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On the regularity of the bilinear term
for solutions to the incompressible
Navier-Stokes equations
Marco Cannone
U.F.R. Math´ematiques,
Universit´e Paris 7,
2 place Jussieu,
75251 Paris Cedex 05, France,
e-mail cannone@math.jussieu.fr
and
Fabrice Planchon
Program in Applied and Computational Mathematics,
Princeton University,
Princeton NJ 08544-1000, USA
e-mail fabrice@math.princeton.edu
Abstract
We derive various estimates for strong solutions to the Navier-
3 3Stokes equations in C([0,T),L ( )) that allow us to prove some reg-
ularity results on the kinematic bilinear term.
Introduction and deﬁnitions
The Cauchy problem for the Navier-Stokes equations governing the time
evolution of the velocityu(x,t)= (u (x,t),u (x,t),u (x,t))and thepressure1 2 3
1
R3p(x,t) of an incompressible ﬂuid ﬁlling all of is described by the system
∂u = Δu−∇·(u⊗u)−∇p,
∂t
(1) ∇·u = 0, 3u(x,0) = u (x), x∈ , t≥ 0.0
The existence of local solutions to this system which are strongly continuous
p 3in time and take value in Lebesgue spaces L ( ) is a well known result for
p ≥ 3 (see [2]). In the critical case, p = 3, for which solutions of (1) are
invariant by rescaling, one can construct strong solutions in a subclass of
3 3 3C (L ) = C([0,T),L ( )) (see [5, 20, 7, 9]), but their uniqueness withint
the natural class was proved only recently ([6]). The key tool in obtaining
3−1,∞
q˙this uniqueness result was the use of the Besov spaces B , for q < 3.q
These spaces have been used previously, but mainly withq≥ 3, in obtaining
global existence results (see [12],[2],[18]). In addition, it was already noticed
invariouscontexts (see[2],[17])howthebilinearterm, which isthediﬀerence
between the solution and the solution to the linear heat equation (with same
initialdata),behavesbetterthanthesolutionitself. Weimprovetheseresults
in the present paper, and show how this gain in regularity is related to the
uniqueness problem, the main estimates involved being of the same kind.
Moreover, this allows to extend the decay estimates on the gradient of the
solution to (1) obtained by T. Kato in [9].
In order to simplify our study let us introduce the projection operator
onthedivergence freevectorﬁelds. Weremarkthat isapseudo-diﬀerential
operator of order 0 which will be continuous on all spaces subsequently used
p(primarilybecauseitiscontinuousonallLebesguespacesL ,for1<p<∞).
A common method solving (1) is to reduce the system to an integral
equation,
Z t
(2) u(x,t) =S(t)u (x)− S(t−s)∇·(u⊗u)(x,s)ds,0
0
tΔwhereS(t)=e is the heat kernel, and then to solve it via a ﬁxed point ar-
gument in a suitable Banach space (see [2],[9],[10]). Following [2], we remark
that the bilinear term in (2) can be reduced to a scalar operator,
Z
t 1 ·
√(3) B(f,g)= G ∗(fg)ds,
2(t−s) t−s0
2
R
R
R
P
P
R
Pwhere G is analytic, such that
C
(4) |G(x)| ≤
41+|x|
C
(5) |∇G(x)| ≤ .
41+|x|
This can be derived easily from the study of the operator under the integral
sum, S(t−s)∇·, since its symbol consists of terms like
ξ ξ ξ 2j k l −(t−s)|ξ|(6) − e
2|ξ|
2−(t−s)|ξ|outside the diagonal, with another term ξ e on it. For the sake ofj
2−|ξ|simplicity, we will take G as the inverse Fourier transform of|ξ|e .
As we mentioned previously, Besov spaces are a useful tool in studying
the bilinear operator B. In what follows we will use spaces of functions on
3, so henceforth the reference to the domain space will be omitted. Let us
recall the following deﬁnition. The reader will ﬁnd equivalent deﬁnitions of
Besov spaces in [1], [16],[19].
Definition 1
∞Let θ(x)∈C be such that
2−|ξ|ˆθ(ξ)=|ξ|e .
s,q˙Let p,q∈ (1,+∞),s∈ , s< 1. Then, f ∈B if and only ifp
1Z ∞ qdt−s q(7) kt θ ∗fk <+∞,pt L t0
1 ·where θ is the rescaled function θ( ), and this norm is equivalent to thet 3t t
usual dyadic norm.
We will also make use of the homogeneous version of the Sobolev-Bessel
ss p˙ 2spaces,deﬁnedsimplybyf ∈H ifandonlyifΔ f ∈L . Thereaderfamiliarp
22 −|ξ|ˆ ˆwith Besov spaces will note that by replacing θ(ξ) with θ(ξ) =|ξ| e one
obtains the usual characterization via the Gauss-Weierstrass kernel. We will
use this fact further in the paper. Among various embeddings between these
p 0,2˙spaces and the Lebesgue and Sobolev ones, we recall that L = H , forp
s,2 s s˙ ˙ ˙1<p<∞, and B =H =H , the usual homogeneous Sobolev space.2 2
3
R
R
PTheorems and proofs
Let us start with the aforementioned result on the regularity of the solution.
Theorem 1
3 3Let u(x,t) be a solution of (2) in C([0,T),L ), with initial data u ∈L and0
denote by w the function w =u−S(t)u , then0
1˙(8) w∈C([0,T),H ).3/2
3/2In other words, the gradient of w is continuous in time with value in L .
1 3˙Ofcourse,thisestimatemakessense,for,viaSobolevembedding,H ֒→L .
3/2
This regularity result can be seen in connection with an estimate derived by
T. Kato in [9] that assures that u(x,t), the solution of Theorem 1, is such
1−3/2q qthat t ∇u(x,t)∈L , for q≥ 3. Therefore the estimate (8) extends this
last estimate to the value q ≥ 3/2 for the bilinear term alone, as if u only0
3belongs to L , the linear part in general doesn’t verify (8).
Let’s postpone the proof of the theorem for a moment, and comment
furtheronthemeaningofthisresult. In[17],itwasshownthatforself-similar
3
−(1− ),∞
q3 ˙solutions (forwhich theinitialdatawasn’t inL , butinsomeB withq
1,2˙q > 3), the bilinear term was in B , and it is a simple matter to obtain
3/2
1 s s,2˙ ˙ ˙H instead. This is slightly better, as H ֒→ B for p < 2. Now, in3/2 p p
order to obtain this result, one makes use of the special structure of a self-
similar solution. For such solutions, the time regularity is intimately related
√ √
to the space regularity because of the scaling u(x,t) =1/ tU(x/ t). On
p qthe other hand, using L (L ) estimates, it was proved in [17] that for at x
0,23 3 3˙solution in C (L ) with initial data u ∈ L the function w ∈ B ֒→ L .t 0 3
0,21˙ ˙One remarks that H ֒→B . The proof of that result was a consequence33/2
of the following proposition applied with q = 6.
Proposition 1
Let 3 ≤ q ≤ 6. Then the bilinear operator B(f,g) is bicontinuous from
q2 2 6−1,
1−3/q 1−3/q q q−3q q ∞ ˙L (L )×L (L ) into L (B ).t tx x t q/2
∞ 3 ∞ 3In particular, if q = 3, B(f,g) is bicontinuous from L (L )×L (L ) intot x t x
1,∞∞ ˙L (B ). This last estimate for q = 3 was used in [6]. The proof we are3t
2
4giving here for 3 ≤ q ≤ 6 is nothing but paraphrasing the case q = 6 dealt
with in [17]. More precisely, we will prove the estimate by duality. To this
end, let 0<T <∞. By hypothesis,
Z T 1
1−3/q
(9) kfg(x,t)k dt<∞,
q/2
0
where the integral in time is replaced by a sup = sup if q = 3. Thent t∈[0,T]
∞for an arbitrary test function ϕ(x)∈C , we consider the functional0
(10) I =hB(f,g),ϕi.t
We ﬁnd
Z tD E1 ·
(11) I = G √ ∗(fg),ϕ dst 2(t−s) t−s0
√Z tD E1 ·2 ˇ= 2 fg(t−s ), G ∗ϕ ds
3s s0
ˇwhere G(x) = G(−x), and we made a change of variable. Applying H¨older
inequality both in time and space variables, we get
Z 1−3/qt 1
1−3/q
(12) |I| ≤ kfg(t−s)k dsqt L
0
!√ 3/qZ t dsq/36/q−1ˇ× ks G (·)∗ϕ ks q/(q−2) s0
1 ·ˇ ˇwhere G = G . Using Deﬁnition 1, the second integral is found to bes 3s s
6 q 6 q1− , −1,
q 3 q q−3˙ ˙lessthanthenormofϕinB ,whichisexactlythedualofB (Theq/(q−2) q/2
restriction q ≤ 6 is mainly because we are interested in positive regularity
indices). We see that with this proposition we are far from the actual result
pof Theorem 1, because the third index is greater than 2. Nevertheless, if
p−3
we think of uniqueness, we can make a parallel with a recent result, proved
(among other things, and in a more general framework) in [8]. We state it
here in a pure analytical frame instead of a stochastic one and applying the
Besov formalism once again. In order to proceed we need to introduce the
following deﬁnition involving pseudo-measures.
Definition 2
3A tempered distribution ψ is called a pseudo-measure on if
ˆ(13) sup|ψ(ξ)|<∞.
ξ
5
RThe set of pseudo-measures will be denoted by PM.
Then, we can deﬁne Besov type spaces based on pseudo-measures, by re-
pplacing the L norm by the norm on PM in Deﬁnition 1. In the context of
the Navier-Stokes equations, this was done before, for example in [12] with
Morrey-Campanato spaces. Note that with respect to scaling, thePM norm
1behaves like an L norm, so given the scaling invariance of (1) it is natural
2,∞˙to introduce the space B , which we deﬁne by
PM
2,∞ 2˙ ˆ(14) ψ∈B iﬀ sup|ξ| |ψ(ξ)|<∞,
PM
ξ
3whose norm behaves like theL norm with respect to the scaling invariance.
Now we can state
Theorem 2 ([8])
2,∞ 2,∞∞ ∞˙ ˙ThebilinearoperatorB(f,g)isbicontinuousfromL (B )×L (B )intot PM t PM
2,∞ 2,∞∞ ∞˙ ˙L (B ). Thereforethereexistsa uniqueglobalsolution to (2)inL (B )t PM t PM
2,∞˙provided the initial data is suﬃciently small inB . Moreover, this solutionPM
is self-similar if the initial data is homogeneous of degree −1.
This result is, to the best of our knowledge, the ﬁrst where the uniqueness is
obtained in the natural space where the initial data is to be taken, instead
of a given subclass. Note that the authors state their theorem within the
class ofweak solutions, but this assumption is unnecessary in orderto obtain
Theorem 2. Moreover, the functional class used in [8] obliges to use rather
subtle techniques to obtain the continuity of the bilinear operator. However,
in the particular case of the pseudo-measures, it is straightforward.
Let us simply see why the bilinear operator is actually continuous. We
ˆworkinFourierspace, withf andgˆinsteadoff andg. Astandardargument
(rotationnal invariance and homogeneity) shows that
1 1 C
(15) ∗ = .
2 2|ξ| |ξ| |ξ|
Thus Z t
2−(t−s)|ξ| ˆ(16) F(B(f,g))(t,ξ)= |ξ|e f∗gˆds,
0
6and, upon using (15)
Z t
22 2 2 2 −(t−s)|ξ|ˆ ˆ(17) sup(|ξ| |B|)≤Csup(|ξ| |f|)sup(|ξ| |gˆ|)sup |ξ| e ds.
t,ξ t,ξ t,ξ t,ξ 0
This last integral is in turn less than unity, which concludes the proof once
the ﬁxed point algorithm is recalled. The main point here is that we are
working within Besov spaces with their third index being equal to ∞, and
therefore we are allowed to make estimates with the frequencyξ being ﬁxed.
3−1,∞
q˙Thislaststatementledtheauthorsin[6]toinvestigatethespacesB ,q
for q < 3, and derive the needed estimates to obtain uniqueness. Y. Meyer
proved later that the bilinear operator [14] is actually bicontinuous on weak
3 3L , which gives another way to obtain uniqueness in L . Recall that the
3bilinear operator is not continuous on L (see [15] for details). Y. Meyer’s
pestimates are based on a direct characterization of weak L spaces, and we
present here a diﬀerent proof, which provides a better estimate and relates
both [6] and [14] results. The crucial point here is the embedding, for q < 3
(see [1])
3−1,∞
q 3,∞˙B ֒→L .q
For the sake of completeness, we add that such a uniqueness result can be
derived from earlier estimates on the pressure for the Navier-Stokes system
obtained by P.L. Lions ([13]). It should be noted that beyond its usefulness
for the uniqueness problem, the forthcoming estimate in itself is yet another
variantoftherepeatedstatementontheimprovedregularityofthenon-linear
term.
Proposition 2
Let 3/2 < q < 3. The bilinear operator B(f,g) is bicontinuous from
3
−1,∞
q∞ 3,∞ ∞ 3,∞ ∞ ˙L (L )×L (L ) into L (B ).qt x t x t
1Again, we prove the estimate by duality, and ﬁx q = 2, which gives as
2
∞the regularity index. For an arbitrary test function ϕ(x)∈C , we recall0
functional
(18) I =hB(f,g),ϕi.t
We now have
Z tD E1 ·
√(19) I = G ∗(fg),ϕ dst 2(t−s) t−s0
7√Z tD E
2˜ ˇ= 2 G ∗fg(t−s ), H (·)∗ϕ dss s
0
1 ·ˇwhere H(x) =H(−x), F = F( ), ands 3s s
2 2|ξ| |ξ| ˆ− −ˆ ˆ ˜2 2G(ξ)=|ξ|e e =H(ξ)G(ξ).
Then,
√Z
t
2˜ ˇ2(20) |I|≤ kG ∗fg(t−s )k kH ∗ϕk dst s s 2L
0
Using the generalized Young inequality,
12˜ 2(21) kG fg(t−s )k 2 ≤Cs kfgk 3 ,s L ,∞
2L
we ﬁnd Z ∞√ ds˜(22) |I|≤Csup(kgk 3,∞kfk 3,∞ skG∗ϕk .t L L 2
st 0
1Thelastintegralisnothingbutkϕk ,which,byinvokingduality,achieves− ,1˙ 2B
2
the proof.
One could see proposition 2 as a counterpart of theorem 1 for solutions
3,∞with an initial data in L . However, we already know that for such so-
∞ 3lutions, the bilinear term is in L (L ) ([17]). This and the aformentioned
result on the bilinear term for self-similar solution tend to indicate we can
do better than proposition 2. We recall that a (global) solution u with an
3,∞initial data in L veriﬁes, for all p> 3
1 3−
2 2p(23) supt ku(x,t)k ,p
t
as proved in [18]. This allows to prove
Theorem 3
∞ 3,∞Let f(x,t)∈L (L ) andg(x,t) verifying (23). Then the bilinear operatort
1
∞ ˙ 2B(f,g) belongs to L (H ).t
1˙ 2We chose H , but a simple modiﬁcation of the argument would lead to
3
−1
q˙H , for 3/2 < p < 3. The important point is how the structure of g canp
be usefully exploited here, and it is indeed such considerations which lead to
8the proof of theorem 1. But let us deal with the proposition: g veriﬁes (23).
For example, if we denote
1
8(24) kg| =supt kg(x,t)k ,4 4
t
then
kg(x,t)| ≤C.4
We have therefore
kg|42kg(x,t−s )k ≤ .4 1
2 8(t−s )
Upon returning to the estimates of the previous lemma, we obtain, start-
ing from (20)
√Z t
2˜ ˇ|I|≤ kG ∗fg(t−s )k 2kH ∗ϕk dst s L s 2
0
Now we chose diﬀerent exponents, to get
12 −˜ 4(25) kG ∗fg(t−s )k 2 ≤Cs kfgk 12 ,s L ,∞
7L
we ﬁnd √Z t ˜kG∗ϕk2
(26) |I|≤Csup(kg| kfk 3,∞) ds.t 4 L 1 1
2 8 4t 0 (t−s ) s
Applying Cauchy-Schwarz to this last integral, we bound it from above by
1 !√1Z Z 21 t2dθ
2˜kG∗ϕk ds ≤Ckϕk 12 −˙2 1/4 1/2 2H(1−θ ) θ0 0
which concludes the proof of proposition 3.
It is worth noting that from the two propositions 2 and 3 we can deduce
3the important uniqueness result in C (L ) from Furioli-Lemari´e-Terraneo.t
Theorem 4 ([6])
3Let u ∈ L . Then there exists a unique local strong solution of (2) in the0
3class C (L ).t
The existence of such a solution was proved in [20, 7, 9], where it is also
shown that this solution is global if the initial data is small enough (or small
9enough in a larger space, see [2, 3]). As far as its uniqueness is concerned,
considertwo solutionsu(x,t)andv(x,t)withthesameinitialdatau andfor0
whichu is actually the solution constructed via the ﬁxed point method ([9]).
Wedenotew =u−S(t)u andw˜ =v−S(t)u . Then,ifwetemporarilyforget0 0
thatthebilinear operatorappearingin(1)isvectorial andnon-commutative,
we may abuse the notation and write (as in the scalar case)
w−w˜ =2B(S(t)u ,w−w˜)+B(w+w˜,w−w˜).0
By applying Lemma 3 to the ﬁrst term, and Lemma 2 to the second, we
obtain
3,∞ 3,∞supkw−w˜k ≤C(kS(t)u | +supkw+w˜k )supkw−w˜k0 4,T 3L L
t t t
where kf| indicates we take the sup over (0,T) in (24). We then deduce4,T t
thatw =w˜ atleast onasmall intervalintime, asbothquantitieskS(t)u |0 4,T
and sup kw+w˜k go to zero when T goes to zero (the ﬁrst by density and3t
3the second by the strong continuity in L of the solutions). We conclude
by a simple continuation argument, as if the time on which the solution
agree is strictly less than the time on which they are deﬁned, we would get
a contradiction. And this achieves the proof of Theorem 4. It is interesting
3to remark somehow that the construction of solutions in C (L ) and theirt
subsequent uniqueness are somewhat disconnected. Besides the functional
3class used in [9], other subclasses of C (L ) can be used to construct uniquet
solutions as shown for example in [4] where the class arises naturally from
the energy inequality.
We are ﬁnally in the position to prove Theorem 1. It is useful to rewrite
B in a more suitable form, namely
√Z t
2(27) B(f,g)= 2 G (x)∗fg(x,t−s )dss
0
1 xwhere, as usual, G (x) = G( ). In addition, it is useful to work with thes 3s s √
following operator, A(f,g) = ΛB(f,g), where Λ = −Δ is the Calder´on
operator, with symbol |ξ|. Then
√Z t ds2˜(28) A(f,g)=2 G (x)∗fg(x,t−s ) ,s
s0
22 −|ξ|˜and F(G)(ξ) = |ξ| e . As we have already noted, if f and g were time
tΔindependent, then A would reduced to I −e applied to the product fg.
10
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