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：本文是我们论文的第二部分，旨在证明具有有限能量与边界的光滑初值条件下，二维Boussinesq方程与三维轴对称Euler方程存在有限时间爆破现象。在论文第一部分[ChenHou2023a]中，我们建立了一个分析框架，通过加权$L^\infty$范数与加权$C^{1/2}$范数的组合，证明了近似自相似爆破解的稳定性。在依赖于近似稳态解的稳定性常数满足稳定性引理中所述不等式的假设下，我们证明了具有光滑初值与边界的二维Boussinesq方程和三维Euler方程存在稳定的近似自相似爆破。在本文第二部分中，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。此外，我们借助计算机辅助方法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条件，从而完成了对具有光滑初值与边界的轴对称Euler方程有限时间奇点的完整分析。