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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, (1992), 83.
|
[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. |
[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. |
[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. |
[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.
|
[7] |
N. Kalton and L. Weis, The $H^\infty$-calculus and sums of closed operators,, Math. Ann., 321 (2001), 319.
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, 1855 (2004), 65.
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.
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.
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.
doi: 10.1137/070700632. |
[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. |
[14] |
H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland, (1978).
|
[15] |
H. Triebel, "Theory of Function Spaces,", 78 of Monographs in Mathematics, 78 (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.
|
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, (1992), 83.
|
[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. |
[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. |
[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. |
[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.
|
[7] |
N. Kalton and L. Weis, The $H^\infty$-calculus and sums of closed operators,, Math. Ann., 321 (2001), 319.
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, 1855 (2004), 65.
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.
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.
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.
doi: 10.1137/070700632. |
[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. |
[14] |
H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland, (1978).
|
[15] |
H. Triebel, "Theory of Function Spaces,", 78 of Monographs in Mathematics, 78 (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.
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