| 词条 | Ancient solution |
| 释义 |
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)}}."[1] The term was introduced in Grigori Perelman's research on the Ricci flow,[1] and has since been applied to other geometric flows[2][3][4][5] as well as to other systems such as the Navier–Stokes equations[6][7] and heat equation.[8] References1. ^1 {{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}}. 2. ^{{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 | url = http://projecteuclid.org/euclid.jdg/1197320604 | volume = 78 | year = 2008}}. 3. ^{{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 | url = http://projecteuclid.org/euclid.jdg/1279114297 | volume = 84 | year = 2010| bibcode = 2008arXiv0806.1757D}}. 4. ^{{citation | last = You | first = Qian | publisher = University of Wisconsin–Madison | series = Ph.D. thesis | title = Some Ancient Solutions of Curve Shortening | url = http://search.proquest.com/docview/1641120538 | year = 2014}}. 5. ^{{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 | url = http://projecteuclid.org/euclid.jdg/1442364652 | volume = 101 | year = 2015}}. 6. ^{{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}}. 7. ^{{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 }}. 8. ^{{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}}. 1 : Differential equations |
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