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Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case. (English) Zbl 1198.35027
Summary: Extending previous results of M. Oh and K. Zumbrun [Arch. Ration. Mech. Anal. 196, No. 1, 1–23 (2010; Zbl 1197.35075)] and Johnson and 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.

MSC:
35B35 Stability in context of PDEs
35C07 Traveling wave solutions
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35L65 Hyperbolic conservation laws
35L45 Initial value problems for first-order hyperbolic systems
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