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On a Cahn-Hilliard type phase field system related to tumor growth
1. | Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia |
2. | Laboratoire de Mathématiques, CNRS et Université de Paris-Sud, 91405 Orsay |
References:
[1] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach spaces, Noordhoff International Publishing, Leyden, 1976. |
[2] |
H. Brezis, Opérateurs Maximaux Monotones Et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973. |
[3] |
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
doi: 10.1063/1.1744102. |
[4] |
V. Cristini, H. B. Frieboes, X. Li, J. S. Lowengrub, P. Macklin, S. Sanga, S. M. Wise and X. Zheng, Nonlinear modeling and simulation of tumor growth, in Selected Topics in Cancer Modeling: Genesis, Evolution, Immune Competition, and Therapy, N. Bellomo, M. Chaplain, and E. De Angelis, eds., Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, (2008), 113-181. |
[5] |
V. Cristini, X. Li, J. S. Lowengrub and S. M. Wise, Nonlinear simulations of solid tumor growth using a mixture model: Invasion and branching, J. Math. Biol., 58 (2009), 723-763.
doi: 10.1007/s00285-008-0215-x. |
[6] |
V. Cristini and J. S. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, Cambridge University Press, Cambridge, 2010.
doi: 10.1017/CBO9780511781452. |
[7] |
V. Cristini, J. S. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth, J. Math. Biol., 46 (2003), 191-224.
doi: 10.1007/s00285-002-0174-6. |
[8] |
R. Denk, M. Hieber and J. Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.
doi: 10.1007/s00209-007-0120-9. |
[9] |
C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation. II. Analysis, J. Differential Equations, 128 (1996), 387-414.
doi: 10.1006/jdeq.1996.0101. |
[10] |
C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357.
doi: 10.1007/BF00251803. |
[11] |
S. Frigeri, M. Grasselli and E. Rocca, On a diffuse interface model of tumor growth, preprint arXiv:1405.3446 [math.AP] (2014), 1-27. |
[12] |
A. Hawkins-Daarud, K. G. van der Zee and J. T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Methods Biomed. Eng., 28 (2012), 3-24.
doi: 10.1002/cnm.1467. |
[13] |
D. Hilhorst, J. Kampmann, T. N. Nguyen and K. G. van der Zee, Formal asymptotic limit of a diffuse-interface tumor-growth model, preprint (2013), 1-28, to appear in Math. Models Methods Appl. Sci.. |
[14] |
J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969. |
[15] |
J. S. Lowengrub, H. B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modeling of cancer: Bridging the gap between cells and tumors, Nonlinearity, 23 (2010), R1-R91.
doi: 10.1088/0951-7715/23/1/R01. |
[16] |
J. T. Oden, A. Hawkins and S. Prudhomme, General diffuse-interface theories and an approach to predictive tumor growth modeling, Math. Models Methods Appl. Sci., 20 (2010), 477-517.
doi: 10.1142/S0218202510004313. |
[17] |
R. E. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, Math. Surveys Monogr. 49, American Mathematical Society, Providence, RI, 1997. |
[18] |
J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[19] |
S. Zheng, Nonlinear Evolution Equations, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math. 133, Chapman & Hall/CRC, Boca Raton, FL, 2004.
doi: 10.1201/9780203492222. |
show all references
References:
[1] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach spaces, Noordhoff International Publishing, Leyden, 1976. |
[2] |
H. Brezis, Opérateurs Maximaux Monotones Et Semi-groupes de Contractions Dans Les Espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973. |
[3] |
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
doi: 10.1063/1.1744102. |
[4] |
V. Cristini, H. B. Frieboes, X. Li, J. S. Lowengrub, P. Macklin, S. Sanga, S. M. Wise and X. Zheng, Nonlinear modeling and simulation of tumor growth, in Selected Topics in Cancer Modeling: Genesis, Evolution, Immune Competition, and Therapy, N. Bellomo, M. Chaplain, and E. De Angelis, eds., Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, (2008), 113-181. |
[5] |
V. Cristini, X. Li, J. S. Lowengrub and S. M. Wise, Nonlinear simulations of solid tumor growth using a mixture model: Invasion and branching, J. Math. Biol., 58 (2009), 723-763.
doi: 10.1007/s00285-008-0215-x. |
[6] |
V. Cristini and J. S. Lowengrub, Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, Cambridge University Press, Cambridge, 2010.
doi: 10.1017/CBO9780511781452. |
[7] |
V. Cristini, J. S. Lowengrub and Q. Nie, Nonlinear simulation of tumor growth, J. Math. Biol., 46 (2003), 191-224.
doi: 10.1007/s00285-002-0174-6. |
[8] |
R. Denk, M. Hieber and J. Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.
doi: 10.1007/s00209-007-0120-9. |
[9] |
C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation. II. Analysis, J. Differential Equations, 128 (1996), 387-414.
doi: 10.1006/jdeq.1996.0101. |
[10] |
C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357.
doi: 10.1007/BF00251803. |
[11] |
S. Frigeri, M. Grasselli and E. Rocca, On a diffuse interface model of tumor growth, preprint arXiv:1405.3446 [math.AP] (2014), 1-27. |
[12] |
A. Hawkins-Daarud, K. G. van der Zee and J. T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Methods Biomed. Eng., 28 (2012), 3-24.
doi: 10.1002/cnm.1467. |
[13] |
D. Hilhorst, J. Kampmann, T. N. Nguyen and K. G. van der Zee, Formal asymptotic limit of a diffuse-interface tumor-growth model, preprint (2013), 1-28, to appear in Math. Models Methods Appl. Sci.. |
[14] |
J.-L. Lions, Quelques Méthodes de Résolution Des Problèmes Aux Limites Non Linéaires, Dunod; Gauthier-Villars, Paris, 1969. |
[15] |
J. S. Lowengrub, H. B. Frieboes, F. Jin, Y.-L. Chuang, X. Li, P. Macklin, S. M. Wise and V. Cristini, Nonlinear modeling of cancer: Bridging the gap between cells and tumors, Nonlinearity, 23 (2010), R1-R91.
doi: 10.1088/0951-7715/23/1/R01. |
[16] |
J. T. Oden, A. Hawkins and S. Prudhomme, General diffuse-interface theories and an approach to predictive tumor growth modeling, Math. Models Methods Appl. Sci., 20 (2010), 477-517.
doi: 10.1142/S0218202510004313. |
[17] |
R. E. Showalter, Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, Math. Surveys Monogr. 49, American Mathematical Society, Providence, RI, 1997. |
[18] |
J. Simon, Compact sets in the space $L^p(0,T; B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[19] |
S. Zheng, Nonlinear Evolution Equations, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math. 133, Chapman & Hall/CRC, Boca Raton, FL, 2004.
doi: 10.1201/9780203492222. |
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