Nonnegative Weak Solutions of Thin Film Equations Related to Viscous Flows in Cylindrical Geometries
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.
Keywords
Cite
@article{arxiv.1902.08685,
title = {Nonnegative Weak Solutions of Thin Film Equations Related to Viscous Flows in Cylindrical Geometries},
author = {Jeremy L. Marzuola and Sterling Swygert and Roman Taranets},
journal= {arXiv preprint arXiv:1902.08685},
year = {2019}
}
Comments
16 pages, comments welcome!! Updated to address some questions that arose in a referee report