In this paper, we present strong numerical evidences that the 3D  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.
In this paper, we present strong numerical evidences that the 3D incompressible axisymmetric Navier-Stokes equations with degenerate diffusion coefficients and smooth initial data of finite energy develop a potential finite time locally self-similar singularity at the origin. The spatial part of the degenerate diffusion coefficient is a smooth function of r and z independent of the solution and vanishes like O(r^2)+O(z^2) near the origin. This potential singularity is induced by a potential singularity of the 3D Euler equations. 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 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 . Without the viscous regularization, the 3D Euler equations develop an additional small scale characterizing the thickness of the sharp front. On the other hand, the Navier-Stokes equations with a constant diffusion coefficient regularize the two-scale solution structure and do not develop a finite time singularity for the same initial data. 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.
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incompressible
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variable
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The spatial part of the degenerate diffusion coefficient is a smooth function of r and z independent of the solution and vanishes like O(r^2)+O(z^2) near the origin. This potential singularity is induced by a potential singularity of the 3D Euler equations.
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to
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, which transport the solution first towards z=0 and then towards the symmetry axis r=0.
. Without the viscous regularization, the 3D Euler equations develop an additional small scale characterizing the thickness of the sharp front. On the other hand, the Navier-Stokes equations with a constant diffusion coefficient regularize the two-scale solution structure and do not develop a finite time singularity for the same initial data.