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

2019/05/21 by Yang, Huanyu, Zhu, Rongchan
#Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1905.09182

Abstract

We study the asymptotic limit, as ε\searrow 0, of solutions of the stochastic Cahn-Hilliard equation: ∂_t u^ε=Δ(-εΔu^ε+(1)/(ε)f(u^ε))+W^ε_t,
where W^ε=ε^σW or W^ε=ε^σW^ε, W is a Q-Wiener process and W^ε is smooth in time and converges to W as ε\searrow 0. In the case that W^ε=ε^σW, we prove that for all σ>(1)/(2), the solution u^ε 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≥(1)/(2), u^ε converges to the deterministic Hele-Shaw model. In the case that W^ε=ε^σW^ε, we prove that for all σ>0, u^ε converges to the weak solution to the deterministic limit Cahn-Hilliard equation. In radial symmetric case we prove that u^ε converges to deterministic Hele-Shaw model when σ>0 and converges to a stochastic model related to stochastic Hele-Shaw model when σ=0.

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