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 two-dimensional Boussinesq and three-dimensional axisymmetric Euler equations with smooth initial data of finite energy and boundary. In Part I of our paper [Chen and Hou, preprint, arXiv:2210.07191, 2022], we establish an analytic framework to prove the nonlinear stability of an approximate self-similar blowup profile using a combination of weighted L∞ and weighted C1/2 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 velocity in the regular case using numerical integration with computer assistance. These results enable us to verify that the constants in the energy estimate obtained in Part I [Chen and Hou, preprint, arXiv:2210.07191, 2022] indeed satisfy the inequalities for nonlinear stability. The nonlinear stability further implies the finite time singularity of the axisymmetric three-dimensional Euler equations with smooth initial data and boundary.

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