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 εσẆ , where ε>0 is an interfacial width parameter. We prove that, for a 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 Lᵖ -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+8/d+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 H¹ -norm and also provide improved convergence estimates for the sharp interface limit of the deterministic problem.