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

Ludovic Goudenège (1) , Bin Xie (2)

1 Fédération de Mathématiques de CentraleSupélec
2 Department of Mathematical Sciences, Faculty of Science, Shinshu University

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.

Mots clés: Stochastic Cahn-Hilliard equation; Asymptotic log-Harnack inequality; Harnack inequality with power; Logarithmic free energy; Essentially elliptic condition

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]; Probabilités [math.PR]

Ludovic Goudenège, Bin Xie. Ergodicity of stochastic Cahn-Hilliard equations with logarithmic potentials driven by degenerate or nondegenerate noises. Journal of Differential Equations, 2020, 269 (9), pp.6988-7014. ⟨10.1016/j.jde.2020.04.047⟩. ⟨hal-03101180⟩