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

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.