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Limiting Problems for a Nonstandard Viscous Cahn–Hilliard System with Dynamic Boundary Conditions

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Trends in Applications of Mathematics to Mechanics

Part of the book series: Springer INdAM Series ((SINDAMS,volume 27))

Abstract

This note is concerned with a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by boundary and initial conditions. The system arises from a model of two-species phase segregation on an atomic lattice and was introduced by Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105–118. The two unknowns are the phase parameter and the chemical potential. In contrast to previous investigations about this PDE system, we consider here a dynamic boundary condition for the phase variable that involves the Laplace-Beltrami operator and models an additional nonconserving phase transition occurring on the surface of the domain. We are interested in some asymptotic analysis and first discuss the asymptotic limit of the system as the viscosity coefficient of the order parameter equation tends to 0: the convergence of solutions to the corresponding solutions for the limit problem is proven. Then, we study the long-time behavior of the system for both problems, with positive or zero viscosity coefficient, and characterize the omega-limit set in both cases.

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Acknowledgements

PC and GG gratefully acknowledge some financial support from the MIUR-PRIN Grant 2015PA5MP7 “Calculus of Variations”, the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica) and the IMATI—C.N.R. Pavia.

Author information

Authors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università di Pavia, Pavia, Italy

    Pierluigi Colli & Gianni Gilardi

  2. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Jürgen Sprekels

  3. Department of Mathematics, Humboldt-Universität zu Berlin, Berlin, Germany

    Jürgen Sprekels

Authors
  1. Pierluigi Colli
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  2. Gianni Gilardi
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  3. Jürgen Sprekels
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Corresponding author

Correspondence to Pierluigi Colli .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, Pavia, Pavia, Italy

    Elisabetta Rocca

  2. Fakultät für Mathematik, Universität Wien, Wien, Austria

    Ulisse Stefanelli

  3. CNRS, Paris, Paris, France

    Lev Truskinovsky

  4. Dipartimento di Matematica, Università degli Studi di Trento, Povo di Trento, Trento, Italy

    Augusto Visintin

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Colli, P., Gilardi, G., Sprekels, J. (2018). Limiting Problems for a Nonstandard Viscous Cahn–Hilliard System with Dynamic Boundary Conditions. In: Rocca, E., Stefanelli, U., Truskinovsky, L., Visintin, A. (eds) Trends in Applications of Mathematics to Mechanics. Springer INdAM Series, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-319-75940-1_11

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