmetric Euler equations in d greater than or equal to 3 with initial data in C to the 1,alpha intersect smooth away from origin for alpha less than 1-2/d.
-
Asymptotically Self-Similar Blowup for 3D Incompressible Euler with $C^{1, 1/3-}$ Velocity I: $C^{\infty}$ 1D Limiting Profiles
Constructs C^∞ self-similar blowup profiles for 1D models of 3D Euler at α=1/3 using fixed-point around a numerical approximation, plus nearby exact profiles for α slightly below 1/3.
-
On the blowup rate of vorticity for the Euler equations in a bounded domain
For first-time blowup solutions of the 3D incompressible Euler equations in a bounded domain, the L^infty norms of vorticity derivatives satisfy explicit pointwise-in-time lower bounds, and the associated Gronwall inequality admits wildly oscillating solutions.