Convergence of Cahn–Hilliard systems to the Stefan problem with dynamic boundary conditions

Article type: Research Article

Authors: Fukao, Takeshi

Affiliations: Department of Mathematics, Faculty of Education, Kyoto University of Education, 1 Fujinomori, Fukakusa, Fushimi-ku, Kyoto 612-8522, Japan. E-mail: fukao@kyokyo-u.ac.jp

Abstract: This paper examines the well-posedness of the Stefan problem with a dynamic boundary condition. To show the existence of the weak solution, the original problem is approximated by the limit of an equation and a dynamic boundary condition of Cahn–Hilliard-type. Using this Cahn–Hilliard approach, the enthalpy formulation of the Stefan problem is characterized by the asymptotic limit of a fourth-order system that has a double-well structure. The main result obtained for the Stefan problem can also be applied to a wider class of degenerate parabolic equations by setting the nonlinearity to give the general maximal monotone graph.

Keywords: Stefan problem, dynamic boundary condition, weak solution, Cahn–Hilliard system

DOI: 10.3233/ASY-161373

Journal: Asymptotic Analysis, vol. 99, no. 1-2, pp. 1-21, 2016