Higher moments for the Stochastic Cahn - Hilliard Equation with multiplicative Fourier noise

Abstract

We consider in dimensions $d=1,2,3$ the $\eps$-dependent stochastic Cahn-Hilliard equation with a multiplicative and sufficiently regular in space infinite dimensional Fourier noise with strength of order $\mathcal{O}(\eps^\gamma)$, $\gamma>0$. The initial condition is non-layered and independent from $\eps$. Under general assumptions on the noise diffusion $\sigma$, we prove moment estimates in $H^1$ (and in $L^\infty$ when $d=1$). Higher $H^2$ regularity $p$-moment estimates are derived when $\sigma$ is bounded, yielding as well space H\"older and $L^\infty$ bounds for $d=2,3$, and path a.s. continuity in space. All appearing constants are expressed in terms of the small positive parameter $\eps$. As in the deterministic case, in $H^1$, $H^2$, the bounds admit a negative polynomial order in $\eps$. Finally, assuming layered initial data of initial energy uniformly bounded in $\eps$, as proposed by X.F. Chen in \cite{chenjdg}, we use our $H^1$ $2$d-moment estimate and prove the stochastic solution's convergence to $\pm 1$ as $\eps\rightarrow 0$ a.s., when the noise diffusion has a linear growth.