Remarks on the smoothness of the C1,α asymptotically self-similar singularity in the 3D Euler and 2D Boussinesq equations

Abstract

We show that the constructions of C1,α asymptotically self-similar singularities for the three-dimensional (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 u∈C1,α 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 C1,α singularity for the 3D axisymmetric Euler equations without swirl and with velocity u∈C∞(R3∖{0}) .