# 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 and 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.

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.
 [1] Harald Garcke, Kei Fong Lam. Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis. Discrete and Continuous Dynamical Systems, 2017, 37 (8) : 4277-4308. doi: 10.3934/dcds.2017183 [2] Andrea Signori. Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme. Mathematical Control and Related Fields, 2020, 10 (2) : 305-331. doi: 10.3934/mcrf.2019040 [3] Pierluigi Colli, Gianni Gilardi, Danielle Hilhorst. On a Cahn-Hilliard type phase field system related to tumor growth. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2423-2442. doi: 10.3934/dcds.2015.35.2423 [4] Anouar El Harrak, Hatim Tayeq, Amal Bergam. A posteriori error estimates for a finite volume scheme applied to a nonlinear reaction-diffusion equation in population dynamics. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2183-2197. doi: 10.3934/dcdss.2021062 [5] Alp Eden, Varga K. Kalantarov. 3D convective Cahn--Hilliard equation. Communications on Pure and Applied Analysis, 2007, 6 (4) : 1075-1086. doi: 10.3934/cpaa.2007.6.1075 [6] Jaemin Shin, Yongho Choi, Junseok Kim. An unconditionally stable numerical method for the viscous Cahn--Hilliard equation. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1737-1747. doi: 10.3934/dcdsb.2014.19.1737 [7] Pierluigi Colli, Gianni Gilardi, Paolo Podio-Guidugli, Jürgen Sprekels. An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 353-368. doi: 10.3934/dcdss.2013.6.353 [8] Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182 [9] Svetlana Matculevich, Pekka Neittaanmäki, Sergey Repin. A posteriori error estimates for time-dependent reaction-diffusion problems based on the Payne--Weinberger inequality. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2659-2677. doi: 10.3934/dcds.2015.35.2659 [10] Hongyong Cui, Yangrong Li. Asymptotic $H^2$ regularity of a stochastic reaction-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021290 [11] Alain Miranville, Ramon Quintanilla, Wafa Saoud. Asymptotic behavior of a Cahn-Hilliard/Allen-Cahn system with temperature. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2257-2288. doi: 10.3934/cpaa.2020099 [12] Junping Shi, Jimin Zhang, Xiaoyan Zhang. Stability and asymptotic profile of steady state solutions to a reaction-diffusion pelagic-benthic algae growth model. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2325-2347. doi: 10.3934/cpaa.2019105 [13] M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079 [14] Jan-Phillip Bäcker, Matthias Röger. Analysis and asymptotic reduction of a bulk-surface reaction-diffusion model of Gierer-Meinhardt type. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1139-1155. doi: 10.3934/cpaa.2022013 [15] Annalisa Iuorio, Stefano Melchionna. Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3765-3788. doi: 10.3934/dcds.2018163 [16] Tiberiu Harko, Man Kwong Mak. Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach. Mathematical Biosciences & Engineering, 2015, 12 (1) : 41-69. doi: 10.3934/mbe.2015.12.41 [17] John W. Barrett, Harald Garcke, Robert Nürnberg. On sharp interface limits of Allen--Cahn/Cahn--Hilliard variational inequalities. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 1-14. doi: 10.3934/dcdss.2008.1.1 [18] Laurence Cherfils, Madalina Petcu, Morgan Pierre. A numerical analysis of the Cahn-Hilliard equation with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1511-1533. doi: 10.3934/dcds.2010.27.1511 [19] Danielle Hilhorst, Hideki Murakawa. Singular limit analysis of a reaction-diffusion system with precipitation and dissolution in a porous medium. Networks and Heterogeneous Media, 2014, 9 (4) : 669-682. doi: 10.3934/nhm.2014.9.669 [20] Wanli Yang, Jie Sun, Su Zhang. Analysis of optimal boundary control for a three-dimensional reaction-diffusion system. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 325-344. doi: 10.3934/naco.2017021

2020 Impact Factor: 2.425