Incompressible navier‐stokes and euler limits of the boltzmann equation

A. de Masi, R. Esposito, J. L. Lebowitz

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Abstract

We consider solutions of the Boltzmann equation, in a d‐dimensional torus, d = 2, 3, (Formula Presented.) For macroscopic times τ = t/ϵN, ϵ « 1, t ≧ 0, when the space variations are on a macroscopic scale x = ϵN−1r, N ≧ 2, x in the unit torus. Let u(x, t) be, for t ≦ t0, 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 ≦ t0 which are close, to O(ϵ2) 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 languageEnglish (US)
Pages (from-to)1189-1214
Number of pages26
JournalCommunications on Pure and Applied Mathematics
Volume42
Issue number8
DOIs
StatePublished - Dec 1989

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics

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