6 research outputs found
Motion of a droplet for the mass-conserving stochastic Allen-Cahn equation
We study the stochastic mass-conserving Allen-Cahn equation posed on a
bounded two-dimensional domain with additive spatially smooth space-time noise.
This equation associated with a small positive parameter describes the
stochastic motion of a small almost semicircular droplet attached to domain's
boundary and moving towards a point of locally maximum curvature. We apply
It\^o calculus to derive the stochastic dynamics of the droplet by utilizing
the approximately invariant manifold introduced by Alikakos, Chen and Fusco for
the deterministic problem. In the stochastic case depending on the scaling, the
motion is driven by the change in the curvature of the boundary and the
stochastic forcing. Moreover, under the assumption of a sufficiently small
noise strength, we establish stochastic stability of a neighborhood of the
manifold of droplets in and , which means that with overwhelming
probability the solution stays close to the manifold for very long time-scales
Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model
We present a computer-assisted proof of heteroclinic connections in the
one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a
fourth-order parabolic partial differential equation subject to homogeneous
Neumann boundary conditions, which contains as a special case the celebrated
Cahn-Hilliard equation. While the attractor structure of the latter model is
completely understood for one-dimensional domains, the diblock copolymer
extension exhibits considerably richer long-term dynamical behavior, which
includes a high level of multistability. In this paper, we establish the
existence of certain heteroclinic connections between the homogeneous
equilibrium state, which represents a perfect copolymer mixture, and all local
and global energy minimizers. In this way, we show that not every solution
originating near the homogeneous state will converge to the global energy
minimizer, but rather is trapped by a stable state with higher energy. This
phenomenon can not be observed in the one-dimensional Cahn-Hillard equation,
where generic solutions are attracted by a global minimizer
Topological Microstructure Analysis Using Persistence Landscapes
International audiencePhase separation mechanisms can produce a variety of complicated and intricate microstructures, which often can be difficult to characterize in a quantitative way. In recent years, a number of novel topological metrics for microstructures have been proposed, which measure essential connectivity information and are based on techniques from algebraic topology. Such metrics are inherently computable using computational homology, provided the microstructures are discretized using a thresholding process. However, while in many cases the thresholding is straightforward, noise and measurement errors can lead to misleading metric values. In such situations, persistence landscapes have been proposed as a natural topology metric. Common to all of these approaches is the enormous data reduction, which passes from complicated patterns to discrete information. It is therefore natural to wonder what type of information is actually retained by the topology. In the present paper, we demonstrate that averaged persistence landscapes can be used to recover central system information in the Cahn-Hilliard theory of phase separation. More precisely, we show that topological information of evolving microstructures alone suffices to accurately detect both concentration information and the actual decomposition stage of a data snapshot. Considering that persistent homology only measures discrete connectivity information, regardless of the size of the topological features, these results indicate that the system parameters in a phase separation process affect the topology considerably more than anticipated. We believe that the methods discussed in this paper could provide a valuable tool for relating experimental data to model simulations