Invariant Manifolds for the Thin Film Equation

TL;DR

This paper studies the long-term behavior of solutions to the thin film equation, constructing invariant manifolds to understand higher order asymptotics and convergence towards self-similar solutions.

Contribution

It introduces a method to construct finite-dimensional invariant manifolds for the thin film equation, enabling detailed analysis of solution asymptotics.

Findings

Solutions converge to self-similar Smyth--Hill solutions

Invariant manifolds approximate solutions to arbitrary order

Higher order asymptotics are characterized under symmetry assumptions

Abstract

The large-time behavior of solutions to the thin film equation with linear mobility in the complete wetting regime on $\mathbb{R}^N$ is examined: We investigate the higher order asymptotics of solutions converging towards self-similar Smyth--Hill solutions under certain symmetry assumptions on the initial data. The analysis is based on a construction of finite-dimensional invariant manifolds that solutions approximate to an arbitrarily prescribed order.