zbMATH — the first resource for mathematics

All time \(C^\infty\)-regularity of the interface in degenerate diffusion: A geometric approach. (English) Zbl 1017.35052
Authors’ abstract: We study the connection between the geometry and all time regularity of the interface in degenerated diffusion. Our model considers the porous medium equation \(u_t=\Delta u^m\), \(m>1\), with initial data \(u_0\) nonnegative, integrable, and compactly supported. We show that if the initial pressure \(f_0=u_0^{m-1}\) is smooth up to the interface and in addition it is root-concave and also satisfies the nondegeneracy condition \(|Df_0 |\neq 0\) at \(\partial\overline {\text{supp}} f_0\), then the pressure \(f= u^{m-1}\) remains \(C^\infty\)-smooth up to the interface and root-concave, for all time \(0<t <\infty\). In particular, the free boundary is \(C^\infty\)-smooth for all time.

35K65 Degenerate parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
35A30 Geometric theory, characteristics, transformations in context of PDEs
35K15 Initial value problems for second-order parabolic equations
35K55 Nonlinear parabolic equations
53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
Full Text: DOI
[1] S. B. Angenent and D. G. Aronson, The focusing problem for the radially symmetric porous medium equation , Comm. Partial Differential Equations 20 (1995), 1217–1240. MR 96c:35074 · Zbl 0830.35062
[2] D. G. Aronson and P. Bénilan, Régularité des solutions de l’équation des milieux poreux dans \(\mathbf R^n\) , C. R. Acad. Sci. Paris Ser. A-B 288 (1979), 103–105. MR 82i:35090 · Zbl 0397.35034
[3] D. G. Aronson and L. A. Caffarelli, The initial trace of a solution of the porous medium equation , Trans. Amer. Math. Soc. 280 (1983), 351–366. MR 85c:35042 JSTOR: · Zbl 0556.76084
[4] L. A. Caffarelli and A. Friedman, Regularity of the free boundary of a gas flow in an n-dimensional porous medium , Indiana Univ. Math. J. 29 (1980), 361–391. MR 82a:35096 · Zbl 0439.76085
[5] L. A. Caffarelli, J. L. Vázquez and N. I. Wolanski, Lipschitz continuity of solutions and interfaces of the \(N\)-dimensional porous medium equation , Indiana Univ. Math. J. 36 (1987), 373–401. MR 88k:35221 · Zbl 0644.35058
[6] L. A. Caffarelli and N. I. Wolanski, \(C^1,\alpha\) regularity of the free boundary for the \(N\)-dimensional porous media equation , Comm. Pure Appl. Math. 43 (1990), 885–902. MR 91h:35332 · Zbl 0728.76103
[7] P. Daskalopoulos and R. Hamilton, The free boundary for the n-dimensional porous medium equation , Internat. Math. Res. Notices 1997 , 817–831. MR 98m:35227 · Zbl 0886.35114
[8] –. –. –. –., Regularity of the free boundary for the porous medium equation , J. Amer. Math. Soc. 11 (1998), 899–965. MR 99d:35182 JSTOR: · Zbl 0910.35145
[9] H. Koch, Non-Euclidean singular integrals and the porous medium equation , University of Heidelberg, Habilitation thesis, 1999, http://www.iwr.uni- heidelberg.de/groups/amj/koch.html P. Sacks, Continuity of solutions of a singular parabolic equation , Nonlinear Anal. 7 (1983), 387–409. MR 84d:35081 · Zbl 0511.35052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.