Related papers: Tightness for a stochastic Allen--Cahn equation
We consider the stochastic Cahn-Hilliard equation with additive space-time white noise $\epsilon^{\gamma}\dot{W}$ in dimension $d=2,3$, where $\epsilon>0$ is an interfacial width parameter. We study numerical approximation of the equation…
We study the the sharp interface limit of $\varepsilon$-dependent two dimensional stochastic Cahn-Hilliard equation driven by space-time white noise and conservative noise as $\varepsilon\to 0$. In the case when the noise is sufficiently…
We revisit the interface fluctuation problem for the $1$D Allen-Cahn equation perturbed by a small space-time white noise. We show that if the initial data is a standing wave solution to the deterministic equation, then under proper long…
We consider the sharp interface limit for the scalar-valued and vector-valued Allen-Cahn equation with homogeneous Neumann boundary condition in a bounded smooth domain $\Omega$ of arbitrary dimension $N\geq 2$ in the situation when a…
We show convergence of solutions of a convective Allen-Cahn equation for a given smooth and divergence free velocity field to a transport equation for an evolving interface in the case when the thickness of the diffuse interface tends to…
We study the two and three dimensional stochastic Cahn-Hilliard equation in the sharp interface limit, where the positive parameter $\epsilon$ tends to zero, which measures the width of transition layers generated during phase separation.…
We present a convergence result for solutions of the vector-valued Allen-Cahn Equation. In the spirit of the work of Luckhaus and Sturzenhecker we establish convergence towards a distributional formulation of multi-phase mean-curvature flow…
This paper studies the 1D stochastic Allen--Cahn equation on a bounded domain driven by localized white noise. We prove that the associated Markov process admits a unique invariant measure and is exponential mixing. The main challenge lies…
We prove existence and uniqueness of a solution for the stochastic Allen-Cahn equation with logarithmic potential and multiplicative Wiener noise, under homogeneous Neumann boundary condition. The existence of a solution is obtained in the…
We study an $\ep$-dependent stochastic Allen--Cahn equation with a mild random noise on a bounded domain in $\mathbb{R}^n$, $n\geq 2$. Here $\ep$ is a small positive parameter that represents formally the thickness of the solution…
We study the Allen-Cahn equation with respect to a transport term in a bounded domain. We prove that the limit interface is the mean curvature flow with the transport term, given the condition that the energy is uniformly bounded with…
This paper is concerned with the sharp interface limit for the Allen-Cahn equation with a nonlinear Robin boundary condition in a bounded smooth domain $\Omega\subset\mathbb{R}^2$. We assume that a diffuse interface already has developed…
We are concerned with solutions to the parabolic Allen-Cahn equation in Riemannian manifolds. For a general class of initial condition we show non positivity of the limiting energy discrepancy. This in turn allows to prove almost…
We prove convergence of solutions to the parabolic Allen-Cahn equation to Brakke's motion by mean curvature in space forms, generalizing previous results from [15] in Euclidean space. We show that a sequence of measures, associated to…
The Cahn-Hilliard/Allen-Cahn equation with noise is a simplified mean field model of stochastic microscopic dynamics associated with adsorption and desorption-spin flip mechanisms in the context of surface processes. For such an equation we…
We investigate the sharp interface limit of a diffuse interface system that couples the Allen--Cahn equation with the instationary Navier--Stokes system in a bounded domain in $\mathbb{R}^d$ with $d \in \{2,3\}$. This model is used to…
We consider the Allen-Cahn equation with nonlinear anisotropic diffusion and derive anisotropic direction-dependent curvature flow under the sharp interface limit. The anisotropic curvature flow was already studied, but its derivation is…
We analyze the sharp interface limit for the Allen-Cahn equation with an anisotropic, spatially periodic mobility coefficient and prove that the large-scale behavior of interfaces is determined by mean curvature flow with an effective…
The aim of this contribution is to address the convergence study of a time and space approximation scheme for an Allen-Cahn problem with constraint and perturbed by a multiplicative noise of It\^o type. The problem is set in a bounded…
We define a (mean curvature flow) entropy for Radon measures in $\mathbb{R}^n$ or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen-Cahn equations. If the…