Nonnegative solutions for a long-wave unstable thin film equation with convection

TL;DR

This paper studies a complex thin film equation modeling liquid flow on a rotating cylinder, proving the existence of nonnegative 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 nonlinear 4th-order 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

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.