Concentration diffusion Effects in Viscous Incompressible Flows

Indiana University

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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

Sujets

• University
• Navier?Stokes equations

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 10
• Langue: English

Extrait

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