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Regularity of solutions and interfaces of the porous medium equation via local estimates*

Published online by Cambridge University Press:  14 November 2011

Juan Luis Vazquez
Juan Luis Vazquez
Affiliation:
Departamento de Matemáticas, Universidad Autonóma de Madrid, 28049 Madrid, Spain
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Synopsis

We prove C∞ regularity for moving interfaces of local solutions of the porous medium equation as well as C∞ lateral regularity for the pressure function near such an interface.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 112 , Issue 1-2 , 1989 , pp. 1 - 13
Copyright
Copyright © Royal Society of Edinburgh 1989

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References

1Alikakos, N. D. and Rostamian, R.. Large Time Behavior of Solutions of Neumann Boundary Value Problem for the Porous Medium Equation. Indiana Univ. Math. J. 30 (1981), 749–785.Google Scholar
2Angenent, S.. Analyticity of the interface of the porous media equation after waiting time. Proc. Amer. Math. Soc. 102 (1988), 329–336.Google Scholar
3Aronson, D. G.. Regularity properties of flows through porous media. SIAM J. Appl. Math. 17 (1969), 461–467.Google Scholar
4Aronson, D. G.. Regularity properties of flows through porous media: The interface. Arch. Rational Mech. Anal. 37 (1970), 1–10.CrossRefGoogle Scholar
5Aronson, D. G.. Regularity properties of flows through porous media: A counterexample. SIAM J. Appl. Math. 19 (1970), 299–307.Google Scholar
6Aronson, D. G. and Bénilan, Ph.. Régularité des solutions de l'équation des milieux poreux dans ∝N. C. R. Acad. Sci. Paris. Ser. A-B 288 (1979), 103–105.Google Scholar
7Aronson, D. G., Caffarelli, L. A. and Kamin, S.. How an initially stationary interface begins to move in porous medium flow. SIAM J. Math. Anal. 14 (1983), 639–658.Google Scholar
8Aronson, D. G., Caffarelli, L. A. and Vazquez, J. L.. Interfaces with a corner-point in one-dimensional porous medium flow. Comm. Pure Appl. Math. 38 (1985), 375–404.Google Scholar
9Aronson, D. G., Crandall, M. G. and Peletier, L. A.. Stabilization of solutions of a degenerate nonlinear diffusion problem. Nonlinear Anal. 6 (1982), 1001–1022.Google Scholar
10Aronson, D. G. and Peletier, L. A.. Large-time behaviour of solutions of the porous media equation in bounded domains. J. Differential Equations 39 (1981), 378–412.Google Scholar
11Aronson, D. G. and Vazquez, J. L.. Eventual C∞-regularity and concavity for flows in one-dimensional porous media. Arch. Rational Mech. Anal. 99 (1987), 329–348.Google Scholar
12Bénilan, Ph.. Thesis, University of Paris XI, Orsay, 1969.Google Scholar
13Bénilan, Ph.. Evolution Equations and Accretive Operators (Lecture Notes, University of Kentucky, 1981).Google Scholar
14Bénilan, Ph. and Vazquez, J. L.. Concavity of solutions of the porous medium equation. Trans. Amer. Math. Soc. 299 (1987), 81–93.Google Scholar
15Caffarelli, L. A. and Friedman, A.. Regularity of the free boundary for the one-dimensional flow of a gas in a porous medium. Amer. J. Math. 101 (1979), 1193–1218.Google Scholar
16Caffarelli, L. A. and Friedman, A.. Continuity of the density of a gas flow in a porous medium. Trans. Amer. Math. Soc. 252 (1979), 99–113.CrossRefGoogle Scholar
17Caffarelli, L. A. and Friedman, A.. Regularity of the free boundary of a gas flow in an n-dimensional porous medium. Indiana Univ. Math. J. 29 (1980), 361–391.Google Scholar
18Caffarelli, L. A., Vazquez, J. L. and Wolanski, N. E.. Lipschitz continuity of solutions and interfaces to the N-dimensional porous medium equation. Indiana Univ. Math. J. 36 (1987), 373–401.Google Scholar
19Dahlberg, B. E. J. and Kenig, C. E.. Non-negative solutions of the porous medium equation. Comm. Partial Differential Equations 9 (1984), 409–437.Google Scholar
20Herrero, M. A. and Vazquez, J. L.. The one-dimensional nonlinear heat equation with absorption: regularity of solutions and interfaces. SIAM J. Math. Anal. 18 (1987), 149–167.Google Scholar
21Hollig, K. and Kreiss, H. O.. C∞-Regularity for the Porous Medium Equation. Math. Z. 192 (1986), 217–224.Google Scholar
22Knerr, B. F.. The porous medium equation in one dimension. Trans. Amer. Math. Soc. 234 (1977), 381–415.Google Scholar
23Lacey, A. A., Ockendon, J. R. and Tayler, A. B.. ‘Waiting-time’ solutions of a nonlinear diffusion equation. SIAM J. Appl. Math. 42 (1982), 1252–1264.CrossRefGoogle Scholar
24Muskat, M.. The Flow of Homogeneous Fluids Through Porous Media (New York: McGraw-Hill, 1937).Google Scholar
25Oleinik, O. A., Kalashnikov, A. S. and Czhou, Y. L.. The Cauchy problem and boundary problems for equations of the type of nonstationary filtration. Izv. Akad. Nauk. SSSR Ser. Mat. 22 (1958), 667–704 (in Russian).Google Scholar
26Peletier, L. A.. The porous media equation. In Application of Nonlinear Analysis in the Physical Sciences, eds. Amann, H.et al., 229–241 (London: Pitman, 1981).Google Scholar
27Vazquez, J. V.. Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium. Trans. Amer. Math. Soc. 277 (1983), 507–527.Google Scholar
28Vazquez, J. L.. The interfaces of one-dimensional flows in porous media. Trans. Amer. Math. Soc. 285 (1984), 717–737.Google Scholar
29Vazquez, J. L.. Hyperbolic aspects in the theory of the porous medium equation. In Metastability and Incompletely Posed Problems, eds. Antman, S.et al., IMA Volumes in Mathematics 3 (New York: Springer, 1987).Google Scholar
30Zeldovich, Y. B. and Raizer, Y. P.. Physics of Shock-waves and High-temperature Hydrodynamic Phenomena, Vol. 2 (New York: Academic Press, 1966).Google Scholar