August  2022, 15(8): 2305-2329. doi: 10.3934/dcdss.2022008

On the Cahn-Hilliard-Darcy system with mass source and strongly separating potential

Dipartimento di Matematica, Università di Pavia, Via Ferrata, 5, I-27100 Pavia, Italy

Dedicated to Maurizio Grasselli on the occasion of his 60th birthday, with friendship and admiration

Received  September 2021 Published  August 2022 Early access  January 2022

Fund Project: This research was supported by the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018-2022), Department of Mathematics "F. Casorati", University of Pavia. The present paper also benefits from the support of the MIUR-PRIN Grant 2015PA5MP7 "Calculus of Variations" and of the GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica)

We study an evolutionary system of Cahn-Hilliard-Darcy type including mass source and transport effects. The system may arise in a number of physical situations related to phase separation phenomena with convection, with the main and most specific application being related to tumoral processes, where the variations of the mass may correspond to growth, or shrinking, of the tumor. We prove existence of weak solutions in the case when the configuration potential for the order parameter $ \varphi $ is designed in such a way to keep $ \varphi $ in between the reference interval $ (-1, 1) $ despite the occurrence of mass source effects. Moreover, in the two-dimensional case, we obtain existence and uniqueness of strong (i.e., more regular) solutions.

Citation: Giulio Schimperna. On the Cahn-Hilliard-Darcy system with mass source and strongly separating potential. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2305-2329. doi: 10.3934/dcdss.2022008
References:
[1]

H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, 1984.

[2]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leiden, 1976.

[3]

H. Brézis, Opérateurs Maximaux Monotones et Sémi-Groupes de Contractions dans Les Éspaces de Hilbert, North-Holland Math. Studies 5, North-Holland, Amsterdam, 1973.

[4]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. 

[5]

P. ColliG. 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.

[6] V. Cristini and J. Lowengrub, Multiscale Modeling of Cancer. An Integrated Experimental and Mathematical Modeling Approach, Cambridge Univ. Press, 2010. 
[7]

V. CristiniX. LiJ. 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.

[8]

H. Engler, An alternative proof of the Brezis-Wainger inequality, Comm. Partial Differential Equations, 14 (1989), 541-544. 

[9]

S. FrigeriM. Grasselli and E. Rocca, On a diffuse interface model of tumor growth, European J. Appl. Math., 26 (2015), 215-243.  doi: 10.1017/S0956792514000436.

[10]

S. FrigeriK. F. LamE. Rocca and G. Schimperna, On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials, Commun. Math. Sci., 16 (2018), 821-856.  doi: 10.4310/CMS.2018.v16.n3.a11.

[11]

H. Garcke and K. F. Lam, Global weak solutions and asymptotic limits of a Cahn-Hilliard-Darcy system modelling tumour growth, AIMS Mathematics, 1 (2016), 318-360. 

[12]

H. Garcke and K. F. Lam, Analysis of a Cahn-Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis, Discrete Contin. Dyn. Syst., 37 (2017), 4277-4308.  doi: 10.3934/dcds.2017183.

[13]

H. GarckeK. F. LamE. Sitka and V. Styles, A Cahn-Hilliard-Darcy model for tumour growth with chemotaxis and active transport, Math. Models Methods Appl. Sci., 26 (2016), 1095-1148.  doi: 10.1142/S0218202516500263.

[14]

A. GiorginiM. Grasselli and H. Wu, The Cahn-Hilliard-Hele-Shaw system with singular potential, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1079-1118.  doi: 10.1016/j.anihpc.2017.10.002.

[15]

A. Giorgini, K. F. Lam, E. Rocca and G. Schimperna, On the Existence of Strong Solutions to the Cahn-Hilliard-Darcy system with mass source, SIAM J. Math. Anal., arXiv: 2009.13344.

[16]

J. JiangH. Wu and S. Zheng, Well-posedness and long-time behavior of a non-autonomous Cahn-Hilliard-Darcy system with mass source modeling tumor growth, J. Differential Equations, 259 (2015), 3032-3077.  doi: 10.1016/j.jde.2015.04.009.

[17]

J. S. LowengrubE. Titi and K. Zhao, Analysis of a mixture model of tumor growth, European J. Appl. Math., 24 (2013), 691-734.  doi: 10.1017/S0956792513000144.

[18]

A. MiranvilleE. Rocca and G. Schimperna, On the long time behavior of a tumor growth model, J. Differential Equations, 267 (2019), 2616-2642.  doi: 10.1016/j.jde.2019.03.028.

[19]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582.  doi: 10.1002/mma.464.

[20]

J. T. OdenA. 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.

[21]

A. Signori, Optimal distributed control of an extended model of tumor growth with logarithmic potential, Appl. Math. Optim., 82 (2020), 517-549.  doi: 10.1007/s00245-018-9538-1.

[22]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[23]

S. M. WiseJ. S. LowengrubH. B. Frieboes and V. Cristini, Three-dimensional multispecies nonlinear tumor growth-I: Model and numerical method, J. Theoret. Biol., 253 (2008), 524-543.  doi: 10.1016/j.jtbi.2008.03.027.

show all references

References:
[1]

H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, 1984.

[2]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leiden, 1976.

[3]

H. Brézis, Opérateurs Maximaux Monotones et Sémi-Groupes de Contractions dans Les Éspaces de Hilbert, North-Holland Math. Studies 5, North-Holland, Amsterdam, 1973.

[4]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. 

[5]

P. ColliG. 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.

[6] V. Cristini and J. Lowengrub, Multiscale Modeling of Cancer. An Integrated Experimental and Mathematical Modeling Approach, Cambridge Univ. Press, 2010. 
[7]

V. CristiniX. LiJ. 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.

[8]

H. Engler, An alternative proof of the Brezis-Wainger inequality, Comm. Partial Differential Equations, 14 (1989), 541-544. 

[9]

S. FrigeriM. Grasselli and E. Rocca, On a diffuse interface model of tumor growth, European J. Appl. Math., 26 (2015), 215-243.  doi: 10.1017/S0956792514000436.

[10]

S. FrigeriK. F. LamE. Rocca and G. Schimperna, On a multi-species Cahn-Hilliard-Darcy tumor growth model with singular potentials, Commun. Math. Sci., 16 (2018), 821-856.  doi: 10.4310/CMS.2018.v16.n3.a11.

[11]

H. Garcke and K. F. Lam, Global weak solutions and asymptotic limits of a Cahn-Hilliard-Darcy system modelling tumour growth, AIMS Mathematics, 1 (2016), 318-360. 

[12]

H. Garcke and K. F. Lam, Analysis of a Cahn-Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis, Discrete Contin. Dyn. Syst., 37 (2017), 4277-4308.  doi: 10.3934/dcds.2017183.

[13]

H. GarckeK. F. LamE. Sitka and V. Styles, A Cahn-Hilliard-Darcy model for tumour growth with chemotaxis and active transport, Math. Models Methods Appl. Sci., 26 (2016), 1095-1148.  doi: 10.1142/S0218202516500263.

[14]

A. GiorginiM. Grasselli and H. Wu, The Cahn-Hilliard-Hele-Shaw system with singular potential, Ann. Inst. H. Poincaré Anal. Non Linéaire, 35 (2018), 1079-1118.  doi: 10.1016/j.anihpc.2017.10.002.

[15]

A. Giorgini, K. F. Lam, E. Rocca and G. Schimperna, On the Existence of Strong Solutions to the Cahn-Hilliard-Darcy system with mass source, SIAM J. Math. Anal., arXiv: 2009.13344.

[16]

J. JiangH. Wu and S. Zheng, Well-posedness and long-time behavior of a non-autonomous Cahn-Hilliard-Darcy system with mass source modeling tumor growth, J. Differential Equations, 259 (2015), 3032-3077.  doi: 10.1016/j.jde.2015.04.009.

[17]

J. S. LowengrubE. Titi and K. Zhao, Analysis of a mixture model of tumor growth, European J. Appl. Math., 24 (2013), 691-734.  doi: 10.1017/S0956792513000144.

[18]

A. MiranvilleE. Rocca and G. Schimperna, On the long time behavior of a tumor growth model, J. Differential Equations, 267 (2019), 2616-2642.  doi: 10.1016/j.jde.2019.03.028.

[19]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582.  doi: 10.1002/mma.464.

[20]

J. T. OdenA. 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.

[21]

A. Signori, Optimal distributed control of an extended model of tumor growth with logarithmic potential, Appl. Math. Optim., 82 (2020), 517-549.  doi: 10.1007/s00245-018-9538-1.

[22]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[23]

S. M. WiseJ. S. LowengrubH. B. Frieboes and V. Cristini, Three-dimensional multispecies nonlinear tumor growth-I: Model and numerical method, J. Theoret. Biol., 253 (2008), 524-543.  doi: 10.1016/j.jtbi.2008.03.027.

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