Remarks on the smoothness of the C^1,α asymptotically self-similar singularity in the 3D Euler and 2D Boussinesq equations

2023/08/31 by Chen, Jiajie · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.00150

Abstract

We show that the constructions of C^1,α 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 u ∈ C^1,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 C^1,α singularity for the 3D axisymmetric Euler equations without swirl and with velocity u ∈ C^∞(ℝ³ \backslash \0\).

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• Finite-time singularity via multi-layer degenerate pendula for the 2D Boussinesq equation with uniform C^{1,√((4)/(3))-1-ε}∩ L² force

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