Regularity for solutions of the two-phase Stefan problem

Marianne Korten; Charles N. Moore

doi:10.3934/cpaa.2008.7.591

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2008, Volume 7, Issue 3: 591-600. Doi: 10.3934/cpaa.2008.7.591

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Regularity for solutions of the two-phase Stefan problem

Marianne Korten^1, and
Charles N. Moore^1,

1.
Department of Mathematics, Kansas State University, Manhattan, KS 66506

Received: February 28, 2007

Revised: September 30, 2007

Published: April 2008

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Abstract

We consider the two-phase Stefan problem $u_t=\Delta\alpha(u)$ where $\alpha(u) =u+1$ for $u<-1$, $\alpha(u) =0$ for $-1 \leq u \leq 1$, and $\alpha(u)=u-1$ for $u \geq 1$. We show that if $u$ is an $L_{l o c}^2$ distributional solution then $\alpha(u)$ has $L_{l o c}^2$ derivatives in time and space. We also show $|\alpha(u)|$ is subcaloric and conclude that $\alpha(u)$ is continuous.

Keywords:

Stefan problem,
degenerate parabolic equation.

Mathematics Subject Classification: Primary: 35K65; Secondary: 80A22.

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