Nearly self-similar blowup of generalized axisymmetric Navier-Stokes equations

We numerically investigate the nearly self-similar blowup of the generalized axisymmetric Navier--Stokes equations. First, we rigorously derive the axisymmetric Navier--Stokes equations with swirl in both odd and even dimensions, marking the first such derivation for dimensions greater than three. Building on this, we generalize the equations to arbitrary positive real-valued dimensions, preserving many known properties of the 3D axisymmetric Navier--Stokes equations. To address scaling instability, we dynamically vary the space dimension to balance advection scaling along the r and z directions. A major contribution of this work is the development of a novel two-scale dynamic rescaling formulation, leveraging the dimension as an additional degree of freedom. This approach enables us to demonstrate a one-scale self-similar blowup with solution-dependent viscosity. Notably, the self-similar profile satisfies the axisymmetric Navier-Stokes equations with constant viscosity. We observe that the effective dimension is approximately 3.188 and appears to converge toward 3 as background viscosity diminishes. Furthermore, we introduce a rescaled Navier--Stokes model derived by dynamically rescaling the axial velocity in 3D. This model retains essential properties of 3D Navier-Stokes. Our numerical study shows that this rescaled Navier--Stokes model with two constant viscosity coefficients exhibits a nearly self-similar blowup with maximum vorticity growth on the order of O(10^{30}).