Nearly self-similar blowup of generalized axisymmetric Navier-Stokes and Boussinesq equations
<|reference_start|>Nearly self-similar blowup of generalized axisymmetric Navier-Stokes and Boussinesq equations: 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}).<|reference_end|>
@article{hou2024nearly,
    title={Nearly self-similar blowup of generalized axisymmetric Navier-Stokes and
  Boussinesq equations},
    author={Thomas Y. Hou},
    journal={arXiv preprint arXiv:2405.10916},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.10916},
    primaryClass={math.AP cs.NA math.NA}
}