Geodesic
The order of convergence in the Stefan problem with vanishing specific heat
Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 41, Tome 385 (2010), pp. 206-223 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper is concerned with a two-phase Stefan problem with a small parameter $\varepsilon$ which coresponds to the specific heat of the material. We assume that the initial condition does not coincide with the value at $t=0$ of the solution to the limit problem related to $\varepsilon=0$. To remove this discrepancy, we introduce an auxiliary boundary layer type function. We prove that the solution to the two-phase Stefan problem with parameter $\varepsilon$ differs from the sum of the solution to the limit Hele–Shaw problem and the boundary layer type function by quantities of the order $O(\varepsilon)$. The estimates are obtained in Hölder norms. Bibl. 13 titles. 
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     author = {E. V. Frolova},
     title = {The order of convergence in the {Stefan} problem with vanishing specific heat},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2010_385_a9/}
}
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E. V. Frolova. The order of convergence in the Stefan problem with vanishing specific heat. Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 41, Tome 385 (2010), pp. 206-223. http://geodesic.mathdoc.fr/item/ZNSL_2010_385_a9/

[1] E. V. Frolova, V. A. Solonnikov,, “Justification of the quasistationary approximation for the Stefan problem”, Zap. Nauchn. Semin. POMI, 348, 2007, 209–253

[2] E. V. Frolova, “Two-phase Stefan problem with vanishing specific heat”, Zap. Nauchn. Semin. POMI, 362, 2008, 337–363 | Zbl

[3] E. V. Frolova, “Quasistationary approximation for the Stefan problem”, Probl. Mat. Anal., 31 (2005), 167–178 | MR

[4] A. M. Meirmanov, The Stefan Problem, Nauka, Novosibirsk, 1986 | MR

[5] V. S. Belonosov, T. I. Zelenyak, Nonlocal Problems in the Theory of Quasilinear Parabolic Equations, NGU, Novosibirsk, 1975 | MR

[6] E. I. Hanzawa, “Classical solutions of the Stefan problem”, Tohoku Math. J., 33 (1981), 297–335 | DOI | MR | Zbl

[7] G. I. Bizhanova, V. A. Solonnikov, “Free boundary problems for second order parabolic equations”, Algebra Analiz, 12:6 (2000), 98–139 | MR | Zbl

[8] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic problems, Birkhauser, Basel–Boston–Berlin, 1995 | MR | Zbl

[9] S. N. Kruzhkov, A. Castro, M. Lopes, “Schauder type estimates and theorems on the existence of the solution of fundamental problems for linear and nonlinear parabolic equations”, Dokl. Akad. Nauk SSSR, 220:2 (1975), 277–28 | MR

[10] E. Sinestrari, W. von Wahl, “On the solutions of the first boundary value problem for the linear parabolic equations”, Proc. Royal Soc. Edinburg A, 108 (1988), 339–355 | DOI | MR | Zbl

[11] V. A. Solonnikov, “Lectures on evolution free boundary problems: classical solutions”, Lect. Notes Math., 1812, Springer, 2003, 123–175 | DOI | MR | Zbl

[12] V. A. Solonnikov, “Estimates of solutions of the second order initial-boundary value problem for the Stokes system in the spaces of functions with Hölder continuous derivatives with respect to spartial variables”, Zap. Nauchn. Semin. POMI, 259, 1999, 254–279 | MR | Zbl

[13] V. A. Solonnikov, “On the justification of the quasistationary approximation in the problem of motion of a viscous capillary drop”, Interfaces and free boundaries, 1 (1999), 125–173 | DOI | MR | Zbl