Affiliations: [a] 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 | [b] School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Han Dan Road 220, Shanghai 200433, China | [c] Key Laboratory of Mathematics for Nonlinear Science (Fudan University), Ministry of Education, Han Dan Road 220, Shanghai 200433, China
Abstract: We consider a class of Cahn–Hilliard equation that models phase separation process of binary mixtures involving nontrivial boundary interactions in a bounded domain with non-permeable wall. The system is characterized by certain dynamic type boundary conditions and the total mass, in the bulk and on the boundary, is conserved for all time. For the case with physically relevant singular (e.g., logarithmic) potential, global regularity of weak solutions is established. In particular, when the spatial dimension is two, we show the instantaneous strict separation property such that for arbitrary positive time any weak solution stays away from the pure phases ±1, while in the three dimensional case, an eventual separation property for large time is obtained. As a consequence, we prove that every global weak solution converges to a single equilibrium as t→∞, by the usage of an extended Łojasiewicz–Simon inequality.
Keywords: Cahn–Hilliard equation, dynamic boundary condition, singular potential, separation from pure states, convergence to equilibrium
