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On the longtime behavior of a viscous Cahn–Hilliard system with convection and dynamic boundary conditions

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Abstract

In this paper, we study the longtime asymptotic behavior of a phase separation process occurring in a three-dimensional domain containing a fluid flow of given velocity. This process is modeled by a viscous convective Cahn–Hilliard system, which consists of two nonlinearly coupled second-order partial differential equations for the unknown quantities, the chemical potential and an order parameter representing the scaled density of one of the phases. In contrast to other contributions, in which zero Neumann boundary conditions were assumed for both the chemical potential and the order parameter, we consider the case of dynamic boundary conditions, which model the situation when another phase transition takes place on the boundary. The phase transition processes in the bulk and on the boundary are driven by free energy functionals that may be nondifferentiable and have derivatives only in the sense of (possibly set-valued) subdifferentials. For the resulting initial-boundary value system of Cahn–Hilliard type, general well-posedness results have been established in a recent contribution by the same authors. In the present paper, we investigate the asymptotic behavior of the solutions as times approaches infinity. More precisely, we study the \(\omega \)-limit (in a suitable topology) of every solution trajectory. Under the assumptions that the viscosity coefficients are strictly positive and that at least one of the underlying free energies is differentiable, we prove that the \(\omega \)-limit is meaningful and that all of its elements are solutions to the corresponding stationary system, where the component representing the chemical potential is a constant.

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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. The authors declare that they have no conflict of interest.

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Authors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università di Pavia and Research Associate at the IMATI-C.N.R. Pavia, via Ferrata 5, 27100, Pavia, Italy

    Pierluigi Colli & Gianni Gilardi

  2. Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Jürgen Sprekels

  3. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117, Berlin, Germany

    Jürgen Sprekels

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  1. Pierluigi Colli
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  2. Gianni Gilardi
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  3. Jürgen Sprekels
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Correspondence to Pierluigi Colli.

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Dedicated to our friend Prof. Dr. Alexander Mielke on the occasion of his 60th birthday with best wishes.

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Colli, P., Gilardi, G. & Sprekels, J. On the longtime behavior of a viscous Cahn–Hilliard system with convection and dynamic boundary conditions. J Elliptic Parabol Equ 4, 327–347 (2018). https://doi.org/10.1007/s41808-018-0021-6

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  • DOI: https://doi.org/10.1007/s41808-018-0021-6

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