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A doubly nonlinear Cahn-Hilliard system with nonlinear viscosity

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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

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  • 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.

    Mathematics Subject Classification: Primary: 35G31, 35K52, 35D35, 74N20.


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  •   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. 
      V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976.
      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. 
      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. 
      H. Brezis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Math. Stud., 5 North-Holland, Amsterdam, 1973.
      J. W. Cahn  and  J. E. Hilliard , Free energy of a nonuniform system Ⅰ. Interfacial free energy, J. Chem. Phys., 2 (1958) , 258-267. 
      P. Colli  and  T. Fukao , Nonlinear diffusion equations as asymptotic limits of Cahn-Hilliard systems, J. Differential Equations, 260 (2016) , 6930-6959. 
      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. 
      P. Colli  and  L. Scarpa , From the viscous Cahn-Hilliard equation to a regularized forward-backward parabolic equation, Asymptot. Anal., 99 (2016) , 183-205. 
      P. Colli  and  A. Visintin , On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations, 15 (1990) , 737-756. 
      C. M. Elliott  and  H. Garcke , On the Cahn-Hilliard equation with degenerate mobility, SIAM J. Math. Anal., 27 (1996) , 404-423. 
      C. M. Elliott  and  A. M. Stuart , Viscous Cahn-Hilliard equation. Ⅱ. Analysis, J. Differential Equations, 128 (1996) , 387-414. 
      C. M. Elliott  and  S. Zheng , On the Cahn-Hilliard equation, Arch. Rational Mech. Anal., 96 (1986) , 339-357. 
      M. Frémond, Non-Smooth Thermomechanics, Springer-Verlag, Berlin, 2002.
      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. 
      M. Gurtin , Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D, 92 (1996) , 178-192. 
      M. Latroche , Structural and thermodynamic properties of metallic hydrides used for energy storage, J. Phys. Chem. Solids, 65 (2004) , 517-522. 
      A. Miranville  and  G. Schimperna , On a doubly nonlinear Cahn-Hilliard-Gurtin system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010) , 675-697. 
      A. Miranville  and  S. Zelik , Doubly nonlinear Cahn-Hilliard-Gurtin equations, Hokkaido Math. J., 38 (2009) , 315-360. 
      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.
      A. Novick-Cohen  and  R. L. Pego , Stable patterns in a viscous diffusion equation, Trans. Amer. Math. Soc., 324 (1991) , 331-351. 
      B. Schweizer , The Richards equation with hysteresis and degenerate capillary pressure, J. Differential Equations, 252 (2012) , 5594-5612. 
      J. Simon , Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987) , 65-96. 
      G. Tomassetti , Smooth and non-smooth regularizations of the nonlinear diffusion equation, Discrete Contin. Dyn. Syst. Ser. S, 10 (2017) , 1519-1537. 
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