Potential Singularity Formation of Incompressible Axisymmetric Euler Equations with Degenerate Viscosity Coefficients
- Creators
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Hou, Thomas Y.
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Huang, De
Abstract
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)½) 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.
Additional Information
© 2023 Society for Industrial and Applied Mathematics. This research was in part supported by NSF grants DMS-1907977 and DMS-1912654. The second author was supported by the National Key R&D Program of China under grant 2021YFA1001500 and received support from the Choi Family Postdoc Gift Fund.Attached Files
Published - 22m1470906.pdf
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Additional details
- Eprint ID
- 121597
- Resolver ID
- CaltechAUTHORS:20230530-441187700.9
- NSF
- DMS-1907977
- NSF
- DMS-1912654
- National Key Research and Development Program of China
- 2021YFA1001500
- Choi Family Gift Fund
- Created
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2023-06-21Created from EPrint's datestamp field
- Updated
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2023-06-21Created from EPrint's last_modified field