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 $gammaapprox 1/3$ is uniformly bounded up to the singularity time.
تحميل البحث