We perform numerical investigation of nearly self-similar blowup of generalized axisymmetric Navier-Stokes equations and Boussinesq system with a time-dependent fractional dimension. The dynamic change of the space dimension is proportional to the ratio R(t)/Z(t), where (R(t),Z(t)) is the position at which the maximum vorticity achieves its global maximum. This choice of space dimension is to ensure that the advection along the r-direction has the same scaling as that along the z-direction, thus preventing formation of two-scale solution structure. For the generalized axisymmetric Navier-Stokes equations with solution dependent viscosity, we show that the solution develops a self-similar blowup with dimension equal to 3.188 and the self-similar profile satisfies the axisymmetric Navier-Stokes equations with constant viscosity. We also study the nearly self-similar blowup of the axisymmetric Boussinesq system with constant viscosity. The generalized axisymmetric Boussinesq system preserves almost all the known properties of the 3D Navier-Stokes equations except for the conservation of angular momentum. We present convincing numerical evidence that the generalized axisymmetric Boussinesq system develops a stable nearly self-similar blowup solution with maximum vorticity increased by O(10^{30}).

翻译：我们对具有时间依赖分形维数的广义轴对称Navier-Stokes方程和Boussinesq方程组的近自相似爆破进行了数值研究。空间维数的动态变化与比值R(t)/Z(t)成正比，其中(R(t),Z(t))为最大涡量达到全局最大值时的位置。该空间维数的选取旨在确保沿r方向的平流与沿z方向具有相同的尺度缩放，从而避免形成双尺度解结构。对于具有解依赖黏性的广义轴对称Navier-Stokes方程，我们证明了当维数等于3.188时解会发展出自相似爆破，且该自相似剖面满足常黏性轴对称Navier-Stokes方程。此外，我们研究了常黏性轴对称Boussinesq方程组的近自相似爆破。广义轴对称Boussinesq方程组保留了三维Navier-Stokes方程几乎所有已知性质，仅角动量守恒定律除外。我们提出了令人信服的数值证据，表明广义轴对称Boussinesq方程组会发展出稳定的近自相似爆破解，其最大涡量可增大O(10^{30})倍。