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Invariant Manifolds for PDEs and the motion of particles driven by Allen-Cahn dynamics.
by
Peter Bates
Michigan State University
Coauthors: Kening Lu and Chongchun Zeng
We consider particles described as peak-like solutions to a singularly perturbed nonlinear parabolic partial differential equation. Minimal energy stationary states were shown to exist by Ni and Takagi in a series of papers, where detailed qualitative properties of these states were also derived. Taking the gradient flow of the energy functional leads to a nonlinear parabolic equation and it is natural to ask about the motion of particles as dynamic peak-like solutions away from equilibrium. By proving an abstract theorem about the existence of a true invariant manifold in the neighborhood of an approximately invariant, approximately normally hyperbolic invariant manifold, we are able to answer this question, giving the global dynamics of a particle on the boundary of a smooth domain.
Date received: April 1, 2009
Copyright © 2009 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cayt-34.