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

TL;DR

This paper proves finite time singularity formation for the Hou-Luo (HL) model of 3D Euler equations with boundary, demonstrating asymptotic self-similarity and stability of the blowup profile from smooth initial data.

Contribution

It establishes the nonlinear stability of a self-similar blowup profile for the HL model, confirming finite time singularity with smooth initial data and analyzing its asymptotic behavior.

Findings

Finite time singularity from smooth initial data.

Asymptotically self-similar blowup profile.

Bounded $C^rac{1}{3}$ norm of density up to singularity.

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 $C^\gamma$ norm of the density $\theta$ with $\gamma\approx 1/3$ is uniformly bounded up to the singularity time.