We consider solutions of the Boltzmann equation, in a d-dimensional torus, d = 2,3, for macroscopic times T = t / e N , e a: 1, t 2 0, when the space variations are on a macroscopic scale x = eN-'r, N 2 2, x in the unit torus. Let u ( x , r ) be, for t to, 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 r 6 to which are close, to O(e2) in a suitable norm, to the local Maxwellian [6/(2~T)~/']exp( -[ ueu(x, t)]'/2T} with constant density 6 and temperature 7. 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 2 3 these equations correspond to variable density IE while for N = 2 they involve higher-order derivatives of the density.It is now possible to make various formal expansions of f, in powers of E about some local or global Maxwellian. In particular the well-known Hilbert-Chapman-Enskog method [l] leads to a sequence of equations for the hydrodynamical variables corresponding, respectively, to the (non-dissipative) Euler equations, the Navier-Stokes equations, etc. From now on we always consider a collision operator Q corresponding, a la Grad [2], to cut-off hard potentials. An important step in the mathematical development of the theory was taken by Caflisch [7] (and by Caflisch and Papanicolau [8] for a discrete velocities model) who proved the following: let p, u, T be smooth solutions of Euler equations in a torus for r E [0, to]. Then there exists a solution f E ( r , T ; u ) of the
