### 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 language | English |
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Pages (from-to) | 1043-1065 |

Number of pages | 23 |

Journal | SIAM Journal on Applied Dynamical Systems |

Volume | 8 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2009 |

### Keywords

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

## Cite this

*SIAM Journal on Applied Dynamical Systems*,

*8*(3), 1043-1065. https://doi.org/10.1137/08073651X