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

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journal contribution
posted on 01.01.1997, 00:00 authored by Sergio A. Alvarez, Chun Liu
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 γ contained in a tubular neighborhood of a fixed simple closed curve γ₀ 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[superscript h] of functions f whose Fourier transform (f̂[subscript k])k[element of]Z decays faster than [absolute value of k][superscript -h], for h > 5. Our results are based on the machinery developed in [1], [2], [3], which allows the application of the method of maximal regularity [11], [14], [4] in the spaces E[superscript h]."


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