Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case

作者：Johnson Mathew A; Zumbrun Kevin^*

来源：Journal of Differential Equations, 2010, 249(5): 1213-1240.

DOI：10.1016/j.jde.2010.04.015

• Periodic traveling waves
• Bloch decomposition
• Modulated waves

Extending previous results of Oh-Zumbrun and Johnson-Zumbrun, we show that spectral stability implies linearized and nonlinear stability of spatially periodic traveling wave solutions of viscous systems of conservation laws for systems of generic type, removing a restrictive assumption that wave speed be constant to first order along the manifold of nearby periodic solutions. Key to our analysis is a nonlinear cancellation estimate observed by Johnson and Zumbrun, along with a detailed understanding of the Whitham averaged system. The latter motivates a careful analysis of the Bloch perturbation expansion near zero frequency and suggests factoring out an appropriate translational modulation of the underlying wave, allowing us to derive the sharpened low-frequency estimates needed to close the nonlinear iteration arguments.

• 出版日期2010-9-1