### Abstract

The Chapman-Enskog-Hilbert expansion is a method for describing a gas in the "hydrodynamical stage," beginning from the Boltzmann equation. The present paper is devoted to the analog of this expansion for the problem ∂p/∂t + e∂p/∂x = p(-e) - p(+e), where e = ±1 and x ∈ R^{1}. Though the situation is vastly simpler than in the Boltzmann case, new and amusing mathematical phenomena are encountered. One studies solutions of ∂p/∂t + e∂p/∂x = ∈ ^{-1}[p(-e) - p(+e)] which are (formal) power series in ∈ (Hilbert solutions): such a solution solves ∂p/∂t = ∈^{-1}[(1 + ∈^{2}∂^{2}/∂x^{2})^{1/2} - 1]p (hydrodynamical equation) and is completely determined by the initial value of p(-e) + p(+e) (Hilbert paradox). Also, every solution of the original problem comes very rapidly close to a Hilbert solution which is actually convergent (hydrodynamical stage).

Original language | English (US) |
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Pages (from-to) | 547-552 |

Number of pages | 6 |

Journal | Journal of Mathematical Physics |

Volume | 8 |

Issue number | 3 |

DOIs | |

State | Published - 1967 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Journal of Mathematical Physics*,

*8*(3), 547-552. https://doi.org/10.1063/1.1705230