Optimal relaxation of bump-like solutions of the one-dimensional Cahn-Hilliard equation

TL;DR

This paper derives optimal relaxation rates for bump-like solutions of the one-dimensional Cahn-Hilliard equation, extending previous methods from single transition layers to two-layer bumps, with results on convergence and metastability.

Contribution

It extends the relaxation analysis of the Cahn-Hilliard equation to two transition layers, providing quantitative convergence rates and metastability insights.

Findings

On the torus, exponential convergence to the bump state.

On the line, initial algebraic relaxation with metastability.

Energy gap decay is translation invariant, requiring new arguments for the bump.

Abstract

REVISED VERSION INCORPORATING THE ERRATUM ON LEMMA 2.1 AND WITH A CORRECTION TO LEMMA 2.8 In this paper we derive optimal relaxation rates for the Cahn-Hilliard equation on the one-dimensional torus and the line. We consider initial conditions with a finite (but not small) $L^1$-distance to an appropriately defined bump. The result extends the relaxation method developed previously for a single transition layer (the ``kink'') to the case of two transition layers (the ``bump''). As in the previous work, the tools include Nash-type inequalities, duality arguments, and Schauder estimates. For both the kink and the bump, the energy gap is translation invariant and its decay alone cannot specify to which member of the family of minimizers the solution converges. Whereas in the case of the kink, the conserved quantity singles out the longtime limit, in the case of a bump, a new argument is needed. On the torus, we quantify the (initially algebraic and ultimately exponential) convergence to the bump that is the longtime limit; on the line, the bump-like states are merely metastable and we quantify the initial algebraic relaxation behavior.