Stochastic Cahn–Hilliard Equation with Double Singular Nonlinearities and Two Reflections

Abstract

We consider a stochastic partial differential equation with two logarithmic nonlinearities, two reflections at 1 and −1, and a constraint of conservation of the space average. The equation, driven by the derivative in space of a space-time white noise, contains a bi-Laplacian in the drift. The lack of a maximum principle for the bi-Laplacian generates difficulties for the classical penalization method, which uses a crucial monotonicity property. Being inspired by the works of Debussche, Goudenège, and Zambotti, we obtain existence and uniqueness of a solution for initial conditions in the interval (−1, 1). Finally, we prove that the unique invariant measure is ergodic, and we give a result of exponential mixing.Introduction and main results. The Cahn-Hilliard-Cook equation is a model to describe phase separation in a binary alloy (see [6], [7], and [8]) in the presence of thermal fluctuations (see [11] and [27]). It takes the form

Asymptotic properties of stochastic Cahn–Hilliard equation with singular nonlinearity and degenerate noise

“…However the choice of smooth-enough space-time noise permits to work with an Itô's formula. Following the approach of [16], we shall consider the problem with a polynomial approximation of the nonlinearity ψ. We obtain a priori estimates on the approximated solutions which are sufficient to prove tightness.…”

The Stochastic Viscous Cahn–Hilliard Equation: Well-Posedness, Regularity and Vanishing Viscosity Limit

“…Existence, uniqueness and regularity have been investigated in the works [22,24,27] for the stochastic pure Cahn-Hilliard equation with a polynomial double-well potential, and in [25,39,62] for the pure case with a possibly singular double-well potential. The reader can also refer to [1] for a study of a stochastic Cahn-Hilliard equation with unbounded noise and [25,26,39] dealing with stochastic Cahn-Hilliard equations with reflections. To the best of our knowledge, the only available results dealing with the stochastic viscous Cahn-Hilliard equation seem to be [41,45]: here, the authors prove existence of mild solution and attractors under the classical case of a polynomial double-well potential.…”

Optimal control of stochastic phase-field models related to tumor growth

“…The mathematical literature on the stochastic Cahn-Hilliard and Allen-Cahn equations is quite developed: we refer for example to [3,47] for the stochastic Allen-Cahn equation, and to [12,15,16,20,31,50,51]) for the stochastic Cahn-Hilliard equation. For completeness, let us quote also [21,22] for a study on a stochastic diffuse interface model involving the Cahn-Hilliard and Navier-Stokes equations.…”