Asymptotically Self-Similar Blowup for 3D Incompressible Euler with $C^{1, 1/3-}$ Velocity I: $C^{\infty}$ 1D Limiting Profiles

TL;DR

This paper constructs self-similar blowup profiles for 3D Euler equations using 1D models, revealing different blowup behaviors depending on the regularity parameter, and sets the stage for further 3D blowup analysis.

Contribution

It introduces a novel fixed-point approach to construct smooth and unbounded self-similar blowup profiles for 1D models of 3D Euler near the axis, depending on the regularity parameter.

Findings

Constructed a $C^{ abla}$ self-similar blowup profile with unbounded stream function.

Perturbed the critical profile to obtain bounded stream function profiles for $eta<1/3$.

Set up the framework for lifting 1D profiles to 3D Euler blowup scenarios.

Abstract

We consider a one-parameter family of 1D models for the 3D axisymmetric incompressible Euler equation with $C^{\alpha}$ vorticity and without swirl near the symmetry axis. For $\alpha = \frac13$, we impose a crucial normalization and construct a $C^{\infty}$ self-similar blowup profile with unbounded 1D stream function and infinite spatial blowup rate, using a fixed-point argument around a numerically constructed approximate profile. For $\alpha < \frac13$ sufficiently close to $\frac13$, we perturb the $\frac13$-profile and analytically construct exact smooth 1D profiles with bounded stream function and finite spatial blowup rate. In the companion work~\cite{chen2026eulerII}, for any $\alpha \in (0,\frac13)$, we lift these 1D blowup profiles to construct exact $C^{1,\alpha}$ self-similar blowup profiles for 3D Euler, and build on them to prove sharp asymptotically self-similar blowup for 3D axisymmetric Euler without swirl from $C_c^\alpha$ initial vorticity and $C^{1,\alpha} \cap L^2$ initial velocity.