Incompressible navier‐stokes and euler limits of the boltzmann equation
Abstract We consider solutions of the Boltzmann equation, in a d ‐dimensional torus, d = 2, 3, magnified image 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 − ϵ u ( x,t )] 2 /2 T } 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.