DEPARTMENT OF MATHEMATICAL SCIENCES

APPLIED AND COMPUTATIONAL MATHEMATICS SEMINAR

**Speaker:** Peter Bates, Mathematics, Michigan State University

**Title: ***
The motion of particles driven by Allen-Cahn dynamics on the boundary of a smooth domain
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**Abstract:**
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. Questions concerning the dynamics of particles in the interior of the domain or driven by Cahn-Hilliard or other evolution laws may possibly be addressed by this approach.

**Time:** Friday, April 17, 2009, 1:30-2:30 p.m.

**Place:** Science and Tech I, Room 242

Department of Mathematical Sciences

George Mason University

4400 University Drive, MS 3F2

Fairfax, VA 22030-4444

http://math.gmu.edu/

Tel. 703-993-1460, Fax. 703-993-1491