191,343 research outputs found
Stationary solutions for the Cahn-Hilliard equation
No full textWe study the Cahn-Hilliard equation in a bounded domain without any symmetry assumptions. We assume that the mean curvature of the boundary
has a nongenerate critical point. Then we show that there exists a spike-like stationary solution whose global maximum lies on the boundary. Our method is based on Lyapunov-Schmidt reduction and the Brouwer fixed-point theorem
On the stationary Cahn-Hilliard equation: Interior spike solutions
No full textWe study solutions of the stationary Cahn-Hilliard equation in a bounded smooth domain which have a spike in the interior. We show that a large class of interior points (the "nondegenerate peak" points) have the following property: there exist such solutions whose spike lies close to a given nondegenerate peak point. Our construction uses among others the methods of viscosity solution, weak convergence of measures and Liapunov-Schmidt reduction
On the Stationary Cahn-Hilliard Equation: Bubble Solutions
No full textWe study
stationary solutions of the Cahn--Hilliard equation in a bounded
smooth domain which have an interior spherical interface (bubbles).
We show that a large class of interior points
(the ``nondegenerate peak'' points)
have the following property: there exists such a
solution whose bubble center lies close to a given nondegenerate peak point.
Our construction uses among others the Liapunov-Schmidt
reduction method and exponential asymptotics
Solutions for the Cahn-Hilliard Equation With Many Boundary Spike Layers
No full textIn this paper we
construct new classes of stationary solutions for the Cahn-Hilliard
equation
by a novel approach.
One of the results is as follows:
Given a positive integer K and a (not necessarily nondegenerate) local
minimum point of the mean curvature of the boundary then there are
boundary
K-spike solutions
whose peaks all approach this point.
This implies that for any smooth and bounded domain there
exist boundary K-spike solutions.
The central ingredient of our analysis is the novel derivation and
exploitation of a reduction of the energy to finite dimensions (Lemma 3.5),
where the variables are closely related to the peak loations
Triviality of the 2D stochastic Allen-Cahn equation
No full textWe consider the stochastic Allen-Cahn equation driven by mollified space-time white noise. We show that, as the mollifier is removed, the solutions converge weakly to
0, independently of the initial condition. If the intensity of the noise simultaneously converges to 0 at a sufficiently fast rate, then the solutions converge to those of the
deterministic equation. At the critical rate, the limiting solution is still deterministic, but it exhibits an additional damping term
Multi-interior-spike solutions for the Cahn-Hilliard equation with arbitrarily many peaks
No full textWe study the Cahn-Hilliard equation in a bounded smooth
domain without any symmetry
assumptions. We prove that for any fixed positive integer K there
exist interior --spike solutions
whose peaks have maximal possible distance from the boundary and
from one another. This implies that for any bounded and smooth
domain there
exist interior K-peak solutions.
The central ingredient of our analysis is the novel derivation
and exploitation of a reduction of the energy to finite dimensions
(Lemma 5.5) with variables which are closely related to the location of
the peaks.
We do not assume nondegeneracy of the points of
maximal distance to the boundary but can do with a global condition instead
which in many cases is weaker
Numerical analysis of a coupled pair of Cahn-Hilliard equations
No full textA mathematical analysis has been carried out for a coupled pair of Cahn-Hilliard equations, which appear in modelling a phase separation on a thin film of binary liquid mixture coating substrate, which is wet by one component. Existence and uniqueness are proved for a weak formulation of the problem, which possesses a Lyapunov functional. Regularity results are presented for the weak formulation. A fully practical piecewise linear finite element approximation is proposed where existence and uniqueness of the numerical solution, and its convergence to the solution of the continuous problem are proven. An error bound between the discrete and continuous solutions is given in three space dimensions. A practical algorithm for solving the resulting algebraic problem at each time step is suggested and its convergence is proven. Finally, linear stability analysis for one space dimension is presented, and some numerical simulations in one and two spaces dimension are exhibited
Numerical analysis of a coupled pair of Cahn-Hilliard equations with non-smooth free energy
No full textA mathematical analysis has been carried out for a coupled pair of Cahn-Hilliard equations with a double well potential function with infinite walled free energy, which appears in modelling a phase separation on a thin film of binary liquid mixture coating substrate, which is wet by one component. Existence and uniqueness are proved for a weak formulation of the problem, which possesses a Lyapunov functional. Regularity results for the weak formulation are presented. Semi and fully discrete finite element approximations are proposed where existence and uniqueness of their solutions are proven. Their convergence to the solution of the continuous solutions are presented. Error bound between semi-discrete and continuous solutions, between semi-discrete and fully discrete solutions, and between fully discrete and continuous solutions are all investigated. A practical algorithm to solve the fully discrete finite element formulation at each time step is introduced and its convergence is shown. Finally, a linear stability analysis of the equations in one dimension space is presented and some numerical simulations in one and two dimension spaces are preformed
Finite Element Approximations for a linear Cahn-Hilliard-Cook equation driven by the space derivative of a space-time white noise
No full textWe consider an initial- and Dirichlet boundary- value problem for
a linear Cahn-Hilliard-Cook equation, in one space dimension,
forced by the space derivative of a space-time white noise.
First, we propose an approximate regularized stochastic parabolic
problem discretizing the noise using linear splines. Then
fully-discrete approximations to the solution of the
regularized problem are constructed using, for the discretization
in space, a Galerkin finite element method based on
piecewise polynomials, and, for time-stepping, the Backward
Euler method.
Finally, we derive strong a priori estimates for the modeling error and
for the numerical approximation error to the solution of the regularized problem
Mathematical and Numerical Analysis of a Pair of Coupled Cahn-Hilliard Equations with a Logarithmic Potential
No full textMathematical and numerical analysis has been undertaken for a pair of coupled Cahn-Hilliard equations with a logarithmic potential and with homogeneous Neumann boundary conditions. This pair of coupled equations arises in a phase separation model of thin film of binary liquid mixture. Global existence and uniqueness of a weak solution to the problem is proved using Faedo-Galerkin method. Higher regularity results of the weak solution are established under further regular requirements on the initial data. Further, continuous dependence on the initial data is presented.
Numerically, semi-discrete and fully-discrete piecewise linear finite element approximations to the continuous problem are proposed for which existence, uniqueness and various stability estimates of the approximate solutions are proved. Semi-discrete and fully-discrete error bounds are derived where the time discretisation error is optimal. An iterative method for solving the resulting nonlinear algebraic system is introduced and linear stability analysis in one space dimension is studied. Finally, numerical experiments illustrating some of the theoretical results are performed in one and two space dimensions
- …
