Nonlinear Stability of Viscous Roll Waves

Extending results of Oh--Zumbrun and Johnson--Zumbrun for parabolicconservation laws, we show that spectral stability implies nonlinear stabilityfor spatially periodic viscous roll wave solutions of the one-dimensional St.Venant equations for shallow water flow down an inclined ramp. The main newissues to be overcome are incomplete parabolicity and the nonconservative formof the equations, which leads to undifferentiated quadratic source terms thatcannot be handled using the estimates of the conservative case. The first isresolved by treating the equations in the more favorable Lagrangiancoordinates, for which one can obtain large-amplitude nonlinear dampingestimates similar to those carried out by Mascia--Zumbrun in the related shockwave case, assuming only symmetrizability of the hyperbolic part. The second isresolved by the observation that, similarly as in the relaxation and detonationcases, sources occurring in nonconservative components experience greater thanexpected decay, comparable to that experienced by a differentiated source.