On the stability of a forward-backward heat equation

TL;DR

This paper investigates the spectral properties and stability of a class of non-self-adjoint periodic Sturm-Liouville problems arising from fluid dynamics, focusing on resolvent operator properties and evolution equation regularity.

Contribution

It provides new insights into the spectral stability and Schatten class inclusions of the resolvent for a family of singular, non-self-adjoint Sturm-Liouville problems.

Findings

Spectral properties include purely real spectrum despite non-self-adjointness.

Determined Schatten class inclusions for the resolvent operator.

Analyzed regularity properties of the associated evolution equation.

Abstract

In this paper we examine spectral properties of a family of periodic singular Sturm-Liouville problems which are highly non-self-adjoint but have purely real spectrum. The problem originated from the study of the lubrication approximation of a viscous fluid film in the inner surface of a rotating cylinder and has received a substantial amount of attention in recent years. Our main focus will be the determination of Schatten class inclusions for the resolvent operator and regularity properties of the associated evolution equation.