Incompressible navier‐stokes and euler limits of the boltzmann equation

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−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.