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arXiv · 2106.05422

Asymptotically self-similar blowup of the Hou-Luo model for the 3D Euler equations

Jiajie Chen·Thomas Y. Hou·De Huang
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Abstract

Inspired by the numerical evidence of a potential 3D Euler singularity \cite{luo2014potentially,luo2013potentially-2}, we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in \cite{luo2014potentially,luo2013potentially-2} for the 3D Euler equations with boundary. Our finite time blowup solution for the HL model and the singular solution considered in \cite{luo2014potentially,luo2013potentially-2} share some essential features, including similar blowup exponents, symmetry properties of the solution, and the sign of the solution. We use a dynamical rescaling formulation and the strategy proposed in our recent work in \cite{chen2019finite} to establish the nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the HL model with smooth initial data and finite energy will develop a focusing asymptotically self-similar singularity in finite time. Moreover the self-similar profile is unique within a small energy ball and the $C^γ$ norm of the density $θ$ with $γ\approx 1/3$ is uniformly bounded up to the singularity time.

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Jiajie Chen, Thomas Y. Hou, De Huang. 2021-06-12. Asymptotically self-similar blowup of the Hou-Luo model for the 3D Euler equations. https://arxiv.org/abs/2106.05422

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