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Stefan problem

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Abstract

We prove the existence of a global classical solution of the multidimensional two-phase Stefan problem. The problem is reduced to a quasilinear parabolic equation with discontinuous coefficients in a fixed domain. With the help of a small parameter ε, we smooth coefficients and investigate the resulting approximate solution. An analytical method that enables one to obtain the uniform estimates of an approximate solution in the cross-sections t = const is developed. Given the uniform estimates, we make the limiting transition as ε → 0. The limit of the approximate solution is a classical solution of the Stefan problem, and the free boundary is a surface of the class H 2+α,1+α/2.

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Authors and Affiliations

  1. Donetsk National University, 24, Universitetskaya Str., Donetsk, 83055, Ukraine

    Mikhail A. Borodin

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  1. Mikhail A. Borodin
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Correspondence to Mikhail A. Borodin.

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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 8, No. 1, pp. 17–54, January–February, 2011.

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Borodin, M.A. Stefan problem. J Math Sci 178, 13–40 (2011). https://doi.org/10.1007/s10958-011-0524-2

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