Remarks on the smoothness of the $C^{1,\alpha}$ asymptotically self-similar singularity in the 3D Euler and 2D Boussinesq equations

TL;DR

This paper extends the construction of $C^{1,eta}$ asymptotically self-similar singularities in fluid equations, showing such singularities can be non-smooth at only one point, with detailed criteria and perturbation methods.

Contribution

It demonstrates the extension of existing singularity constructions to include localized non-smoothness at a single point in 3D Euler and 2D Boussinesq equations.

Findings

Constructed singularities with localized non-smoothness.

Developed a BKM-type criterion for one-point nonsmoothness.

Used weighted Hölder estimates near the singular point.

Abstract

We show that the constructions of $C^{1,\alpha}$ asymptotically self-similar singularities for the 3D Euler equations by Elgindi, and for the 3D Euler equations with large swirl and 2D Boussinesq equations with boundary by Chen-Hou can be extended to construct singularity with velocity $\mathbf{u} \in C^{1,\alpha}$ that is not smooth at only one point. The proof is based on a carefully designed small initial perturbation to the blowup profile, and a BKM-type continuation criterion for the one-point nonsmoothness. We establish the criterion using weighted H\"older estimates with weights vanishing near the singular point. Our results are inspired by the recent work of Cordoba, Martinez-Zoroa and Zheng that it is possible to construct a $C^{1,\alpha}$ singularity for the 3D axisymmetric Euler equations without swirl and with velocity $\mathbf{u} \in C^{\infty}(\mathbb{R}^3 \backslash \{0\})$.