ancient solution
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 {{math|(−∞, T)}}."{{citation
| last = Perelman | first = Grigori | author-link = Grigori Perelman
| arxiv = math/0211159
| title = The entropy formula for the Ricci flow and its geometric applications
| year = 2002| bibcode = 2002math.....11159P}}.
The term was introduced by Richard Hamilton in his work on the Ricci flow.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 It has since been applied to other geometric flows{{citation
| last1 = Loftin | first1 = John
| last2 = Tsui | first2 = Mao-Pei
| issue = 1
| journal = Journal of Differential Geometry
| mr = 2406266
| pages = 113–162
| title = Ancient solutions of the affine normal flow
| volume = 78
| year = 2008| doi = 10.4310/jdg/1197320604
| arxiv = math/0602484
| s2cid = 420652
}}.{{citation
| last1 = Daskalopoulos | first1 = Panagiota | author1-link = Panagiota Daskalopoulos
| last2 = Hamilton | first2 = Richard | author2-link = Richard S. Hamilton
| last3 = Sesum | first3 = Natasa | author3-link = Nataša Šešum
| arxiv = 0806.1757
| issue = 3
| journal = Journal of Differential Geometry
| mr = 2669361
| pages = 455–464
| title = Classification of compact ancient solutions to the curve shortening flow
| volume = 84
| year = 2010| bibcode = 2008arXiv0806.1757D| doi = 10.4310/jdg/1279114297 | s2cid = 18747005 }}.{{citation
| last = You | first = Qian
| publisher = University of Wisconsin–Madison
| series = Ph.D. thesis
| title = Some Ancient Solutions of Curve Shortening
| year = 2014| id = {{ProQuest|1641120538}}
}}.{{citation
| last1 = Huisken | first1 = Gerhard | author1-link = Gerhard Huisken
| last2 = Sinestrari | first2 = Carlo
| issue = 2
| journal = Journal of Differential Geometry
| mr = 3399098
| pages = 267–287
| title = Convex ancient solutions of the mean curvature flow
| volume = 101
| year = 2015| doi = 10.4310/jdg/1442364652 | doi-access = free
| arxiv = 1405.7509
}}. as well as to other systems such as the Navier–Stokes equations{{citation
| last = Seregin | first = Gregory A.
| contribution = Weak solutions to the Navier-Stokes equations with bounded scale-invariant quantities
| mr = 2827878
| pages = 2105–2127
| publisher = Hindustan Book Agency, New Delhi
| title = Proceedings of the International Congress of Mathematicians
| volume = III
| year = 2010| title-link = Proceedings of the International Congress of Mathematicians
}}.{{citation
| last1 = Barker | first1 = T.
| last2 = Seregin | first2 = G.
| doi = 10.1007/s00021-015-0211-z
| issue = 3
| journal = Journal of Mathematical Fluid Mechanics
| mr = 3383928
| pages = 551–575
| title = Ancient solutions to Navier-Stokes equations in half space
| volume = 17
| year = 2015| arxiv = 1503.07428
| bibcode = 2015JMFM...17..551B
| s2cid = 119138067
}}. and heat equation.{{citation
| last = Wang | first = Meng
| doi = 10.1090/S0002-9939-2011-11170-5
| issue = 10
| journal = Proceedings of the American Mathematical Society
| mr = 2813381
| pages = 3491–3496
| title = Liouville theorems for the ancient solution of heat flows
| volume = 139
| year = 2011| doi-access = free
}}.