Application of the Renormalization-group Method to the Reduction of Transport Equations

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.