June  2018, 38(6): 3033-3054. doi: 10.3934/dcds.2018132

A singular cahn-hilliard-oono phase-field system with hereditary memory

1. 

Politecnico di Milano, Dipartimento di Matematica "F. Brioschi", Via Bonardi 9, I-20133 Milano, Italy

2. 

Università di Modena e Reggio Emilia, Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Via Campi 213/B, I-41125 Modena, Italy

3. 

Université de Poitiers, Laboratoire de Mathématiques et Applications, UMR CNRS 7348, SP2MI, Boulevard Marie et Pierre Curie -Téléport 2, F-86962 Chasseneuil Futuroscope Cedex, France

4. 

Xiamen University, School of Mathematical Sciences, 361005 Xiamen, Fujian, China

* Corresponding author: Stefania Gatti

Received  September 2017 Published  April 2018

We consider a phase-field system modeling phase transition phenomena, where the Cahn-Hilliard-Oono equation for the order parameter is coupled with the Coleman-Gurtin heat law for the temperature. The former suitably describes both local and nonlocal (long-ranged) interactions in the material undergoing phase-separation, while the latter takes into account thermal memory effects. We study the well-posedness and longtime behavior of the corresponding dynamical system in the history space setting, for a class of physically relevant and singular potentials. Besides, we investigate the regularization properties of the solutions and, for sufficiently smooth data, we establish the strict separation property from the pure phases.

Citation: Monica Conti, Stefania Gatti, Alain Miranville. A singular cahn-hilliard-oono phase-field system with hereditary memory. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3033-3054. doi: 10.3934/dcds.2018132
References:
[1]

D. Brochet, Maximal attractor and inertial sets for some second and fourth order phase field models, in Pitman Res. Notes Math. Ser. , Longman Sci. Tech., Harlow, 296 (1993), 77–85.  Google Scholar

[2]

D. BrochetD. Hilhorst and A. Novick-Cohen, Maximal attractor and inertial sets for a conserved phase field model, Adv. Diff. Eqns., 1 (1996), 547-578.   Google Scholar

[3]

G. Caginalp, A conserved phase field system; implications for kinetic undercooling, Phys. Rev. B, 38 (1988), 789-791.   Google Scholar

[4]

G. Caginalp, The dynamics of a conserved phase-field system: Stefan-like, Hele-Shaw and Cahn-Hilliard models as asymptotic limits, IMA J. Appl. Math., 44 (1990), 77-94.   Google Scholar

[5]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801.   Google Scholar

[6]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.   Google Scholar

[7]

B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208.   Google Scholar

[8]

M. Conti and A. Giorgini, The three-dimensional Cahn-Hilliard-Brinkman system with unmatched viscosities, preprint. Google Scholar

[9]

M. ContiV. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215.   Google Scholar

[10]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.   Google Scholar

[11]

C. G. GalA. Giorgini and M. Grasselli, The nonlocal Cahn-Hilliard equation with singular potential: Well-posedness, regularity and strict separation property, J. Differential Equations, 263 (2017), 5253-5297.   Google Scholar

[12]

S. GattiM. Grasselli and V. Pata, Exponential Attractors for a conserved phase-field system with memory, Phys. D, 189 (2004), 31-48.   Google Scholar

[13]

S. GattiA. MiranvilleV. Pata and S. Zelik, Continuous families of exponential attractors for singularly perturbed equations with memory, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 329-366.   Google Scholar

[14]

C. GiorgiM. Grasselli and V. Pata, Uniform attractors for a phase-field model with memory and quadratic nonlinearity, Indiana Univ. Math. J., 48 (1999), 1395-1445.   Google Scholar

[15]

A. GiorginiM. Grasselli and A. Miranville, The Cahn-Hilliard-Oono equation with singular potential, Math. Models Methods Appl. Sci., 27 (2017), 2485-2510.   Google Scholar

[16]

M. Grasselli and V. Pata, Attractors of phase-field systems with memory, in Mathematical methods and models in phase transitions, (eds. A. Miranville), Nova Science Publishers, 2005,157–175.  Google Scholar

[17]

M. GrasselliV. Pata and F. M. Vegni, Longterm dynamics of a conserved phase-field model with memory, Asymptot. Anal., 33 (2003), 261-320.   Google Scholar

[18]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126.   Google Scholar

[19]

A. Haraux, Systémes Dynamiques Dissipatifs et Applications, Recherches en Mathématiques Appliquées, Vol. 17, Masson, Paris, 1991.  Google Scholar

[20]

A. Miranville, Asymptotic behavior of the Cahn-Hilliard-Oono equation, J. Appl. Anal. Comput., 1 (2011), 523-536.   Google Scholar

[21]

A. Miranville, On the conserved phase-field system, J. Math. Anal. Appl., 400 (2013), 143-152.   Google Scholar

[22]

A. Miranville, The Cahn-Hilliard equation and some of its variants, AIMS Math., 2 (2017), 479-544.   Google Scholar

[23]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582.   Google Scholar

[24]

Y. Oono and S. Puri, Computationally efficient modeling of ordering of quenched phases, Phys. Rev. Lett., 58 (1987), 836-839.   Google Scholar

[25]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486.   Google Scholar

[26]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar

[27]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997.  Google Scholar

[28]

S. Villain-Guillot, Phases Modulées et Dynamique de Cahn-Hilliard, Habilitation thesis, Université Bordeaux I, 2010. Google Scholar

show all references

References:
[1]

D. Brochet, Maximal attractor and inertial sets for some second and fourth order phase field models, in Pitman Res. Notes Math. Ser. , Longman Sci. Tech., Harlow, 296 (1993), 77–85.  Google Scholar

[2]

D. BrochetD. Hilhorst and A. Novick-Cohen, Maximal attractor and inertial sets for a conserved phase field model, Adv. Diff. Eqns., 1 (1996), 547-578.   Google Scholar

[3]

G. Caginalp, A conserved phase field system; implications for kinetic undercooling, Phys. Rev. B, 38 (1988), 789-791.   Google Scholar

[4]

G. Caginalp, The dynamics of a conserved phase-field system: Stefan-like, Hele-Shaw and Cahn-Hilliard models as asymptotic limits, IMA J. Appl. Math., 44 (1990), 77-94.   Google Scholar

[5]

J. W. Cahn, On spinodal decomposition, Acta Metall., 9 (1961), 795-801.   Google Scholar

[6]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.   Google Scholar

[7]

B. D. Coleman and M. E. Gurtin, Equipresence and constitutive equations for rigid heat conductors, Z. Angew. Math. Phys., 18 (1967), 199-208.   Google Scholar

[8]

M. Conti and A. Giorgini, The three-dimensional Cahn-Hilliard-Brinkman system with unmatched viscosities, preprint. Google Scholar

[9]

M. ContiV. Pata and M. Squassina, Singular limit of differential systems with memory, Indiana Univ. Math. J., 55 (2006), 169-215.   Google Scholar

[10]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.   Google Scholar

[11]

C. G. GalA. Giorgini and M. Grasselli, The nonlocal Cahn-Hilliard equation with singular potential: Well-posedness, regularity and strict separation property, J. Differential Equations, 263 (2017), 5253-5297.   Google Scholar

[12]

S. GattiM. Grasselli and V. Pata, Exponential Attractors for a conserved phase-field system with memory, Phys. D, 189 (2004), 31-48.   Google Scholar

[13]

S. GattiA. MiranvilleV. Pata and S. Zelik, Continuous families of exponential attractors for singularly perturbed equations with memory, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 329-366.   Google Scholar

[14]

C. GiorgiM. Grasselli and V. Pata, Uniform attractors for a phase-field model with memory and quadratic nonlinearity, Indiana Univ. Math. J., 48 (1999), 1395-1445.   Google Scholar

[15]

A. GiorginiM. Grasselli and A. Miranville, The Cahn-Hilliard-Oono equation with singular potential, Math. Models Methods Appl. Sci., 27 (2017), 2485-2510.   Google Scholar

[16]

M. Grasselli and V. Pata, Attractors of phase-field systems with memory, in Mathematical methods and models in phase transitions, (eds. A. Miranville), Nova Science Publishers, 2005,157–175.  Google Scholar

[17]

M. GrasselliV. Pata and F. M. Vegni, Longterm dynamics of a conserved phase-field model with memory, Asymptot. Anal., 33 (2003), 261-320.   Google Scholar

[18]

M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal., 31 (1968), 113-126.   Google Scholar

[19]

A. Haraux, Systémes Dynamiques Dissipatifs et Applications, Recherches en Mathématiques Appliquées, Vol. 17, Masson, Paris, 1991.  Google Scholar

[20]

A. Miranville, Asymptotic behavior of the Cahn-Hilliard-Oono equation, J. Appl. Anal. Comput., 1 (2011), 523-536.   Google Scholar

[21]

A. Miranville, On the conserved phase-field system, J. Math. Anal. Appl., 400 (2013), 143-152.   Google Scholar

[22]

A. Miranville, The Cahn-Hilliard equation and some of its variants, AIMS Math., 2 (2017), 479-544.   Google Scholar

[23]

A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582.   Google Scholar

[24]

Y. Oono and S. Puri, Computationally efficient modeling of ordering of quenched phases, Phys. Rev. Lett., 58 (1987), 836-839.   Google Scholar

[25]

V. Pata and S. Zelik, A result on the existence of global attractors for semigroups of closed operators, Commun. Pure Appl. Anal., 6 (2007), 481-486.   Google Scholar

[26]

V. Pata and A. Zucchi, Attractors for a damped hyperbolic equation with linear memory, Adv. Math. Sci. Appl., 11 (2001), 505-529.   Google Scholar

[27]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Applied Mathematical Sciences, Vol. 68, Springer-Verlag, New York, 1997.  Google Scholar

[28]

S. Villain-Guillot, Phases Modulées et Dynamique de Cahn-Hilliard, Habilitation thesis, Université Bordeaux I, 2010. Google Scholar

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