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The Stefan problem with surface tension as the limit of a phase field model. (English. Russian original) Zbl 0802.35165

Differ. Equations 29, No. 3, 395-404 (1993); translation from Differ. Uravn. 29, No. 3, 461-471 (1993).
The authors consider one of the versions of the phase field model where the state of a medium is described by its temperature \(\theta_ \varepsilon\) and a phase function \(\varphi_ \varepsilon\). The pair \((\theta_ \varepsilon, \varphi_ \varepsilon)\) is regarded as a solution to some initial and boundary value problem of a system of partial differential equations with a small parameter \(\varepsilon\). It is shown that this problem has a family of generalized solutions that converge to a generalized solution of the Stefan problem with surface tension as \(\varepsilon \to 0\).
The solvability of the Stefan problem with surface tension has been considered in S. Luckhaus [Eur. J. Appl. Math. 1, No. 2, 101–111 (1990; Zbl 0734.35159)].

MSC:

35R35 Free boundary problems for PDEs
35D99 Generalized solutions to partial differential equations
35B25 Singular perturbations in context of PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Citations:

Zbl 0734.35159
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