Matched asymptotic analysis of self-similar blow-up profiles of the thin film equation

TL;DR

This paper performs a matched asymptotic analysis of self-similar blow-up profiles in the thin film equation, revealing detailed structure near criticality and supporting findings with numerical continuation.

Contribution

It provides a novel asymptotic description of blow-up profiles near the critical exponent, connecting inner and outer regions through matched asymptotics.

Findings

Profiles have a peak and a decaying tail near criticality

Asymptotic relationship between peak height and criticality distance

Numerical continuation supports asymptotic analysis

Abstract

We consider asymptotically self-similar blow-up profiles of the thin film equation consisting of a stabilising fourth order and destabilising second order term. It has previously been shown that blow up is only possible when the exponent in the second order term is above a certain critical value (dependent on the exponent in the fourth order term). We show that in the limit that the critical value is approached from above, the primary branch of similarity profiles exhibits a well-defined structure consisting of a peak near the origin, and a thin, algebraically decaying tail, connected by an inner region equivalent (to leading order) to a generalised version of the Landau--Levich `drag-out' problem in lubrication flow. Matching between the regions ultimately gives the asymptotic relationship between a parameter representing the height of the peak and the distance from the criticality threshold. The asymptotic results are supported by numerical computations found using continuation.