Construction of Invariant Manifold by Renormalization-group Method: reduction of dynamical systems and its applications

We have first given a comprehensive review of the renormalization group(RG) method for global and asymp-totic analysis on the basis of the following articles[l, 2, 3, 4, 5, 6]: An emphasis is put on the relevance to the classical theory of envelopes and the existence of invariant manifolds of the dynamics under consideration. We $cla\dot{n}6^{r} $ that an oesential point of the method is to convert the problem from solving differential equations to ob-taining 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 relativistic Navier-Stokes equation from the Boltz-mann equation$[7, 8] $ , as an example of the reduction of dynamics; the non-relativistc case was already treated successfully in $[1, 6] $. It tums out that the derived equation in the par-ticle frame is a stable dissipative relativistic hydrodynamic equation [8]. We indicate that the usual constraint on the dissipative part of the energy-momentum tensor $\delta T^{\mu\nu} $ after Eckart in the particle frame is not compatible with the underlying relativistic Boltzmann equation. We demonstrate that the solution around the thermal equilibrium state ob