Ancient solution
From HandWiki
In mathematics, an ancient solution to a differential equation is a solution that can be extrapolated backwards to all past times, without singularities. That is, it is a solution "that is defined on a time interval of the form (−∞, T)."[1]
The term was introduced by Richard Hamilton in his work on the Ricci flow.[2] It has since been applied to other geometric flows[3][4][5][6] as well as to other systems such as the Navier–Stokes equations[7][8] and heat equation.[9]
References
- ↑ The entropy formula for the Ricci flow and its geometric applications, 2002, Bibcode: 2002math.....11159P.
- ↑ Hamilton, Richard S. The formation of singularities in the Ricci flow. Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 7–136, Int. Press, Cambridge, MA, 1995
- ↑ Loftin, John; Tsui, Mao-Pei (2008), "Ancient solutions of the affine normal flow", Journal of Differential Geometry 78 (1): 113–162, doi:10.4310/jdg/1197320604.
- ↑ "Classification of compact ancient solutions to the curve shortening flow", Journal of Differential Geometry 84 (3): 455–464, 2010, doi:10.4310/jdg/1279114297, Bibcode: 2008arXiv0806.1757D.
- ↑ You, Qian (2014), Some Ancient Solutions of Curve Shortening, Ph.D. thesis, University of Wisconsin–Madison, ProQuest 1641120538.
- ↑ "Convex ancient solutions of the mean curvature flow", Journal of Differential Geometry 101 (2): 267–287, 2015, doi:10.4310/jdg/1442364652.
- ↑ Seregin, Gregory A. (2010), "Weak solutions to the Navier-Stokes equations with bounded scale-invariant quantities", Proceedings of the International Congress of Mathematicians, III, Hindustan Book Agency, New Delhi, pp. 2105–2127.
- ↑ Barker, T.; Seregin, G. (2015), "Ancient solutions to Navier-Stokes equations in half space", Journal of Mathematical Fluid Mechanics 17 (3): 551–575, doi:10.1007/s00021-015-0211-z, Bibcode: 2015JMFM...17..551B.
- ↑ Wang, Meng (2011), "Liouville theorems for the ancient solution of heat flows", Proceedings of the American Mathematical Society 139 (10): 3491–3496, doi:10.1090/S0002-9939-2011-11170-5.
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