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Existence and continuity of a weak solution to the problem of a free boundary in a concentrated capacity. (English) Zbl 0591.35086

A concentrated capacity is a heat conducting body having infinite thermal conductivity along the lines of a prescribed vector field. As a consequence, the heat flux entering the body from outside (e.g. from an adjacent heat conductor) acts on it as a volumetric source. The author considers the overall heat conduction problem with the external body occupying the region \(0<x<1\), \(0<z<1\) and the heat capacity lying below the side \(x=0\), \(0<z<1\), with infinite thermal conductivity in the z direction, so that the temperature in the heat capacity is a function of x only.
First the classical formulation of the problem with change of phase occurring in the heat capacity is recalled. Then a weak formulation is proposed and existence and continuity properties are proved. The author stresses the possibility that a mushy region can develop and he sketches how the classical formulation should be modified in the presence of a mushy region.
The problem is interesting and also appealing from the mathematical point of view. The reviewer notes that it is always assumed that enthalpy in the concentrated capacity depends on x and t only, likewise temperature. Although it cannot be said a priori that this is actually the case, his point of view is quite correct if enthalpy at a point (x,t) is interpreted as an averaged quantity along z.
Reviewer: A.Fasano

MSC:

35R35 Free boundary problems for PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
76S05 Flows in porous media; filtration; seepage
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References:

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