Potential Singularity Formation of 3D Axisymmetric Navier-Stokes Equations with Degenerate Variable Diffusion Coefficients

In this paper, we present strong numerical evidences that the $3$D axisymmetric Navier-Stokes equations with degenerate variable diffusion coefficients and smooth initial data of finite energy develop a potential finite time locally self-similar singularity at the origin. An important feature of this potential singularity is that the solution develops a two-scale traveling wave that travels towards the origin. The two-scale feature is characterized by the property that the center of the traveling wave approaches to the origin at a slower rate than the rate of the collapse of the singularity. The driving mechanism for this potential singularity is due to two antisymmetric vortex dipoles that generate a strong shearing layer in both the radial and axial velocity fields, which transport the solution first towards $z=0$ and then towards the symmetry axis $r=0$. The initial condition is designed in such a way that it generates a positive feedback loop that enforces a strong nonlinear alignment of vortex stretching, leading to a stable locally self-similar blowup at the origin. We perform careful resolution study and asymptotic scaling analysis to provide further support of the potential finite time locally self-similar blowup.