Layer Dynamics for the one dimensional $\eps$-dependent Cahn-Hilliard / Allen-Cahn Equation
Affiliation
University of Chester; University of Crete; IACM/FORTHPublication Date
2021-08-27
Metadata
Show full item recordAbstract
We study the dynamics of the one-dimensional ε-dependent Cahn-Hilliard / Allen-Cahn equation within a neighborhood of an equilibrium of N transition layers, that in general does not conserve mass. Two different settings are considered which differ in that, for the second, we impose a mass-conservation constraint in place of one of the zero-mass flux boundary conditions at x = 1. Motivated by the study of Carr and Pego on the layered metastable patterns of Allen-Cahn in [10], and by this of Bates and Xun in [5] for the Cahn-Hilliard equation, we implement an N-dimensional, and a mass-conservative N−1-dimensional manifold respectively; therein, a metastable state with N transition layers is approximated. We then determine, for both cases, the essential dynamics of the layers (ode systems with the equations of motion), expressed in terms of local coordinates relative to the manifold used. In particular, we estimate the spectrum of the linearized Cahn-Hilliard / Allen-Cahn operator, and specify wide families of ε-dependent weights δ(ε), µ(ε), acting at each part of the operator, for which the dynamics are stable and rest exponentially small in ε. Our analysis enlightens the role of mass conservation in the classification of the general mixed problem into two main categories where the solution has a profile close to Allen-Cahn, or, when the mass is conserved, close to the Cahn-Hilliard solution.Citation
Antonopoulou, D., Karali, G., & Tzirakis, K. (2021). Layer dynamics for the one dimensional $\eps$-dependent Cahn-Hilliard / Allen-Cahn Equation. Calculus of Variations and Partial Differential Equations, 60, 207. https://doi.org/10.1007/s00526-021-02085-4Publisher
SpringerAdditional Links
https://www.springer.com/journal/526https://link.springer.com/article/10.1007/s00526-021-02085-4
Type
ArticleDescription
The final publication is available at Springer via http://dx.doi.org/10.1007/s00526-021-02085-4ISSN
0944-2669EISSN
1432-0835ae974a485f413a2113503eed53cd6c53
10.1007/s00526-021-02085-4
Scopus Count
Collections
The following license files are associated with this item:
- Creative Commons
Except where otherwise noted, this item's license is described as Attribution-NonCommercial-NoDerivatives 4.0 International