Motion of a closed curve by minus the surface Laplacian of curvature (Q5933768)

This paper is related to the phenomenon of surface diffusion modelled by a quasilinear parabolic partial differential equation associated with the negative of the surface Laplacian curvature operator. The authors study the following problem: Let \(\Gamma_0\) be a closed curve in the plane, NEWLINE\[NEWLINEV_n=-\triangle_{\Gamma (t)}\kappa (\Gamma (t)) \text{ for } t\in (0,T), \quad \Gamma (0)=\Gamma_0, \tag{1}NEWLINE\]NEWLINE where \(V_n\) is the normal velocity of the family of closed curves \(\{\Gamma (t)\}\) at time \(t,\) \(\triangle_{\Gamma (t)}\) is the Laplacian operator relative to the arc-length metric on \(\Gamma (t),\) and \(\kappa(\Gamma(t))\) is the curvature of \(\Gamma (t).\)NEWLINENEWLINENEWLINEHere \(\Gamma (t)\) is the interface parametrized by a function \(d(t)\in E^h,\) where \(E^h\), \(h>0\) is the space of distributions \(f: S^1\rightarrow \mathbb R,\) whose Fourier transform \(\widehat{f}:\mathbb Z \rightarrow \mathbb R,\) satisfies \(\widehat{f}_k =o (|k|^{-h})\), \(k\rightarrow\infty,\) equipped with the Banach norm \( \|f\|_{E^h} = \sup \{ (1 + |K|^h)|\widehat{f}_k|\), \(k\in\mathbb Z\}\).NEWLINENEWLINENEWLINEThe main result of the paper is the following:NEWLINENEWLINENEWLINETheorem 1.1. Let \(h >5.\) Assume that the curve parametrized by \(d =0\) is smooth. Then there exist \(R >0\) and \(T>0,\) such that if \(d_0\) is an element of \(E^h\) having \(E^h\) norm less than \(R,\) then there is a map \(d\in C([0, T], E^h)\) solving the initial-value problem given above in equation \((1),\) with \(\Gamma (t) =\Gamma (d(t))\) and \(\Gamma_0 =\Gamma (d_0)\). Furthermore the map \(d_0\mapsto d\) is analytic from the space \(E^h\) to the space \( C([0, T], E^h)\).