Weak solutions of the porous medium equation
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- by Björn E. J. Dahlberg and Carlos E. Kenig PDF
- Trans. Amer. Math. Soc. 336 (1993), 711-725 Request permission
Abstract:
We show that if $u \geq 0$, $u \in L_{{\text {loc}}}^m(\Omega )$, $\Omega \subset {{\mathbf {R}}^{n + 1}}$ solves $\partial u/\partial t = \Delta {u^m}$, $m > 1$ , in the sense of distributions, then $u$ is locally Hölder continuous in $\Omega$.References
- D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc. 280 (1983), no. 1, 351–366. MR 712265, DOI 10.1090/S0002-9947-1983-0712265-1
- Philippe Bénilan, Michael G. Crandall, and Michel Pierre, Solutions of the porous medium equation in $\textbf {R}^{N}$ under optimal conditions on initial values, Indiana Univ. Math. J. 33 (1984), no. 1, 51–87. MR 726106, DOI 10.1512/iumj.1984.33.33003
- Luis A. Caffarelli and Avner Friedman, Regularity of the free boundary of a gas flow in an $n$-dimensional porous medium, Indiana Univ. Math. J. 29 (1980), no. 3, 361–391. MR 570687, DOI 10.1512/iumj.1980.29.29027
- Björn E. J. Dahlberg and Carlos E. Kenig, Nonnegative solutions of the porous medium equation, Comm. Partial Differential Equations 9 (1984), no. 5, 409–437. MR 741215, DOI 10.1080/03605308408820336
- Björn E. J. Dahlberg and Carlos E. Kenig, Nonnegative solutions of generalized porous medium equations, Rev. Mat. Iberoamericana 2 (1986), no. 3, 267–305. MR 908054
- Emmanuele DiBenedetto and Avner Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math. 357 (1985), 1–22. MR 783531, DOI 10.1515/crll.1985.357.1
- Jürgen Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591. MR 159138, DOI 10.1002/cpa.3160140329
- Paul E. Sacks, Continuity of solutions of a singular parabolic equation, Nonlinear Anal. 7 (1983), no. 4, 387–409. MR 696738, DOI 10.1016/0362-546X(83)90092-5
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Additional Information
- © Copyright 1993 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 336 (1993), 711-725
- MSC: Primary 35D05; Secondary 35K55, 76S05
- DOI: https://doi.org/10.1090/S0002-9947-1993-1085939-X
- MathSciNet review: 1085939