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

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 ${\Omega }_{\ast ,\alpha }^{\theta }$ 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 $\alpha \ge 1/3$. 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 ${\mathbb{R}}^3$.

Minor edits. 133 pages