Analytical and Numerical Results on the Positivity of Steady State Solutions of a Thin Film Equation

TL;DR

This paper investigates the positivity of steady state solutions in a thin film equation on a rotating cylinder, providing analytical proofs, a spectral algorithm for computation, and exploring non-existence conditions.

Contribution

It offers new analytical proofs of positivity, introduces an iterative spectral method for steady state computation, and examines non-existence inequalities for solutions.

Findings

Weak and classical steady states are strictly positive when rotation speed is nonzero

An iterative spectral algorithm effectively computes steady states

A non-existence inequality for steady states is explored

Abstract

We consider an equation for a thin-film of fluid on a rotating cylinder and present several new analytical and numerical results on steady state solutions. First, we provide an elementary proof that both weak and classical steady states must be strictly positive so long as the speed of rotation is nonzero. Next, we formulate an iterative spectral algorithm for computing these steady states. Finally, we explore a non-existence inequality for steady state solutions from the recent work of Chugunova, Pugh, & Taranets.