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

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 $\alpha =\frac{1}{3}$, 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 $\alpha <\frac{1}{3}$ sufficiently close to $\frac{1}{3}$, we perturb the $\frac{1}{3}$-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 $\alpha \in (0,\frac{1}{3})$, 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.

92 pages