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 [ChenHou2023a], we establish an analytic framework to prove stability of an approximate self-similar blowup profile by a combination of a weighted $L^\infty$ norm and a weighted $C^{1/2}$ norm. Under the assumption that the stability constants, which depend on the approximate steady state, satisfy certain inequalities stated in our stability lemma, we prove stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. In Part II of our paper, we provide sharp stability estimates of the linearized operator by constructing space-time solutions with rigorous error control. We also obtain sharp estimates of the velocity in the regular case using computer assistance. These results enable us to verify that the stability constants obtained in Part I [ChenHou2023a] indeed satisfy the inequalities in our stability lemma. This completes the analysis of the finite time singularity of the axisymmetric Euler equations with smooth initial data and boundary.

翻译：本文为系列论文的第二部分，证明了具有有限能量和边界的二维Boussinesq方程与三维轴对称Euler方程在光滑初值条件下的有限时间爆破。在第一部分[ChenHou2023a]中，我们建立了结合加权$L^\infty$范数与加权$C^{1/2}$范数的分析框架，用于证明近似自相似爆破剖面的稳定性。在稳定性引理中提出的依赖于近似稳态的稳定性常数满足特定不等式的假设下，我们证明了具有光滑初值与边界的二维Boussinesq方程与三维Euler方程的稳定近似自相似爆破。在第二部分中，我们通过构造具有严格误差控制的时空解，给出了线性化算子的精确稳定性估计。同时利用计算机辅助获得了正则情形下速度场的精确估计。这些结果使我们能够验证第一部分[ChenHou2023a]中得到的稳定性常数确实满足稳定性引理中的不等式，从而完成了对具有光滑初值与边界的轴对称Euler方程有限时间奇点问题的分析。