Communications in Mathematical Sciences

Volume 16 (2018)

Number 3

Thin film flow dynamics on fiber nets

Pages: 763 – 775



Roman M. Taranets (Institute of Applied Mathematics and Mechanics of the NASU, Sloviansk, Ukraine)

Marina Chugunova (Institute of Mathematical Sciences, Claremont Graduate University, Claremont, California, U.S.A.)


We analyze existence and qualitative behavior of non-negative weak solutions for fourth order degenerate parabolic equations on graph domains with Kirchhoff’s boundary conditions at the inner nodes and Neumann boundary conditions at the boundary nodes. The problem is originated from industrial constructions of spray coated meshes which are used in water collection and in oil-water separation processes. For a certain range of parameter values we prove convergence toward a constant steady state that corresponds to the uniform distribution of coating on a fiber net.


thin film equation, graph theory, steady states, existence of weak solutions, energy method, entropy estimates, Galerkin method

2010 Mathematics Subject Classification

05C21, 35K25, 35K35, 35Q35, 76A20

This work was partially supported by a grant from the Simons Foundation (#275088 to Marina Chugunova) and by a grant from Ministry of Education and Science of Ukraine (0118U003138 to Roman Taranets).

Received 24 May 2017

Received revised 27 January 2018

Accepted 27 January 2018

Published 30 August 2018