August  2022, 15(8): 2135-2172. doi: 10.3934/dcdss.2022001

Well-posedness and optimal control for a Cahn–Hilliard–Oono system with control in the mass term

1. 

Dipartimento di Matematica "F. Casorati", Università di Pavia, and Research Associate at the IMATI – C.N.R. Pavia, via Ferrata 5, 27100 Pavia, Italy

2. 

Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany

3. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany

*Corresponding author: Elisabetta Rocca

Dedicated to our dear friend Maurizio Grasselli on the occasion of his 60th birthday

Received  August 2021 Revised  November 2021 Published  August 2022 Early access  January 2022

The paper treats the problem of optimal distributed control of a Cahn–Hilliard–Oono system in $ {{\mathbb{R}}}^d $, $ 1\leq d\leq 3 $, with the control located in the mass term and admitting general potentials that include both the case of a regular potential and the case of some singular potential. The first part of the paper is concerned with the dependence of the phase variable on the control variable. For this purpose, suitable regularity and continuous dependence results are shown. In particular, in the case of a logarithmic potential, we need to prove an ad hoc strict separation property, and for this reason we have to restrict ourselves to the case $ d = 2 $. In the rest of the work, we study the necessary first-order optimality conditions, which are proved under suitable compatibility conditions on the initial datum of the phase variable and the time derivative of the control, at least in case of potentials having unbounded domain.

Citation: Pierluigi Colli, Gianni Gilardi, Elisabetta Rocca, Jürgen Sprekels. Well-posedness and optimal control for a Cahn–Hilliard–Oono system with control in the mass term. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2135-2172. doi: 10.3934/dcdss.2022001
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 Monographs in Mathematics. Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[3]

A. L. BertozziS. 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]

J. F. Blowey and C.M. Elliott, The Cahn–Hilliard gradient theory for phase separation with nonsmooth free energy. I. Mathematical analysis, European J. Appl. Math., 2 (1991), 233-280.  doi: 10.1017/S095679250000053X.

[5]

S. BosiaM. Grasselli and A. Miranville, On the long-time behavior of a 2D hydrodynamic model for chemically reacting binary fluid mixtures, Math. Methods Appl. Sci., 37 (2014), 726-743.  doi: 10.1002/mma.2832.

[6]

H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973.

[7]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801.  doi: 10.1016/0001-6160(61)90182-1.

[8]

J. W. Cahn and J. E. Hilliard, Free Energy of a Nonuniform System. Ⅰ. Interfacial Free Energy, J. Chem. Phys., 28 (1958), 258-267.  doi: 10.1063/1.1744102.

[9]

C. CavaterraE. 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.

[10]

P. ColliG. 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.

[11]

P. ColliG. GilardiE. Rocca 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.

[12]

P. ColliG. GilardiE. Rocca and J. Sprekels, Asymptotic analyses and error estimates for a Cahn–Hilliard type phase field system modelling tumor growth, Discret. Contin. Dyn. Syst. Ser. S, 10 (2017), 37-54.  doi: 10.3934/dcdss.2017002.

[13]

P. ColliG. GilardiE. 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.

[14]

P. ColliG. 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.

[15]

P. ColliG. Gilardi and J. Sprekels, A boundary control problem for the pure Cahn–Hilliard equation with dynamic boundary conditions, Adv. Nonlinear Anal., 4 (2015), 311-325.  doi: 10.1515/anona-2015-0035.

[16]

P. Colli, G. Gilardi and J. Sprekels, On a Cahn–Hilliard system with convection and dynamic boundary conditions, Ann. Mat. Pura Appl. (4), 197 (2018), 1445–1475. doi: 10.1007/s10231-018-0732-1.

[17]

P. ColliG. 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.

[18]

P. ColliG. Gilardi and J. Sprekels, A distributed control problem for a fractional tumor growth model, Mathematics, 7 (2019), 792.  doi: 10.3390/math7090792.

[19]

P. ColliG. 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. 

[20]

P. ColliG. Gilardi and J. Sprekels, Recent results on well-posedness and optimal control for a class of generalized fractional Cahn–Hilliard systems, Control Cybernet., 48 (2019), 153-197. 

[21]

P. ColliG. Gilardi and J. Sprekels, Optimal distributed control of a generalized fractional Cahn–Hilliard system, Appl. Math. Optim., 82 (2020), 551-589.  doi: 10.1007/s00245-018-9540-7.

[22]

P. ColliG. Gilardi and J. Sprekels, Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 243-271.  doi: 10.3934/dcdss.2020213.

[23]

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.

[24]

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.

[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 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.  doi: 10.3934/Math.2016.3.318.

[27]

H. GarckeK.-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.

[28]

G. GilardiA. 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 and E. Rocca, Well posedness and long time behaviour for a singular phase field system of conserved type, IMA J. Appl. Math., 72 (2007), 498-530.  doi: 10.1093/imamat/hxm015.

[30]

G. Gilardi and J. Sprekels, Asymptotic limits and optimal control for the Cahn–Hilliard system with convection and dynamic boundary conditions, Nonlinear Anal., 178 (2019), 1-31.  doi: 10.1016/j.na.2018.07.007.

[31]

A. GiorginiM. 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.

[32]

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, preprint arXiv: 2009.13344 [math.AP] (2020), 1–30.

[33]

E. Khain and L. M. Sander, Generalized Cahn–Hilliard equation for biological applications, Phys. Rev. E, 77 (2008), 051129.  doi: 10.1103/PhysRevE.77.051129.

[34]

S. Melchionna and E. Rocca, On a nonlocal Cahn–Hilliard equation with a reaction term, Adv. Math. Sci. Appl., 24 (2014), 461-497. 

[35]

A. Miranville, Asymptotic behavior of the Cahn–Hilliard–Oono equation, J. Appl. Anal. Comput., 1 (2011), 523-536.  doi: 10.11948/2011036.

[36]

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.

[37]

A. Miranville and R. Temam, On the Cahn–Hilliard–Oono–Navier–Stokes equations with singular potentials, Appl. Anal., 95 (2016), 2609-2624.  doi: 10.1080/00036811.2015.1102893.

[38]

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.

[39]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.

[40]

Y. Oono and S. Puri, Computationally efficient modeling of ordering of quenched phases, Phys. Rev. Lett., 58 (1987), 836-839.  doi: 10.1103/PhysRevLett.58.836.

[41]

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.  doi: 10.1103/PhysRevA.38.434.

[42]

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.  doi: 10.1103/PhysRevA.38.1542.

[43]

A. Signori, Optimal treatment for a phase field system of Cahn–Hilliard type modeling tumor growth by asymptotic scheme, Math. Control Relat. Fields, 10 (2020), 305-331.  doi: 10.3934/mcrf.2019040.

[44]

A. Signori, Vanishing parameter for an optimal control problem modeling tumor growth, Asymptot. Anal., 117 (2020), 43-66.  doi: 10.3233/ASY-191546.

[45]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl. (4), 146 (1987), 65–96. doi: 10.1007/BF01762360.

show all references

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 Monographs in Mathematics. Springer, New York, 2010. doi: 10.1007/978-1-4419-5542-5.

[3]

A. L. BertozziS. 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]

J. F. Blowey and C.M. Elliott, The Cahn–Hilliard gradient theory for phase separation with nonsmooth free energy. I. Mathematical analysis, European J. Appl. Math., 2 (1991), 233-280.  doi: 10.1017/S095679250000053X.

[5]

S. BosiaM. Grasselli and A. Miranville, On the long-time behavior of a 2D hydrodynamic model for chemically reacting binary fluid mixtures, Math. Methods Appl. Sci., 37 (2014), 726-743.  doi: 10.1002/mma.2832.

[6]

H. Brezis, Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Math. Stud. 5, North-Holland, Amsterdam, 1973.

[7]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801.  doi: 10.1016/0001-6160(61)90182-1.

[8]

J. W. Cahn and J. E. Hilliard, Free Energy of a Nonuniform System. Ⅰ. Interfacial Free Energy, J. Chem. Phys., 28 (1958), 258-267.  doi: 10.1063/1.1744102.

[9]

C. CavaterraE. 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.

[10]

P. ColliG. 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.

[11]

P. ColliG. GilardiE. Rocca 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.

[12]

P. ColliG. GilardiE. Rocca and J. Sprekels, Asymptotic analyses and error estimates for a Cahn–Hilliard type phase field system modelling tumor growth, Discret. Contin. Dyn. Syst. Ser. S, 10 (2017), 37-54.  doi: 10.3934/dcdss.2017002.

[13]

P. ColliG. GilardiE. 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.

[14]

P. ColliG. 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.

[15]

P. ColliG. Gilardi and J. Sprekels, A boundary control problem for the pure Cahn–Hilliard equation with dynamic boundary conditions, Adv. Nonlinear Anal., 4 (2015), 311-325.  doi: 10.1515/anona-2015-0035.

[16]

P. Colli, G. Gilardi and J. Sprekels, On a Cahn–Hilliard system with convection and dynamic boundary conditions, Ann. Mat. Pura Appl. (4), 197 (2018), 1445–1475. doi: 10.1007/s10231-018-0732-1.

[17]

P. ColliG. 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.

[18]

P. ColliG. Gilardi and J. Sprekels, A distributed control problem for a fractional tumor growth model, Mathematics, 7 (2019), 792.  doi: 10.3390/math7090792.

[19]

P. ColliG. 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. 

[20]

P. ColliG. Gilardi and J. Sprekels, Recent results on well-posedness and optimal control for a class of generalized fractional Cahn–Hilliard systems, Control Cybernet., 48 (2019), 153-197. 

[21]

P. ColliG. Gilardi and J. Sprekels, Optimal distributed control of a generalized fractional Cahn–Hilliard system, Appl. Math. Optim., 82 (2020), 551-589.  doi: 10.1007/s00245-018-9540-7.

[22]

P. ColliG. Gilardi and J. Sprekels, Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 243-271.  doi: 10.3934/dcdss.2020213.

[23]

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.

[24]

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.

[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 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.  doi: 10.3934/Math.2016.3.318.

[27]

H. GarckeK.-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.

[28]

G. GilardiA. 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 and E. Rocca, Well posedness and long time behaviour for a singular phase field system of conserved type, IMA J. Appl. Math., 72 (2007), 498-530.  doi: 10.1093/imamat/hxm015.

[30]

G. Gilardi and J. Sprekels, Asymptotic limits and optimal control for the Cahn–Hilliard system with convection and dynamic boundary conditions, Nonlinear Anal., 178 (2019), 1-31.  doi: 10.1016/j.na.2018.07.007.

[31]

A. GiorginiM. 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.

[32]

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, preprint arXiv: 2009.13344 [math.AP] (2020), 1–30.

[33]

E. Khain and L. M. Sander, Generalized Cahn–Hilliard equation for biological applications, Phys. Rev. E, 77 (2008), 051129.  doi: 10.1103/PhysRevE.77.051129.

[34]

S. Melchionna and E. Rocca, On a nonlocal Cahn–Hilliard equation with a reaction term, Adv. Math. Sci. Appl., 24 (2014), 461-497. 

[35]

A. Miranville, Asymptotic behavior of the Cahn–Hilliard–Oono equation, J. Appl. Anal. Comput., 1 (2011), 523-536.  doi: 10.11948/2011036.

[36]

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.

[37]

A. Miranville and R. Temam, On the Cahn–Hilliard–Oono–Navier–Stokes equations with singular potentials, Appl. Anal., 95 (2016), 2609-2624.  doi: 10.1080/00036811.2015.1102893.

[38]

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.

[39]

J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.  doi: 10.1512/iumj.1971.20.20101.

[40]

Y. Oono and S. Puri, Computationally efficient modeling of ordering of quenched phases, Phys. Rev. Lett., 58 (1987), 836-839.  doi: 10.1103/PhysRevLett.58.836.

[41]

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.  doi: 10.1103/PhysRevA.38.434.

[42]

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.  doi: 10.1103/PhysRevA.38.1542.

[43]

A. Signori, Optimal treatment for a phase field system of Cahn–Hilliard type modeling tumor growth by asymptotic scheme, Math. Control Relat. Fields, 10 (2020), 305-331.  doi: 10.3934/mcrf.2019040.

[44]

A. Signori, Vanishing parameter for an optimal control problem modeling tumor growth, Asymptot. Anal., 117 (2020), 43-66.  doi: 10.3233/ASY-191546.

[45]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl. (4), 146 (1987), 65–96. doi: 10.1007/BF01762360.

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