Renormalizing Partial Differential Equations

We explain how to apply Renormalization Group ideas to the analysis of the long-time asymptotics of solutions of partial differential equations. We illustrate the method on several examples of nonlinear parabolic equations. We discuss many applications, including the stability of profiles and fronts in the Ginzburg-Landau equation, anomalous scaling laws in reaction-diffusion equations, and the shape of a solution near a blow-up point. 1 Introduction. The development of a qualitative theory of infinite dimensional dynamical systems is a major scientific challenge. Such systems are expressed through (nonlinear) partial differential equations, and we shall concentrate on equations of the form u t = \Deltau + F (u; ru; rru): (1) where, u(x; t) 2 R (or C ), x 2 R d , t 2 R + and F is nonlinear or random. Usually, the following problems are considered: 1. Existence and regularity of the solution over a finite time interval. 2. Extension of the first problem to an infinite time interval..