Periodic-coefficient damping estimates, and stability of large-amplitude roll waves in inclined thin film flow

TL;DR

This paper addresses the nonlinear stability of large-amplitude roll waves in inclined thin film flow by replacing a problematic slope condition with an averaged version, extending the theory to larger Froude numbers relevant in engineering.

Contribution

It introduces an averaged slope condition that replaces the original pointwise condition, enabling stability analysis for higher Froude numbers in thin film flow models.

Findings

The averaged slope condition always holds, allowing stability results for Froude numbers 3 to 5.

The approach extends the nonlinear stability theory to larger Froude numbers.

The analysis connects to Kawashima-type damping estimates used in viscous shock wave stability.

Abstract

A technical obstruction preventing the conclusion of nonlinear stability of large-Froude number roll waves of the St. Venant equations for inclined thin film flow is the "slope condition" of Johnson-Noble-Zumbrun, used to obtain pointwise symmetrizability of the linearized equations and thereby high-frequency resolvent bounds and a crucial H s nonlinear damping estimate. Numerically, this condition is seen to hold for Froude numbers 2 \textless{} F 3.5, but to fail for 3.5 F. As hydraulic engineering applications typically involve Froude number 3 F 5, this issue is indeed relevant to practical considerations. Here, we show that the pointwise slope condition can be replaced by an averaged version which holds always, thereby completing the nonlinear theory in the large-F case. The analysis has potentially larger interest as an extension to the periodic case of a type of weighted "Kawashima-type" damping estimate introduced in the asymptotically-constant coefficient case for the study of stability of large-amplitude viscous shock waves.