# 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, 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
 [1] Donatella Danielli, Marianne Korten. On the pointwise jump condition at the free boundary in the 1-phase Stefan problem. Communications on Pure & Applied Analysis, 2005, 4 (2) : 357-366. doi: 10.3934/cpaa.2005.4.357 [2] Marianne Korten, Charles N. Moore. Regularity for solutions of the two-phase Stefan problem. Communications on Pure & Applied Analysis, 2008, 7 (3) : 591-600. doi: 10.3934/cpaa.2008.7.591 [3] Mauro Garavello. Boundary value problem for a phase transition model. Networks & Heterogeneous Media, 2016, 11 (1) : 89-105. doi: 10.3934/nhm.2016.11.89 [4] Anna Lisa Amadori. Contour enhancement via a singular free boundary problem. Conference Publications, 2007, 2007 (Special) : 44-53. doi: 10.3934/proc.2007.2007.44 [5] Francesca Marcellini. Existence of solutions to a boundary value problem for a phase transition traffic model. Networks & Heterogeneous Media, 2017, 12 (2) : 259-275. doi: 10.3934/nhm.2017011 [6] Chérif Amrouche, Yves Raudin. Singular boundary conditions and regularity for the biharmonic problem in the half-space. Communications on Pure & Applied Analysis, 2007, 6 (4) : 957-982. doi: 10.3934/cpaa.2007.6.957 [7] V. S. Manoranjan, Hong-Ming Yin, R. Showalter. On two-phase Stefan problem arising from a microwave heating process. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1155-1168. doi: 10.3934/dcds.2006.15.1155 [8] Chifaa Ghanmi, Saloua Mani Aouadi, Faouzi Triki. Recovering the initial condition in the one-phase Stefan problem. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021087 [9] Mauro Garavello, Francesca Marcellini. The Riemann Problem at a Junction for a Phase Transition Traffic Model. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5191-5209. doi: 10.3934/dcds.2017225 [10] Giovanni Gravina, Giovanni Leoni. On the behavior of the free boundary for a one-phase Bernoulli problem with mixed boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4853-4878. doi: 10.3934/cpaa.2020215 [11] Yang Zhang. A free boundary problem of the cancer invasion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021092 [12] Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10 [13] Xavier Fernández-Real, Xavier Ros-Oton. On global solutions to semilinear elliptic equations related to the one-phase free boundary problem. Discrete & Continuous Dynamical Systems, 2019, 39 (12) : 6945-6959. doi: 10.3934/dcds.2019238 [14] Huiqiang Jiang. Regularity of a vector valued two phase free boundary problems. Conference Publications, 2013, 2013 (special) : 365-374. doi: 10.3934/proc.2013.2013.365 [15] Feifei Tang, Suting Wei, Jun Yang. Phase transition layers for Fife-Greenlee problem on smooth bounded domain. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1527-1552. doi: 10.3934/dcds.2018063 [16] Brahim Bougherara, Jacques Giacomoni, Jesus Hernández. Some regularity results for a singular elliptic problem. Conference Publications, 2015, 2015 (special) : 142-150. doi: 10.3934/proc.2015.0142 [17] Michael L. Frankel, Victor Roytburd. Fractal dimension of attractors for a Stefan problem. Conference Publications, 2003, 2003 (Special) : 281-287. doi: 10.3934/proc.2003.2003.281 [18] Lincoln Chayes, Inwon C. Kim. The supercooled Stefan problem in one dimension. Communications on Pure & Applied Analysis, 2012, 11 (2) : 845-859. doi: 10.3934/cpaa.2012.11.845 [19] Piotr B. Mucha. Limit of kinetic term for a Stefan problem. Conference Publications, 2007, 2007 (Special) : 741-750. doi: 10.3934/proc.2007.2007.741 [20] Norbert Požár, Giang Thi Thu Vu. Long-time behavior of the one-phase Stefan problem in periodic and random media. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 991-1010. doi: 10.3934/dcdss.2018058

2020 Impact Factor: 1.392