3,587 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
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
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
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
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
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
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
Analysis of the viscous Cahn–Hilliard equation in RN
No full textAbstractSolvability of Cauchyʼs problem in RN for an extended viscous Cahn–Hilliard equation is studied. The problem is considered first in a standard Sobolev space H1(RN), next a notion of the ‘H-solution’ is introduced that is well adapted to the structure of the viscous Cahn–Hilliard equation. Several properties of the solutions are reported, in particular those connected with their asymptotic behavior. We collect here also known properties of the unbounded operator (−Δ)−1 in RN that are needed in our considerations
On the Cahn–Hilliard equation in H1(RN)
No full textAbstractIn this paper we exhibit the dissipative mechanism of the Cahn–Hilliard equation in H1(RN). We show a weak form of dissipativity by showing that each individual solution is attracted, in some sense, by the set of equilibria. We also indicate that strong dissipativity, that is, asymptotic compactness in H1(RN), cannot be in general expected. Then we consider two types of perturbations: a nonlinear perturbation and a small linear perturbation. In both cases we show that, for the resulting equations, the dissipative mechanism becomes strong enough to obtain the existence of a compact global attractor
Multi-Peak Solutions for a Wide Class of Singular Perturbation Problems
No full textIn this paper we are
concerned with a wide class of singular perturbation problems arising
from such diverse fields as phase transitions,
chemotaxis, pattern formation,
population dynamics and chemical reaction theory.
We study the corresponding elliptic
equations in a bounded domain without any symmetry
assumptions. We assume that the
mean curvature of the boundary
has \overline{M} isolated, non-degenerate critical points.
Then we show that for any positive integer m\leq \overline{M}
there exists a stationary
solution with M local peaks which are attained on the boundary and
which lie close to these critical points.
Our method is based on Liapunov-Schmidt reduction
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