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Stability and Asymptotic Behavior of Periodic Traveling Wave Solutions of Viscous Conservation Laws in Several Dimensions

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An Erratum to this article was published on 26 January 2010

Abstract

Under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem, we determine sharp L p estimates on the linearized solution operator about a multidimensional planar periodic wave of a system of conservation laws with viscosity, yielding linearized L 1 ∩ L p → L p stability for all \({p \geqq 2}\) and dimensions \({d \geqq 1}\) and nonlinear L 1 ∩ H s → L p ∩ H s stability and L 2-asymptotic behavior for \({p\geqq 2}\) and \({d\geqq 3}\) . The behavior can in general be rather complicated, involving both convective (that is, wave-like) and diffusive effects.

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Authors and Affiliations

  1. Department of Mathematics, University of Kansas, Kansas, USA

    Myunghyun Oh

  2. Department of Mathematics, Indiana University, Indianapolis, USA

    Kevin Zumbrun

Authors
  1. Myunghyun Oh
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  2. Kevin Zumbrun
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Correspondence to Kevin Zumbrun.

Additional information

Communicated by A. Bressan

Research of M. Oh was partially supported under NSF grant No. DMS-0708554 and K. Zumbrun was partially supported under NSF grants no. DMS-0070765 and DMS-0300487.

An erratum to this article can be found at http://dx.doi.org/10.1007/s00205-010-0291-0

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Oh, M., Zumbrun, K. Stability and Asymptotic Behavior of Periodic Traveling Wave Solutions of Viscous Conservation Laws in Several Dimensions. Arch Rational Mech Anal 196, 1–20 (2010). https://doi.org/10.1007/s00205-009-0229-6

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  • DOI: https://doi.org/10.1007/s00205-009-0229-6

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