Asymptotically Self-Similar Blowup for 3D Incompressible Euler with $C^{1, 1/3-}$ Velocity II: 3D Profiles, Blowup, and Limiting behavior

TL;DR

This paper constructs and analyzes exact self-similar blowup profiles for the 3D incompressible Euler equations, characterizing their limiting behavior as the regularity parameter approaches a critical value, and establishing stability and blowup results.

Contribution

It develops a novel method to lift 1D blowup profiles to 3D Euler solutions and characterizes the asymptotic behavior as the regularity approaches 1/3.

Findings

Constructed $C^{eta}$ self-similar blowup profiles for Euler vorticity.

Proved asymptotic convergence of profiles as $eta o 1/3^-$.

Established stability of the blowup profiles in a low-regularity setting.

Abstract

For any $\alpha \in (0,1/3)$, we construct exact $C^{\alpha}$ self-similar blowup profiles for the vorticity of the 3D incompressible Euler equation without swirl, and build on them to prove asymptotically self-similar blowup from $C_c^\alpha$ initial vorticity and $C^{1,\alpha}\cap L^2$ initial velocity. Moreover, we provide a complete characterization of the limiting behavior of the $C^{\alpha}$ vorticity profiles and the associated blowup solutions as $\alpha\to(1/3)^-$. Specifically, as $\alpha \to(1/3)^-$, the spatial blowup rate $\mathsf{c}_{\mathsf{x},\alpha}$ diverges to $\infty$, while the $C^{\alpha}$ vorticity profile $\Omega_{*,\alpha}^{\theta}$ asymptotically factorizes and converges strongly in a weighted $L^\infty$ norm to a nonzero constant multiple of $r^{1/3}\bar W_{1/3}(z)$, where $\bar W_{1/3}$ is a $C^\infty$ 1D blowup profile. Our construction is inspired by the Hou--Zhang blowup scenario. Using a fixed-point argument, we lift the $C^\infty$ blowup profiles for a 1D model constructed in the companion work [11] to exact 3D blowup profiles. To overcome the lack of $r$-directional decay in the approximate profile and capture the anisotropic structure, we develop a family of anisotropic weighted estimates and introduce a crucial integration-by-parts method along trajectories that exploits the equation twice. We then develop a finite codimension stability argument in a low-regularity setting to prove stability of the 3D profiles and establish asymptotically self-similar blowup. This blowup result is sharp in view of the global regularity theory for axisymmetric Euler without swirl with $ C_c^{\alpha}$ initial vorticity for all $\alpha \geq 1/3$. To the best of our knowledge, our results provide the first example in which a singularity from a 1D nonlocal fluid model is lifted to construct blowup for incompressible fluid equations in $\mathbb{R}^2$ or $\mathbb{R}^3$.