Improved estimates for the sharp interface limit of the stochastic Cahn-Hilliard equation with space-time white noise
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We study the sharp interface limit of the stochastic Cahn-Hilliard equation with cubic double-well potential and additive space-time white noise ϵ^σẆ where ϵ>0 is an interfacial width parameter. We prove that, for sufficiently large scaling constant σ >0, the stochastic Cahn-Hilliard equation converges to the deterministic Mullins-Sekerka/Hele-Shaw problem for ϵ→ 0. The convergence is shown in suitable fractional Sobolev norms as well as in the L^p-norm for p∈ (2, 4] in spatial dimension d=2,3. This generalizes the existing result for the space-time white noise to dimension d=3 and improves the existing results for smooth noise, which were so far limited to p∈(2, frac2d+8d+2] in spatial dimension d=2,3. As a byproduct of the analysis of the stochastic problem with space-time white noise, we identify minimal regularity requirements on the noise which allow convergence to the sharp interface limit in the ℍ^1-norm and also provide improved convergence estimates for the sharp interface limit of the deterministic problem.
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