  We first give a comprehensive review of the renormalization group method for
global and asymptotic analysis, putting an emphasis on the relevance to the
classical theory of envelopes and on the importance of the existence of
invariant manifolds of the dynamics under consideration. We clarify that an
essential point of the method is to convert the problem from solving
differential equations to obtaining suitable initial (or boundary)
conditions:The RG equation determines the slow motion of the would-be integral
constants in the unperturbative solution on the invariant manifold. The RG
method is applied to derive the Navier-Stokes equation from the Boltzmann
equation, as an example of the reduction of dynamics. We work out to obtain the
transport coefficients in terms of the one-body distribution function.
