5,618 research outputs found
Coalescence in the 1D Cahn-Hilliard model
We present an approximate analytical solution of the Cahn-Hilliard equation
describing the coalescence during a first order phase transition. We have
identified all the intermediate profiles, stationary solutions of the noiseless
Cahn-Hilliard equation. Using properties of the soliton lattices, periodic
solutions of the Ginzburg-Landau equation, we have construct a family of ansatz
describing continuously the processus of destabilization and period doubling
predicted in Langer's self similar scenario
Global solvability and blow up for the convective Cahn-Hilliard equations with concave potentials
We study initial boundary value problems for the convective Cahn-Hilliard
equation \Dt u +\px^4u +u\px u+\px^2(|u|^pu)=0. It is well-known that without
the convective term, the solutions of this equation may blow up in finite time
for any . In contrast to that, we show that the presence of the convective
term u\px u in the Cahn-Hilliard equation prevents blow up at least for
. We also show that the blowing up solutions still exist if is
large enough (). The related equations like
Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard
equation, are also considered
Weak Solutions to the Degenerate Viscous Cahn-Hilliard Equation
The Cahn--Hilliard equation is a common model to describe phase separation
processes of a mixture of two components. In this paper, we study the viscous
Cahn--Hilliard equation with degenerate phase-dependent mobility. We define a
notion of weak solutions and prove the existence of such weak solutions by
considering the limits of the viscous Cahn--Hilliard equation with positive
mobility. Also, we prove that such weak solutions satisfy an energy dissipation
inequality under some additional conditions
On a Cahn-Hilliard system with convection and dynamic boundary conditions
This paper deals with an initial and boundary value problem for a system
coupling equation and boundary condition both of Cahn-Hilliard type; an
additional convective term with a forced velocity field, which could act as a
control on the system, is also present in the equation. Either regular or
singular potentials are admitted in the bulk and on the boundary. Both the
viscous and pure Cahn-Hilliard cases are investigated, and a number of results
is proven about existence of solutions, uniqueness, regularity, continuous
dependence, uniform boundedness of solutions, strict separation property. A
complete approximation of the problem, based on the regularization of maximal
monotone graphs and the use of a Faedo-Galerkin scheme, is introduced and
rigorously discussed.Comment: Key words: Cahn-Hilliard system, convection, dynamic boundary
condition, initial-boundary value problem, well-posedness, regularity of
solution
From the viscous Cahn-Hilliard equation to a regularized forward-backward parabolic equation
A rigorous proof is given for the convergence of the solutions of a viscous
Cahn-Hilliard system to the solution of the regularized version of the
forward-backward parabolic equation, as the coefficient of the diffusive term
goes to 0. Non-homogenous Neumann boundary condition are handled for the
chemical potential and the subdifferential of a possible non-smooth double-well
functional is considered in the equation. An error estimate for the difference
of solutions is also proved in a suitable norm and with a specified rate of
convergence.Comment: Key words and phrases: Cahn-Hilliard system, forward-backward
parabolic equation, viscosity, initial-boundary value problem, asymptotic
analysis, well-posednes
On a Cahn--Hilliard--Darcy system for tumour growth with solution dependent source terms
We study the existence of weak solutions to a mixture model for tumour growth
that consists of a Cahn--Hilliard--Darcy system coupled with an elliptic
reaction-diffusion equation. The Darcy law gives rise to an elliptic equation
for the pressure that is coupled to the convective Cahn--Hilliard equation
through convective and source terms. Both Dirichlet and Robin boundary
conditions are considered for the pressure variable, which allows for the
source terms to be dependent on the solution variables.Comment: 18 pages, changed proof from fixed point argument to Galerkin
approximatio
- …