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Interface motion by interface diffusion driven by bulk energy: justification of a diffusive interface model

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Abstract

We construct an asymptotic solution of a system consisting of the partial differential equations of linear elasticity theory coupled with a degenerate parabolic equation, and show that when a regularity parameter tends to zero, this solution converges to a solution of a sharp interface model, which describes the phase interface in an elastically deformable solid moving by interface diffusion. Therefore, the coupled system can be used as diffusive interface model. Differently from diffusive interface models based on the Cahn–Hilliard equation, the interface diffusion is solely driven by the bulk energy, hence the Laplacian of the curvature is not part of the driving force. Also, no rescaling of the parabolic equation is necessary. Since the asymptotic solution does not solve the system exactly, the proof is formal.

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289, Darmstadt, Germany

    Hans-Dieter Alber

  2. Basque Center for Applied Mathematics (BCAM), Building 500, Bizkaia Technology Park, 48160, Derio, Spain

    Peicheng Zhu

  3. IKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain

    Peicheng Zhu

Authors
  1. Hans-Dieter Alber
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  2. Peicheng Zhu
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Correspondence to Hans-Dieter Alber.

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Communicated by Prof. Epifanio Virga.

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Alber, HD., Zhu, P. Interface motion by interface diffusion driven by bulk energy: justification of a diffusive interface model. Continuum Mech. Thermodyn. 23, 139–176 (2011). https://doi.org/10.1007/s00161-010-0162-9

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  • DOI: https://doi.org/10.1007/s00161-010-0162-9

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