# American Institute of Mathematical Sciences

February  2017, 10(1): 37-54. doi: 10.3934/dcdss.2017002

## 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

Received  March 2015 Revised  May 2015 Published  December 2016

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 [7] and [9] from the viewpoint of well-posedness, long-time behavior and asymptotic convergence as two positive viscosity coefficients tend to zero at the same time. Here, we continue the analysis performed in [9] by showing two independent sets of results as just one of the coefficents tends to zero, the other remaining fixed. We prove convergence results, uniqueness of solutions to the two resulting limit problems, and suitable error estimates.

Citation: Pierluigi Colli, Gianni Gilardi, Elisabetta Rocca, Jürgen Sprekels. Asymptotic analyses and error estimates for a Cahn-Hilliard type phase field system modelling tumor growth. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 37-54. doi: 10.3934/dcdss.2017002
##### 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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##### 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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