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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Regularity of the free boundary for the porous medium equation
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by P. Daskalopoulos and R. Hamilton PDF
J. Amer. Math. Soc. 11 (1998), 899-965 Request permission

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
  • MR Author ID: 353551
  • Email: pdaskalo@math.uci.edu
  • R. Hamilton
  • Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0001
  • Received by editor(s): January 19, 1998
  • © Copyright 1998 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 11 (1998), 899-965
  • MSC (1991): Primary 35Jxx
  • DOI: https://doi.org/10.1090/S0894-0347-98-00277-X
  • MathSciNet review: 1623198