renormalization of the kdv-burgers and kuramoto

CANADIAN APPLIEDMATHEMATICS QUARTERLYVolume 9, Number 3, Fall 2001

RENORMALIZATION OF THE KdV-BURGERSAND KURAMOTO-SIVASHINSKY EQUATIONS

CHIU-YA LAN AND CHI-KUN LIN

ABSTRACT. In this article we study the turbulent dif-fusion of the KdV-Burgers and Kuramoto-Sivashinsky equa-tions. Using the concept of renormalization we prove that thehomogenized equation is the diffusion equation for short-rangecorrelation while for long-range correlation it is the super-diffusion equation.

1. Introduction. The advection-diffusion of a passive scalar by anincompressible velocity is described by the equation

(1.1)∂T

∂t+ (v · ∇)T = κ∆T,

where the incompressible velocity field, v(x, t) satisfies div v = ∇ · v =0. The coefficient κ ≥ 0 is the molecular diffusion coefficient. Itcorresponds to the Prandtl number when the passive scalar is thetemperature.

This important problem arises commonly in the study of turbulentdiffusion in the atmosphere, in dispersion of tracers in Benard conven-tion rolls, in turbulent transport of heat in Rayleigh-Benard convectionand the transport of magnetic fields in the kinematic dynamo problemas well as in elementary models in plasma physics.

The idea that molecular transport can be enhanced by turbulentmotion is at the very core of turbulent modeling. Transport of scalarquantities on scales much larger than the (energy) scale of the turbulentflow, indeed, typically leads to enhanced diffusion. It is well known thatthe motion of a diffusive particle advected by a fluctuating velocityfield is equivalent at large spatial scales and long times to an effectiveenhanced diffusive motion (Taylor).

The basic result is that the presence of a stirred ambient fluidenhances the rate of dispersion of particles and, in this regard, two

Key words and phrases. Advection, turbulence, diffusion, dispersion, renormali-zation.

Copyright c©2001 Rocky Mountain Mathematics Consortium

249

250 C.-Y. LAN AND C.-K. LIN

generic universality classes of random velocities have been identified.These universality classes correspond to either diffusive behavior atlarge time scales or super-diffusive behavior.

The renormalization group (RNG) method has recently had somesuccess when applied to problems in critical phenomena. Such phe-nomena are characterized by fluctuations on all length scales, rangingfrom the molecular to the macroscopic. The RNG method might be ap-plied in principle to fluid turbulence. Avellaneda and Majda have stud-ied several renormalization procedures for this equation. We refer totheir works [1 9] for details. They obtained different limiting regimes:among them, the mean field and the so-called anomalous regimes. Infact, Majda and Avellaneda have introduced a family of simple modelproblems with turbulent velocity statistics where the above issues canbe understood and clarified in an unambiguous and mathematicallyrigorous fashion. These simple models have an exactly solvable renor-malization theory for statistical behaviors at large scales and long timesand exhibit various “phase transitions” from convectional diffusion the-ories with scale-separated velocity statistics to more complex equationsinvolving anomalous super-diffusion as the velocity statistics vary toallow infrared divergences and long-range correlation without separa-tion of scales. The rigorous exact renormalization theory is developedthrough tools involving Fourier analysis and the Feynman-Kac formula.

For the model of (1.1) with the special form

(1.2)∂T

∂t+ (v(t) · ∇)T = κ∆T,

where v(t) is a random velocity field, such simple model problems wereintroduced by Kubo [19]. When the correlations for v(t) are sufficientlyshort range, the well-known Kubo theory is valid; however, if the veloc-ity field exhibits sufficiently long-range correlation, a “phase transition”occurs together with super-diffusion and a different structure for theeffective diffusion equation for 〈T 〉 at large scales and long times, see[1 9] for details.

In this paper we consider an equation which represents a combinationof the Korteweg-de Vries and Burgers equation (KdVB), namely,

(1.3)∂T

∂t+ T

∂T

∂x− κ∂

2T

∂x2+ µ

∂3T

∂x3= 0.

RENORMALIZATION OF KdV-BURGERS AND K-S EQUATIONS 251

This type of equation occurs in some classes of nonlinear dispersivesystems with dissipation which was derived by Su and Gardner [28]for a wide class of nonlinear systems in weak nonlinearity and long-wave length approximations. Physical considerations require that thedissipative parameter κ must always be positive, while the dispersiveparameter µ may be either positive or negative.

Similar to the theory of eddy diffusivity we consider the linearizedKdVB equation with the natural initial data varying on the integrallength scale, i.e., it involves only long wavelengths:

(1.4)∂T δ

∂t= κ

∂2T δ

∂x2− ∂

3T δ

∂x3,

T δ|t=0 = T0(δx), δ � 1,

where T0(x) has a Fourier transform of compact support. In orderto study (1.4) at large scales and long times, we introduce the scaledvariables

(1.5) x′ = δx, t′ = ρ2(δ)t.

Then the limit x, t → ∞ is equivalent to the limit δ, ρ → 0. Afterdropping the primes in (1.4), we obtain the rescaled equation

(1.6)

∂T δ

∂t= κ

(δ2

ρ2(δ)

)∂2T δ

∂x2−

(δ3

ρ2(δ)

)∂3T δ

∂x3,

T δ|t=0 = T0(x).

Comparing the first and second terms of the righthand side of (1.6), itis easy to check that as δ → 0 there is a unique scaling with nontriviallimiting behavior, namely,

(1.7) ρ(δ) = δ.

Therefore, the large-scale long-time limit equation of (1.6) is the heatequation

(1.8)∂T

∂t= κ

∂2T

∂x2, T |t=0 = T0(x).


In this case the rescaling function ρ(δ) = δ corresponds to the usual dif-fusive scaling. The determination of this large-scale rescaling functionρ(δ) as “phase transition” occurs is one of the goals of renormalizedtheories for eddy diffusivity.

The aim of this work is to study the turbulent diffusion of thelinearized KdV-Burgers equation:

(1.9)∂T δ

∂t+ v(t)

∂T δ

∂x= κ

∂2T δ

∂x2− ∂

3T δ

∂x3,

where v(t) is a random velocity field. It is assumed to be Gaussian.A random flow field is said to be Gaussian if all its finite dimensionaldistributions are Gaussian. We restrict our attention to homogeneousturbulence fields. This means that statistical quantities do not dependupon their absolute position in space. In the case of the velocity field,this implies that the mean velocity is a constant. Accordingly, wenormally work in a system of coordinates in which the constant meanvelocity is zero. Using the concept of renormalization as investigatedby Majda and Avellaneda we prove that the homogenized equation isthe diffusion equation for short-range correlation while for long-rangecorrelation it is the super-diffusion equation.

We are now ready to explain the organization of this paper. First,in Section 2 we develop the Kubo theory for the Korteweg-de Vries-Burgers equation. When the correlations for the velocity field aresufficiently short range, we show the Kubo’s theory is valid. Thedispersion effect (third derivative term) is not important in the limitingprocess and the effective equation is a diffusion equation. The effectivediffusion coefficient is given by the integral of the correlation.

Section 3 is devoted to the long-range correlation. We prove that thelimiting equation is the diffusion equation for short-range correlationwhile for long-range correlation it is the super-diffusion equation. Thephase transition occurs from normal diffusion to super-diffusion as theparameter varies.

Then, in Section 4, we apply the results of Sections 2 and 3 to studysome interesting equations, especially the ones similar to the Kuramoto-Sivashinsky equation.

2. Kubo theory for the Korteweg de Vries Burgers equa-tion. We begin this section by briefly mentioning some properties of


homogeneous turbulence. The energy spectrum E(k) is calculated bythe integration of the Fourier transform of the trace of the velocity cor-relation tensor over the sphere of radius k = |κ|, where κ is the wavevector dual to the separation r. In other words, let the velocity at apoint x be u(x) = (u1, u2, u3). The velocity correlation tensor is

(2.1) Rij(r) = 〈ui(x)uj(x+ r)〉

where the brackets denote an ensemble average. This terminology dif-fers from the usual statistics terminology and is peculiar to turbulencetheory. The Fourier transform of Rij(r) is

(2.2) φij(κ) = (2π)−3/2

∫eiκ·rRij(r) dr

and the energy spectrum is defined as

(2.3) E(k) =∫|κ|=k

φii(κ) dκ.

Clearly the mean energy at a point is

(2.4)12〈u2〉 =

∫ ∞

0

E(k) dk.

Similarly, the vorticity correlation tensor is

(2.5) Qij(r) = 〈ξi(x)ξj(x+ r)〉

where ξ = (ξ1, ξ2, ξ3) is the vorticity. Its Fourier transform is ψij(κ),and the vorticity spectrum is

(2.6) Z(k) =∫|κ|=k

ψii(κ) dκ,

with

(2.7) 〈ξ2〉 =∫ ∞

0

Z(k) dk.


The relation Z(k) = k2E(k) follows from the definition ξ = curlu =∇ × u. It is generally believed that, in the inertial range, i.e., in therange of scales intermediate between the stirring and the dissipationscales,

(2.8) E(k) ∼ k−γ

where γ is the inertial (Kolmogorov) exponent. A widely acceptedvalue for γ is γ = 5/3. A random function is called stationary ifall its moments and joint moments are independent of the choice oftime origin. The concept of a stationary random function represents asignificant simplification, since then, for example, an average like 〈v〉will be independent of time, as will all the averages of functions of v,including the averages of the powers of v.

Now we are in a position to consider the model problem

(2.9)∂T

∂t+ v(t)

∂T

∂x= κ

∂2T

∂x2− ∂

3T

∂x3, T |t=0 = T0(x).

Moreover, we assume that v(t) is a stationary mean-zero Gaussianrandom statistic, [30] with correlation

(2.10) R(t) = 〈v(t+ τ )v(τ )〉,which gives information about the average time dependence of a pro-cess. Note that 〈·〉 represents the ensemble average over all possiblerealizations of v. The Fourier transform of the time correlation func-tion R(t) is called the power spectral density, defined by

(2.11) ϕ(ω) =12π

∫ ∞

−∞eiωtR(t) dt.

Applying the Fourier’s integral theorem to the power spectral density,one has

(2.12) R(t) =∫ ∞

−∞e−iωtϕ(ω) dω.

We also describe several fundamental results of stationary Gaussianprocess that we shall use. For more details and comment, we refer thereader to Avellaneda-Majda [3] and Yaglom [30].


Lemma 2.1. If the velocity field v is a stationary mean-zeroGaussian random process, then

(2.13)⟨exp[−i

∫ t

0

v(s)ξ ds]⟩= exp

[− 12ξ2

∫ t

0

∫ t

0

R(|s− s′|) ds ds′],

(2.14)12

∫ t

0

∫ t

0

R(|s− s′|) ds ds′ =∫ t

0

(t− s)R(s) ds.

If we take the Fourier transform with respect to x, then the solutionsof the model equation (2.9) satisfy the transformed equation

(2.15)∂T

∂t+ (2πiξ)v(t)T = κ(2πiξ)2T − (2πiξ)3T ,

where the space Fourier transform is defined by

f(x) ≡∫e2πixξ f(ξ) dξ.

The solution of (2.15) is given explicitly by

(2.16) T = T0 exp[− 2πi

∫ t

0

ξv(s) ds− 4π2ξ2κt+ i8π3ξ3t

].

Then applying the Fourier inverse transform to (2.16), the solution ofthe model problem is given by

(2.17) T (x, t) =∫e2πixξ exp(−4π2ξ2κt+ i8π3ξ3t)

× exp[− 2πi

∫ t

0

v(s)ξ ds]T0(ξ) dξ.

The random function that occurs in (2.17) is

exp[− 2πi

∫ t

0

v(s)ξ ds].


To overcome the principal difficulty which comes from the randomvariation of the velocity we are forced to the statistical methods. Infact, the ensemble average allows one to form averages for the time-dependent process. Taking the ensemble average of (2.17) over thevelocity statistics and using Lemma 2.1, we obtain

(2.18) 〈T 〉 =∫e2πixξ exp(−4π2ξ2κt+ i8π3ξ3t)

× exp[− 4π2|ξ|2

∫ t

0

(t− s)R(s) ds]T0(ξ) dξ.

From (2.18) we find that 〈T 〉 satisfies the differential equation

(2.19)∂

∂t〈T 〉 = K∗ ∂

2

∂x2〈T 〉+ κ ∂

2

∂x2〈T 〉 − ∂3

∂x3〈T 〉

with

(2.20) K∗ =∫ t

0

〈v(s)v(0)〉 ds =∫ t

0

R(s) ds.

For (2.19) (2.20) defining a well-posed initial value problem for anyt0 ≥ 0, it is necessary and sufficient that

(2.21) κ+K∗ = κ+∫ t0

0

R(s) ds ≥ 0, ∀ t0 ≥ 0.

The well-posedness condition of this initial value problem is equivalentto that the velocity v(t) has correlation satisfying (2.21). The corre-lation is allowed to be negative over some region. On the other hand,if the velocity v has negative correlation over a sufficient wide rangeof integration, then the differential equation in (2.19) (2.20), which isthe unstable backward heat equation, for the mean 〈T 〉 is an ill-posedproblem.

Next we consider the large scale initial value problem

(2.22)∂T δ

∂t+ v(t)

∂T δ

∂x= κ

∂2T δ

∂x2− ∂

3T δ

∂x3,

T δ|t=0 = T0(δx), δ � 1.


We introduce the large-scale variables with diffusive scaling given by

(2.23) x′ = δx, t′ = δ2t,

i.e., the problem is viewed as coarse grained. Then the coarse-grainedmean, T (x′, t′) is defined by

(2.24) T (x′, t′) ≡ limδ→0

⟨T δ

(x′

δ,t′

δ2

)⟩.

Using (2.13), (2.14) and (2.16) and dropping the primes in (2.24) andthe same procedure as above, we find that

(2.25)⟨T δ

(x

δ,t

δ2

)⟩=

∫e2πixξ exp(−4π2ξ2κt+ i8π3ξ3δt)

×⟨exp

[− 2πi

∫ tδ2

0

v(s)δξ ds]⟩T0(ξ) dξ

=∫e2πixξ exp(−4π2ξ2κt+ i8π3ξ3δt)

× exp[− 4π2ξ2

(t

∫ t

δ2

0

R(s) ds−δ2∫ t

δ2

0

sR(s) ds)]

× T0(ξ) dξ.

We assume that the correlation satisfies

(2.26)∫ ∞

0

R(s) ds <∞ and limT→∞

(1T

∫ T

0

sR(s) ds)= 0.

Taking the limit of (2.25), as δ → 0, and using (2.26) the coarse-grainedmean, T (x, t) is given by

(2.27)

T (x, t) =∫e2πixξe−4π2ξ2κt exp

(− 4π2ξ2t

∫ ∞

0

R(s) ds)T0(ξ) dξ,

and satisfies the differential equation

∂T

∂t= κ

∂2T

∂x2+K∗ ∂

2T

∂x2,(2.28)


where

K∗ =∫ ∞

0

R(s) ds ≥ 0.(2.29)

Thus the convention by a random velocity field has a diffusion effecton the mean. Clearly, for any velocity statistics with K∗ + κ > 0, theeffective equation in (2.28) for the ensemble average, T , at large scalesand long times is stable and any randomness at all has a tendency tomake the averaged problem more stable. Thus the problem becomeswell-posed with enhanced diffusion after suitable coarse graining. Thusthe coarse-grained problem in (2.28) is always well-posed with enhanceddiffusion due to the randomness because

K∗ + κ > 0.

3. Super-diffusion with long-range correlations. Now we con-sider a family of Gaussian velocity statistics depending on a parameter,ε, as suggested in [9], with power spectrum given by

(3.1) |ω|−εϕ∞(|ω|), −∞ < ε < 1,

where ϕ∞(s) is a smooth rapidly decreasing even function of s withϕ∞ identically one in a neighborhood of the origin and ϕ∞(s) ≥ 0, see(2.12). Here ε with −∞ < ε < 1 is a parameter characterizing thevelocity spectrum and is familiar from renormalization theory. It isclear that as the parameter ε increases, longer-range correlations in thevelocity statistics build up in time. To choose the proper Gaussianrandom statistic vε(t) for further discussion we need the followingtheorem which is due to Khinchin.

Theorem 3.1 (Khinchin). For a function R(t), −∞ < t <∞, to bethe correlation function of a field which has translation invariant meansand correlation functions and also satisfies the condition

(3.2) 〈|v(t+ τ )− v(t)|2〉 −→ 0 as τ → 0

it is necessary and sufficient that it has a representation of the form

(3.3) R(t) =∫ ∞

−∞e2πitω dF (ω)


where F (ω) is a nondecreasing function of ω.

F (ω) is called the spectral distribution function of v. If F (ω) isdifferentiable, F ′(ω) = ϕ(ω), and the function ϕ is called the spectraldensity of v. Given F (ω), or ϕ(ω), R(t) can be constructed with thehelp of the Fourier integral. It follows from (3.1) and the Khinchintheorem that a stationary mean zero Gaussian random statistic vε(t)exists with correlation

(3.4) Rε(t) = 〈vε(t+ τ )vε(τ )〉,

given by

(3.5) Rε(t) = R∫e2πiωt|ω|−εϕ∞(|ω|) dω, R > 0.

Using the scaling and the method of stationary phase, we have thefollowing proposition which gives the large time behavior of the corre-lation functions in (3.5), see [9].

Proposition 3.2. The correlation function in (3.4) is a smoothfunction of t with the following behavior:

(3.6) |Rε(t)| ≤ Cε(1 + |t|)−1+ε;

for 0 < ε < 1 and |t| ≥ 1, the correlation function has the asymptoticexpansion

(3.7) Rε(t) = RAε|t|−1+ε + Eε(t)

with

(3.8) Aε = (2π)ε sin(12επ

)Γ(−ε+ 1)

and Eε satisfying

(3.9) |Eε(t)| ≤ CN (1 + |t|)−N for any n > 0.


We consider the large-scale long-time behavior of the mean statisticsfor the model problem

(3.10)∂T δ

∂t+ vε(t)

∂T δ

∂x= κ

∂2T δ

∂x2− ∂

3T δ

∂x3

T δ|t=0 = T0(δx), δ � 1,

as ε varies for −∞ < ε < 1. For ε ≤ 0, (3.6) guarantees that theconditions in (2.26) for the Kubo theory in Section 2 are satisfied sothat with the simple diffusion scaling in (2.23) the averaged equationfor the coarsed-grained limit is given by (2.28) (2.29).

∂T

∂t= κ

∂2T

∂x2+K∗ ∂

2T

∂x2,

where

K∗ =∫ ∞

0

R(s) ds ≥ 0.

For ε > 0, the asymptotic behavior of the correlation function in (3.7)guarantees that the correlations decay slowly so the integrals in (2.26)diverge and the theory in Section 2 is no longer valid. Thus the problemneeds to be renormalized on a different timescale. We introduce thenew large-scale long-time scaling variables

(3.11) x′ = δx, t′ = ρ2(δ)t.

Dropping the primes in (3.11), we find

(3.12)⟨T δ

(x

δ,t

ρ2(δ)

)⟩

=∫

exp(2πixξ − 4π2ξ2κ

(δ2

ρ2

)t+ i8π3ξ3

(δ3

ρ2

)t)

× exp(− 4π2

[ξ2

(δ2

ρ2t

∫ t/ρ2

0

Rε(s) ds− δ2∫ t/ρ2

0

sRε(s) ds)])

T0(ξ) dξ

=∫

exp(2πixξ − 4π2ξ2κ

(δ2

ρ2

)t+ i8π3ξ3

(δ3

ρ2

)t)

× exp(− 4π2ξ2

(δ2

ρ2+2ε

)Aεt

1+ε

ε(ε+ 1)R

)T0(ξ) dξ


where

(3.13) Aε = (2π)ε sin(12επ

)Γ(−ε+ 1).

The coarse-grained mean T (x, t) is defined by

(3.14) T (x, t) = limδ→0

⟨T δ

(x

δ,t

ρ2(δ)

)⟩.

From the final term of (3.12) we may choose the scaling

(3.15) ρ(δ) = δ1/1+ε, 0 < ε < 1,

which corresponds to short time scales of nontrivial activity than theusual diffusive scaling. Then direct computation as in the previoussection involving Fourier transform yields

(3.16) T (x, t) =∫e2πixξ exp

(− 4π2ξ2

Aεt1+ε

ε(ε+ 1)R

)T0(ξ) dξ.

Thus the effective equation for the ensemble average, T , in the large-scale long-time limit satisfies the super-diffusion equation

(3.17)∂T

∂t= ε−1Aεt

εR∂2T

∂x2, T |t=0 = T0(x),

for 0 < ε < 1. Direct computation by using (3.17) yields the followingmoments

( ∫ ∞

−∞x2nT dx

)(t) =

n∑k=0

(2n)!αk

(2n−2k)!k!

∫ ∞

−∞x2n−2kT0 dx

( ∫ ∞

−∞x2n+1T dx

)(t) =

n∑k=0

(2n+1)!αk

(2n+1−2k)!k!

∫ ∞

−∞x2n+1−2kT0 dx,(3.18)

where

α =RAε

ε(ε+1)tε+1.


In particular, the second moment of T in time is given by

(3.19)(∫ ∞

−∞x2T dx

)(t) = 2α

∫ ∞

−∞T0 dx+

∫ ∞

−∞x2T0 dx.

Thus, with T0 = δ(x) the Dirac delta function, the particles diffuse inx at a super-diffusion rate proportional to

√α ∼ t(ε+1)/2 in contrast to

the standard spreading√t of the heat equation.

Remark 3.3. The exponent of the time rescaling function ρ(δ) playsthe role of an order parameter characterizing the “phase transition”which occurs from normal diffusion to super-diffusion as ε crossesthrough zero with different effective equations at large scales and longtimes. In fact, let

(3.20) f(ε) ≡ log(ρ(δ))log δ

={1, if ε < 0;1/(1 + ε), if 0 < ε < 1,

then the graph of f(ε) reveals the classical behavior of an orderparameter in a first-order phase transition. We may consider δ to be afunction of ρ. The asymptotic scaling behavior of x is determined from

(3.21) γ =12limρ→0

d log δ(ρ)d log ρ

={(1/2), if ε ≤ 0;(1 + ε)/2, if 0 < ε < 1.

In the asymptotic regime x scales as tγ . When the scaling exponentequals (1/2), the diffusion is normal. Any other value corresponds toanomalous diffusion.

Remark 3.4. The effective renormalized equation in (3.17) is invariantunder the space-time symmetry group associated with (3.11), i.e.,solutions of (3.17) are invariant under the transformations

(3.22) (x, t) −→ (λx, λ2/(1+ε)t).

4. Some generalizations. From the discussion in Sections 2 and3 we find that the effect of the dispersion term ∂3T δ/∂x3 vanishes inthe ensemble average at large scales and long time dynamics. This


is because the velocity fields are zero-mean Gaussian statistics andthe dispersion is oscillation basically. With the previous sections asbackground, we consider the more general problem

(4.1)∂T

∂t+ v(t)

∂T

∂x=

N∑n=1

an∂nxT ≡ P (∂x)T,

T |t=0 = T0(x),

where P (z) = a1z + · · · + aNzN is a polynomial of Nth degree. The

Fourier transform T of T satisfies

(4.2)∂T

∂t+ (2πiξ)v(t)T = P (2πiξ)T ,

which is a first order differential equation in the t variable. Its solutionis given by

(4.3) T (ξ, t) = T0(ξ) exp[− 2πi

∫ t

0

ξv(s) ds+ tP (2πiξ)].

Thus, by Fourier inverse transform, the solution of (4.1) is given

(4.4) T (x, t) =∫

exp[2πixξ + tP (2πiξ)]

× exp[− 2πi

∫ t

0

ξv(s) ds]T0(ξ) dξ,

then the same computations as in Section 2 can be repeated and theeffective equation for the ensemble average 〈T 〉 is given by

(4.5)∂〈T 〉∂t

= K∗ ∂2〈T 〉∂x2

+ P (∂x)〈T 〉.

Next we consider the linearized equation of (4.1) with large-scaleinitial data

(4.6)∂T δ

∂t=

N∑n=1

an∂nxT

δ ≡ P (∂x)T δ,

T δ|t=0 = T0(δx), δ � 1.


To study (4.6) at large scales and long times, we introduce the scaling

(4.7) x′ = δx, t′ = ρ2(δ)t,

then the rescaled equation of (4.6) is given by (after dropping theprimes)

(4.8)∂T δ

∂t=

1ρ2P (δ∂x)T δ =

δ

ρ2a1∂T δ

∂x+δ2

ρ2a2∂2T δ

∂x2+ · · ·

T δ|t=0 = T0(x).

It is easy to check that, as δ ↓ 0 there is a unique scaling withnontrivial limiting behavior, namely ρ2(δ) = δ (the convective scale ofhyperscaling) and the large-scale long-time limit equation is the lineartransport equation

(4.9)∂T

∂t= a1

∂T

∂x, T |t=0 = T0(x),

which is a typical hyperbolic equation with finite propagation velocitya1. Now we consider the model equation

(4.10)∂T δ

∂t+ v(t)

∂T δ

∂x= P (∂x)T δ, T δ|t=0 = T0(δx),

where v(t) satisfies the assumption in (2.26). If we consider the limitof the ensemble average at large scales and long times

(4.11) T (x, t) = limδ→0

⟨T δ

(x

δ,t

δ

)⟩,

then the same procedure shows that the effective equation for T is

(4.12)∂T

∂t= a1

∂T

∂x.

If a1 = 0 but a2 �= 0, then the rescaling function is diffusion scalingρ(δ) = δ. If we consider the limit of the ensemble average at largescales and long times,

(4.13) T (x, t) = limδ→0

⟨T δ

(x

δ,t

δ2

)⟩,



(4.14)∂T

∂t= K∗ ∂

2T

∂x2+ a2

∂2T

∂x2.

Next, we follow the same terminology as Section 3 and consider thelarge-scale long-time behavior of the mean statistics for the modelproblem

(4.15)∂T δ

∂t+ vε(t)

∂T δ

∂x= P (∂x)T δ

T δ|t=0 = T0(δx), δ � 1,

as ε varies for −∞ < ε < 1. We introduce the large-scale long-timescaling variables

(4.16) x′ = δx, t′ = ρ2(δ)t.

Then (3.12) is replaced by

(4.17)⟨T δ

(x

δ,t

ρ2(δ)

)⟩

=∫e2πixξ exp

(t

ρ2P (δ2πiξ)

)exp

(− 4π2ξ2

(δ2

ρ2+2ε

)Aεt

1+ε

ε(ε+1)R

+O(δ2

ρ2

))T0(ξ) dξ.

Note that

(4.18)t

ρ2P (δ2πiξ) =

t

ρ2[a1δ(2πiξ) + a2δ2(2πiξ)2 + · · ·+ aNδ

N (2πiξ)N ],

then the unique scaling is the convective scale ρ2(δ) = δ for a1 �= 0. Inthis case

(4.19) T (x, t) = limδ→0

⟨T δ

(x

δ,t

δ

)⟩=

∫e2πixξ+ta12πiξT0(ξ) dξ,



(4.20)∂T

∂t= a1

∂T

∂x.

However, for a1 = 0, a2 �= 0, then the unique scaling is the same as(3.15)

ρ(δ) = δ1/(1+ε), 0 < ε < 1,

then

(4.21)

T (x, t) = limδ→0

⟨T δ

(x

δ,t

ρ2(δ)

)⟩

=∫e2πixξ exp

(− 4π2ξ2

Aεt1+ε

ε(ε+ 1)R

)T0(ξ) dξ,

and T satisfies the super-diffusion equation

(4.22)∂T

∂t= ε−1Aεt

εR∂2T

∂x2, T |t=0 = T0(x),

for 0 < ε < 1.

A particular example of (4.1) is the Kuramoto-Sivashinsky equation

(4.23) ∂tT + v(t)∂xT + ∂2xT + ∂4

xT = 0, T (0, x) = T0(x).

By using renormalization group ideas, Avellaneda and Majda gavethe rigorous proof that the effect of randomness does not curtail thegrowth of instability for individual realizations but nevertheless thelarge-scale long-time effect equation for the ensemble average satisfiesa well-posed effective equation. We will investigate a similar equationwith a different dissipation at small scales, see [31]:

(4.24)T δ

t + v(t)T δx + T δ

xx − T δx6 = 0,

T δ0 = T0(δx), δ � 1,

where T δx6 denotes ∂6T δ/∂x6 and T0(x) is a smooth deterministic

function with a Fourier transform of compact support. We shall seethat the solutions of this equation have the same large scale behavioras Kuramoto-Sivashinsky equations.


First we consider the linearized equation of (4.24) with large-scaleinitial data, i.e.,

(4.25) T δt + T δ

xx − T δx6 = 0, T δ

0 = T0(δx), δ � 1.

In order to study (4.25) at large scales and long times, we introducethe scaled variables

(4.26) x′ = δx, t′ = ρ2(δ)t.

Then the limit x, t → ∞ is equivalent to the limit δ, ρ → 0. Afterdropping the primes in (4.25), we obtain the rescaled equation

(4.27)

∂T δ

∂t= −

(δ2

ρ2(δ)

)∂2T δ

∂x2+

(δ2

ρ2(δ)

)∂6T δ

∂x6,

T δ|t=0 = T0(x).

Comparing the first and second terms of the righthand side of (4.27), itis easy to check that as δ → 0 there is a unique scaling with nontriviallimiting behavior, namely,

(4.28) ρ(δ) = δ.

In this case the rescaling function ρ(δ) = δ corresponds to the usualdiffusive scaling. Therefore, the large-scale long-time limit equation of(4.27) is the unstable backward heat equation,

(4.29)∂T

∂t= −∂

2T

∂x2, T |t=0 = T0(x).

However, we consider the model equation (4.24) with velocity v(t)satisfying the assumption (2.13) (2.14). Proceeding as in Section 2,the limit of the ensemble average at large scales and long times of thediffusive scaling is

(4.30)T (x, t) ≡ lim

δ→0

⟨T δ

(x

δ,t

δ2

)⟩

=∫e2πixξe4π2ξ2t exp

[− 4π2ξ2t

∫ ∞

0

R(s) ds]T0(ξ) dξ.


Then the effective equation is given by

(4.31) T t = −T xx +D∗T xx, D∗ =∫ ∞

0

R(s) ds,

which is stable if D∗ > 1 and any randomness at all has a tendency tomake the averaged problem more stable. Moreover, we may choose thevelocity field v(t) properly such that the correlation R(s) satisfies

(4.32) D∗ =∫ ∞

0

R(s) ds = 1,

then the effective equation (4.31) is the bi-diffusion equation

(4.33) T t = −T xx + T xx = 0.

Thus T is a conservative quantity.

On the other hand, we parametrize the velocity statistics so that themean field theory occurs in the model from (4.24) with ε < 0 whileanomalous phenomena requiring renormalization occur for ε > 0. Weconsider the velocity fields vε satisfying (3.1) and (3.5)

(4.34)T δ

t + vε(t)T δx + T δ

xx − T δx6 = 0,

T δ0 = T0(δx), δ � 1,

where vε(t) is a stationary mean zero Gaussian random field with longrange correlations which satisfy the assumptions in (3.5) in Section 3for 0 < ε < 1. Then the computation as in Section 3 can be repeatedand the effective equation for T is the super-diffusion equation

(4.35) T t =Aεt

ε

εR T xx,

where, choosing ρ(δ) = δ1/(1+ε),

(4.36)

T (x, t) ≡ limδ→0

⟨T δ

(x

δ,t

ρ2

)⟩

=∫e2πixξ exp

[− 4π2ξ2

Aεt1+ε

ε(ε+ 1)R

]T0(ξ) dξ,

Aε = (2π)ε sin(12επ

)Γ(−ε+ 1), 0 < ε < 1.


For ε < 0, the effective equation is the same as (4.31).

Finally we consider a similar problem as (4.34):

(4.37)T δ

t + vε(t)T δx =

∣∣∣∣ ∂∂x∣∣∣∣1+γ

T δ + T δx6 ,

T δ0 = T0(δx), δ � 1,

vε(t) satisfies the same assumption. The operator |∂/∂x|1+γ is definedthrough the Fourier transform, via

(4.38)∣∣∣∣ ∂∂x

∣∣∣∣1+γ

f =∫e2πix·ξ|2πξ|1+γ f(ξ) dξ, 0 < γ < 1.

The deterministic problem of (4.37) is

(4.39) T δt =

∣∣∣∣ ∂∂x∣∣∣∣1+γ

T δ + T δx6 , T δ

0 = T0(δx).

Introducing the same space-time scaling as (4.26), we obtain therescaled equation

(4.40)∂T δ

∂t=

(δ1+γ

ρ2(δ)

)∣∣∣∣ ∂∂x∣∣∣∣1+γ

T δ +(δ6

ρ2(δ)

)∂6T δ

∂x6,

T δ|t=0 = T0(x).

The only nontrivial scaling of (4.40) is ρ(δ) = δ(1+γ)/2, and thecorresponding large-scale limit equation is given by

(4.41) T t =∣∣∣∣ ∂∂x

∣∣∣∣1+γ

T , T 0 = T0(x).

This equation is unstable at all wave numbers like the backward heatequation. However, for the model equation (4.37), if we equate the timescaling ρ(δ) above (4.41) and ρ(δ) in (3.15) to obtain

(4.42) 1 + γ =2

1 + ε, 0 < ε < 1.


With ρ(δ) = δ1/(1+ε) the limit of the ensemble average at large scalesand long times is

(4.43) T (x, t) ≡ limδ→0

⟨T δ

(x

δ,t

ρ2(δ)

)⟩.

Then the effective equation is given by

(4.44) T t =∣∣∣∣ ∂∂x

∣∣∣∣2/(1+ε)

T +Aεt

ε

εRT xx,

where Aε is the same as (4.36). This equation is well-posed but has aninfinite band of unstable modes at t = 0 and a finite band of unstablemodes for any t > 0 that narrows with times and vanishes in the limitas t→ ∞, see [9].

Acknowledgments. We thank C.D. Levermore for helpful discus-sions and suggesting that we look at the Kuramoto-Sivashinsky equa-tion and its similar equations.

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Department of Mathematics, National Cheng Kung University, Tainan70101, Taiwan

Department of Mathematics, National Cheng Kung University, Tainan70101, TaiwanE-mail address: [email protected]

renormalization of the kdv-burgers and kuramoto

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