Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data II: Rigorous Numerics

Abstract

This is Part II of our paper in which we prove finite time blowup of the 2D Boussinesq and 3D axisymmetric Euler equations with smooth initial data of finite energy and boundary. In Part I of our paper \cite{ChenHou2023a}, we establish an analytic framework to prove nonlinear stability of an approximate self-similar blowup profile using a combination of weighted $L^\infty$ and weighted $C^{1/2}$ energy estimates. We reduce proving nonlinear stability to verifying several inequalities for the constants in the energy estimate which depend on the approximate steady state and the weights in the energy functional only. In Part II of our paper, we construct approximate space-time solutions with rigorous error control, which are used to obtain sharp stability estimates of the linearized operator in Part I. We also obtain sharp estimates of the regular part of the velocity using numerical integration with computer assistance. These results enable us to verify that the constants in the energy estimate obtained in Part I \cite{ChenHou2023a} indeed satisfy the inequalities for nonlinear stability. The nonlinear stability further implies the finite time singularity of the axisymmetric 3D Euler equations with smooth initial data and boundary.