Abstract
In this paper, we derive the time-fractional Cahn-Hilliard equation from continuum mixture theory with a modification of Fick's law of diffusion. This model describes the process of phase separation with nonlocal memory effects. We analyze the existence, uniqueness, and regularity of weak solutions of the time-fractional Cahn-Hilliard equation. In this regard, we consider degenerating mobility functions and free energies of Landau, Flory--Huggins and double-obstacle type. We apply the Faedo-Galerkin method to the system, derive energy estimates, and use compactness theorems to pass to the limit in the discrete form. In order to compensate for the missing chain rule of fractional derivatives, we prove a fractional chain inequality for semiconvex functions. The work concludes with numerical simulations and a sensitivity analysis showing the influence of the fractional power. Here, we consider a convolution quadrature scheme for the time-fractional component, and use a mixed finite element method for the space discretization.
Original language | English |
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Pages (from-to) | 66-87 |
Number of pages | 22 |
Journal | Computers and Mathematics with Applications |
Volume | 108 |
Early online date | 14 Jan 2022 |
DOIs | |
Publication status | Published - 15 Feb 2022 |
Bibliographical note
Acknowledgements:The authors gratefully acknowledge the support from DFG through TUM IGSSE, GSC 81. MLR acknowledges support from the Laura Bassi Postdoctoral Fellowship (Technical University of Munich). MF and BW were partially funded by DFG, WO-671 11-1.
Keywords
- time-fractional PDE
- Cahn-Hilliard equation
- well-posedness
- weak solutions
- degenerate mobility
- fractional chain inequality