ON a CERTAIN RENORMALIZATION GROUP METHOD M. Ziane

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 Poincaré-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]. The renormalization group method, as developed by Goldenfeld and his group, is based on a diﬀerent trick: the naive perturbation is independent on the parameter σ, hence the approximate solution should not depend on σ (at least up to order O(ε)). Therefore, we have formally ∂ (e tAuε(t)) = O(ε2), ∀σ, ∂σ which implies that ∂ ∂ ∂v(σ) v(σ) − εR(v(σ)) + ε R(v(σ))(t − σ) ∂σ ∂v ∂σ n i t(λi−(Λ,α)) ∂v(σ) ∂ Cαe α 2 + ε v (σ) ei = O(ε ). ∂σ ∂v i λi − (Λ, α) Xi=1 α/X∈Nr 6 M. ZIANE

∂ Hence, we must have ∂σ v(σ) = O(ε), and then obviously ∂ v(σ) = εR(v(σ)) + O(ε2), ∀σ. ∂σ

This is the renormalization group (RG) equation as called by Chen, Goldenfeld and Ono in [CGO1,2,3]. An alternative way, which is simpler, for deriving the (RG) equation follows from the following observation: In order to kill the secular term εR(v(σ))(t − σ), we look at the term v(σ) + εR(v(σ))(t − σ) as the Taylor expansion of order 1 of some function V ε(t) around t = σ. Hence, we introduce the following renormalization group diﬀerential system:

dV ε = εR(V ε(t)), (1.19)  dt  ε V |t=t0 = v(t0).  The renormalized equation written above coincides with the one obtained in [CGO1,2,3]. Further- more, in the case where the real parts of the λis, ∀i =1,...,n, are zero, the RG equations are just the averaged equation in the theory of averaging, see [BM]. Note also that the nonlinear term in the (RG) equation is the the ﬁrst order Poincar´enormal form of the nonlinear term of (1.1), see [A]. Suppose now that we can solve the RG equation. Then, setting σ = t in (1.18), we obtain the following ansatz approximate solution:

n i −t(Λ,α) ε −tA ε Cαe ε α 2 (1.20)u ˜ (t)=e V (t) + (V ) ei + O(ε ), i λi − (Λ, α) Xi=1 α/X∈Nr

A straightforward calculation gives

ε n i −(Λ,α)t du˜ ε −tA ε Cαe d ε α 2 (1.21) + Au˜ = εF e V (t) + ε (V ) ei + O(ε ). dt i λi − (Λ, α) dt Xi=1 α/X∈Nr

Let wε(t) = uε(t) − u˜ε(t). We have thanks to (1.1) and (1.21)

ε n i −(Λ,α)t dw ε ε −tA ε Cαe d ε α (1.22) + Aw = ε F (u ) − F e V (t) − ε (V ) ei. dt i λi − (Λ, α) dt Xi=1 α/X∈Nr

Therefore,

t ε −(t−t0)A ε −tA sA ε −sA ε w (t)=e w (t0) + εe e F u (s) − F e V (s) ds, Z t0 (1.23) t − εe−tA esAN ε(s)ds, Zt0 RENORMALIZATION GROUP METHOD 7 where

n i −(Λ,α)s ε Cαe d ε α (1.24) N (s) = (V ) (s)ei. i λi − (Λ, α) ds Xi=1 α/X∈Nr

So far no assumptions have been made on the complex eigenvalues of A. We will treat two cases:

• Case I. There exists a constant K, such that

etA ≤ 1, ∀ t ∈ R, (1.25) T T and, for some T > 0, V ε( t ) ≤ K, ∀ t ∈ − , . ε ε

Note here that, since we are in the ﬁnite dimensional case, the norm · is any of the equivalent C norms on Mn( ), the space of n × n matrices with complex entries. • Case II. There exist two constants K > 0 and κ > 0, such that

Re λi ≥ κ, ∀ i ∈{1,...,n}, (1.26) and e−tAV ε(t) ≤ K, ∀ t ≥ 0.

In Case I, we are assuming that Re λi =0, ∀i =1,...,n, we may have some of the eigenvalues equal to zero. One may look at the problem of finding asymptotics in this case in terms of the “small time” τ and notice that we have a highly oscillatory problem; only weak convergence may be obtained from studying the system (1.1) directly. However, using the technique described above and the Theorem 1 below, we are able to control the oscillations and obtain an oscillatory corrector.This can be done even in the case of partial differential equations, see [MZ]. Uniform estimates are obtained for the difference between the actual solution and the oscillatory corrector. In case II, the situation is completely different, since we have dissipation. We do not have oscillatory terms (in general), but we have terms presenting a boundary layer type behavior. We include this case to cover some of the examples on boundary layers in [CGO1,2,3].

2. The estimates

The estimates for Case I. We have the following:

Theorem 1. Let uε be the solution of

duε 1 + Auε = F (uε),  dτ ε  ε ε u |τ=0 = u0, .  8 M. ZIANE and assume that

τ e ε A ≤ 1, ∀ τ ∈ R,

(H1) and for some arbitrarie large T > 0, V ε(τ) ≤ K, ∀ τ ∈ − T, T ; where V ε(τ) is the solution of dV ε = R(V ε(τ)),  dτ  ε V |τ=0 = v0. Let 

τ n i (λi−(Λ,α)) τ C e ε ε − ε A ε α ε α (1.27)u ˜ (τ) = e V (τ) + (V ) (τ) ei, i λi − (Λ, α) Xi=1 α/X∈Nr

Then there exists a constant C depending on K, T , on the spectrum of A and the solution of the renormalization equation, such that

(1.28) sup ||uε(τ) − u˜ε(τ)|| ≤ Cε. −T ≤τ≤T

Proof. We will work in terms of the time t. Thanks to (1.23) and H1, we have

t t ε ε ε (s−t)A ε (1.29) ||w (t)|| ≤ ||w (t0)|| + ε ||G (s)||ds + ε e N (s)ds , 0 Z Z t0 t0 where Gε(s) = F u˜ε + wε − F e−sAV ε(s) . We writeu ˜ε as

n i −t(Λ,α) ε −sA ε ε ε Cαe ε α u˜ (s)=e V (s) + εvN (s), where vN (s) = (V ) (s)ei. i λi − (Λ, α) Xi=1 α/X∈Nr

Thanks to assumption (H1), there exists a constant C1 depending on K2, n and the nonlinear term F , such that

T T (1.30) ||R(V ε(t))|| ≤ C , ∀ t ∈ − , , 1 ε ε n i −t(Λ,α) ε Cαe ε α T T (1.31) ||vN (t)|| = (V ) (t)ei ≤ C1, ∀ t ∈ − , , i λi − (Λ, α) ε ε Xi=1 α/X∈Nr

d T T (1.32) V α(t) ≤ εC , ∀ t ∈ − , . dt 1 ε ε

RENORMALIZATION GROUP METHOD 9

Note that under assumption (H1) and (1.31), there exists a constant C2, independent of ε, such that

T T (1.33) ||u˜ε(t)|| ≤ C , ∀ t ∈ − , . 2 ε ε

Hence, there exists a constant C3 depending on C1 such that

T T (1.34) ||N ε(t)|| ≤ C ε, ∀ t ∈ − , . 3 ε ε Furthermore,

||Gε(s)|| ≤ ||F (˜uε + wε) − F (˜uε)|| + ||F (˜uε) − F e−sAV ε(s) || ≤ sup ||D(F )(˜uε(s)+(1 − β)wε(s))|| ||wε(s)|| 0≤β≤1

(1.35) ε ε ε + ε sup ||D(F )(˜u (s)+(1 − β)vN (s))|| ||vN (s)|| 0≤β≤1

ε ε ε ≤ sup ||D(F )(˜u (s)+(1 − β)w (s))|| ||w (s)|| + C4ε, 0≤β≤1 where C4 is a constant independent of ε. Combining (1.30)-(1.35), we obtain

t ε ε 2 ε T T (1.36) ||w (t)|| ≤ ||w (t0)|| + C5ε t + ε ηε(s)||w (s)||ds, − ≤ t ≤ , Zt0 ε ε where

ε ε (1.37) ηε(s)= sup ||D(F )(˜u (s)+(1 − β)w (s))||. 0≤β≤1

T T Now, assume ||wε(t )|| ≤ K ε; we have for − ≤ t ≤ , 0 0 ε ε

t ε ε (1.38) ||w (t)|| ≤ K0ε + +C5εT + ε ηε(s)||w (s)||ds. Zt0

Deﬁne

t ε ε (1.39) y (t) = ηε(s)||w (s)||ds. Zt0

¿From (1.38), we obtain

′ (1.40) y (t) ≤ (K0 + C5εT )εηε(t) + εηε(t)y(t), 10 M. ZIANE which implies that

t t (1.41) y(t) ≤ (K0 + C5εT )ε ηε(s)exp ε ηε(r)dr ds. Zt0 Zs

T T Let I =] − T , T [⊂] − , [ be the maximal interval such that 1 2 ε ε

ε (1.42) ||w (t)|| ≤ 1, ∀ t ∈] − T1, T2[.

Then, there exists a constant K6 such that ηε(s) ≤ K6 for s ∈ I, and

C6T (1.43) y(t) ≤ (K0 + C5εT )εe C6|t|, ∀ t ∈] − T1, T2[.

Hence

ε C6T 2 (1.44) ||w (t)|| ≤ (K0 + C5εT )ε +(K0 + C5εT )e C6|t|ε , which implies that

T T (1.45) |wε(t)|| ≤ Cε, ∀t ∈] − T , T [⊂] − , [, 1 2 ε ε where C is obviously independent on T1 and T2. This is in contradiction with the maximality condition of I given in (1.42). Therefore,

T T (1.46) ||wε(t)|| ≤ 1, ∀ t ∈] − , [, ε ε and the argument above shows that

T T (1.47) ||wε(t)|| ≤ Cε, ∀ t ∈] − , [. ε ε

Going back to the time τ = εt, we conclude the proof of the theorem. RENORMALIZATION GROUP METHOD 11

• The estimates for Case II.

Theorem 2. Let uε be the solution of

duε 1 + Auε = F (uε),  dτ ε  ε ε u |τ=0 = u0,  and assume that there exist constants K3 and κ > 0, such that

Re λi ≥ κ, ∀ i ∈{1,...,n}, (1.48) τ − ε A ε and e V (τ) ≤ K3, ∀ τ ≥ 0.

where V ε(τ) is the solution of dV ε = R(V ε(τ)),  dτ  ε V |τ=0 = v0. Then, if 

τ n i (λi−(Λ,α)) τ C e ε ε − ε A ε α ε α (1.49)u ˜ (τ) = e V (τ) + (V ) (τ) ei, i λi − (Λ, α) Xi=1 α/X∈Nr there exists a constant K4 depending on K3, n, the spectrum of A and the nonlinear term F such that

ε ε (1.50) sup ||u (τ) − u˜ (τ)|| ≤ K4ε. τ≥0

Proof. Thanks to (1.17) and (1.48), we have

−tA ||e R(V (t))|| ≤ C2, ∀t ≥ 0, −tA ε −κt ||e vN (t)|| ≤ C3e , ∀ t ≥ 0, d ||e −tA V α(t)|| ≤ εC e −κt, ∀t ≥ 0, dt 4 ||N(t)|| ≤ εC5(F, A), ∀t ≥ 0, ε ||u˜ (t)|| ≤ C6, ∀t ≥ 0, ε ||Gε(t)|| ≤ c7||w (t)|| + c5ε. ∀t ≥ 0.

Hence, following the same lines as in the proof of Theorem 1, we conclude the proof. 12 M. ZIANE

Acknowledgment

This work was partially supported by the National Science Foundation under Grant NSF-DMS- 9705229, and by the Research Fund of Indiana University. The author would like to thank Ciprian Foias and Oscar Manley for bringing [CGO3] to his attention, for the several interesting discussions and for the critical reading of the manuscript.

References

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