Well-posedness and optimal control for a viscous Cahn–Hilliard–Oono system with dynamic boundary conditions

Gianni Gilardi; Elisabetta Rocca; Andrea Signori

doi:10.3934/dcdss.2023127

Article Contents

2023, Volume 16, Issue 12: 3573-3605. Doi: 10.3934/dcdss.2023127

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Well-posedness and optimal control for a viscous Cahn–Hilliard–Oono system with dynamic boundary conditions

1.
Dipartimento di Matematica "F. Casorati", Università di Pavia, and Research Associate at the IMATI – C.N.R. Pavia, via Ferrata 5, I-27100 Pavia, Italy
2.
Dipartimento di Matematica, Politecnico di Milano, via Bonardi 9, I-20133 Milano, Italy

^*Corresponding author: Gianni Gilardi
^*Corresponding author: Gianni Gilardi

Dedicated to our dear friend Pierluigi Colli on the occasion of his 65th birthday

Received: January 2023

Revised: June 2023

Early access: June 2023

Published: December 2023

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Abstract

In this paper we consider a nonlinear system of PDEs coupling the viscous Cahn–Hilliard–Oono equation with dynamic boundary conditions enjoying a similar structure on the boundary. After proving well-posedness of the corresponding initial boundary value problem, we study an associated optimal control problem related to a tracking-type cost functional, proving first-order necessary conditions of optimality. The controls enter the system in the form of a distributed and a boundary source. We can account for general potentials in the bulk and in the boundary part under the common assumption that the boundary potential is dominant with respect to the bulk one. For example, the regular quartic potential as well as the logarithmic potential can be considered in our analysis.

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Mathematics Subject Classification: Primary: 35K55, 49J20, 49K20; Secondary: 35K61, 49J50.

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References

[1]	M. Bahiana and Y. Oono, Cell dynamical system approach to block copolymers, Phys. Rev. A, 41 (1990), 6763-6771.
[2]	V. Barbu, Nonlinear Differential Equations of Monotone Type in Banach Spaces, Springer, London, New York, 2010. doi: 10.1007/978-1-4419-5542-5.
[3]	A. L. Bertozzi, S. Esedoglu and A. Gillette, Inpainting of binary images using the Cahn–Hilliard equation, IEEE Trans. Image Process., 16 (2007), 285-291. doi: 10.1109/TIP.2006.887728.
[4]	E. Bonetti, W. Dreyer and G. Schimperna, Global solution to a generalized Cahn–Hilliard equation with viscosity, Adv. Differential Equations, 8 (2003), 231-256.
[5]	H. Brézis, Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert, North-Holland Math. Stud., Vol. 5, North-Holland, Amsterdam, 1973.
[6]	L. Calatroni and P. Colli, Global solution to the Allen–Cahn equation with singular potentials and dynamic boundary conditions, Nonlinear Anal., 79 (2013), 12-27. doi: 10.1016/j.na.2012.11.010.
[7]	C. Cavaterra, E. Rocca and H. Wu, Long-time dynamics and optimal control of a diffuse interface model for tumor growth, Appl. Math. Optim., 83 (2021), 739-787. doi: 10.1007/s00245-019-09562-5.
[8]	P. Colli and T. Fukao, Cahn-Hilliard equation with dynamic boundary conditions and mass constraint on the boundary, J. Math. Anal. Appl., 429 (2015), 1190-1213. doi: 10.1016/j.jmaa.2015.04.057.
[9]	P. Colli, G. Gilardi and D. Hilhorst, On a Cahn-Hilliard type phase field system related to tumor growth, Discret. Cont. Dyn. Syst., 35 (2015), 2423-2442. doi: 10.3934/dcds.2015.35.2423.
[10]	P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), 2518-2546. doi: 10.1088/1361-6544/aa6e5f.
[11]	P. Colli, G. Gilardi, E. Rocca and J. Sprekels, Well-posedness and optimal control for a Cahn–Hilliard–Oono system with control in the mass term, Discret. Contin. Dyn. Syst. Ser. S, 15 (2022), 2135-2172. doi: 10.3934/dcdss.2022001.
[12]	P. Colli, G. Gilardi and J. Sprekels, On the Cahn-Hilliard equation with dynamic boundary conditions and a dominating boundary potential, J. Math. Anal. Appl., 419 (2014), 972-994. doi: 10.1016/j.jmaa.2014.05.008.
[13]	P. Colli, G. Gilardi and J. Sprekels, Recent results on the Cahn–Hilliard equation with dynamic boundary conditions, Vestn. Yuzhno-Ural. Gos. Univ., Ser. Mat. Model. Program., 10 (2017), 5-21.
[14]	P. Colli, G. Gilardi and J. Sprekels, On a Cahn–Hilliard system with convection and dynamic boundary conditions, Ann. Mat. Pura Appl., 197 (2018), 1445-1475. doi: 10.1007/s10231-018-0732-1.
[15]	P. Colli, G. Gilardi and J. Sprekels, Optimal velocity control of a viscous Cahn–Hilliard system with convection and dynamic boundary conditions, SIAM J. Control Optim., 56 (2018), 1665-1691. doi: 10.1137/17M1146786.
[16]	P. Colli, G. Gilardi and J. Sprekels, Optimal velocity control of a convective Cahn–Hilliard system with double obstacles and dynamic boundary conditions: A 'deep quench' approach, J. Convex Anal., 26 (2019), 485-514.
[17]	P. Colli and A. Signori, Boundary control problem and optimality conditions for the Cahn–Hilliard equation with dynamic boundary conditions, Internat. J. Control, 94 (2021), 1852-1869. doi: 10.1080/00207179.2019.1680870.
[18]	P. Colli, A. Signori and J. Sprekels, Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis, ESAIM Control Optim. Calc. Var., 27 (2021), Paper No. 73, 46 pp. doi: 10.1051/cocv/2021072.
[19]	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.
[20]	H. P. Fischer, P. Maass and W. Dieterich, Novel surfs in spinodal decomposition, Phys. Rev. Letters, 79 (1997), 893-896.
[21]	H. P. Fischer, P. Maass and W. Dieterich, Diverging time and length scales of spinodal decomposition modes in thin flows, Europhys. Letters, 42 (1998), 49-54.
[22]	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.
[23]	T. Fukao and Y. Noriaki, A boundary control problem for the equation and dynamic boundary condition of Cahn-Hilliard type, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs: In Honour of Prof. Gianni Gilardi, (2017), 255-280.
[24]	H. Garcke and P. Knopf, Weak solutions of the Cahn-Hilliard system with dynamic boundary conditions: A gradient flow approach, SIAM J. Math. Anal., 52 (2020), 340-369. doi: 10.1137/19M1258840.
[25]	H. Garcke and K. F. Lam, Well-posedness of a Cahn–Hilliard system modelling tumour growth with chemotaxis and active transport, European. J. Appl. Math., 28 (2017), 284-316. doi: 10.1017/S0956792516000292.
[26]	H. Garcke, K. F. Lam and E. Rocca, Optimal control of treatment time in a diffuse interface model of tumor growth, Appl. Math. Optim., 78 (2018), 495-544. doi: 10.1007/s00245-017-9414-4.
[27]	H. Garcke, K. F. Lam, E. Sitka and V. Styles, A Cahn–Hilliard–Darcy model for tumour growth with chemotaxis and active transport, Math. Models Methods in Appl. Sci., 26 (2016), 1095-1148. doi: 10.1142/S0218202516500263.
[28]	G. Gilardi, A. Miranville and G. Schimperna, On the Cahn–Hilliard equation with irregular potentials and dynamic boundary conditions, Commun. Pure Appl. Anal., 8 (2009), 881-912. doi: 10.3934/cpaa.2009.8.881.
[29]	G. Gilardi, A. Miranville and G. Schimperna, Long-time behavior of the Cahn-Hilliard equation with irregular potentials and dynamic boundary conditions, Chinese Annals of Mathematics Series B, 31 (2010), 679-712. doi: 10.1007/s11401-010-0602-7.
[30]	A. Giorgini, M. Grasselli and A. Miranville, The Cahn–Hilliard–Oono equation with singular potential, Math. Models Methods Appl. Sci., 27 (2017), 2485-2510. doi: 10.1142/S0218202517500506.
[31]	R. Kenzler, F. Eurich, P. Maass, B. Rinn, J. Schropp, E. Bohl and W. Dieterich, Phase separation in confined geometries: Solving the Cahn–Hilliard equation with generic boundary conditions, Comput. Phys. Comm., 133 (2001), 139-157. doi: 10.1016/S0010-4655(00)00159-4.
[32]	E. Khain and L. M. Sander, Generalized Cahn–Hilliard equation for biological applications, Phys. Rev. E, 77 (2008), 051129-1-051129-7.
[33]	P. Knopf and A. Signori, Existence of weak solutions to multiphase Cahn–Hilliard–Darcy and Cahn–Hilliard–Brinkman models for stratified tumor growth with chemotaxis and general source terms, Comm. Partial Differential Equations, 47 (2022), 233-278. doi: 10.1080/03605302.2021.1966803.
[34]	P. Knopf and A. Signori, On the nonlocal Cahn–Hilliard equation with nonlocal dynamic boundary condition and boundary penalization, J. Differential Equations, 280 (2021), 236-291. doi: 10.1016/j.jde.2021.01.012.
[35]	J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod; Gauthier-Villars, Paris, 1969.
[36]	C. Liu and H. Wu, An energetic variational approach for the Cahn–Hilliard equation with dynamic boundary conditions: Model derivation and mathematical analysis, Arch. Ration. Mech. Anal., 233 (2019), 167-247. doi: 10.1007/s00205-019-01356-x.
[37]	A. Makki, A. Miranville and M. Petcu, A numerical analysis of the coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions, Int. J. Numer. Anal. Model., 19 (2022), 630-655.
[38]	S. Melchionna and E. Rocca, On a nonlocal Cahn–Hilliard equation with a reaction term, Adv. Math. Sci. Appl., 24 (2014), 461-497.
[39]	A. Miranville, Asymptotic behavior of the Cahn–Hilliard–Oono equation, J. Appl. Anal. Comput., 1 (2011), 523-536.
[40]	A. Miranville, E. 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.
[41]	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.
[42]	A. Novick-Cohen, Energy methods for the Cahn–Hilliard equation, Quart. Appl. Math., 46 (1988), 681-690. doi: 10.1090/qam/973383.
[43]	Y. Oono and S. Puri, Computationally efficient modeling of ordering of quenched phases, Phys. Rev. Lett., 58 (1987), 836-839.
[44]	Y. Oono and S. Puri, Study of phase-separation dynamics by use of cell dynamical systems. I. Modeling, Phys. Rev. A, 38 (1988), 434-453.
[45]	S. Puri and Y. Oono, Study of phase-separation dynamics by use of cell dynamical systems. II. Two-dimensional demonstrations, Phys. Rev. A, 38 (1988), 1542-1565.
[46]	E. Rocca, G. Schimperna and A. Signori, On a Cahn–Hilliard–Keller–Segel model with generalized logistic source describing tumor growth, J. Differential Equations, 343 (2023), 530-578. doi: 10.1016/j.jde.2022.10.026.
[47]	L. Scarpa and A. Signori, On a class of non-local phase-field models for tumor growth with possibly singular potentials, chemotaxis, and active transport, Nonlinearity, 34 (2021), 3199-3250. doi: 10.1088/1361-6544/abe75d.
[48]	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.