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

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 Hölder 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 $\mathbf{u}\in C^{\infty }({\mathbb{R}}^3\backslash \{0\})$.

In the previous version, the initial data is not in the weighted Holder space. We modify the space and show that the initial data is in the new weighted Holder space. 20 pages