Scaling limits of waves in convex scalar conservation laws under random initial perturbations

Jan Wehr, Jack Xin

Research output: Contribution to journalArticlepeer-review

Abstract

We study waves in convex scalar conservation laws under noisy initial perturbations. It is known that the building blocks of these waves are shock and rarefaction waves, both are invariant under hyperbolic scaling. Noisy perturbations can generate complicated wave patterns, such as diffusion process of shock locations. However we show that under the hyperbolic scaling, the solutions converge in the sense of distribution to the unperturbed waves. In particular, randomly perturbed shock waves move at the unperturbed velocity in the scaling limit. Analysis makes use of the Hopf formula of the related Hamilton-Jacobi equation and regularity estimates of noisy processes.

Original languageEnglish (US)
Pages (from-to)361-370
Number of pages10
JournalJournal of Statistical Physics
Volume122
Issue number2
DOIs
StatePublished - Jan 1 2006

Keywords

  • Convex scalar conservation laws
  • Noise
  • Scaling limit
  • Waves

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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