2021/06/09 by Jiajie Chen, Thomas Y. Hou, Chen, Jiajie +3 · 1 voice · 1 citation
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Computational Fluid Dynamics and Aerodynamics #Navier-Stokes equation solutions #math.AP
paper · pdf · doi:10.48550/arxiv.2106.05422
openalex publication_date 2021/06/09 · openalex created_date 2022/11/18 · openalex updated_date 2026/07/28
Inspired by the numerical evidence of a potential 3D Euler singularity \citeluo2014potentially,luo2013potentially-2, we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in \citeluo2014potentially,luo2013potentially-2 for the 3D Euler equations with boundary. Our finite time blowup solution for the HL model and the singular solution considered in \citeluo2014potentially,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 \citechen2019finite 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 Cγ norm of the density θ with γ≈ 1/3 is uniformly bounded up to the singularity time.