Well-posedness and stability for a mixed order system arising in thin film equations with surfactant

TL;DR

This paper establishes well-posedness and asymptotic stability for a complex, coupled thin film equation with surfactant, filling a significant gap in the mathematical analysis of such systems.

Contribution

It provides the first well-posedness result for a mixed order, strongly coupled thin film system with surfactant without smallness restrictions, and proves stability of the flat equilibrium.

Findings

Proved well-posedness of the system.

Established asymptotic stability of the flat equilibrium.

Extended analytical understanding of thin film equations with surfactant.

Abstract

The objective of the present work is to provide a well-posedness result for a capillary driven thin film equation with insoluble surfactant. The resulting parabolic system of evolution equations is not only strongly coupled and degenerated, but also of mixed orders. To the best of our knowledge the only well-posedness result for a capillary driven thin film with surfactant is provided in [4] by the same author, where a severe smallness condition on the surfactant concentration is assumed to prove the result. Thus, in spite of an intensive analytical study of thin film equations with surfactant during the last decade, a proper well-posedness result is still missing in the literature. It is the aim of the present paper to fill this gap. Furthermore, we apply a recently established result on asymptotic stability in interpolation spaces [15] to prove that the flat equilibrium of our system is asymptotically stable.