Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Strict contractivity of the 2-Wasserstein distance for the porous medium equation by mass-centering


Authors: J. A. Carrillo, M. Di Francesco and G. Toscani
Journal: Proc. Amer. Math. Soc. 135 (2007), 353-363
MSC (2000): Primary 35K65; Secondary 35B40
DOI: https://doi.org/10.1090/S0002-9939-06-08594-7
Published electronically: August 21, 2006
MathSciNet review: 2255281
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that the Euclidean Wasserstein distance between two compactly supported solutions of the one-dimensional porous medium equation having the same center of mass decays to zero for large times. As a consequence, we detect an improved $ L^1$-rate of convergence of solutions of the one-dimensional porous medium equation towards well-centered self-similar Barenblatt profiles, as time goes to infinity.


References [Enhancements On Off] (What's this?)

  • 1. S. Angenent, Large time asymptotics for the porous media equation, Nonlinear diffusion equations and their equilibrium states, I (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., vol. 12, Springer, New York, 1988, pp. 21-34. MR 0956056 (90k:35028)
  • 2. D. G. Aronson, The porous medium equation, Nonlinear diffusion problems (Montecatini Terme, 1985), Lecture Notes in Math., vol. 1224, Springer, Berlin, 1986, pp. 1-46. MR 0877986 (88a:35130)
  • 3. J. A. Carrillo, M. P. Gualdani, and G. Toscani, Finite speed of propagation in porous media by mass transportation methods, C. R. Math. Acad. Sci. Paris 338 (2004), no. 10, 815-818. MR 2059493 (2005a:35155)
  • 4. J. A. Carrillo, R. J. McCann, and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana 19 (2003), no. 3, 971-1018. MR 2053570 (2005a:35126)
  • 5. J. A. Carrillo and G. Toscani, Asymptotic $ L\sp 1$-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J. 49 (2000), no. 1, 113-142. MR 1777035 (2001j:35155)
  • 6. -, Long-time asymptotics for strong solutions of the thin film equation, Comm. Math. Phys. 225 (2002), no. 3, 551-571. MR 1888873 (2002m:35115)
  • 7. J. A. Carrillo, R. J. McCann, and C. Villani, Contractions in the $ 2$-Wasserstein length space and thermalization of granular media, to appear in Archive for Rational Mechanics and Analysis (2005).
  • 8. J. A. Carrillo and G. Toscani, Wasserstein metric and large-time asymptotics of nonlinear diffusion equations, New Trends in Mathematical Physics (In Honour of the Salvatore Rionero 70th Birthday), World Scientific, Singapore, 2005, pp. 234-244. MR 2163983 (2006c:35147)
  • 9. J. A. Carrillo and J. L. Vázquez, Fine asymptotics for fast diffusion equations, Comm. Partial Differential Equations 28 (2003), 1023-1056. MR 1986060 (2004a:35118)
  • 10. S. K. Chua and R. L. Wheeden, Sharp conditions for weighted $ 1$-dimensional Poincaré inequalities, Indiana Univ. Math. J. 49 (2000), no. 1, 143-175. MR 1777034 (2001h:26021)
  • 11. D. Cordero-Erausquin and R. J. McCann, Accelerated diffusion to minimum entropy, personal communication (2005).
  • 12. J. Denzler and R. J. McCann, Fast diffusion to self-similarity: complete spectrum, long time asymptotics, and numerology, Arch. Rational Mech. Anal. 175 (2005), 301-342. MR 2126633 (2005k:35214)
  • 13. J. Duoandikoetxea and E. Zuazua, Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 6, 693-698. MR 1183805 (94a:42046)
  • 14. T. Goudon, S. Junca, and G. Toscani, Fourier-based distances and Berry-Esseen like inequalities for smooth densities, Monatsh. Math. 135 (2002), no. 2, 115-136. MR 1894092 (2003d:60054)
  • 15. Y. J. Kim and R. J. McCann, Potential theory and optimal convergence rates in fast nonlinear diffusion, Preprint, 2004.
  • 16. Y. J. Kim and W. M. Ni, Higher order approximations in the heat equation and the truncated moment problem, Preprint, 2005.
  • 17. R. J. McCann and D. Slepcev, Nearly optimal convergence rates for centered solutions to the fast-diffusion equations, Preprint, 2005.
  • 18. F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26 (2001), no. 1-2, 101-174. MR 1842429 (2002j:35180)
  • 19. J. L. Vázquez, Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ. 3 (2005), 67-118. MR 1977429 (2004d:35138)
  • 20. -, Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium, Trans. Amer. Math. Soc. 277 (1983), no. 2, 507-527. MR 0694373 (84h:35014)
  • 21. -, Behaviour of the velocity of one-dimensional flows in porous media, Trans. Amer. Math. Soc. 286 (1984), no. 2, 787-802. MR 0760987 (86b:35016)
  • 22. -, The interfaces of one-dimensional flows in porous media, Trans. Amer. Math. Soc. 285 (1984), no. 2, 717-737. MR 0752500 (85h:35229)
  • 23. -, An introduction to the mathematical theory of the porous medium equation, Shape optimization and free boundaries (Montreal, PQ, 1990), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 380, Kluwer Acad. Publ., Dordrecht, 1992, pp. 347-389. MR 1260981 (95b:35101)
  • 24. C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics, vol. 58, American Mathematical Society, Providence, RI, 2003. MR 1964483 (2004e:90003)
  • 25. T. P. Witelski and A. J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations, Stud. Appl. Math. 100 (1998), no. 2, 153-193. MR 1491842 (99d:35081)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35K65, 35B40

Retrieve articles in all journals with MSC (2000): 35K65, 35B40


Additional Information

J. A. Carrillo
Affiliation: Departament de Matemàtiques - ICREA, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
Email: carrillo@mat.uab.es

M. Di Francesco
Affiliation: Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences (ÖAW), A-4040 Linz, Austria
Address at time of publication: Sezione di Matematica per L’Ingegneria, Universita di L’Aquila, Piazzale Pontieri, Monteluco di Roio, I-67100 L’Aquila, Italy
Email: marco.difrancesco@oeaw.ac.at

G. Toscani
Affiliation: Dipartimento di Matematica, Università di Pavia, I-27100 Pavia, Italy
Email: giuseppe.toscani@unipv.it

DOI: https://doi.org/10.1090/S0002-9939-06-08594-7
Keywords: Porous medium equation, Barenblatt solutions, Wasserstein distance.
Received by editor(s): July 27, 2005
Published electronically: August 21, 2006
Communicated by: Walter Craig
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society