In this paper the homogeneous, incompressible Navier-Stokes equations areconsidered, and a number of results are reviewed which are related to thescaling of the equations. More specifically the initial value problem is studied inscale-invariant function spaces, insisting on the special role of the “largest” scale-invariant function space; the specificity of two space dimensions is recalled, interms of the velocity field and the vorticity. Some examples of arbitrarily largeinitial data giving rise to a global solution are also provided, as well as a studyof the long-time behavior of global solutions and their behavior at blow-up time(supposing such a time exists).
1 Introduction
1.1 The Equations
The initial value problem for the homogeneous, incompressible Navier-Stokessystem writes
.NS/
8<
:
@tuC u � ru ��u D �rp in RC � R
d
div u D 0ujtD0 D u0;
where p D p.t; x/ and u D .u1; : : : ; ud /.t; x/ are, respectively, the pressure andvelocity of an incompressible, viscous fluid in d space dimensions. The viscosityis chosen equal to one to simplify the notation, and all exterior forcing terms areneglected. In the sequel the physical situations d D 2 or 3 are considered. Thereason for considering the equations in the whole space only, and not with physicalboundary conditions, has to do with the critical nature of the study, as explainedfurther down.
As is well-known, the Navier-Stokes system enjoys two important features. Firstit formally conserves the energy, in the sense that any smooth solution, decaying tozero as jxj goes to infinity, satisfies the following energy equality for all times t � 0:
1
2ku.t/k2
L2.Rd /C
ˆ t
0
kru.t 0/k2L2.Rd /
dt 0 D1
2ku0k
2L2.Rd /
: (1)
The energy equality (1) can easily be obtained (formally) by noticing that thanks tothe divergence-free condition, the nonlinear term is skew-symmetric in L2: one hasindeed if u and p are smooth enough and decaying at infinity,
�u.t/ � ru.t/Crp.t/ju.t/
�
L2.Rd /D 0:
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 3
Second, (NS) enjoys a scaling invariance property: defining the scaling and transla-tion operators, for any positive real number � and any point x0 of Rd , by
ƒ�;x0�.t; x/ WD1
��� t
�2;x � x0
�
�; (2)
if u solves (NS) with data u0, then ƒ�;x0u solves (NS) with data ƒ�;x0u0.This is the main property this chapter focusses on: the aim is to analyze the
wellposedness of (NS) in critical spaces, meaning spaces invariant under theoperators ƒ�;x0 (see the coming Sect. 1.2 for a definition).
1.2 Critical Spaces
A family X0 of distributions defined on Rd is critical if
8� > 0; 8x0 2 Rd ; u0 2 X0” ƒ�;x0u0 2 X0 with ku0kX0 D kƒ�;x0u0kX0:
Similarly a family .XT /T>0 of spaces of distributions defined over .0; T / � Rd is
scale invariant if for all T > 0 one has, with notation (2),
(3)Some examples of critical spaces follow. Defining (homogeneous) Sobolevspaces by
kf k PHs.Rd / WD
�ˆRd
j Of .�/j2j�j2s d�
� 12
where Of is the Fourier transform of f
Of .�/ WD
ˆRd
f .x/e�ix�� dx;
it is easy to see that PHd2�1.Rd / is critical. Similarly the (larger) Lebesgue
space Ld.Rd / is critical and so are the (yet larger again if p > d ) Besov
spaces PBdp�1
p;q .Rd /, defined as follows, using the Littlewood-Paley decomposition.
Definition 1. Let O� be a radial function in D.R/ such that O�.t/ D 1 for jt j 6 1
and O�.t/ D 0 for jt j > 2. For j 2 Z, Fourier truncations are defined by
bSj f .�/ WD O��2�j .j�j/
�Of .�/ and �j WD SjC1 � Sj :
4 I. Gallagher
For all p; q in Œ1;1� and s in R with s < d=p (or s 6 d=p if q D 1), thehomogeneous Besov space PBs
p;q is defined as the space of tempered distributions fsuch that
kf k PBsp;q.Rd /WD���2jsk�jf kLp.Rd /
���`q<1:
In all other cases of indexes s, the Besov space is defined similarly, up to taking thequotient with polynomials.
An equivalent norm is given for all s 2 R by
kf k PBsp;q.Rd /���k�1�s=2@�e
��f kLp��Lq.RC; d�� /
;
and when s < 0
kf k PBsp;q.Rd /���k��s=2e��f kLp
��Lq.RC; d�� /
: (4)
Note that in dimension 2, the energy spaceL2.R2/ is critical. The equation is said tobe critical. On the opposite in dimensions 3 and higher, the equation is supercritical.
Let us now define a solution to (NS).
Definition 2. A vector field u is said a (scaled) solution to (NS) associated withthe data u0 if it is a solution in the sense of distributions, belonging to a family ofscale-invariant spaces.
Remark 1. The pressure is not mentioned in the definition of a solution. This is dueto the fact that it may be recovered from the velocity field by solving the equation
��p D div .u � ru/;
as can be seen by applying the divergence operator to (NS).
Remark that since one is considering translations and changes of scale in thespace variable, it is natural to restrict our attention to the case of the whole space Rd .The (NS) system makes sense of course (in fact physically more sense) when set ina domain with boundaries, but scaling is less relevant in that setting so the focus ison the whole space here.
1.3 Plan of the Paper
Section 2 is devoted to the presentation of two general methods of solving (NS) incritical spaces: the first one, in Sect. 2.1, by an energy-type estimate and the other, inSect. 2.2, by a fixed-point argument. In this chapter those techniques are developed
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 5
in two different functional settings, namely, in the Sobolev space PHd2 �1 and the
Besov space PBdp�1
p;1 , respectively, which are both scale-invariant spaces. One canthen ask whether it is possible to solve (NS) in other scale-invariant spaces by suchtechniques and if so in “how many” such spaces. The answer is provided in Sect. 2.3,where it is shown that there is a largest possible critical space, namely, PB�11;1.Unfortunately (NS) is not wellposed in that space; however there is a slightly smallerspace in which (NS) is wellposed, which is BMO�1; that space is presented also inthat paragraph.
In Sect. 3 the special case of two space dimensions is studied, where the equationsare critical and shown to be globally wellposed. Two frameworks are studied inparticular: on the one hand, the case when the velocity lies in the energy spaceor in larger scale-invariant spaces (Sects. 3.1 and 3.2) and, on the other hand,measure-valued vorticity (Sect. 3.3). Contrary to Sect. 2, in this part not only scalingproperties are used, but also the energy conservation for the velocity, or the transportof vorticity. This is important as it can be shown (examples are provided in Sect. 2.4)that some equations with the same scale invariance as (NS) blow up in finite time.
In Sect. 4 some examples are provided showing that smallness of critical normsis not strictly necessary in order to prove the existence and uniqueness of globalsolutions.
Section 5 is devoted to the large-time behavior of global solutions.Finally Sect. 6 studies the behavior of solutions at possible blow-up time,
supposing such a time exists.As this text is intended as a survey rather than a research article, very few rigorous
proofs are to be found here. Some results will be stated with no proofs at all, oftendue to their technical nature, and others will be presented with a rough sketch ofproof, whose goal is to give a flavor of the methods involved. However precisereferences to the literature are given all along the text for the interested reader.
2 The Initial Value Problem in Critical Spaces
In this section some classical methods of solving the initial value problem for (NS)in critical spaces are reviewed. The first results in that direction go back to thepioneering works of J. Leray [68, 69] which will be referred to constantly in thesequel. However more modern versions of those results are presented, in particularthe Fujita-Kato theorem in Sect. 2.1, the Cannone-Meyer-Planchon theorem inSect. 2.2, and the Koch-Tataru theorem in Sect. 2.3. Some remarks on the specialrole of the nonlinear term are provided in Sect. 2.4, in particular in the constructionof weak solutions.
2.1 Wellposedness in PHd2�1
In this paragraph the proof of the following important result is sketched, originallydue to H. Fujita and T. Kato [31].
6 I. Gallagher
Theorem 1 ([31]). There is a constant c > 0 such that if u0 2 PHd2 �1.Rd / is a
divergence-free vector field satisfying
ku0k PH d2 �1.Rd /
6 c; (5)
then there is only one solution associated with u0, which satisfies for all t � 0
ku.t/k2PHd2 �1.Rd /
C
ˆ t
0
kru.t 0/k2PHd2 �1.Rd /
dt 0 6 ku0k2PHd2 �1.Rd /
:
Without the smallness assumption (5), existence holds at least for a short time, timeat which the solution ceases to belong to L2.Œ0; T �I PH
d2 .Rd //.
Sketch of proof. One can write a (formal) energy estimate in PHd2 �1.Rd / on the (NS)
system. Denoting by .� j �/ PHs.Rd / the scalar product in PHs.Rd /,
1
2ku.t/k2
PHd2 �1.Rd /
C
ˆ t
0
kru.t 0/k2PHd2 �1.Rd /
dt 0 �1
2ku0k
2
PHd2 �1.Rd /
Cˇˇˇ
ˆ t
0
.u � ruju/PHd2 �1.Rd /
.t 0/ dt 0ˇˇˇ:
Note that the pressure has disappeared, thanks to the fact that u is divergence-freewhich implies that
.u j rp/PHd2 �1.Rd /
D 0:
Then writing
ˇˇ.ajb/
PHd2 �1.Rd /
ˇˇ 6 Ckak
PHd2 �2.Rd /
krbkPHd2 �1.Rd /
one infers
1
2ku.t/k2
PHd2 �1.Rd /
C
ˆ t
0
kru.t 0/k2PHd2 �1.Rd /
dt 0 6 12ku0k
2
PHd2 �1.Rd /
C C
ˆ t
0
ku � ru.t 0/kPHd2 �2.Rd /
kru.t 0/kPHd2 �1.Rd /
dt 0:
(6)
If d D 2 then one can use the fact that u is divergence-free to write
ku � ru.t 0/k PH�1.R2/ D kdiv .u˝ u/.t 0/k PH�1.R2/
6 Cku˝ u.t 0/kL2.R2/
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 7
where C is a constant which may change from line to line. Then by Hölder’sfollowed by Gagliardo-Nirenberg’s inequality, one finds
Similarly if d D 3 then by Sobolev’s embeddings, one has
ku � ru.t 0/kPH�
12 .R3/6 Cku � ru.t 0/k
L32 .R3/
and then by Hölder’s inequality and Sobolev’s embeddings, again one infers
ku � ru.t 0/kPH�
12 .R3/6 Cku.t 0/kL3.R3/kru.t 0/kL3.R3/
6 Cku.t 0/kPH12 .R3/kru.t 0/k
PH12 .R3/
:
So in both cases, one finds finally
ku � ru.t 0/kPHd2 �2.Rd /
6 Cku.t 0/kPHd2 �1.Rd /
kru.t 0/kPHd2 �1.Rd /
: (7)
Returning to (6) one has therefore
1
2ku.t/k2
PHd2 �1.Rd /
C
ˆ t
0
kru.t 0/k2PHd2 �1.Rd /
dt 0 6 12ku0k
2
PHd2 �1.Rd /
C C
ˆ t
0
ku.t 0/kPHd2 �1.Rd /
kru.t 0/k2PHd2 �1.Rd /
dt 0:
One concludes (formally) that as long as ku.t/kPHd2 �1.Rd /
6 2c 6 1=2C then
ku.t/kPHd2 �1.Rd /
6 ku0k PH d2 �1.Rd /
6 c;
so a continuity argument gives the result in the case of small enough initial data.In the case when the initial data is not small, then one writes
u D uL C v with uL WD et�u0
and the equation for v is solved:
8<
:
@tv C v � rv C uL � rv C v � ruL ��v D �rp � uL � ruL in RC � R
d
div v D 0vjtD0 D 0:
8 I. Gallagher
Again an PHd2�1.Rd / energy estimate gives formally, exactly as above,
1
2kv.t/k2
PHd2 �1.Rd /
C
ˆ t
0
krv.t 0/k2PHd2 �1.Rd /
dt 0
6 Cˆ t
0
kv.t 0/kPHd2 �1.Rd /
krv.t 0/k2PHd2 �1.Rd /
dt 0
C C
ˆ t
0
kv.t 0/kPHd2 �1.Rd /
krv.t 0/kPHd2 �1.Rd /
kruL.t0/kPHd2 �1.Rd /
dt 0
C C
ˆ t
0
krv.t 0/kPHd2 �1.Rd /
kuL.t0/k2PHd�12 .Rd /
dt 0;
where in the last line the inequality
kuL � ruL.t0/kPHd2 �2.Rd /
6 CkuL.t 0/k2PHd�12
has been used, which can be proved by a similar argument to (7) above. Nowconsider the largest time interval on which kv.t 0/k
PHd2 �1.Rd /
6 2c 6 1=4C . Then
on that time interval
kv.t/k2PHd2 �1.Rd /
C
ˆ t
0
krv.t 0/k2PHd2 �1.Rd /
dt 0
6ˆ t
0
kv.t 0/k2PHd2 �1.Rd /
kruL.t0/k2PHd2 �1.Rd /
dt 0
C
ˆ t
0
kuL.t0/k4PHd�12dt 0;
and Gronwall’s inequality again�noting that kruLk
L2.RCI PHd2 �1.Rd //
6Cku0k PH d2 �1.Rd /
�
gives the result as long as
ˆ T
0
kuL.t0/k4PHd�12dt 0 6 c (8)
for c small enough. It is easy to see that kuLkL4.RCI PH
d�12 .Rd //
6 Cku0k PHd2 �1.Rd /
so one recovers again the fact that small data gives a global solution, butif ku0k PH d
2 �1.Rd /is not small, then requirement (8) holds if T is small enough.
This concludes the proof of Theorem 1. �
Remark 2. The size of T su that (8) holds is actually a complicated function of theinitial data: typically one can compute T as follows. One splits the initial data u0into a small, high-frequency part u]0 and a smooth, low-frequency part u[0 by setting
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 9
u0 D u]0 C u[0; Ou]0.�/ WD 1j�j�N Ou0.�/:
If N is large enough, then ku]0k PH d2 �1.Rd /
is smaller than 1=8C . Then one writes
ˆ T
0
ku[L.t0/k4PHd�12 .Rd /
dt 0 6 T ku[Lk4L1.RCI PH
d�12 .Rd //
6 TN2ku0k4
PHd2 �1.Rd /
which may be made smaller than 1=8C if T is chosen small enough (with respectto N and ku0k PH d
2 �1.Rd /).
Remark 3. In two space dimensions, the conservation of energy (1) implies that forany initial data, the solution is global and unique since theL2..0; T /I PH1.R2// normremains under control for all T > 0. More on this is provided in Sect. 3.
2.2 Wellposedness in PBdp�1
p;1
In this section a different method to prove a result similar to Theorem 1 is presented,using a fixed-point approach. The idea is to write (NS) under the following form(where P WD Id���1rdiv denotes the projector onto divergence-free vector fields)
u.t/ D uL.t/C B�u; u
�.t/
with uL.t/ WD et�u0 and B.a; b/.t/ WD �
ˆ t
0
e.t�t0/�
Pdiv.a˝ b/.t 0/ dt 0
and to look for a scale-invariant Banach spaceXT of distributions defined on Œ0; T ��Rd such that uL belongs to X1 and
kB.a; a/kXT 6 Ckak2XT (9)
for some constant C (independent of T since XT is scale invariant). If one can findsuch a space, then by the Banach fixed-point theorem, as long as
kuLkXT 61
4C(10)
there is a unique solution u in XT of size less than 1=2C .This procedure can be carried out in the context of the Sobolev space PH
d2 �1.Rd /,
choosing, for instance,XT WD L1..0; T /I PHd2 �1.Rd //\L2..0; T /I PH
d2 .Rd //. One
can implement this idea with the following space: fix p > d and define
10 I. Gallagher
XpT WD
nu 2 C..0; T /ILp.Rd // = sup
t2.0;T /
t12 .1�
dp /ku.t/kLp.Rd / <1
o: (11)
It is an exercise to check that XpT is scale invariant, in the sense of (3). The fact
that (9) holds is a consequence of Young’s inequalities. Indeed it is not difficult tosee that
ket�f kLp.Rd / 6C
td2 .
1q�
1p /kf kLq.Rd / and sup
j˛jD1
k@˛et�f kLp.Rd /
6 C
t12C
d2 .
1q�
1p /kf kLq.Rd /:
Then for any couple .r; p/ of real numbers such that 1=r 6 2=p, one gets
kB.u; u/.t/kLr .Rd / 6 Cˆ t
0
1
.t � t 0/12C
d2 .
2p�
1r /kP.u.t 0/˝ u.t 0//k
Lp2 .Rd /
dt 0:
Since the Leray projector is continuous over Lq�Rd�
for all 1 6 q < 1, Hölder’sinequality gives
kB.u; u/.t/kLr .Rd / 6 Cˆ t
0
1
.t � t 0/12C
d2 .
2p�
1r /ku.t 0/k2
Lp.Rd /dt 0:
It follows that
t12 .1�
dr /kB.u; u/.t/kLr .Rd / 6 Ct
12 .1�
dr /kuk2Xt
ˆ t
0
dt 0
.t � t 0/12C
d2 .
2p�
1r /t01� dp
6 Ckuk2Xt ;
since d < p < C1, and inequality (9) follows. Noticing that by (4) the space Xp1
is nothing but the Besov space PBdp�1
p;1 , the smallness condition (10) for T D1 is in
fact a smallness condition on the initial data in PBdp�1
p;1 . The following result thereforeholds.
Theorem 2 ([13, 85]). Let d < p < 1 be given. There is a constant c > 0 such
that if u0 is a divergence-free vector field in PBdp�1
p;1 .Rd / satisfying
ku0kPBdp�1
p;1 .Rd /
6 c; (12)
then there is only one solution to (NS) associated with u0, which belongs to XpT
defined in (11) for all times T > 0.
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 11
For large initial data, the idea is to find T small enough so that (10) holds.Unfortunately, linked to the fact that the space of smooth functions is not dense
in PBdp�1
p;1 .Rd /, that requires adopting a slightly smaller space for the initial data,
namely, PBdp�1
p;q .Rd / with finite q. Then the result is the following:
Theorem 3 ([13,85]). Let d < p <1 and q <1 be given. If u0 is a divergence-
free vector field in PBdp�1
p;q .Rd /, then there is a time T and a unique solution to (NS)
associated with u0, which lies in XpT defined in (11).
Note that Theorems 2 and 3 follow a series of works generalizing Theorem [31]to the Lebesgue space Ld.Rd /; the interested reader can consult [50, 57, 78, 103]for more details. For the case of domains, we refer, for instance, to [28] (with noexterior force) and [29] (with an exterior force).
2.3 The Largest Critical Space
2.3.1 Definition of the Largest Critical SpaceOne may want to try to implement the fixed-point method introduced above in thelargest possible space for the initial data. Indeed choosing a larger space meansactually shrinking the size of the initial data, so the larger the space in which onemeasures the data, the more likely one is to obtain a global unique solution. Forinstance, if an initial data presents oscillations, then this makes its PH
d2�1.Rd / norm
large, whereas its PBdp�1
p;q .Rd / norm is small (because it measures negative regularity
since p > d ). Typically if ' is a function in the Schwartz class S.Rd /, then thefunction
'".x/ WD '.x/ cos�x
"
�
satisfies k'"k PH d2 �1.Rd /
� "1�d2 � 1 whereas k'"k
PBdp�1
p;q
� "�1C d
p 1 if " 1
and d < p.One may then wonder whether there is any end to the chain of function spaces in
which one can implement the fixed-point algorithm. The answer is provided in thefollowing proposition, due to Y. Meyer [78]:
Proposition 1 ([78]). Any critical Banach space of tempered distributions embedsin the space PB�11;1.R
d /.
Proof. The proof of that result is actually quite straightforward: let X be a scale-invariant Banach space of tempered distributions and let f belong to X . Thendenoting the duality bracket between S 0.Rd / and S.Rd / by h�; �i and the Gaussian
12 I. Gallagher
G.x/ WD1
.4/d2
e�jxj2
4
one has
jhf;Gij 6 Ckf kX :
It suffices to use the invariance of X by scale and translation to find directly theresult. �
It follows that the Calderón space PB�11;1.Rd / is the largest critical space in which
a fixed-point algorithm may be implemented for (NS).
2.3.2 Illposedness in the Largest Critical SpaceIt is worth noticing that Theorem 2 just falls short of solving (NS) in PB�11;1 for
small data, since the initial data must be in PB�1C d
pp;1 .Rd / for finite p. Actually it was
proved later that (NS) is ill-posed in PB�11;1. More precisely P. Germain [45] provesthat the map which associates with the data a solution of (NS) cannot be of class C2
from PB�11;q to S 0 as soon as q > 2. This result is obtained by proving that the firstiterate of the Picard fixed-point scheme
F�1�bP.�/e�j�j
2
ˆ 1
0
ˆet0.j�j2�jj2�j��j2/.Ou./ � .� � /Ou.� � // ddt 0
�
is not bounded from PB�11;q � PB�11;q to S 0. The idea to prove that result is to construct
sequences of functions which violate the boundedness inequality, which are closelyrelated to the counterexample by Montgomery-Smith [79] and the counterexampleby Stein [96] on operators not bounded in L2 although their symbols are in S01;1.
A more general result is obtained independently by J. Bourgain and N. Pavlovicin [7]: they prove that initial data in the Schwartz class S that are arbitrarily smallin PB�11;1 can produce solutions arbitrarily large in PB�11;1 after an arbitrarily shorttime. This “norm inflation” result relies again on the study of the first iterate and aspecial choice of initial data.
Other related results have been obtained since: for instance, Yoneda [105] showsthe solution map is discontinuous in PB�11;q for any q > 2 and Wang [102] obtains a
norm inflation result in PB�11;q for any q � 1.
2.3.3 The Space BMO�1
In view of the results presented in the previous paragraph if one wants to implementthe fixed-point algorithm in a scale-invariant space, one needs to choose the initialdata in a space strictly contained in PB�11;1. To this day the best result in that directionis due to H. Koch and D. Tataru [62]. They are able to solve (NS) globally for initialdata small enough in BMO�1, where
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 13
ku0kBMO�1.Rd / WD supt>0
t12 ket�u0kL1.Rd /
C supx2RdR>0
1
Rd2
�ˆ.0;R2/�B.x;R/
j.et�u0/.t; y/j2 dydt
� 12
:
If the initial data is not small, then the solution may be constructed (and is unique)on a small time interval, provided the data belongs to the closure of the Schwartzspace for the BMO�1 norm.
Remark 4. The results on the initial value problem presented in Sects. 2.1, 2.2,and 2.3 above provide a unique solution in some function space XT , speciallytailored to the space in which the initial data lies. However thanks to the regularizingeffect of the Laplacian operator, it can be proved that those solutions are actuallysmooth and in fact analytic or even Gevrey. One can refer, for instance, to [2,15,48]and the references therein for more on the subject.
2.4 The Special Role of the Nonlinear Term
2.4.1 On the Initial Value ProblemIt is interesting to notice that none of the wellposedness results presented so far in thecontext of critical spaces use the specific structure of the nonlinear term, except forthe Fujita-Kato theorem in two space dimensions (see Remark 3). Indeed one maycheck easily by reading the proofs that those results hold as soon as the nonlinearterm is of the type
X
16i;j6dQij .D/.uiuj / with Qij .D/ a Fourier multiplier of order
one. However it is possible to construct operators Qij such that some smooth initialdata produce a solution to the associate equation blowing up in finite time. Thiswas first performed in a one-dimensional model by Montgomery-Smith in [79]. Themodel was extended to two and three space dimensions, with the divergence-freecondition, in [39]. Unfortunately in the example of [39], it is impossible to maintainthe conservation of energy; on the other hand, it is worthwhile to notice that theblowing-up initial data of [39] actually leads to a global solution for (NS), thanks toa result in [18] (described in Sect. 4 further down). Recently T. Tao [99] was able toconstruct an example of an equation for which both the divergence-free conditionand the energy inequality
1
2ku.t/k2
L2.Rd /C
ˆ t
0
kru.t 0/k2L2.Rd /
dt 0 6 12ku0k
2L2.Rd /
: (13)
hold and for which blow-up in finite time can also hold for an open set of smoothinitial data.
14 I. Gallagher
Note also that D. Li and Ya. Sinai in [70] prove the blow-up in finite time ofsolutions to the Navier-Stokes equations for complex initial data.
If one’s goal is to solve the Navier-Stokes equations in critical spaces, it istherefore crucial to go beyond the fixed-point algorithm and to use in a deeperway the structure of the nonlinear term (and not only the energy conservation, dueto [99]). Sections 3 and 4 provide some examples of results where the structure ofthe nonlinear term is used.
2.4.2 Weak SolutionsExcept in two space dimensions (see the next paragraph), weak (distributional)solutions to (NS) are not constructed for initial data in critical spaces: they involveindeed the energy bound (1), which corresponds to initial data inL2.Rd /. This is thereason why such solutions are not discussed in this survey. However it is interestingto know whether both theories coincide if the initial data lies both in L2.Rd / and ina critical space such as described in this paragraph. Let us first recall the theory ofweak solutions, which goes back to J. Leray [68].
Theorem 4 ([68]). Associated with any divergence-free vector field in L2.R3/,there is a global in time solution in L1.RCIL2.R3// \ L2.RCI PH1.R3//, whichsatisfies the energy inequality (13).
The question of weak-strong uniqueness is the following: assume the initial datalies in both in L2.Rd / and in a critical space, then do all weak solutions coincidewith the unique solution obtained by fixed point (for instance), as long as the latterexists? W. von Wahl proves the result for the critical space L3.Rd / in [101] (seealso [30, 63]). Generalizations may be found in [16, 40, 47].
3 The Case of Two Space Dimensions
This section gathers a number of results concerning the global wellposedness ofthe Navier-Stokes equations in two space dimensions. Section 3.1 recalls the well-known Leray theorem in L2, and Sects. 3.2 and 3.3 consist in extensions of thatresult to more general critical spaces.
3.1 Global Wellposedness in L2.R2/
As noted in Remark 3, the blow-up criterion of the Fujita-Kato theorem (Theorem 1above) in dimension 2 joint with the energy equality (1) – which can be proved tobe valid in this setting of strong solutions – implies that all solutions associated withan L2 initial data are in fact global.
Theorem 5 ([69]). Associated with any divergence-free vector field in L2.R2/,there is a unique, global-in-time solution in L1.RCIL2.R2// \ L2.RCI PH1.R2//.
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 15
This theorem is proved by J. Leray in [69], along with a fundamental result ofglobal wellposedness of weak solutions in L2.Rd / in any space dimensions (withno uniqueness however) in [68], as recalled in Sect. 2.4.2.
3.2 Global Wellposedness in Critical Spaces
Following the Leray theorem of global, unique solutions in the energy space indimension 2, one may ask whether all local-in-time results in critical spaces asdescribed in Sect. 2 could be extended to global-in-time results in two spacedimensions. This is far from being obvious a priori, since the key feature enablingthis to be true in L2.R2/ is the energy estimate, which fails in other scale-invariant
spaces such as PB�1C 2
pp;q .R2/ for p > 2, or BMO�1.R2/. In [40] the following
theorem is nevertheless proved.
Theorem 6 ([40]). Let r and q be two real numbers such that 2 6 r <1 and 2 <
q < 1. Let u0 be a divergence-free vector field in PB2r �1r;q .R2/. Then there exists a
unique global solution to (NS) such that u 2 C.Œ0;1/; PB2r �1r;q .R2//. Moreover, if
2
rC2
q� 1, then there exists a constant Cr;q such that
8t � 0; ku.t/kPB2r �1r;q .R2/
6 Cr;qku0k1C rC1
2
PB2r �1r;q .R2/
: (14)
Theorem 6 is extended to initial data in the closure of the space of Schwartzfunctions for the BMO�1 norm in [46].
Sketch of proof of Theorem 6. The proof relies on a method introduced by C.Calderón in [11] to prove the global existence of weak solutions in Lp . The ideais to split the initial data into
u0 D v0Cw0; with v0 2 L2.R2/ and w0 small in PB
2Qr�1
Qr;Qq.R2/; Qr > r; Qq > q:
It is known that there is a unique global solution w to (NS) associated with w0 thanksto Theorem 2 so it remains to solve the equation for v WD u � w:
@tv ��v C Pr � .v ˝ w/C Pr � .w˝ v/C Pr � .v ˝ v/ D 0: (15)
The idea is to first use a fixed-point argument to solve the equation locally in timein L2.R2/ and then to check that the energy of v satisfies a global a priori boundin L2.R2/; that will show that the solution may be extended for all times in L2.R2/.Let us detail that part of the proof. Formally, one can multiply (15) by v and integrateover x and t (with t ranging from t0 > 0 to T ) to get, using the fact that v isdivergence free (therefore here the structure of the nonlinear term is used)
16 I. Gallagher
kv.T /k2L2C 2
ˆ T
t0
krv.t/k2L2dt C
ˆ T
t0
ˆR2
.v � r/v �wdxds 6 kv.t0/k2L2 : (16)
It can be checked that the small solution w satisfies
supt
ptkwkL1 < "0;
which allows to write by Hölder’s inequality,
ˇˇˇˇ
ˆ T
t0
ˆR2
.v � r/vw dxdt
ˇˇˇˇ 6 C"0
ˆ T
t0
krv.t/k2L2dt C
ˆ T
t0
kv.t/k2L2
tdt
!
:
This yields the expected bound after applying the Gronwall Lemma. Note that theformal computation (16) is justified since one applies the energy inequality froma time t > t0 > 0, all terms are smooth and there is no difficulty in definingthe various quantities. The local solution v may thus be extended globally, and theglobal existence result follows. The a priori bound (14) is obtained using a nonlinearreal interpolation method which is not detailed here. �
3.3 Measure-Valued Vorticity
In dimension 2, the vorticity plays a crucial role. Defined in general by ! WD curl u,it satisfies in dimension 2 the transport-diffusion equation
@t! C u � r! ��! D 0: (17)
In dimension 3 there is an additional stretching term �! � ru on the left-hand side,which destroys the conservations that can be seen on (17), namely, the fact thatany Lp.R2/ norm of ! is formally bounded by that of the initial data. The criticalsetting for the initial data in terms of the vorticity is L1. Global existence for largedata was proved for measure-valued vorticity !0 (see the works of G.-H. Cottet [26]and Y. Giga, T. Miyakawa and H. Osada [51]): define
k�kM WD sup
ˆ� d�
ˇˇˇ� 2 C0.R
2/; k�kL1 6 1
:
The initial velocity field u0 given by the Biot-Savart Law
u0.x/ D1
2
ˆ ˆ.x � y/?
jx � yj2!0.y/ dy ; x 2 R
2 ; t > 0 ;
is known to be in the Lorentz space L2;1, which is strictly larger than L2, but notall u0 2 L2;1 can be paired with a measure-valued vorticity.
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 17
Uniqueness (and continuous dependence on the data) was proved in [51, 57]under a smallness assumption on the atomic part of the measure. The case of alarge Dirac mass was solved in [41] using the following strategy: one can rewritethe vorticity equation (17) in self-similar variables and show that the Oseen vortices˛G (˛ 2 R), with
G.y/ WD1
4e�jyj2
4 ; (18)
are the only equilibria of the rescaled equation. By compactness arguments, one thendeduces that all solutions converge in L1.R2/ to Oseen vortices as t ! C1; as aby-product one finds that (18) is the unique solution of (17) such that k!.�; t /kL1 6K for all t > 0 and !.�; t / * ˛ı0 as t ! 0, where ı0 is the Dirac mass at the origin.
Uniqueness in the case of general measures, with no smallness assumption, isproved in [33] (see [34] for another proof of the special case of a Dirac mass). Thefinal result is the following:
Theorem 7 ([33, 51, 57]). For any � 2 M.R2/, the vorticity equation (17) has aunique global solution
! 2 C0..0;1/; L1.R2/ \ L1.R2//
such that k!.�; t /kL1 6 k�kM for all t > 0 and !.�; t / * � as t ! 0. Thissolution depends continuously on the initial measure � in the norm topology ofM.R2/, uniformly in time on compact intervals. Moreover,
ˆ!.t; x/dx D ˛ WD �.R2/ ; for all t > 0 ;
and
limt!1
t1� 1
p
���!.t; x/ �
˛
tG� xpt
����Lp.R2/
D 0 ; for all p 2 Œ1;1� :
Let us give a very rough idea of the proof of the uniqueness result. Since previousworks on the subject assume that the initial vorticity � either has a small atomicpart [51,57] or consists of a single Dirac mass [41], the idea is to decompose � intoa finite sum of mutually singular Dirac masses and a remainder whose atomic partis arbitrarily small (depending on the number of terms in the previous sum). Theidea is then to use the methods of [41] to deal with the large Dirac masses and theargument of [51, 57] to treat the remainder. The difficulty is of course that Eq. (17)is nonlinear so that the interactions between the various terms have to be controlled,but in the end, one can show that the solution ! also admits a natural decompositioninto a sum of Oseen vortices and a remainder, and this enables one to conclude theproof.
18 I. Gallagher
4 Examples of Large Data in Critical Spaces Giving Riseto a Global Solution
The results presented in Sect. 2 suggest that in order to obtain global solutionsin critical spaces to (NS) in dimension 3, the initial data must be small in somecritical space. Actually it is not necessarily so, and it is possible to construct initialdata as large as wanted (the size of the initial data is measured in PB�11;1, which ismeaningful due to Proposition 1) which nevertheless gives rise to a global solution.All results in that framework use in some way the structure of the nonlinear term:this is necessary as explained above in Sect. 2.4.
4.1 Geometrical Constraints
One can assume some additional geometrical constraints on the flow, which implythe conservation of quantities beyond scaling (namely, spherical, helicoidal, oraxisymmetric conditions). One can refer, for instance, to [66, 75, 86], or [100] forsuch studies.
One other possibility is to use the fact that the 2D equations are wellposed (asrecalled in Sect. 3) to deduce results on the 3D case in some special situations forthe initial data. An important example where a unique global in time solution existsfor large initial data is the case where the domain is thin in the vertical direction (inthree space dimensions): that is proved by G. Raugel and G. Sell in [88] (see alsothe paper [54] by D. Iftimie, G. Raugel, and G. Sell). The authors obtain the globalexistence of a strong solution for initial data which are allowed to have a large two-dimensional part (the vertical mean of the initial data) and a small three-dimensionalpart. Another example of large initial data generating a global solution is obtainedby A. Mahalov and B. Nicolaenko in [74]: in that case, the initial data is chosen soas to transform the equation into a rotating fluid equation (for which it is known thatglobal solutions exist for a sufficiently strong rotation).
4.2 Anisotropic Oscillations
In [17] an example of periodic initial data is presented, which is strongly oscillatingand large in PB�11;1 but yet generates a global solution. Such an initial data is givenby the following formula, writing uh;N for .u1;N ; u2;N /
where kuh;N kL2.T2/ 6 C.lnN/14 , and its PB�11;1 norm is typically of the same size.
One assumes that .x1; x2; x3/ lies in the three-dimensional torus T3, where T
d isthe d -dimensional torus. The main idea is to prove a global existence result under a(nonlinear) smallness assumption on the first iterate of the Picard scheme instead of
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 19
a smallness condition directly on et�u0. That result will not be stated here but ratherits counterpart in the whole space, which is proved in [18].
Theorem 8 ([18]). Let p 2�3;1Œ be given. There is a constant C0 such that thefollowing result holds. Let u0 2 PH
12 .R3/ be a divergence-free vector field. Suppose
that���P�et�u0 � re
t�u0����E6 C�10 exp
��C0ku0k
2PB�11;2
�; (19)
where
kf kE WD kf kL1.RCI PB�11;1/CX
j2Z
2�j���k�jf .t/kL1
���L2.RCItdt/
:
Then there is a unique, global solution to (NS) associated with u0, satisfying
u 2 Cb�RCI PH
12
�\ L2
�RCI PH
32
�:
Condition (19) is a nonlinear smallness condition on the initial data. The proof ofTheorem 8 consists in writing the solution u (which exists for a short time at least),as
u D et�u0 CR
and in proving a global wellposedness result for the perturbed Navier-Stokesequation satisfied by R thanks to the smallness condition (19). We shall not enterinto any detail here.
Now let us give an example of large initial data satisfying the assumptions ofTheorem 8.
Theorem 9 ([18]). Let � 2 S.R3/ be a given function, and consider two realnumbers " and ˛ in �0; 1Œ. Define
'".x/ WD.� log "/
13
"1�˛cos
�x3
"
���x1;
x2
"˛; x3
�:
Then for any p > 3, there is a constant C > 0 such that for " small enough, thesmooth, divergence-free vector field
u0;".x/ WD .@2'".x/;�@1'".x/; 0/
satisfies
C�1.� log "/13 6 ku0;"k PB�1
1;16 C.� log "/
13 ;
20 I. Gallagher
and
ket�u0;" � ret�u0;"kE 6 C"
˛3 : (20)
In particular for " small enough, the vector field u";0 generates a unique, globalsolution to .NS/.
The proof of this result, which will very briefly be sketched, relies heavily on thespecial structure of the nonlinear term. One starts by noticing that estimating et�u0;"�ret�u0;" boils down to estimating
et�u10;"@1et�u10;" C e
t�u20;"@2et�u10;"
and
et�u10;"@1et�u20;" C e
t�u20;"@2et�u20;":
But
et�u10;"@1et�u10;" C e
t�u20;"@2et�u10;" D
1
"2.� log "/
25 et�f"e
t�g" and
et�u10;"@1et�u20;" C e
t�u20;"@2et�u20;" D
1
"2�˛.� log "/
25 et� Qf"e
t� Qg";
where f , Qf , g, and Qg are smooth functions and
f".x/ WD eix3" f
�x1;
x2
"˛; x3
�:
The conclusion comes from the fact that for any functions f and g in PB�11;2 \PH�1.R3/, one has the interpolation-type inequality
kP.et�fet�g/kE 6 C�kf k PB�1
1;2kgk PB�1
1;2
� 23�kf k PH�1kgk PH�1
� 13
along with the estimate, for � 2�0; 3
�1 � 1
p
��and p � 1,
kf"k PB��p;16 C"�C
˛p :
4.3 Slow Variations in One Direction
In [19], the global wellposedness of the two-dimensional equation is used to provea global existence result for large data which is slowly varying in one direction.
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 21
Theorem 10 ([19]). Let vh0 D .v10; v
20/ be a horizontal, smooth divergence-free
vector field on R3 and let w0 be a smooth divergence-free vector field on R
3. Thenthere exists a positive "0 such that, if 0 < " 6 "0, the initial data
u"0.x/ WD .vh0 C "w
h0;w
30/.x1; x2; "x3/ (21)
generates a unique, global solution u" of (NS).
Remark 5. One can check that the initial data in (21) may be chosen as large aswanted: indeed if .f; g/ are in S.R2/� S.R/ and if h".x/ WD f .x1; x2/g."x3/ thenif " is small enough there holds
kh"k PB�11;1.R
3/ �1
4kf k PB�1
1;1.R2/kgkL1.R/:
Sketch of proof of Theorem 10. The idea of the proof of Theorem 10 is the follow-ing: one defines the solution vh D .v1; v2/ to the two-dimensional Navier-Stokesequation associated with the data vh
0 , which is known to be global and unique thanksto Theorem 5. Next one solves the linear transport equation
@tw" C vh � rw" ��hw" � "
2@23w" D �.rhp; "2@3p/
with initial data w0, where �h WD @21 C @22 and rh WD .@1; @2/. Finally one defines
the approximate solution
uapp" WD .v
h C "wh";w
3"/.t; x1; x2; "x3/
and the proof of the result is complete if it can be proved that
R" WD u" � uapp"
exists globally in time, where u" is the solution (defined a priori on a finite timeinterval) associated with u0;". This is possible due to the fact that R" solves
8<
:
@tR" CR" � rR" ��R" C uapp" � rR" CR" � ruapp
" D F" � rq"divR" D 0R"jtD0 D 0
where the force F" depends on vh and w" and can be proved to be small: this pointis linked to the fact that all errors made between the 2D and the 3D equation involvepartial derivatives in the x3 direction, which all produce a factor ". This impliesthat R" exists globally and concludes the proof. �
Using the language of the weak compressible limit or fast rotating fluids, thecase studied in Theorem 10 may be qualified as a “well-prepared” case. Indeed
22 I. Gallagher
the initial data converges uniformly for x3 in any compact subset of R to a two-dimensional vector field which generates global smooth solutions. The result of [19]is generalized in [20] to the “ill-prepared” initial data
u0;".x/ WD�vh0.x1; x2; "x3/;
1
"v30.x1; x2; "x3/
�
as soon as (writing eajD3j for the Fourier multiplier eaj�3j in Fourier space)
keajD3jv0k PH4 6 ;
for some a > 0 and for small enough: then for " small enough, u0;" generates aglobal smooth solution u" of (NS) on T
2 � R. The result [20] is further generalizedin [82]. The result of [19] is also extended, in [21], to the case when one adds to thedata (21) any vector field giving rise to a global solution.
Theorem 11. Let u0, vh0 D .v
10; v
20/, and w0 be three smooth divergence-free vector
fields defined on R3, satisfying
• u0 belongs to PH12 .R3/ and generates a unique global solution to the Navier-
Stokes equations;• vh
0.x1; x2; 0/ D w30.x1; x2; 0/ D 0 for all .x1; x2/ 2 R2:
Then there exists a positive number "0 depending on u0 and on norms of vh0 and w0
such that for any " 2 .0; "0�, there is a unique, global solution to the Navier-Stokesequations with initial data
u0;".x/ WD u0.x/C .vh0 C "w
h0;w
30/.x1; x2; "x3/:
Remark 6. Let u0 be any element of the (open) set G of PH12 divergence-free vector
fields generating a global smooth solution to (NS), and let N be an arbitrarilylarge number. Then for any smooth divergence-free vector field f h (over R2) andscalar function g (over R) satisfying kf hk PB�1
1;1.R2/kgkL1.R/ � 4N , and such that
g.0/ D 0, Theorem 11 implies that there is "N depending on u0 and on norms of f h
and g such that u0 C .f h ˝ g; 0/.x1; x2; "N x3/ belongs to G, where f h ˝ g.x/ D
.f 1.x1; x2/g.x3/; f2.x1; x2/g.x3//. Since "N only depends on norms of f h and g,
that implies that for any � 2 Œ�1; 1�, the initial data u0C�.f h˝g; 0/.x1; x2; "N x3/
also belongs to G. One concludes that through u0 passes an uncountable number ofsegments of length N included in G.
The proof of Theorem 11 begins, like the proof of Theorem 10, by defining anapproximate solution associated with the data u0;". In this case it is of the form
uapp" .t; x/ WD u.t; x/C .vh C "wh
";w3"/.t; x1; x2; "x3/
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 23
where u is the global solution associated with u0, and vh and w" are defined exactlyas in the proof of Theorem 11. The fact that vh
0.x1; x2; 0/ D w30.x1; x2; 0/ D 0 is ofcrucial importance, as it can be proved that under that assumption, vh.t; x1; x2; 0/
and w".t; x1; x2; 0/ are small. This means that the support in x3 of vh.t; x1; x2; "x3/
and w".t; x1; x2; "x3/ lies essentially at x3 � O.1="/, and since u does notdepend on ", it follows that the supports in x3 of u and vh.t; x1; x2; "x3/ C
".wh";w
3"/.t; x1; x2; "x3/ are essentially disjoint. So this sum can be proved to be
an approximate solution to (NS).
5 Large-Time Behavior of Global Solutions
In this section the large-time behavior of global solutions is studied in two and threespace dimensions: results on the velocity are presented first and then on the vorticity.
5.1 Behavior of the Velocity
In this section the large-time behavior of global solutions (whatever their initial size)is analyzed, in scale-invariant spaces. The link between large-time behavior in spaceand time is developed in another chapter (see [8], for instance).
In two space dimensions, it is known since the work of Wiegner [104] that finiteenergy solutions decay to zero in L2 for large times (see also [77] and [92] amongothers). Here strong solutions (as constructed in Sect. 2) are studied, in three spacedimensions. The main result is the following (where VMO�1.R3/ is the closure ofthe space of Schwartz class functions for the BMO�1.R3/ norm).
Theorem 12 ([1]). Let u0 2 VMO�1.R3/ give rise to a unique, global solution ubelonging to C.Œ0;C1Œ;BMO�1/ (constructed, for instance, by a fixed-pointargument). Then
limt!C1
ku.t/kBMO�1 D 0:
This result follows the work [36] where the Besov setting is considered. Here theproof of the easier case when the initial data lies in PH
12 is sketched.
Sketch of proof in the PH12 setting. This proof may be found in [35], and it is based
on the following remark: assume that u0 belongs to L2 and not only to PH12 .R3/.
Then it can be proved, by a weak-strong uniqueness property as described inSect. 2.4.2, that the solution remains in L2.R3/ for all times and satisfies the energyinequality (13). By interpolation between L1.RCIL2.R3// and L2.RCI PH1.R3//,it is known that u belongs to L4.RCI PH
12 .R3//. For all "0 > 0, one can therefore
find a time t0 such that ku.t0/k PH 12 .R3/6 "0, and Theorem 1 then implies the result.
24 I. Gallagher
Now if u0 belongs to PH12 .R3/ alone, one can use the method introduced in [11]
(in the context of building weak, infinite energy solutions) and recalled in Sect. 3.2,which consists in decomposing
u0 D v0 C w0
with w0 small in PH12 .R3/ and v0 2 L2 \ PH
12 .R3/. This may be done, for instance,
by cutting off the low frequencies of u0, since the resulting vector field v0 will bein L2 \ PH
12 .R3/, while the part with the small frequencies w0 can be made as
small as needed provided the cutoff is small enough. Then one starts by solvingthe Navier-Stokes system with data w0. This produces a global solution denoted w,which satisfies for all t � 0
kw.t/kPH12 .R3/6 kw0k PH 1
2 .R3/(22)
thanks to Theorem 1. Then v WD u � w satisfies the equation
@tv C v � rv C v � rwC w � rv ��v D �rp; vjtD0 D v0:
One writes the formal energy estimate
kv.t/k2L2.R3/
C2
ˆ t
0
krvk2L2.R3/
dt 0 6 kv0k2L2.R3/C2ˇˇˇˇ
ˆ t
0
ˆR3
.v � rw/ � v.t 0/ dxdt 0ˇˇˇˇ ;
(23)and a product estimate along with (22) implies that
ˇˇˇˇ
ˆ t
0
ˆR3
.v � rw/ � v.t 0/ dxdt 0ˇˇˇˇ 6 Ckw0k PH 1
2 .R3/
ˆ t
0
krv.t 0/k2L2.R3/
dt 0:
If Ckw0k PH 12 .R3/6 1
2one concludes from (23) that the energy of v remains bounded
for all times. One can deduce that v belongs to L4.RCI PH12 .R3// so as above there
is a time t0 > 0 for which kv.t0/k PH 12 .R3/
6 kw0k PH 12 .R3/
. In particular one infers
that ku.t0/k PH 12 .R3/
6 2kw0k PH12 .R3/
which concludes the proof since kw0k PH 12 .R3/
can be chosen arbitrarily small. �
5.2 Behavior of the Vorticity
In this section a result on the large-time behavior of the vorticity in two spacedimensions is presented, in scale-invariant spaces.
As recalled in Sect. 3.3, the Oseen vortex (18) plays an important role in thedynamics of the vorticity equation (17) in two space dimensions: it is the solutionwith a single Dirac mass as initial data. It also describes the long-time asymptotics of
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 25
solutions to (17), as stated in the following result due to [41]. Recall the notation (18)for the Gaussian function.
Theorem 13 ([41]). For all initial data !0 2 M.R2/, the solution ! 2
C.R�IL1.R2// satisfies
limt!1
����! �
˛
tG� xpt
�����L1.R2/
;with ˛ D
ˆd!0:
Sketch of proof. Let us briefly sketch the proof of this result. The idea is to use self-similar variables
y Dxpt; � D log
t
T
and to write
!.t; x/ D1
tw
�xpt; log
t
T
�
and u.t; x/ D1ptv
�xpt; log
t
T
�
:
Then w satisfies
@�wC v � ryw D Lw; with L WD �y C1
2y � ry C 1:
Oseen vortices are equilibria of the rescaled system. The interest of this formulationis that it can be shown that the operator L has better spectral properties than theLaplacian, when acting on weighted function spaces. In particular the trajectoriesin L1 are compact, in the sense that the set w.�/��0 is relatively compact in L1.R2/if w0 2 L1.R2/. The next important step of the proof consists in proving that the!-limit set of w0 is reduced to one point ˛G: this Liouville-type theorem is provedby exhibiting a new Lyapunov functional, other than the L1 norm. This functionalis based on the relative entropy of w
H.w/ WDˆR2
w.y/ log
�w.y/
G.y/
�
dy;
which is strictly decreasing on the trajectories of the equation, except on thestationary solutions. By the La Salle principle, the !-limit set of w0 can only beone point, namely, the stationary solution ˛G. The convergence result follows byrescaling back to the original variables. �
26 I. Gallagher
6 Behavior at Blow-Up Time
In this section the space variable is taken in R3 and the following question is
addressed: suppose that there do exist solutions blowing up in finite time; whatis their behavior at blow-up time? In this section it is assumed that blowing-upsolutions do exist.
6.1 Blow-Up of Scale-Invariant Norms
A natural question is to ask whether at blow-up time T �, scale-invariant norms ofthe solution blow up. The fixed-point methods described in Sect. 2 provide someinformation as to what types of norms should blow up at T �. For instance, it canbe proved (see, for instance, [68]) that solutions can be continued as long as theylie in Lp..0; T /ILq.R3// with 2=p C 3=q D 1 and p < 1. This is often knownas the “Ladyzhenskaya-Prodi-Serrin criterion.” J. Leray also proves in [68] that forany q > 3, there is a constant C such that
ku.t/kLq.R3/ �C
.T � � t /12 .1�
3q /�
We refer also, for instance, to [49] for a discussion in the case of domains.The much harder question (as it is not a consequence of the fixed-point method)
concerns the limiting case p D 1 can be stated as follows: if the initial databelongs to some critical space X and if the solution blows up at time T �, thendoes sup
0<t<T �ku.t/kX D 1‹ This corresponds, for instance (when X D L3), to the
limiting case p D1 in the Ladyzhenskaya-Prodi-Serrin criterion mentioned aboveand is left open in [68] and for many years following that pioneering paper. In [68],J. Leray actually suggests a blow-up profile of the type
u.t; x/ D1
pT � � t
U
�x
pT � � t
�
;
which is self-similar and for which the L1..0; T /IL3.R3// norm remains constant.Many years later, by studying the equation satisfied by U , J. Necas, M. Ruzicka, andV. Šverák were able to rule out such a profile (see [80]). This was later generalizedto different types of “pseudo-self-similar” solutions, as in [76] among others.
The case X D L3 was finally settled in full generality by L. Escauriaza, G. A.Seregin, and V. Šverák in [27]. They were able to prove that if the solution blowsup in finite time, then a subsequence of times tn converging to blow up time exists,such that the L3 norm of the solution measured at that sequence of times convergesto infinity. This was later generalized to any sequence of times in [93]. The ideaof the proof in [27] is to rescale the solution at blow-up time and to analyze thecorresponding blow-up profile; this profile is in the end shown not to exist thanks to
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 27
a backward uniqueness/unique continuation argument joint with a Caffarelli-Kohn-Nirenberg criterion. This will be discussed in more detail below. The result of [27]was re-obtained by different techniques in [58] for the smaller space X D PH
12 , then
in [37] for X D L3 using the profile decomposition technique of [58], and finally
the general PB3p�1
p;q case is dealt with in [38] by improving the methods of [37]. Letus remark that a nice intermediate result can be found in [22], for the same spacesin a certain range of values of q < 3 and with an additional regularity assumptionon the data. Finally [84] extends [27] to the Lorentz space L3;q , with 3 < q < C1,in the context of Leray-Hopf weak solutions. Let us state the result [38] and give asketch of its proof.
Theorem 14 ([38]). Let p; q 2 .3;1/ be given, and consider a divergence-free
vector field u0 in PB3p�1
p;q .R3/. Let u be the unique strong Navier-Stokes solution of
(NS) as constructed in Sect. 2.2 with maximal time of existence T �. If T � < 1,then
lim supt!T �
ku.t/kPB3p�1
p;q
D1:
Sketch of proof. Let us describe the main ideas of the proof of this result, whichis inspired from [58] and [37] and follows the “road map” initiated in [59]. In thefollowing NS.u0/ denotes the solution associated with u0. The first step consists indefining
Ac WD supnA > 0= sup
t2Œ0;T �/
kNS.u0/.t/kPB3p�1
p;q
6 A H) T � D1 8u0 2 PB3p�1
p;q
o:
Note that Ac is well-defined by small data results. Moreover, if Ac is finite, then
Ac D infn
supt2Œ0;T �/
kNS.u0/.t/kPB3p�1
p;q
= u0 2 PB3p�1
p;q with T � <1o:
In the case when Ac < 1, introduce the (possibly empty) set of initial datagenerating “critical elements” as follows:
Dc WDnu0 2 PB
3p�1
p;q = T � <1 and supt2Œ0;T �/
kNS.u0/.t/kPB3p�1
p;q
D Ac <1o:
Then it suffices to prove the three following results:
1. (Existence of a critical element) If Ac <1, then the set Dc is nonempty.2. (Compactness at blow-up time of critical elements) IfAc <1, then any u0 in Dc
satisfies
28 I. Gallagher
u.t/! 0 in S 0; as t % T �:
3. (Rigidity of critical elements) If u0 belongs to PB3p�1
p;q with
supt2Œ0;T �/
ku.t/kPB3p�1
p;q
<1
and if u.t/! 0 in S 0 as t % T �, then T � D1.
The proofs of the first two statements rely on “profile decomposition” results. Thethird one uses, as the proof of [27] mentioned above, backward uniqueness, uniquecontinuation, and “"-regularity” results detailed below.
6.1.1 Profile Decompositions and the Navier-Stokes EquationsProfile decompositions originate in the study of the defect of compactness inSobolev embeddings, which goes back to [43,72,73,98]; actually earlier decompo-sitions of bounded sequences into a sum of “profiles” (or “bubbles”) can be foundalso in [9, 97]. Our source of inspiration here is the work [44] in which the defectof compactness of the critical Sobolev embedding PHs Lp is analyzed. This wasgeneralized to other Sobolev spaces in [55], to Besov spaces in [61], and finally togeneral critical embeddings in [4] (see also [90] for an abstract, functional analyticpresentation of the concept in various settings).
In the pioneering works [5] (for the critical 3D wave equation) and [71] (forthe critical 2D Schrödinger equation), this type of decomposition was introducedin the study of nonlinear partial differential equations. The ideas of [5] wererevisited in [60] and [32] in the context of the Schrödinger equations and Navier-Stokes equations, respectively, with an aim at describing the structure of boundedsequences of solutions to those equations. These profile decomposition techniqueshave since then been succesfully used in order to study the possible blow-up ofsolutions to nonlinear partial differential equations, in various contexts; one canrefer, for instance, to [37, 53, 56, 58, 59, 87, 91].
Now let us present this theory in the context of the Navier-Stokes equations(see [37]). Before doing so introduce some notation, where N D f0; 1; : : : g. Thefollowing definition means that sequences of cores and scales of concentration aresaid orthogonal if either the scales are asymptotically moving away from eachother (�1;n=�2;n ! 0 or 1), or the cores are very far apart at one of the scales(jx1;n � x2;nj=�1;n/ ! 1. If two initial data are supported on orthogonal scales orcores, then essentially the associate solutions do not interact, and the sum of the twosolutions is a solution associated with the sum of the initial data. That is in essencethe proof of the following theorem.
Definition 3. Two sequences .�j;n; xj;n/n2N 2 ..0;1/ � R3/N for j 2 f1; 2g are
orthogonal, which is written .�1;n; x1;n/n2N ? .�2;n; x2;n/n2N; if
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 29
limn!C1
�1;n
�2;nC�2;n
�1;nCjx1;n � x2;nj
�1;nD C1: (24)
Similarly a set of sequences .�j;n; xj;n/n2N, for j 2 N, j � 1, is (pairwise)orthogonal if for all j ¤ j 0, .�j;n; xj;n/n2N ? .�j 0;n; xj 0;n/n2N:
In the following the shorthand notation
ƒj;n WD ƒ�j;n;xj;n
will be used for any set of sequences .�j;n; xj;n/n2N (for j � 1) and whereƒ�j;n;xj;n
was defined in (2).
Theorem 15 (NS Evolution of Profile Decompositions, [38]). Fix p; q with 3 <p < q 6 1. Let .u0;n/n�1 be a bounded sequence of divergence-free vector fields
in PB3p�1
p;p .R3/, and let �1 be any weak limit point of .u0;n/. Then up to extracting
a subsequence the following holds. There is a sequence of divergence-free vector
fields .�j /j�2 in PB3p�1
p;p .R3/ and a set of sequences .�j;n; xj;n/n�1 for j 2 N with
.�1;n; x1;n/ � .1; 0/ which are orthogonal in the sense of Definition 3, such thatdenoting Uj WD NS.�j / on Œ0; T �j / and un WD NS.u0;n/, the following propertieshold:
• there is a finite (possibly empty) subset I of N such that T �j < 1 if and only
if j 2 I . For all j 2 I fix any Tj < T �j and define �n WD minj2I
�2j;nTj if I is
nonempty and �n WD 1 otherwise.• setting wJn WD e
t� Jn , there exists some J0 2 N and N.J / 2 N for each J > J0
such that rJn given by
un D U1 CJX
jD2
ƒj;nUj C wJn C rJn (25)
is well-defined for J > J0, n > N.J /, t < �n and x 2 R3, and moreover wJn
and rJn are small remainders in the sense that
limJ!1
�lim supn!1
kwJn kXq1
�D lim
J!1
�lim supn!1
krJn kXq�n
�D 0; (26)
where XqT is defined in (11).
To prove this theorem, one starts by writing a profile decomposition forthe sequence of initial data (using the results of [61]); then one defines rJn asin (25) and the difficulty is to check that rJn exists and is unique on the time
30 I. Gallagher
set Œ0; �n� (at least if n and J are large enough). This uses the orthogonality ofthe sequences .�j;n; xj;n/n�1. We shall not give more details here.
6.1.2 Proof of Theorem 14,(1)To prove this result, an element of Dc is explicitly constructed: this turns out tobe a profile of the initial data of a minimizing sequence of Ac . Let us consider a
sequence u0;n, bounded in the space PB3p�1
p;p , such that its life span satisfies T �.u0;n/ <1 for each n 2 N and such that An WD supt2Œ0;T �.u0;n// kNS.u0;n/.t/k
PB3p�1
p;p
satisfies
Ac 6 An and An ! Ac; n!1:
Applying Theorem 15 above to u0;n, one finds that up to a subsequence, for all t <�n, the solutions un D NS.u0;n/ satisfy
un.t/ DJX
jD1
ƒj;nUj .t/C wJn .t/C rJn .t/
with Uj D NS.�j / (as introduced in Theorem 15) and (26) holds. Defining T �j WDT �.�j / to be the life span of Uj D NS.�j /, Theorem 15 also ensures that thereis j0 2 N such that T �j0 < 1, and hence the profiles may be reordered so that withthe new ordering
T �j <1 ” 1 6 j 6 J � (27)
and for n0 D n0.J �/ sufficiently large,
8n � n0; 1 6 j 6 j 0 6 J � H) �2j;nT�j 6 �2j 0;nT
�j 0 : (28)
Notice that in particular T �1 <1, hence by definition of Ac
sups2Œ0;T �1 /
kU1.s/kPB3p�1
p;p
� Ac: (29)
Then an orthogonality argument implies that for any s 2 .0; T �1 /, setting tn WD �21;ns,
An WD supt2Œ0;T �.u0;n//
kun.t/kPB3p�1
p;p
� kun.tn/kPB3p�1
p;p
� k�ƒ1;nU1
�.tn/k
PB3p�1
p;p
C ".n; s/
D kU1.s/kPB3p�1
p;p
C ".n; s/
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 31
which with (29) and the fact that An ! Ac as n ! 1 implies that �1 belongsto Dc . That proves the expected result.
6.1.3 Proof of Theorem 14,(2)The argument is similar: one chooses u0;c 2 Dc and one picks a sequence of times snsuch that sn % T �.u0;c/. One then defines the sequence u0;n WD uc.sn/, where uc WDNS.u0;c/, which is bounded and to which one applies Theorem 15. As in the previousparagraph, one may rearrange the first J � terms of the profile decomposition sothat (27) and (28) hold and one has clearly
�21;nT�1 6 T �.u0;n/ D T �.u0;c/ � sn
for large n, and hence �1;n ! 0 as n!1. Let us denote by j0 the (unique) index,after this renumbering, satisfying �j0;n � 1 and xj0;n � 0, so that �j0 is the weaklimit of u0;n. Note that j0 ¤ 1, and one needs to show that �j0 � 0.
As in the previous argument by orthogonality, one can prove that NS.�1/ is acritical element. Now let " > 0 be fixed, and choose s 2 .0; T �.u0;c// such that,writing U1 WD NS.�1/,
Apc � kU1.s/kp
PB3p�1
p;p
< "=2:
Then defining un WD NS.u0;n/ and tn WD �21;ns, one can show that
Apc � kun.tn/kp
PB3p�1
p;p
� kU1.s/kp
PB3p�1
p;p
C��
JX
jD2
.ƒj;nUj /.tn/C wJn .tn/C rJn .tn/
��p
PB3p�1
p;p
C .n; s/
� kU1.s/kp
PB3p�1
p;p
C Ck
JX
jD2
ƒj;nUj .tn/C wJn .tn/C rJn .tn/k
p
PB3q�1
q;q
C .n; s/
since q � p, where .n; s/! 0 as n!1. Choosing J large enough so that
CkwJn .tn/C rJn .tn/k
p
PB3q�1
q;q
6 "=2;
for sufficiently large n, one finds that
��
JX
jD2
.ƒj;nUj /.tn/��q
PB3q�1
q;q
. " � .n; s/: (30)
But orthogonality arguments (see the proof of [37, Lemma 3.6]) show that
32 I. Gallagher
JX
jD2
k.ƒj;nUj /.tn/kq
PB3q�1
q;q
D k
JX
jD2
.ƒj;nUj /.tn/kq
PB3q�1
q;q
C ".J; n/ (31)
where for each J , ".J; n/! 0when n!1. In particular for j D j0, (30) and (31)imply
k�j0kPB3q�1
q;q
D kUj0.0/kPB3q�1
q;q
. "
since tn ! 0 as n!1, and hence �j0 � 0 which proves the result.
6.1.4 Proof of Theorem 14,(3)This part is based on a backward uniqueness argument similar to that in [27](see also [37, 58]). However in order to implement this argument, some positiveregularity on the solution near blow-up time needs to be recovered. This is the maindifficulty in the proof and the purpose of the next statement. We shall not prove thatstatement here as it is rather involved, but the main idea, which can also be foundin [22], is the use of “self-improving bounds”: it is well-known that the Duhamelterm is in some sense more regular than the linear heat flow, and this fact can beiterated up to positive regularity.
Proposition 2 (Positive regularity at blow-up). For u0 divergence free belonging
to PB3p�1
p;p with 3 < p < 1, define the associate solution u WD NS.u0/ on Œ0; T �Œ.
If T � < 1 and if u belongs to L1.Œ0; T ��I PB3p�1
p;p /, then there exist v;w definedon Œ0; T �Œ such that
u D v C w in XT �
as defined in (11) and such that moreover, for some " 2 .0; T �/,
v 2 L1.ŒT � � "; T ��ILp.R3// and w 2 L3..0; T �/IL3.R3//:
Let us apply Proposition 2, to u D NS.u0/: as T � <1, moreover
v 2 Lp.ŒT � � "; T �� � R3/:
Fix any R > 0 and set
Q";R.x/ WD f.y; t/ 2 R � R3 = jy � xj < R; t 2 ŒT � � "; T ��g:
As p > 3, for fixed ";R > 0
kukL3.Q";R.x// . kvkLp.Q";R.x// C kwkL3.Q";R.x// �! 0 as jxj ! 1:
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 33
This (and a similar property on the pressure) is the key to the “"-regularity” theoryfor “suitable weak solutions.” Thanks to the Caffarelli-Kohn-Nirenberg theory [10],one can conclude as in [58] that u is a suitable weak solution and is smooth at andnear the time T � outside of some large compact set K R
3. Hence if u.t/ ! 0
in S 0 as t % T �, one can conclude that actually u.x; T �/ � 0 for all x 2 Kc , andbackward uniqueness and unique continuation applied to the vorticity ! WD r � uas in [27] allow us to conclude that in fact u.t/ � 0 for some t 2 .0; T �/; one canrefer to [58] for more details, including the statements of the backward uniquenessand unique continuation results. Therefore T � D 1 by small data results, contraryto assumption, which proves the result, hence the theorem. �
6.2 Qualitative Behavior
In this section a number of other results concerning the behavior of solutions atpossible blow-up time are collected, without proofs.
6.2.1 On the Size of the Singular SetLet the singular set for (NS) be defined as follows (in three space dimensions):
S.u/ WD˚.t; x/ 2 R
C � R3 = u is not bounded in a neighborhood of .t; x/
�:
It can be proved by a rather classical (in the theory of parabolic equations) bootstrapargument that if u is bounded in a (parabolic) space-time ball of radius R, thenit is C1 in the space-time ball of radius R=2. Caffarelli, Kohn, and Nirenbergprove in [10] that the singular set of a suitable weak solution (a weak solutionsatisfying a certain generalization of the conservation of energy) to (NS) hasparabolic Hausdorff dimension at most 1, meaning that the singular set must besmaller than a curve in space-time. Note that this result is generalized in [89] torelate the Hausdorff dimension to the Ladyzhenskaya-Prodi-Serrin criterion recalledabove. Note also that the theory of partial regularity for suitable weak solutionsholds for general, bounded or unbounded domains.
6.2.2 Minimal Blowing-Up SolutionsAssuming that there do exist blowing-up solutions, it is interesting to characterizethe set of such initial data and in particular the set of “minimal” initial data givingrise to a solution blowing up in finite time. This is the object of the followingtheorem, due to [91].
Theorem 16 ([91]). Define � WD inf˚ku0k PH
12 .R3/
j T �.u0/ < C1�. Then there
exists u0 in PH12 .R3/ such that T �.u0/ < 1 and ku0k PH 1
2 .R3/D �. Moreover the
set of such minimal u0 is compact in PH12 .R3/, up to the invariances of the equation
(namely, space translations and dilations).
34 I. Gallagher
This theorem was later generalized to larger scale-invariant spaces using profiledecomposition techniques as presented above in Sect. 6.1 (see [37, 38]). Note thatin [87], a similar result is proved in spaces that are not scale invariant; in that casethe compactness holds up to translations only, with no change of scale.
6.2.3 Directions of the VorticityIn [24], the authors prove that the solution remains smooth as long as the followingholds: denote by �.t; x/ the direction of vorticity �.t; x/ D curl u.t; x/ andby '.t; x; y/ the angle between �.t; x/ and �.t; y/. Then there are constants�0 > 0
and � > 0 if the magnitude of �.t; x/ and �.t; y/ is larger than � then
j sin'.t; x; y/j �jx � yj
��
In other words, in regions of high vorticity, one must have good control on thedirection of the vorticity. One can refer to [6, 52, 106] for related results and morereferences.
6.2.4 Specializing Components of the Velocity FieldA number of articles are devoted to understanding under what minimal possibleconditions one can ensure that blow up in finite time occurs. Many results are of thefollowing type:
if T � <1 then ku3kLp.Œ0;T ��ILq.R3// D1
or
if T � <1 then k@j u3kLp.Œ0;T ��ILq.R3// D1
with various relations (which are not scale invariant) between p and q. One canrefer, among other references, to [14, 64, 81, 83, 94, 107].
In [65], the authors are able to prove the following scale-invariant criterion:
if T � <1 then k@3ukLp.Œ0;T ��ILq.R3// D1 with2
pC3
qD 2; q 2 Œ9=4; 3�:
Finally in [23], J.-Y. Chemin and P. Zhang prove that if the initial vorticity belongsto L
32 .R3/ (which is slightly stronger than u0 2 PH
12 .R3/) then
if T � <1 then ku3kLp.Œ0;T ��I PH
12C
2p .R3//
D1 with p 2 .4; 6/:
The main idea of the proof is to write an equation for the horizontal vorticity, andan equation for u3, and to use the structure of the equation as much as possibleto eliminate quadratic nonlinearities in the equation on the vorticity and to trade
Critical Function Spaces for the Wellposedness of the Navier-Stokes Initial. . . 35
off some vertical derivatives for horizontal ones, thanks to the divergence-freecondition.
7 Conclusion
This text is a (far from exhaustive) presentation of some results that can be provedon the initial value problem for the homogeneous, incompressible Navier-Stokessystem in the whole space, using critical function spaces – meaning spaces invariantthrough the scaling of the system. Particular emphasis is made on the resolution ofthe initial value problem, on large-time asymptotics, as well as on the behavior ofsolutions at the possible blow-up time. For more on the Navier-Stokes equations inthat setting, the interested reader can refer, for instance, to the books [3, 12, 25, 42,67, 78, 95].
Cross-References
�Blow Up Criteria in the Barotropic Case and in the Complete Navier-Stokes–Fourier System
�Equations for Incompressible Viscous Fluids in Geophysics�Equations for Viscoelastic Fluids� Finite Time Blow Up of Smooth Solutions to the Compressible Navier-Stokes
Equations� Fourier Analysis Methods for the Compressible Navier-Stokes Equations�Global Existence of Smooth Solutions with Large Oscillations to the Compress-
ible Navier-Stokes Equations�Large Time Behavior of the Navier-Stokes Flow�Leray’s Problem on Existence of Steady State for the Navier-Stokes Flow�Regularity Criteria for Navier-Stokes Solutions� Self-Similar Solutions to the Nonstationary Navier-Stokes Equations�Time Periodic Solutions to the Navier-Stokes Equations�Weak Solutions to 2D and 3D Compressible Navier-Stokes Equations in Critical
Cases
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