Motion of a closed curve by minus the
surface Laplacian of curvature
Differential and Integral Equations,
vol. 13, no. 10-12, Oct.-Dec. 2000, pp. 1583-1594
Sergio A. Alvarez and Chun Liu
Center for Nonlinear Analysis and
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213-3890
Abstract
The phenomenon of surface diffusion is of interest in a variety
of physical situations, see e.g. Cahn-Taylor, Acta Metallurgica
et Materialia, 42 (1994), 1045-1063.
Surface diffusion is modelled by a fourth-order quasilinear parabolic
partial differential equation associated with the negative of the
surface Laplacian of curvature operator.
We address the well-posedness of the corresponding initial value problem
in the case in which the interface is a smooth closed curve Gamma
contained in a tubular neighborhood of a fixed simple closed curve
Gamma0 in the plane.
We prove existence and uniqueness, as well as analytic dependence
on the initial data of classical solutions of this problem
locally in time,
in the spaces Eh of functions f whose Fourier transform
(f^k), {k in Z} decays faster than |k|-h, for
h > 5.
Our results are based on the machinery developed by Alvarez and Pego
which allows the application of the method of maximal regularity
of DaPrato-Grisvard, Lunardi, Angenent, and others
in the spaces Eh.
AMS Subject Classifications: 35K22, 35K30, 35K55, 35Q72.