Global large solutions to 3-D inhomogeneous Navier-Stokes system with one slow variable

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by  Jean-Yves Chemin

GLOBAL LARGE SOLUTIONS TO 3-D INHOMOGENEOUS
NAVIER-STOKES SYSTEM WITH ONE SLOW VARIABLE
JEAN-YVESCHEMIN, MARIUSPAICU,ANDPING ZHANG
Abstract. In this paper, we are concerned with the global wellposedness of 3-D inhomo-
3
geneous incompressible Navier-Stokes equations (1.2) in the critical Besov spaces with the
1
norm of which are invariant by the scaling of the equations and under a nonlinear small-
0
2 ness condition on the isentropic critical Besov norm to the fluctuation of theinitial density
and the critical anisotropic Besov norm of the horizontal components of the initial velocity
n
which haveto beexponentially small compared with thecritical anisotropic Besov norm to
a
thethirdcomponent of theinitial velocity. Thenoveltyof this resultsis that theisentropic
J
space structure to the homogeneity of the initial density function is consistent with the
7
propagation ofanisotropic regularity forthevelocityfield. Inthesecond part,weapplythe
2
same idea to provethe global wellposedness of (1.2) with some large data which are slowly

] varyingin one direction.
P
Keywords: Inhomogeneous Navier-Stokes Equations, Littlewood-Paley Theory, Anisotropic
A
Besov spaces
.
h
AMS Subject Classification (2000): 35Q30, 76D03
t
a
m
1. Introduction
[

1 In this paper, we consider the global wellposeness to the following 3-D incompressible
v inhomogeneous Navier-Stokes equations with initial data in the critical Besov spaces and
3
with the third component of the initial velocity being large:
1
3 ∂ ρ+div(ρu) = 0, (t,x) ∈ R+×R3,
t
6
 ∂ (ρu)+div(ρu⊗u)−µ∆u+∇Π= 0,
1. (1.1)  dtivu = 0,
0 
ρ| = ρ , ρu| = ρ u ,
3 t=0 0 t=0 0 0
1 
where ρ,u = (u ,u ,u )stand for the density and velocity of the fluid respectively, Π is a
: 1 2 3
v
scalar pressure function. Such system describes a fluid which is obtained by mixing two im-
i
X miscible fluids that are incompressible and that have different densities. It may also describe
r a fluid containing a melted substance.
a
In [19], O. Ladyˇzenskaja and V. Solonnikov first addressed the question of unique re-
solvability of (1.1). More precisely, they considered the system (1.1) in bounded domain Ω
with homogeneous Dirichlet boundary condition for u. Under the assumption that u be-
0
2−2,p
longs to W p (Ω) with p greater than d, is divergence free and vanishes on ∂Ω and that ρ0
is C1(Ω), bounded and away from zero, then they proved
• Global well-posedness in dimension d= 2;
2−2,p
• Local well-posedness in dimension d = 3. If in addition u0 is small in W p (Ω),
then global well-posedness holds true.
Date: 10/11/2012.
1
2 J.-Y.CHEMIN,M.PAICU,ANDP.ZHANG
Similar results were obtained by R. Danchin [14] in Rd with initial data in the almost critical
Sobolev spaces. In general, the global existence of weak solutions with finite energy was
established by P.-L. Lions in [20] (see also the reference therein, and the monograph [6]). H.
Abidi, G. Gui and the last author established in [3]the large time decay and stability to any
given global smooth solutions of (1.1).
When the initial density is away from zero, we denote by a d=ef 1 −1, and then (1.1) can
ρ
be equivalently formulated as
∂ a+u·∇a = 0, (t,x) ∈ R+×R3,
t
 ∂ u+u·∇u+(1+a)(∇Π−µ∆u)= 0,
(1.2) t
 divu= 0,

(a,u)| = (a ,u ).
t=0 0 0


Notice that just as the classical Navier-Stokes system (which corresponds to a = 0 in (1.2)),
the inhomogeneous Navier-Stokes system (1.2) also has a scaling. Indeed if (a,u) solves (1.2)
with initial data (a ,u ), then for ∀ℓ> 0,
0 0
(1.3) (a,u) d=ef (a(ℓ2·,ℓ·),ℓu(ℓ2·,ℓ·)) and (a ,u ) d=ef (a (ℓ·),ℓu (ℓ·))
ℓ 0 0 ℓ 0 0
(a,u) is also a solution of (1.2) with initial data (a ,u ) .
ℓ 0 0 ℓ
d −1+d
It is easy to check that the norm of Bp (Rd) × B p(Rd) is scaling invariant under
p,1 p,1
the scaling transformation (a ,u ) given by (1.3). In [1], H. Abidi proved in general space
0 0 ℓ
dimension d that: if 1 < p < 2d, 0 < µ < µ(ρ), given initial data (a ,u ) sufficiently small
0 0
d −1+d
in Bp (Rd)×B p(Rd), (1.2) has a global solution. Moreover, this solution is unique if p
p,1 p,1
is in ]1,d[. This result generalized the wellposedness results of R. Danchin in [13] and [14],
which corresponds to the celebrated results by Fujita and Kato [16] devoted to the classical
Navier-Stokes system, and was improved by H. Abidi and the second author in [2] with a
0
d −1+d
in Bq (Rd) and u in B p(Rd) for p,q satisfying some technical assumptions. H. Abidi,
q,1 0 p,1
G. Gui and the last author removed the smallness condition for a in [4, 5]. Notice that the
0
main feature of the density space is to be a multiplier on the velocity space and this allows to
define the nonlinear terms in the system (1.1). Recently, R. Danchin and P. Mucha proved
in [15] a more general wellposedness result of (1.1) by considering very rough densities in
−1+d
some multiplier spaces on the Besov spaces B p(Rd) for p in ]1,2d[ which in particular
p,1
completes the uniqueness result in [1] for p in ]d,2d[ in the constant viscosity case.
Motivated by [18, 22, 24] concerning the global wellposedness of 3-D incompressible aniso-
tropic Navier-Stokes system with the third component of the initial velocity field being large,
the last two authors relaxed in [23] the smallness condition in [2] so that (1.2) still has a
unique global solution (see Theorem 1.1 below for details). We emphasize that the proof in
[23] used in a fundamental way the algebraical structure of (1.2). The first step is to obtain
energy estimates on the horizontal components of the velocity field on the one hand and then
on the vertical component on the other hand. Compared with [18, 22, 24], the additional
difficulties with this strategy are that: there appears a hyperbolic type equation in (1.2)
and due to the appearance of a in the momentum equation of (1.2), the pressure term is
more difficult to be handled. We remark that the equation on the vertical component of the
velocity field is a linear equation with coefficients dependingon the horizontal components of
the velocity field and a. Therefore, the equation on the vertical component does not demand
any smallness condition. While the equations on the horizontal components of the velocity
LARGE SOLUTIONS OF 3−D INHOMOGENEOUS NS EQUATIONS 3
field contain bilinear terms in the horizontal components and also terms taking into account
the interactions between the horizontal components and the vertical one. In order to solve
thisequation, weneedasmallnessconditiononaandthehorizontalcomponent(amplifiedby
the vertical component) of the initial data. The purpose of this paper is to prove the global
wellposedness of (1.2) with initial data, a ,u = (uh,u3), satisfying some nonlinear smallness
0 0 0 0
condition on the critical isentropic Besov norm to a and the critical anisotropic Besov norm
0
to uh which have to be exponentially small in contrast with the critical anisotropic Besov
0
norm to u3. Then we apply the same idea to prove the global wellposedness of (1.2) with
0
some large data which are slowly varying in one direction.
Before going further, we recall the functional space framework we are going to use. As
in[9],[12]and[21],thedefinitionsofthespaceswearegoingtoworkwithrequiresanisotropic
dyadic decomposition of the Fourier variables. Let us recall from [7] that
∆ha = F−1(ϕ(2−k|ξ |)a), ∆va = F−1(ϕ(2−ℓ|ξ |)a),
k h ℓ 3
(1.4) Sha = F−1(χ(2−k|ξ |)a), Sva = F−1(χ(2−ℓ|ξ |)a) and
k h b ℓ 3 b
∆ a = F−1(ϕ(2−j|ξ|)a), S a = F−1(χ(2−j|ξ|)a),
j b j b
where ξ = (ξ ,ξ ), Fa and a denote the Fourier transform of the distribution a, χ(x)
h 1 2 b b
and ϕ(τ) are smooth functions such that
b
3 8
Supp ϕ ⊂ τ ∈ R / ≤ |τ|≤ and ∀τ > 0, ϕ(2−jτ) = 1,
4 3
n o Xj∈Z
4
Supp χ ⊂ τ ∈ R / |τ|≤ and χ(τ)+ ϕ(2−jτ)= 1.
3
n o Xj≥0
Definition 1.1. Let (p,r) ∈ [1,+∞]2, s ∈ R and u ∈ S′(R3), which means that u ∈ S′(R3)
h
and limj→−∞kSjukL∞ = 0, we set
def
kukBs = 2qsk∆qukLp .
p,r (cid:16) (cid:17)ℓr
def
• For s < 3 (or s = 3 if r = 1), we define Bs (R3) = u ∈S′(R3) kuk < ∞ .
p p p,r h Bps,r
• If k ∈ N and 3 +k ≤ s < 3 +k +1 (or s = 3 +k(cid:8)+1 if r = 1(cid:12)), then Bs (R(cid:9)3) is
p p p (cid:12) p,r
definedas the subsetof distributions u∈ S′(R3) such that ∂βu∈ Bs−k(R3)whenever
h p,r
|β| = k.
Notations In all that follows, we shall denote
def
Bs = Bs .
p p,1
The following theorem was proved by the last two authors in [23]:
Theorem 1.1. Let p be in ]1,6[. There exist positive constants c and C such that, for any
0 0
3 −1+3
data a in Bp(R3) and u = (uh,u3) in B p(R3) verifying
0 p 0 0 0 p
def C
(1.5) η = µka k +kuhk exp 0ku3k2 ≤ c µ,
(cid:0) 0 Bpp3 0 Bp−1+p3(cid:1) (cid:16)µ2 0 Bp−1+p3 (cid:17) 0
the system (1.2) has a unique global solution (a,u) in the space
3 −1+3 1+3
C ([0,∞[;Bp(R3))× C ([0,∞);B p(R3))∩L1(R+,B p(R3)) .
b p b p p
(cid:0) (cid:1)
4 J.-Y.CHEMIN,M.PAICU,ANDP.ZHANG
We want to prove here an anisotropic version of the above theorem. Let us define the
anisotropic Besov space that we are going to use.
Definition 1.2. Let p be in [1,+∞], s ≤ 2, s ≤ 1 and u in S′(R3), we set
1 p 2 p h
def
kukBs1,s2 = 2js12ks2k∆hj∆vkukLp .
p (cid:16) (cid:17)ℓ1
The case when s > 2 or s > 1 can be similarly modified as that in Definition 1.1.
1 p 2 p
Notations In all that follows, we shall denote
B0 d=ef B−1+p2,p1 , B1 d=ef Bp2,p1 ∩B−1+p2,1+p1 and B2 d=ef B1+p2,p1 ∩Bp2,1+p1
p p p p p p p p
Our first result in this paper is as follows:
Theorem 1.2. Let p be in ]3,4[ and r in [p,6[. Let us consider an intial data (a ,u ) in the
0 0
space Bp3 ×B0∩B−1+r3. Then there exist positive constants c and C such that if
p p r 0 0
def C
(1.6) η = (cid:0)µka0kBpp3 +kuh0kB0p(cid:1)exp(cid:16)µ20ku30k2B0p(cid:17) ≤ c0µ,
the system (1.2) has a unique global solution
(1.7) a ∈ C ([0,∞);B3p(R3)) and u ∈ C ([0,∞);B−1+3r(R3))∩L1(R+;B1+r3(R3)).
b p b r r
Moreover, there holds
kuhk +µ kak +kuhk ≤ Cη,
(1.8) ku3kLLe∞∞((RR++;;BB0p0p))+µk(cid:0)u3kLLe1∞(R(R++;B;B2pp0))≤ 2ku30kLB1(0pR++;Bc22pµ)(cid:1).
e
Remark 1.1. (1) We emphasize that for any givenfunctiona,φ inthe Schwartz space S(R3),
any p in ]3,4[, Theorem 1.2 implies the global wellposedness of (1.2) with initial data of the
form
aε0(x) = (−lnε)δε1+1pa(x1,x2,εx3) and
(1.9)
uǫ0 = ε0(−lnε)δ ε−(1−2p)sin xε1 0,−ε∂3φ,∂2φ (x1,x2,εx3),
(cid:16) (cid:17)
(cid:0) (cid:1)
for 0 < δ < 1, and ε,ε being sufficiently small. Indeed it is well-known that
2 0
x1 1−2
sin ∇φ(x1,x2,x3) ≤ Cφε p,
(cid:13) (cid:16) ε (cid:17) (cid:13)B0p
(cid:13) (cid:13)
ka(cid:13)(x1,x2,εx3)k 3 ≤ Cε−p1(cid:13)kak1L−p3pk∇akL3pp,
Bpp
which ensures that
C
(cid:0)µkaε0kBpp3 +kuε0,hkB0p(cid:1)exp(cid:16)µ20kuε0,3k2B0p(cid:17) ≤ Cε(−lnε)δexp(cid:0)(−lnε)2δ(cid:1) → 0
which tends to 0 when ε tends to 0. Hence Theorem 1.2 implies that (1.2) with initial data
(aε,uε) has a unique global solution (aε,uε).
0 0
LARGE SOLUTIONS OF 3−D INHOMOGENEOUS NS EQUATIONS 5
(2) In the case when δ = 0 in (1.9), the homogeneity of the initial density aε could be much
0
larger. In fact, it follows from the same line as the proof of part (1) that (1.2) with the data
1
aε0(x) = εpa(x1,x2,εx3) and
uǫ0 = ε0ε−(1−2p)sin xε1 0,−ε∂3φ,∂2φ (x1,x2,εx3),
(cid:16) (cid:17)
(cid:0) (cid:1)
also has a unique global solution for ε ,ka k and ε being sufficiently small.
0 0 3
Bpp
Theorem 1.2 also ensures the global wellposedness of (1.2) with data of the form:
a (x ,x ),(εuh(x ,εx ),u3(x ,εx ))
0 h 3 0 h 3 0 h 3
for any smooth divergen(cid:0)ce free vector field u = (uh,u3) and(cid:1) with ε, ka k , for some p
0 0 0 0 3
Bpp
in ]3,4[ being sufficiently small. Notice that the authors [10] proved the global existence of
smooth solutions to 3-D classical Navier-Stokes system for some large data which are slowly
varying in one direction. The main idea behind the proof in [10] is that the solutions to 3-D
Navier-Stokes equations slowly varying in one space variable can be well approximated by
solutionsof2-DNavier-Stokseequation. Yetjustastheclassical2-DNavier-Stokes system,2-
D inhomogeneous Navier-Stokes equations is also globally wellposed with general initial data
(see [14, 19] for instance). This motivates us to study the global wellposedness of (1.2) with
large data which are slowly variable in one direction and which do not satisfy the nonlinear
smallness condition (1.6).
3 −1+3
Theorem 1.3. Let σ be a real number greater than 1/4 and a a function of Bp ∩B q
0 p q
for some p in ]3,4[ and q in ]3,2[. Let vh = (v1,v2) be a horizontal, smooth divergence free
2 0 0 0
vector field onR3, belonging, as well as allits derivatives, to L2(R ;H˙−1(R2)).Furthermore,
x3
we assume that for any α in N3, ∂α∂ vh belongs to B−1,21(R3). Then there exists a positive
3 0 2
ε such that if ε ≤ ε , the initial data
0 0
(1.10) aε(x) = εσa (x ,εx ), uε(x) = (vh(x ,εx ),0)
0 0 h 3 0 0 h 3
generates a unique global solution (aε,uε) of (1.2).
Remark 1.2. (1) With vh being given by Theorem 1.3 and w a smooth divergence free
0 0
vector field on R3, I. Gallagher and the first author proved in [10] that there exists a positive
ε such that if 0 < ε≤ ε , the classical Navier-Stokes system (which corresponds to a = 0 in
0 0
(1.2)) with the initial data
(1.11) uε(x) = (vh+εwh,w3)(x ,εx )
0 0 0 0 h 3
has a unique global solution.
(2) G. Gui, J. Huang and and the last author proved in [17] similar global wellposedness result
for (1.2) with initial data aε(x) = εδ0a (x ,εx ) and initial velocity given by (1.11) provided
0 0 h 3
that a ∈ W1,p∩H2 for some p ∈ (1,2) and δ > 1. We should point out that one difficulty
0 0 p
in [17] is to derive L∞(R+;B1) estimate for the solution a of the free transport equation in
p
(1.2). Toward this, the authors in [17] assumed more regularities for a and then use an
0
interpolation argument to get this estimate. The advantage of the argument used in the proof
of Theorem 1.3 is that: as observed from the proof of Theorem 1.2, the isentropic regularities
of a is matched with the anisotropic regularities of u, so that we can still work this problem
in the scaling invariant spaces, which leads to the improvement of the index σ > 1 in [17] to
2
be σ > 1 here.
4
6 J.-Y.CHEMIN,M.PAICU,ANDP.ZHANG
(3) It follows from the proof of Theorem 1.3 that we can prove similar wellposedness result
for (1.2) with data (aε,uε) given by (1.11) provided that ε ≤ ε and kaεk +εkaεk
0 0 0 0 Bpp3 0 Bq−1+q3
being sufficiently small and for some p,q satisfying p in ]3,4[ and q in ]3,2[. Nevertheless, as
2
w part in (1.11) satisfies our nonlinear smallness condition (1.6), we choose to investigate
0
the case (1.10) here.
The organization of this paper is as follows:
Inthesecond section, weprove some lemmas usingLittlewood-Paley theory in particular a
lemmaof product,alemma whichexplains how tocomputethepressureinthecase whenais
3
small in Bp and a lemma of propagation for the transport equation which takes into account
p
some anisotropy.
In the third section, we prove Theorem 1.2.
In the forth section, we prove Theorem 1.3
Let us complete this section by the notations of the paper:
Let A,B be two operators, we denote [A;B] = AB−BA, the commutator between A and
B. For a . b, wemeanthatthereisauniformconstantC,whichmay bedifferentondifferent
lines, such that a ≤ Cb. We denote by (a|b) the L2(R3) inner product of a and b, (dj)j∈Z
(resp. (d ) ) will be a generic element of ℓ1(Z) (resp. ℓ1(Z2)) so that d = 1
j,k j,k∈Z2 j∈Z j
(resp. j,k∈Z2dj,k = 1). P
For XPa Banach space and I an interval of R, we denote by C(I; X) the set of continuous
functions on I with values in X, and by C (I; X) the subset of boundedfunctions of C(I; X).
b
For q ∈ [1,+∞], the notation Lq(I; X) stands for the set of measurable functions on I with
values in X, such that t 7−→ kf(t)k belongs to Lq(I).
X
2. Some estimates related to Littlewood-Paley analysis
Asweshallfrequentlyusetheanisotropic Littlewood-Paley theory, andinparticular aniso-
tropic Bernstein inequalities. For the convenience of the readers, we first recall the following
Bernstein type lemma from [12, 21]:
Lemma 2.1. Let B (resp. B ) a ball of R2 (resp. R ), and C (resp. C ) a ring of R2
h v h v h v h
(resp. R ); let 1 ≤ p ≤ p ≤∞ and 1 ≤ q ≤ q ≤ ∞. Then there holds:
v 2 1 2 1
If the support of a is included in 2kB , then
h
b k∂xαhakLph1(Lqv1) . 2k(cid:16)|α|+2(cid:16)p12−p11(cid:17)(cid:17)kakLph2(Lqv1).
If the support of a is included in 2ℓB , then
v
β ℓ(β+( 1 − 1 ))
b k∂3akLph1(Lqv1) .2 q2 q1 kakLph1(Lqv2).
If the support of a is included in 2kC , then
h
b kakLph1(Lqv1) . 2−kN |αs|u=pNk∂xαhakLph1(Lqv1).
If the support of a is included in 2ℓC , then
v
b kakLph1(Lqv1) . 2−ℓNk∂3NakLph1(Lqv1).
To consider the product of a distribution in the isentropic Besov space with a distribution
in the anisotropic Besov space, we need the following result which allows to embed isotropic
Besov spaces into the anisotropic ones.
LARGE SOLUTIONS OF 3−D INHOMOGENEOUS NS EQUATIONS 7
Lemma 2.2. Let s and t be positive real numbers. Then for any p in [1,∞], one has
kfk . kfk .
Bsp,t Bps+t
Proof. Thanks to Definition 1.2, one has
kfkBs,t = 2js2ktk∆hj∆vkfkLp.
p
j,Xk∈Z2
We separate the above sum into two parts, depending on whether k < j or k ≥ j and we
shall only detail the first case (the second one is identical). We notice that if k < j,
k∆hj∆vkfkLp ≤ k∆ℓ∆hj∆vkfkLp . k∆ℓfkLp.
Xℓ∈Z |ℓ−Xj|≤N0
Then we infer from the fact that t > 0
2js2ktk∆hj∆vkfkLp . 2jsk∆ℓfkLp 2kt
Xj∈Z j,Xℓ∈Z2 Xk<j
k<j |j−ℓ|≤N0
. 2j(s+t)k∆jfkLp . kfkBps+t.
Xj∈Z
And the result follows. (cid:3)
In order to obtain a better description of the regularizing effect of the transport-diffusion
equation, we will use Chemin-Lerner type spaces Lλ(Bs (R3)) (see [7] for instance).
T p,r
To study product laws between distributions in the anisotropic Besov spaces, we need to
e
modify the isotropic para-differential decomposition of Bony [8] to the setting of anisotropic
version. We first recall the isotropic para-differential decomposition from [8]: let a and b be
in S′(R3),
ab= T(a,b)+R(a,b), or ab= T(a,b)+T¯(a,b)+R(a,b), where
T(a,b) = S a∆ b, T¯(a,b) = T(b,a), R(a,b) = ∆ aS b, and
j−1 j j j+2
(2.1) Xj∈Z Xj∈Z
j+1
R(a,b) = ∆ a∆˜ b, with ∆˜ b = ∆ a.
j j j ℓ
Xj∈Z ℓ=Xj−1
In what follows, we shall also use the anisotropic version of Bony’s decomposition for both
horizontal and vertical variables.
As an application of the above basic facts on Littlewood-Paley theory, we present the
following product laws in the anisotropic Besov spaces.
Lemma 2.3. Let p≥ q ≥ 1 with 1+1 ≤ 1, and s ≤ 2, s ≤ 2 with s +s > 0. Let σ ≤ 1,
p q 1 q 2 p 1 2 1 q
σ ≤ 1 with σ + σ > 0. Then for a in Bs1,σ1(R3) and b in Bs2,σ2(R3), the product ab
2 p 1 2 q p
belongs to Bs1+s2−2q,σ1+σ2−1q(R3), and
p
kabkBs1+s2−q2,σ1+σ2−1q . kakBsq1,σ1kbkBps2,σ2.
p
8 J.-Y.CHEMIN,M.PAICU,ANDP.ZHANG
Proof. We first get by applying Bony’s decomposition (2.1) in both horizontal and vertical
variables that
ab =(Th +T¯h+Rh)(Tv +T¯v +Rv)(a,b)
(2.2) =ThTv(a,b)+ThT¯v(a,b)+ThRv(a,b)+T¯hTv(a,b)
+T¯hT¯v(a,b)+T¯hRv(a,b)+RhTv(a,b)+RhT¯v(a,b)+RhRv(a,b).
In what follows, we shall detail the estimates to some typical terms above, the other cases
can be followed along the same line. Note that σ +σ > 0, we get, by applying Lemma 2.1,
1 2
that
k∆hj∆vk(ThRv(a,b))kLp .2kq kSjh′−1∆vk′akL∞h (Lqv)k∆hj′∆vk′bkLp
|j′X−j|≤4
k′≥k−N0 e
.2kq dj′,k′2−j′(s1+s2−2q)2−k′(σ1+σ2)kakBs1,σ1kbkBs2,σ2
q p
|j′X−j|≤4
k′≥k−N0
.dj,k2−j(s1+s2−q2)2−k(σ1+σ2−1q)kakBs1,σ1kbkBs2,σ2.
q p
The same estimate holds for ThTv(a,b) and ThT¯v(a,b).
Along the same lines, we obtain
k∆hj∆vk(T¯hRv(a,b))kLp .22j(1q−p1)2kq k∆hj′∆vk′akLqkSjh−1∆vk′bkL∞h (Lpv)
|j′X−j|≤4
k′≥k−N0 e
.22j(1q−p1)2kq dj′,k′2−j′(s1+s2−p2)2−k′(σ1+σ2)kakBs1,σ1kbkBs2,σ2
q p
|j′X−j|≤4
k′≥k−N0
.dj,k2−j(s1+s2−2q)2−k(σ1+σ2−q1)kakBs1,σ1kbkBs2,σ2.
q p
The same estimate holds for T¯hTv(a,b) and T¯hT¯v(a,b). Finally applying Lemma 2.1 once
again and using the fact that s +s > 0, σ +σ > 0, gives rise to
1 2 1 2
k∆hj∆vk(RhRv(a,b))kLp . 22qj2kq k∆hj′∆vk′akLqk∆hj′∆vk′bkLp
kj′′≥≥Xjk−−NN00 e e
. 22qj2kq dj′,k′2−j′(s1+s2)2−k′(σ1+σ2)kakBs1,σ1kbkBs2,σ2
q p
j′≥Xj−N0
k′≥k−N0
. dj,k2−j(s1+s2−2q)2−k(σ1+σ2−q1)kakBs1,σ1kbkBs2,σ2.
q p
The same estimate holds for RhTv(a,b) and RhT¯v(a,b). This together with (2.2) completes
the proof of Lemma 2.3. (cid:3)
As an application of the laws of product, we state a lemma which will describe the way
how to compute the pressure in the case when a is small.
Lemma 2.4. Let p ∈ (1,4), we consider a function a such that kak is small enough. If Π
3
Bpp
satisfies
(D) div((1+a)∇Π−f)= 0
LARGE SOLUTIONS OF 3−D INHOMOGENEOUS NS EQUATIONS 9
with f in B0, then (D) has a unique solution which satisfies
p
k∇ΠkB0 . kfkB0 and thus k(1+a)∇ΠkB0 . kfkB0.
p p p p
Proof. We first write (D) as
∆Π = −div(a∇Π)+divf.
Applying now the operator ∇∆−1 to this identity implies that
∇Π= −M (∇Π)+∇∆−1divf with −M (g) d=ef ∇∆−1div(ag).
a a
Laws of productfrom Lemma2.3 together with Lemma 2.2 implies that kMakL(B0) . kak 3
p Bpp
because p < 4. Thus, if kak is small enough, the operator (Id−M )−1 is well defined as
3 a
Bpp
an element of L(B0) by the formula
p
∞
(Id−M )−1 = Mk.
a a
Xk=0
As ∇∆−1div is a homogenenous Fourier multiplier of degree 0, the lemma is proved. (cid:3)
Now, we are going the prove a lemma which is a variation about the classical propagation
lemma for regularity of index less than 1.
3
Lemma 2.5. Let a be in Bp(R3), and u = (uh,u3) be a divergence free vector field such
0 p
that ∇u belongs to L1([0,T],L∞(R3)). Let f be in L1([0,T]) with k∇u3(t)kL∞ ≤ Cf(t) for
all t in [0,T]. We denote
def t
a = aexp −λ f(t′)dt′ .
λ
(cid:16) Z0 (cid:17)
Then, the unique solution a of
(2.3) ∂ a+u·∇a = 0, a| = a
t t=0 0
satisfies, for any tin [0,T] and λ large enough,
λ t t
(2.4) kaλk 3 + f(t′)kaλ(t′)k 3dt′ ≤ ka0k 3 +Ckaλk 3 k∇uh(t′)kL∞dt′.
L∞t (Bpp) 2 Z0 Bpp Bpp L∞t (Bpp)Z0
e e
Proof. The proof of this lemma basically follows from that of Proposition 3.1 in [23]. The
novelty of our observation here is that the L1(Lip(R3)) estimate of the convection velocity
T
3
enables us to propagate the Bp regularity for (2.3) when p > 3.
p
As both the existence and uniqueness of solutions to (2.3) essentially follows from the
estimate (2.4) for some appropriate approximate solutions to (2.3). For simplicity, here we
just present the a priori estimate (2.4) for smooth enough solutions of (2.3). In this case,
thanks to (2.3), we have
∂ a +λf(t)a +u·∇a = 0.
t λ λ λ
Applying∆ totheaboveequationandthentakingL2 innerproductoftheresultingequation
j
with |∆ a |p−2∆ a , we obtain
j λ j λ
1 d
(2.5) k∆ a (t)kp +λf(t)k∆ a (t)kp + ∆ (u·∇a ) | |∆ a |p−2∆ a = 0.
qdt j λ Lp j λ Lp j λ j λ j λ
(cid:0) (cid:1)
10 J.-Y.CHEMIN,M.PAICU,ANDP.ZHANG
While as divu= 0, we get, by using Bony’s decomposition (2.1),
u·∇a = T(u,∇a )+R(u,∇a ),
λ λ λ
and a standard commutator’s argument, that
∆j(T(u,∇aλ)) | |∆ja|p−2∆ja = [∆j;Sj′−1u]∆j′∇aλ | |∆jaλ|p−2∆jaλ
(cid:0) (cid:1) |j′X−j|≤5(cid:16)(cid:0) (cid:1)
+ (Sj′−1u−Sj−1u)∆j∆j′∇aλ | |∆jaλ|p−2∆jaλ .
(cid:17)
(cid:0) (cid:1)
Then we deduce from (2.5) that
t
k∆jaλ(t)kLp +λ f(t′)k∆jaλ(t′)kLpdt′
Z
0
(2.6) ≤ k∆ja0kLp +C k[∆j;Sj′−1u]∆j′∇aλkL1(Lp)
(cid:16)|j′X−j|≤4(cid:0) t
+k(Sj′−1u−Sj−1u)∆j∆j′∇aλkL1(Lp) +kR(u,∇aλ)kL1(Lp) .
t t (cid:17)
(cid:1)
Applying the classical estimate on commutator (see [7] for instance) leads to
k[∆j;Sj′−1u]∆j′∇aλkL1(Lp)
t
|j′X−j|≤4
t
. kSj′−1∇uhkL1(L∞)k∆j′aλkL∞(Lp)+ kSj′−1∇u3(t′)kL∞k∆j′aλ(t′)kLp dt′
|j′X−j|≤4(cid:16) t t Z0 (cid:17)
′ t
. dj′2−3pj k∇uhkL1(L∞)kaλk 3 + k∇u3(t′)kL∞k∆j′aλ(t′)kLpdt′
|j′X−j|≤5(cid:16) t L∞t (Bpp) Z0 (cid:17)
e
t
. dj2−3pj k∇uhkL1(L∞)kaλk 3 + f(t′)kaλ(t′)k 3 dt′ .
(cid:0) t L∞t (Bpp) Z0 Bpp (cid:1)
e
Similarly we get, by applying Lemma 2.1, that
k(Sj′−1u−Sj−1u)∆j∆j′∇aλkL1(Lp)
t
|j′X−j|≤4
. k(Sj′−1∇uh−Sj−1∇uh)kL1(L∞)k∆jaλkL∞(Lp)
|j′X−j|≤4(cid:16) t t
t
+ k(Sj′−1∇u3−Sj−1∇u3)(t′)kL∞k∆jaλ(t′)kLp dt′
Z0 (cid:17)
t
. dj2−3pjk∇uhkL1(L∞)kaλk 3 + k∇u3(t′)kL∞k∆jaλ(t′)kLpdt′
t L∞t (Bpp) |j′X−j|≤4Z0
e
t
. dj2−3pj k∇uhkL1(L∞)kaλk 3 + f(t′)kaλ(t′)k 3 dt′ .
(cid:0) t L∞t (Bpp) Z0 Bpp (cid:1)
e