3,017 research outputs found

    Nonlinear Stability of Periodic Traveling Wave Solutions of Viscous Conservation Laws in Dimensions One and Two

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    This is the published version, also available here: http://dx.doi.org/10.1137/100781808.Extending results of Oh and Zumbrun in dimensions d≥3d\geq3, we establish nonlinear stability and asymptotic behavior of spatially periodic traveling-wave solutions of viscous systems of conservation laws in critical dimensions d=1,2d=1,2, under a natural set of spectral stability assumptions introduced by Schneider in the setting of reaction diffusion equations. The key new steps in the analysis beyond that in dimensions d≥3d\geq3 are a refined Green function estimate separating off translation as the slowest decaying linear mode and a novel scheme for detecting cancellation at the level of the nonlinear iteration in the Duhamel representation of a modulated periodic wave

    Space-modulated Stability and Averaged Dynamics

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    In this brief note we give a brief overview of the comprehensive theory, recently obtained by the author jointly with Johnson, Noble and Zumbrun, that describes the nonlinear dynamics about spectrally stable periodic waves of parabolic systems and announce parallel results for the linearized dynamics near cnoidal waves of the Korteweg-de Vries equation. The latter are expected to contribute to the development of a dispersive theory, still to come.Comment: Proceedings of the "Journ\'ees \'Equations aux d\'eriv\'ees partielles", Roscoff 201

    Whitham Averaged Equations and Modulational Stability of Periodic Traveling Waves of a Hyperbolic-Parabolic Balance Law

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    In this note, we report on recent findings concerning the spectral and nonlinear stability of periodic traveling wave solutions of hyperbolic-parabolic systems of balance laws, as applied to the St. Venant equations of shallow water flow down an incline. We begin by introducing a natural set of spectral stability assumptions, motivated by considerations from the Whitham averaged equations, and outline the recent proof yielding nonlinear stability under these conditions. We then turn to an analytical and numerical investigation of the verification of these spectral stability assumptions. While spectral instability is shown analytically to hold in both the Hopf and homoclinic limits, our numerical studies indicates spectrally stable periodic solutions of intermediate period. A mechanism for this moderate-amplitude stabilization is proposed in terms of numerically observed "metastability" of the the limiting homoclinic orbits.Comment: 27 pages, 5 figures. Minor changes throughou

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

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    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.Comment: Fixed minor typo

    On the Modulation Equations and Stability of Periodic GKdV Waves via Bloch Decompositions

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    In this paper, we complement recent results of Bronski and Johnson and of Johnson and Zumbrun concerning the modulational stability of spatially periodic traveling wave solutions of the generalized Korteweg-de Vries equation. In this previous work it was shown by rigorous Evans function calculations that the formal slow modulation approximation resulting in the Whitham system accurately describes the spectral stability to long wavelength perturbations. Here, we reproduce this result without reference to the Evans function by using direct Bloch-expansion methods and spectral perturbation analysis. This approach has the advantage of applying also in the more general multi-periodic setting where no conveniently computable Evans function is yet devised. In particular, we complement the picture of modulational stability described by Bronski and Johnson by analyzing the projectors onto the total eigenspace bifurcating from the origin in a neighborhood of the origin and zero Floquet parameter. We show the resulting linear system is equivalent, to leading order and up to conjugation, to the Whitham system and that, consequently, the characteristic polynomial of this system agrees (to leading order) with the linearized dispersion relation derived through Evans function calculation.Comment: 19 pages

    Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions

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    Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem, we determine sharp LpL^p estimates on the linearized solution operator about a multidimensional planar periodic wave of a system of conservation laws with viscosity, yielding linearized L1∩Lp→LpL^1\cap L^p\to L^p stability for all p≥2p \ge 2 and dimensions d≥1d \ge 1 and nonlinear L1∩Hs→Lp∩HsL^1\cap H^s\to L^p\cap H^s stability and L2L^2-asymptotic behavior for p≥2p\ge 2 and d≥3d\ge 3. The behavior can in general be rather complicated, involving both convective (i.e., wave-like) and diffusive effects

    Nonlinear stability of viscous roll waves

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    Extending results of Oh--Zumbrun and Johnson--Zumbrun for parabolic conservation laws, we show that spectral stability implies nonlinear stability for spatially periodic viscous roll wave solutions of the one-dimensional St. Venant equations for shallow water flow down an inclined ramp. The main new issues to be overcome are incomplete parabolicity and the nonconservative form of the equations, which leads to undifferentiated quadratic source terms that cannot be handled using the estimates of the conservative case. The first is resolved by treating the equations in the more favorable Lagrangian coordinates, for which one can obtain large-amplitude nonlinear damping estimates similar to those carried out by Mascia--Zumbrun in the related shock wave case, assuming only symmetrizability of the hyperbolic part. The second is resolved by the observation that, similarly as in the relaxation and detonation cases, sources occurring in nonconservative components experience greater than expected decay, comparable to that experienced by a differentiated source.Comment: Revision includes new appendix containing full proof of nonlinear damping estimate. Minor mathematical typos fixed throughout, and more complete connection to Whitham averaged system added. 42 page
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