Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects

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June 2015, 35(6): 2565-2590. doi: 10.3934/dcds.2015.35.2565

Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects

Christian Heinemann ^1, and Christiane Kraus ^1,

1.	Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117 Berlin, Germany, Germany

Received December 2013 Revised February 2014 Published December 2014

Abstract
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In this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolic-parabolic system. To this end, we first adopt the notion of weak solutions introduced in [12]. Then we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques.

Keywords: weak solutions, phase separation, hyperbolic-parabolic systems, Cahn-Hilliard system, rate-dependent systems., linear elasticity, doubly nonlinear differential inclusions, existence results, energetic solutions.

Mathematics Subject Classification: 35L20, 35L51, 35K86, 35K55, 49J40, 49S05, 74A45, 74G25, 34A12, 82B26, 82C26, 35K92, 35K3.

Citation: Christian Heinemann, Christiane Kraus. Existence of weak solutions for a PDE system describing phase separation and damage processes including inertial effects. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2565-2590. doi: 10.3934/dcds.2015.35.2565

References:

[1]	L. Bartkowiak and I. Pawlow, The Cahn-Hilliard-Gurtin system coupled with elasticity, Control Cybernet., 34 (2005), 1005-1043.
[2]	J. Bernard, Density results in Sobolev spaces whose elements vanish on a part of the boundary, Chin. Ann. Math., Ser. B, 32 (2011), 823-846. doi: 10.1007/s11401-011-0682-z.
[3]	T. Böhme, W. Dreyer, F. Duderstadt and W. Müller, A Higher Gradient Theory of Mixtures for Multi-component Materials with Numerical Examples for Binary Alloys, WIAS-Preprint No. 1286, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, 2007.
[4]	T. Böhme, W. Dreyer and W. Müller, Determination of stiffness and higher gradient coefficients by means of the embedded atom method: An approach for binary alloys, Contin. Mech. Thermodyn., 18 (2007), 411-441. doi: 10.1007/s00161-006-0037-2.
[5]	E. Bonetti, P. Colli, W. Dreyer, G. Gilardi, G. Schimperna and J. Sprekels, On a model for phase separation in binary alloys driven by mechanical effects, Phys. D, 165 (2002), 48-65. doi: 10.1016/S0167-2789(02)00373-1.
[6]	E. Bonetti and G. Schimperna, Local existence for Frémond's model of damage in elastic materials, Contin. Mech. Thermodyn., 16 (2004), 319-335. doi: 10.1007/s00161-003-0152-2.
[7]	E. Bonetti, G. Schimperna and A. Segatti, On a doubly nonlinear model for the evolution of damaging in viscoelastic materials, J. Differential Equations, 218 (2005), 91-116. doi: 10.1016/j.jde.2005.04.015.
[8]	M. Carrive, A. Miranville and A. Piétrus, The Cahn-Hilliard equation for deformable elastic media, Adv. Math. Sci. Appl., 10 (2000), 539-569.
[9]	M. Frémond, Non-smooth Thermomechanics, Berlin: Springer, 2002. doi: 10.1007/978-3-662-04800-9.
[10]	M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual power, Int. J. Solids Structures, 33 (1996), 1083-1103. doi: 10.1016/0020-7683(95)00074-7.
[11]	H. Garcke, On Mathematical Models for Phase Separation in Elastically Stressed Solids, Habilitation thesis, University Bonn, 2000.
[12]	C. Heinemann and C. Kraus, Existence of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage, Adv. Math. Sci. Appl., 21 (2011), 321-359.
[13]	C. Heinemann and C. Kraus, A degenerating Cahn-Hilliard system coupled with complete damage processes, WIAS preprint no. 1759.
[14]	C. Heinemann and C. Kraus, Existence of weak solutions for a hyperbolic-parabolic phase field system with mixed boundary conditions on non-smooth domains, WIAS preprint no. 1890.
[15]	C. Heinemann and C. Kraus, Existence results for diffuse interface models describing phase separation and damage, European J. Appl. Math., 24 (2013), 179-211. doi: 10.1017/S095679251200037X.
[16]	D. Knees, R. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model, Math. Models Methods Appl. Sci., 23 (2013), 565-616. doi: 10.1142/S021820251250056X.
[17]	A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity, Math. Models Methods Appl. Sci., 16 (2006), 177-209. doi: 10.1142/S021820250600111X.
[18]	A. Mielke and M. Thomas, Damage of nonlinearly elastic materials at small strain - Existence and regularity results, ZAMM Z. Angew. Math. Mech, 90 (2010), 88-112. doi: 10.1002/zamm.200900243.
[19]	I. Pawlow and W. M. Zajączkowski, Measure-valued solutions of a heterogeneous Cahn-Hilliard system in elastic solids, Colloquium Mathematicum, 112 (2008), 313-334. doi: 10.4064/cm112-2-7.
[20]	E. Rocca and R. Rossi, A degenerating PDE system for phase transitions and damage, Math. Models Methods Appl. Sci., 24 (2014), 1265-1341. arXiv:1205.3578v1. doi: 10.1142/S021820251450002X.
[21]	G. Schimperna and I. Pawłow, A Cahn-Hilliard equation with singular diffusion, J. Differential Equations, 254 (2013), 779-803. doi: 10.1016/j.jde.2012.09.018.
[22]	G. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion, SIAM J. Math. Anal., 45 (2013), 31-63. doi: 10.1137/110835608.
[23]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1986), 65-96. doi: 10.1007/BF01762360.

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References:

[1]	L. Bartkowiak and I. Pawlow, The Cahn-Hilliard-Gurtin system coupled with elasticity, Control Cybernet., 34 (2005), 1005-1043.
[2]	J. Bernard, Density results in Sobolev spaces whose elements vanish on a part of the boundary, Chin. Ann. Math., Ser. B, 32 (2011), 823-846. doi: 10.1007/s11401-011-0682-z.
[3]	T. Böhme, W. Dreyer, F. Duderstadt and W. Müller, A Higher Gradient Theory of Mixtures for Multi-component Materials with Numerical Examples for Binary Alloys, WIAS-Preprint No. 1286, Weierstrass Institute for Applied Analysis and Stochastics, Berlin, 2007.
[4]	T. Böhme, W. Dreyer and W. Müller, Determination of stiffness and higher gradient coefficients by means of the embedded atom method: An approach for binary alloys, Contin. Mech. Thermodyn., 18 (2007), 411-441. doi: 10.1007/s00161-006-0037-2.
[5]	E. Bonetti, P. Colli, W. Dreyer, G. Gilardi, G. Schimperna and J. Sprekels, On a model for phase separation in binary alloys driven by mechanical effects, Phys. D, 165 (2002), 48-65. doi: 10.1016/S0167-2789(02)00373-1.
[6]	E. Bonetti and G. Schimperna, Local existence for Frémond's model of damage in elastic materials, Contin. Mech. Thermodyn., 16 (2004), 319-335. doi: 10.1007/s00161-003-0152-2.
[7]	E. Bonetti, G. Schimperna and A. Segatti, On a doubly nonlinear model for the evolution of damaging in viscoelastic materials, J. Differential Equations, 218 (2005), 91-116. doi: 10.1016/j.jde.2005.04.015.
[8]	M. Carrive, A. Miranville and A. Piétrus, The Cahn-Hilliard equation for deformable elastic media, Adv. Math. Sci. Appl., 10 (2000), 539-569.
[9]	M. Frémond, Non-smooth Thermomechanics, Berlin: Springer, 2002. doi: 10.1007/978-3-662-04800-9.
[10]	M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual power, Int. J. Solids Structures, 33 (1996), 1083-1103. doi: 10.1016/0020-7683(95)00074-7.
[11]	H. Garcke, On Mathematical Models for Phase Separation in Elastically Stressed Solids, Habilitation thesis, University Bonn, 2000.
[12]	C. Heinemann and C. Kraus, Existence of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage, Adv. Math. Sci. Appl., 21 (2011), 321-359.
[13]	C. Heinemann and C. Kraus, A degenerating Cahn-Hilliard system coupled with complete damage processes, WIAS preprint no. 1759.
[14]	C. Heinemann and C. Kraus, Existence of weak solutions for a hyperbolic-parabolic phase field system with mixed boundary conditions on non-smooth domains, WIAS preprint no. 1890.
[15]	C. Heinemann and C. Kraus, Existence results for diffuse interface models describing phase separation and damage, European J. Appl. Math., 24 (2013), 179-211. doi: 10.1017/S095679251200037X.
[16]	D. Knees, R. Rossi and C. Zanini, A vanishing viscosity approach to a rate-independent damage model, Math. Models Methods Appl. Sci., 23 (2013), 565-616. doi: 10.1142/S021820251250056X.
[17]	A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity, Math. Models Methods Appl. Sci., 16 (2006), 177-209. doi: 10.1142/S021820250600111X.
[18]	A. Mielke and M. Thomas, Damage of nonlinearly elastic materials at small strain - Existence and regularity results, ZAMM Z. Angew. Math. Mech, 90 (2010), 88-112. doi: 10.1002/zamm.200900243.
[19]	I. Pawlow and W. M. Zajączkowski, Measure-valued solutions of a heterogeneous Cahn-Hilliard system in elastic solids, Colloquium Mathematicum, 112 (2008), 313-334. doi: 10.4064/cm112-2-7.
[20]	E. Rocca and R. Rossi, A degenerating PDE system for phase transitions and damage, Math. Models Methods Appl. Sci., 24 (2014), 1265-1341. arXiv:1205.3578v1. doi: 10.1142/S021820251450002X.
[21]	G. Schimperna and I. Pawłow, A Cahn-Hilliard equation with singular diffusion, J. Differential Equations, 254 (2013), 779-803. doi: 10.1016/j.jde.2012.09.018.
[22]	G. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion, SIAM J. Math. Anal., 45 (2013), 31-63. doi: 10.1137/110835608.
[23]	J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1986), 65-96. doi: 10.1007/BF01762360.

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