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 $E^h$.