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

Abstract: For any $α\in (0, 1/3)$, we construct exact $C^{1,α}$ self-similar blowup profiles for the 3D incompressible Euler equation without swirl, and build on them to prove asymptotically self-similar blowup from $C_c^α$ initial vorticity and $C^{1,α}\cap L^2$ initial velocity. Moreover, we provide a complete characterization of the limiting behavior of the $C^{1,α}$ blowup profiles and the associated $C^{1,α}$ blowup solutions as $α\to(1/3)^-$. Specifically, as $α\to(1/3)^-$, the spatial blowup rate $c_{x,α}$ diverges to $\infty$, while the $C^{1,α}$ blowup profile $Ω{*,α}^θ$ 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 $C^{1,1/3+}\cap L^2$ initial velocity. 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 $\mathbb{R}^3$.