American Institute of Mathematical Sciences

Advanced
  • Previous Article
    On the representation of hysteresis operators acting on vector-valued, left-continuous and piecewise monotaffine and continuous functions
  • DCDS Home
  • This Issue
  • Next Article
    Robust exponential attractors for the modified phase-field crystal equation
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

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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[2]

J. Bernard, Density results in Sobolev spaces whose elements vanish on a part of the boundary,, Chin. Ann. Math., 32 (2011), 823.  doi: 10.1007/s11401-011-0682-z.  Google Scholar

[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, (1286).   Google Scholar

[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.  doi: 10.1007/s00161-006-0037-2.  Google Scholar

[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.  doi: 10.1016/S0167-2789(02)00373-1.  Google Scholar

[6]

E. Bonetti and G. Schimperna, Local existence for Frémond's model of damage in elastic materials,, Contin. Mech. Thermodyn., 16 (2004), 319.  doi: 10.1007/s00161-003-0152-2.  Google Scholar

[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.  doi: 10.1016/j.jde.2005.04.015.  Google Scholar

[8]

M. Carrive, A. Miranville and A. Piétrus, The Cahn-Hilliard equation for deformable elastic media,, Adv. Math. Sci. Appl., 10 (2000), 539.   Google Scholar

[9]

M. Frémond, Non-smooth Thermomechanics,, Berlin: Springer, (2002).  doi: 10.1007/978-3-662-04800-9.  Google Scholar

[10]

M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual power,, Int. J. Solids Structures, 33 (1996), 1083.  doi: 10.1016/0020-7683(95)00074-7.  Google Scholar

[11]

H. Garcke, On Mathematical Models for Phase Separation in Elastically Stressed Solids,, Habilitation thesis, (2000).   Google Scholar

[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.   Google Scholar

[13]

C. Heinemann and C. Kraus, A degenerating Cahn-Hilliard system coupled with complete damage processes,, WIAS preprint no. 1759., (1759).   Google Scholar

[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., (1890).   Google Scholar

[15]

C. Heinemann and C. Kraus, Existence results for diffuse interface models describing phase separation and damage,, European J. Appl. Math., 24 (2013), 179.  doi: 10.1017/S095679251200037X.  Google Scholar

[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.  doi: 10.1142/S021820251250056X.  Google Scholar

[17]

A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity,, Math. Models Methods Appl. Sci., 16 (2006), 177.  doi: 10.1142/S021820250600111X.  Google Scholar

[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.  doi: 10.1002/zamm.200900243.  Google Scholar

[19]

I. Pawlow and W. M. Zajączkowski, Measure-valued solutions of a heterogeneous Cahn-Hilliard system in elastic solids,, Colloquium Mathematicum, 112 (2008), 313.  doi: 10.4064/cm112-2-7.  Google Scholar

[20]

E. Rocca and R. Rossi, A degenerating PDE system for phase transitions and damage,, Math. Models Methods Appl. Sci., 24 (2014), 1265.  doi: 10.1142/S021820251450002X.  Google Scholar

[21]

G. Schimperna and I. Pawłow, A Cahn-Hilliard equation with singular diffusion,, J. Differential Equations, 254 (2013), 779.  doi: 10.1016/j.jde.2012.09.018.  Google Scholar

[22]

G. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion,, SIAM J. Math. Anal., 45 (2013), 31.  doi: 10.1137/110835608.  Google Scholar

[23]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1986), 65.  doi: 10.1007/BF01762360.  Google Scholar

show all references

References:
[1]

L. Bartkowiak and I. Pawlow, The Cahn-Hilliard-Gurtin system coupled with elasticity,, Control Cybernet., 34 (2005), 1005.   Google Scholar

[2]

J. Bernard, Density results in Sobolev spaces whose elements vanish on a part of the boundary,, Chin. Ann. Math., 32 (2011), 823.  doi: 10.1007/s11401-011-0682-z.  Google Scholar

[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, (1286).   Google Scholar

[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.  doi: 10.1007/s00161-006-0037-2.  Google Scholar

[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.  doi: 10.1016/S0167-2789(02)00373-1.  Google Scholar

[6]

E. Bonetti and G. Schimperna, Local existence for Frémond's model of damage in elastic materials,, Contin. Mech. Thermodyn., 16 (2004), 319.  doi: 10.1007/s00161-003-0152-2.  Google Scholar

[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.  doi: 10.1016/j.jde.2005.04.015.  Google Scholar

[8]

M. Carrive, A. Miranville and A. Piétrus, The Cahn-Hilliard equation for deformable elastic media,, Adv. Math. Sci. Appl., 10 (2000), 539.   Google Scholar

[9]

M. Frémond, Non-smooth Thermomechanics,, Berlin: Springer, (2002).  doi: 10.1007/978-3-662-04800-9.  Google Scholar

[10]

M. Frémond and B. Nedjar, Damage, gradient of damage and principle of virtual power,, Int. J. Solids Structures, 33 (1996), 1083.  doi: 10.1016/0020-7683(95)00074-7.  Google Scholar

[11]

H. Garcke, On Mathematical Models for Phase Separation in Elastically Stressed Solids,, Habilitation thesis, (2000).   Google Scholar

[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.   Google Scholar

[13]

C. Heinemann and C. Kraus, A degenerating Cahn-Hilliard system coupled with complete damage processes,, WIAS preprint no. 1759., (1759).   Google Scholar

[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., (1890).   Google Scholar

[15]

C. Heinemann and C. Kraus, Existence results for diffuse interface models describing phase separation and damage,, European J. Appl. Math., 24 (2013), 179.  doi: 10.1017/S095679251200037X.  Google Scholar

[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.  doi: 10.1142/S021820251250056X.  Google Scholar

[17]

A. Mielke and T. Roubíček, Rate-independent damage processes in nonlinear elasticity,, Math. Models Methods Appl. Sci., 16 (2006), 177.  doi: 10.1142/S021820250600111X.  Google Scholar

[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.  doi: 10.1002/zamm.200900243.  Google Scholar

[19]

I. Pawlow and W. M. Zajączkowski, Measure-valued solutions of a heterogeneous Cahn-Hilliard system in elastic solids,, Colloquium Mathematicum, 112 (2008), 313.  doi: 10.4064/cm112-2-7.  Google Scholar

[20]

E. Rocca and R. Rossi, A degenerating PDE system for phase transitions and damage,, Math. Models Methods Appl. Sci., 24 (2014), 1265.  doi: 10.1142/S021820251450002X.  Google Scholar

[21]

G. Schimperna and I. Pawłow, A Cahn-Hilliard equation with singular diffusion,, J. Differential Equations, 254 (2013), 779.  doi: 10.1016/j.jde.2012.09.018.  Google Scholar

[22]

G. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion,, SIAM J. Math. Anal., 45 (2013), 31.  doi: 10.1137/110835608.  Google Scholar

[23]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1986), 65.  doi: 10.1007/BF01762360.  Google Scholar

[1]

Tohru Nakamura, Shinya Nishibata, Naoto Usami. Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space. Kinetic & Related Models, 2018, 11 (4) : 757-793. doi: 10.3934/krm.2018031
[2]

Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027
[3]

Elena Bonetti, Pierluigi Colli, Luca Scarpa, Giuseppe Tomassetti. A doubly nonlinear Cahn-Hilliard system with nonlinear viscosity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1001-1022. doi: 10.3934/cpaa.2018049
[4]

Peter Howard, Bongsuk Kwon. Spectral analysis for transition front solutions in Cahn-Hilliard systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 125-166. doi: 10.3934/dcds.2012.32.125
[5]

Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630
[6]

Tian Ma, Shouhong Wang. Cahn-Hilliard equations and phase transition dynamics for binary systems. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 741-784. doi: 10.3934/dcdsb.2009.11.741
[7]

Irena Pawłow, Wojciech M. Zajączkowski. Regular weak solutions to 3-D Cahn-Hilliard system in elastic solids. Conference Publications, 2007, 2007 (Special) : 824-833. doi: 10.3934/proc.2007.2007.824
[8]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Phase transition and separation in compressible Cahn-Hilliard fluids. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 73-88. doi: 10.3934/dcdsb.2014.19.73
[9]

Kun Li, Jianhua Huang, Xiong Li. Traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2091-2119. doi: 10.3934/dcdsb.2018227
[10]

Alain Miranville, Giulio Schimperna. On a doubly nonlinear Cahn-Hilliard-Gurtin system. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 675-697. doi: 10.3934/dcdsb.2010.14.675
[11]

Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020325
[12]

Alberto Farina. Some symmetry results for entire solutions of an elliptic system arising in phase separation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2505-2511. doi: 10.3934/dcds.2014.34.2505
[13]

Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033
[14]

Pierluigi Colli, Gianni Gilardi, Danielle Hilhorst. On a Cahn-Hilliard type phase field system related to tumor growth. Discrete & Continuous Dynamical Systems - A, 2015, 35 (6) : 2423-2442. doi: 10.3934/dcds.2015.35.2423
[15]

Ming-Po Chen, Li-hong Huang. Existence of solutions in the future for a class of nonlinear differential systems. Conference Publications, 1998, 1998 (Special) : 138-147. doi: 10.3934/proc.1998.1998.138
[16]

Tohru Nakamura, Shinya Nishibata. Energy estimate for a linear symmetric hyperbolic-parabolic system in half line. Kinetic & Related Models, 2013, 6 (4) : 883-892. doi: 10.3934/krm.2013.6.883
[17]

Francesca R. Guarguaglini. Global solutions for a chemotaxis hyperbolic-parabolic system on networks with nonhomogeneous boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1057-1087. doi: 10.3934/cpaa.2020049
[18]

Riccarda Rossi, Giuseppe Savaré. A characterization of energetic and $BV$ solutions to one-dimensional rate-independent systems. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 167-191. doi: 10.3934/dcdss.2013.6.167
[19]

Yanni Zeng. LP decay for general hyperbolic-parabolic systems of balance laws. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 363-396. doi: 10.3934/dcds.2018018
[20]

Fuqin Sun, Mingxin Wang. Non-existence of global solutions for nonlinear strongly damped hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 949-958. doi: 10.3934/dcds.2005.12.949

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences