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On microscopic origins of generalized gradient structures
Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth
1. | Dipartimento di Matematica “F. Casorati”, Università di Pavia, via Ferrata 1,27100 Pavia, Italy |
2. | Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39,10117 Berlin, Germany |
3. | Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6,10099 Berlin, Germany |
This paper is concerned with a phase field system of Cahn-Hilliard type that is related to a tumor growth model and consists of three equations in terms of the variables order parameter, chemical potential and nutrient concentration. This system has been investigated in the recent papers [
References:
[1] |
V. Barbu,
Nonlinear Differential Equations of Monotone Types in Banach spaces Springer Monographs in Mathematics, 2010.
doi: 10.1007/978-1-4419-5542-5. |
[2] |
H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud., $\mathbf{5}$, 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., 2 (1958), 258-267.
|
[4] |
G. Canevari and P. Colli,
Solvability and asymptotic analysis of a generalization of the Caginalp phase field system, Commun. Pure Appl. Anal., 11 (2012), 1959-1982.
doi: 10.3934/cpaa.2012.11.1959. |
[5] |
G. Canevari and P. Colli,
Convergence properties for a generalization of the Caginalp phase field system, Asymptot. Anal., 82 (2013), 139-162.
|
[6] |
P. Colli, G. Gilardi and M. Grasselli,
Asymptotic analysis of a phase field model with memory for vanishing time relaxation, Hiroshima Math. J., 29 (1999), 117-143.
|
[7] |
P. Colli, G. Gilardi and D. Hilhorst,
On a Cahn-Hilliard type phase field system related to tumor growth, Discrete Contin. Dyn. Syst., 35 (2015), 2423-2442.
doi: 10.3934/dcds.2015.35.2423. |
[8] |
P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels,
An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 353-368.
|
[9] |
P. Colli, G. Gilardi, E. Rocca Sprekels and J. Sprekels,
Vanishing viscosities and error estimate for a Cahn-Hilliard type phase field system related to tumor growth, Nonlinear Anal. Real World Appl., 26 (2015), 93-108.
doi: 10.1016/j.nonrwa.2015.05.002. |
[10] |
P. Colli and J. Sprekels,
Stefan problems and the Penrose-Fife phase field model, Adv. Math. Sci. Appl., 7 (1997), 911-934.
|
[11] |
A. Damlamian, N. Kenmochi and N. Sato,
Subdifferential operator approach to a class of nonlinear systems for Stefan problems with phase relaxation, Nonlinear Anal., 23 (1994), 115-142.
doi: 10.1016/0362-546X(94)90255-0. |
[12] |
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. |
[13] |
C. M. Elliott and S. Zheng,
On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357.
doi: 10.1007/BF00251803. |
[14] |
S. Frigeri, M. Grasselli and E. Rocca,
On a diffuse interface model of tumor growth, European J. Appl. Math., 26 (2015), 215-243.
doi: 10.1017/S0956792514000436. |
[15] |
M. Girotti,
Vanishing time relaxation for a phase-field model with entropy balance, Adv. Math. Sci. Appl., 22 (2012), 553-575.
|
[16] |
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. Meth. Biomed. Engng., 28 (2012), 3-24.
doi: 10.1002/cnm.1467. |
[17] |
D. Hilhorst, J. Kampmann, T. N. Nguyen and K. G. Van der Zee,
Formal asymptotic limit of a diffuse-interface tumor-growth model, Math. Models Methods Appl. Sci., 25 (2015), 1011-1043.
doi: 10.1142/S0218202515500268. |
[18] |
J. -L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969. |
[19] |
E. Rocca,
Asymptotic analysis of a conserved phase-field model with memory for vanishing time relaxation, Adv. Math. Sci. Appl., 10 (2000), 899-916.
|
[20] |
R. Rossi,
Asymptotic analysis of the Caginalp phase-field model for two vanishing time relaxation parameters, Adv. Math. Sci. Appl., 13 (2003), 249-271.
|
[21] |
R. Rossi,
Well-posedness and asymptotic analysis for a Penrose-Fife type phase field system, Math. Methods Appl. Sci., 27 (2004), 1411-1445.
doi: 10.1002/mma.510. |
[22] |
G. Schimperna,
Singular limit of a transmission problem for the parabolic phase-field model, Appl. Math., 45 (2000), 217-238.
doi: 10.1023/A:1023070928404. |
[23] |
X. Wu, G. J. van Zwieten and K. G. van der Zee,
Stabilized second-order convex splitting schemes for Cahn-Hilliard models with applications to diffuse-interface tumor-growth models, Int. J. Numer. Meth. Biomed. Engng., 30 (2014), 180-203.
doi: 10.1002/cnm.2597. |
show all references
References:
[1] |
V. Barbu,
Nonlinear Differential Equations of Monotone Types in Banach spaces Springer Monographs in Mathematics, 2010.
doi: 10.1007/978-1-4419-5542-5. |
[2] |
H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud., $\mathbf{5}$, 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., 2 (1958), 258-267.
|
[4] |
G. Canevari and P. Colli,
Solvability and asymptotic analysis of a generalization of the Caginalp phase field system, Commun. Pure Appl. Anal., 11 (2012), 1959-1982.
doi: 10.3934/cpaa.2012.11.1959. |
[5] |
G. Canevari and P. Colli,
Convergence properties for a generalization of the Caginalp phase field system, Asymptot. Anal., 82 (2013), 139-162.
|
[6] |
P. Colli, G. Gilardi and M. Grasselli,
Asymptotic analysis of a phase field model with memory for vanishing time relaxation, Hiroshima Math. J., 29 (1999), 117-143.
|
[7] |
P. Colli, G. Gilardi and D. Hilhorst,
On a Cahn-Hilliard type phase field system related to tumor growth, Discrete Contin. Dyn. Syst., 35 (2015), 2423-2442.
doi: 10.3934/dcds.2015.35.2423. |
[8] |
P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels,
An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 353-368.
|
[9] |
P. Colli, G. Gilardi, E. Rocca Sprekels and J. Sprekels,
Vanishing viscosities and error estimate for a Cahn-Hilliard type phase field system related to tumor growth, Nonlinear Anal. Real World Appl., 26 (2015), 93-108.
doi: 10.1016/j.nonrwa.2015.05.002. |
[10] |
P. Colli and J. Sprekels,
Stefan problems and the Penrose-Fife phase field model, Adv. Math. Sci. Appl., 7 (1997), 911-934.
|
[11] |
A. Damlamian, N. Kenmochi and N. Sato,
Subdifferential operator approach to a class of nonlinear systems for Stefan problems with phase relaxation, Nonlinear Anal., 23 (1994), 115-142.
doi: 10.1016/0362-546X(94)90255-0. |
[12] |
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. |
[13] |
C. M. Elliott and S. Zheng,
On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357.
doi: 10.1007/BF00251803. |
[14] |
S. Frigeri, M. Grasselli and E. Rocca,
On a diffuse interface model of tumor growth, European J. Appl. Math., 26 (2015), 215-243.
doi: 10.1017/S0956792514000436. |
[15] |
M. Girotti,
Vanishing time relaxation for a phase-field model with entropy balance, Adv. Math. Sci. Appl., 22 (2012), 553-575.
|
[16] |
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. Meth. Biomed. Engng., 28 (2012), 3-24.
doi: 10.1002/cnm.1467. |
[17] |
D. Hilhorst, J. Kampmann, T. N. Nguyen and K. G. Van der Zee,
Formal asymptotic limit of a diffuse-interface tumor-growth model, Math. Models Methods Appl. Sci., 25 (2015), 1011-1043.
doi: 10.1142/S0218202515500268. |
[18] |
J. -L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod; Gauthier-Villars, Paris, 1969. |
[19] |
E. Rocca,
Asymptotic analysis of a conserved phase-field model with memory for vanishing time relaxation, Adv. Math. Sci. Appl., 10 (2000), 899-916.
|
[20] |
R. Rossi,
Asymptotic analysis of the Caginalp phase-field model for two vanishing time relaxation parameters, Adv. Math. Sci. Appl., 13 (2003), 249-271.
|
[21] |
R. Rossi,
Well-posedness and asymptotic analysis for a Penrose-Fife type phase field system, Math. Methods Appl. Sci., 27 (2004), 1411-1445.
doi: 10.1002/mma.510. |
[22] |
G. Schimperna,
Singular limit of a transmission problem for the parabolic phase-field model, Appl. Math., 45 (2000), 217-238.
doi: 10.1023/A:1023070928404. |
[23] |
X. Wu, G. J. van Zwieten and K. G. van der Zee,
Stabilized second-order convex splitting schemes for Cahn-Hilliard models with applications to diffuse-interface tumor-growth models, Int. J. Numer. Meth. Biomed. Engng., 30 (2014), 180-203.
doi: 10.1002/cnm.2597. |
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