Hyperbolic relaxation of a sixth-order Cahn-Hilliard equation

TL;DR

This work analyzes a hyperbolic relaxation of a $6$-th order Cahn–Hilliard equation with an inertial term $\tau \partial_t^2\varphi$, incorporating curvature effects through higher-order derivatives and a source term that breaks mean conservation. The authors establish well-posedness of the relaxed problem via a Faedo–Galerkin approach, deriving uniform stability estimates for the solution triplet $(\varphi,\mu,w)$ and proving convergence to the unrelaxed ($\tau=0$) model as $\tau\to0$, with a quantified $O(\tau^{1/2})$ error rate. They extend the analysis to rigorous asymptotics, proving convergence in appropriate topologies and uniform convergence of nonlinearities, and provide detailed proofs split into the convergence (4.1) and the error estimate (4.2) components. Overall, the paper delivers a robust weak-solution framework for the hyperbolic relaxation, alongside precise convergence and error estimates that illuminate the relation between the relaxed and classical sixth-order Cahn–Hilliard dynamics, with potential implications for modeling non-equilibrium phase separation and curvature-driven interface phenomena.

Abstract

This work explores the solvability of a sixth-order Cahn--Hilliard equation with an inertial term, which serves as a relaxation of a higher-order variant of the classical Cahn--Hilliard equation. The equation includes a source term that disrupts the conservation of the mean value of the order parameter. The incorporation of additional spatial derivatives allows the model to account for curvature effects, leading to a more precise representation of isothermal phase separation dynamics. We establish the existence of a weak solution for the associated initial and boundary value problem under the assumption that the double-well-type nonlinearity is globally defined. Additionally, we derive uniform stability estimates, which enable us to demonstrate that any family of solutions satisfying these estimates converges in a suitable topology to the unique solution of the limiting problem as the relaxation parameter approaches zero. Furthermore, we provide an error estimate for specific norms of the difference between solutions in terms of the relaxation parameter.

Theorems & Definitions (11)

• Remark 2.1
• Remark 2.2
• Theorem 2.3
• Remark 2.4
• Theorem 2.5
• Theorem 2.6
• Remark 3.1
• Lemma 4.1
• proof
• Lemma 4.2