Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials

The well-posedness of a system of partial differential equations and dynamic boundary conditions, both of Cahn-Hilliard type, is discussed. The existence of a weak solution and its continuous dependence on the data are proved using a suitable setting for the conservation of a total mass in the bulk plus the boundary. A very general class of double-well like potentials is allowed. Moreover, some further regularity is obtained to guarantee the strong solution

On a coupled bulk-surface Allen-Cahn system with an affine linear transmission condition and its approximation by a Robin boundary condition

We study a coupled bulk-surface Allen-Cahn system with an affine linear transmission condition, that is, the trace values of the bulk variable and the values of the surface variable are connected via an affine relation, and this serves to generalize the usual dynamic boundary conditions. We tackle the problem of well-posedness via a penalization method using Robin boundary conditions. In particular, for the relaxation problem, the strong well-posedness and long-time behavior of solutions can be shown for more general and possibly nonlinear relations. New difficulties arise since the surface variable is no longer the trace of the bulk variable, and uniform estimates in the relaxation parameter are scarce. Nevertheless, weak convergence to the original problem can be shown. Using the approach of Colli and Fukao (Math. Models Appl. Sci. 2015), we show strong existence to the original problem with affine linear relations, and derive an error estimate between solutions to the relaxed and original problems.Comment: 34 page

Separation property and convergence to equilibrium for the equation and dynamic boundary condition of Cahn-Hilliard type with singular potential

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 and -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 the time goes to infinity, by the usage of an extended Lojasiewicz-Simon inequality.Comment: 34 page

Heat equation on the hypergraph containing vertices with given data

This paper is concerned with the Cauchy problem of a multivalued ordinary differential equation governed by the hypergraph Laplacian, which describes the diffusion of ``heat'' or ``particles'' on the vertices of hypergraph. We consider the case where the heat on several vertices are manipulated internally by the observer, namely, are fixed by some given functions. This situation can be reduced to a nonlinear evolution equation associated with a time-dependent subdifferential operator, whose solvability has been investigated in numerous previous researches. In this paper, however, we give an alternative proof of the solvability in order to avoid some complicated calculations arising from the chain rule for the time-dependent subdifferential. As for results which cannot be assured by the known abstract theory, we also discuss the continuous dependence of solution on the given data and the time-global behavior of solution.Comment: 19 pages, 3 figure

Singular limit of Allen--Cahn equation with constraints and its Lagrange multiplier

We consider the Allen-Cahn equation with constraint. Our constraint is the subdifferential of the indicator function on the closed interval, which is the multivalued function. In this paper we give the characterization of the Lagrange multiplier to our equation. Moreover, we consider the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier to our problem