# Nonnegative Weak Solutions of Thin Film Equations Related to Viscous Flows in Cylindrical Geometries

## TL;DR

This paper proves the existence of weak solutions for nonlinear, degenerate fourth-order PDEs modeling thin film flows on cylinders, with results on local and global in time solutions under certain conditions.

## Contribution

It establishes the existence of weak solutions for thin film equations in cylindrical geometries, extending previous numerical and theoretical work.

## Key findings

- Existence of local in time weak solutions for all cylinder lengths.
- Global in time weak solutions exist under length constraints.
- Results are motivated by and consistent with numerical studies.

## Abstract

Motivated by models for thin films coating cylinders in two physical cases proposed by V.I. Kerchman and A.L. Frenkel, we analyze the dynamics of corresponding thin film models. The models are governed by nonlinear, fourth-order, degenerate, parabolic PDEs. We prove, given positive and suitably regular initial data, the existence of weak solutions in all length scales of the cylinder, where all solutions are only local in time. We also prove that given a length constraint on the cylinder, long-time and global in time weak solutions exist. This analytical result is motivated by numerical work on related models in the Ph.D. Thesis of R. Ogrosky in conjunction with multiple further works jointly worked on by combinations of Camassa, Forest, Lee, the first author, Ogrosky, Olander, and Vaughn.