Potential singularity formation of incompressible axisymmetric Euler equations with degenerate viscosity coefficients. (English) Zbl 1514.35318
Summary: In this paper, we present strong numerical evidence that the incompressible axisymmetric Euler equations with degenerate viscosity 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 toward the origin. The two-scale feature is characterized by the scaling property that the center of the traveling wave is located at a ring of radius \(O((T-t)^{1/2})\) surrounding the symmetry axis while the thickness of the ring collapses at a rate \(O(T-t)\). The driving mechanism for this potential singularity is due to an antisymmetric vortex dipole that generates a strong shearing layer in both the radial and axial velocity fields. Without the viscous regularization, the three-dimensional Euler equations develop a sharp front and some shearing instability in the far field. On the other hand, the Navier-Stokes equations with a constant viscosity coefficient regularize the two-scale solution structure and do not develop a finite-time singularity for the same initial data.
Keywords:
potential singularity; Navier-Stokes equations; Euler equations; multiscale blowup; degenerate viscosityReferences:
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