On the global well-posedness of 2-D density-dependent Navier-Stokes system with variable viscosity

0.36 MB·English

by  Hammadi Abidi

ON THE GLOBAL WELL-POSEDNESS OF 2-D DENSITY-DEPENDENT
NAVIER-STOKES SYSTEM WITH VARIABLE VISCOSITY
HAMMADIABIDI ANDPING ZHANG
Abstract. Givensolenoidalvectoru ∈H−2δ∩H1(R2),ρ −1∈L2(R2),andρ ∈L∞∩W˙ 1,r(R2)
3 0 0 0
with a positive lower bound for δ ∈ (0,1) and 2 < r < 2 , we prove that 2-D incompressible
1 2 1−2δ
0 inhomogeneous Navier-Stokes system (1.1) has a unique global solution provided that the viscous
2 coefficient µ(ρ0) is close enough to 1 in the L∞ norm compared to the size of δ and the norms
of the initial data. With smoother initial data, we can prove the propagation of regularities for
n
such solutions. Furthermore, for 1 < p < 4, if (ρ −1,u ) belongs to the critical Besov spaces
a 0 0
J B˙pp2,1(R2)×(cid:0)B˙p−,11+p2 ∩L2(R2)(cid:1) and the B˙pp2,1(R2) norm of ρ0−1 is sufficiently small compared to
11 theexponential of ku0k2L2 +ku0kB˙p−,11+p2, we provethe global well-posedness of (1.1) in thescaling
invariant spaces. Finally for initial data in the almost critical Besov spaces, we prove the global
] well-posedness of (1.1) underthe assumption that the L∞ norm of ρ −1 is sufficiently small.
P 0
A Keywords: Inhomogeneous Navier-Stokes systems, Littlewood-Paley Theory, critical
. regularity
h
t AMS Subject Classification (2000): 35Q30, 76D03
a
m
[
1. Introduction

1
v The purpose of this paper is to investigate the global well-posedness of the following two-
1 dimensional incompressible inhomogeneous Navier-Stokes equations with variable viscous coeffi-
7
cient
3
2 ∂ ρ+div(ρu) = 0, (t,x) R+ R2,
t
01. (1.1)  ∂dtiv(ρuu)=+0,div(ρu⊗u)−div(2µ∈(ρ)d)×+∇Π= 0,

3
ρ = ρ , ρu = m ,
1  |t=0 0 |t=0 0
: where ρ,u= (u ,u ) stand for the density and velocity of the fluid respectively, and d= 1(∂ u +
v 1 2 2 i j
i ∂ u ) denotesthedeformationtensor,Πisascalarpressurefunction,andtheviscousc(cid:0)oefficient
X j i 2×2
µ(ρ) (cid:1)is a smooth, positive and non-decreasing function on [0, ). Such a system describes for
r ∞
a instance a fluid that is incompressible but has nonconstant density owing to the complex structure
of the flow due to a mixture (e.g. blood flow) or pollution (e.g. model of rivers). It may also
describe a fluid containing a melted substance.
Whenµ(ρ)isapositiveconstant,andtheinitialdensityhasapositivelowerbound,Ladyˇzenskaja
and Solonnikov [19] first addressed the question of unique solvability of (1.1). More precisely, they
consideredthesystem(1.1)inaboundeddomainΩwithhomogeneousDirichletboundarycondition
2−2,p
for u. Under the assumptions that u0 W p (Ω) (p > d) is divergence free and vanishes on ∂Ω
∈
and that ρ C1(Ω) is bounded away from zero, then they [19] proved
0
∈
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: Jan. 10, 2013.
1
2 H.ABIDIANDP.ZHANG
Danchin [11] proved similar well-posedness result of (1.1) in the whole space and with the initial
data in the almost critical Sobolev spaces. In particular, in two space dimensions, he proved the
the global well-poseness of (1.1) with µ(ρ) = µ > 0 provided that the initial data (ρ ,u ) satisfies
0 0
ρ 1 H1+α(R2) ( ρ L∞(R2) if α = 1), ρ b > 0, and
0 0 0
− ∈ ∇ ∈ ≥
u Hβ(R2) for any α > 0, β (0,α) (α 1,α+1).
0
∈ ∈ ∩ −
Very recently, Paicu, Zhang and Zhang [22] proved the global well-posedness of (1.1) with µ(ρ) =
µ > 0 for initial data: ρ L∞(R2) with a positive lower bound and u Hs(R2) for any s > 0.
0 0
∈ ∈
This result improves the former interesting well-posedness theorem of Danchin and Mucha [14] by
removing the smallness assumption on the fluctuation to the initial density and also with much less
regularity for the initial velocity.
In general, Lions [20] (see also the references therein, and the monograph [5]) proved the global
existence of finite energy weak solutions to (1.1). Yet the uniqueness and regularities of such weak
solutions are big open questions even in two space dimensions, as was mentioned by Lions in [20]
(see page 31-32 of [20]). Except under the additional assumptions that
(1.2) µ(ρ0) 1 L∞(T2) ε0 and u0 H1(T2),
k − k ≤ ∈
Desjardins [15] proved that the global weak solution, (ρ,u, Π), constructed in [20] satisfies u
L∞((0,T);H1(T2)) and ρ L∞((0,T) T2) for any T < ∇. Moreover, with additional regularit∈y
assumptions on the initial d∈ata, hecoul×d also prove that u∞L2((0,τ);H2(T2)) for some shorttime
∈
τ (see Theorem 1.1 below).
To understand the system (1.1) further, the second author to this paper proved the global well-
posedness to a modified 2-D model system, which coincides with the 2-D inhomogeneous Navier-
Stokes equations with µ(ρ) = µ > 0 , with general initial data in [26]. Gui and Zhang [16] proved
the global well-posedness of (1.1) with initial data satisfying ρ 1 being sufficiently small
0 Hs+1
k − k
and u Hs H˙−2δ(R2) for some s > 2 and 0 < δ < 1. Yet the exact size of ρ 1 was not
0 ∈ ∩ 2 k 0− kHs+1
presented in [16]. Huang, Paicu and Zhang [17] basically proved that as long as
(1.3) η d=ef ρ 1 exp C 1+µ2(1) exp C0 u 2 c0µ(1) ,
k 0− kBpp2,1 n 0(cid:0) (cid:1) (cid:0)µ2(1)k 0kBp−,11+p2(cid:1)o ≤ 1+µ(1)
2
for some sufficiently small c , (1.1) has a global solution so that ρ 1 ([0, );Bp (R2)) and
0 − ∈ Cb ∞ p,1
−1+2 1+2
u ([0, );B p(R2)) L1(R+;B p(R2)) for 1 < p < 4. In a recent preprint [18], Huang
∈ Cb ∞ p,1 ∩ p,1
and Paicu can prove the global existence of solution of (1.1) with much weaker assumption than
(1.3). Yet as there is no L1((0,T);Lip(R2)) estimate for the velocity field, the uniqueness of such
solutions is not clear in [18].
Let be the usual Riesz transform, d=ef (∆)−1div, and P d=ef I be the Leray projection
R Q ∇ −Q
operator on the space of divergence-free vector fields, we first recall the following result from
Desjardins [15]:
Theorem 1.1. Letρ L∞(T2),u H1(T2)withdivu = 0.Thenthereexistsapositiveconstant
0 0
∈ ∈
ε suchthatundertheassumptionof (1.2), Lionsweak solutions ([20])to (1.1) satisfy thefollowing
0
regularity properties for all T > 0:
u L∞((0,T); H1(T2)) and √ρ∂tu L2((0,T) T2),
• ∈ ∈ ×
ρ and µ(ρ) L∞((0,T) T2) ([0,T]; Lp(T2)) for all p [1, ),
• (Π ∈(2µd )) and× (P∩C (2µd)) L2((0,T) T∈2), ∞
i j ij ij
• ∇ −R R ∇ Q ∈ ×
Π may be renormalized in suchNa way that for some universal constant C0 > 0,
•
(1.4) Π and u L2((0,T); Lp(T2)) for all p [4,p∗],
∇ ∈ ∈
WELL-POSEDNESS OF 2-D INHOMOGENEOUS NAVIER-STOKES SYSTEM 3
where
1
(1.5) p∗ = 2C0kµ(ρ0)−1kL∞.
Moreover, if µ(ρ ) µ and log(µ(ρ )) W1,r(T2) for some r > 2, there exists some positive time
0 0
≥ ∈
τ so that u L2((0,τ);H2(T2)) and µ(ρ) C([0,τ];W1,r¯(T2)) for any r¯< r.
∈ ∈
In what follows, we shall always assume that
0< µ µ(ρ ), µ() W2,∞(R+) and µ(1) = 1.
0
≤ · ∈
Notations: In the rest of this paper, we always denote a to be any number strictly bigger than
+
a and a any number strictly less then a. We shall denote [Y] the integer part of Y, and C¯ to be a
−
uniform constant depending only m,M in (1.6) below and µ′ L∞, which may change from line to
k k
line.
Our first purpose in this paper is to prove the following global well-posedness result for (1.1).
Theorem 1.2. Let m,M be two positive constants and δ (0, 1), 2 < r < 2 . Let u
∈ 2 1−2δ 0 ∈
H−2δ H1(R2) be a solenoidal vector filed, and ρ 1 L2 L∞ W˙ 1,r(R2) satisfy
0
∩ − ∈ ∩ ∩
(1.6) m ρ0 M, µ(ρ0) 1 L∞ ε0,
≤ ≤ k − k ≤
and for some q (1/δ,p∗],
∈
def
(1.7) C = u 2 + ρ 1 4 + u 4 + u 2 exp C u 4 .
0 k 0kH−2δ k 0− kL2 k 0kL2 k∇ 0kL2 k 0kL2
(cid:0) (cid:1)
there holds
1
(1.8) kµ(ρ0)−1kL∞(cid:16)δ +4C¯2C0 1+kρ0kB∞(2/,∞q)+ exp C¯C0 (cid:17) ≤ ε0
(cid:0) (cid:1) (cid:0) (cid:1)
for some sufficiently small ε . Then (1.1) has a unique global solution (ρ,u, Π) with ρ 1
0
∇ − ∈
C ([0, );L2 L∞ W˙ 1,r(R2)),u ([0, );H1(R2)) L1(R+;H2(R2)),∂ u, Π L1(R+;L2(R2)),
b b t
∞ ∩ ∩ ∈ C ∞ ∩ ∇ ∈
and
(1.9) k∇ukL1(B˙∞0 ,1) ≤ 2C¯C0 1+kρ0kB∞(2/,∞q)+ exp C¯C0 .
(cid:0) (cid:1) (cid:0) (cid:1)
If in addition, µ() W2+[s],∞(R+), ρ 1 H1+s(R2) and u Hs(R2) for some s > 1, then the
0 0
· ∈ − ∈ ∈
global solution ρ 1 ([0, );H1+s(R2)), u ([0, );Hs(R2)) L1 (R+;H˙2+s(R2)).
− ∈ C ∞ ∈C ∞ ∩ loc
Remark 1.1. Without the assumptions that ρ 1 L2(R2) and u e H−2δ(R2) in Theorem 1.2,
0 0
our proof of Theorem 1.2 ensures that (1.1) has−a u∈nique solution (ρ∈,u) on a time interval [0,T]
with T being determined by
T C(m,M, ρ0 Lr, u0 H1) µ(ρ0) 1 L∞ −1
≥ k∇ k k k k − k
and ρ L∞((0,T);L∞ W˙(cid:0)1,r(R2)), u L∞((0,T);H1(R2)) L2((0,(cid:1)T);H2(R2)).
∈ ∩ ∈ ∩
Remark 1.2. We should point out that the reason why Desjardin [15] only proved (1.4) for p
∈
[2,p∗] with p∗ being determined by (1.5) is because of the fact that the Riesz transform maps
R
continuously fromLp(Rd)toLp(Rd)with theoperator norm(see Theorem 3.1.1. of [8]forinstance):
C p
L(Lp→Lp) 0
kRk ≤
for some uniform constant C .Our main observation used in the proof of Theorem 1.2 is that: Riesz
0
transform maps continuously from homogeneous Besov spaces B˙s (Rd) (see Definition A.1) to
p,r
R
B˙s (Rd) with the operator norm
p,r
C ,
kRkL(B˙ps,r→B˙ps,r) ≤ 0
which enables us to prove the a priori estimate for k∇ukL1T(L∞). This is in fact the most important
ingredient used in the proof of Theorem 1.2.
4 H.ABIDIANDP.ZHANG
The other important ingredient used in the proof of (1.9) is the time decay estimates (2.12) and
(2.13), which is a slight generalization of the decay estimates obtained by Huang and Paicu in [18].
The proof of such decay estimates is a direct application of Schonbek’s frequency splitting method as
well as the strategy of Wiegner [25] to prove the time decay estimate for classical 2-D Navier-Stokes
system.
In the particular case when µ(ρ) is a positive constant, the proof of Theorem 1.2 yields the
following corollary, which does not require any low frequency assumption on u .
0
Corollary 1.1. Let α (0,1) and m,M be positive constants. Let u B˙0 (R2) be a solenoidal
∈ 0 ∈ 2,1
vector filed and ρ 1 B˙1 B˙α (R2) with m ρ M. Then (1.1) with µ(ρ) = 1 has a
0 − ∈ 2,1 ∩ ∞,∞ ≤ 0 ≤
unique global solution (ρ,u) so that ρ 1 ([0, );B˙1 B˙α (R2)), u ([0, );B˙0 (R2))
− ∈ C ∞ 2,1 ∩ ∞,∞ ∈ C ∞ 2,1 ∩
L1 (R+;B˙2 (R2)).
loc 2,1
Another importantfeature of (1.1) is thescaling invariant property, namely, if (ρ,u) is asolution
of (1.1) associated to the initial data (ρ ,u ), then
0 0
(1.10) ρ (t,x),u (t,x) d=ef ρ(λ2t,λx),λu(λ2t,λx) ρ (x),u (x) d=ef ρ (λx),λu (λx) ,
λ λ 0,λ 0,λ 0 0
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
(ρ (t,x),u (t,x)) is a also a solution of (1.1) associated with the initial data (ρ (x),u (x)). A
λ λ 0,λ 0,λ
functional space for the data (ρ ,u ) or for the solution (ρ,u) is said to be at the scaling of the
0 0
equation if its norm is invariant under the transformation (1.10). In the very interesting paper
[13], Danchin and Mucha proved the global well-posedness of (1.1) with µ(ρ) = µ > 0 in d space
d −1+d
dimensions and with small initial data in the critical spaces ρ 1 B˙ p (Rd) and u B˙ p(Rd)
0− ∈ p,1 0 ∈ p,1
−1+d
forp [1,2d). Infact, they[13]onlyrequireρ 1tobesmallinthemultiplierspaceofB˙ p(Rd).
∈ 0− p,1
One may check [13] and the references therein for more details in this direction.
2 −1+2
It is easy to check that B˙ p (R2) B˙ p L2(R2) is at the scaling of (1.1). When ρ 1 is
p,1 × p,1 ∩ 0−
2(cid:0) (cid:1)
small enough in the critical space B˙ p (R2), we have the following global well-posedness result for
p,1
(1.1), which in particular improves the smallness condition (1.3) in [17] to (1.11) below (with only
one exponential), and completes the uniqueness gap for p (2,4) in [17].
∈
2 −1+2
Theorem 1.3. Let 1 <p < 4, ρ 1 B˙ p (R2) and u B˙ p L2(R2) which satisfy divu = 0
0− ∈ p,1 0 ∈ p,1 ∩ 0
and
(1.11) ρ 1 exp C ( u + u 2 ) ε
k 0− kB˙p2 0 k 0kB˙−1+2p k 0kL2 ≤ 0
p,1 (cid:8) p,1 (cid:9)
forsomeuniformconstantC andε beingsufficientlysmall. Then(1.1)hasauniqueglobalsolution
0 0
2 −1+2 1+2
(ρ,u, Π) so that ρ ([0, );B˙ p (R2)), u ([0, );B˙ p L2(R2)) L1(R+;B˙ p(R2)),
∇ ∈ Cb ∞ p,1 ∈ Cb ∞ p,1 ∩ ∩ p,1
−1+2
and ∂ u, Π L1(R+;B˙ p(R2)).
t ∇ ∈ p,1
Finally, in the case when the initial data is in the almost scaling invariant spaces and ρ0 1 L∞
k − k
is sufficiently small, we have the following global well-posedness result for (1.1):
2 2+ε
Theorem 1.4. Let 1 < p < 4 and 0 < ε < 4 1. Let ρ 1 B˙ p B˙ p (R2) and u
p − 0 − ∈ p,1 ∩ p,1 0 ∈
−1+2 −1+2−ε
B˙ p B˙ p L2(R2) be a solenoidal vector filed. Then (1.1) has a unique global solution
p,1 ∩ p,1 ∩
2 2+ε −1+2−ε −1+2
(ρ,u, Π) so that ρ 1 ([0, );B˙ p B˙ p (R2)), u ([0, );B˙ p B˙ p L2(R2))
∇ − ∈ Cb ∞ p,1∩ p,1 ∈ Cb ∞ p,1 ∩ p,1 ∩ ∩
1+2 1+2−ε −1+2 −1+2−ε
L1(R+;B˙ p B˙ p (R2)), and ∂ u, Π L1(R+;B˙ p B˙ p (R2)) provided that
p,1 ∩ p,1 t ∇ ∈ p,1 ∩ p,1
(1.12) ρ0 1 L∞ ε0
k − k ≤
WELL-POSEDNESS OF 2-D INHOMOGENEOUS NAVIER-STOKES SYSTEM 5
for some small enough ε .
0
Remark 1.3. One may check (8.8) for the exact size of ε in (1.12).
0
Scheme of the proof and organization of the paper. In the second section, we shall present
the a priori time decay estimate for u(t) and u(t) which leads to the crucial estimate for
L2 L2
k k k∇ k
u for q satisfying qδ > 1. Based on these estimates and the observation in Remark 1.2,
k∇ kL1(R+;Lq)
in Section 3, we shall present the a priori L1(R+;B˙1 ) estimate for velocity field. In Section 4,
∞,1
we present a blow-up criterion for smooth enough solutions of (1.1). We then present the proofs of
Theorem 1.2 in Section 5 and Corollary 1.1 in Section 6. Finally we present the proofs of Theorem
1.3 in Section 7 and Theorem 1.4 in Section 8. For the convenience of the readers, we collect
some basic facts on Littlewood-Paley analysis, which has been used throughout the paper, in the
Appendix A.
Let us complete this section with the notations we are going to use in this context.
Notations: Let A,B be two operators, we denote [A,B] = AB BA, the commutator between A
−
and B. For a . b, we mean that there is a uniform constant C, which may be different on different
lines, such that a Cb. We shall denote by (a b) (or (a b) ) the L2(R2) inner product of a and b.
L2
For X a Banac≤h space and I an interval |of R, we|denote by (I; X) the set of continuous
C
functions on I with values in X, and by (I; X) the subset of bounded functions of (I; X). For
b
C C
q [1,+ ], the notation Lq(I; X) stands for the set of measurable functions on I with values
∈ ∞
in X, such that t f(t) belongs to Lq(I). For any vector field v = (v ,v ), we denote
X 1 2
d(v) = 21 ∂ivj+∂jvi7−→i,j=k1,2. Fkinally, (dj)j∈Z (resp. (cj)j∈Z) will be a generic element of ℓ1(Z) (resp.
ℓ2(Z)) so(cid:0)that (cid:1)d = 1 (resp. c2 = 1).
j∈Z j j∈Z j
P P
2. Basic Estimates
In this section, we shall improve the a priori estimate of u , which was obtained by
k∇ kL2(R+;Lp)
Desjardins [15] in the case of T2, to be that of u for any p (1/δ,p∗] with p∗ being
k∇ kL1(R+;Lp) ∈
determinedby (1.5). Thiswillbeoneofthecrucial ingredientforusto provetheL1(R+;B˙1 (R2))
∞,1
estimate of the velocity field in Section 3. The main idea to achieve the estimate of u
k∇ kL1(R+;Lp)
is to use the decay estimate for velocity field in [18, 24, 25] and the energy method in [15].
Proposition 2.1. Let f(t) be a positive smooth function, let (ρ,u) be a smooth enough solution
of (1.1) on [0,T∗) for some positive time T∗. Then under the assumption (1.6), one has
d
f(t) µ(ρ)d :ddx +f(t) ∂ u2dt′
t
dt(cid:16) ZR2 (cid:17) ZR2| |
(2.1)
4f′(t) µ(ρ)d : ddx+C f(t)(1+ u 2 ) u 4 for t [0,T∗),
≤ ZR2 m,M(cid:16) k kL2 k∇ kL2(cid:17) ∈
where C is a positive constant depending on m,M in (1.6).
m,M
Proof. The proof of this proposition basically follows from that of Theorem 1 in [15]. For com-
pleteness, we outline its proof here. Indeed thanks to (1.6), one has
(2.2) m ρ(t,x) M for t [0,T∗).
≤ ≤ ∈
In what follows, the uniform constant C always depends on m,M and sometimes on µ′ L∞ also,
k k
yet we neglect the subscripts m,M for simplicity.
6 H.ABIDIANDP.ZHANG
By taking L2 inner product of the momentum equation of (1.1) with ∂ u and using integration
t
by parts, we deduce from the derivation of (29) in [15] that
d
f(t) ρ∂ u2dx+ f(t) µ(ρ)d :ddx
t
ZR2 | | dt(cid:16) ZR2 (cid:17)
= f′(t) µ(ρ)d :ddx f(t) ∂ u ρu u dx f(t) (u )u div 2µ(ρ)d dx
t
ZR2 − ZR2 | ·∇ − ZR2 ·∇ |
(cid:0) (cid:1) (cid:0) (cid:1)
= f′(t) µ(ρ)d :ddx 2f(t) ∂ u ρu u dx f(t) ρu u2dx
t
ZR2 − ZR2 | ·∇ − ZR2 | ·∇ |
(cid:0) (cid:1)
f(t) u u Πdx,
− ZR2 ·∇ | ∇
where in the last step we used the momentum equation of (1.1) so that div 2µ(ρ)d = ρ∂ u+
t
ρu u+ Π. This gives rise to (cid:0) (cid:1)
·∇ ∇
d
f(t) ρ∂ u2dx+ f(t) µ(ρ)d :ddx
t
ZR2 | | dt(cid:16) ZR2 (cid:17)
2 f′(t) µ(ρ)d :ddx+f(t) √ρu u 2 f(t) u u Πdx
(2.3) ≤ (cid:16) ZR2 k ·∇ kL2 − ZR2 ·∇ | ∇ (cid:17)
2
2f′(t) µ(ρ)d :ddx+Cf(t) u 2 u 2 + Π∂ uk∂ uidx
≤ ZR2 (cid:16)k kL4k∇ kL4 (cid:12)(cid:12)iX,k=1ZR2 i k (cid:12)(cid:12)(cid:17)
(cid:12) (cid:12)
TodealwiththepressurefunctionΠ,weget, bytakingspacedivergencetothemomentumequation
of (1.1), that
(2.4) Π = ( ∆)−1div ρ∂ u+ρu u ( ∆)−1div div 2µ(ρ)d ,
t
− ·∇ − − ⊗
(cid:0) (cid:1) (cid:0) (cid:1)
from which, we deduce
2
Π∂ uk∂ uidx . u u 2
(cid:12)(cid:12)iX,k=1ZR2 i k (cid:12)(cid:12) k∇ kL2k∇ kL4
(cid:12) (cid:12)
2
+ ( ∆)−1div ρ∂ u+ρ(u )u ∂ uk∂ ui ,
k − t ·∇ kBMO i k H1
(cid:0) (cid:1) (cid:13)iX,k=1 (cid:13)
(cid:13) (cid:13)
where f H1 denotes the Hardy norm of f. Yet as divu= 0, it follows from [10] that
k k
2
∂ uk∂ ui . u 2 ,
i k H1 k∇ kL2
(cid:13)iX,k=1 (cid:13)
(cid:13) (cid:13)
and f . f , we obtain
k kBMO(R2) k∇ kL2(R2)
2
Π∂ uk∂ uidx . u u 2 + ρ∂ u+ρ(u )u u 2 ,
(cid:12)(cid:12)iX,k=1ZR2 i k (cid:12)(cid:12) k∇ kL2k∇ kL4 k t ·∇ kL2k∇ kL2
(cid:12) (cid:12)
which along with u 2 . u u and (2.3) ensures that
k kL4 k kL2k∇ kL2
d
f(t) ρ∂ u2dx+ f(t) µ(ρ)d : ddx
t
ZR2 | | dt(cid:16) ZR2 (cid:17)
(2.5)
3f′(t) µ(ρ)d :ddx+C f(t) u 4 +f(t)(1+ u ) u u 2 .
≤ ZR2 (cid:16) k∇ kL2 k kL2 k∇ kL2k∇ kL4(cid:17)
To handle u , we write
L4
k∇ k
(2.6) u= ( ∆)−1Pdiv 2(µ(ρ) 1)d ( ∆)−1Pdiv 2µ(ρ)d ,
∇ ∇ − − −∇ −
(cid:0) (cid:1) (cid:0) (cid:1)
WELL-POSEDNESS OF 2-D INHOMOGENEOUS NAVIER-STOKES SYSTEM 7
which together with the following interpolation inequality from [9]
2 1−2
(2.7) f C√r f r f r , 2 r < ,
k kLr(R2) ≤ k kL2(R2)k∇ kL2(R2) ≤ ∞
ensures that for any p [2, )
∈ ∞
2 1−2
k∇ukLp ≤ C0pkµ(ρ0)−1kL∞k∇ukLp +C√pk∇ukLp2kPdiv 2µ(ρ)d kL2p
(cid:0) (cid:1)
with C > 0 being a universal constant. Taking ε sufficiently small in (1.6), we obtain for 2 p
0 0
≤ ≤
p∗ = 1 that
2C0kµ(ρ0)−1kL∞
2 1−2
(2.8) k∇ukLp ≤ C√pk∇ukLp2kρ∂tu+ρ(u·∇)ukL2p
2 1−2 1−2 1−2
C√p u p ∂ u p + u p u p .
≤ k∇ kL2 k t kL2 k kL4 k∇ kL4
(cid:0) (cid:1)
In particular taking p = 4 in (2.8) results in
(2.9) u 2 C u ∂ u + u u 3 .
k∇ kL4 ≤ k∇ kL2k t kL2 k kL2k∇ kL2
(cid:0) (cid:1)
Substituting the above inequality into (2.5), we obtain (2.1). This completes the proof of the
proposition. (cid:3)
Corollary 2.1. Under the assumption of Proposition 2.1, we have
(2.10) kuk2L∞t (L2)+k∇uk2L2t(L2) ≤ Cku0k2L2,
kht′i12∇uk2L∞t (L2)+kht′i12∂tuk2L2t(L2) ≤ Ck∇u0k2L2exp Cku0k4L2 ,
(cid:0) (cid:1)
def
for all t [0,T∗) and where t = e+t.
∈ h i
Proof. We first get, by using standard energy estimate to (1.1), that
1 d
(2.11) ρu2dx+ µ(ρ)d : ddx = 0,
2dt ZR2 | | ZR2
which implies the first inequality of (2.10).
Whereas taking f(t′) = t′ in (2.1) and integrating the resulting inequality over [0,t], we obtain
h i
t t
Z0 ZR2ht′i|∂tu|2dxdt′+kht′i12∇uk2L∞t (L2) ≤ C(cid:0)k∇u0k2L∞t (L2)+(1+ku0k2L2)Z0 ht′ik∇uk4L2dt′(cid:1),
Applying Gronwall’s inequality and using the first inequality of (2.10) gives rise to
t
Z0 ZR2ht′i|∂tu|2dxdt′+kht′i12∇uk2L∞t (L2) ≤Ck∇u0k2L2exp(cid:8)C(1+ku0k2L2)k∇uk2L2t(L2)(cid:9)
C u 2 exp C u 4 .
≤ k∇ 0kL2 k 0kL2
(cid:0) (cid:1)
This completes the proof of (2.10). (cid:3)
Proposition 2.2. With the additional assumption that ρ 1 L2(R2), u H−2δ(R2) for
0 0
− ∈ ∈
δ (0, 1), then under the assumption of Proposition 2.1, we have
∈ 2
(2.12) t′ δu + t′ δ− u C C exp CC ,
kh i kL2 kh i ∇ kL2t(L2) ≤ 0 0
p (cid:0) (cid:1)
and
(2.13) kht′i(12+δ)−∇ukL∞t (L2)+kht′i(12+δ)−utkL2t(L2) ≤ C C0exp CC0 ,
p (cid:0) (cid:1)
for any t [0,T∗) and C being determined by (1.7).
0
∈
8 H.ABIDIANDP.ZHANG
Remark 2.1. Large time decay estimates for u(t) and u(t) were obtained by Gui and
L2 L2
k k k∇ k
the authors in [3] for 3-D inhomogeneous Navier-Stokes system with constant viscosity. Gui and
the second author proved the time decay estimate for u(t) in (2.12) for 2-D inhomogeneous
L2
k k
Navier-Sttokes system with variable density in [16]. Similar time decay estimates as (2.12) and
(2.13) were obtained by Huang and Paicu in [18]. Note that for p [1,2) and δ = 1 1, Lp(R2) can
∈ p−2
be continuously imbedded into H−2δ(R2), the decay estimates (2.12) and (2.13) are slightly general
than that in [18], where the authors require the low frequency assumption for u that u Lp(R2)
0 0
∈
for p [1,2). For completeness, here we shall outline the proof which basically follows from the
∈
corresponding argument in [25] for the classical 2-D Navier-Stokes system.
According to [25] for classical Navier-Stokes system, the key ingredient used in the proof of the
decay estimate for u(t) in (2.12) is the following Lemma:
L2
k k
Lemma 2.1. Under the assumption of Proposition 2.2, we have
1
(2.14) u(t) C C for any t [0,T∗).
L2 0
k k ≤ ln t ∈
p h i
Proof. Following the proofs of Theorem 3.1 of [18] and Lemma 4.4 of [16], we first deduce from
(2.11) that
d
(2.15) √ρu 2 +2µ u 2 0.
dtk kL2 k∇ kL2 ≤
Applying Schonbek’s strategy in [24], by splitting the phase space R2 into two time-dependent
domain: R2 = S(t) S(t)c, where S(t) d=ef ξ : ξ Mg(t) for some g(t), which will be chosen
∪ | | ≤ 2µ
later on. Then we deduce from (2.15) that(cid:8) q (cid:9)
d
(2.16) √ρu(t) 2 +g2(t) √ρu(t) 2 Mg2(t) uˆ(t,ξ)2dξ.
dtk kL2 k kL2 ≤ Z | |
S(t)
To deal with the low frequency part of u on the right hand side of (2.16), we write
t
u(t) = et∆u + e(t−t′)∆P div (µ(ρ) 1)d(u) +(1 ρ)(u +u u) u u (s)dt′.
0 t
Z0 (cid:16) (cid:0) − (cid:1) − ·∇ − ·∇ (cid:17)
Taking Fourier transform with respect to x variables gives rise to
t
uˆ(t,ξ) .e−t|ξ|2 uˆ (ξ) + e−(t−t′)|ξ|2 ξ [(µ(ρ) 1)d(u)] + (u u)
0 x x
| | | | Z0 (cid:16)| |(cid:0)(cid:12)F − (cid:12) (cid:12)F ⊗ (cid:12)(cid:1)
(cid:12) (cid:12) (cid:12) (cid:12)
+ [(1 ρ)(u +u u)] (t′)dt′,
x t
F − ·∇
(cid:12) (cid:12)(cid:17)
(cid:12) (cid:12)
so that
t
uˆ(t,ξ)2dξ . e−t|ξ|2 uˆ (ξ)2dξ+g4(t) [(µ(ρ) 1)d(u)]
(2.17) ZS(t)| | ZS(t) | 0 | (cid:16)Z0 (cid:0)(cid:13)Fx − (cid:13)L∞ξ
2 t (cid:13) (cid:13) 2
+ (u u) dt′ +g2(t) [(1 ρ)(u +u u)] dt′ .
(cid:13)Fx ⊗ (cid:13)L∞ξ (cid:1) (cid:17) (cid:16)Z0 (cid:13)Fx − t ·∇ (cid:13)L∞ξ (cid:17)
(cid:13) (cid:13) (cid:13) (cid:13)
It is easy to observe that
e−t|ξ|2 uˆ (ξ)2dξ t −2δ u 2 ,
Z | 0 | ≤ h i k 0kH−2δ
S(t)
WELL-POSEDNESS OF 2-D INHOMOGENEOUS NAVIER-STOKES SYSTEM 9
and
t 2
[(µ(ρ) 1)d(u)] + (u u) dt′
(cid:16)Z0 (cid:0)(cid:13)Fx − (cid:13)L∞ξ (cid:13)Fx ⊗ (cid:13)L∞ξ (cid:1) (cid:17)
(cid:13) t (cid:13) (cid:13) (cid:13) 2
. (µ(ρ) 1) (u) + u u dt′
L1 L1
(cid:16)Z0 (cid:0)k − M k k ⊗ k (cid:1) (cid:17)
t 2
.kµ(ρ)−1k2L∞t (L2)k∇uk2L2t(L2)t+(cid:16)Z0 ku(t′)k2L2dt′(cid:17) ,
Finally thanks to (2.9) and (2.10), we have
t 2 t
u (t′) dt′ Cln t t′ u (t′) 2 dt′ C u 2 exp C u 4 ln t ,
(cid:16)Z0 k t kL2 (cid:17) ≤ h iZ0 h ik t kL2 ≤ k∇ 0kL2 (cid:0) k 0kL2(cid:1) h i
and
t t
1 1
u u dt′ . u 2 u u 2 + u u 2 dt′
Z k kL4k∇ kL4 Z k kL2k∇ kL2k tkL2 k kL2k∇ kL2
0 0 (cid:0) (cid:1)
.kukL12∞t (L2)k∇ukL2t(L2)khti12utkL212ln14hti+kukL∞t (L2)k∇uk2L2t(L2)
≤Ck∇u0kL212exp Cku0k4L2 ln14hti,
(cid:0) (cid:1)
which leads to
t 2
[(1 ρ)(u +u u)] (t′)dt′
(cid:16)Z0 (cid:13)Fx − t ·∇ (cid:13)L∞ξ (cid:17)
(cid:13) t (cid:13) 2 t 2
≤k(1−ρ)k2L∞t (L2)h(cid:16)Z0 kut(t′)kL2dt′(cid:17) +(cid:16)Z0 kukL4k∇ukL4dt′(cid:17) i
C ρ 1 2 u 2 exp C u 4 ln t .
≤ k 0− kL2k∇ 0kL2 k 0kL2 h i
(cid:0) (cid:1)
Resuming the above estimates into (2.17) and then using (2.16), we obtain
d t 2
√ρu(t) 2 +g2(t) √ρu(t) 2 Mg6(t) u(t′) 2 dt′
(2.18) dtk kL2 k kL2 ≤ (cid:16)Z0 k kL2 (cid:17)
+CC g2(t) t −2δ +g6(t) t +g4(t)ln t ,
0
h i h i h i
(cid:16) (cid:17)
for C given by (1.7). Taking g2(t) = 3 in the above inequality and then integrating the
0 htilnhti
resulting inequality over [0,t] resulting (2.14). (cid:3)
We now turn to the proof of Proposition 2.2.
Proof of Proposition 2.2. With Lemma 2.1 and (2.18), the decay estimate of u(t) in (2.12)
L2
k k
follows by an standard argument as [25] for the classical 2-D Navier-Stokes system (One may check
page 310-311 of [25] for details). Whereas multiplying (2.15) by t (2δ)− and then integrating the
h i
resulting inequality over [0,t], we obtain
t
t δ−u 2 +2µ t′ δ− u 2 C u 2 + t′ (2δ−1)− u(t′) 2 dt′
(2.19) kh i kL2 kh i ∇ kL2t(L2) ≤ (cid:0)k 0kL2 Z0 h i k kL2 (cid:1)
CC exp CC ),
0 0
≤
(cid:0)
for C given by (1.7). This proves (2.12).
0
On the other hand, taking f(t) = t (1+2δ)− in (2.1), and then using (2.19) and Gronwall’s
h i
inequality, we obtain (2.13). This completes the proof of the Proposition. (cid:3)
10 H.ABIDIANDP.ZHANG
Notation: In all that follows, for C given by (1.7), we already denote
0
def
(2.20) C = C C exp CC .
1 0 0
p (cid:0) (cid:1)
We now present the key estimate in this section:
Proposition 2.3. Under the assumptions of Proposition 2.2, for p [2,p∗] with p∗ being deter-
∈
mined by (1.5), we have for any t [0,T∗)
∈
(2.21) kht′i(12+δ−p1)−∇ukL2t(Lp) ≤ √pC12−p2.
Proof. We first, get by resuming (2.9) into (2.8), that
2 1−2 1(1−2) 1(1−2) 1−2 2−2
k∇ukLp ≤ C√p k∇ukLp2kutkL2p +kukL22 p k∇ukL2kutkL22 p +kukL2pk∇ukL2p .
(cid:16) (cid:17)
Notice that p 2, multiplying t′ (21+δ−p1)− to the above inequality and then taking L2(0,t) norm
≥ h i
of the resulting inequality, we obtain
kht′i(12+δ−p1)−∇ukL2t(Lp) ≤ C√p(cid:16)kht′iδ−∇ukLp22t(L2)kht′i(12+δ)−utk1L−2t(2pL2)
+kht′iδukL12∞t−(p1L2)kht′i(12+δ)−∇ukL12∞t−(p1L2)kht′iδ−∇ukL122t+(Lp12)kht′i(21+δ)−utkL122t−(Lp12)
+kuk1L−∞t 2p(L2)kht′i(12+δ)−∇uk1L−∞t p2(L2)kht′iδ−∇ukL2t(L2)(cid:17).
Then we get, by resuming (2.12) and (2.13) into the above inequality, that
kht′i(12+δ−p1)−∇ukL2t(Lp) ≤ √pC12−p2 1+ku0kL1−2p2 ,
(cid:0) (cid:1)
which together with (1.7) and (2.20) leads to (2.21). (cid:3)
3. The L1(R+;B˙1 ) estimate for the velocity field
∞,1
Thegoal of thethis section is to present thea priori L1(R+;B˙1 )estimate for the velocity field,
∞,1
which is the most important ingredient used in the proof of Theorem 1.2.
Lemma 3.1. Letq (1/δ,p∗]withp∗ beingdeterminedby (1.5) andε > 0suchthat 2+ε< 1.Let
∈ q
(ρ,u, Π)beasmoothenoughsolutionof (1.1)on[0,T∗).ThenundertheassumptionsProposition
∇
2.2, one has
(3.1) u u +CC2 1+ ρ for any t < T∗,
k kLe1t(B˙q1,+∞2q+ε) ≤ k 0kB˙q−,∞1+q2+ε 1(cid:0) k kL∞t (B∞2q+,∞ε)(cid:1)
where the norm u is given by Definition A.2 and the constant C by (2.20).
k kLe1t(B˙q1,+∞2q+ε) 1
Proof. Let P d=ef I (∆)−1div be Leray projection operator. We get, by first dividing the mo-
mentum equation o−f (∇1.1) by ρ and then applying the resulting equation by P, that
1
∂ u+P u u P div(2µ(ρ)d) Π = 0.
t
·∇ − ρ −∇
(cid:8) (cid:9) (cid:8) (cid:0) (cid:1)(cid:9)
Applying ∆˙ to the above equation and using a standard commutator’s process yields
j
ρ∂ ∆˙ u+ρu ∆˙ u ∆∆˙ u 2div (µ(ρ) 1)Pd(∆˙ u)
t j j j j
·∇ − − −
(3.2) 1 (cid:0) (cid:1)
= ρ[∆˙ P;u ]u+ρ[∆˙ P; ] div(2µ(ρ)d) Π +2div[∆˙ P;µ(ρ)]d.
j j j
− ·∇ ρ −∇
(cid:0) (cid:1)
Throughout this paper, we always denote d(v) d=ef 1(∂ v +∂ v ) , and abbreviate d(u) as d.
2 i j j i 2×2
(cid:0) (cid:1)