Ancient solution

Today, Ancient solution is a topic that generates great interest and debate in different areas of society. For years, this topic has acquired significant relevance, awakening the interest of experts, academics, professionals and the general public. The importance of Ancient solution lies in its impact on various aspects of daily life, as well as its influence on decision-making at a political, social and economic level. Therefore, it is essential to understand in depth the aspects surrounding Ancient solution, its implications and its short- and long-term effects. That is why in this article we will comprehensively and objectively address the different aspects related to Ancient solution, with the aim of providing a clear and complete vision of this topic that is so relevant today.

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

  1. ^ Perelman, Grigori (2002), The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159, Bibcode:2002math.....11159P.
  2. ^ 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
  3. ^ Loftin, John; Tsui, Mao-Pei (2008), "Ancient solutions of the affine normal flow", Journal of Differential Geometry, 78 (1): 113–162, arXiv:math/0602484, doi:10.4310/jdg/1197320604, MR 2406266, S2CID 420652.
  4. ^ 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, doi:10.4310/jdg/1279114297, MR 2669361, S2CID 18747005.
  5. ^ You, Qian (2014), Some Ancient Solutions of Curve Shortening, Ph.D. thesis, University of Wisconsin–Madison, ProQuest 1641120538.
  6. ^ Huisken, Gerhard; Sinestrari, Carlo (2015), "Convex ancient solutions of the mean curvature flow", Journal of Differential Geometry, 101 (2): 267–287, arXiv:1405.7509, doi:10.4310/jdg/1442364652, MR 3399098.
  7. ^ Seregin, Gregory A. (2010), "Weak solutions to the Navier-Stokes equations with bounded scale-invariant quantities", Proceedings of the International Congress of Mathematicians, vol. III, Hindustan Book Agency, New Delhi, pp. 2105–2127, MR 2827878.
  8. ^ 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, S2CID 119138067.
  9. ^ 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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