Renormalization in dispersive and dissipative equation

Applied Mathematics Applied Mathematics

Letters

15 (2002)

Letters

71-75

www.elsevier.com/locate/aml

Renormalization in Dispersive and Dissipative Equation CHIU-YA LAN AND CHI-KUN LIN Department of Mathematics, National Cheng Kung University Tainan 70101, Taiwan, R.O.C. cklinQmail.ncku.edu.tw (Received

and accepted

Communicated

January

2001)

by P. Markowich

Abstract-In this article, we study the turbulent diffusion of the dispersion-dissipation equations, especially the KdV-Burgers equation. Using the concept of renormalization, we prove that the renormalized equation is the diffusion equation for short-range correlation, while for long-range correlation, it is the superdiffusion equation. @ 2001 Elsevier Science Ltd. All rights reserved. Keywords-Advection,

Turbulence,

Diffusion,

Dispersion,

Renormalization

1. INTRODUCTION In this paper, we consider an equation which represents a combination of the Korteweg-deVries and Burgers equation (KdVB), namely,

$z++g

d3T +pm=o.

Physical considerations require that the dissipative parameter h: must always be positive, while the dispersive parameter I_Lmay be either positive or negative. Similar to the theory of eddy diffusivity, we consider the linearized KdVB equation with the natural initial data vary on the integral length scale, i.e., it involves only long wavelengths: dT6 a

-zz

d3T6

d2T6

QF

-

ax3 ’

T”l,=, = To(Sxc), 6 K

1,

where To(x) has a Fourier transform of compact support. In order to study (2) at large scales and long times, we introduce the scaled variables x’ = 6x, t’ = p2(6)t. Then the limit x, t + 00 is equivalent to the limit 6, fi 4 0. After dropping the primes in (2), we obtain the resealed equation

(3) Comparing the first and second terms of the right-hand side of (3), it is easy to check that as b --+ 0 there is a unique scaling with nontrivial limiting behavior, namely, p(S) = 6. Therefore, 0893-9659/01/$ - see front matter PII: SO893-9659(01)00095-7

@ 2001 Elsevier

Science Ltd. All rights reserved.

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by _&$-‘Ij$

72

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

the large-scale long-time limit equation of (3) is the heat equation aT d2T dt=KGd52

Tl,,,, = To(x).

(4)

In this case, the resealing function p(6) = 6 corresponds to the usual diffusive scaling. The determination of this large-scale resealing function p(b) as “phase transition” occurs as one of the goals of renormalized theories for eddy diffusivity. The aim of this work is to study the turbulent diffusion of the linearized KdV-Burger’s g

6

equation:

d2T6 ST6 = &- -

+ v(t)Z

822

8x3

(5)

'

where v(t) is a random velocity field. It is assumed to be Gaussian and a homogeneous turbulence field. This means that statistical quantities do not depend upon their absolute position in space. In the case of the velocity field, this implies that the mean velocity is a constant. Accordingly, we normally work in a system of coordinates in which the constant mean velocity is zero. Using the concept of renormalization, we show that the homogenized equation is the diffusion equation for short-range correlation, while for long-range correlation, it is the superdiffusion equation.

2. DIFFUSION

FOR SHORT-RANGE

CORRELATION

First, we consider the model problem

(6)

Tl,,O = To(X).

Moreover, we assume that v(t) is a stationary mean-zero Gaussian random statistic with correlation R(t) = (v(t + r)u(r)), which g ives information about the average time dependence of a process. Note that ( .) represents the ensemble average over all possible realization of 21. The Fourier transform of the time correlation function R(t) is called the power spectral density, defined by p(w) = &

I,

_W etwtR(t) dt.

(7)

For problem (6), i f we take the Fourier transform with respect to x, then the solution of the model equation (6) is given explicitly by T(x,t)

=

e2xzzc exp ( -4.rr2J2Kt + i8n3E3t) x exp [-%ilv(s)
J

If the velocity field w is a stationary (exp 1 JJ 20

t

=exp

R(ls - s’l) ds ds’ =

0

s0

[-~F2~~tR(is-s’l)dsds’]

=

Therefore,

J

,pis~exp

I

(9)

t(t - s)R(s) ds.

Taking the ensemble average of (8) over the velocity statistics (T)

(8)

mean-zero Gaussian random process, then

[-ilv(s)Eds])

t

c(<)d[.

( -4x2J2rct + i8n3t3t) x exp [ -47%1z

and using (9),(10), i’

we obtain

(t - s)R(s) ds ] z(c)

dt.

(11)

(T) satisfies the differential equation t

g(T)

= K*-&T)

f&T)

- g(T),

K’

= J0

R(s) ds.

(12)

Dispersive and Dissipative For (12) defining

a, well-posed

initial

value problem

tl1at

Equation

73

for any to > 0, it is necessary

and sufficient

to r;+K*

=tc+

R(s)

ds > 0,

vtl-J > 0.

of this initial value problem

is equivalent

(13)

.i’0

well-posedness

'he

has correlation

condition

satisfying

other

hand,

then

the diEerentia1

mean

(13).

if the velocity

u has negative

to that

is allowed to be negative

correlation

in (12)> which

equation

(T) is an ill-posed

Next we consider

The correlation

over a sufficient

is the unstable

the velocity

over some region.

w(t)

On the

wide range of integration,

backward

heat

equation,

for the

problem.

the large-scale

initial value problem

(14) Lye introduce

t,he large-scale

viewed as coarse grained. t’/fi’)).

Using

variables

with diffusive scaling 2’ = 6x, t’ = S2t, i.e., the problem

Then the coarse-grained

(9);( 10) and tl re F ourier analysis

mean is defined by T(x’, and the same procedure

is

t’) s lim~+a(T’(~‘/6, as above,

we find that

(15) provided

that the corrclat,ion

satisfies

R(s) ds < cc Then

the coarse-grained

and

mean T satisfies

($

=0.

[sR(s)ds)

$rir

the differential

(16)

equation

(17) Thus, the convention suitable

coarse

dilfusion

by a random velocity field has a diffusion elIect on the mean.

graining,

the coarse-grsined

due to the randomness

because

3. SUPER-DIFFUSION Now,WCconsider power spect.rum

a family

problem

pm(s)

is a smooth,

FOR LONG-RANGE

of Gaussian

rapidly

neighborhood

of the origin the velocity

as the parameter

velocity

random

decreasing

and pm(s) spectrum

E increases,

(18), as suggested

zero Gaussian

statistics

CORRELATIONS

depending

on a parameter

E with

--cx,
> 0 (see (7)).

and is familiar

longer-range

statistic

u’(t)

of s with cpoo identically

H ere E with -co in the velocity

t,heorem that

with correlation

R’(t)

there

functions

in (19) are smooth

functions

theory.

statistics exists

= (u’(t + T)u’(T)),

It is clear that

build up in time. a stationary

mean

given by

0.

of t with the following

IRE(t)1 I CE(l + Itl)-l+E;

one in a

< E < 1 is a parameter

from renormalization

correlations

in [l], and the Khinchin

(18)

R> The correlation

with enhanced

given by

characterizing Using

in (17) is always well-posed

after

I<* + E > 0.

l~l-‘u!X(l~l)? where

Therefore,

(19) behaviour:

(20)

C.-Y.

74

for 0 < E < 1 and ItI > 1, the correlation

LAN AND C.-K.

function

has the asymptotic

= RA,Itl-‘+’

R’(t)

LIN

+ E’(t),

with A, = (27r)’ sin(( 1/2)e7r)l?( -6 + 1) and E’ satisfying We consider

the large-scale

long-time

behaviour

expansion

(21)

jE’(t)l

< CN(~ + Itl)-N

of the mean statistics

T61tz0 = To(hx), as E varies for --03 < E < 1. For c < 0, (20) guarantees theory

in Section

averaged

2 are satisfied

equation

for the coarse-grained

For E > 0, the asymptotic correlations large-scale

the problem long-time

behaviour

of the correlation

in (16) for the Kubo

scaling,

function

z’ = Sx, t’ = S”t, the

in (21) guarantees

in (16) diverge and the theory

needs to be renormalized

scaling variables

(22)

limit is given by (17).

decay slowly so the integrals

valid. Thus,

S < 1,

that the conditions

so that with the simple diffusion

for any N > 0.

for the model problem

on a different

x’ = Sx, t’ = p2(6)t,

in Section

time scale.

that

t,he

2 is no longer

Introducing

the new

we find

(T”(~,&))=/‘exp(2~@-4n2<2n($)~+i8~3<3($)~) x exp (-48’6’

The coarse-grained

(A)

(23) SR)

mean T(x, t) is defined by T(z,t)

= lim~_o(T”(x/b,

term of (23), we may choose the scaling p(S) = 6 ‘/(l+‘), time scales of nontrivial the previous

section

activity

t/p”(6))).

From the final

0 < E < 1, which corresponds

than the usual diffusive scaling.

Then

to shorter

direct computation

as in

yields

= /e2”L”Cexp

T(x,t)

Thus, the effective equation the superdiffusion

?$(<)d<.

(-47r2t2$$R)

for the ensemble

E(I)dE.

average, ?;, in the large-scale

(24)

long-time

limit satisfies

equation

(25) for 0 < E < 1. Direct

([l

x2??

(s” -co where (-L:= (RA,/c(c

by using (25) yields the following moments:

computation

xan+‘T

+ l))t’+l.

(t) =

dx

(t) = go (2F;

00

In particular, \

to fi

N t(E+‘)/2 in contrast

x2rL--2kT0dx,

i-00

the particles

to the standard

(26)

.I’:” x2n+1-2kT0 dx, 03

the second moment

(t) = 2a]_mTodx+

with TO = 6(x) the Dirac delta function,

proportional

:i);;;!,!

>

(S_::x2Tdx)

Thus,

go ,2;T’!&lm

dx)

of T in time is given by

I.00

_

j_mx’Todx.

(27)

diffuse in x at a super-diffusion spreading

&

of the heat equation.

rate

Dispersive and Dissipative

Equation

75

REMARK.

(1) The

asymptotic

scaling

behaviour

of x is determined 1

dlog@) 2p+0 y = 1 lim In the asymptotic diffusion

(2) The

regime,

is normal.

effective

dlogp =

1

x scales like t?.

2’

if E 5 0,

1+c -, 2

ifO
group

associated

under

the transformations

equation

(28)

When

Any other value corresponds

renormalized

from

the scaling

to anomalous

in (25) is invariant

under

exponent

l/2,

the

diffusion. the space-time

with the short time scales we choose, i.e., solutions (x, t) -

equals

symmetry

of (25) are invariant

(XX, X2/(‘+‘)t).

REFERENCES 1. M. Avellaneda and A. Majda, Simple examples with features of renormalization for turbulent transport, Phil. nans. R. Sot. Lond. A 346, 205-233 (1994). 2. M. Avellaneda and A. Majda, Mathematical models with exact renormalization for turbulent transport, Common. Math. Phys. 131, 381-429 (1990). 3. L.-Y. Chen, N. Goldenfeld and Y. Oono, Renormalization group theory for global asymptotic analysis, Phys. Rev. Lett. 73, 1311-1315 (1994). 4. L.-Y. Chen, N. Goldenfeld and Y. Oono, The renormalization group and singular perturbations: Multiple scales, boundary layers and reductive perturbation theory, Physical Review E 54, 376-394 (1997). 5. D.L. Koch and J.F. Brady, A nonlocal description of advection-diffusion with application to dispersion in porous media, J. Fluid Me&. 180, 387-403 (1987). 6. H. Kesten and G.C. Papanicolaou, A limit theorem for turbulent diffusion, Commun. Math. Phys. 65, 97-128 (1979). 7. M. Lesieur, Turbulence in Fluid, Second Edition, Kluwer, Boston, (1990). equation and generalization. III: Derivation of the Ko8. C.H. Su and C.S. Gardner, Korteweg-deVries rteweg-de Vries equation and Burgers equation, J. Math. Phys. 10, 536-539 (1969).