Higher moments for the stochastic Cahn–Hilliard equation with multiplicative Fourier noise

Abstract

Abstract We consider in dimensions d = 1 , 2 , 3 the ɛ-dependent stochastic Cahn–Hilliard equation with a multiplicative and sufficiently regular in space infinite dimensional Fourier noise with strength of order , γ > 0. The initial condition is non-layered and independent from ɛ. Under general assumptions on the noise diffusion σ, we prove moment estimates in H 1 (and in L ∞ when d = 1). Higher H 2 regularity p-moment estimates are derived when σ is bounded, yielding as well space Hölder and L ∞ bounds for d = 2 , 3 , and path a.s. continuity in space. All appearing constants are expressed in terms of the small positive parameter ɛ. As in the deterministic case, in H 1, H 2, the bounds admit a negative polynomial order in ɛ. Finally, assuming layered initial data of initial energy uniformly bounded in ɛ, as proposed by Chen (1996 J. Differ. Geom. 44 262–311), we use our H 1 2d-moment estimate and prove the stochastic solution’s convergence to ± 1 as ε → 0 a.s. when the noise diffusion has a linear growth.