A new result for the porous medium equation derived from the Ricci flow
HTML articles powered by AMS MathViewer
- by Lang-Fang Wu PDF
- Bull. Amer. Math. Soc. 28 (1993), 90-94 Request permission
Abstract:
Given ${\mathbb {R}^2}$, with a "good" complete metric, we show that the unique solution of the Ricci flow approaches a soliton at time infinity. Solitons are solutions of the Ricci flow, which move only by diffeomorphism. The Ricci flow on ${\mathbb {R}^2}$ is the limiting case of the porous medium equation when m is zero. The results in the Ricci flow may therefore be interpreted as sufficient conditions on the initial data, which guarantee that the corresponding unique solution for the porous medium equation on the entire plane asymptotically behaves like a "soliton-solution".References
-
D. G. Aronson, The porous medium equations, Some Problems in Nonlinear Diffusion (A. Fasano and M. Primicerio, eds.), Lecture Notes in Maths., vol. 1224, Springer, New York, 1986.
- Bennett Chow and Lang-Fang Wu, The Ricci flow on compact $2$-orbifolds with curvature negative somewhere, Comm. Pure Appl. Math. 44 (1991), no. 3, 275–286. MR 1090433, DOI 10.1002/cpa.3160440302
- Juan R. Esteban, Ana Rodríguez, and Juan L. Vázquez, A nonlinear heat equation with singular diffusivity, Comm. Partial Differential Equations 13 (1988), no. 8, 985–1039. MR 944437, DOI 10.1080/03605308808820566
- Richard S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986) Contemp. Math., vol. 71, Amer. Math. Soc., Providence, RI, 1988, pp. 237–262. MR 954419, DOI 10.1090/conm/071/954419 —, Notes on Harnack’s inequality, preprint.
- Miguel A. Herrero, A limit case in nonlinear diffusion, Nonlinear Anal. 13 (1989), no. 6, 611–628. MR 998508, DOI 10.1016/0362-546X(89)90082-5 —, Singular diffusion on the line (to appear).
- Wan-Xiong Shi, Complete noncompact Kähler manifolds with positive holomorphic bisectional curvature, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 437–440. MR 1044171, DOI 10.1090/S0273-0979-1990-15954-3 J. L. Valazquez, Two nonlinear diffusion equations with finite speed of propagation, Proceedings of the conference in honor of Jack Hale on the occasion of his 60th birthday, preprint.
- Lang-Fang Wu, The Ricci flow on $2$-orbifolds with positive curvature, J. Differential Geom. 33 (1991), no. 2, 575–596. MR 1094470 —, The Ricci flow on complete ${\mathbb {R}^2}$ (The limiting case of the porous medium equations as $m \to 0$), submitted.
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 28 (1993), 90-94
- MSC: Primary 58G30; Secondary 35Q51, 53C21, 58G11, 76S05
- DOI: https://doi.org/10.1090/S0273-0979-1993-00336-7
- MathSciNet review: 1164949