Non-preservation of \(\alpha\)-concavity for the porous medium equation. (English) Zbl 1534.35249
Summary: We show that the porous medium equation does not in general preserve \(\alpha\)-concavity of the pressure for \(0 \leq \alpha < 1 / 2\) or \(1 / 2 < \alpha \leq 1\). In particular, this resolves an open problem of Vázquez on whether concavity of pressure is preserved by the porous medium equation. Our results strengthen an earlier work of Ishige-Salani, who considered the case of small \(\alpha > 0\). Since Daskalopoulos-Hamilton-Lee showed that 1/2-concavity is preserved, our result is sharp.
Our explicit examples show that concavity can be instantaneously broken at an interior point of the support of the initial data. For \(0 \leq \alpha < 1 / 2\), we give another set of examples to show that concavity can be broken at a boundary point.
Our explicit examples show that concavity can be instantaneously broken at an interior point of the support of the initial data. For \(0 \leq \alpha < 1 / 2\), we give another set of examples to show that concavity can be broken at a boundary point.
MSC:
| 35K65 | Degenerate parabolic equations |
| 35K15 | Initial value problems for second-order parabolic equations |
References:
| [1] | Angenent, S. B.; Aronson, D. G., The focusing problem for the radially symmetric porous medium equation, Commun. Partial Differ. Equ., 20, 7-8, 1217-1240, (1995) · Zbl 0830.35062 |
| [2] | Aronson, D.; Crandall, M. G.; Peletier, L. A., Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal., 6, 10, 1001-1022, (1982) · Zbl 0518.35050 |
| [3] | Aronson, D. G.; Graveleau, J., A self-similar solution to the focusing problem for the porous medium equation, Eur. J. Appl. Math., 4, 1, 65-81, (1993) · Zbl 0780.35079 |
| [4] | Bénilan, P.; Crandall, M. G., The continuous dependence on Φ of solutions of \(u_t - \operatorname{\Delta} \operatorname{\Phi} = 0\), Indiana Univ. Math. J., 30, 2, 161-177, (1981) · Zbl 0482.35012 |
| [5] | Bénilan, P.; Vázquez, J. L., Concavity of solutions of the porous medium equation, Trans. Am. Math. Soc., 299, 1, 81-93, (1987) · Zbl 0628.76092 |
| [6] | Borell, C., Brownian motion in a convex ring and quasiconcavity, Commun. Math. Phys., 86, 1, 143-147, (1982) · Zbl 0516.60084 |
| [7] | Brascamp, H. J.; Lieb, E. H., On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal., 22, 4, 366-389, (1976) · Zbl 0334.26009 |
| [8] | Caffarelli, L. A.; Friedman, A., Continuity of the density of a gas flow in a porous medium, Trans. Am. Math. Soc., 252, 99-113, (1979) · Zbl 0425.35060 |
| [9] | Caffarelli, L. A.; Friedman, A., Regularity of the free boundary of a gas flow in an n-dimensional porous medium, Indiana Univ. Math. J., 29, 3, 361-391, (1980) · Zbl 0439.76085 |
| [10] | Caffarelli, L. A.; Vázquez, J. L.; Wolanski, N. I., Lipschitz continuity of solutions and interfaces of the N-dimensional porous medium equation, Indiana Univ. Math. J., 36, 2, 373-401, (1987) · Zbl 0644.35058 |
| [11] | Chau, A.; Weinkove, B., Counterexamples to quasiconcavity for the heat equation, Int. Math. Res. Not., 2020, 22, 8564-8579, (2020) · Zbl 1459.35228 |
| [12] | Chau, A.; Weinkove, B., Strong space-time convexity and the heat equation, Indiana Univ. Math. J., 70, 4, 1189-1210, (2021) · Zbl 1475.35173 |
| [13] | Chau, A.; Weinkove, B., The Stefan problem and concavity, Calc. Var. Partial Differ. Equ., 60, 2, Article 176 pp., (2021), 13 pp. · Zbl 1473.80009 |
| [14] | Chen, C. Q.; Hu, B. W., A microscopic convexity principle for space-time convex solutions of fully nonlinear parabolic equations, Acta Math. Sin. Engl. Ser., 29, 4, 651-674, (2013) · Zbl 1267.35105 |
| [15] | Chen, C.; Ma, X.; Salani, P., On space-time quasiconcave solutions of the heat equation, Mem. Am. Math. Soc., 259, 1244, (2019) · Zbl 1442.35002 |
| [16] | Dahlberg, B. E.J.; Kenig, C. E., Nonnegative solutions of the initial-Dirichlet problem for generalized porous medium equations in cylinders, J. Am. Math. Soc., 1, 2, 401-412, (1988) · Zbl 0699.35155 |
| [17] | Daskalopoulos, P.; Hamilton, R., Regularity of the free boundary for the porous medium equation, J. Am. Math. Soc., 11, 4, 899-965, (1998) · Zbl 0910.35145 |
| [18] | Daskalopoulos, P.; Hamilton, R.; Lee, K., All time \(C^\infty \)-regularity of the interface in degenerate diffusion: a geometric approach, Duke Math. J., 108, 2, 295-327, (2001) · Zbl 1017.35052 |
| [19] | Friedman, A., Variational Principles and Free-Boundary Problems, A Wiley-Interscience Publication. Pure and Applied Mathematics, (1982), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York · Zbl 0564.49002 |
| [20] | Ishige, K.; Salani, P., Is quasi-concavity preserved by heat flow?, Arch. Math. (Basel), 90, 5, 450-460, (2008) · Zbl 1176.35012 |
| [21] | Ishige, K.; Salani, P., Convexity breaking of the free boundary for porous medium equations, Interfaces Free Bound., 12, 1, 75-84, (2010) · Zbl 1189.35387 |
| [22] | Ishige, K.; Salani, P., Parabolic quasi-concavity for solutions to parabolic problems in convex rings, Math. Nachr., 283, 11, 1526-1548, (2010) · Zbl 1206.35020 |
| [23] | Koch, H., Non-Euclidean singular integrals and the porous medium equation, (1999), University of Heidelberg, Habilitation Thesis |
| [24] | Pierre, M., Uniqueness of the solutions of \(u_t - \operatorname{\Delta} \operatorname{\Phi}(u) = 0\) with initial datum a measure, Nonlinear Anal., 6, 2, 175-187, (1982) · Zbl 0484.35044 |
| [25] | Vázquez, J. L., The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, (2007), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press Oxford · Zbl 1107.35003 |
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