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

Abstract

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

43 pages. arXiv admin note: text overlap with arXiv:1904.05930 by other authors

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