Periodic-Coefficient Damping Estimates, and Stability of Large-Amplitude Roll Waves in Inclined Thin Film Flow

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 nonlinear damping estimate. Numerically, this condition is seen to hold for Froude numbers 2 < 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.