Viscosity Solutions for the Two-Phase Stefan Problem

We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and...

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Bibliographic Details
Published in:Communications in partial differential equations 2011-01, Vol.36 (1), p.42-66
Main Authors: Kim, Inwon C., Požár, Norbert
Format: Article
Language:English
Subjects:
Comparison principle
Free-boundary problems
Mathematics
Partial differential equations
Primary 35R35
Secondary 80A22, 35B51, 35D40
Two-phase Stefan problem
Viscosity solutions
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Summary:We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mushy region, viscosity solution is the unique weak solution with the same boundary data.
ISSN:0360-5302
1532-4133
DOI:10.1080/03605302.2010.526980