## Abstract

We consider solutions of the Boltzmann equation, in a d‐dimensional torus, d = 2, 3, (Formula Presented.) For macroscopic times τ = t/ϵ^{N}, ϵ « 1, t ≧ 0, when the space variations are on a macroscopic scale x = ϵ^{N−1}r, N ≧ 2, x in the unit torus. Let u(x, t) be, for t ≦ t_{0}, a smooth solution of the incompressible Navier Stokes equations (INS) for N = 2 and of the Incompressible Euler equation (IE) for N > 2. We prove that (*) has solutions for t ≦ t_{0} which are close, to O(ϵ^{2}) in a suitable norm, to the local Maxwellian [p/(2πT)^{d/2}]exp{−[v − ϵu(x,t)]^{2}/2T} with constant density p and temperature T. This is a particular case, defined by the choice of initial values of the macroscopic variables, of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations. For N ≧ 3 these equations correspond to variable density IE while for N = 2 they involve higher‐order derivatives of the density.

Original language | English (US) |
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Pages (from-to) | 1189-1214 |

Number of pages | 26 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 42 |

Issue number | 8 |

DOIs | |

State | Published - Dec 1989 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics