Improved estimates for the sharp interface limit of the stochastic Cahn–Hilliard equation with space-time white noise

We study the sharp interface limit of the stochastic Cahn–Hilliard equation with cubic double-well potential and additive space-time white noise ^Ẇ, where >0 is an interfacial width parameter. We prove that, for a sufficiently large scaling constant >0, the stochastic Cahn–Hilliard equation converges to the deterministic Mullins–Sekerka/Hele-Shaw problem for 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, 2d+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.