2024/12/02 by Yuan Chen, Chen, Yuan
Engineering · Materials Science · #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Thin Films #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.2412.01633
openalex publication_date 2024/12/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the mass-preserving L2-gradient flow associated with a generalized Cahn--Hilliard equation. Our focus is on the sharp interface regime, where the interface width parameter ε > 0 is small. For well-prepared initial data, we rigorously prove that, as ε → 0, solutions of the diffuse-interface model converge to the volume-preserving Willmore flow in arbitrary spatial dimensions n ≥ 2. The proof incorporates matched asymptotic expansions and energy estimates to establish convergence of the order parameter away from the interface, alongside precise motion law derivation for the limiting interface. This result extends the analysis of Fei and Liu~\citefei2021phase from two-dimensional settings to general n-dimensional domains, and it applies to a broad class of symmetric double-well potentials beyond the classical quartic form. Our work thus provides a general PDE framework linking higher-order phase-field models to volume-preserving curvature flows in the sharp interface limit.