Using Global Invariant Manifolds to Understand Metastability in the Burgers Equation With Small Viscosity

Margaret Beck, C. Eugene Wayne

Research output: Contribution to journalArticle

Abstract

The large-time behavior of solutions to the Burgers equation with small viscosity is described using invariant manifolds. In particular, a geometric explanation is provided for a phenomenon known as metastability, which in the present context means that solutions spend a very long time near the family of solutions known as diffusive N-waves before finally converging to a stable self-similar diffusion wave. More precisely, it is shown that in terms of similarity, or scaling, variables in an algebraically weighted L-2 space, the self-similar diffusion waves correspond to a one-dimensional global center manifold of stationary solutions. Through each of these fixed points there exists a one-dimensional, global, attractive, invariant manifold corresponding to the diffusive N-waves. Thus, metastability corresponds to a fast transient in which solutions approach this "metastable" manifold of diffusive N-waves, followed by a slow decay along this manifold, and, finally, convergence to the self-similar diffusion wave.

Original languageEnglish
Pages (from-to)1043-1065
Number of pages23
JournalSIAM Journal on Applied Dynamical Systems
Volume8
Issue number3
DOIs
Publication statusPublished - 2009

Keywords

  • Burgers equation
  • invariant manifolds
  • metastability
  • scaling variables
  • self-similar
  • LONG-TIME ASYMPTOTICS
  • NAVIER-STOKES
  • PATTERNS
  • SHOCK
  • PROPAGATION
  • CONVERGENCE
  • TURBULENCE
  • DIFFUSION
  • STABILITY
  • MOTION

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