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 (−∞, 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]



References





  1. ^ ab Perelman, Grigori (2002), The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159, Bibcode:2002math.....11159P.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.


  2. ^ Loftin, John; Tsui, Mao-Pei (2008), "Ancient solutions of the affine normal flow", Journal of Differential Geometry, 78 (1): 113–162, MR 2406266.


  3. ^ Daskalopoulos, Panagiota; Hamilton, Richard; Sesum, Natasa (2010), "Classification of compact ancient solutions to the curve shortening flow", Journal of Differential Geometry, 84 (3): 455–464, arXiv:0806.1757, Bibcode:2008arXiv0806.1757D, MR 2669361.


  4. ^ You, Qian (2014), Some Ancient Solutions of Curve Shortening, Ph.D. thesis, University of Wisconsin–Madison.


  5. ^ Huisken, Gerhard; Sinestrari, Carlo (2015), "Convex ancient solutions of the mean curvature flow", Journal of Differential Geometry, 101 (2): 267–287, MR 3399098.


  6. ^ 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, MR 2827878.


  7. ^ Barker, T.; Seregin, G. (2015), "Ancient solutions to Navier-Stokes equations in half space", Journal of Mathematical Fluid Mechanics, 17 (3): 551–575, arXiv:1503.07428, Bibcode:2015JMFM...17..551B, doi:10.1007/s00021-015-0211-z, MR 3383928.


  8. ^ 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, MR 2813381.









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