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Published June 2022 | metadata_only
Journal Article

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

Abstract

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)

Additional Information

The research was in part supported by NSF Grants DMS-1907977 and DMS-1912654. D. Huang would like to acknowledge the generous support from the Choi Family Postdoc Gift Fund.

Additional details

Created:
August 22, 2023
Modified:
August 22, 2023