## Differential and Integral Equations

### Motion of a closed curve by minus the surface Laplacian of curvature

#### Abstract

The phenomenon of surface diffusion is of interest in a variety of physical situations [8]. 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 $\Gamma_0$ 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 $E^h$ of functions $f$ whose Fourier transform $(\hat f_k)_{k \in \bf Z}$ decays faster than $|k|^{-h}$, for $h > 5$. Our results are based on the machinery developed in [1], [2], and [3], which allows the application of the method of maximal regularity ([11], [14], and [4]) in the spaces $E^h$.

#### Article information

Source
Differential Integral Equations, Volume 13, Number 10-12 (2000), 1583-1594.

Dates
First available in Project Euclid: 21 December 2012

https://projecteuclid.org/euclid.die/1356061141

Mathematical Reviews number (MathSciNet)
MR1787083

Zentralblatt MATH identifier
0974.35053

Subjects
Primary: 35R35: Free boundary problems
Secondary: 35K55: Nonlinear parabolic equations

#### Citation

Alvarez, Sergio A.; Liu, Chun. Motion of a closed curve by minus the surface Laplacian of curvature. Differential Integral Equations 13 (2000), no. 10-12, 1583--1594. https://projecteuclid.org/euclid.die/1356061141