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

Inspired by the numerical evidence of a potential 3D Euler singularity [54, 55], we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in [54, 55] for the 3D Euler equations with boundary. Our finite time blowup solution for the HL model and the singular solution considered in [54, 55] 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 [11] 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 ≈ 1/3 is uniformly bounded up to the singularity time. [11] Chen, J., Hou, T.Y., Huang, D.: On the finite time blowup of the De Gregorio model for the 3D Euler equations. Communications on Pure and Applied Mathematics 74(6), 1282–1350 (2021) [54] Luo, G., Hou, T.: Toward the finite-time blowup of the 3D incompressible Euler equations: a numerical investigation. SIAM Multiscale Modeling and Simulation 12(4), 1722–1776 (2014) [55] Luo, G., Hou, T.Y.: Potentially singular solutions of the 3d axisymmetric euler equations. Proceedings of the National Academy of Sciences 111(36), 12968–12973 (2014)