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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 ISABELLE GALLAGHER, FABRICE PLANCHON Abstract This paper studies the bidimensional Navier–Stokes equations with large ini- tial data in the homogeneous Besov space _B 2 r 1 r;q (R 2 ). As long as r; q < +1, global existence and uniqueness of solutions are proved. We also prove that weak– strong uniqueness holds for the d-dimensional equations with data in _B d r 1 r;q (R d )\ L 2 (R d ) for d=r + 2=q 1. 1. Introduction We are interested in solving the 2D incompressible Navier-Stokes system in the whole space, say 8 > < > : @u @t = u u rurp; r u = 0; u(x; 0) = u 0 (x); x 2 R 2 ; t 0: (1.1) The vector field u(t; x) stands for the velocity of the fluid, the scalar field p for its pressure, and r u = 0 means that the fluid is incompressible.

  • vector field

  • leray solutions

  • global existence

  • navier stokes equations

  • only known

  • interpolation between

  • besov space

  • numbers such


Published : Tuesday, June 19, 2012
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Origin : math.unice.fr
Number of pages: 29
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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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