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