# American Institute of Mathematical Sciences

June  2014, 3(2): 257-275. doi: 10.3934/eect.2014.3.257

## A vanishing diffusion limit in a nonstandard system of phase field equations

 1 Dipartimento di Matematica "F. Casorati", Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy 2 Institute of Mathematics, Czech Academy of Sciences, Žitná 25, CZ-11567 Praha 1 3 Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin

Received  January 2013 Revised  February 2014 Published  May 2014

We are concerned with a nonstandard phase field model of Cahn-Hilliard type. The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. It has been recently investigated by Colli, Gilardi, Podio-Guidugli, and Sprekels in a series of papers: see, in particular, SIAM J. Appl. Math. 2011 and Boll. Unione Mat. Ital. 2012. In the latter contribution, the authors can treat the very general case in which the diffusivity coefficient of the parabolic PDE is allowed to depend nonlinearly on both variables. In the same framework, this paper investigates the asymptotic limit of the solutions to the initial-boundary value problems as the diffusion coefficient $\sigma$ in the equation governing the evolution of the order parameter tends to zero. We prove that such a limit actually exists and solves the limit problem, which couples a nonlinear PDE of parabolic type with an ODE accounting for the phase dynamics. In the case of a constant diffusivity, we are able to show uniqueness and to improve the regularity of the solution.
Citation: Pierluigi Colli, Gianni Gilardi, Pavel Krejčí, Jürgen Sprekels. A vanishing diffusion limit in a nonstandard system of phase field equations. Evolution Equations & Control Theory, 2014, 3 (2) : 257-275. doi: 10.3934/eect.2014.3.257
##### References:
 [1] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Editura Academiei Republicii Socialiste România, (1976). Google Scholar [2] H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes De Contractions Dans les Espaces de Hilbert,, (French) North-Holland Mathematics Studies, (1973). Google Scholar [3] P. Colli, G. Gilardi, P. Krejčí and J. Sprekels, A continuous dependence result for a nonstandard system of phase field equations,, to appear in Math. Methods Appl. Sci., (2014). doi: 10.1002/mma.2892. Google Scholar [4] P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Well-posedness and long-time behaviour for a nonstandard viscous Cahn-Hilliard system,, SIAM J. Appl. Math., 71 (2011), 1849. doi: 10.1137/110828526. Google Scholar [5] P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity,, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 353. Google Scholar [6] P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity,, J. Differential Equations, 254 (2013), 4217. doi: 10.1016/j.jde.2013.02.014. Google Scholar [7] P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Global existence for a strongly coupled Cahn-Hilliard system with viscosity,, Boll. Unione Mat. Ital. (9), 5 (2012), 495. Google Scholar [8] P. Colli and J. Sprekels, On a Penrose-Fife model with zero interfacial energy leading to a phase-field system of relaxed Stefan type,, Ann. Mat. Pura Appl. (4), 169 (1995), 269. doi: 10.1007/BF01759357. Google Scholar [9] P. Colli and J. Sprekels, Global solution to the Penrose-Fife phase-field model with zero interfacial energy and Fourier law,, Adv. Math. Sci. Appl., 9 (1999), 383. Google Scholar [10] G. Gilardi, P. Krejčí and J. Sprekels, Hysteresis in phase-field models with thermal memory,, Math. Methods Appl. Sci., 23 (2000), 909. doi: 10.1002/1099-1476(20000710)23:10<909::AID-MMA142>3.0.CO;2-E. Google Scholar [11] P. Krejčí and J. Sprekels, Hysteresis operators in phase-field models of Penrose-Fife type,, Appl. Math., 43 (1998), 207. doi: 10.1023/A:1023276524286. Google Scholar [12] P. Krejčí and J. Sprekels, A hysteresis approach to phase-field models,, Nonlinear Anal., 39 (2000), 569. doi: 10.1016/S0362-546X(98)00222-3. Google Scholar [13] P. Krejčí and J. Sprekels, Phase-field models with hysteresis,, J. Math. Anal. Appl., 252 (2000), 198. doi: 10.1006/jmaa.2000.6974. Google Scholar [14] P. Krejčí, J. Sprekels and S. Zheng, Asymptotic behaviour for a phase-field system with hysteresis,, J. Differential Equations, 175 (2001), 88. doi: 10.1006/jdeq.2001.3950. Google Scholar [15] P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice,, Ric. Mat., 55 (2006), 105. doi: 10.1007/s11587-006-0008-8. Google Scholar [16] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura. Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar

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##### References:
 [1] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces,, Editura Academiei Republicii Socialiste România, (1976). Google Scholar [2] H. Brezis, Opérateurs Maximaux Monotones et Semi-Groupes De Contractions Dans les Espaces de Hilbert,, (French) North-Holland Mathematics Studies, (1973). Google Scholar [3] P. Colli, G. Gilardi, P. Krejčí and J. Sprekels, A continuous dependence result for a nonstandard system of phase field equations,, to appear in Math. Methods Appl. Sci., (2014). doi: 10.1002/mma.2892. Google Scholar [4] P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Well-posedness and long-time behaviour for a nonstandard viscous Cahn-Hilliard system,, SIAM J. Appl. Math., 71 (2011), 1849. doi: 10.1137/110828526. Google Scholar [5] P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, An asymptotic analysis for a nonstandard Cahn-Hilliard system with viscosity,, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 353. Google Scholar [6] P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Global existence and uniqueness for a singular/degenerate Cahn-Hilliard system with viscosity,, J. Differential Equations, 254 (2013), 4217. doi: 10.1016/j.jde.2013.02.014. Google Scholar [7] P. Colli, G. Gilardi, P. Podio-Guidugli and J. Sprekels, Global existence for a strongly coupled Cahn-Hilliard system with viscosity,, Boll. Unione Mat. Ital. (9), 5 (2012), 495. Google Scholar [8] P. Colli and J. Sprekels, On a Penrose-Fife model with zero interfacial energy leading to a phase-field system of relaxed Stefan type,, Ann. Mat. Pura Appl. (4), 169 (1995), 269. doi: 10.1007/BF01759357. Google Scholar [9] P. Colli and J. Sprekels, Global solution to the Penrose-Fife phase-field model with zero interfacial energy and Fourier law,, Adv. Math. Sci. Appl., 9 (1999), 383. Google Scholar [10] G. Gilardi, P. Krejčí and J. Sprekels, Hysteresis in phase-field models with thermal memory,, Math. Methods Appl. Sci., 23 (2000), 909. doi: 10.1002/1099-1476(20000710)23:10<909::AID-MMA142>3.0.CO;2-E. Google Scholar [11] P. Krejčí and J. Sprekels, Hysteresis operators in phase-field models of Penrose-Fife type,, Appl. Math., 43 (1998), 207. doi: 10.1023/A:1023276524286. Google Scholar [12] P. Krejčí and J. Sprekels, A hysteresis approach to phase-field models,, Nonlinear Anal., 39 (2000), 569. doi: 10.1016/S0362-546X(98)00222-3. Google Scholar [13] P. Krejčí and J. Sprekels, Phase-field models with hysteresis,, J. Math. Anal. Appl., 252 (2000), 198. doi: 10.1006/jmaa.2000.6974. Google Scholar [14] P. Krejčí, J. Sprekels and S. Zheng, Asymptotic behaviour for a phase-field system with hysteresis,, J. Differential Equations, 175 (2001), 88. doi: 10.1006/jdeq.2001.3950. Google Scholar [15] P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice,, Ric. Mat., 55 (2006), 105. doi: 10.1007/s11587-006-0008-8. Google Scholar [16] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura. Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar
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