Weak solutions to the sharp interface limit of stochastic Cahn-Hilliard equations

Abstract

We study the asymptotic limit, as $\varepsilon\searrow 0$, of solutions of the stochastic Cahn-Hilliard equation: $$ \partial_t u^\varepsilon=Δ\left(-\varepsilonΔu^\varepsilon+\frac{1}{\varepsilon}f(u^\varepsilon)\right)+\dot{\mathcal{W}}^\varepsilon_t, \\ $$ where $\mathcal{W}^\varepsilon=\varepsilon^σW$ or $\mathcal{W}^\varepsilon=\varepsilon^σW^\varepsilon$, $W$ is a $Q$-Wiener process and $W^\varepsilon$ is smooth in time and converges to $W$ as $\varepsilon\searrow 0$. In the case that $\mathcal{W}^\varepsilon=\varepsilon^σW$, we prove that for all $σ>\frac{1}{2}$, the solution $u^\varepsilon$ converges to a weak solution to an appropriately defined limit of the deterministic Cahn-Hilliard equation. In radial symmetric case we prove that for all $σ\geq\frac{1}{2}$, $u^\varepsilon$ converges to the deterministic Hele-Shaw model. In the case that $\mathcal{W}^\varepsilon=\varepsilon^σW^\varepsilon$, we prove that for all $σ>0$, $u^\varepsilon$ converges to the weak solution to the deterministic limit Cahn-Hilliard equation. In radial symmetric case we prove that $u^\varepsilon$ converges to deterministic Hele-Shaw model when $σ>0$ and converges to a stochastic model related to stochastic Hele-Shaw model when $σ=0$.