# 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 and 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. [2] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. [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. [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. [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. [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. [7] P. Colli and T. Fukao, Nonlinear diffusion equations as asymptotic limits of Cahn-Hilliard systems, J. Differential Equations, 260 (2016), 6930-6959. [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. [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. [10] P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations, 15 (1990), 737-756. [11] C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423. [12] C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation. Ⅱ. Analysis, J. Differential Equations, 128 (1996), 387-414. [13] C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357. [14] M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. [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. [16] M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192. [17] M. Latroche, Structural and thermodynamic properties of metallic hydrides used for energy storage, J. Phys. Chem. Solids, 65 (2004), 517-522. [18] A. Miranville and G. Schimperna, On a doubly nonlinear Cahn-Hilliard-Gurtin system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 675-697. [19] A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations, Hokkaido Math. J., 38 (2009), 315-360. [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. [21] A. Novick-Cohen and R. L. Pego, Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc., 324 (1991), 331-351. [22] B. Schweizer, The Richards equation with hysteresis and degenerate capillary pressure, J. Differential Equations, 252 (2012), 5594-5612. [23] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl.(4), 146 (1987), 65-96. [24] G. Tomassetti, Smooth and non-smooth regularizations of the nonlinear diffusion equation, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 1519-1537.

show all references

##### 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. [2] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. [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. [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. [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. [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. [7] P. Colli and T. Fukao, Nonlinear diffusion equations as asymptotic limits of Cahn-Hilliard systems, J. Differential Equations, 260 (2016), 6930-6959. [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. [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. [10] P. Colli and A. Visintin, On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations, 15 (1990), 737-756. [11] C. M. Elliott and H. Garcke, On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996), 404-423. [12] C. M. Elliott and A. M. Stuart, Viscous Cahn-Hilliard equation. Ⅱ. Analysis, J. Differential Equations, 128 (1996), 387-414. [13] C. M. Elliott and S. Zheng, On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986), 339-357. [14] M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002. [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. [16] M. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996), 178-192. [17] M. Latroche, Structural and thermodynamic properties of metallic hydrides used for energy storage, J. Phys. Chem. Solids, 65 (2004), 517-522. [18] A. Miranville and G. Schimperna, On a doubly nonlinear Cahn-Hilliard-Gurtin system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 675-697. [19] A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations, Hokkaido Math. J., 38 (2009), 315-360. [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. [21] A. Novick-Cohen and R. L. Pego, Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc., 324 (1991), 331-351. [22] B. Schweizer, The Richards equation with hysteresis and degenerate capillary pressure, J. Differential Equations, 252 (2012), 5594-5612. [23] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl.(4), 146 (1987), 65-96. [24] G. Tomassetti, Smooth and non-smooth regularizations of the nonlinear diffusion equation, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017), 1519-1537.
 [1] Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630 [2] L. Chupin. Existence result for a mixture of non Newtonian flows with stress diffusion using the Cahn-Hilliard formulation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 45-68. doi: 10.3934/dcdsb.2003.3.45 [3] Peng Jiang. Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 3015-3037. doi: 10.3934/dcds.2015.35.3015 [4] Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59 [5] Michal Beneš. Mixed initial-boundary value problem for the three-dimensional Navier-Stokes equations in polyhedral domains. Conference Publications, 2011, 2011 (Special) : 135-144. doi: 10.3934/proc.2011.2011.135 [6] Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113 [7] Gilles Carbou, Bernard Hanouzet. Relaxation approximation of the Kerr model for the impedance initial-boundary value problem. Conference Publications, 2007, 2007 (Special) : 212-220. doi: 10.3934/proc.2007.2007.212 [8] Xianpeng Hu, Dehua Wang. The initial-boundary value problem for the compressible viscoelastic flows. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 917-934. doi: 10.3934/dcds.2015.35.917 [9] R.M. Brown, L.D. Gauthier. Inverse boundary value problems for polyharmonic operators with non-smooth coefficients. Inverse Problems and Imaging, 2022, 16 (4) : 943-966. doi: 10.3934/ipi.2022006 [10] Ciprian G. Gal, Mahamadi Warma. Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1279-1319. doi: 10.3934/dcds.2016.36.1279 [11] Xu Liu, Jun Zhou. Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity. Electronic Research Archive, 2020, 28 (2) : 599-625. doi: 10.3934/era.2020032 [12] Zhiyuan Li, Xinchi Huang, Masahiro Yamamoto. Initial-boundary value problems for multi-term time-fractional diffusion equations with $x$-dependent coefficients. Evolution Equations and Control Theory, 2020, 9 (1) : 153-179. doi: 10.3934/eect.2020001 [13] Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3063-3092. doi: 10.3934/dcds.2020398 [14] Jitao Liu. On the initial boundary value problem for certain 2D MHD-$\alpha$ equations without velocity viscosity. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1179-1191. doi: 10.3934/cpaa.2016.15.1179 [15] Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027 [16] Dong Li. A regularization-free approach to the Cahn-Hilliard equation with logarithmic potentials. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2453-2460. doi: 10.3934/dcds.2021198 [17] 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 [18] Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033 [19] Ciprian G. Gal, Maurizio Grasselli. Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145 [20] Xiaoyun Cai, Liangwen Liao, Yongzhong Sun. Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 917-923. doi: 10.3934/dcdss.2014.7.917

2021 Impact Factor: 1.273

## Metrics

• PDF downloads (268)
• HTML views (300)
• Cited by (6)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]