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On singular Navier-Stokes equations and irreversible phase transitions
An abstract existence theorem for parabolic systems
1. | University Bonn, Institute for Applied Mathematics, Endenicher Allee 60, 53115 Bonn, Germany |
References:
[1] |
H. Amann, "Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory," Monograph in Mathematics, Birkhäuser Basel, 1995. |
[2] |
H. W. Alt, "Elliptische Probleme mit freiem Rand," Lecture Notes 21 SFB 256, Bonn, 1991. |
[3] |
H. W. Alt, Partielle Differentialgleichungen III, Vorlesung Winter semester 2003/04, Universität Bonn, unpublished manuscript. |
[4] |
H. W. Alt and E. DiBenedetto, Nonsteady flow of water and oil through inhomogeneous porous media, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., 12 (1985), 335-392. |
[5] |
H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. |
[6] |
H. W. Alt, S. Luckhaus and A. Visintin, On nonstationary flow through porous media, Ann. Mat. Pura Appl., 136 (1984), 303-316. |
[7] |
A.-K. Becher, "Ein abstrakter Existenzsatz für elliptisch-parabolische Systeme," Diplomarbeit 2005, Universität Bonn. |
[8] |
M. S. Berger, "Nonlinearity and Functional Analysis," Lectures on Nonlinear Problems in Mathematical Analysis, Academic Press, 1977. |
[9] |
F. Bernis, Existence results for doubly nonlinear higher order parabolic equations on unbounded domains, Math. Ann., 279 (1988), 373-394. |
[10] |
E. DiBenedetto, "Degenerate Parabolic Equations," Universitext, Springer, 1993. |
[11] |
G. Duvaut and J. L. Lions, "Inequalities in Mechanics and Physics," Grundlehren der mathematischen Wissenschaften 219, Springer-Verlag, 1976. |
[12] |
A. Friedman, "Partial Differential Equations," Holt, Rinehart and Winston New York, 1969. |
[13] |
U. Fermum, "Nichtlineare elliptisch-parabolische Gleichungen mit zeitabhängigen Hindernissen," Diplomarbeit 2005, Universität Bonn. |
[14] |
W. Jäger and J. Kačur, Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes, Math. Modelling Numer. Anal., 29 (1995), 605-627. |
[15] |
N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and application, Bull. Fac. Education Chiba Univ., 30 (1981), 1-87. |
[16] |
N. Kenmochi and I. Pawlow, A class of nonlinear elliptic-parabolic equations with time dependent constraints, Nonlinear Analysis, 10 (1986), 1181-1202. |
[17] |
D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and their Application," Academic Press, 1980. |
[18] |
D. Kröner and S. Luckhaus, Flow of oil and water in a porous medium, J. Differential Equations, 55 (1984), 276-288. |
[19] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, 1968. |
[20] |
I. Müller, "Thermodynamics," Interaction of mechanics and mathematics series, Pitman Boston London Melbourne, 1985. |
[21] |
P. A. Raviart, Sur la résolution de certaines equations paraboliques non linéaires, J. Functional Analysis, 5 (1970), 299-328. |
[22] |
M. Růžička, "Nichtlineare Funktionalanalysis. Eine Einführung," See also the version in http://aam.mathematik.uni-freiburg.de/IAM/homepages/rose/springer.html, Springer, 2004. |
[23] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987). |
[24] |
M. Schmidt, "Ein Existenzsatz für parabolische Systeme zur Beschreibung von chemischen Reaktionen," Diplomarbeit 2006, Universität Bonn. |
[25] |
A. Visintin, "Models of Phase Transition," Birkhäuser, 1996. |
[26] |
G. Duvaut and J. L. Lions, "Inequalities in Mechanics and Physics," Grundlehren der mathematischen Wissenschaften 219, Springer-Verlag, 1976. |
[27] |
E. DiBenedetto, "Partial Differential Equations," 2nd edition, Birkhäuser Boston, 2010. |
show all references
References:
[1] |
H. Amann, "Linear and Quasilinear Parabolic Problems, Volume I: Abstract Linear Theory," Monograph in Mathematics, Birkhäuser Basel, 1995. |
[2] |
H. W. Alt, "Elliptische Probleme mit freiem Rand," Lecture Notes 21 SFB 256, Bonn, 1991. |
[3] |
H. W. Alt, Partielle Differentialgleichungen III, Vorlesung Winter semester 2003/04, Universität Bonn, unpublished manuscript. |
[4] |
H. W. Alt and E. DiBenedetto, Nonsteady flow of water and oil through inhomogeneous porous media, Ann. Scuola Norm. Sup. Pisa, Cl. Sci., 12 (1985), 335-392. |
[5] |
H. W. Alt and S. Luckhaus, Quasilinear elliptic-parabolic differential equations, Math. Z., 183 (1983), 311-341. |
[6] |
H. W. Alt, S. Luckhaus and A. Visintin, On nonstationary flow through porous media, Ann. Mat. Pura Appl., 136 (1984), 303-316. |
[7] |
A.-K. Becher, "Ein abstrakter Existenzsatz für elliptisch-parabolische Systeme," Diplomarbeit 2005, Universität Bonn. |
[8] |
M. S. Berger, "Nonlinearity and Functional Analysis," Lectures on Nonlinear Problems in Mathematical Analysis, Academic Press, 1977. |
[9] |
F. Bernis, Existence results for doubly nonlinear higher order parabolic equations on unbounded domains, Math. Ann., 279 (1988), 373-394. |
[10] |
E. DiBenedetto, "Degenerate Parabolic Equations," Universitext, Springer, 1993. |
[11] |
G. Duvaut and J. L. Lions, "Inequalities in Mechanics and Physics," Grundlehren der mathematischen Wissenschaften 219, Springer-Verlag, 1976. |
[12] |
A. Friedman, "Partial Differential Equations," Holt, Rinehart and Winston New York, 1969. |
[13] |
U. Fermum, "Nichtlineare elliptisch-parabolische Gleichungen mit zeitabhängigen Hindernissen," Diplomarbeit 2005, Universität Bonn. |
[14] |
W. Jäger and J. Kačur, Solution of doubly nonlinear and degenerate parabolic problems by relaxation schemes, Math. Modelling Numer. Anal., 29 (1995), 605-627. |
[15] |
N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and application, Bull. Fac. Education Chiba Univ., 30 (1981), 1-87. |
[16] |
N. Kenmochi and I. Pawlow, A class of nonlinear elliptic-parabolic equations with time dependent constraints, Nonlinear Analysis, 10 (1986), 1181-1202. |
[17] |
D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and their Application," Academic Press, 1980. |
[18] |
D. Kröner and S. Luckhaus, Flow of oil and water in a porous medium, J. Differential Equations, 55 (1984), 276-288. |
[19] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type," Translations of Mathematical Monographs, Vol. 23, 1968. |
[20] |
I. Müller, "Thermodynamics," Interaction of mechanics and mathematics series, Pitman Boston London Melbourne, 1985. |
[21] |
P. A. Raviart, Sur la résolution de certaines equations paraboliques non linéaires, J. Functional Analysis, 5 (1970), 299-328. |
[22] |
M. Růžička, "Nichtlineare Funktionalanalysis. Eine Einführung," See also the version in http://aam.mathematik.uni-freiburg.de/IAM/homepages/rose/springer.html, Springer, 2004. |
[23] |
J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987). |
[24] |
M. Schmidt, "Ein Existenzsatz für parabolische Systeme zur Beschreibung von chemischen Reaktionen," Diplomarbeit 2006, Universität Bonn. |
[25] |
A. Visintin, "Models of Phase Transition," Birkhäuser, 1996. |
[26] |
G. Duvaut and J. L. Lions, "Inequalities in Mechanics and Physics," Grundlehren der mathematischen Wissenschaften 219, Springer-Verlag, 1976. |
[27] |
E. DiBenedetto, "Partial Differential Equations," 2nd edition, Birkhäuser Boston, 2010. |
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