# American Institute of Mathematical Sciences

May  2018, 17(3): 1001-1022. doi: 10.3934/cpaa.2018049

## A doubly nonlinear Cahn-Hilliard system with nonlinear viscosity

 1 Dipartimento di Matematica "F. Enriques", Università degli Studi di Milano, Via Saldini 50,20133 Milano, Italy 2 Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1,27100 Pavia, Italy 3 Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom 4 Dipartimento di Ingegneria -Sezione Ingegneria Civile, Università degli Studi "Roma Tre", Via Vito Volterra 62, Roma, Italy

* Corresponding author

Received  October 2017 Revised  November 2017 Published  January 2018

Fund Project: PC gratefully acknowledges some financial support from the MIUR-PRIN Grant 2015PA5MP7 "Calculus of Variations"; the present paper also benefits from the support of the GNAMPA (Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) and the IMATI – C.N.R. Pavia for EB and PC

In this paper we discuss a family of viscous Cahn-Hilliard equations with a non-smooth viscosity term. This system may be viewed as an approximation of a ''forward-backward'' parabolic equation. The resulting problem is highly nonlinear, coupling in the same equation two nonlinearities with the diffusion term. In particular, we prove existence of solutions for the related initial and boundary value problem. Under suitable assumptions, we also state uniqueness and continuous dependence on data.

Citation: Elena Bonetti, Pierluigi Colli, Luca Scarpa, Giuseppe Tomassetti. A doubly nonlinear Cahn-Hilliard system with nonlinear viscosity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1001-1022. doi: 10.3934/cpaa.2018049
##### References:
 [1] F. Bai, C. M. Elliott, A. Gardiner, A. Spence and A. M. Stuart, The viscous Cahn-Hilliard equation. Ⅰ. Computations, Nonlinearity, 8 (1995), 131-160. Google Scholar [2] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. Google Scholar [3] E. Bonetti, P. Colli and G. Tomassetti, A non-smooth regularization of a forward-backward parabolic equation, Math. Models Methods Appl. Sci., 27 (2017), 641-661. Google Scholar [4] N. D. Botkin, M. Brokate and E. G. El Behi-Gornostaeva, One-phase flow in porous media with hysteresis, Phys. B, 486 (2016), 183-186. Google Scholar [5] H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud., 5 North-Holland, Amsterdam, 1973. Google Scholar [6] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system Ⅰ. Interfacial free energy, J. Chem. Phys., 2 (1958), 258-267. Google Scholar [7] P. Colli and T. Fukao, Nonlinear diffusion equations as asymptotic limits of Cahn-Hilliard systems, J. Differential Equations, 260 (2016), 6930-6959. Google Scholar [8] 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. Google Scholar [9] P. Colli and L. Scarpa, From the viscous Cahn-Hilliard equation to a regularized forward-backward parabolic equation, Asymptot. Anal., 99 (2016), 183-205. Google Scholar [10] P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations, 15 (1990), 737-756. Google Scholar [11] C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423. Google Scholar [12] C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation. Ⅱ. Analysis, J. Differential Equations, 128 (1996), 387-414. Google Scholar [13] C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357. Google Scholar [14] M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. Google Scholar [15] 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. Google Scholar [16] M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192. Google Scholar [17] M. Latroche, Structural and thermodynamic properties of metallic hydrides used for energy storage, J. Phys. Chem. Solids, 65 (2004), 517-522. Google Scholar [18] A. Miranville and G. Schimperna, On a doubly nonlinear Cahn-Hilliard-Gurtin system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 675-697. Google Scholar [19] A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations, Hokkaido Math. J., 38 (2009), 315-360. Google Scholar [20] A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in Material instabilities in continuum mechanics (Edinburgh, 1985–1986), Oxford Sci. Publ., Oxford Univ. Press, New York, 1988, pp. 329–342. Google Scholar [21] A. Novick-Cohen and R. L. Pego, Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc., 324 (1991), 331-351. Google Scholar [22] B. Schweizer, The Richards equation with hysteresis and degenerate capillary pressure, J. Differential Equations, 252 (2012), 5594-5612. Google Scholar [23] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl.(4), 146 (1987), 65-96. Google Scholar [24] G. Tomassetti, Smooth and non-smooth regularizations of the nonlinear diffusion equation, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 1519-1537. Google Scholar

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##### References:
 [1] F. Bai, C. M. Elliott, A. Gardiner, A. Spence and A. M. Stuart, The viscous Cahn-Hilliard equation. Ⅰ. Computations, Nonlinearity, 8 (1995), 131-160. Google Scholar [2] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. Google Scholar [3] E. Bonetti, P. Colli and G. Tomassetti, A non-smooth regularization of a forward-backward parabolic equation, Math. Models Methods Appl. Sci., 27 (2017), 641-661. Google Scholar [4] N. D. Botkin, M. Brokate and E. G. El Behi-Gornostaeva, One-phase flow in porous media with hysteresis, Phys. B, 486 (2016), 183-186. Google Scholar [5] H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud., 5 North-Holland, Amsterdam, 1973. Google Scholar [6] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system Ⅰ. Interfacial free energy, J. Chem. Phys., 2 (1958), 258-267. Google Scholar [7] P. Colli and T. Fukao, Nonlinear diffusion equations as asymptotic limits of Cahn-Hilliard systems, J. Differential Equations, 260 (2016), 6930-6959. Google Scholar [8] 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. Google Scholar [9] P. Colli and L. Scarpa, From the viscous Cahn-Hilliard equation to a regularized forward-backward parabolic equation, Asymptot. Anal., 99 (2016), 183-205. Google Scholar [10] P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations, 15 (1990), 737-756. Google Scholar [11] C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423. Google Scholar [12] C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation. Ⅱ. Analysis, J. Differential Equations, 128 (1996), 387-414. Google Scholar [13] C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357. Google Scholar [14] M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. Google Scholar [15] 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. Google Scholar [16] M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192. Google Scholar [17] M. Latroche, Structural and thermodynamic properties of metallic hydrides used for energy storage, J. Phys. Chem. Solids, 65 (2004), 517-522. Google Scholar [18] A. Miranville and G. Schimperna, On a doubly nonlinear Cahn-Hilliard-Gurtin system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 675-697. Google Scholar [19] A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations, Hokkaido Math. J., 38 (2009), 315-360. Google Scholar [20] A. Novick-Cohen, On the viscous Cahn-Hilliard equation, in Material instabilities in continuum mechanics (Edinburgh, 1985–1986), Oxford Sci. Publ., Oxford Univ. Press, New York, 1988, pp. 329–342. Google Scholar [21] A. Novick-Cohen and R. L. Pego, Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc., 324 (1991), 331-351. Google Scholar [22] B. Schweizer, The Richards equation with hysteresis and degenerate capillary pressure, J. Differential Equations, 252 (2012), 5594-5612. Google Scholar [23] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl.(4), 146 (1987), 65-96. Google Scholar [24] G. Tomassetti, Smooth and non-smooth regularizations of the nonlinear diffusion equation, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 1519-1537. Google Scholar
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