Journal of Dynamics and Differential Equations, Vol. 13, No. 2, 2001 Renormalization Group Method. Applications to Partial Differential Equations I. Moise 1 and M. Ziane 2 Received April 3, 1998; revised September 29, 2000 Our aim in this article is to present a simplified form of the renormalization group (RG) method introduced by Chen, Goldenfeld, and Oono and to derive a rigorous study of the validity in time of the asymptotic solutions furnished by the RG method. We apply the renormalization group method to a slightly com- pressible fluid equation and to the SwiftHohenberg equation. KEY WORDS: Renormalization group method; slightly compressible fluids; NavierStokes equations; SwiftHohenberg equation. 1991 Mathematics Subject Classifications: 35Q10, 35Q72, 81Q15, 81T17. 1. INTRODUCTION Recently, Chen, Goldenfeld, and Oono (see [12, 13]) proposed a simple and unified approach for asymptotic analysis based upon renormalization group (RG) ideas. This newly developed RG method seems to overcome many of the drawbacks of the traditional perturbation methods ([12, 13]). The RG method was originally developed by physicists attempting to understand the divergent higher-order terms in quantum electrodynamics. Its further development by Bricmont and Kupiainen (see [9, 10]) is by now well known. The description provided below is a little different from the standard rendition. The main advantage of the RG method is that it provides an algorithm that can be easily applied to many problems, without too much a priori knowledge. The starting point is a naive perturbation expansion, so that one does not need to guess or to introduce unexpected asymptotic sequences. 275 1040-7294010400-027519.500  2001 Plenum Publishing Corporation 1 Department of Mathematics, University of Texas at Austin, Texas 78712. 2 Department of Mathematics, Texas A 6 M University, College Station, Texas 77843.