Global Existence for a Class of Keyfitz--Kranzer Systems with Application to Thin-Film Flows

TL;DR

This paper proves the existence of global weak entropy solutions for a class of Keyfitz-Kranzer systems, including thin-film flow models, by identifying entropy pairs, invariant regions, and justifying the vanishing-diffusion limit.

Contribution

It introduces a method to establish global solutions for non-symmetric Keyfitz-Kranzer systems with applications to thin-film flows, including entropy analysis and limit justification.

Findings

Established existence of weak entropy solutions for the systems.

Identified entropy/entropy-flux pairs suitable for the models.

Derived a-priori bounds and justified the vanishing-diffusion limit.

Abstract

We prove the existence of global weak entropy solutions for a class of non-symmetric Keyfitz-Kranzer type systems that includes lubrication models for thin-film flow. We identify a family of entropy/entropy-flux pairs for these first-order systems, which is, in particular, admissible for a tailored second-order approximate system. The latter is motivated by higher-order dissipation operators in thin-film flow models. By identifying an invariant region in the state space, it is possible to derive a-priori $L^\infty$-bounds for the sequence of solutions to the approximate system. Exploiting the parabolic and transport structure of the equations associated with the Riemann invariants, we then rigorously justify the vanishing-diffusion limit and establish the existence of weak entropy solutions for the Cauchy problem for the first-order systems.