Volume Preserving Willmore Flow in a Generalized Cahn-Hilliard Flow

We investigate the mass-preserving $L^2$-gradient flow associated with a generalized Cahn--Hilliard equation. Our focus is on the sharp interface regime, where the interface width parameter $\varepsilon > 0$ is small. For well-prepared initial data, we rigorously prove that, as $\varepsilon \to 0$, solutions of the diffuse-interface model converge to the \emph{volume-preserving Willmore flow} in arbitrary spatial dimensions $n \geq 2$. 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~\cite{fei2021phase} 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.