Concentration diffusion Effects in Viscous Incompressible Flows

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Concentration-diffusion Effects in Viscous Incompressible Flows LORENZO BRANDOLESE ABSTRACT. Given a finite sequence of times 0 < t1 < < tN , we construct an example of a smooth solution of the free nonstationnary Navier-Stokes equations in Rd, d ? 2;3, such that: (i) The velocity field u?x; t? is spatially poorly localized at the beginning of the evolution but tends to concentrate until, as the time t approaches t1, it becomes well-localized. (ii) Then u spreads out again after t1, and such concentration-diffusion phenomena are later reproduced near the instants t2, t3, . . . . 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 ?x0; t0? in space-time. This is the goal of the present 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 x0, as the time t approaches t0

  • identities between such

  • ?x? ? d?n?1?

  • datum can

  • navier stokes equations

  • such geometric

  • time ti

  • curl ?

  • ti ?j


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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