Regularity of the free boundary for the porous medium equation

Daskalopoulos, P.; Hamilton, R.

doi:10.1090/S0894-0347-98-00277-X

Regularity of the free boundary for the porous medium equation
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by P. Daskalopoulos and R. Hamilton

J. Amer. Math. Soc. 11 (1998), 899-965

DOI: https://doi.org/10.1090/S0894-0347-98-00277-X

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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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