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- vector field
- leray solutions
- global existence
- navier stokes equations
- only known
- interpolation between
- besov space
- numbers such

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Tuesday, June 19, 2012

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ARMA manuscript No.

(will be inserted by the editor)

On infinite energy solutions to the

Navier–Stokes equations: global 2D existence

and 3D weak-strong uniqueness

I

SABELLE

G

ALLAGHER

, F

ABRICE

P

LANCHON

Abstract

This paper studies the bidimensional Navier–Stokes equations with large ini-

tial data in the homogeneous Besov space

. As long as

,

global existence and uniqueness of solutions are proved. We also prove that weak–

strong uniqueness holds for the

-dimensional equations with data in

for

.

1. Introduction

We are interested in solving the 2D incompressible Navier-Stokes system in

the whole space, say

(1.1)

The vector field

stands for the velocity of the fluid, the scalar field

for its

pressure, and

means that the fluid is incompressible.

Recall that global existence for large data in the energy class is well-known;

that result goes back to J. Leray [19], and states that for any divergence free ini-

tial data

in the space

, there is a unique, global solution

to (1.1).

If

is the homogeneous Sobolev space then the solution

is in the energy

space

, where

stands for the space

of functions which are continuous and bounded on

. Moreover, the solution

satisfies the energy equality

2

I. Gallagher and F. Planchon

More recently, global existence for large data was proved for measure-valued

vorticity (G.-H. Cottet [9] and Y. Giga, T. Miyakawa and H. Osada [14]); unique-

ness is only known under a smallness assumption on the atomic part of the measure

([14,16]). In this situation, the initial velocity field

given by the Biot-Savart Law

is known to be at least in the Lorentz space

, which is strictly larger than

;

but not all

can be paired with a measure-valued vorticity. On the other

hand, global existence holds for almost every conceivable function space under a

smallness assumption. The most recent and almost final result is for data which are

first derivatives of

functions (see the work of H. Koch and D. Tataru [17]);

we will call that space

in the sequel.

In 3D the situation is a lot more complex, and little is known between the

weak

solutions (Leray’s solutions, in

,

which are known to exist with no uniqueness result) and the strong small

solutions (Kato’s solutions [15], which exist and are unique in

(see [11] for uniqueness) for small data). One has however weak solutions for a

large class of initial data: weak

solutions for

were constructed

by C. Calder

´

on in [4] and more recently, P.-G. Lemari

´

e extended those results to

“locally

” data ([18]). Uniqueness is of course an open problem. We refer to the

work of P. Auscher and P. Tchamitchian [1] for the presentation of a large class of

function spaces in which the Navier-Stokes equations can be solved uniquely and

globally, for small data (or locally for large data).

On the other hand, in 2D one expects the small data existence to extend to

large data, even beyond

data, as long as one works with a functional space

which scales like

. Recall that the scaling of the Navier–Stokes equations in

,

with

, is as follows: for any real number

,

is a solution to the Navier–

Stokes equations associated with the data

if the same goes for

associated

with

, with

def

and

def

The space

is clearly invariant under the transformation

.

In order to achieve global existence results, we will follow Calder

´

on’s proce-

dure [4] and perform a (non-linear) interpolation between Leray’s solutions and

Kato’s solutions (or more accurately, their extensions to Besov spaces). Hence,

one expects to get any data which fits into any interpolation space between

and

. The Besov spaces

appear very naturally in this con-

text, for

,

(the case where

is essentially easy, as the

regularity is then positive). We note that by using the different techniques devel-

oped in [18], one could get another class of initial data (roughly the density of the

Schwartz class in the Morrey-Campanato space

), but still miss homoge-

neous data. We emphasize the fact that the most interesting case is for

and

large, for which

is close to

. Indeed, as soon as one gets a

global existence result for

large, it automatically implies global existence for

all

and

, because of the embedding

.

Before stating our result we recall what Besov spaces are, through their char-

acterizations via frequency localization (see [2] for details).

Infinite energy solutions to the Navier–Stokes equations

3

Definition 1.

Let

be such that

for

and

for

. Define, for

, the function

def

, and the Littlewood–

Paley operators

def

and

def

Let

be in

.

If

,

then

belongs to the homogeneous Besov space

if and only if

–

The partial sum

converges towards

as a tempered distribution;

–

The sequence

def

belongs to

.

Theorem 1.1 (2D global existence).

Let

and

be two real numbers such that

and

. Let

be a divergence

free vector field. Then there exists a unique global solution to (1.1) such that

. Moreover, if

, then there exists a con-

stant

such that

(1.2)

R

EMARK

. Note that previous results recalled above, on data whose curl is a

measure, do not include this situation. Indeed such results correspond heuristically

to cases when

and

, whereas in our case the interest lies espe-

cially when

and

are close to infinity. Note moreover that examples of functions

precisely in such Besov spaces can be constructed, either simply by using the def-

inition presented above, or more explicitly for instance as in the book [22]. To get

a sense of perspective, one may imagine plotting spaces in the interpolation square

, for

. Global existence was previously known only at the point

(J. Leray [19]) and (for a subset of) the segment

(Y. Giga, T.

Miyakawa and H. Osada [14]). As the result at one point yields the result for the

upper–right square, that is materialized by a “Leray-square” and a “GMO-square”

on the figure below; the result proved here is the rest of the square, except the

remaining part of its lower side and its left vertical boundary (

or

).

1/2

1/2

1

1

Leray

1/q

1/r

G-M-0

0

4

I. Gallagher and F. Planchon

Let us note that in the situation where

, for

nothing prevents

from choosing

. Indeed all estimates in this situation can be made indepen-

dent of

, and thus one would recover the bottom line result between

and

. We

elected not to do so, at this requires some non-trivial form of limiting procedure,

in the same spirit as for measure-valued vorticities.

Let us sketch the procedure leading to the result:

1. take

. Split

where

and

with small norm (actually, taking

with small

norm will do, assuming

and

).

2. construct the small data solution

to the Navier-Stokes system with initial

data

.

3. write down the equation for

. This becomes a Navier-Stokes-like

system for

, with additional terms containing

, which we then solve to obtain

local in time

solutions.

4. obtain an a priori bound for the energy of

. In order to do so, we need to

control the additional terms containing

, by the energy of

. We then extend

the local solution

to the desired global solution.

5. local existence and uniqueness are known to hold in such a Besov space

, hence the solution

obtained is unique.

The crucial point in the procedure described above is to obtain estimates on

the additional terms in the equation on

containing

(points 4 and 5). Actu-

ally some of those estimates will turn out to be very similar to estimates use-

ful in higher dimensions. As recalled above, if one considers an initial data

in

, for

and

,

, then there is a

unique maximal time

and a unique solution

to

associated with

, in

the space

and such that

with

. Note that one has in fact continuity in time, except at time zero. A

natural question one can ask is then the following: if the initial data is additionally

in

, then it is not difficult to see that

is also a Leray solution associated

with

; but does uniqueness hold in that larger class of Leray solutions? In other

words, as long as one has a “strong” solution to

, which is also in the energy

space

def

do all Leray solutions coincide with that one?

We have called “Leray solution” any weak

solution

of the Navier–Stokes

equations in

, with

, satisfying the energy estimate

(1.3)

Infinite energy solutions to the Navier–Stokes equations

5

The answer to that question is given through the following stability theorem. Be-

fore stating it, we will need the following proposition, which will be proved in the

last section.

Proposition 1.1.

Consider

, and let

and

be two real numbers such

that

. Then for any divergence free initial data

, the unique solution

associated with

satisfies, for all

and all

,

Theorem 1.2 (Stability).

Consider

, and let

and

be two real numbers

such that

. Suppose additionally that

Let

and

be two divergence free vector fields in

, and suppose that

is also an element of

. Let

be any Leray solution associated

with

,

and

let

be

the

unique

solution

associated

with

,

with

for some time

. Then

def

satisfies, for all times

,

Actually, the result holds with

as well, with the Besov norm

inside the

norm above replaced by an

, when

. In-

deed this can be seen somehow as a consequence of Serrin’s criterion [23] and inte-

grability properties of strong solutions with data

in

with

.

Thus, our theorem is really of interest when

, for which one can go up to

. Another reason which makes the result worth stating is its proof, which

divides the crucial trilinear estimate into three different pieces of which only one

requires the restriction on

and

; it does not seem possible to improve on the

continuity Lemma 1.1 below without using in a much deeper way the fact that not

only

and

are in the Leray class

but also solutions of the equation.

That theorem yields in a direct way the following corollary.

Corollary 1.1 (Weak–strong uniqueness).

Let

be a divergence free vector

field in

, with the same restrictions as in Theorem 1.2, and

define the associate solution

, which is unique

in

. Then all Leray solutions associated with

coincide

with

on the time

.

6

I. Gallagher and F. Planchon

Before going on with the proof of the results presented here, let us state the main

lemma for the proof of Theorem 1.2.

Lemma 1.1.

Let

be fixed, and let

and

be two real numbers such that

and

. Then for every

, the

trilinear form

is continuous. In particular the following estimates hold:

(1.4)

and

(1.5)

The rest of the paper is organized as follows. In the first section, we prove

Theorem 1.2, as some of the estimates will be useful in the 2D case as well. Then

in Section 3 we proceed with Theorem 1.1. The last section consists in the proof

of various estimates used in the previous sections.

2. Proof of the weak–strong uniqueness result

The aim of this section is to prove Theorem 1.2. Let us recall the situation: we

consider two divergence free vector fields

and

, with

and

The space dimension here is

, and we have chosen

with

We associate with

and

two Leray solutions

and

, in the space

, with

additionally according to Proposition 1.1,

Infinite energy solutions to the Navier–Stokes equations

7

If

def

, then we wish to prove that for all

,

The method of proof of that type of stability result goes back to J. Leray [19],

and can be found in the book of W. von Wahl [24] (see also the more recent

works [12] and [13]). The idea is as follows: since the vector field

def

is in

by assumption, we can write

where

denotes the scalar product in

. The energy estimate (1.3) re-

called in the introduction then implies that

Let us prove the following result.

Lemma 2.1.

Under the assumptions of Theorem 1.2, we have, for all times

,

P

ROOF OF

L

EMMA

2.1. A formal computation yields the result with no difficulty;

in order to prove it, let us consider two sequences of smooth, divergence free vector

fields

and

such that

in

and

in

Taking the scalar product with

and

of the Navier–Stokes equations on

and

respectively yields, after integration in time and and integration by parts in

the space variables,

8

I. Gallagher and F. Planchon

and

It is now a matter of taking the limit in

, and of summing the limits found. Since

both

and

converge in

towards

and

respectively,

it is clear that

Then Lemma 1.1 implies that

Similarly, since the divergence free condition on

yields

we have, still by Lemma 1.1,

But

in

, where

stands for the Leray projector onto

divergence–free vector fields, so those limits imply in particular that

and similarly

Putting everything together, we find that the limit of

is

Infinite energy solutions to the Navier–Stokes equations

9

Then we just need to notice that

and on the other hand

and Lemma 2.1 is proved.

Now let us go back to the proof of the theorem. Recall that we have obtained

so with Lemma 2.1, that means that

But Lemma 1.1, and in particular estimate (1.5), then yields

and since

is an element of

, the result follows by a

Gronwall inequality, and Theorem 1.2 is proved.

Let us make some additional remarks on the case

: from the prop-

erties of

, namely

and

, it is well-known that

,

(2.1)

where

is the Hardy space. The first part was proved in [8], while the second

follows from (sharp) Sobolev embedding and H

¨

older. Hence, in order to make

sense of the trilinear form in Lemma 1.1, a sufficient condition would be for the

strong solution

to verify

BMO

(2.2)

By interpolation, one is naturally led to the (stronger) Serrin condition

with

(2.3)

Such a condition is automatically verified for data

with

,

see e.g. [6].

10

I. Gallagher and F. Planchon

3. Global

existence

The aim of this section is to prove Theorem 1.1. In the whole of this section,

all space norms will be taken over

, which we will omit to specify from now

on. We start by splitting the data

into two distinct pieces,

and

, with a small norm. We suppose that

and

. This can

always be achieved since our Besov space is an interpolation space between

and

the larger Besov space

. We remark that the worst case scenario is clearly

for

large, and thus we implicitly assume from now on that

for some

large

, unless explicitly mentioned.

3.1. Small solutions in the Besov space

This section corresponds to Part 2 of the procedure explained in the introduc-

tion. In order to simplify notations, we relabel

and

to be

and

: that should

not lead to any confusion, as we will not be considering the function

any longer.

All known results apply to solve

(3.1)

where

stands for the Leray projector onto divergence free vector fields. The

interested reader may consult [5,7,21] for results of this type by different methods.

What we will use is the following result.

Proposition 3.1.

Let

with small norm. Then there exists a unique

global solution

of (3.1) which is such that

.

We remark that the uniqueness part does not follow from the construction of the

solution and is in fact a recent result, [11]. As explained in the above references,

the unique solution

satisfies many additional properties, of which the following

estimate will be the most useful:

for

and

(3.2)

In what follows we will assume that

is very small.

3.2.

solutions to a modified Navier-Stokes system

We shall deal here with points 3 and 4 of the procedure sketched in the intro-

duction. We aim at getting a solution of

(3.3)

Infinite energy solutions to the Navier–Stokes equations

11

where recall that

, and

satisfies the estimates of Section 3.1.

For this step, many different choices are possible. One may proceed by mol-

lifying the data and/or the equation as it is customary for the weak

theory.

Though this can be easily accomplished even with the addition of the

term, we

proceed differently and simply get a local in time solution by fixed point on the

integral equation

(3.4)

That local solution will be made global in time by proving an energy estimate in

the next section.

The result we shall prove is the following. Before stating it, note that we are

going to use Lorentz spaces

, which as far as we are concerned may simply be

seen as the real interpolation spaces

(see [2]).

Proposition 3.2.

Let us define the space

, for

:

def

where

def

(3.5)

Then there exists a time

and a unique solution

to (3.3) in the space

.

P

ROOF OF

P

ROPOSITION

3.2. This follows readily by contraction in

(note

that the choice of

is of course one out of many). As we are going to perform

computations on Lorentz spaces, we refer to the last section for the equivalent of

H

¨

older and Young’s inequalities for those spaces, which we shall refer to as O’Neil

inequalities.

We will denote by

,

,

and

each part of the norm

defined in (3.5). Note that the introduction of Lorentz spaces will turn out useful

to obtain the

estimate on

.

Before proceeding with estimates, we perform the following reduction: we can

replace the bilinear operator in (3.4) by its scalar counterpart, which reads

def

(3.6)

where

(as

). Thus every function is now a scalar

function which should be understood as any coordinate of the velocity field.

We proceed now with proving contraction properties for

for all the norms

of (3.5). We begin with the first one, which is Kato’s weighted norm. If

and

stand for two successive iterates in a fixed point scheme, then we write

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