Passing from bulk to bulk/surface evolution in the Allen-Cahn equation
Please always quote using this URN:urn:nbn:de:0296-matheon-9643
- In this paper we formulate a boundary layer approximation for an Allen--Cahn-type equation involving a small parameter $\eps$. Here, $\eps$ is related to the thickness of the boundary layer and we are interested in the limit when $\eps$ tends to $0$ in order to derive nontrivial boundary conditions. The evolution of the system is written as an energy balance formulation of the L^2-gradient flow with the corresponding Allen--Cahn energy functional. By transforming the boundary layer to a fixed domain we show the convergence of the solutions to a solution of a limit system. This is done by using concepts related to Gamma- and Mosco convergence. By considering different scalings in the boundary layer we obtain different boundary conditions.
| Author: | Matthias Liero |
|---|---|
| URN: | urn:nbn:de:0296-matheon-9643 |
| Referee: | Frank Schmidt |
| Document Type: | Preprint, Research Center Matheon |
| Language: | English |
| Date of first Publication: | 2012/01/10 |
| Release Date: | 2012/01/10 |
| Tag: | Allen-Cahn equation; Dynamic boundary condition; Energy balance; Gamma-convergence; Gradient flow |
| Institute: | Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS) |
| MSC-Classification: | 35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Kxx Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] / 35K20 Initial-boundary value problems for second-order parabolic equations |
| 35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Kxx Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] / 35K55 Nonlinear parabolic equations | |
| 82-XX STATISTICAL MECHANICS, STRUCTURE OF MATTER / 82Cxx Time-dependent statistical mechanics (dynamic and nonequilibrium) / 82C26 Dynamic and nonequilibrium phase transitions (general) | |
| Preprint Number: | 849 |
