Singular limits for the two-phase Stefan problem

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November 2013, 33(11&12): 5379-5405. doi: 10.3934/dcds.2013.33.5379

Singular limits for the two-phase Stefan problem

Jan Prüss ^1, , Jürgen Saal ^2, and Gieri Simonett ^3,

1.	Martin-Luther-Universität Halle-Wittenberg, Institut für Mathematik, Theodor-Lieser-Strasse 5, D-06120 Halle
2.	Technische Universität Darmstadt, Center of Smart Interfaces, 64287 Darmstadt, Germany
3.	Department of Mathematics, Vanderbilt University, Nashville, TN 37240

Received February 2012 Published May 2013

Abstract
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We prove strong convergence to singular limits for a linearized fully inhomogeneous Stefan problem subject to surface tension and kinetic undercooling effects. Different combinations of $\sigma \to \sigma_0$ and $\delta\to\delta_0$, where $\sigma,\sigma_0\ge 0$ and $\delta,\delta_0\ge 0$ denote surface tension and kinetic undercooling coefficients respectively, altogether lead to five different types of singular limits. Their strong convergence is based on uniform maximal regularity estimates.

Keywords: Stefan problem, singular limits, maximal regularity., free boundary problem, phase transition.

Mathematics Subject Classification: Primary: 35R35, 35B65, 80A22; Secondary: 35K2.

Citation: Jan Prüss, Jürgen Saal, Gieri Simonett. Singular limits for the two-phase Stefan problem. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5379-5405. doi: 10.3934/dcds.2013.33.5379

References:

[1]	B. Bazaliy and S. P. Degtyarev, The classical Stefan problem as the limit case of the Stefan problem with a kinetic condition at the free boundary, Free Boundary Problems in Continuum Mechanics (Novosibirsk, 1991), Internat. Ser. Numer. Math., 106, Birkhäuser, Basel, (1992), 83-90.
[2]	R. Denk, M. Hieber and J. Prüss, "$\mathcal R$-Boundedness, Fourier Multipliers, and Problems of Elliptic and Parabolic Type," AMS Memoirs 788, Providence, R.I., 2003.
[3]	R. Denk, J. Prüss and R. Zacher, Maximal $L_p$-regularity of parabolic problems with boundary conditions of relaxation type, J. Funct. Anal., 255 (2008), 3149-3187. doi: 10.1016/j.jfa.2008.07.012.
[4]	R. Denk, J. Saal and J. Seiler, Inhomogeneous symbols, the Newton polygon, and maximal $L^p$-regularity, Russian J. Math. Phys. (2), 15 (2008), 171-192. doi: 10.1134/S1061920808020040.
[5]	J. Escher, J. Prüss and G. Simonett, Analytic solutions for a Stefan problem with Gibbs-Thomson correction, J. Reine Angew. Math., 563 (2003), 1-52. doi: 10.1515/crll.2003.082.
[6]	M. Hieber and J. Prüss, Functional calculi for linear operators in vector-valued $L^p$-spaces via the transference principle, Adv. Differential Equations, 3 (1998), 847-872.
[7]	N. Kalton and L. Weis, The $H^\infty$-calculus and sums of closed operators, Math. Ann., 321 (2001), 319-345. doi: 10.1007/s002080100231.
[8]	P. C. Kunstmann and L. Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus, Functional analytic methods for evolution equations, Lecture Notes in Math., 1855, Springer, Berlin, (2004), 65-311. doi: 10.1007/978-3-540-44653-8_2.
[9]	M. Meyries and R. Schnaubelt, Interpolation, embeddings and traces of anisotropic fractional Sobolev spaces with temporal weights, J. Funct. Anal., 262 (2012), 1200-1229. doi: 10.1016/j.jfa.2011.11.001.
[10]	J. Prüss, J. Saal and G. Simonett, Existence of analytic solutions for the classical Stefan problem, Math. Ann., 338 (2007), 703-755. doi: 10.1007/s00208-007-0094-2.
[11]	J. Prüss and G. Simonett, Stability of equilibria for the Stefan problem with surface tension, SIAM J. Math. Anal., 40 (2008), 675-698. doi: 10.1137/070700632.
[12]	J. Prüss, G. Simonett and M. Wilke, On thermodynamically consistent Stefan problems with variable surface energy, submitted, arXiv:1109.4542.
[13]	J. Prüss, G. Simonett and R. Zacher, Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension, Arch. Ration. Mech. Anal., 207 (2013), 611-667. doi: 10.1007/s00205-012-0571-y.
[14]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978.
[15]	H. Triebel, "Theory of Function Spaces," 78 of Monographs in Mathematics, Birkhäuser, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.
[16]	T. Youshan, The limit of the Stefan problem with surface tension and kinetic undercooling on the free boundary, J. Partial Differential Equations, 9 (1996), 153-168.

show all references

References:

[1]	B. Bazaliy and S. P. Degtyarev, The classical Stefan problem as the limit case of the Stefan problem with a kinetic condition at the free boundary, Free Boundary Problems in Continuum Mechanics (Novosibirsk, 1991), Internat. Ser. Numer. Math., 106, Birkhäuser, Basel, (1992), 83-90.
[2]	R. Denk, M. Hieber and J. Prüss, "$\mathcal R$-Boundedness, Fourier Multipliers, and Problems of Elliptic and Parabolic Type," AMS Memoirs 788, Providence, R.I., 2003.
[3]	R. Denk, J. Prüss and R. Zacher, Maximal $L_p$-regularity of parabolic problems with boundary conditions of relaxation type, J. Funct. Anal., 255 (2008), 3149-3187. doi: 10.1016/j.jfa.2008.07.012.
[4]	R. Denk, J. Saal and J. Seiler, Inhomogeneous symbols, the Newton polygon, and maximal $L^p$-regularity, Russian J. Math. Phys. (2), 15 (2008), 171-192. doi: 10.1134/S1061920808020040.
[5]	J. Escher, J. Prüss and G. Simonett, Analytic solutions for a Stefan problem with Gibbs-Thomson correction, J. Reine Angew. Math., 563 (2003), 1-52. doi: 10.1515/crll.2003.082.
[6]	M. Hieber and J. Prüss, Functional calculi for linear operators in vector-valued $L^p$-spaces via the transference principle, Adv. Differential Equations, 3 (1998), 847-872.
[7]	N. Kalton and L. Weis, The $H^\infty$-calculus and sums of closed operators, Math. Ann., 321 (2001), 319-345. doi: 10.1007/s002080100231.
[8]	P. C. Kunstmann and L. Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus, Functional analytic methods for evolution equations, Lecture Notes in Math., 1855, Springer, Berlin, (2004), 65-311. doi: 10.1007/978-3-540-44653-8_2.
[9]	M. Meyries and R. Schnaubelt, Interpolation, embeddings and traces of anisotropic fractional Sobolev spaces with temporal weights, J. Funct. Anal., 262 (2012), 1200-1229. doi: 10.1016/j.jfa.2011.11.001.
[10]	J. Prüss, J. Saal and G. Simonett, Existence of analytic solutions for the classical Stefan problem, Math. Ann., 338 (2007), 703-755. doi: 10.1007/s00208-007-0094-2.
[11]	J. Prüss and G. Simonett, Stability of equilibria for the Stefan problem with surface tension, SIAM J. Math. Anal., 40 (2008), 675-698. doi: 10.1137/070700632.
[12]	J. Prüss, G. Simonett and M. Wilke, On thermodynamically consistent Stefan problems with variable surface energy, submitted, arXiv:1109.4542.
[13]	J. Prüss, G. Simonett and R. Zacher, Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension, Arch. Ration. Mech. Anal., 207 (2013), 611-667. doi: 10.1007/s00205-012-0571-y.
[14]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978.
[15]	H. Triebel, "Theory of Function Spaces," 78 of Monographs in Mathematics, Birkhäuser, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.
[16]	T. Youshan, The limit of the Stefan problem with surface tension and kinetic undercooling on the free boundary, J. Partial Differential Equations, 9 (1996), 153-168.

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