Convergence Analysis of the Geometric Thin-Film Equation

TL;DR

This paper proves the global existence, uniqueness, and regularity of solutions for the geometric thin-film equation, modeling droplet spreading, using a particle solution approach and ODEs.

Contribution

It introduces a novel approach to analyze the geometric thin-film equation by constructing solutions from particle solutions and proves their well-posedness.

Findings

Solutions exist globally for all time.

Solutions are uniquely determined by initial data.

Solutions are 1/2-Hölder continuous in time.

Abstract

The Geometric Thin-Film equation is a mathematical model of droplet spreading in the long-wave limit, which includes a regularization of the contact-line singularity. We show that the weak formulation of the problem, given initial Radon data, admits solutions that are globally defined for all time and are expressible as push-forwards of Borel measurable functions whose behaviour is governed by a set of ordinary differential equations (ODEs). The existence is first demonstrated in the special case of a finite weighted sum of delta functions whose centres evolve over time -- these are known as `particle solutions'. In the general case, we construct a convergent sequence of particle solutions whose limit yields a solution of the above form. Moreover, we demonstrate that all weak solutions constructed in this way are $1/2$-H\"older continuous in time and are uniquely determined by the initial conditions.