Technical report on a long-wave unstable thin film equation with convection

TL;DR

This technical report analyzes a complex nonlinear thin film equation modeling liquid film dynamics on a rotating cylinder, proving existence of solutions, their bounded growth, and supporting findings with numerical simulations.

Contribution

It establishes the existence of nonnegative periodic weak solutions and their linear growth bounds for a degenerate parabolic PDE modeling thin film flow.

Findings

Existence of nonnegative periodic weak solutions.

Solutions and gradients grow at most linearly in time.

Numerical simulations illustrating solution behavior.

Abstract

In this technical report, we consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. Depending on the initial data and the parameter values, we prove the existence of nonnegative periodic weak solutions. In addition, we prove that these solutions and their gradients cannot grow any faster than linearly in time; there cannot be a finite-time blow-up. Finally, we present numerical simulations of solutions.