# American Institute of Mathematical Sciences

## Cahn-Hilliard equation with capillarity in actual deforming configurations

 1 Mathematical Institute, Math.-Phys. Faculty, Charles University, Sokolovská 83, CZ-186 75 Praha 8, Czech Republic 2 Institute of Thermomechanics,Czech Academy of Sciences, Dolejškova 5, CZ-182 00 Praha 8, Czech Republic

Dedicated to Alexander Mielke on the occasion of his sixtieth birthday.

Received  March 2019 Revised  August 2019 Published  March 2020

Fund Project: * This research has been supported from the grants 17-04301S (regarding the focus on the dissipative evolution of internal variables), 19-04956S (regarding the focus on the dynamic and nonlinear behaviour), and 19-29646L (especially regarding the focus on the large strains in materials science) of Czech Science Foundation, and from the FWF grant I 4052 N3 with BMBWF through the OeAD-WTZ project CZ04/2019, and also from the institutional support RVO: 61388998 (ČR).

The diffusion driven by the gradient of the chemical potential (by the Fick/Darcy law) in deforming continua at large strains is formulated in the reference configuration with both the Fick/Darcy law and the capillarity (i.e. concentration gradient) term considered at the actual configurations deforming in time. Static situations are analysed by the direct method. Evolution (dynamical) problems are treated by the Faedo-Galerkin method, the actual capillarity giving rise to various new terms as e.g. the Korteweg-like stress and analytical difficulties related to them. Some other models (namely plasticity at small elastic strains or damage) with gradients at an actual configuration allow for similar models and analysis.

Citation: Tomáš Roubíček. Cahn-Hilliard equation with capillarity in actual deforming configurations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020303
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