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Existence results for diffuse interface models describing phase separation and damage*

Published online by Cambridge University Press:  09 November 2012

CHRISTIAN HEINEMANN  and
CHRISTIANE KRAUS
CHRISTIAN HEINEMANN
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117 Berlin, Germany emails: christian.heinemann@wias-berlin.de, christiane.kraus@wias-berlin.de
CHRISTIANE KRAUS
Affiliation:
Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstr. 39, 10117 Berlin, Germany emails: christian.heinemann@wias-berlin.de, christiane.kraus@wias-berlin.de
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Abstract

In this paper, we analytically investigate multi-component Cahn–Hilliard and Allen–Cahn systems which are coupled with elasticity and uni-directional damage processes. The free energy of the system is of the form ∫Ω½Γ∇c : ∇c + ½|∇z|2+Wch(c)+Wel(e,c,z)dx with a polynomial or logarithmic chemical energy density Wch, an inhomogeneous elastic energy density Wel and a quadratic structure of the gradient of damage variable z. For the corresponding elastic Cahn–Hilliard and Allen–Cahn systems coupled with uni-directional damage processes, we present an appropriate notion of weak solutions and prove existence results based on certain regularization methods and a higher integrability result for strain e.

Keywords

Cahn–Hilliard systemsAllen–Cahn systemsPhase separationDamageElliptic-parabolic systemsEnergetic solutionWeak solutionDoubly nonlinear differential inclusionsExistence resultsRate-dependent systemsLogarithmic-free energy
Type
Papers
Information
European Journal of Applied Mathematics , Volume 24 , Issue 2 , April 2013 , pp. 179 - 211
Copyright
Copyright © Cambridge University Press 2012

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Footnotes

*

This project is partially supported by the DFG project A 3 ‘Modeling and sharp interface limits of local and non-local generalized Navier–Stokes–Korteweg Systems.’

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