Relative energy estimates for the Cahn-Hilliard equation with concentration dependent mobility

TL;DR

This paper develops stability and error estimates for the Cahn-Hilliard equation with concentration-dependent mobility, providing theoretical guarantees for numerical approximations and establishing a weak-strong uniqueness principle.

Contribution

It introduces relative energy estimates for stability analysis and derives order-optimal a-priori error bounds for discretized schemes of the Cahn-Hilliard equation.

Findings

Established weak-strong uniqueness principle.

Derived sharp bounds for discretization errors.

Proved order-optimal a-priori error estimates.

Abstract

Based on relative energy estimates, we study the stability of solutions to the Cahn-Hilliard equation with concentration dependent mobility with respect to perturbations. As a by-product of our analysis, we obtain a weak-strong uniqueness principle on the continuous level under realistic regularity assumptions on strong solutions. We then show that the stability estimates can be further inherited almost verbatim by appropriate Galerkin approximations in space and time. This allows us to derive sharp bounds for the discretization error in terms of certain projection errors and to establish order-optimal a-priori error estimates for semi- and fully discrete approximation schemes.