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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


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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).
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