Abstract
In this paper, inspired from the study on denoising, segmentation and reconstruction in image processing, and combiningwith the theories of two phase flows, we introduce one class of initial-boundary value problem of the Cahn-Hilliardequation with nonlocal terms. Then, by using the Schauder fixed point theorem, we obtain the existence of weak solutionsto this initial boundary value problem for the nonlocal Cahn-Hilliard equation.
Highlights
In Betes, P.W. and Han, J., 2005, P.235-277, the author has investigated a class of long distance disturb term Cahn-Hilliard equation with nonlocal term, they proved the existence, uniqueness and the stability property of the solution by the energy methods
A.L., Esedoglu, S., & Gillette,2007, P. 285-291 propose the following initial boundary value problems based on Cahn-Hilliard equation
The existence and uniqueness of a solution to the initial boundary value problem(3) is investigated, just like the Gauss denoise used by Witkin, A.P., 1983, P.1019-1021. the edge is smoothed in the time of denoise such that the image loose the fidelity
Summary
In order to describes the complicated phase separation and coarsening phenomena in a melted alloy, Cahn J.W., Hilliard E., 1 958, P. 258-297 proposed the following initial-boundary problem:. 1986, P.339-357, have established the existence and uniqueness of the solution to this initial boundary value problem. Pego, R.L., 1989, P.261-278 has analyzed the asymptotic property of the solution in the case of → 0, and the author proved that the limit of the solution satisfy the Hele-Shaw initial problems with free edge conditions£ Alikakos, N.D., Bates, P.W. and Chen, X., 1994, P.165-205, Alikakos, N.D., Fusco, G., 1993, P.637-674, Chen, X., 1993, P.117-151, 1994, P.1371-1395, and Stoth, B., 1996, P.154-183. Qian, T., Wang, X., Sheng, P., 2003, P.1-15 propose to describe the two-phase fluid property by coupling Cahn-Hilliard equation and Navier-Stokes equation. In Betes, P.W. and Han, J., 2005, P.235-277, the author has investigated a class of long distance disturb term Cahn-Hilliard equation with nonlocal term, they proved the existence, uniqueness and the stability property of the solution by the energy methods
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