Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I: Analysis

Abstract

Inspired by numerical evidence of a potential 3D Euler singularity \cite{luo2014potentially,luo2013potentially-2}, we prove finite-time, nearly self-similar blowup of the 2D Boussinesq and 3D axisymmetric Euler equations with smooth initial data of finite energy and boundary. The proof encounters several essential difficulties. One of the essential difficulties is to control a number of nonlocal terms that do not seem to offer any damping effect. Another essential difficulty is that the strong advection normal to the boundary introduces a large growth factor for the perturbation when using weighted $L^2$ or $H^k$ estimates. We overcome these difficulties by combining weighted $L^\infty$ estimates and weighted $C^{1/2}$ estimates, and by developing sharp functional inequalities using the symmetry properties of the kernels and some techniques from optimal transport. Moreover, we decompose the linearized operator into a leading order operator and a finite-rank operator. We design the leading order operator to obtain sharp stability estimates. The contribution from the finite-rank operator to linear stability is estimated by constructing approximate space-time solutions. These ingredients enable us to establish the nonlinear stability of the approximate self-similar profile and to prove stable nearly self-similar blowup.