## 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 language | English (US) |
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Pages (from-to) | 607-633 |

Number of pages | 27 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 33 |

Issue number | 3 |

DOIs | |

State | Published - 2001 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Applied Mathematics