Concentration diVusion EVects in

Viscous Incompressible Flows

LORENZO BRANDOLESE

ABSTRACT. Given a ﬁnite sequence of times 0<t< <1

t , we construct an example of a smooth solution of the freeN

dnonstationnary Navier Stokes equations inR ,d? 2; 3, such

that: (i) The velocity ﬁeldu x;t? is spatially poorly localized at

the beginning of the evolution but tends to concentrate until,

as the timet approachest , it becomes well localized. (ii) Then1

u spreads out again aftert , and such concentration diVusion1

phenomena are later reproduced near the instantst ,t ,... .2 3

1. INTRODUCTION

One of the most important questions in mathematical Fluid Mechanics, which is

still far from being understood, is to know whether a ﬁnite energy, and initially

smooth, nonstationnary Navier Stokes ﬂow will always remain regular during its

evolution, or can become turbulent in ﬁnite time.

As a ﬁrst step toward the understanding of possible blow up mechanisms, it is

interesting to exhibit examples of smooth and decaying initial data such that, even

if the corresponding solutions remain regular for all time, “something strange”

happens around a given point x ;t?in space time. This is the goal of the present0 0

paper.

Our main result is the construction of a class of (smooth) solutions to the

incompressible Navier Stokes equations such that, in the absence of any external

forces, the motion of the ﬂuid particles tends to be more concentrated around

x , as the time t approaches t . This corresponds precisely to the qualitative0 0

behavior that one would expect in the presence of a singularity, even though such

“concentration of the motion” is not strong enough to imply their formation.

789

cIndiana University Mathematics Journal , Vol. 58, No. 2 (2009)

790 LORENZO BRANDOLESE

d1.1. Statement of the main result. For a ﬂuid ﬁlling the whole space R ,

d 2, the Navier Stokes system can be written as

8

>@u?Pr u ⊗u ??u; t>><

r u?0;

>>>:

u x; 0??a x ;

dwhere u ? u ;:::;u?, a is a divergence free vector ﬁeld in R and1 d

−1P? Id−r? div is the Leray Hopf projector.

Because of the presence of the non local operatorP a velocity ﬁeld that is

spatially well localized (say, rapidly decaying asjxj!1 ) at the beginning of

the evolution, in general, will immediately spread out. A sharp description of

this phenomenon is provided by the two estimates (1.2) below. In order to rule

out the case of somewhat pathological ﬂows (such as two dimensional ﬂows with

radial vorticity, or the three dimensional ﬂows described in [2], which behave

quite diVerently asjxj!1if compared with generic solutions), we will restrict

our attention to data satisfying the following mild non symmetry assumption (for

j,k? 1;:::;d):

Z Z Z

2 2(1.1) 9j,k: a a x? dx, 0; or a x? dx, a x? dx:j k j k

p

Then, for suYciently fast decaying data, we have, forjxj C= t (see [5]):

x−d−1 −d−1 d−1(1.2) t?jxj ju x;t?j t?jxj ; 2S n? ;1 2 a

jxj

these estimates being valid during a small time intervalt2? 0;t?. HereC,t > 01 1

and and are positive functions, independent onx, behaving likec t as1 2 j

d−1 dt!0(j?1, 2). Moreover,S denotes the unit sphere inR and the subset?a

d−1of S represents the directions along which the lower bound may fail to hold:

the result of [5] tells us that? can be taken of arbitrarily small surface measurea

on the sphere. In other words, the lower bound holds true in quasi all directions,

whereas the upper bound is valid along all directions.

Moreover, the upper bound will hold during the whole lifetime of the strong

solutionu (see [12], [13]), whereas the lower bound is valid, a priori, only during

a very short time interval. The main reason for this is that the matrix

Z

u u x;t? dxj k

is non invariant during the Navier Stokes evolution, in a such way that even if the

datum satisﬁes (1.1), it cannot be excluded that at later times the solution features

Concentration diVusion EVects in Viscous Incompressible Flows 791

some kind of creation of symmetry, yielding to a better spatial localization and,

aftert>t, to an improved decay asjxj!1.(Werefere.g.to[4 ], [2], [7], [8],1

for the connection between the symmetry and the decay of solutions).

The purpose of this paper is to show that this indeed can happen. We con

struct an example of a solution of the Navier Stokes equations, with datuma2

dS?R ?(the Schwartz class),d? 2, 3, such that the lower bound

−d−1(1.3) ju x;t?j t?jxj ;1

holds in some interval? 0;t?, but then brakes down att , where a stronger upper1 1

bound can be established. This means that the motion of the ﬂuid concentrates

around the origin at such instant. Then the lower bound (1.3) will hold true again

aftert , until it will break down once more at a timet >t. This diVusion 1 2 1

concentration eVect can be repeated an arbitrarily large number of times.

More precisely, we will prove the following theorem.

Theorem 1.1. Letd ? 2, 3,let0<t< <t be a ﬁnite sequence,1 N

dand">0 . Then there exist a divergence free vector ﬁelda2S?R? and two se

0 0 quences t ;:::;t ?and t ;:::;t ?such that the corresponding unique strong solu 1 N 1 N

tionu x;t? of the Navier Stokes system satisﬁes, for alli? 1;:::;N and alljxj large

enough, the pointwise the lower bound

0 −d−1ju x;t ?j cjxj ;!i

and the stronger upper bound

−d−2ju x;t ?j Cjxj ;i

for a constantC>0independent onx and a constantc independent onjxj,but!

possibly dependent on the projection!?x=jxj ofx on the sphere, and such that

d−1 0c > 0 for a.e.!2S . Moreover,t andt can be taken arbitrarily close tot :! ii i

0jt −tj<" and jt −tj<"; fori? 1;:::;N:i ii i

Remark 1.2. The initial datum can be chosen of the form a ? curl ,

where is a linear combination of dilated and modulated of a single function (or

dvector ﬁeld, ifd? 3)’2S?R?, with compactly supported Fourier transform.

Roughly speaking, our construction works as follows: we look for an initial

datum of the forma? curl , where

d N?1?X

d=2 x?? ’ x? cos? x :j j

j?1

2d N?1? dThe unknown vector ?? ;:::; ?2R of all the phases 2R1 d N?1? j

2d N?1?will be assumed to belong to a suitable subspaceV R of dimension

792 LORENZO BRANDOLESE

d N ?1?in order to ensure, a priori, some nice geometrical properties of the ﬂow.

Such geometric properties consist of a kind of rotational symmetry, similar to that

considered in [4], but less stringent. In this way, the problem can be reduced to

the study of the zeros of the real function

Z Zt

t, u u x;s? dx ds:1 2

0

By an analyticity argument, this in turn is reduced to the study of the sign of the

function Z Zt

s? s?t, e a e a x;s? dx ds:1 2

0

This last problem is ﬁnally reduced to a linear system that can be solved with

elementary linear algebra.

The spatial decay at inﬁnity of the velocity ﬁeld is known to be closely relatedZ

to special algebraic relations in terms of the moments x curl u x;t? dx of the

vorticity curl u of the ﬂow, see [6]. Thus, one could restate the theorem in an

equivalent way in terms of identities between such moments for diVerent values

d 0of 2N , which are satisﬁed at the timet but brake down whent?t .i i

1.2. A concentration eVect of a diVerent nature. The concentration diffu

sion eVects described in Theorem 1.1 genuinely depend on the very special struc

ture of the nonlinearityPr u ⊗u , more than to the presence of the?u term.

Even though this result is not known for inviscid ﬂows yet, it can be expected that

a similar property should be observed also for the Euler equation.

On the other hand, the Laplace operator, commonly associated with diVusion

eVects, can be responsible also of concentration phenomena, of a diVerent nature.

For example, it can happen thata x? is a non decaying (or very slowly decaying)

vector ﬁeld, but such that the unique strong solutionu x;t? of the Navier Stokes

−d−1system have a quite fast pointwise decay asjxj!1(say, jxj ). This is

typically the case whena has rapidly increasing oscillations in the far ﬁeld. We

will discuss this issue in Section 3. Though elementary, the examples of ﬂows

presented in that section have some interest, being closely related to a problem

p daddressed by Kato about strong solutions inL ?R ?whenp<d , in his well

known paper [9].

1.3. Notation. Troughout the paper, ifu? u ;:::;u?is a vector ﬁeld1 d

dwith components in a linear spaceX,wewillwriteu2X, instead ofu2X .We

will adopt a similar convention for the tensors of the formu⊗u. We denote with

t?e the heat semigroup.

d dLetB 0;1?betheunitballinR and’2S?R? a function satisfying

Z

11 d 2ˆ ˆ ˆ ˆ ˆ(1.4) ’2C ?R ; supp’ B 0;1 ; ’ radial; ’ 0; j’j ? .0

d

Concentration diVusion EVects in Viscous Incompressible Flows 793

Z

−i xˆOur deﬁnition for the Fourier transform is’ ? ? ’ x?e dx. Then we

set

ˆ’ =?

ˆ(1.5) ’ ??? ; >0:

d=2

d dNext we deﬁne the orthogonal transformation ˜: R !R ,by

2˜?? ; ; if ?? ; ?2R;2 1 1 2

(1.6)

3˜ ?? ; ; ; if ?? ; ; ?2R:2 3 1 1 2 3

We deﬁne the curl??operator by

2curl ??−@;@ ; if : R !R;2 1

and by

0 1

@ −@ 2 3 3 2

B C

B C 3 3curl ?B@ −@ C if : R !R :3 1 1 3

@ A

@ −@ 1 2 2 1

−The notationf x;t? O ?jxj ? asjxj!1means thatf satisﬁes, for larget

−jxj, a bound of the formjf x;t?j A t?jxj , for some functionA locally

?bounded inR .

We shall make use of the usual Kronecker symbol, ?1or0,ifj?korj;k

j,k.

2. NONLINEAR CONCENTRATION-DIFFUSION EFFECTS

2.1. The analyticity of the ﬂow map. In this subsection we recall a few

well known facts.

B be the Navier Stokes bilinear operator, deﬁned byLet

Zt

t−s ?B?u;v t? − e Pr u ⊗v s? ds:

0

Then the Navier Stokes equations can be written in the following integral form

t?(2.1) u?u ?B?u;u ; u ?e a; div a ? 0:0 0

Even though in the sequel we will only deal with “concrete” functional spaces,

the problematic is better understood in an abstract setting: we will present it as

formulated in the paper by P. Auscher and Ph. Tchamitchian [1]. LetF be a

Banach space,u 2F,andletB:F F!F be a continuous bilinear operator,0

with operator normkBk.

794 LORENZO BRANDOLESE

Let us introduce the nonlinear operatorsT :F!F,k?1;2:::, deﬁnedk

through the formulae

T ? Id1 F

k−1X

T v? B T v ;T v ; k 2:k ‘ k−‘

‘?1

Then the following estimate holds:

C k−3=2kT u ?k k 4kBkkuk :k 0 F 0F

kBk

kMoreover, T is the restriction to the diagonal ofF ?FF of ak-k

kmultilinear operator :F !F.

Under the smallness assumption

1

(2.2) kuk ;0 F 4kBk

the series

1X

(2.3) ? u ? T u ;0 k 0

k?1

is absolutely convergent inF and its sum ? u ?is a solution of the equation0

u?u ?B?u;u . Furthermore, ? u ?is the only s in the closed ball0 0

B ? 0; 1= 2kBk?? . The proof of these claims would be a straightforward appli F

cation of the contraction mapping theorem under the more restrictive condition

kuk < 1= 4kBk? . The proof of the more subtle version stated here can be0 F

found in [1], [11].

Coming back to Navier Stokes, in the proof of Theorem 1.1 we will need to

write the solution of the Navier Stokes system asu?? a , where

1X

t?(2.4) ? a t? T e a ;k

k?1

2 dthe series being absolutely convergent inC?? 0;1 ;L ?R ?? . There are several

ways to achieve this, and one of the simplest (which goes through in all dimen

siond 2) is the following: we chooseF as the space of all functionsf in

2 dC?? 0;1 ;L ?R ?? , such thatkfk <1, whereF

d?1 d?1 = 2(2.5) kfk ess sup ? 1 j xj? jf x;t?j ess sup ? 1?t jf x;t?j:F x;t x;t

The bicontinuity of the bilinear operatorB is easily proved in this spaceF (see

[12]). Indeed, one can prove this only using the well known scaling relations and

t?pointwise estimates on the kernelF x;t? of the operatore P div:

p

− d?1 = 2 −d−1F x;t??t F x= t;1 ; jF x; 1?j C 1 jxj? :

Concentration diVusion EVects in Viscous Incompressible Flows 795

We can conclude that there is a constant > 0, depending only ond,suchd

that if

t?(2.6) ke ak < ;F d

then there is a solutionu?? a 2Fof the Navier Stokes equations such that

the series (2.4) is absolutely convergent in theF norm. The absolute convergence

2 dof such series inC?? 0;1 ;L ?R ??is then straightforward under the smallness

assumption (2.6).

t?The ﬁniteness ofke ak can be ensured, e.g. by the two conditionsF

Z

d?1ja x?j? 1 jxj?dx<1 and ess sup ? 1 jxj? ja x?j<1:dx2R

The smallness condition (2.6) could be be slightly relaxed, see [5].

2.2. The construction of the initial datum. This section devoted to a con

structive proof of the following lemma.

Lemma 2.1. Letd? 2, 3,and">0 . Let alsoN2N and 0<t< <t1 N

be a ﬁnite sequence. Then there exists a divergence free vector ﬁelda? a ;:::;a?21 d

dS?R?, such that

d˜ ˜(2.7) a x??a x ; x 2R

?(see Section 1.3 for the notation) and such that the function E a t? : R ! R,

deﬁned by

ZZt

s? s?(2.8) E a t? − e a x e a x? dx ds;1 2

0

changes sign inside t −";t ?" ,fori?1;:::;N.i i

Proof. It is convenient to separate the two and three dimensional cases

2The cased? 2. We start setting, for each 2R ,

ˆ ˆ ˆ ˆ ˜ ˆ ˜(2.9) ?? ’ −? ?’ ?? −’ −? −’ ? ;

2for some’2S?R? satisfying conditions (1.4). Next we introduce the divergence

free vector ﬁelda x?, through the relation

!

ˆ−i ??2ˆ(2.10) a ??? :ˆi ??1

796 LORENZO BRANDOLESE

2 ˆNote thata 2S?R? is real valued (because is real valued and such that

ˆ ˆ ??? ?−? ) and satisﬁes the fundamental symmetry condition

˜ ˜(2.11) a x??a ?x :

ˆNext we deﬁne , a as before, by simply replacing’ˆ with’ˆ in the corre-

sponding deﬁnitions. The vector ﬁeldsa will be our “building blocks” of our

initial datum.

Applying the Plancherel theorem to the right hand side of (2.8) and using the

symmetry relations

ˆ ˆ ??? ?−?;

˜ˆ ˆj ??j j ??j;

we get

Z

2 1 2−2tj j 2ˆ(2.12) E a t? ? ? 1−e ? j ??j d 22j j

Z

2 1 2−2tj j 2ˆ? 2 ? 1−e ? j ??j d :

2j j j j1 2

2From now on, the components of 2 R will be assumed to satisfy the

following conditions

8

> >j j;< 1 2

(2.13)

>: , 0:2

p

This guarantees that for a suYciently small > 0 (i.e., when >jj 2),1 2

we have

Z

2 1 2 −2tj j 2ˆ(2.14) E a t? ? 2 ? 1−e ? j’ ? −?j d : 2j j

If we set

2app 1 2−2tj j(2.15) aE t? ? 1−e ? ;

2j j

then we immediately obtain

appE a t? !E t ; as ! 0

uniformly with respect tot 0.

Concentration diVusion EVects in Viscous Incompressible Flows 797

We now associate to t ;:::;t ?two more sequences (to be chosen later)1 N

N?1 2 N?1?? ;:::; ?2R and? ;:::; ?2R , where ?? ; ? .1 N?1 1 N?1 j j;1 j;2?

First we require that the components and satisfy condition (2.13)forallj;1 j;2

0 0

0j?1;:::;N?1andthat , ,forj,j andj,j ? 1;:::;N?1. Thisj j

ˆ ˆsecond requirement ensures that the supports ofa anda are disjoint when 0j j

is suYciently small.

We now consider the initial data of the form

N?1X

(2.16) a x? a x :j j

j?1

ˆOwing to the condition on the supports ofa , we see that for > 0smal j

enough,

N?1X

2 (2.17) E a t? ? E a t :j j

j?1

Let

2 j;1 j;2 2j −2j jj(2.18) ? and A ?e :j j2j jj

appThus, as ! 0, we getE a t? !E t?, uniformly in? 0;1? , where

N?1X

app t(2.19) E t?? ? 1−A :j j

j?1

Let us observe that

N?1app XdE t(2.20) t? − log A A :j j jdt

j?1

We want to determine ? ;:::; ?and ? ;:::; ?in such a way that1 N?1 1 N?1

appE t? vanishes at t;:::;t , changing sign at those points. This leads us to1 N

study the system ofN equalities andN ‘non equalities’,

8

app>E t ?? 0;> i<

(2.21) i? 1;:::;N:app>dE: t ?, 0;idt

2Let us choosej j ?γj, for an arbitraryγ>0 (this choice is not essential,j

−2γtibut will greatly simplify the calculations) and setT ?e . Recalling (2.18),i

t jiwe getA ?T and log A ? −2γj. In order to study the system (2.21), wejj i

2introduce the N ? 1? matrix

798 LORENZO BRANDOLESE

0 1

2 N?11−T 1−T 1−T1 1 1

B C

B C

B : : : C: : :B C: : :B C

(2.22) M BB C:

B C

2 N?1B C1−T 1−T 1−TNB N N C

@ A

2 N?1T 2T N ? 1 T1 1 1

We claim that detM , 0. Indeed, by an explicit computation,

N NY Y Y

det M? −T?1−T? ?1−T? T −T? T 0−T :1 1 i 1 i i i

0i?1 i?2 1 i<i N

−2γtiRecalling thatT ?e 2? 0;1?and thatt ,t 0 proves our claim. The abovei i i

formula can be checked by induction. Otherwise, one can reduceM after elemen

tary factorizations to a Vandermonde type matrix (see [10] for explicit formulae

on determinants).

Then, for anyc, 0, the linear system with unknown ?? ;:::; ?,1 N?1

0 1

0

B C:B C::B C(2.23) M ?B C

@ 0 A

c

N?1 has a unique solution 2 R , , 0. By our construction, the function

app E t? obtained taking ? satisﬁes the N equations and the ﬁrst ‘non

equality’ of the system (2.21). More precisely, we get

d cappE t ?? , 0:1dt 2γ

The otherN− 1 ‘non equalities’ of the system (2.21) are then automatically ful

appﬁlled. Indeed, if otherwise, we had? d=dt E t ?? 0, for somei? 2;:::;N,i

then the matrix obtained replacing in (2.22) the last line with

2 N?1T 2T N ? 1 Ti i i

would have been of determinant zero, thus contradicting our preceding formula

for det M?.

By conditions (2.18), for allj? 1;:::;N?1, the real number deﬁnes (inj

a non unique way) a real and a vector? ; ?with components satisfyingj j;1 j;2

(2.13), such that? ;:::; ?,?0;:::;0?.1 N?1