131 research outputs found
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
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