Ergodicity of stochastic Cahn-Hilliard equations with logarithmic potentials driven by degenerate or nondegenerate noises

TL;DR

This paper investigates the long-term behavior of stochastic Cahn-Hilliard equations with logarithmic potentials under different noise types, establishing dimension-free Harnack inequalities and analyzing their implications for ergodicity.

Contribution

It introduces new Harnack inequalities for the stochastic Cahn-Hilliard equation with logarithmic free energy under various noise conditions, advancing understanding of its ergodic properties.

Findings

Established log-Harnack inequality for degenerate noise

Proved Harnack inequality with power for non-degenerate noise

Analyzed the impact of singularities and mass conservation on solutions

Abstract

We study the asymptotic properties of the stochastic Cahn-Hilliard equation with the logarithmic free energy by establishing different dimension-free Harnack inequalities according to various kinds of noises. The main characteristics of this equation are the singularities of the logarithmic free energy at 1 and --1 and the conservation of the mass of the solution in its spatial variable. Both the space-time colored noise and the space-time white noise are considered. For the highly degenerate space-time colored noise, the asymptotic log-Harnack inequality is established under the so-called essentially elliptic conditions. And the Harnack inequality with power is established for non-degenerate space-time white noise.