Existence and stability of nonmonotone hydraulic shocks for the Saint Venant equations of inclined thin-film flow

TL;DR

This paper rigorously analyzes the existence and stability of hydraulic shock profiles in inclined thin-film flows described by Saint Venant equations, extending previous work and confirming dynamics through numerical experiments.

Contribution

It provides a complete classification of hydraulic shock existence and stability for the Saint Venant equations, including explicit calculations and stability diagrams.

Findings

Existence and stability diagrams are rigorously derived.

Numerical experiments confirm asymptotic dynamics in unstable regimes.

Hydrodynamic instability leads to stable shocks or wave patterns.

Abstract

Extending work of Yang-Zumbrun for the hydrodynamically stable case of Froude number F < 2, we categorize completely the existence and convective stability of hydraulic shock profiles of the Saint Venant equations of inclined thin-film flow. Moreover, we confirm by numerical experiment that asymptotic dynamics for general Riemann data is given in the hydrodynamic instability regime by either stable hydraulic shock waves, or a pattern consisting of an invading roll wave front separated by a finite terminating Lax shock from a constant state at plus infinity. Notably, profiles, and existence and stability diagrams are all rigorously obtained by mathematical analysis and explicit calculation.