Behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations

Abstract

Abstract. We establish nonlinear stability and asymptotic behavior of traveling periodic waves of viscous conservation laws under localized perturbations or nonlocalized perturbations asymptotic to constant shifts in phase, showing that long-time behavior is governed by an associated secondorder formal Whitham modulation system. A key point is to identify the way in which initial perturbations translate to initial data for this formal system, a task accomplished by detailed estimates on the linearized solution operator about the background wave. Notably, our approach gives both a common theoretical treatment and a complete classification in terms of "phase-coupling" or "-decoupling" of general systems of conservation or balance laws, encompassing cases that had previously been studied separately or not at all. At the same time, our refined description of solutions gives the new result of nonlinear asymptotic stability with respect to localized perturbations in the phase-decoupled case, further distinguishing behavior in the different cases. An interesting technical aspect of our analysis is that for systems of conservation laws the Whitham modulation description is of system rather than scalar form, as a consequence of which renormalization methods such as have been used to treat the reaction-diffusion case in general do not seem to apply.

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

“…More generally, by adapting the previous computations to higher-order estimates, along the lines of the method expounded in next subsection, one proves the following result required by the analysis of [JZN11,JNRZ14].…”

“…The purpose of the present brief note is to show, by a refined version of the energy estimates of [JZN11,BJRZ11], that the pointwise condition (1.3) can be replaced by an averaged version that is always satisfied, while still retaining the high-frequency resolvent and nonlinear damping estimates needed for the nonlinear analysis of [JZN11,JNRZ14], thus effectively completing the nonlinear stability theory.…”

“…High-frequency resolvent bound. An important part of the proofs in [JZN11,JNRZ14] is dedicated to estimates of semigroups generated by linearization around a given wave, to be used in an integral formulation of the original nonlinear systems. These estimates are deduced from spectral considerations and the noncritical part of the linearized evolution is directly controlled by an abstract spectral gap argument that only requires uniform bounds on certain resolvents.…”

L1(R)-Nonlinear Stability of Nonlocalized Modulated Periodic Reaction-Diffusion Waves

“…Stable diffusive mixing of periodic reaction-diffusion waves has been obtained in [6] based on a nonlinear decomposition of phase and amplitude variables and renormalization techniques. Johnson, Zumbrun, and their collaborators also showed (R) ( ≥ 2) nonlinear modulational stability of periodic traveling waves of systems of reaction-diffusion equations and of conservation under both localized and nonlocalized perturbations in [1,[7][8][9]. By using pointwise linear estimates together with a nonlinear iteration scheme developed by Johnson-Zumbrun, pointwise nonlinear stability of such 2 Advances in Mathematical Physics waves has been also studied in [2,3,10].…”

“…For 2 ≤ ≤ ∞, (R)-estimates of such nonlocalized modulated perturbations have been already established in [1] and [9] for systems of reaction-diffusion equations and of conservation laws, respectively, by using the generalized Hausdorff-Young inequality…”