# American Institute of Mathematical Sciences

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

## Singular limits for the two-phase Stefan problem

 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.
Citation: Jan Prüss, Jürgen Saal, Gieri Simonett. Singular limits for the two-phase Stefan problem. Discrete & Continuous Dynamical Systems - A, 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, (1992), 83.   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, 788 (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.  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.  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.  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.   Google Scholar [7] N. Kalton and L. Weis, The $H^\infty$-calculus and sums of closed operators,, Math. Ann., 321 (2001), 319.  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, 1855 (2004), 65.  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.  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.  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.  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.  doi: 10.1007/s00205-012-0571-y.  Google Scholar [14] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland, (1978).   Google Scholar [15] H. Triebel, "Theory of Function Spaces,", 78 of Monographs in Mathematics, 78 (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.   Google Scholar

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##### 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, (1992), 83.   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, 788 (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.  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.  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.  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.   Google Scholar [7] N. Kalton and L. Weis, The $H^\infty$-calculus and sums of closed operators,, Math. Ann., 321 (2001), 319.  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, 1855 (2004), 65.  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.  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.  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.  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.  doi: 10.1007/s00205-012-0571-y.  Google Scholar [14] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland, (1978).   Google Scholar [15] H. Triebel, "Theory of Function Spaces,", 78 of Monographs in Mathematics, 78 (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.   Google Scholar
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