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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Regularity of the free boundary
for the porous medium equation


Authors: P. Daskalopoulos and R. Hamilton
Journal: J. Amer. Math. Soc. 11 (1998), 899-965
MSC (1991): Primary 35Jxx
MathSciNet review: 1623198
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Abstract: We study the regularity of the free boundary for solutions of the porous medium equation $u_{t}=\Delta u^{m}$, $m >1$, on ${\mathcal{R}}^{2} \times [0,T]$, with initial data $u^{0}=u(x,0)$ nonnegative and compactly supported. We show that, under certain assumptions on the initial data $u^{0}$, the pressure $f=m\, u^{m-1}$ will be smooth up to the interface $\Gamma = \partial \{ u >0 \}$, when $0<t\leq T$, for some $T >0$. As a consequence, the free-boundary $\Gamma $ is smooth.


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Additional Information

P. Daskalopoulos
Affiliation: Department of Mathematics, University of California, Irvine, California 92697-3875
Email: pdaskalo@math.uci.edu

R. Hamilton
Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0001

DOI: http://dx.doi.org/10.1090/S0894-0347-98-00277-X
PII: S 0894-0347(98)00277-X
Keywords: Porous medium equation, free-boundary, $C^{\infty }$-regularity
Received by editor(s): January 19, 1998
Article copyright: © Copyright 1998 American Mathematical Society