The Stefan problem in several space variables
1968; American Mathematical Society; Volume: 133; Issue: 1 Linguagem: Inglês
10.1090/s0002-9947-1968-0227625-7
ISSN1088-6850
Autores Tópico(s)Advanced Mathematical Modeling in Engineering
ResumoAVNER FRIEDMAN^)Introduction.The Stefan problem is a free boundary problem for parabolic equations.The solution is required to satisfy the usual initial-boundary conditions, but a part of the boundary is free.Naturally, an additional condition is imposed at the free boundary.A two-phase problem is such that on both sides of the free boundary there are given parabolic equations and initial-boundary conditions, and neither of the solutions is identically constant.In case the space-dimension is one, there are numerous results concerning existence, uniqueness, stability, and asymptotic behavior of the solution; we refer to [1] and the literature quoted there (see also [8]).In the case of several space variables the problem is much harder.The difficulty is not merely due to mathematical shortcomings but also to complications in the physical situation.Thus, even if the data are very smooth the solution need not be smooth, in general.For example, when a body of ice having the shape keeps growing, the interfaces AB and CD may eventually coincide.Then, in the next moment the whole joint boundary will disappear.Thus the free boundary varies in a discontinuous manner.This example motivates one to look for "weak" solutions.In [4] the concept of a weak solution is defined.Furthermore, existence and uniqueness theorems are proved.The existence proofs are based on a finite-difference approximation.In the present work we give a simpler derivation of the existence theorems of [4].Our method has also the advantages that (i) it yields better inequalities on the solution and on its first derivatives than in [4], and (ii) it enables us to find certain
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