ON A CERTAIN RENORMALIZATION GROUP METHOD M. Ziane Department of Mathematics, Texas A & M University, College Station, TX 77843. Abstract In this note, a mathematical study of the renormalization group method , recently intro- duced by Chen, Goldenfeld and Oono [CGO1,2,3], is given for the case of autonomous nonlinear systems of differential equations. We also observe that the approximation results obtained by this method in [CGO3] are valid over long time intervals. Moreover, a connection between this method and the classical Poincar´e-Dulac normal forms and the averaging method is briefly discussed. 0. Introduction Recently, a perturbative renormalization group method was developed by Chen, Goldenfeld and Oono as a unified tool for asymptotic analysis. The origin of the CGO-RG method goes back to perturbative quantum field theory and critical phenomena, see Goldenfeld [G] and Zinn-Justin [Z], and the notion of intermediate asymptotics developed by Barenblatt [B]. Its effectiveness was illustrated in several examples of ordinary differential equations involving multiple scales, boundary layers and WKB analysis, see [CGO1,2,3]. The CGO-renormalization group method, as observed in [CGO1,2,3], does not require ad hoc assumptions about the structure of the perturbation series nor the use of asymptotic matching. The rationale for the present work is CGO’s explicit plea for a mathematical study of the RG method. Thus, our aim in this article is to give a contribution towards the understanding of the CGO-RG method. We will study in this paper the following system of differential equations, which covers most of the examples illustrated in [CGO1,2,3]. duε 1 + Auε = F (uε), dτ ε ε ε u |τ=0 = u0, Key words and phrases. Renormalization, Resonances, Oscillations, Poincar´enormal forms, Averaging method, Multiple-scale technique. Typeset by AMS-TEX 2 M. ZIANE where A is a complex matrix, assumed for simplicity to be diagonalizable, F is a polynomial nonlinear term, and ε is a small parameter. We will follow the three steps of the CGO-RG method and derive the renormalization group equation when F (u) is a polynomial in u. First, we write a naive perturbation expansion, which contains (in general) secular terms. Then, following [CGO1,2,3], we introduce a free parameter σ in order to remove the “possible” secular terms. Finally, the RG equations are derived using the fact that the approximate solution of the perturbed problem should not depend on the free parameter σ. We show, in Section 1, that the RG equation is given by du¯ = R(¯u), (RG) dτ u¯|τ=0 = u0, where R(u) is the resonant part of F (u) relative to the operator A, i.e., etAR(e−tAv) = R(v), ∀v. Hence (RG) can be written in the equivalent form dU¯ + AU¯ = R(U¯), dτ U¯|τ=0 = u0, where U¯ =e−tAu¯. τ In section 2, we derive estimates on the difference between uε and e− ε Au¯ over time intervals independent of ε. Finally, in Section 3, we show how this general setting applies to the examples of [CGO1,2,3] and we also estimate the error. In this paper, we are mainly concerned with the finite dimensional case (a system of nonlinear differential equations), but the approach given herein extens as well to the infinite dimensional case, for instance the method is applied in [MZ] to partial differential equations involving multiple scale phenomena, such as the Navier-Stokes equations of slightly compressible fluids and the Navier- Stokes equations of rotating fluids. Numerical investigations based on the RG method are given in [MSTZ] 1. The CGO-RG method and the formal derivation of the RG equation We consider the following system of differential equations: duε 1 + Auε = F (uε), (1.1) dτ ε ε ε u |τ=0 = u0. RENORMALIZATION GROUP METHOD 3 ε ε ε ε Our aim is to study the asymptotic behavior of the solution u (τ)=(u1(τ), u2(τ),...,un(τ)) of (1.1) as ε → 0. We denote by λ1, λ2,...,λn, with λi ∈ C, the eigenvalues of A and assume that A is diagonalizable; the general case of any “smooth” time-dependent matrix A can be treated using a decomposition of A as A = A1 + A2, where A1 is semisimple and A2 is nilpotent with A1A2 = A2A1. This will be done elsewhere. Since we are assuming that A is diagonalizable, we may suppose without loss of generality that A = diag (λ1, λ2,...,λn) and we write the nonlinear term in the basis of eigenvectors of A as α (1.2) F (u) = Cαu , |αX|≤m t 1 n α α1 αn where Cα = [Cα,...,Cα ] and u = u1 ...un , α = (α1,...,αn). Note here that F (0) and DF (0) are not necessarily zero. τ Introducing a new time t = , we rewrite equation (1.1) in the form of a weakly nonlinear ε problem duε + Auε = εF (uε), (1.3) dt ε ε u |t=0 = u0. Step1. The naive expansion. We write the naive perturbation expansion: (1.4) uε = u(0) + εu(1) + ε2u(2) + . and derive formally du(0) (1.5) + Au(0) =0, i.e. u(0)(t)=e −tAv(t ), dt 0 where v(t0) is some vector function; its relation to uε(0) will be made precise later on. Then, at next order: du(1) (1.6) + Au(1) = F (u(0)). dt Solving (1.6), we find a solution t (1) −tA −tA sA −sA (1.7) u (t)=e w(t0)+e e F (e v(t0))ds. Zt0 ε Now we write the approximation of u found so far to be valid locally in some neighborhood of t0. We have ε (0) (1) U1 (t) = u (t) + εu (t) t −tA sA −sA =e v(t0) + εw(t0) + ε e F e v(t0) ds . Z t0 4 M. ZIANE We note that, since we are interested in approximations up to order O(ε), the vector w(t0) is ε irrelevant and may be taken to be zero. In fact, letv ˜(t0) = v(t0) + εw(t0). Remembering that U1 (t) is sought to be an approximation valid up to order O(ε) and any term O(ε2) may be neglected, we have t ε −tA sA −sA −sA U (t)=e v˜(t0) + ε e F e v˜(t0) − εe w(t0) ds 1 Z t0 t −tA sA −sA (1.8) =e v˜(t0) + ε e F e v˜(t0) + O(ε) ds Z t0 t −tA sA −sA 2 =e v˜(t0) + ε (e F e v˜(t0) ds + O(ε ). Z t0 Hence we may take w(t0) = 0 and write t ε −tA sA −sA (1.9) U (t)=e v(t0) + ε e F e v(t0) ds . 1 Z t0 Step2. The introduction of a free parameter. We introduce a free parameter σ, t0 ≤ σ ≤ t and use the additivity of integrals; we find σ t ε −tA sA −sA sA −sA (1.10) U (t)=e v(t0) + ε e F e v(t0) ds + ε e F e v(t0) ds . 1 Z Z t0 σ Set then σ sA −sA (1.11) v(σ) = v(t0) + ε e F e v(t0) ds Z t0 to obtain t (1.12) U ε(t)=e−tA v(σ) + ε esAF e−sAv(σ) ds + O(ε2), 1 Z σ and σ is a free parameter. One needs to control secular terms, which appear because of the presence of resonances in the spectrum of A. More precisely, in order for (1.5) to be valid, we need to have the size of u(1) to be of the same order as that of u(0). However, in the expression of u(1) we have the term t sA −sA e F (e v(t0))ds, which may contain (and usually it does) secular terms, which are the Zt0 sA −sA terms proportional to t. These terms are generated by the constant part of e F (e v(t0)). Thus we write sA −sA (1.13) e F (e v(t0)) = R(v(t0)) + Q(t, v(t0)), RENORMALIZATION GROUP METHOD 5 where n i α Rn (1.14) R(v) = Cαv ei, v ∈ , i Xi=1 αX∈Nr n i t(λi−(Λ,α)) α (1.15) Q(t, v) = Cαe v ei, i Xi=1 α/X∈Nr with n (1.16) N i = α ∈ Nn, |α| ≤ m; (Λ, α) := α λ = λ , r k k i Xk=1 and (ei)1≤i≤n is the set of eigenvectors of the matrix A. Note that tA −tA (1.17) e R(e v(t0)) = R(v(t0)). Using (1.7) and (1.13)-(1.15), we can write n i t(λi−(Λ,α)) tA 1 Cαe α e u (t) = R(v(t0))(t − t0) + v ei + W (t0). i λi − (Λ, α) Xi=1 α/X∈Nr Again, we claim here that W (t0) is irrelevant and may be taken equal to zero, and we can still perform the RG method (see the argument given in (1.8)-(1.10)). Taking into account (1.5) and (1.17), we rewrite (1.12) as n i t(λi−(Λ,α)) tA ε Cαe α 2 (1.18) e u (t) = v(σ) + εR(v(σ))(t − σ) + ε v (σ) ei + O(ε ). i λi − (Λ, α) Xi=1 α/X∈Nr The shortcoming of the naive perturbation is the presence of the secular term v(σ)+εR(v(σ))(t−σ). Several Methods have been proposed to overcome this problem, for instance Poincar´e- Lindstedt method [P], multiple scale method [CK], and the method of averaging [BM].