-
Previous Article
On the manifold of closed hypersurfaces in $\mathbb{R}^n$
- DCDS Home
- This Issue
-
Next Article
Global existence via a multivalued operator for an Allen-Cahn-Gurtin equation
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 |
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, ().
|
[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, ().
|
[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. |
[1] |
Donatella Danielli, Marianne Korten. On the pointwise jump condition at the free boundary in the 1-phase Stefan problem. Communications on Pure and 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 and 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 and 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 and 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 and Applied Analysis, 2007, 6 (4) : 957-982. doi: 10.3934/cpaa.2007.6.957 |
[7] |
Mauro Garavello, Francesca Marcellini. The Riemann Problem at a Junction for a Phase Transition Traffic Model. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5191-5209. doi: 10.3934/dcds.2017225 |
[8] |
V. S. Manoranjan, Hong-Ming Yin, R. Showalter. On two-phase Stefan problem arising from a microwave heating process. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1155-1168. doi: 10.3934/dcds.2006.15.1155 |
[9] |
Chifaa Ghanmi, Saloua Mani Aouadi, Faouzi Triki. Recovering the initial condition in the one-phase Stefan problem. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1143-1164. doi: 10.3934/dcdss.2021087 |
[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 and Applied Analysis, 2020, 19 (10) : 4853-4878. doi: 10.3934/cpaa.2020215 |
[11] |
Toyohiko Aiki. A free boundary problem for an elastic material. Conference Publications, 2007, 2007 (Special) : 10-17. doi: 10.3934/proc.2007.2007.10 |
[12] |
Yang Zhang. A free boundary problem of the cancer invasion. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1323-1343. doi: 10.3934/dcdsb.2021092 |
[13] |
Xavier Fernández-Real, Xavier Ros-Oton. On global solutions to semilinear elliptic equations related to the one-phase free boundary problem. Discrete and 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 and 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] |
Hayk Mikayelyan, Henrik Shahgholian. Convexity of the free boundary for an exterior free boundary problem involving the perimeter. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1431-1443. doi: 10.3934/cpaa.2013.12.1431 |
[18] |
Norbert Požár, Giang Thi Thu Vu. Long-time behavior of the one-phase Stefan problem in periodic and random media. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 991-1010. doi: 10.3934/dcdss.2018058 |
[19] |
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 |
[20] |
Lincoln Chayes, Inwon C. Kim. The supercooled Stefan problem in one dimension. Communications on Pure and Applied Analysis, 2012, 11 (2) : 845-859. doi: 10.3934/cpaa.2012.11.845 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]