We 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 limi...

A finite element method for the Cahn-Hilliard equation (a semilinear parabolic equation of fourth order) is analyzed, both in a spatially semidisCrete case and in a completely discrete case based on the backward Euler method. Error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data. A detailed study of the regularity of the exac...

We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes model and describes two phase very viscous flows in porous media. The scheme is...

Diblock copolymers are a class of materials formed by the reaction of two linear polymers. The different structures taken on by these polymers grant them special properties, which can prove useful in applications such as the development of new adhesives and asphalt additives. We consider a model for the formation of diblock copolymers first proposed by Ohta and Kawasaki [26]. Their model yields...

Abstract: This paper presents some numerical methods for Allen-Cahn equation using different time stepping and space discretization methods with non-periodic boundary conditions. In space the equation is discretized by Chebyshev spectral method, while in time the exponential time differencing fourth-order Runge-Kutta (ETDRK4) and implicit-explicit scheme is used. For comparison we also use the ...

We present an efficient scheme within the phase field framework for imposing dynamic contact angle boundary conditions for wall-bounded flows of two immiscible incompressible fluids with large density ratios. First, we develop an algorithm for imposing the dynamic contact angle boundary conditions to the Cahn–Hilliard equation. Our algorithm consists of two components: (i) we ignore the boundar...

This paper proposes an abstract theory concerned with dynamical systems generated by doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations in a reflexive Banach space setting. In order to construct global attractors, an approach based on the notion of generalized semiflow is employed instead of the usual semi-group approach, since solutions o...

We 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 limi...

We prove the global well-posedness of the so-called hyperbolic relaxation of the Cahn-Hilliard-Oono equation in the whole space R with the non-linearity of the sub-quintic growth rate. Moreover, the dissipativity and the existence of a smooth global attractor in the naturally defined energy space is also verified. The result is crucially based on the Strichartz estimates for the linear Scröding...

We investigate the dynamics of a nonequilibrium interface between coexisting phases in a system described by a Cahn-Hilliard equation with an additional driving term. By means of a matched asymptotic expansion we derive equations for the interface motion. A linear stability analysis of these equations results in a condition for the stability of a flat interface. We find that the stability prope...