Stability of Curved Interfaces in the Perturbed Two-Dimensional Allen-Cahn System
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Date
2009Author
Iron, David
Kolokolonikov, Theodore
Rumsey, John
Wei, Juncheng
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We consider the singular limit of a perturbed Allen-Cahn model on a bounded two-dimensional domain: $\left\{\begin{array}{@{}ll@{}} u_t = \varepsilon^2 \Delta u - 2 (u - \varepsilon a) (u^2 - 1), & x \in \Omega \subset \mathbb{R}^2 \ \partial_n u = 0, & x \in \partial \Omega \end{array} \right.$ where $\varepsilon$ is a small parameter and $a$ is an $O(1)$ quantity. We study equilibrium solutions that have the form of a curved interface. Using singular perturbation techniques, we fully characterize the stability of such an equilibrium in terms of a certain geometric eigenvalue problem, and give a simple geometric interpretation of our stability results. Full numerical computations of the time-dependent PDE as well as of the associated two-dimensional eigenvalue problem are shown to be in excellent agreement with the analytical predictions.
Citation
Iron, David, Theodore Kolokolonikov, John Rumsey, and Juncheng Wei. 2009. "Stability of Curved Interfaces in the Perturbed Two-Dimensional Allen-Cahn System." SIAM Journal on Applied Mathematics 69(5): 1228-16.