2023/04/28 by Banas Lubomir, Baňas, Ľubomír, Mukam, Jean Daniel · 1 citation
Mathematics · Physics and Astronomy · Materials Science · #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.2304.14785
We study the sharp interface limit of the stochastic Cahn-Hilliard equation with cubic double-well potential and additive space-time white noise εσW 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 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.