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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

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 & 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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. Prüss, G. Simonett and M. Wilke, On thermodynamically consistent Stefan problems with variable surface energy,, submitted, ().   Google Scholar

[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.  Google Scholar

[14]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978.  Google Scholar

[15]

H. Triebel, "Theory of Function Spaces," 78 of Monographs in Mathematics, Birkhäuser, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

J. Prüss, G. Simonett and M. Wilke, On thermodynamically consistent Stefan problems with variable surface energy,, submitted, ().   Google Scholar

[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.  Google Scholar

[14]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978.  Google Scholar

[15]

H. Triebel, "Theory of Function Spaces," 78 of Monographs in Mathematics, Birkhäuser, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[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.  Google Scholar

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