Spectral Stability of Periodic Solutions of Viscous Conservation Laws: Large Wavelength Analysis

Abstract

We complete and unify the works by Oh and Zumbrun (2003a) and by the author (1994), about the spectral stability of traveling waves that are spatially periodic, in systems of n conservation laws. Our context is one-dimensional. These systems are of order larger than one, in general. For instance, they could be viscous approximations of first-order systems that are not everywhere hyperbolic. However, modelling considerations often lead to higher order terms, like capillarity in fluid dynamics; our framework remains valid in this more general setting. We make generic assumptions, saying in particular that the set of periodic traveling waves is a manifold of maximal dimension, under the restrictions given by the conserved quantities. The spectral stability of a periodic traveling wave is studied through Floquet's theory. Following Gardner (1993), we introduce an Evans function D(λ, θ), being λ the Laplace frequency and θ the phase shift. The large wavelength analysis is the description of the zero set of D around the origin. Our main result is that this zero set is described, at the leading order, by a characteristic equation This formula involves a flux F, which enters into a first-order system of conservation laws, describing the slow modulation of the periodic traveling waves. Its size N is in practice larger than n. The important consequence is that hyperbolicity of the latter system is a necessary condition for spectral stability of periodic traveling waves. Finally, we show that a similar treatment works for coupled map lattices obtained by discretizing systems of conservation laws.