Diffusive N-waves and metastability in the burgers equation

Yong Jung Kim*, Athanasios Tzavaras

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

34 Scopus citations

Abstract

We study the effect of viscosity on the large time behavior of the viscous Burgers equation by using a transformed version of Burgers (in self-similar variables) that captures efficiently the mechanism of transition to the asymptotic states and allows us to estimate the time of evolution from an N-wave to the final stage of a diffusion wave. Then we construct certain special solutions of diffusive N-waves with unequal masses. Finally, using a set of similarity variables and a variant of the Cole-Hopf transformation, we obtain an integrated Fokker-Planck equation. The latter is solvable and provides an explicit solution of the viscous Burgers equation in a series of Hermite polynomials. This format captures the long-time-small-viscosity interplay, as the diffusion wave and the diffusive N-waves correspond, respectively, to the first two terms in the Hermite polynomial expansion.

Original languageEnglish (US)
Pages (from-to)607-633
Number of pages27
JournalSIAM Journal on Mathematical Analysis
Volume33
Issue number3
DOIs
StatePublished - Jan 1 2001

Keywords

  • Convection-diffusion
  • Diffusion waves
  • Diffusive N-waves
  • Metastability

ASJC Scopus subject areas

  • Analysis
  • Computational Mathematics
  • Applied Mathematics

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