Global wellposdeness to incompressible inhomogeneous
fluid system with bounded density and non-Lipschitz
velocity
Jingchi Huang, Marius Paicu, Ping Zhang
To cite this version:
Jingchi Huang, Marius Paicu, Ping Zhang. Global wellposdeness to incompressible inhomogeneous fluid system with bounded density and non-Lipschitz velocity. 2012.
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GLOBAL WELLPOSDENESS TO INCOMPRESSIBLE INHOMOGENEOUS
FLUID SYSTEM WITH BOUNDED DENSITY AND NON-LIPSCHITZ
VELOCITY
JINGCHI HUANG, MARIUS PAICU, AND PING ZHANG
Abstract. In this paper, we first prove the global existence of weak solutions to the d-dimensional
incompressible inhomogeneous Navier-Stokes equations with initial data a0 ∈ L∞ (Rd ), u0 =
−1+ d
(uh0 , ud0 ) ∈ Ḃp,r p (Rd ), which satisfies µka0 kL∞ + kuh0 k −1+ d exp Cr µ−2r kud0 k2r−1+ d ≤ c0 µ
Ḃp,r
p
Ḃp,r
p
for some positive constants c0 , Cr and 1 < p < d, 1 < r < ∞. The regularity of the initial velocity is
critical to the scaling of this system and is general enough to generate non-Lipschitz velocity field.
Furthermore, with additional regularity assumption on the initial velocity or on the initial density,
we can also prove the uniqueness of such solution. We should mention that the classical maximal
Lp (Lq ) regularity theorem for the heat kernel plays an essential role in this context.
Keywords: Inhomogeneous Navier-Stokes equations, maximal Lp (Lq ) regularity for heat kernel,
Littlewood-Paley theory.
AMS Subject Classification (2000): 35Q30, 76D03
1. Introduction
In this paper, we consider the global wellposedness to the following d-dimensional incompressible
inhomogeneous Navier-Stokes equations with the regularity of the initial velocity being almost
critical and the initial density being a bounded positive function, which satisfies some nonlinear
smallness condition,


∂ ρ + div(ρu) = 0,
(t, x) ∈ R+ × Rd ,

 t
∂t (ρu) + div(ρu ⊗ u) − µ∆u + ∇Π = 0,
(1.1)
div u = 0,



ρ|t=0 = ρ0 , ρu|t=0 = ρ0 u0 ,
where ρ, u = (uh , ud ) stand for the density and velocity of the fluid respectively, Π is a scalar
pressure function, and µ the viscosity coefficient. Such system describes a fluid which is obtained
by mixing two immiscible fluids that are incompressible and that have different densities. It may
also describe a fluid containing a melted substance. We remark that our hypothesis on the density
is of physical interest, which corresponds to the case of a mixture of immiscible fluids with different
and bounded densities.
In particular, we shall focus on the global wellposedness of (1.1) with small homogeneity for
the initial density function in L∞ (Rd ) and small horizontal components of the velocity compared
with its vertical component. This approach was already applied by Paicu and Zhang [26, 27]
for 3-D anisotropic Navier-Stokes equations and for inhomogeneous Navier-Stokes system in the
framework of Besov spaces. The main novelty of the present paper is to consider the initial density
function in L∞ (Rd ), which is close enough to some positive constant. Then in order to handle the
nonlinear terms appearing in (1.1), we need to use the maximal regularity effect for the classical
heat equation. We should mention that the initial data have scaling invariant regularities and the
Date: 12/Oct/2012.
1
2
J. HUANG, M. PAICU, AND P. ZHANG
global weak solutions obtained here, under a nonlinear-type smallness condition, also belong to
the critical spaces. Moreover, the regularity of the velocity field obtained in this paper is general
enough to include the case of non-Lipschitz vector fields.
When ρ0 is bounded away from 0, Kazhikov [22] proved that: the inhomogeneous Navier-Stokes
equations (1.1) has at least one global weak solutions in the energy space. In addition, he also proved
the global existence of strong solutions to this system for small data in three space dimensions and
all data in two dimensions. However, the uniqueness of both type weak solutions has not be solved.
Ladyženskaja and Solonnikov [23] first addressed the question of unique resolvability of (1.1). More
precisely, they considered the system (1.1) in a bounded domain Ω with homogeneous Dirichlet
2− 2 ,p
boundary condition for u. Under the assumption that u0 ∈ W p (Ω) (p > d) is divergence free
and vanishes on ∂Ω and that ρ0 ∈ C 1 (Ω) is bounded away from zero, then they [23] 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.
Similar results were obtained by Danchin [11] in Rd with initial data in the almost critical Sobolev
spaces. Abidi, Gui and Zhang [3] investigated the large time decay and stability to any given global
smooth solutions of (1.1), which in particular implies the global wellposedness of 3-D inhomogeneous
Navier-Stokes equations with axi-symmetric initial data provided that there is no swirl part for the
initial velocity field and the initial density is close enough to a positive constant. In general, when
the viscosity coefficient, µ(ρ), depends on ρ, Lions [24] proved the global existence of weak solutions
to (1.1) in any space dimensions.
def 1
In the case when the density function ρ is away from zero, we denote by a =
system (1.1) can be equivalently reformulated as

∂ a + u · ∇a = 0,
(t, x) ∈ R+ × Rd ,


 t
∂t u + u · ∇u + (1 + a)(∇Π − µ∆u) = 0,
(1.2)
div
u = 0,



(a, u)|t=0 = (a0 , u0 ).
ρ
− 1, then the
Notice that just as the classical Navier-Stokes system, the inhomogeneous Navier-Stokes system
(1.2) also has a scaling. More precisely, if (a, u) solves (1.2) with initial data (a0 , u0 ), then for
∀ ℓ > 0,
def
def
(a, u)ℓ = (a(ℓ2 ·, ℓ·), ℓu(ℓ2 ·, ℓ·))
(1.3)
and (a0 , u0 )ℓ = (a0 (ℓ·), ℓu0 (ℓ·))
(a, u)ℓ is also a solution of (1.2) with initial data (a0 , u0 )ℓ .
In [10], Danchin studied in general space dimension d the unique solvability of the system (1.2)
in scaling invariant homogeneous Besov spaces, which generalized the celebrated results by Fujita
and Kato [16] devoted to the classical Navier-Stokes system. In particular, the norm of (a, u) ∈
d
d
−1
2
2
Ḃ2,∞
(Rd ) ∩ L∞ (Rd ) × Ḃ2,1
(Rd ) is scaling invariant under the change of scale of (1.3). In this case,
d
d
−1
2
2
(Rd )∩L∞ (Rd )× Ḃ2,1
(Rd ) with a0 sufficiently
Danchin proved that if the initial data (a0 , u0 ) ∈ Ḃ2,∞
d
2
small in Ḃ2,∞
(Rd ) ∩ L∞ (Rd ), then the system (1.2) has a unique local-in-time solution. Abidi [1]
d
−1
d
p
p
proved that if 1 < p < 2d, 0 < µ < µ̃(a), u0 ∈ Ḃp,1
(Rd ) and a0 ∈ Ḃp,1
(Rd ), then (1.2) has a global
solution provided that ka0 k pd + ku0 k dp −1 ≤ c0 for some c0 sufficiently small. Furthermore, such a
Ḃp,1
Ḃp,1
solution is unique if 1 < p ≤ d. This result generalized the corresponding results in [10, 11] and was
improved by Abidi and Paicu in [2] when µ̃(a) is a positive constant by using different Lebesgue
d
−1+ dp
q
(Rd ) and u0 ∈ Ḃp,1
indices for the density and for the velocity. More precisely, for a0 ∈ Ḃq,1
(Rd )
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
3
with | p1 − 1q | < d1 and p1 + 1q > d1 , they obtained the existence of solutions to (1.2) and under a
more restrictive condition: 1p + 1q ≥ d2 , they proved the uniqueness of this solution. In particular,
with a well prepared regularity for the density function, this result implies the global existence of
solutions to (1.2) for any 1 < p < ∞ and the uniqueness of such solution when 1 < p < 2d. Very
recently, Danchin and Mucha [13] filled the gap for the uniqueness result in [1] with p ∈ (d, 2d)
through Langrage approach, and Abidi, Gui and Zhang relaxed the smalness condition for a0 in
[4, 5].
On the other hand, when the initial density ρ0 ∈ L∞ (Ω) with positive lower bound and u0 ∈
H 2 (Ω), Danchin and Mucha [14] proved the local wellposedness of (1.1). They also proved the
global wellposedness result provided that the fluctuation of the initial density is sufficiently small,
2− 2
and initial velocity is small in Bq,p q (Ω) for 1 < p < ∞, d < q < ∞ in 3-D and any velocity in
1 (Ω) ∩ L2 (Ω) in 2-D. Motivated by Proposition 2.1 below concerning the alternative definition of
B4,2
Besov spaces (see Definition A.1) with negative indices and [17], where Kato solved the local (resp.
global) wellposedness of 3-D classical Navier-Stokes system through elementary Lp approach, we
shall investigate the global existence of weak solutions to (1.2) with initial data a0 ∈ L∞ (Rd ) and
−1+ pd
u0 ∈ Ḃp,r
(Rd ) for p ∈ (1, d) and r ∈ (1, ∞), which satisfies the nonlinear smallness condition
−1+ dp +ε
(1.6). Furthermore, if we assume, in addition, u0 ∈ Ḃp,r
we can also prove the uniqueness of such solution.
(Rd ) for some sufficiently small ε > 0,
Definition 1.1. We call (a, u, ∇Π) a global weak solution of (1.2) if
• for any test function φ ∈ Cc∞ ([0, ∞) × Rd ), there holds
Z
Z ∞Z
φ(0, x)a0 (x) dx = 0,
a(∂t φ + u · ∇φ) dx dt +
Rd
0
Rd
Z ∞Z
(1.4)
div uφ dx dt = 0,
Rd
0
• for any vector valued function Φ = (Φ1 , · · · , Φd ) ∈ Cc∞ ([0, ∞) × Rd ), one has
Z
Z ∞Z n
o
u0 · Φ(0, x) dx = 0.
u · ∂t Φ − (u · ∇u) · Φ + (1 + a)(µ∆u − ∇Π) · Φ dx dt +
(1.5)
0
Rd
Rd
We denote the vector field by u = (uh , ud ) where uh = (u1 , u2 , ..., ud−1 ). Our first main result in
this paper is as follows:
−1+ d
Theorem 1.1. Let p ∈ (1, d) and r ∈ (1, ∞). Let a0 ∈ L∞ (Rd ) and u0 ∈ Ḃp,r p (Rd ). Then there
exist positive constants c0 , Cr so that if
o
n
def
(1.6)
η = µka0 kL∞ + kuh0 k −1+ pd exp Cr µ−2r kud0 k2r−1+ d ≤ c0 µ,
Ḃp,r
Ḃp,r
p
(1.2) has a global weak solution (a, u) in the sense of Definition 1.1, which satisfies
dr
(1) when p ∈ (1, 3r−2
],
1
1
µ r k∆uh k
Lr (R
+
dr
;L 3r−2 )
+ µ 2r k∇uh k
dr
+ µ 2r k∇ud k
µ r k∆ud k
Lr (R+ ;L 3r−2 )
1
µ r k∇Πk
≤ Cη,
dr
≤ Ckud0 k
1
1
(1.7)
dr
L2r (R+ ;L 2r−1 )
dr
Lr ;(R+ ;L 3r−2 )
L2r (R+ ;L 2r−1 )
≤ Cη kud0 k
−1+ d
Ḃp,r p
+ cµ ;
d
−1+ p
Ḃp,r
+ cµ,
4
J. HUANG, M. PAICU, AND P. ZHANG
dr
, d),
(2) when p ∈ ( 3r−2
ktβ1 ∇uh kL2r (R+ ;Lp2 ) + ktβ2 ∇uh kL∞ (R+ ;Lp2 )
1
(1− pd )
3
ktγ1 uh kL∞ (R+ ;Lp3 ) + ktγ2 uh kL2r (R+ ;Lp3 ) ≤ Cη,
+ µ2
1
(3− pd ) α1
1− d
β2
d
1 kt
µ2
∆ud kL2r (R+ ;Lp1 ) + µ 2p2 ktβ1 ∇ud kL2r
∇u
k
p2 ) + kt
∞ (R+ ;Lp2 )
(L
L
t
1
d
(1−
)
γ
d
γ
d
p3
+ µ2
kt 1 u kL∞ (R+ ;Lp3 ) + kt 2 u kL2r (R+ ;Lp3 ) ≤ 2Ckud0 k −1+ pd + cµ,
1
µ2
(1.8)
(3− pd )
1
ktα1 ∆uh kL2r (R+ ;Lp1 ) + µ
1− 2pd
2
Ḃp,r
and
1
µ2
(1.9)
1
µ2
(3− pd )
1
(3− pd )
1
η
(kud0 k −1+ pd + cµ + η) ,
µ
Ḃp,r
ktα1 ∇ΠkL2r (R+ ;Lp1 ) + ktα2 ∇ΠkLr (R+ ;Lp1 ) ≤ Cη kud0 k −1+ dp + cµ ,
ktα2 ∆ukLr (R+ ;Lp1 ) ≤ C ku0 k
−1+ d
Ḃp,r p
+
Ḃp,r
dr
) < p1 < d, and
for some small enough constant c, where p1 , p2 , p3 satisfy max(p, 2r−1
1
1
1
so that p2 + p3 = p1 , the indices α1 , α2 , β1 , β2 , γ1 , γ2 are determined by
1
(3 −
2
1
α2 = (3 −
2
α1 =
(1.10)
d
)−
p1
d
)−
p1
1
,
2r
1
,
r
dr
r−1
< p3 < ∞
1
d
1
1
d
(2 − ) −
and γ1 = (1 − ),
2
p2
2r
2
p3
1
d
1
d
1
β2 = (2 − ) and γ2 = (1 − ) − .
2
p2
2
p3
2r
β1 =
−1+ pd +ε
Furthermore, if we assume, in addition, that u0 ∈ Ḃp,r
then such a global solution is unique.
(Rd ) for 0 < ε < min{ 1r , 1 − 1r , dp − 1},
Remark 1.1. The main idea to prove Theorem 1.1 is to use the maximal Lp (Lq ) regularizing effect
for heat kernel (see Lemma 2.1). In fact, similar to the classical Navier-Stokes equations ([17]), we
first reformulate (1.2) as
Z t
µt∆
eµ(t−s)∆ −u · ∇u + µa∆u − (1 + a)∇Π ds.
(1.11)
u = e u0 +
0
Then we can prove appropriate approximate solutions to (1.11) satisfies the uniform estimate (1.71.9). With these estimates, the existence part of Theorem 1.1 follows by a compactness argument.
−1+ dp
We remark that given initial data u0 ∈ Ḃp,r
(Rd ), the maximal regularity we can expect for u
1+ dp
e 1t (Ḃp,r ). With this regularity for u and a ∈ L∞ (R+ × Rd ), we do not know how to define the
is L
product a∆u in the sense of distribution if p < d. This explains in some sense why we can only
prove Theorem 1.1 for p ∈ (1, d).
Remark 1.2. The smallness condition (1.6) is motivated by the one in [27] (see also [18, 26, 29] for
the related works on 3-D incompressible anisotropic Navier-Stokes system), where we prove that:
for 1 < q ≤ p < 6 with
verifying
(1.12)
1
q
−
−1+ p3
3
q
(R3 ) and u0 = (uh0 , u30 ) ∈ Ḃp,1
≤ 13 , given any data a0 ∈ Ḃq,1
1
p
def
η = µka0 k
3
q
Ḃq,1
+ kuh0 k
−1+ 3
Ḃp,1 p
(R3 )
. o
n
exp C0 ku30 k2 −1+ 3 µ2 ≤ c0 µ,
Ḃp,1
p
3
q
for some positive constants c0 and C0 , (1.2) has a unique global solution a ∈ C([0, ∞); Ḃq,1
(R3 ))
−1+ 3p
and u ∈ C([0, ∞); Ḃp,1
1+ 3
(R3 )) ∩ L1 (R+ ; Ḃp,1 p (R3 )). Similar wellposedness result ([20]) holds with
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
ka0 k
3
q
Ḃq,1
in (1.12) being replaced by ka0 k
−1+ 3
M(Ḃp,1 p )
5
−1+ p3
, the norm to the multiplier space of Ḃp,1
(R3 ).
We emphasize that our proof in [27, 20] uses in a fundamental way the algebraical structure of
(1.2), namely, div u = 0, which will also be one of the key ingredients in the proof of Theorem 1.1
and Theorem 1.2 below.
Remark 1.3. We should also mention the recent interesting result by Danchin and Mucha [14] that
with more regularity assumption on the initial velocity field, namely, m ≤ ρ0 < M and u0 ∈ H 2 (Rd )
for d = 2, 3, they can prove the local wellposedness of (1.1) for large data and global wellposedness for
−1+ d
small data. We emphasize that here we work our initial velocity field in the critical space Ḃp,r p (Rd )
for p ∈ (1, d) and r ∈ (1, ∞) and also the fact that Theorem 1.1 remains to be valid in the case
of bounded smooth domain with Dirichlet boundary conditions for the velocity field. Moreover, our
uniqueness result in Theorem 1.1 is strongly inspired by the Lagrangian approach in [14], but with
an almost critical regularity for the velocity, the proof here will be much more complicated. In fact,
−1+ d +ε
the small extra regularity compared to the scaling (1.3), namely, u0 ∈ Bp,r p (Rd ) for some small
ε > 0, is useful to obtain that ∆u ∈ L1loc (Ld+η ) for some η > 0, which combined with ∆u ∈ L1loc (Lp1 )
with p1 < d (see (1.8)) implies that u ∈ L1loc (Lip) and this allows us to reformulate (1.2) in the
Lagrangian coordinates. One may check Theorem 4.1 One may check Theorem 4.1 below for more
information about this solution.
A different approach to recover the uniqueness of the solution is to impose more regularity on
d
+ε
q
(Rd ), for some small positive ε,
the density function. Indeed, if the density is such that a0 ∈ Bq,∞
we can also prove the global wellposedness of (1.2) under the nonlinear smallness condition (1.13):
Theorem 1.2. Let r ∈ (1, ∞), 1 < q ≤ p < 2d with
1
q
d
+ε
q
−
1
p
1
2d
d and ε ∈ (0, p − 1) be
−1+ d −ε
−1+ d
Ḃp,r p (Rd ) ∩ Ḃp,r p (Rd ).
≤
real number. Let a0 ∈ L∞ (Rd ) ∩ Bq,∞ (Rd ) and u0 ∈
positive constants c0 , Cr,ε so that if
n
def
h
(1.13) δ = µka0 k
exp
Cr,ε µ−2r kud0 k2r−1+ d −ε
d +ε + ku0 k −1+ d −ε
d
−1+ p
q
p
L∞ ∩Bq,∞
∩Ḃp,r
Ḃp,r
Ḃp,r
p
any positive
−1+ d
p
∩Ḃp,r
There exist
o
≤ c0 µ,
(1.2) has a global solution (a, u) so that
d
+ε
q
2
(Rd )),
a ∈ C([0, ∞); L∞ (Rd ) ∩ Bq,∞
−1+ dp −ε
u ∈ C([0, ∞); Ḃp,r
and there holds
kuh k
−1+ d
p −ε
e ∞ (R+ ;Ḃp,r
L
(1.14)
−1+ pd
)
+ kuh k
1+ d −ε
e 1 (R+ ;Ḃp,r p )
L
−1+ d
p −ε
e ∞ (R+ ;Ḃp,r
L
d
+ ku k
)
−1+ d
p
e ∞ (R+ ;Ḃp,r
L
+ kuh k
kud k
d
1+ d
e 1 (R+ ;Ḃp,r p )
L
)
+ kuh k
+ kud k
+ µ kak
1+ d
e 1 (R+ ;Ḃp,r p )
L
−1+ d
p
e ∞ (R+ ;Ḃp,r
L
≤
d
1+ p −ε
1+
e 1 (R+ ; Ḃp,r
(Rd )) ∩ L
(Rd ) ∩ Ḃp,r p (Rd )),
(Rd ) ∩ Ḃp,r
)
d+ε
q 2
e ∞ (R+ ;Bq,∞
)
L
≤ Cδ,
+ µ kud k
2kud0 k −1+ dp −ε −1+ dp
∩Ḃp,r
Ḃp,r
1+ d −ε
e 1 (R+ ;Ḃp,r p
L
)
+ c2 µ,
for some c2 sufficiently small, and the norm k · kX∩Y = k · kX + k · kY . Furthermore, this solution
is unique if 1q + p1 ≥ d2 .
Remark 1.4. We point out that Haspot [19] proved the local well-posedness of (1.2) under similar
conditions of Theorem 1.2. Our novelty here is the global existence of solutions to (1.2) under the
smallness condition (1.13). We should also mention that: to overcome the difficulty that one can
6
J. HUANG, M. PAICU, AND P. ZHANG
not use Gronwall’s inequality in the framework of Chemin-Lerner spaces, motivated by [26, 27], we
introduced the weighted Chemin-Lerner type Besov norms in Definition A.3, which will be one of
key ingredients used in the proof of Theorem 1.2.
Remark 1.5. We remark that in the previous works on the global wellposedness of (1.2), the third
index, r, of the Besov spaces, to which the initial data belong, always equals to 1. In this case,
the regularizing effect of heat equation allows the velocity field to be in L1 (R+ , Lip(Rd )), which
is very useful to solve the transport equation without losing any derivative of the initial data. In
both Theorem 1.1 and Theorem 1.2, the regularity of the velocity field is general enough to include
non-Lipschitz vector-fields.
The organization of the paper. In the second section, we present the proof to the existence
dr
. In Section 3, we shall present the proof to
part of Theorem 1.1 in the case when 1 < p ≤ 3r−2
dr
< p < d. In Section 4, we shall
the existence part of Theorem 1.1 for the remaining case: 3r−2
present the proof to the uniqueness part of Theorem 1.1 with an extra regularity on the initial
velocity. In Section 5, we prove Theorem 1.2 which gives the uniqueness in the case where we have
an additional regularity on the density. Finally in the appendix, we collect some basic facts on
Littlewood-Paley theory and Besov spaces, which have been used throughout this paper.
Let us complete this section by the notations we shall use in this context:
Notation. For X a 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 Lq (I; X) stands for the set of measurable functions on I
with values in X, such that t 7−→ kf (t)kX belongs to Lq (I). For a . b, we mean that there is a
uniform constant C, which may be different on different lines, such
P that a ≤ Cb. We shall denote
by (cj,r )j∈Z to be a generic element of ℓr (Z) so that cj,r ≥ 0 and j∈Z crj,r = 1.
2. Proof to the existence part of Theorem 1.1 for 1 < p ≤
dr
3r−2
One of the key ingredients used in the proof to the existence part of Theorem 1.1 lies in Proposition 2.1 below:
Proposition 2.1 (Theorem 2.34 of [6]). Let s be a negative real number and (p, r) ∈ [1, ∞]2 . A
constant C exists such that
s
s
s
C −1 µ 2 kf kḂ s ≤ kt− 2 et∆ f kLp Lr (R+ ; dt ) ≤ Cµ 2 kf kḂ s .
p,r
p,r
t
−2
In particular, for r ≥ 1, we deduce from Proposition 2.1 that f ∈ Ḃp,rr (Rd ) is equivalent to
eµt∆ f ∈ Lr (R+ ; Lp (Rd )).
−1+ d
Notice that given u0 ∈ Ḃp,r p (Rd ) with 1 < p ≤
can always find some q1 ≥ p and r1 ≥ r such that
−3 +
dr
3r−2
(Rd ), we
d
d
2
2
≥ −3 +
=− ≥− .
p
q1
r1
r
Choosing r1 = r in the above inequality leads to q1 =
dr
−3+ pd
and r ∈ (1, ∞), △u0 ∈ Ḃp,r
dr
3r−2 .
Then △u0 ∈ Ḃ
− r2
dr
,r
3r−2
(Rd ), we infer
that △eµt∆ u0 ∈ Lr (R+ ; L 3r−2 (Rd )). Similarly, we can choose some q2 ≥ p and r2 ≥ r with
−2 +
q2 =
−2+ qd
−2+ dp
d
2
d
2
=
−
,
so
that
∇u
∈
Ḃ
(R
)
֒→
Ḃ
(Rd ).
p,r
q
0
2 ,r2
q2
r2
dr
dr
dr
µt∆ u ∈ L2r (R+ ; L 2r−1 (Rd )). And
0
2r−1 > 3r−2 ≥ p and ∇e
dr
d
2r
+
r−1
∈ L (R ; L (R )) in this case.
Choosing r2 = 2r gives rise to
Sobolev embedding ensures that
et∆ u0
The other key ingredient used in the proof of Theorem 1.1 is the following lemma (see [25] for
instance), which is called maximal Lp (Lq ) regularity for the heat kernel.
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
7
Rt
Lemma 2.1. (Lemma 7.3 of [25]) The operator A defined by f (t, x) 7→ 0 △eµ(t−s)△ f ds is bounded
from Lp ((0, T ); Lq (Rd )) to Lp ((0, T ); Lq (Rd )) for every T ∈ (0, ∞] and 1 < p, q < ∞. Moreover,
there holds
C
kAf kLp (Lq ) ≤ kf kLp (Lq ) .
T
T
µ
Rt
Lemma 2.2. Let 1 < r < ∞. The operator B defined by f (t, x) 7→ 0 ∇eµ(t−s)△ f ds is bounded
dr
dr
from L2r ((0, T ); L 2r−1 (Rd )) to Lr ((0, T ); L 3r−2 (Rd )) for every T ∈ (0, ∞], and there holds
Z
t
C
∇eµ(t−s)△ f ds
≤ 2r−1 kf k r
.
(2.1)
dr
dr
2r (L 2r−1 )
LT (L 3r−2 )
L
0
T
µ 2r
Proof. Notice that
√
π
∇eµ(t−s)△ f (s, x) =
Z
d+1
n |x − y|2 o
(x − y)
p
exp −
f (s, y) dy
4µ(t − s)
2 µ(t − s)
(4πµ(t − s)) 2 Rd
(2.2)
√
π
·
def
K( p
) ∗ f (s, x).
=
d+1
4µ(t − s)
(4πµ(t − s)) 2
Applying Young’s inequality in the space variables yields
k∇eµ(t−s)△ (1[0,T ] (s)f )(s, ·)k
dr
L 2r−1
≤C(µ(t − s))−
d+1
2
≤C(µ(t − s))−
2r−1
2r
kK( p
·
k1[0,T ] (s)f (s)k dr
)k
dr
L 3r−2
4πµ(t − s) L dr−r+1
k1[0,T ] (s)f (s)k
dr
L 3r−2
,
where 1[0,t] (s) denotes the characteristic function on [0, t], from which and Hardy-LittlewoodSobolev inequality, we conclude the proof of (2.1).
In what follows, we shall seek a solution (a, u) of (1.1) in the following space:
dr
dr
def (2.3) X = (a, u) : a ∈ L∞ (R+ × Rd ), ∇u ∈ L2r (R+ ; L 2r−1 (Rd )), △u ∈ Lr (R+ ; L 3r−2 (Rd )) .
We first mollify the initial data (a0 , u0 ), and then construct the approximate solutions (an , un )
via

∂t an + un · ∇an = 0,



∂ u + u · ∇u − µ∆u + ∇Π = a (µ∆u − ∇Π )
t n
n
n
n
n
n
n
n
(2.4)

div
u
=
0
n



(an , un )|t=0 = (SN +n a0 , SN +n u0 ),
where N is a large enough positive integer, and Sn+N a0 denotes the partial sum of a0 (see the
Appendix for its definition).
We have the following proposition concerning the uniform bounds of (an , un ).
Proposition 2.2. Under the assumptions of Theorem 1.1, (2.4) has a unique global smooth solution
(an , un , ∇Πn ) which satisfies
1
(2.5)
1
µ r k∆uhn k
dr
Lr (R+ ;L 3r−2 )
µ 2r k∇uhn k
dr
L2r (R+ ;L 2r−1 )
≤ Cη,
1
1
µ r k∆udn k
dr
Lr (R+ ;L 3r−2 )
+ µ 2r k∇ + udn k
dr
L2r (R+ ;L 2r−1 )
≤ Ckud0 k
and
(2.6)
1
µ r k∇Πn k
dr
Lr (R+ ;L 3r−2 )
≤ Cη kud0 k
−1+ d
Ḃp,r p
−1+ d
p
Ḃp,r
+ cµ
+ cµ,
8
J. HUANG, M. PAICU, AND P. ZHANG
for some small enough constant c and η given by (1.6).
Proof. For N large enough, it is easy to prove that (2.4) has a unique local smooth solution
(an , un , ∇Πn ) on [0, Tn∗ ) for some positive time Tn∗ . Without loss of generality, we may assume that
Tn∗ is the lifespan to this solution. It is easy to observe that
kan kL∞ ((0,T ∗ )×Rd ) ≤ ka0 kL∞ .
(2.7)
n
Next, for λ1 , λ2 > 0, we denote
def
f1,n (t) = k∇udn (t)k2r
def
f2,n (t) = k∆udn (t)kr dr ,
L 3r−2
n Z t
o
def
λ1 f1,n (t′ ) + λ2 f2,n (t′ ) dt′ ,
uλ,n (t, x) = un (t, x) exp −
dr
L 2r−1
(2.8)
,
0
and similar notation for Πλ,n (t, x). To deal with the pressure function Πn in (2.4), we get, by taking
divergence to the momentum equation of (2.4), that
−△Πλ,n = div(an ∇Πλ,n ) − µ div(an △uλ,n ) + div(un · ∇uλ,n ),
from which, (2.7), and the fact div un = 0, so that
div(un · ∇uλ,n ) = divh (uhn · ∇h uhλ,n ) + divh (udn ∂d uhλ,n )
+ ∂d (uhλ,n · ∇h udn ) − ∂d (uhn divh uhλ,n ),
we deduce that
k∇Πλ,n (t)k
dr
L 3r−2
n
≤C ka0 kL∞ k∇Πλ,n (t)k
+ kuhn k
(2.9)
+ µka0 kL∞ k△uλ,n k dr
L 3r−2
d
h
+ kun k dr k∇uλ,n k dr
L r−1
L 2r−1
o
d
k∇un k dr .
dr
L 3r−2
dr
L r−1
+ kuhλ,n k
dr
L r−1
L 2r−1
In particular, if η in (1.6) is so small that Cka0 kL∞ ≤ 21 , we infer from (2.9) that
n
k∇Πλ,n (t)k dr ≤C µka0 kL∞ k△uλ,n k dr + kuhn k dr k∇uhλ,n k dr
L 3r−2
L 3r−2
L r−1
L 2r−1
o
(2.10)
+ kudn k dr k∇uhλ,n k dr + kuhλ,n k dr k∇udn k dr .
L r−1
L 2r−1
L r−1
L 2r−1
Notice on the other hand that we can also equivalently reformulate the momentum equation of
(2.4) as
Z t
eµ(t−s)∆ −un ∇uin + µan ∆uin − (1 + an )∂i Πn ds for i = 1, 2, 3,
(2.11)
uin = uin,L +
0
def
and un,L = eµt∆ Sn+N u0 , from which and (2.8), we write
o
n Z t
h
h
(λ1 f1,n (t′ ) + λ2 f2,n (t′ )) dt′
uλ,n =un,L exp −
0
Z t
o
n Z t
(2.12)
eµ(t−s)∆ exp − (λ1 f1,n (t′ ) + λ2 f2,n (t′ )) dt′
+
s
0
× −un ∇uhλ,n + µan ∆uhλ,n − (1 + an )∇h Πλ,n ds.
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
9
Applying Lemma 2.1, Lemma 2.2, (2.7), and (2.9) leads to
1
µ1− 2r k∇uhλ,n k
+ µk∆uhλ,n k
dr
2r−1 )
L2r
t (L
dr
Lrt (L 3r−2 )
1
≤µ1− 2r k∇uhn,L k
dr
2r−1 )
L2r
t (L
+ µk∆uhL k
dr
Lrt (L 3r−2 )
n
+ C µka0 kL∞ k∆uhλ,n k
dr
Lrt (L 3r−2 )
+ µka0 kL∞ k∆udλ,n k r dr + kuhn k 2r dr k∇uhλ,n k 2r dr
Lt (L 2r−1 )
Lt (L r−1 )
Lt (L 3r−2 )
Z t
Rt
′
′
′
+
e−r s (λ1 f1,n (t )+λ2 f2,n (t )) dt kudn (s)kr dr k∇uhλ,n (s)kr dr
(2.13)
L 2r−1
L r−1
0
+ kuhλ,n kr
k∇udn (s)kr
dr
L r−1
dr
L 2r−1
ds
1 o
r
for t ≤ Tn∗ .
Then as Cka0 kL∞ ≤ 12 , and
kun k
dr
r−1 )
L2r
t (L
Z
t
e−λ1 r
Rt
s
≤ Ck∇un k
dr
2r−1 )
L2r
t (L
f1,n (t′ ) dt′
kudn−1 (s)kr
dr
L r−1
0
,
k∆udλ,n k
k∇uhλ,n (s)kr
dr
L 2r−1
1
≤
1 ,
(λ2 r) r
1
1
r
h
ds ≤
,
dr
1 k∇uλ,n (s)k 2r
Lt (L 2r−1 )
(λ1 r) 2r
dr
Lrt (L 3r−2 )
so that we infer from (2.13) that
1
µ1− 2r k∇uhλ,n k
+ µk∆uhλ,n k r dr
Lt (L 3r−2 )
n µ
1
h
h
∞ + ku k
≤Cµ1− r kuh0 k −1+ pd + C
dr k∇uλ,n k
dr
1 ka0 kL
n 2r r−1
2r−1 )
L
(L
L2r
)
r
Ḃp,r
t
t (L
(λ2 r)
o
1
h
(s)k
k∇u
.
+
dr
1
λ,n
2r−1 )
L2r
t (L
(λ1 r) 2r
dr
2r−1 )
L2r
t (L
Taking λ1 =
(2.14)
(4C)2r
µ2r−1 r
and λ2 =
Cr
rµr−1
in the above inequality, we obtain
3 1− 1
µ 2r k∇uhλ,n k 2r dr + µk∆uhλ,n k r dr
Lt (L 2r−1 )
Lt (L 3r−2 )
4
1
1
≤Cµ1− r kuh0 k
−1+ d
Ḃp,r p
+ µ2− r ka0 kL∞ + kuhn k
dr
r−1 )
L2r
t (L
k∇uhλ,n k
dr
2r−1 )
L2r
t (L
.
Let c1 be a small enough positive constant, which will be determined later on, we denote
(2.15)
1
def
T̄n = sup T ≤ Tn∗ ; µ 2r k∇uhn k
1
dr
r−1 )
L2r
T (L
+ µ r k∆uhn k
dr
LrT (L 3r−2 )
≤ c1 µ .
Then it follows from (2.8) and (2.14) that for t ≤ T̄n
(2.16)
1
1 1
µ 2r k∇uhn k 2r dr + µ r k∆uhn k r dr
2r−1
)
Lt (L
Lt (L 3r−2 )
2
≤ C kuh0 k −1+ dp + µka0 kL∞
Ḃp,r
n
× exp −Cr
Z
t
0
µ1−2r k∇udn (s)k2r
dr
L 2r−1
+ µ1−r k∆udn (s)kr
dr
L 3r−2
o
ds .
10
J. HUANG, M. PAICU, AND P. ZHANG
On the other hand, it follows from a similar derivation of (2.13) that
1
µ1− 2r k∇udn k
dr
2r−1 )
L2r
t (L
+ µk∆udn k
dr
Lrt (L 3r−2 )
n
+ µk∆udn,L k r dr + C µka0 kL∞ k∆un k r dr
L (L 3r−2 )
Lt (L 3r−2 )
+ kuhn k 2r dr + kudn k 2r dr k∇uhn k 2r dr
Lt (L r−1 )
Lt (L 2r−1 )
Lt (L r−1 )
o
for t ≤ Tn∗ ,
+ kuhn k 2r dr k∇udn k 2r dr
1
≤µ1− 2r k∇udn,L k
dr
2r−1 )
L2r
t (L
Lt (L 2r−1 )
Lt (L r−1 )
from which and (2.15), we deduce that
1
1
µ r k∆udn k
dr
Lrt (L 3r−2 )
(2.17)
≤2Ckud0 k
+ µ 2r k∇udn k
dr
2r−1 )
L2r
t (L
d
−1+ p
Ḃp,r
+ 2Cc1 µ(1 + ka0 kL∞ )
for t ≤ T̄n .
Substituting (2.17) into (2.16) leads to
1
(2.18)
µ r k∆uhn k
dr
1 1
+ µ 2r k∇uhn k 2r dr
Lt (L 2r−1 )
2
o c
n
1
+ µka0 kL∞ exp Cr µ−2r kud0 k2r−1+ d ≤ µ
p
2
Ḃp,r
Lrt (L 3r−2 )
≤ C kuh0 k
d
−1+ p
Ḃp,r
for
t ≤ T̄n ,
as long as Cr is sufficiently large and c0 is small enough in (1.6). This shows that T̄n = Tn∗ . Then
thanks to (2.17), (2.18) and Theorem 1.3 in [21], we conclude that Tn∗ = ∞ and there holds (2.5).
By virtue of (2.5) and (2.9), we infer
n
k∇Πn k r dr ≤C µka0 kL∞ k∆un k r dr + k∇uhn k 2r dr
Lt (L 3r−2 )
Lt (L 3r−2 )
Lt (L 2r−1 )
o
+ k∇udn k 2r dr k∇uhn k 2r dr
Lt (L 2r−1 )
≤Cµ
− r1
η kud0 k
d
−1+ p
Lt (L 2r−1 )
+ η + cµ ,
Ḃp,r
for some small constant c, which leads to (2.6). This completes the proof of the proposition.
We now present the proof to the existence part of Theorem 1.1 in the case when 1 < p ≤
Proof to the existence part of Theorem 1.1 for 1 < p ≤
dr
3r−2 . Indeed
dr
+
d
3r−2
dr
3r−2 .
thanks to (2.4) and
(2.5), (2.6), we infer that {∂t un } is uniformly bounded in Lr (R ; L
(R )), from which, (2.5),
(2.6), and Ascoli-Arzela Theorem, we conclude that there exists a subsequence of {an , un , ∇Πn },
dr
which we still denote by {an , un , ∇Πn } and some (a, u, ∇Π) with a ∈ L∞ (R+ × Rd ), ∇u ∈ L2r (R+ ; L 2r−1 (Rd ))
dr
and ∆u, ∇Π ∈ Lr (R+ ; L 3r−2 (Rd )), such that
an ⇀ a
+
d
weak ∗ in L∞
loc (R × R ),
un → u
d
r−1
+
strongly in L2r
loc (R ; Lloc (R )),
dr
(2.19)
dr
∇un → ∇u
d
2r−1
+
strongly in L2r
loc (R ; Lloc (R )),
∆un ⇀ ∆u
and ∇Πn ⇀ ∇Π weakly
dr
in Lr (R+ ; L 3r−2 (Rd )).
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
11
Obviously, (an , un , ∇Πn ) satisfies
Z
Z ∞Z
Sn+N a0 (x)φ(0, x) dx = 0,
an (∂t φ + un · ∇φ) dx dt +
Rd
0
Rd
Z ∞Z
div un φ dx dt = 0 and
Rd
Z ∞Z n 0
(2.20)
o
un · ∂t Φ − (un · ∇un ) · Φ + (1 + an ) µ∆un − ∇Πn · Φ dx dt
0
Rd
Z
Sn+N u0 · Φ(0, x) dx = 0,
+
Rd
for all the test function φ, Φ given by Definition 1.1.
Therefore thanks to (2.19), to prove that (a, u, ∇Π) obtained in (2.19) is indeed a global weak
1
1
solution of (1.2) in the sense of Definition 1.1, we only need to show that 1+a
→ 1+a
almost
n
+
d
everywhere in R × R as n → ∞. Toward this, we shall follow the compactness argument in [24]
+
d
to prove that {an } strongly converges to a in Lm
loc (R × R ) for any m < ∞. In fact, it is easy to
observe from the transport equation of (2.4) that
∂t a2n + div(un a2n ) = 0,
from which, we deduce that
∂t a2 + div(ua2 ) = 0,
(2.21)
where we denote a2 to be the weak ∗ limit of {a2n }.
While thanks to (2.19) and (2.20), there holds
∂t a + div(ua) = 0
dr
in the sense of distributions. Moreover, as ∇u ∈ Lr (R+ ; L 2r−1 (Rd )), we infer by a mollifying
argument as that in [15] that
∂t a2 + div(ua2 ) = 0.
(2.22)
Subtracting (2.22) from (2.21), we obtain
∂t (a2 − a2 ) + div(u(a2 − a2 )) = 0.
(2.23)
Notice that {Sn+N a0 } converges to a0 almost everywhere in Rd , which implies (a2 − a2 )(0, x) = 0
for a. e. x ∈ Rd . Whence it follows from the uniqueness theorem for the transport equation in [15]
that
(a2 − a2 )(t, x) = 0 for
a. e. (t, x) ∈ R+ × Rd ,
which together with kan kL∞ ≤ ka0 kL∞ implies that
(2.24)
an → a
strongly in
+
d
Lm
loc (R × R ) for any m < ∞.
Thanks to (2.19) and (2.24), we can take n → ∞ in (2.20) to verify that (a, u, ∇Π) obtained
in (2.19) satisfies (1.4) and (1.5). Moreover, thanks to (2.5) and (2.6), there holds (1.7). This
dr
].
completes the proof to the existence part of Theorem 1.1 for p ∈ (1, 3r−2
12
J. HUANG, M. PAICU, AND P. ZHANG
3. Proof to the existence part of Theorem 1.1 for
In this case, as −3 +
d
p
<p<d
< − 2r , it is impossible to find some p1 ≥ p and r1 ≥ r so that −3 +
− r21 . However, notice that for all p1 ≥ p and r1 ≥ r, ∆u0 ∈ Ḃ
Proposition 2.1 that t
dr
3r−2
1
(3− pd )− r1
2
1
1
−3+ pd
1
p1 ,r1
d
p1
=
(Rd ), then we deduce from
△et△ u0 ∈ Lr1 (R+ ; Lp1 (Rd )). Similarly, we choose p2 >
1
(1− pd )
2
3
1
(2− pd )− r1
2
2
1
d
2 and
t△
e u0 ∈
p3 > d with p12 + p13 = p11 , there holds t
∇et△ u0 ∈ Lr1 (R+ ; Lp2 (Rd )) and t
L∞ (R+ ; Lp3 (Rd )). With these time weights before et∆ u0 , ∇et∆ u0 and ∆et∆ u0 , to prove the existence
dr
< p < d, we need to use the following time-weighted version of maximal
part of Theorem 1.1 for 3r−2
p
q
L (L ) regularizing effect for the heat kernel:
Lemma 3.1. Let 1 < p, q < ∞ and α a non-negative real number satisfying α + p1 < 1. Let A
be the operator defined by Lemma 2.1. Then if tα f ∈ Lp ((0, T ); Lq (Rd )) for some T ∈ (0, ∞],
tα Af ∈ Lp ((0, T ); Lq (Rd )), and there holds
ktα Af kLp (Lq ) ≤
(3.1)
T
C α
kt f kLp (Lq ) .
T
µ
Proof. We first split the operator A as
Z 2t Z t def
△eµ(t−s)△ f ds = A1 f + A2 f.
+
(3.2)
Af =
t
2
0
Note that when s ∈ [ 2t , t], tα is comparable to sα , so that it follows from the proof of Lemma 7.3
in [25] that
C
ktα A2 f kLp (Lq ) ≤ ktα f kLp (Lq ) .
T
T
µ
To handle A1 f in (3.2), we write
α
t A1 f (t) =
Z
t
2
0
tα
µ(t − s)△eµ(t−s)△ (sα f ) ds.
µ(t − s)sα
As µt△eµt△ is a bounded operator from Lq (Rd ) to Lq (Rd ), we have
Z t
tα
C 2
ktα A1 f (t)kLq (Rd ) ≤
ksα f (s)kLq (Rd ) ds.
µ 0 (t − s)sα
def
Let Fα (s) = ksα f (s)kLq (Rd ) and using the change of variable that s = tτ, we obtain
α
kt A1 f (t)kLq (Rd )
C
≤
µ
Z
1
2
0
1
Fα (tτ ) dτ,
(1 − τ )τ α
from which and Minkowski inequality, we infer
Z 1
1
C 2
α
p
kt A1 f kL (Lq ) ≤
dτ kFα kLp ,
T
T
µ 0 (1 − τ )τ α+ 1p
which along with the fact: α +
1
p
< 1, concludes the proof of (3.1).
In order to heal with the estimate of u and ∇u, we need the following lemmas, which will be
used in the proof of both the existence part and uniqueness part of Theorem 1.1.
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
13
Rt
Lemma 3.2. Let the operator B be defined by f (t, x) 7→ 0 ∇eµ(t−s)△ f ds and C by f (t, x) 7→
R t µ(t−s)△
d
1
< q1 < 1−ε
,
f ds. Let ε ≥ 0 be a small enough number, r1 > 1, and q1 , q2 , q3 satisfy 2rdr
0 e
1 −2
1
1
1
d < q3 < ∞ and q2 + q3 = q1 . We denote
(3.3)
ε
α
eε =
d
1
1
(3 −
− ε) − ,
2
q1
r1
d
1
1
βe1ε = (2 −
− ε) −
2
q2
r1
and γ
e1ε =
eε
d
1
(1 −
− ε).
2
q3
ε
Then if tαe f ∈ Lr1 ((0, T ); Lq1 (Rd )) for some T ∈ (0, ∞], tβ1 Bf ∈ Lr1 ((0, T ); Lq2 (Rd )) and tγe1 Cf ∈
L∞ ((0, T ); Lq3 (Rd )), and there holds
eε
ktβ1 Bf kLr1 (Lq2 ) ≤
(3.4)
T
Cε
µ
ε
1
+ d
2 2q3
ktαe f kLr1 (Lq1 ) ,
T
and
Cε
ε
q3 ≤
ktγe1 Cf kL∞
T (L )
(3.5)
µ
ε
ktαe f kLr1 (Lq1 ) .
d
2q2
T
Proof. We first get, by applying Young’s inequality to (2.2), that
d+1
·
k∇eµ(t−s)△ f (s, ·)kLq2 ≤C(µ(t − s))− 2 kK( p
)k q q−1
3 kf (s)kLq1
L 3
4µ(t
−
s)
(3.6)
−( 12 + 2qd )
≤C(µ(t − s))
def
kf (s)kLq1 .
ε
Then let F ε (s) = ksαe f (s)kLq1 , we have
eε
3
tβ1 kBf (t)kLq2 ≤ Cµ
−( 21 + 2qd ) βeε
1
3
t
Z
0
t
−( 21 + 2qd ) −e
αε
3
s
(t − s)
F ε (s) ds.
eε + 1 = 0, we obtain
Using change of variable with s = tτ and the fact that βe1ε − ( 12 + 2qd3 ) − α
Z 1
ε
eε
−( 1 + d )
−( 1 + d )
tβ1 kBf (t)kLq2 ≤ Cµ 2 2q3
(1 − τ ) 2 2q3 τ −eα F ε (tτ ) dτ.
0
Applying Minkowski inequality leads to
Z 1
−( 1 + d ) −e
αε − r1
−( 12 + 2qd )
ε
βe1ε
r
3
1 dτ kF k r1
(1 − τ ) 2 2q3 τ
kt Bf kL 1 (Lq2 ) ≤Cµ
LT
T
0
Z 1
ε
−( 1 + d )
−( 1 + d ) − 1 (3− qd −ε)
1
≤Cµ 2 2q3
(1 − τ ) 2 2q3 τ 2
dτ ktαe f kLr1 (Lq1 ) .
T
0
Note that the assumption: q3 > d, which together with q1 <
d
q1 − ε) < 1. This proves (3.4).
To deal with Cf , we write
− 2qd
eµ(t−s)△ f = (µ(t − s))
2
d
1−ε
implies that 0 <
d
− 2qd
e−µ(t−s)△ (−µ(t − s)△) 2q2 (−△)
2
1
2
+
d 1
2q3 , 2 (3
−
f.
Observing that for any δ > 0, e△ (−△)δ is a bounded linear operator from Lq3 (Rd ) to Lq3 (Rd ), and
(−△)−δ f = | · |−(d−2δ) ∗ f for 0 < 2δ < d, so that applying Hardy-Littlewood-Sobolev inequality
gives rise to
− 2qd
(3.7)
keµ(t−s)△ f (s)kLq3 ≤C(µ(t − s))
2
− 2qd
2
≤C(µ(t − s))
− 2qd
k(−△)
2
kf (s)kLq1 .
f (s)kLq3
14
J. HUANG, M. PAICU, AND P. ZHANG
Let r1′ be the conjugate index of r1 , we get, by applying Hölder’s inequality, that
Z t
ε
− d
− d
(t − s) 2q2 s−eα F ε (s) ds
kCf (t)kLq3 ≤Cµ 2q2
0
(3.8)
Z t
1′
ε ′
ε
− 2qd
− d r′
≤Cµ 2
(t − s) 2q2 1 s−eα r1 ds r1 ktαe f kLr1 (Lq1 ) .
T
0
While using once again the change of variable that s = tτ , we obtain
Z t
Z 1
ε r′
ε ′
− 2qd r1′ −e
− d r′
1− 2qd r1′ −e
αε r1′
α
1 ds =t
2
(t − s) 2 s
(1 − τ ) 2q2 1 τ −eα r1 dτ
0
0
Z 1
ε ′
ε
′
− d r′
=t−eγ1 r1
(1 − τ ) 2q2 1 τ −eα r1 dτ.
0
Recalling the assumptions that q2 > q1 >
This together with (3.8) ensures that
dr1
2r1 −2
kCf (t)kLq3 ≤ Cε µ
d
1−ε ,
and q1 <
− 2qd
2
ε
which imply
d ′
2p2 r1
< 1 and α
eε r1′ < 1.
ε
t−eγ1 ktαe f kLr1 (Lq1 ) .
T
This concludes the proof of (3.5) and Lemma 3.2.
In order to deal with the term ud ∂d u in (2.11), we need also the following lemma.
Lemma 3.3. Let ε, r1 , q1 , α
eε and the operators B, C be given by Lemma 3.2. Let r1 > 2 and q2 ,
dr1
1
1
1
q3 satisfy q2 + q3 = q1 , r1 −2 < q3 < ∞, we denote
(3.9)
ε
1
d
βe2ε = (2 −
− ε) and
2
q2
d
1
1
e2ε = (1 −
γ
− ε) − .
2
q3
r1
eε
ε
Then if tαe f ∈ Lr1 ((0, T ); Lq1 (Rd )) for some T ∈ (0, ∞], tβ2 Bf ∈ L∞ ((0, T ); Lq2 (Rd )), tγe2 Cf ∈
Lr1 ((0, T ); Lq3 (Rd )), and there holds
ε
Cε
eε
q2 ≤
ktαe f kLr1 (Lq1 ) ,
(3.10)
ktβ2 Bf kL∞
1
d
T (L )
T
+
µ 2 2q3
and
Cε
ε
ε
(3.11)
ktγe2 Cf kLr1 (Lq3 ) ≤ d ktαe f kLr1 (Lq1 ) .
T
T
µ 2q2
Proof. We first get, by a similar derivation of (3.6), that
Z t
ε
−( 21 + 2qd )
−( 1 + d )
3
kBf (t)kLq2 ≤Cµ
(t − s) 2 2q3 s−eα F ε (s) ds
0
Z t
1′
1
d ε
−( 2 + 2q )
−( 21 + 2qd )r1′ −e
r
αε r1′
3
3
≤Cµ
s
ds 1 ktαe f kLr1 (Lq1 ) .
(t − s)
T
0
Whereas using the change of variable: s = tτ, leads to
Z t
Z 1
ε ′
−( 12 + 2qd )r1′ −e
−( 1 + d )r ′
1−( 21 + 2qd )r1′ −e
αε r1′
αε r1′
3
3
(t − s)
(1 − τ ) 2 2q3 1 τ −eα r1 dτ
s
ds =t
0
0
Z 1
ε ′
eε ′
−( 1 + d )r ′
=t−β2 r1
(1 − τ ) 2 2q3 1 τ −eα r1 dτ.
0
By virtue of the assumptions: q3 >
consequence, we obtain
dr1
r1 −2
and q1 <
kBf (t)kLq2 ≤ Cε µ
d
1−ε ,
−( 12 + 2qd ) −βeε
2
3
t
we have 0 < ( 21 +
ε
ktαe f kLr1 (Lq1 ) ,
T
d
′
2q3 )r1 ,
αε r1′ < 1. As a
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
15
which yields (3.10).
On the other hand, it follows the same line of (3.7) that
− 2qd
keµ(t−s)△ f (s)kLq3 ≤ C(µ(t − s))
which implies
γ
e2ε
t kCf (t)k
Lq3
≤ Cµ
− 2qd
2
t
γ
e2ε
Z
0
t
2
kf (s)kLq1 ,
− 2qd
(t − s)
2
ε
s−eα F ε (s) ds.
Using changing of variable with s = tτ and the fact: e
γ2ε − 2qd2 − α
eε + 1 = 0, we obtain
Z 1
ε
− d
− 2qd
γ
e2ε
(1 − τ ) 2q2 τ −eα F ε (tτ ) dτ,
t kCf (t)kLq3 ≤ Cµ 2
0
from which, we deduce
ε
ktγe2 Cf kLr1 (Lq3 ) ≤Cµ
− 2qd
2
T
≤Cµ
Z
1
0
− 2qd
2
Z
0
− 2qd
(1 − τ )
1
2
− 2qd
(1 − τ )
2
τ
τ
−e
αε − r1
1
dτ kF ε kLr1
T
− 12 (3− qd −ε)
1
ε
dτ ktαe f kLr1 (Lq1 ) ,
T
1
> d2 , ensures the integral above is finite. This gives
which along with the facts: q2 > q1 > 2rdr
1 −2
(3.11) and we complete the proof of the lemma.
dr
In what follows, we take r1 = 2r > 2 and p1 , p2 and p3 satisfying max(p, 2r−1
) < p1 < d, and
dr
1
1
1
r−1 < p3 < ∞ so that p2 + p3 = p1 . We shall seek a solution (a, u, ∇Π) of (1.2) in the following
functional space:
n
X = u : a ∈ L∞ (R+ × Rd ), tγ1 u ∈ L∞ (R+ ; Lp3 (Rd )),
tγ2 u ∈ L2r (R+ ; Lp3 (Rd )),
(3.12)
tβ1 ∇u ∈ L2r (R+ ; Lp2 (Rd )),
tβ2 ∇u ∈ L∞ (R+ ; Lp2 (Rd ))
o
tα1 △u ∈ L2r (R+ ; Lp1 (Rd )) ,
with the indices α1 , β1 , β2 , γ1 , γ2 being determined by (1.10).
We construct the approximate solution (an , un , ∇Πn ) through (2.4). Similar to Proposition 2.2,
we have the following proposition concerning the uniform time-weighted bounds of (an , un ).
Proposition 3.1. Under the assumptions of Theorem 1.1, (2.4) has a unique global smooth solution
(an , un , ∇Πn ) which satisfies
1
(3− pd ) α1
1− d
1 kt
µ2
∆uhn kL2r (R+ ;Lp1 ) + µ 2p2 ktβ1 ∇uhn kL2r (R+ ;Lp2 ) + ktβ2 ∇uhn kL∞ (R+ ;Lp2 )
1
(1− pd )
3
ktγ1 uhn kL∞ (R+ ;Lp3 ) + ktγ2 uhn kL2r (R+ ;Lp3 ) ≤ Cη,
+ µ2
(3.13)
1
1− d
(3− pd ) α1
1 kt
µ2
∆udn kL2r (R+ ;Lp1 ) + µ 2p2 ktβ1 ∇udn kL2r (R+ ;Lp2 ) + ktβ2 ∇udn kL∞ (R+ ;Lp2 )
1
(1− pd )
3
+ µ2
ktγ1 udn kL∞ (R+ ;Lp3 ) + ktγ2 udn kL2r (R+ ;Lp3 ) ≤ Ckud0 k −1+ pd + cµ,
Ḃp,r
and
1
µ2
(3.14)
1
µ2
(3− pd )
1
(3− pd )
1
η
(kud0 k −1+ pd + cµ + η) ,
µ
Ḃp,r
ktα1 ∇Πn kL2r (R+ ;Lp1 ) + ktα2 ∇Πn kLr (R+ ;Lp1 ) ≤ Cη kud0 k −1+ dp + cµ
ktα2 ∆un kLr (R+ ;Lp1 ) ≤ C ku0 k
−1+ d
Ḃp,r p
+
Ḃp,r
for some small enough constant c, η being given by (1.6) and α2 by (1.10).
16
J. HUANG, M. PAICU, AND P. ZHANG
Proof. For N large enough, it is easy to prove that (2.4) has a unique local smooth solution (an , un )
on [0, Tn∗ ). Without loss of generality, we may assume that Tn∗ is the lifespan to this solution. For
λ1 , λ2 , λ3 > 0, we denote
def
(3.15)
def
def
f1,n (t) = ktβ1 ∇udn (t)k2r
f2,n (t) = ktγ2 udn (t)k2r
f3,n (t) = ktα1 ∆udn (t)k2r
L p2 ,
L p3 ,
L p1 ,
o
n Z t
def
uλ,n (t, x) = un (t, x) exp −
λ1 f1,n (t′ ) + λ2 f2,n (t′ ) + λ3 f3,n (t′ ) dt′ ,
0
and similar notation for Πλ,n (t, x). Then we deduce by a similar derivation of (2.9) that
n
k∇Πλ,n (t)kLp1 ≤C ka0 kL∞ k∇Πλ,n (t)kLp1 + µka0 kL∞ k△uλ,n kLp1 + kuhn kLp3 k∇uhλ,n kLp2
o
+ kudn kLp3 k∇uhλ,n kLp2 + kuhλ,n kLp3 k∇udn kLp2 .
In particular, if η in (1.6) is so small that Cka0 kL∞ ≤ 12 , we obtain
n
k∇Πλ,n (t)kLp1 ≤C µka0 kL∞ k△uλ,n kLp1 + kuhn kLp3 k∇uhλ,n kLp2
o
(3.16)
+ kudn kLp3 k∇uhλ,n kLp2 + kuhλ,n kLp3 k∇udn kLp2 .
While it follows form (2.11) and (3.15) that
o
n Z t
h
h
uλ,n =un,L exp − (λ1 f1,n (t′ ) + λ2 f2,n (t′ ) + λ3 f3,n (t′ )) dt′
0
Z t
o
n Z t
(3.17)
µ(t−s)∆
(λ1 f1,n (t′ ) + λ2 f2,n (t′ ) + λ3 f3,n (t′ )) dt′
e
exp −
+
s
0
h
× −un ∇uλ,n + µan ∆uhλ,n − (1 + an )∇h Πλ,n ds.
Applying Lemma 3.2 for ε = 0 and r1 = 2r gives
1
d
p3 + µ 2
µ 2p2 ktγ1 uhλ,n kL∞
t (L )
+ 2pd
3
1
d
ktβ1 ∇uhλ,n kL2r
p2
t (L )
+
d
β1
p3 + µ 2 2p3 kt
∇uhn,L kL2r
≤µ 2p2 ktγ1 uhn,L kL∞
p2
t (L )
t (L )
Z t
+ exp − (λ1 f1,n (t′ ) + λ2 f2,n (t′ ) + λ3 f3,n (t′ )) dt′
s
α1
× s −un ∇uhλ,n + µan ∆uhλ,n − (1 + an )∇h Πλ,n L2r (Lp1 ) ,
t
for indices β1 , γ1 given by (1.10).
Similarly, it follows from Lemma 3.1 and Lemma 3.3 for ε = 0 and r1 = 2r that
1
d
2
µ 2p2 ktγ2 uhλ,n kL2r
p3 + µ
t (L )
+ 2pd
d
3
1
α1
p2 + µkt
ktβ2 ∇uhλ,n kL∞
∆uhλ,n kL2r
p1
t (L )
t (L )
+
d
α1
2 2p3
p2 + µkt
ktβ2 ∇uhn,L kL∞
∆uhn,L kL2r
≤µ 2p2 ktγ2 uhn,L kL2r
p3 + µ
p1
t (L )
t (L )
t (L )
Z t
+ exp − (λ1 f1,n (t′ ) + λ2 f2,n (t′ ) + λ3 f3,n (t′ )) dt′
s
α1
× s −un ∇uhλ,n + µan ∆uhλ,n − (1 + an )∇h Πλ,n L2r (Lp1 ) ,
t
for indices α1 , β2 , γ2 given by (1.10).
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
As a consequence, we obtain, by using (2.7), that
1
d
+ d
γ2 h
2 2p3
p3 + kt
uλ,n kL2r
ktβ1 ∇uhλ,n kL2r
µ 2p2 ktγ1 uhλ,n kL∞
p3 ) + µ
p2
(L
t (L )
t
t (L )
p2
+ µktα1 ∆uhλ,n kL2r
+ ktβ2 ∇uhλ,n kL∞
p1
t (L )
t (L )
n
− 1 (1− pd )
α
h
h
1 ku k
∞ kt 1 ∆u
≤µ 2
p1
λ,n kL2r
−1+ pd + C µka0 kL
0
t (L )
Ḃp
1 ,2r
1
β1
p3 kt
+ ktα1 ∆udλ,n kL2r
+ ktγ1 uhn kL∞
∇uhλ,n kL2r
p1
p2
t (L )
t (L )
t (L )
Z t
Rt
′
′
′
β2
h
2r
+
e−2r s (λ1 f1,n (t )+λ2 f2,n (t )) dt ksγ2 udn (s)k2r
Lp3 ks ∇uλ,n (s)kLp2
(3.18)
0
2r1 o
β1
d
2r
.
+ ksγ1 uhλ,n k2r
ks
∇u
(s)k
p
p
L 3
n
L 2 ds
Then as Cka0 kL∞ ≤ 21 , and
ksα1 ∆udλ,n kL2r
p1 ≤
t (L )
Z
Z
t
e−2λ1 r
e−2λ2 r
1
(2λ3 r) 2r
,
Rt
s
f1,n (t′ ) dt′
β1
d
2r
ksγ1 uhλ,n k2r
Lp3 kt ∇un (s)kLp2 ds
1
≤
Rt
f2,n (t′ ) dt′
2r
β2
h
ksγ2 udn (s)k2r
Lp3 kt ∇uλ,n (s)kLp2 ds
1
0
t
1
s
0
2r
2r
1
γ1 h
p3 ,
uλ,n kL∞
1 ks
t (L )
(2λ1 r) 2r
1
β2
p2 ,
∇uhλ,n kL∞
≤
1 ks
t (L )
(2λ2 r) 2r
we infer from (3.18) that
1
d
+ d
γ2 h
2 2p3
p3 ) + kt
p2
k
u
ktβ2 ∇uhλ,n kL∞
µ 2p2 ktγ1 uhλ,n kL∞
2r (Lp3 ) + µ
(L
λ,n
L
t
t (L )
t
+ ktβ1 ∇uhλ,n kL2r
+ µktα1 ∆uhλ,n kL2r
p2
p1
t (L )
t (L )
n
µ
− 1 (1− pd )
γ h
β
h
1 ku0 k
∞ + kt 1 u kL∞ (Lp3 ) kt 1 ∇u
≤µ 2
p2
−1+ pd + C
1 ka0 kL
n t
λ,n kL2r
t (L )
2r
Ḃp1 ,r 1
(2λ3 r)
o
1
1
γ1 h
β2
h
∞ (Lp3 ) +
∞ (Lp2 ) .
k
k
kt
u
kt
∇u
+
L
L
1
1
λ,n t
λ,n t
(2λ1 r) 2r
(2λ2 r) 2r
Taking λ1 =
(4C)2r
d r
2rµ p2
, λ2 =
(4C)2r
(1+ pd )r
3
2rµ
and λ3 =
C 2r
( d −1)r
2rµ p1
in the above inequality results in
1
3 γ1 h
+ d
γ2 h
2 2p3 ktβ1 ∇uh k
p3 + kt
kt uλ,n kL∞
uλ,n kL2r
p2
λ,n L2r
(Lp3 ) + µ
t (L )
t (L )
t
4
3
α1
p2 ) + µkt
+ ktβ2 ∇uhλ,n kL∞
∆uhλ,n kL2r
p1
(L
t
t (L )
4
1 d
( −1)
β1
p3 kt
∇uhλ,n kL2r
kuh0 k −1+ pd + µka0 kL∞ + Cktγ1 uhn kL∞
≤Cµ 2 p1
p2 .
t (L )
t (L )
d
µ 2p2
(3.19)
Ḃp,r
Let c1 be a small enough positive constant, which will be determined later on, we denote
(3.20)
n
1
def
1− 2pd
− d
β
h
2 kt 1 ∇u k 2r
p3 ) + µ
T̄n = sup T ≤ Tn∗ ; µ 2 2p3 ktγ1 uhn kL∞
(L
n LT (Lp2 )
T
o
3
(1− pd ) α
h
1 kt ∆u k 2r
+ µ2
n L (Lp1 ) ≤ c1 µ .
T
17
18
J. HUANG, M. PAICU, AND P. ZHANG
We shall prove that T̄n = Tn∗ = ∞ as long as we take Cr sufficiently large and c0 sufficiently small
in (1.6). In fact, if T̄n < Tn∗ , it follows from (3.15) and (3.19) that
1
µ2
1− 2pd
γ h
β
h
2 kt 1 ∇u k 2r
p3 ) + kt 2 u k 2r
ktγ1 uhn kL∞
p3 ) + µ
(L
n
n Lt (Lp2 )
(L
L
t
t
1
d
(3− p ) α1
1 kt
p2
+ ktβ2 ∇uhn kL∞
+ µ2
∆uhn kL2r
p1
t (L )
t (L )
Z
n
t
− d r
µ p2 ksβ1 ∇udn (s)k2r
≤C kuh0 k −1+ dp + µka0 kL∞ exp −Cr
L p2
(1− pd )
3
(3.21)
Ḃp,r
0
+µ
−(1+ pd )r
3
ksγ2 udn (s)k2r
L p3 + µ
(1− pd )r
1
o
ksα ∆udn (s)k2r
ds .
p
1
L
On the other hand, it follows from a similar derivation of (3.18) that
(3.22)
1
d
+ d
γ2 d
2 2p3
p3 ) + kt
ktβ1 ∇udn kL2r
u
k
2r (Lp3 ) + µ
µ 2p2 ktγ1 udn kL∞
p2
(L
n
L
t
t (L )
t
α1
d
β2
d
p2
+ µkt ∆un kL2r
+ kt ∇un kL∞
p1
t (L )
t (L )
n
− 1 (1− pd )
α1
d
1 ku k
≤µ 2
+ C µka0 kL∞ ktα1 ∆uhn kL2r
∆udn kL2r
p1 + kt
p1
−1+ d
0
t (L )
t (L )
p
Ḃp,r
β
1
p3 ) kt
∇uhn kL2r
+ ktγ1 udn kL∞
p2
(L
t
t (L )
o
β1
p3 kt
+ ktγ1 uhn kL∞
for t < Tn∗ ,
∇udn kL2r
p2
t (L )
t (L )
+
p3
ktγ1 uhn kL∞
t (L )
from which, (3.20) and taking c0 small enough in (1.6) so that Cka0 kL∞ ≤ 21 , we deduce
d
1
+ d
γ d
2 2p3
p3 + kt 2 u k 2r
µ 2p2 ktγ1 udn kL∞
ktβ1 ∇udn kL2r
p2
n Lt (Lp3 ) + µ
t (L )
t (L )
p2
+ ktβ2 ∇udn kL∞
+ µktα1 ∆udn kL2r
p1
t (L )
t (L )
≤ 2Cµ
− 21 (1− pd )
1
kud0 k
1
−1+ d
Ḃp,r p
+ 2Cc1 µ 2
(1+ pd )
1
(1 + ka0 kL∞ ) for
t ≤ T̄n .
which implies
1
µ2
(1− pd )
3
(3.23)
1− d
γ2 d
p3 + kt
+ µ 2p2 ktβ1 ∇udn kL2r
un kL2r
ktγ1 udn kL∞
p3
p2
t (L )
t (L )
t (L )
1
d
(3− p ) α1
1 kt
p2
+ ktβ2 ∇udn kL∞
+ µ2
∆udn kL2r
p1
t (L )
t (L )
≤ 2Ckud0 k
d
−1+ p
Ḃp,r
+ cµ
for t ≤ T̄n .
Substituting (3.23) into (3.21) leads to
1
µ2
(3.24)
1− 2pd
γ h
β
h
2 kt 1 ∇u k 2r
p3 + kt 2 u k 2r
ktγ1 uhn kL∞
n Lt (Lp3 ) + µ
n Lt (Lp2 )
t (L )
1
(3− pd ) α1
1 kt
p2
∆uhn kL2r
+ ktβ2 ∇uhn kL∞
+ µ2
p1
t (L )
t (L )
o c
n
1
≤C kuh0 k −1+ pd + µka0 kL∞ exp Cr µ−2r kud0 k2r−1+ d ≤ µ for t ≤ T̄n ,
p
2
Ḃp,r
Ḃp,r
(1− pd )
3
as long as Cr is sufficiently large and c0 small enough in (1.6). This contradicts with (3.20) and it
in turn shows that T̄n = Tn∗ . Then thanks to (3.23), (3.24) and Theorem 1.3 in [21], we conclude
that Tn∗ = ∞ and there holds (3.13). It remains to prove (3.14). Indeed, similar to (3.22), we get,
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
by applying Lemma 3.1, that
µktα2 ∆un kLr (R+ ;Lp1 ) ≤µ
− 21 (1− pd )
1
ku0 k
−1+ d
Ḃp,r p
19
n
+ C µka0 kL∞ ktα2 ∆un kLr (R+ ;Lp1 )
+ ktγ2 uhn kL2r (R+ ;Lp3 ) + ktγ2 udn kL2r (R+ ;Lp3 ) ktβ1 ∇uhn kL2r (R+ ;Lp2 )
o
,
+ ktγ2 uhn kL2r (R+ ;Lp3 ) ktβ1 ∇udn kL2r
p
2
t (L )
which along with (3.13) and the fact: Cka0 kL∞ ≤ 12 , gives rise to the first inequality of (3.14).
Along the same line, one gets the estimate of ktα2 ∇Πn kLr (R+ ;Lp1 ) in (3.14). Whereas it follows
from and (3.16) that
n
ktα1 ∇Πn kL2r (R+ ;Lp1 ) ≤ C µka0 kL∞ ktα1 ∆un kL2r (R+ ;Lp1 )
o
+ ktγ1 uhn kL∞ (R+ ;Lp3 ) ktβ1 ∇un kL2r (R+ ;Lp2 ) + ktγ2 udn kL2r (R+ ;Lp3 ) ktβ2 ∇uhn kL∞ (R+ ;Lp2 ) ,
from which and (3.13), we obtain the estimate of ktα1 ∇Πn kL2r (R+ ;Lp1 ) in (3.14). This completes
the proof of the proposition.
Now we are in a position to complete the proof to the existence part of Theorem 1.1 for the
remaining case.
dr
, d). Notice that p1 < d ensures
Proof to the existence part of Theorem 1.1 for p ∈ ( 3r−2
′
α2 r < 1, so that for any T > 0, we deduce from (3.14) that ∆un = t−α2 (tα2 ∆un ) is uniformly
bounded in Lτ1 ((0, T ); Lp1 (Rd )) for some τ1 ∈ (1, ∞). Similarly we infer from (1.10) and (3.13)
2
, and {un } is uniformly
that {∇un } is uniformly bounded in Lτ2 ((0, T ); Lp2 (Rd )) for any τ2 < 2p2p
2 −d
2 −d
3 −d
3
. Moreover as 2p2p
+ p2p
= 32 − 2pd1 and
bounded in Lτ3 ((0, T ); Lp3 (Rd )) for any τ3 < p2p
3 −d
2
3
p1 < d, we can choose τ2 and τ3 so that τ12 + τ13 = τ14 < 1. This together with (2.4) implies
that {∂t un } is uniformly bounded in Lτ1 ((0, T ); Lp1 (Rd )) + Lτ4 ((0, T ); Lp1 (Rd )) for any T < ∞,
dp1
, we conclude that there exists a subsequence of
from which, Ascoli-Arzela Theorem and p2 < d−p
1
{an , un , ∇Πn }, which we still denote by {an , un , ∇Πn } and some (a, u, ∇Π) with a ∈ L∞ (R+ × Rd ),
3
2
1
u ∈ Lτloc
(R+ ; Lp3 (Rd )) with ∇u ∈ Lτloc
(R+ ; Lp2 (Rd )) and ∆u, ∇Π ∈ Lτloc
(R+ ; Lp1 (Rd )), such that
+
d
weak ∗ in L∞
loc (R × R ),
an ⇀ a
3
3
strongly in Lτloc
(R+ ; Lploc
(Rd )),
un → u
(3.25)
2
2
∇un → ∇u strongly in Lτloc
(R+ ; Lploc
(Rd )),
∆un ⇀ ∆u and
1
∇Πn ⇀ ∇Π weakly in Lτloc
(R+ ; Lp1 (Rd )).
With (3.13), (3.14) and (3.25), we can repeat the argument at the end of Section 2 to complete the
dr
proof to the existence part of Theorem 1.1 for the case when 3r−2
< p < d.
4. The uniqueness part of Theorem 1.1
−1+ d +ε
With a little bit more regularity on the initial velocity, namely, u0 ∈ Ḃp,r p (Rd ) for some small
enough ε > 0, we can prove the uniqueness of the solution constructed in the last two sections. This
result is strongly inspired by the Lagrangian approach in [13, 14]. Nevertheless, with an almost
critical regularity for the initial velocity field, the proof here is more challenging. The main result
is listed as follows:
Theorem 4.1. Let r ∈ (1, ∞), p ∈ (1, d) and 0 < ε < min{ 1r , 1 − 1r , dp − 1}. Let a0 ∈ L∞ (Rd )
−1+ dp
and u0 ∈ Ḃp,r
−1+ dp +ε
(Rd ) ∩ Ḃp,r
(Rd ), which satisfies the nonlinear smallness condition (1.6). Then
20
J. HUANG, M. PAICU, AND P. ZHANG
(1.2) has a unique global weak solution (a, u) which satisfies (1.7-1.9) and
1
µ2
(3− qd −ε)
1
ε
1− 2qd − 2ε
ε
ε
ktβ1 ∇ukL2r (R+ ;Lq2 ) + ktβ2 ∇ukL∞ (R+ ;Lq2 )
1
ε
ε
(1− qd −ε)
3
+ µ2
ktγ1 ukL∞ (R+ ;Lq3 ) + ktγ2 ukL2r (R+ ;Lq3 )
≤ Cku0 k −1+ pd +ε exp Cr µ−2r kud0 k2r−1+ d ,
ktα1 ∆ukL2r (R+ ;Lq1 ) + µ
(4.1)
2
Ḃp,r
Ḃp,r
and
1
µ2
(4.2)
µ
(3− qd −ε)
1
ε
ktα2 ∆ukLr (R+ ;Lq1 ) ≤ Cku0 k
1
(3− qd −ε)
2
1
−1+ d +ε
Ḃp,r p
ε
p
exp Cr µ−2r kud0 k2r−1+ d ,
ε
ktα1 ∇ΠkL2r (R+ ;Lq1 ) + ktα2 ∇ΠkLr (R+ ;Lq1 )
≤ Cηku0 k −1+ pd +ε exp Cr µ−2r kud0 k2r−1+ d ,
dr
) < q1 <
where q1 , q2 , q3 satisfy max(p, 2r−1
αε1 , αε2 , β1ε , β2ε , γ1ε , γ2ε are determined by
(4.3)
p
Ḃp,r
Ḃp,r
1
αε1 = (3 −
2
1
αε2 = (3 −
2
d
− ε) −
q1
d
− ε) −
q1
1
,
2r
1
,
r
d
1−ε ,
and
p
Ḃp,r
dr
r−1
< q3 < ∞ so that
1
d
1
(2 −
− ε) − ,
2
q2
2r
1
d
β2ε = (2 −
− ε),
2
q2
β1ε =
1
q2
+ q13 =
1
q1 ,
the indices
1
d
γ1ε = (1 −
− ε),
2
q3
1
d
1
γ2ε = (1 −
− ε) − .
2
q3
2r
d
Remark 4.1. To prove the uniqueness part of Theorem 4.1, we shall choose p1 = 1+ε
in (1.10).
dr
) < p1 < d with 0 < ε < min(1 − 1r , dp − 1). Then by virtue
This choice of p1 satisfies max(p, 2r−1
of (1.8), ∆u = t−α2 (tα2 ∆u) ∈ Lτ1 ((0, T ); Lp1 (Rd )) for any T < ∞ and τ1 satisfying α2 < τ11 − 1r ,
2
8
d
which implies τ1 < 2−ε
, we thus take τ1 = 8−ε
. While as q1 < 1−ε
in Theorem 4.1, we can choose
1
∞
q1 > d in order to get ∇u ∈ Lloc (L ) (see Lemma 4.1 below), and this is the reason we need an
d
2d
< 1−ε
additional regularity on u0 for the uniqueness part of Theorem 1.1. We shall choose q1 = 2−ε
ε
d
1
1
p
ε
τ
in (4.3) later on. Then in order that ∆u ∈ L 1 ((0, T ); L 1 (R )), we have α2 < τ ε − r , and hence
we take τ1ε =
8
8−ε
<
1
4
4−ε .
It follows from the existence proof of Theorem 1.1 that: in order to prove the solution constructed
in the last two sections satisfies (4.1) and (4.2), we only need to prove that the same inequalities
hold for the approximate solutions (an , un , ∇Πn ) determined by (2.4).
−3+ dp +ε
We now turn to the uniform estimate of (un , ∇Πn )n∈N when the initial velocity u0 ∈ Ḃp,r
for some ε ∈ (0, min{ 1r , 1 − 1r , dp − 1}). Notice that for all q1 ≥ p and r1 ≥ r, ∆u0 ∈ Ḃ
1
then we deduce from Proposition 2.1 that t 2
we choose q2 >
d
2−ε
and q3 >
1
(1−
d
d
1−ε
with
1
q2
(3− qd −ε)− r1
1
+
1
q3
1
=
−3+ qd +ε
1
q1 ,r1
(Rd )
(Rd ),
△et△ u0 ∈ Lr1 (R+ ; Lq1 (Rd )). Similarly,
1
q1 ,
1
there holds t 2
(2− qd −ε)− r1
2
−ε)
1
∇et△ u0 ∈
q3
et△ u0 ∈ L∞ (R+ ; Lq3 (Rd )). We thus take r1 = 2r > 2 and q1 , q2
Lr1 (R+ ; Lq2 (Rd )) and t 2
dr
d
dr
and q3 satisfying max(p, 2r−1
) < q1 < 1−ε
, r−1
< q3 < ∞, and q12 + q13 = q11 . We shall investigate
the uniform estimate to the solutions (an , un , ∇Πn ) of (2.4) in the following functional space:
n
ε
X = u : a ∈ L∞ (R+ × Rd ), tγ1 u ∈ L∞ (R+ ; Lq3 (Rd )),
(4.4)
ε
tγ2 u ∈ L2r (R+ ; Lq3 (Rd )),
ε
tβ2 ∇u ∈ L∞ (R+ ; Lq2 (Rd ))
ε
tβ1 ∇u ∈ L2r (R+ ; Lq2 (Rd )),
o
ε
tα1 △u ∈ L2r (R+ ; Lq1 (Rd )) ,
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
21
with the indices αε1 , β1ε , β2ε , γ1ε , γ2ε being determined by (4.3). Note that αε1 6= βiε + γiε , but
1
1
and γ̄2 = 12 (2 − qd2 ) − 2r
. Then we can apply
αε1 = β̄1 + γ1ε = β2ε + γ̄2 with β̄1 = 12 (1 − qd3 ) − 2r
Proposition 3.1 to prove the following proposition concerning the uniform time-weighted bounds of
(an , un , ∇Πn ).
Proposition 4.1. Under the assumptions of Theorem 4.1, (2.4) has a unique global smooth solution
(an , un , ∇Πn ) which satisfies
1
ε
ε
(3− qd −ε) αε1
1− d − ε
1
µ2
kt ∆un kL2r (R+ ;Lq1 ) + µ 2q2 2 ktβ1 ∇un kL2r (R+ ;Lq2 ) + ktβ2 ∇un kL∞ (R+ ;Lq2 )
1
ε
ε
(1− qd −ε)
3
+ µ2
ktγ1 un kL∞ (R+ ;Lq3 ) + ktγ2 un kL2r (R+ ;Lq3 )
(4.5)
≤ Cku0 k −1+ pd +ε exp Cr µ−2r kud0 k2r−1+ d ,
Ḃp,r
Ḃp,r
and
1
µ2
(4.6)
1
µ2
(3− qd −ε)
1
(3− qd −ε)
1
ε
ktα2 ∆un kLr (R+ ;Lq1 ) ≤ Cku0 k
−1+ d +ε
Ḃp,r p
p
exp Cr µ−2r kud0 k2r−1+ d ,
p
Ḃp,r
ε
ε
ktα1 ∇Πn kL2r (R+ ;Lq1 ) + ktα2 ∇Πn kLr (R+ ;Lq1 )
≤ Cηku0 k −1+ pd +ε exp Cr µ−2r kud0 k2r−1+ d ,
Ḃp,r
Ḃp,r
p
for some constant C and αεi , βiε , γiε , i = 1, 2, given by (4.3).
Proof. For N large enough, we already proved in Proposition 3.1 that (2.4) has a unique global
smooth solution (an , un , ∇Πn ). It remains to prove (4.5) and (4.6). In order to do so, for λ1 , λ2 > 0,
we denote
def
def
g1,n (t) = ktβ̄1 ∇udn (t)k2r
g2,n (t) = ktγ̄2 udn (t)k2r
Lq2 ,
Lq3 ,
Z
n
t
o
def
λ1 g1,n (t′ ) + λ2 g2,n (t′ ) dt′ ,
uλ,n (t, x) = un (t, x) exp −
(4.7)
0
and similar notation for Πλ,n (t, x). Then it follows from a similar derivation of (2.10) that
n
k∇Πλ,n (t)kLq1 ≤C µka0 kL∞ k△uλ,n kLq1 + kuhλ,n kLq3 k∇uhn kLq2
o
(4.8)
+ kudn kLq3 k∇uhλ,n kLq2 + kuhλ,n kLq3 k∇udn kLq2 .
Whereas by virtue of (2.11) and (4.7), we write
o
n Z t
uλ,n =un,L exp − (λ1 g1,n (t′ ) + λ2 g2,n (t′ )) dt′
0
Z t
o
n Z t
(4.9)
eµ(t−s)∆ exp − (λ1 g1,n (t′ ) + λ2 g2,n (t′ )) dt′
+
s
0
× −un · ∇uλ,n + µan ∆uλ,n − (1 + an )∇Πλ,n ds.
For γ1ε , β1ε given by (4.3), we get, by applying Lemma 3.2 for r1 = 2r, that
d
1
ε
q3 + µ 2
µ 2q2 ktγ1 uλ,n kL∞
t (L )
d
ε
+ 2qd
ε
3
1
ktβ1 ∇uλ,n kL2r
q2
t (L )
+
d
ε
β
q3 + µ 2 2q3 kt 1 ∇un,L k 2r
≤µ 2q2 ktγ1 un,L kL∞
Lt (Lq2 )
t (L )
Z t
+ exp − (λ1 g1,n (t′ ) + λ2 g2,n (t′ )) dt′
s
×s
αε1
−un · ∇uλ,n + µan ∆uλ,n − (1 + an )∇Πλ,n L2r (Lq1 ) .
t
22
J. HUANG, M. PAICU, AND P. ZHANG
Similarly for γ2ε , β2ε and αε1 , we deduce from Lemma 2.1 and Lemma 3.3 for r1 = 2r that
d
1
ε
2
µ 2q2 ktγ2 uλ,n kL2r
q3 + µ
t (L )
d
+ 2qd
ε
3
1
ε
ε
α
q2 + µkt 1 ∆uλ,n k 2r
ktβ2 ∇uλ,n kL∞
Lt (Lq1 )
t (L )
+
d
ε
ε
α
2
2q3
q2 + µkt 1 ∆un,L k 2r
ktβ2 ∇un,L kL∞
≤µ 2q2 ktγ2 un,L kL2r
q3 + µ
Lt (Lq1 )
t (L )
t (L )
Z t
(λ1 g1,n (t′ ) + λ2 g2,n (t′ )) dt′
+ exp −
s
αε1
× s −un · ∇uλ,n + µan ∆uλ,n − (1 + an )∇Πλ,n L2r (Lq1 ) .
t
Hence, by using (2.7), Proposition 2.1 and (4.8), we obtain
d
1
ε
ε
+ d
γε
2 2q3
q3 ) + kt 2 uλ,n k 2r
µ 2q2 ktγ1 uλ,n kL∞
ktβ1 ∇uλ,n kL2r
q3 ) + µ
q2
(L
(L
L
t
t
t (L )
ε
ε
q2
+ ktβ2 ∇uλ,n kL∞
+ µktα1 ∆uλ,n kL2r
q1
t (L )
t (L )
n 1
d
ε
− (1− q −ε)
1
ku0 k −1+ qd +ε + µka0 kL∞ ktα1 ∆uλ,n kL2r
≤C µ 2
q1
t (L )
Ḃq
1 ,2r
(4.10)
1
ε
β̄1
q3 kt
+ ktγ1 uhλ,n kL∞
∇uhn kL2r
q2
t (L )
t (L )
Z t
Rt
′
′
′
β2ε
h
2r
+
e−2r s (λ1 g1,n (t )+λ2 g2,n (t )) dt ksγ̄2 udn (s)k2r
Lq3 ks ∇uλ,n (s)kLq2
0
2r1 o
ε
β̄1
d
2r
.
+ ksγ1 uhλ,n (s)k2r
ks
ds
∇u
(s)k
q
q
3
2
L
n
L
Then as Cka0 kL∞ ≤ 21 , and
Z t
1
Rt
′
′
ε
2r
β̄1
d
2r
≤
q
kt
e−2λ1 r s g1,n (t ) dt ksγ1 uhλ,n k2r
∇u
(s)k
q
ds
L 3
n
L 2
0
Z
t
e−2λ2 r
Rt
s
g2,n (t′ ) dt′
0
ε
β2
h
2r
ksγ̄2 udn (s)k2r
Lq3 kt ∇uλ,n (s)kLq2 ds
1
2r
1
(2λ1 r)
≤
ε
1
2r
q3 ,
ksγ1 uhλ,n kL∞
t (L )
1
(2λ2 r)
ε
1
2r
q2 ,
ksβ2 ∇uhλ,n kL∞
t (L )
we infer from (4.10) that
(4.11)
1
d
ε
ε
+ d
γε
2 2q3
q3 + kt 2 uλ,n k 2r
q2
µ 2q2 ktγ1 uλ,n kL∞
ktβ2 ∇uλ,n kL∞
Lt (Lq3 ) + µ
t (L )
t (L )
ε
ε
+ ktβ1 ∇uλ,n kL2r
+ µktα1 ∆uλ,n kL2r
q2
q1
t (L )
t (L )
n 1
ε
− (1− qd −ε)
β̄1
1
q3 kt
ku0 k −1+ qd +ε + ktγ1 uhλ,n kL∞
≤C µ 2
∇uhn kL2r
q2
t (L )
t (L )
1
Ḃq1 ,r
+
1
(2λ1 r)
ε
1
2r
q3 +
ktγ1 uhλ,n kL∞
t (L )
1
(2λ2 r)
1
2r
o
ε
q2 ) .
ktβ2 ∇uhλ,n kL∞
(L
t
Recalling from (3.13) that
d
2q
ktβ̄1 ∇uhn kL2r
q2 ≤ Cc0 µ 2 ,
t (L )
so that as long as c0 is small enough in (1.6), taking λ1 =
in
d
µ 2q2
(4.12)
(4C)2r
d r
2rµ q2
, λ2 =
(4C)2r
(1+ qd )r
3
in (4.11) results
2rµ
1
1 γ1ε h
ε
+ d
γε h
2 2q3
q3 ) + kt 2 u
ktβ1 ∇uhλ,n kL2r
k
kt uλ,n kL∞
q2
2r (Lq3 ) + µ
(L
λ,n
L
t
t (L )
t
2
3 ε
αε1
h
+ ktβ2 ∇uhλ,n kL∞
q1
(Lq2 ) + µkt ∆uλ,n kL2r
t
t (L )
4
1
≤Cµ 2
( qd −1−ε)
1
ku0 k
d +ε
−1+ p
Ḃp,r
.
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
23
From (3.13), (4.7), and (4.12), we infer
d
1
ε
ε
+ d
γε
2 2q3
q3 + kt 2 un k 2r
µ 2q2 ktγ1 un kL∞
ktβ1 ∇un kL2r
q2
Lt (Lq3 ) + µ
t (L )
t (L )
ε
ε
q2
+ µktα1 ∆un kL2r
+ ktβ2 ∇un kL∞
q1
t (L )
t (L )
Z t
n
1 d
−dr
( −1−ε)
µ q2 ksβ̄1 ∇udn (s)k2r
ku0 k −1+ dp +ε exp Cr
≤Cµ 2 q1
Lq2
Ḃp,r
0
−(1+ qd )r
ksγ̄2 udn (s)k2r
Lq3 ds
o
n
1 d
( −1−ε)
ku0 k −1+ dp +ε exp Cr µ−2r kud0 k2r−1+ d ,
≤Cµ 2 q1
+µ
Ḃp,r
3
Ḃp,r
o
p
which implies (4.5). It remains to prove (4.6). In fact, we get, by applying Lemma 2.1 and (4.8),
that
n
ε
ε
− 1 (1− qd −ε)
1
ku0 k −1+ dp +ε + C µka0 kL∞ ktα2 ∆un kLr (R+ ;Lq1 )
µktα2 ∆un kLr (R+ ;Lq1 ) ≤µ 2
Ḃp,r
γ2ε
+ kt un kL2r (R+ ;Lq3 ) ktβ̄1 ∇uhn kL2r (R+ ;Lq2 )
o
ε
+ ktγ̄2 uhn kL2r (R+ ;Lq3 ) ktβ1 ∇udn kL2r (R+ ;Lq2 ) ,
which along with (3.13), (4.5) and the fact: Cka0 kL∞ ≤ 21 , gives rise to the first inequality of (4.6).
The second inequality of (4.6) follows along the same line. This completes the proof of Proposition
3.1.
To prove the uniqueness part of Theorem 4.1, we need the following lemma:
Proposition 4.2. Let α1 , β1 , β2 , γ1 , γ2 , and p1 , p2 , p3 be given by Theorem 1.1, if tα1 f, tα1 ∇g, tα1 R ∈
L2r ((0, T ); Lp1 (Rd )). Then the system

∂ v − △v + ∇P = f,


 t
div v = g,
(4.13)
∂t g = div R,



v|t=0 = 0,
has a unique solution (v, ∇P ) so that
(4.14)
γ2
p2
p3 + kt
ktγ1 vkL∞
vkL2r (Lp3 ) + ktβ1 ∇vkL2r (Lp2 ) + ktβ2 ∇vkL∞
T (L )
T (L )
T
T
+ k(tα1 ∂t v, tα1 ∇2 v, tα1 ∇P )kL2r (Lp1 ) ≤ Ck(tα1 f, tα1 ∇g, tα1 R)kL2r (Lp1 ) .
T
T
Proof. We first get, by taking space divergence on (4.13), that
∆P = div(f + ∇g − R),
which implies
k∇P kLq (Rd ) ≤ Ck(f, ∇g, R)kLq (Rd ) ,
for any q ∈ (1, ∞). On the other hand, we have
Z t
e(t−s)∆ (f − ∇P ) ds,
(4.15)
v=
0
from which, (4.13), Lemma 3.1, and Lemma 3.2 and Lemma 3.3 for ε = 0 and r1 = 2r, we deduce
(4.14).
24
J. HUANG, M. PAICU, AND P. ZHANG
Lemma 4.1. Let (a, u) be a global weak solution of (1.2), which satisfies (1.8-1.9) and (4.1-4.2).
Then for any ε ∈ (0, min{1 − 1r , dp − 1}), one has
(4.16)
k∇2 uk
+ k∇uk
8
LT8−ε (Ld )
8
LT8−ε (L∞ )
≤ CT,ε ,
ktα1 ∇ukL2r (L∞ ) + ktα1 ∇2 ukL2r (Ld ) ≤ CT,ε
for any
T
T
8
T < ∞.
8
d
2d
Proof. We first deduce from Remark 4.1 that ∇2 u ∈ L 8−ε ((0, T ); L 1+ε (Rd ))∩L 8−ε ((0, T ); L 2−ε (Rd ))
for any T < ∞. Then applying Hölder’s inequality yields
(4.17)
8
LT8−ε (Ld )
≤ k∇2 uk 3
8
d
LT8−ε (L 1+ε )
Whereas by virtue of Lemma A.1, one has
X
˙ j uk 8
k∇uk 8
.
k∇∆
LT8−ε (L∞ )
LT8−ε (L∞ )
j≤0
.
2
1
k∇2 uk
X
j≤0
+
j>0
8
d
LT8−ε (L 1+ε )
. k∇2 uk
8
d
LT8−ε (L 1+ε )
+ k∇2 uk
8
2d
LT8−ε (L 2−ε )
X
˙ j uk
2 k∇ ∆
2
jε
k∇2 uk 3
˙ j uk
k∇∆
+
≤ CT,ε .
8
LT8−ε (L∞ )
X
j>0
8
ε
˙ j uk
2−j 2 k∇2 ∆
8
2d
LT8−ε (L 2−ε )
2d
LT8−ε (L 2−ε )
≤ CT,ε ,
which together with (4.17) proves the first line of (4.16). The second line of (4.17) follows along
the same lines.
Thanks to Lemma 4.1, we can taking T small enough so that
Z T
1
k∇u(t)kL∞ dt ≤ .
(4.18)
2
0
As in [13, 14], we shall prove the uniqueness part of Theorem 4.1 by using the Lagrangian
formulation of (1.2). Toward this, we first recall some basic facts concerning Lagrangian coordinates
from [13, 14]. By virtue of (4.18), for any y ∈ Rd , the following ordinary differential equation has
a unique solution on [0, T ]
dX(t, y)
def
= u(t, X(t, y)) = v(t, y),
X(t, y)|t=0 = y.
dt
This leads to the following relation between the Eulerian coordinates x and the Lagrangian coordinates y:
Z t
(4.20)
x = X(t, y) = y +
v(τ, y) dτ.
(4.19)
0
Let Y (t, ·) be the inverse mapping of X(t, ·). Then Dx Y (t, x) = (Dy X(t, y))−1 for x = X(t, y).
Providing Dy X − Id is small enough, we have
Z t
∞
X
k
−1
(−1) ( Dy v(τ, y)dτ )k .
Dx Y = (Id + (Dy X − Id)) =
0
k=0
def
We denote A(t, y) = (∇X(t, y))−1 = ∇x Y (t, x), then we have
(4.21)
∇x u(t, x) =T A(t, x)∇y v(t, y)
By the chain rule, we also have
(4.22)
and
div u(t, x) = div(A(t, y)v(t, y)).
divy A · =T A : ∇y .
Here and in what follows, we always denote T A the transpose matrix of A.
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
25
As in [13, 14], we denote
def
(4.23)
∇u = T A · ∇y ,
def
divu = div(A·)
def
and
def
b(t, y) = a(t, X(t, y)),
def
∆u = divu ∇u ,
v(t, y) = u(t, X(t, y))
def
and P (t, y) = Π(t, X(t, y)).
Notice that for any t > 0, the solution of (1.2) obtained in Theorem 4.1 satisfies the smoothness
assumption of Proposition 2 in [14], so that (b, v, ∇P ) defined by (4.23) fulfils

b = 0,


 t
∂t v − (1 + b)(µ△u v − ∇u P ) = 0,
(4.24)
divu v = 0,



(b, v)|t=0 = (a0 , u0 ),
which is the Lagrangian formulation of (1.2). For the sake of simplicity, we shall take µ = 1 in
what follows.
We now present the proof of Theorem 4.1.
Proof of Theorem 4.1. We first deduce from the proof to the existence part of Theorem 1.1, (4.5)
and (4.6) that the global weak solution (a, u, ∇Π) constructed in Theorem 1.1 satisfies (4.1) and
(4.2). It remains to prove the uniqueness part of Theorem 4.1.
Let (ai , ui , Πi ), i = 1, 2, be two solutions of (1.2) which satisfies (1.8-1.9) and (4.1-4.2). Let
Xi , (vi , Pi ), Ai , i = 1, 2 be given by (4.20) and (4.23). We denote
def
def
δv = v2 − v1 ,
then (δv, δP ) solves
δP = P2 − P1 ,

∂ δv − △δv + ∇δP = a0 (△δv − ∇δP ) + δf1 + δf2 ,


 t
div δv = δg,
∂t δg = div δR,



δv|t=0 = 0,
(4.25)
with
def
δf1 = (1 + a0 )[(Id −T A2 )∇δP − δA∇P1 ],
def
δf2 = µ(1 + a0 ) div[(AT2 A2 − Id)∇δv + (AT2 A2 − AT1 A1 )∇v1 ],
def
δg = (Id − A2 ) : Dδv − δA : Dv1 ,
def
δR = ∂t [(Id − A2 )δv] − ∂t [δAv1 ].
In what follows, we will use repeatedly the following fact (see [13] for instance) that
Z t
Z t
X X j k−1−j
def
Dvi dτ.
Ci C2
) with Ci (t) =
(4.26)
δA(t) = ( Dδvdτ )(
0
0
k≥1 0≤j<k
Let the indices α1 , β1 , β2 , γ1 and γ2 be given by (1.10). As in Remark 4.1, we take p1 =
dr
p2 , p3 satisfying r−1
< p3 < ∞ and p12 + p13 = p11 . We denote
d
1+ε
and
def
(4.27)
β1
γ2
p3 + kt
∇δvkL2r
G(t) = ktγ1 δvkL∞
δvkL2r
p2
p3 + kt
t (L )
t (L )
t (L )
α1
p2 + k(t
+ ktβ2 ∇δvkL∞
∂t δv, tα1 ∇2 δv, tα1 ∇δP )k
t (L )
d
1+ε )
L2r
t (L
.
Then we deduce from Proposition 4.2 and (4.25) that
(4.28) G(t) ≤ C ktα1 δf1 k
d
1+ε )
L2r
t (L
+ ktα1 δf2 k
d
1+ε )
L2r
t (L
+ ktα1 ∇δgk
d
1+ε )
L2r
t (L
+ ktα1 δRk
d
1+ε )
L2r
t (L
26
J. HUANG, M. PAICU, AND P. ZHANG
as long as Cka0 kL∞ ≤ 21 .
Let us now estimate term by term on the right-hand side of (4.28). We first get, by using (1.8)
and (4.18), that
ktα1 δf1 k
α1
∞ kt
. kId −T A2 kL∞
∇δP k
t (L )
d
1+ε )
L2r
t (L
d
1+ε )
L2r
t (L
. k∇v2 kL1t (L∞ ) G(t) + k∇δvk
d
L1t (L ε )
for θ determined by
1
d
θ
1−θ
p1 + q1 .
=
d
+ kδAk
d
ε
L∞
t (L )
ktα1 ∇P1 kL2r
d
t (L )
θ
α1
1−θ
(ktα1 ∇P1 kL2r
∇P1 kL2r
,
p1 ) (kt
q1 )
t (L )
t (L )
d
However by virtue of Sobolev embedding theorem, W 1, 1+ε (Rd ) ֒→
L ε (Rd ), one has
(4.29)
k∇δvk
d
L1t (L ε
)
1
. k∇2 δvk
. t2
d
L1t (L 1+ε )
( pd −1)
1
ktα1 ∇2 δvk
d
1+ε )
L2r
t (L
,
we obtain
ktα1 δf1 k
(4.30)
d
1+ε )
L2r
t (L
. η(t)G(t),
for some positive continuous function η(t) which tends to 0 as t → 0. Along the same line, we
deduce
ktα1 ∇((Id − A2 ) : Dδv)k
d
1+ε )
L2r
t (L
. kDA2 ⊗ tα1 Dδvk
d
1+ε )
L2r
t (L
. k∇2 v2 kL1t (Ld ) ktα1 ∇δvk
+ k(Id − A2 ) ⊗ tα1 D 2 δvk
d
ε
L2r
t (L )
. ktα1 ∇2 δvk
d
1+ε )
L2r
t (L
d
1+ε )
L2r
t (L
k∇v2 kL1 (L∞ ) ktα1 ∇2 δvk 2r d
Lt (L 1+ε )
+
(k∇2 v2 kL1t (Ld ) + k∇v2 kL1 (L∞ ) ),
and
ktα1 ∇(δA : Dv1 )k 2r d
L (L 1+ε )
Z tτ
|∇2 δv|dτ ′ k
. ktα1 |∇v1 |
d
1+ε )
L2r
t (L
0
2
. ktα1 ∇v1 kL2r
∞ k∇ δvk
t (L )
d
L1t (L 1+ε )
1
. t2
( pd −1)
1
ktα1 ∇2 δvk
d
1+ε )
L2r
t (L
2
α1
+ kt |∇ v1 |
Z
τ
0
|∇δv|dτ ′ k
d
1+ε )
L2r
t (L
+ ktα1 ∇2 v1 kL2r
d k∇δvk
t (L )
d
L1t (L ε )
α1 2
∇ v1 kL2r
(ktα1 ∇v1 kL2r
d ).
∞ + kt
t (L )
t (L )
So that for all t ∈ [0, T ], we get, by using (4.16), that
ktα1 ∇δgk
(4.31)
d
1+ε )
L2r
t (L
. η(t)G(t).
The estimate to the term δf2 can be handled along the same line.
To deal with δR, we denote Dv1,2 to be the components of Dv1 and Dv2 . Then we get , by using
(4.26) once again, that
ktα1 ∂t ((Id − A2 )δv)k
d
1+ε )
L2r
t (L
. ktβ1 Dv2 tγ1 δvk
d
1+ε )
L2r
t (L
+ k(Id − A2 )tα1 ∂t δvk
d
1+ε )
L2r
t (L
γ1
p3
δvkL∞
. ktβ1 ∇v2 kL2r
p2 kt
t (L )
t (L )
+ k∇v2 kL1t (L∞ ) ktα1 ∂t δvk
d
1+ε )
L2r
t (L
,
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
27
and
ktα1 ∂t (δAv1 )k
d
1+ε )
L2r
t (L
.ktγ2 v1 tβ2 Dδvk 2r d + kδAtα1 ∂t v1 k 2r d
Lt (L 1+ε )
Lt (L 1+ε )
Z τ
|Dδv|dτ ′ tβ2 |Dv1,2 |tγ2 |v1 |k 2r d
+k
Lt (L 1+ε )
0
γ2
p2 kt
.ktβ2 ∇δvkL∞
v1 kL2r
p3 + k∇δvk
t (L )
t (L )
d
L1t (L ε )
γ2
p2 kt
ktβ2 ∇v1,2 kL∞
v1 kL2r
p3
t (L )
t (L )
1−θ
γ2
q2 kt
× ktβ2 ∇v1,2 kL∞
v1 kL2r
,
q3 )
(L
t (L )
t
+ k∇δvk
d
L1t (L ε )
for θ given by
1
d
=
θ
p1
+
1−θ
q1 .
ktα1 ∂t v1 kL2r
d
t (L )
θ
Hence it follows from (1.8) and (4.1) that
ktα1 δRk
(4.32)
d
1+ε )
L2r
t (L
. η(t)G(t).
Substituting (4.30-4.32) into (4.28) results in
G(t) . η(t)G(t),
(4.33)
which implies the uniqueness of the solution to (1.2) on a sufficiently small time interval. Then
uniqueness part of Theorem 4.1 can be completed by a bootstrap method. This concludes the proof
of Theorem 4.1.
5. Proof of Theorem 1.2
The goal of this section is to present the proof of Theorem 1.2. Indeed given a0 ∈ L∞ (Rd ) ∩
d
+ε
q
−1+ pd −ε
Bq,∞ (Rd ), u0 ∈ Ḃp,r
−1+ pd
(Rd ) ∩ Ḃp,r
(Rd ) with ka0 k
d +ε
q
L∞ ∩Bq,∞
being sufficiently small and p, q, ε
satisfying the conditions listed in Theorem 1.2, we deduce from [19] that there exists a positive
time T so that (1.2) has a unique solution (a, u, ∇Π) with
d
+ε
q
2
(Rd )),
a ∈ C([0, T ]; L∞ (Rd ) ∩ Bq,∞
(5.1)
d
and
−1+ p −s
e 1T (Ḃp,r
∇Π ∈ L
)
d
d
−1+ p −s
1+ p −s
e ∞ (Ḃp,r
e1 (Ḃp,r
u∈L
)∩L
)
T
T
for s = 0 and ε.
We denote T ∗ to be the lifespan of this solution. Then the proof of Theorem 1.2 is reduced to prove
that T ∗ = ∞.
d
d
−1+ p −ε
−1+ p
e1 (Ḃp,r
e1 (Ḃp,r
5.1. The estimate of the density. Notice from (5.1) that u ∈ L
)∩ L
), which
T
T
is not Lipschitz in the space variables, the regularity of the solution a to (1.2) may be coarsen for
positive time. In order to applying the losing derivative estimate in [6], we first need to prove that
1+ pd
1+ dp −ε
e 1 (Ḃp,r
e1 (Ḃp,r
)∩L
)
u ∈ L1 (Cµ ) for µ(r) = r(− log r)α and some α ∈ (0, 1). Indeed, let u ∈ L
T
T
def
T
for some ε > 0, we denote θ(t, x, y) = |u(t, x) − u(t, y)|. Then similar to the proof of Proposition
2.1 in [8], for any positive integer N and ε1 > 0, one has
1
θ(t, x, y) ≤|x − y|(2 + N )1− r +ε1
+2
X
j>N
X
−1≤j≤N
1
2−(2+j) (2 + j)1− r +ε1
k∇∆j u(t)kL∞
1
(2 + j)1− r +ε1
22+j k∆j u(t)kL∞
1
(2 + j)1− r +ε1
.
28
J. HUANG, M. PAICU, AND P. ZHANG
Note that 2x xα with 0 < α < 1 is a decreasing function, we get, by taking N = [1 − log |x − y|] − 2
in the above inequality, that
X k∇∆j u(t)kL∞
1
,
θ(t, x, y) ≤ C|x − y|(1 − log |x − y|)1− r +ε1
1− 1r +ε1
(2
+
j)
j≥−1
from which, we infer
Z T
sup
(5.2)
θ(t, x, y)
dt
1
|x − y|(1 − log |x − y|)1− r +ε1
X
1
1 X
1
) r′ ≤ Cε1 k∇ukLe 1 (B 0 ) ,
≤ C(
k∇∆j ukrL1 (L∞ ) ) r (
1
′
(1−
+ε
)r
∞,r
T
T
1
r
j≥−1
j≥−1 (2 + j)
0
0<|x−y|<1
where r ′ denotes the conjugate number of r.
On the other hand, it follows from Lemma A.1 that
k∇∆−1 ukL1 (L∞ ) ≤ Cε kuk
1+ d −ε
e 1 (Ḃp,r p
L
T
T
,
)
so that
k∇ukLe 1 (B 0
T
∞,r )
≤ Cε kuk
1+ d −ε
e 1 (Ḃp,r p )
L
T
+ kuk
1+ d
e 1 (Ḃp,r p )
L
T
,
which together with (5.2) and Theorem 3.33 of [6] (see also [12]) implies that
o
n
+
kuk
)
,
exp
C
(kuk
(5.3)
kak
≤
C
ka
k
d −ε
d+ε
d +ε
ε
ε 0
1+ d
1+ p
q
q
p
e ∞ (Bq,∞ 2 )
L
t
for any t < T,
d
q
<1+
d
p
e 1 (Ḃp,r
L
t
Bq,∞
and 0 < ε < 1 +
d
p
)
e 1 (Ḃp,r )
L
t
− dq .
5.2. The estimate of the pressure. We first get, by taking space divergence to the momentum
equation of (1.2), that
−∆Π = div(a∇Π) − µ div(a∆u) + divh divh (uh ⊗ uh ) + divh (ud ∂d uh ) + ∂d (uh · ∇h ud ) + ∂d2 (ud )2 .
Thanks to div u = 0, one has
∂d (uh · ∇h ud ) =∂d uh · ∇h ud + uh · ∇h ∂d ud
= divh (ud ∂d uh ) − ud ∂d divh uh + divh (uh ∂d ud ) − divh uh ∂d ud
= divh (ud ∂d uh ) − ∂d (ud divh uh ) − divh (uh divh uh ),
which gives rise to
(5.4)
−∆Π = div(a∇Π) − µ div(a∆u) + divh divh (uh ⊗ uh ) + 2 divh (ud ∂d uh )
− 3∂d (ud divh uh ) − divh (uh divh uh ).
The following proposition concerning the estimate of the pressure will be the key ingredient used
in the estimate of the horizontal component of the velocity.
Proposition 5.1. Let 1 < q ≤ p < 2d, r > 1 and 0 < ε <
−1+ d −s
1+ d −s
e ∞ (Ḃp,r p ) ∩ L
e 1 (Ḃp,r p
u∈L
T
T
(5.1). We denote
(5.5)
def
p
d
∞
e∞ q
− 1. let a ∈ L∞
T (L ) ∩ LT (Bq,∞ ),
), for s = 0 and ε, be the unique local solution of (1.2) given by
f (t) = kud (t)k2r−1+ d + 1
Ḃp,r
2d
p
r
Z t
o
n
def
for
f (t′ ) dt′
and Πλ = Π exp −λ
0
λ > 0,
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
29
d
d
−1+ p
−1+ p −ε
e1 (Ḃp,r
e 1 (Ḃp,r
)∩L
)
and similar notation for uλ . Then (5.4) has a unique solution ∇Π ∈ L
T
T
which decays to zero when |x| → ∞ so that for all t ∈ [0, T ], there holds
n
1
C
1− 1
kuhλ k 2r 1+ d −s kuhλ k 2r
≤
k∇Πλ k
d
−1+ p −s
−1+ d −s
e 1 (Ḃp,r
1 − Ckak
d
e 1 (Ḃp,r p )
e 1 (Ḃp,r p )
)
L
L
L
t
q
∞
e∞
L∞
t (L )∩Lt (Bq,∞ )
(5.6)
µkak
d
q
∞
e∞
L∞
t (L )∩Lt (Bq,∞ )
t
k
kuhλ k
−1+ d
1+ d −s
p
∞
e
e 1 (Ḃp,r p )
Lt (Ḃp,r
)
L
t
+ ku
+ µkak
d
∞
e∞ q
L∞
t (L )∩Lt (Bq,∞ )
t,f
h
kud k
1+ d −s
e 1 (Ḃp,r p )
L
t
o
for s = 0, ε,
d
provided that Ckak
d
∞
e∞ q
L∞
T (L )∩LT (Bq,∞ )
d
1+ p −s
1+ p −s
e 1 (Ḃp,r
e1 (Ḃp,r
) and L
)
≤ 21 , and where the norms of L
t
t,f
are given by Definitions A.2 and A.3 respectively.
The proof of this proposition will mainly be based on the following lemmas:
Lemma 5.1. For s < dp , one has
kghk
d −s
p
e 1 (Ḃp,r
L
)
t
. kgk
−1+ d
p
e ∞ (Ḃp,r
L
t
)
khk
1+ d −s
e 1 (Ḃp,r p
L
t
)
+ kgk
1+ d
e 1 (Ḃp,r p )
L
t
khk
d −s
−1+ p
e ∞ (Ḃp,r
L
t
.
)
Proof. Applying Bony’s decomposition ([7]) gives
X
˙ j (gh) =
˙ j Ṡj ′ g∆
˙ j ′ g Ṡj ′ +1 h ,
˙ j′ h + ∆
∆
∆
j ′ ≥j−N0
from which and Lemma A.1, we infer
X
˙ j ′ gk 1 p kṠj ′ +1 hkL∞ (L∞ )
˙ j ′ hk 1 p + k∆
˙ j (gh)k 1 p .
∞ k∆
k∆
kṠj ′ gkL∞
Lt (L )
Lt (L )
Lt (L )
t
t (L )
j ′ ≥j−N0
.
X
j ′ ≥j−N
−j ′ ( dp −s)
cj ′ ,r 2
kgk
d
−1+ p
e ∞ (Ḃp,r
L
t
0
)
khk
1+ d −s
e 1 (Ḃp,r p
L
t
+ kgk
−j( dp −s)
.cj,r 2
1+ d
e 1 (Ḃp,r p )
L
t
kgk
−1+ d
e ∞ (Ḃp,r p
L
t
)
khk
1+ d −s
e 1 (Ḃp,r p )
L
t
)
khk
−1+ d −s
e ∞ (Ḃp,r p )
L
t
+ kgk
1+ d
e 1 (Ḃp,r p )
L
t
khk
−1+ d −s
e ∞ (Ḃp,r p )
L
t
,
where
and in what follows, we always denote (cj,r )j∈Z as a generic element of ℓr (Z) so that
P
r
j∈Z cj,r = 1. Then by virtue of Definition A.2, we complete the proof of the lemma.
Lemma 5.2. Let 1 ≤ p < 2d, s ∈ − 1r , 2d
p −1) and f be given by (5.5). Then under the assumptions
of Proposition 5.1, one has
kud ∇uh k
(5.7)
−1+ d −s
e 1 (Ḃp,r p )
L
t
1
1
1− 2r
. kuh k
1+ d −s
e 1 (Ḃp,r p )
L
t
kuh k 2r
−1+ d
p −s
e 1 (Ḃp,r
L
t,f
,
)
and
(5.8)
kud ∇uh k
−1+ d
p −s
e 1 (Ḃp,r
L
t
)
. kuh k
−1+ d
p
e ∞ (Ḃp,r
L
t
)
kud k
1+ d −s
e 1 (Ḃp,r p
L
t
)
+ kud k
Proof. We first get, by applying Bony’s decomposition (A.5), that
(5.9)
−1+ d
p −s
e ∞ (Ḃp,r
L
t
ud ∇uh = Ṫud ∇uh + Ṫ∇uh ud + Ṙ(ud , ∇uh ).
)
kuh k
1+ d
e 1 (Ḃp,r p )
L
t
.
30
J. HUANG, M. PAICU, AND P. ZHANG
Applying Lemma A.1 gives
X
′
˙ j ′ uh (t′ )kLp
˙ j Tud ∇uh (t′ )kLp .
k∆
2j kṠj ′ −1 ud (t′ )kL∞ k∆
|j ′ −j|≤5
X
.
1
′
|j ′ −j|≤5
2j (2− r ) kud (t′ )k
1
−1+ d
p+r
Ḃp,r
˙ j ′ uh (t′ )kLp ,
k∆
integrating the above inequality over [0, t] and using Definition A.3, one has
˙ j Ṫud ∇uh k 1 p
k∆
Lt (L )
o1
nZ t
X
1
1
′
˙ j ′ uh (t′ )k1−1 2rp
˙ j ′ uh (t′ )kLp dt′ 2r k∆
kud (t′ )k2r−1+ d + 1 k∆
.
2j (2− r )
L (L )
0
|j ′ −j|≤5
p
Ḃp,r
t
1
1
1− 2r
d
.cj,r 2−j(−1+ p −s) kuh k
r
1+ d −s
e 1 (Ḃp,r p
L
t
kuh k 2r
d −s
−1+ p
e 1 (Ḃp,r
L
t,f
)
.
)
It follows from the same line that
X
˙ j ′ ud (t′ )kLp
˙ j Ṫ∇uh ud (t′ )kLp .
kṠj ′ −1 ∇uh (t′ )kL∞ k∆
k∆
|j ′ −j|≤5
X
−j(−1+ dp + r1 )
.2
ℓ(1+ dp )
2
ℓ≤j+4
kud (t′ )k
−1+ d + 1
Ḃp,r p r
˙ ℓ uh (t′ )kLp ,
k∆
from which and s > − 1r , we infer
˙ j Ṫ∇uh ud k 1 p
k∆
Lt (L )
o1
X ℓ(1+ d ) nZ t
1
d
1
˙ ℓ uh (t′ )k1−1 2rp
˙ ℓ uh (t′ )kLp dt′ 2r k∆
p
.2−j(−1+ p + r )
kud (t′ )k2r−1+ d + 1 k∆
2
L (L )
0
ℓ≤j+4
d
1
1− 2r
p
Ḃp,r
r
t
1
.cj,r 2−j(−1+ p −s) kuh k
1+ d −s
e 1 (Ḃp,r p )
L
t
kuh k 2r
d −s
−1+ p
e 1 (Ḃp,r
L
t,f
.
)
Finally, we deal with the remaining term in (5.9). Firstly for 2 ≤ p < 2d, as s <
applying Lemma A.1, that
d
˙ j Ṙ(ud , ∇uh ) k 1 p .2j p
k∆
Lt (L )
X
j′
2
0
j ′ ≥j−N0
X
j pd
.2
Z
t
2
Z
0
j ′ ≥j−N0
.cj,r 2
− 1, we get, by
ė ′ ud (t′ )k p k∆
˙ j ′ uh (t′ )kLp dt′
k∆
L
j
−j ′ (−2+ dp + r1 )
−j(−1+ dp −s)
2d
p
t
kud (t′ )k
1
1− 2r
kuh k
1+ d −s
e 1 (Ḃp,r p )
L
t
d+1
−1+ p
r
Ḃp,r
˙ j ′ uh (t′ )kLp dt′
k∆
1
kuh k 2r
d −s
−1+ p
e 1 (Ḃp,r
L
t,f
.
)
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
31
For 1 ≤ p < 2, we get, by applying Lemma A.1 once again, that
˙ j Ṙ(ud , ∇uh ) k 1 p
k∆
Lt (L )
Z t
X
1
′
jd(1− p )
ė ′ ud (t′ )k p k∆
˙ j ′ uh (t′ )kLp dt′
k∆
2j
.2
j
p−1
(5.10)
X
jd(1− 1p )
.2
L
0
j ′ ≥j−N0
j ′ (2+ dp −d− r1 )
2
j ′ ≥j−N
Z
0
−j(−1+ pd −s)
.cj,r 2
t
0
kud (t′ )k
d+1
−1+ p
r
Ḃp,r
˙ j ′ uh (t′ )kLp dt′
k∆
1
1
1− 2r
kuh k
1+ d −s
e 1 (Ḃp,r p )
L
t
kuh k 2r
d −s
−1+ p
e 1 (Ḃp,r
L
t,f
.
)
Whence thanks to (5.9), we finish the proof of (5.7).
On the other hand, it is easy to observe that
˙ j (Ṫ d ∇uh + Ṫ h ud )k 1 p
k∆
u
∇u
Lt (L )
X
d ∞
˙ j ′ ud k 1 p )
˙ j ′ ∇uh k 1 p + kṠj ′ −1 ∇uh kL∞ (L∞ ) k∆
.
(kṠj ′ −1 u kLt (L∞ ) k∆
Lt (L )
Lt (L )
t
|j ′ −j|≤5
−j(−1+ dp −s)
kuh k
. cj,r 2
Whereas as s <
−1+ d
e ∞ (Ḃp,r p )
L
t
kud k
1+ d −s
e 1 (Ḃp,r p )
L
t
+ kud k
−1+ d −s
e ∞ (Ḃp,r p )
L
t
kuh k
1+ d
e 1 (Ḃp,r p )
L
t
.
2d
p
− 1, for 2 ≤ p < 2d, we get, by applying Lemma A.1, that
X
d
′
ė ′ uh k ∞ p
˙ j Ṙ(ud , ∇uh ) k 1 p .2j p
˙ j ′ ud k 1 p k∆
k∆
2j k∆
j
Lt (L )
Lt (L )
Lt (L )
j ′ ≥j−N0
−j(−1+ pd −s)
.cj,r 2
kuh k
−1+ d
p
e ∞ (Ḃp,r
L
t
)
kud k
1+ d −s
e 1 (Ḃp,r p
L
t
.
)
Along the same line to the proof of (5.10), we can prove the same estimate holds for 1 ≤ p < 2.
This proves (5.8) and Lemma 5.2.
d
−1+ p −s
e 1 (Ḃp,r
). Then under the assumptions
Lemma 5.3. Let 1 < q ≤ p < 2d, s < pd + dq −1 and g ∈ L
T
of Proposition 5.1, one has
kagk
d −s
−1+ p
e 1 (Ḃp,r
L
t
)
. kak
d
∞
e∞ q
L∞
t (L )∩Lt (Bq,∞ )
kgk
−1+ d
p −s
e 1 (Ḃp,r
L
t
.
)
Proof. Again thanks to Bony’s decomposition (A.5), we have
(5.11)
ag = Ṫa g + Ṫg a + Ṙ(a, g).
Applying Lemma A.1 gives
˙ j (Ṫa g)k 1 p .
k∆
Lt (L )
X
|j ′ −j|≤5
˙ j ′ gk 1 p
∞ k∆
kṠj ′ −1 akL∞
Lt (L )
t (L )
−j(−1+ pd −s)
.cj,r 2
∞ kgk
kakL∞
t (L )
d −s
−1+ p
e 1 (Ḃp,r
L
t
.
)
While as p ≥ q, applying Lemma A.1 once again gives rise to
X
˙ j (Ṫg a)k 1 p .
˙ j ′ akL∞ (Lp )
k∆
kṠj ′ −1 gkL1t (L∞ ) k∆
Lt (L )
t
|j ′ −j|≤5
−j(−1+ pd −s)
.cj,r 2
kak
d
q
e ∞ (Bq,∞
)
L
t
kgk
d −s
−1+ p
e 1 (Ḃp,r
L
t
.
)
32
J. HUANG, M. PAICU, AND P. ZHANG
Finally as s <
d
p
+
d
q
− 1, for
1
p
+
1
q
≤ 1, we get, by applying Lemma A.1, that
X
ė ′ ak ∞ q k∆
˙ j ′ gk 1 p
k∆
j
Lt (L )
Lt (L )
d
˙ j (Ṙ(a, g))k 1 p .2j q
k∆
Lt (L )
j ′ ≥j−N0
X
j dq
.2
j ′ ≥j−N
0
.cj,r 2
1
p
1
q
+
d
q
e ∞ (Bq,∞
)
L
t
−j(−1+ dp −s)
For the case when
d
d
′
cj ′ ,r 2j (1− p − q +s) kak
kak
d
q
e ∞ (Bq,∞
)
L
t
kgk
kgk
−1+ d
p −s
e 1 (Ḃp,r
L
t
d −s
−1+ p
e 1 (Ḃp,r
L
t
)
.
)
> 1, we have
X
1
˙ j (Ṙ(a, g))k 1 p .2jd(1− p )
k∆
Lt (L )
j ′ ≥j−N
X
jd(1− 1p )
.2
0
j ′ ≥j−N0
ė ′ ak
k∆
j
p
p−1 )
L∞
t (L
˙ j ′ gk 1 p
k∆
Lt (L )
′
cj ′ ,r 2j (1−d+s) kak
d
q
e ∞ (Bq,∞
)
L
t
d
.cj,r 2−j(−1+ p −s) kak
d
q
e ∞ (Bq,∞
)
L
t
kgk
kgk
d −s
−1+ p
e 1 (Ḃp,r
L
t
d −s
−1+ p
e 1 (Ḃp,r
L
t
)
.
)
Whence thanks to (5.11), we prove Lemma 5.3.
We now turn to the proof of Proposition 5.1.
Proof of Proposition 5.1. Again as both the proof of the existence and uniqueness of solutions
to (5.4) essentially follows from the estimates (5.6) for some appropriate approximate solutions.
For the sake of simplicity, we just prove (5.6) for smooth enough solutions of (5.4). Indeed thanks
to (5.4) and (5.5), we write
∇Πλ = ∇(−∆)−1 div(a∇Πλ ) + divh divh (uh ⊗ uhλ ) + 2 divh (ud ∂d uhλ ) − 3∂d (ud divh uhλ )
− divh (uh divh uhλ ) − µ divh (a∆uhλ ) − µ∂d (a∆udλ ) .
˙ j to the above equation and using Lemma A.1 leads to
Acting ∆
˙ j (uh divh uh )k 1 p
˙ j (∇Πλ )k 1 p .k∆
˙ j (a∇Πλ )k 1 p + 2j k∆
˙ j (uh ⊗ uh )k 1 p + k∆
k∆
λ Lt (L )
λ Lt (L )
Lt (L )
Lt (L )
(5.12)
d
h
d
h
˙ j (a∆u )k 1 p ,
˙ j (a∆u )k 1 p + µk∆
˙ j (u ∇u )k 1 p + µk∆
+ k∆
λ Lt (L )
λ Lt (L )
λ Lt (L )
from which, and Lemma 5.1 to Lemma 5.3, we deduce that for s = 0 and ε
n
d
d
˙ j (∇Πλ )k 1 p . cj,r 2−j(−1+ p −s) kak
µkuhλ k
k∆
d −s + µku k
d
1+ p
Lt (L )
q
+ k∇Πλ k
d −s
−1+ p
e 1 (Ḃp,r
L
t
)
∞
e∞
L∞
t (L )∩Lt (Bq,∞ )
+ kuh k
−1+ d
p
e ∞ (Ḃp,r
L
t
)
kuhλ k
1+ d −s
e 1 (Ḃp,r p
L
t
Therefore (5.6) follows as long as Ckak
5.1.
d
∞
e∞ q
L∞
T (L )∩LT (Bq,∞ )
+
)
e 1 (Ḃp,r
L
t
)
1+ d −s
e 1 (Ḃp,r p
L
t
1
1− 1
kuhλ k 2r 1+ d −s kuhλ k 2r
−1+ d −s
e 1 (Ḃp,r p )
e 1 (Ḃp,r p )
L
L
t
t,f
)
o
.
≤ 21 . This finishes the proof of Proposition
To deal with the estimate of ud , we also need the following proposition:
Proposition 5.2. Under the assumptions of Proposition 5.1, one has for s = 0 and ε
n
C
d
kuh k
≤
k∇Πk
d ku k
1+ p
−1+ d −s
−1+ d
p −s
1
1
e
e ∞ (Ḃp,r p )
e
1
−
Ckak
d
Lt (Ḃp,r )
)
L
Lt (Ḃp,r
t
q
∞
∞
∞
e
Lt (L )∩Lt (Bq,∞ )
(5.13)
o
h
h
d
+
ku
k
ku
k
+
ku
k
+ µkak
d −s
d
−1+ d
1+ p
1+ d
q
p
p −s
∞
e∞
L∞
t (L )∩Lt (Bq,∞ )
e ∞ (Ḃp,r
L
t
)
e 1 (Ḃp,r
L
t
)
e 1 (Ḃp,r
L
t
)
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
for t ≤ T provided that Ckak
d
q
∞
e∞
L∞
T (L )∩LT (Bq,∞ )
33
≤ 21 .
Proof. The proof of this proposition exactly follows from that of Proposition 5.1. In fact, taking
λ = 0 in (5.12), and then applying Lemma 5.1, Lemma 5.3, we arrive at
n
d
k∇Πk
≤C
kak
k∇Πk
+ kuh k
d
d ku k
−1+ d
−1+ d
1+ p
−1+ d −s
−s
p
q
p −s
1
∞
∞
1
1
∞
e
e
e
e
e ∞ (Ḃp,r p )
Lt (Ḃp,r
)
Lt (L )∩Lt (Bq,∞ )
Lt (Ḃp,r
)
Lt (Ḃp,r )
L
t
o
h
+ µkak
+
ku
k
kuk
d
d
d −s
−1+
1+
q
p
p
∞
e∞
L∞
t (L )∩Lt (Bq,∞ )
e ∞ (Ḃp,r
L
t
e 1 (Ḃp,r
L
t
)
for t ≤ T, from which and the fact that Ckak
d
∞
e∞ q
L∞
T (L )∩LT (Bq,∞ )
(5.13).
≤
1
2,
)
we conclude the proof of
5.3. The estimate of uh . We first deduce from the transport equation of (1.2) that
(5.14)
kakL∞
∞ ≤ ka0 kL∞
t (L )
for t < T ∗ .
Let f (t), uλ , Πλ be given by (5.5). Then thanks to (1.2), we write
∂t uhλ + λf (t)uhλ − µ∆uhλ = −u · ∇uhλ − (1 + a)∇h Πλ + µa∆uhλ .
˙ j to the above equation and then taking the L2 inner product of the resultApplying the operator ∆
h
˙ j u |p−2 ∆
˙ j uh (in the case when p ∈ (1, 2), we need to make some modification
ing equation with |∆
λ
λ
as that in [9]), we obtain
Z
1 d ˙ h
p
p
h
˙ j uh dx
˙ j uh |∆
˙ j uh |p−2 ∆
˙
∆∆
k∆j uλ (t)kLp + λf (t)k∆j uλ (t)kLp − µ
λ
λ
λ
p dt
Rd
Z
˙ j uh |p−2 ∆
˙ j (u · ∇uh ) + ∆
˙ j ((1 + a)∇h Πλ ) − µ∆
˙ j (a∆uh ) |∆
˙ j uh dx.
∆
=−
λ
λ
λ
λ
Rd
However thanks to [9] (see also [28]), there exists a positive constant c̄ so that
Z
˙ j uh |∆
˙ j uh |p−2 ∆
˙ j uh dx ≥ c̄22j k∆
˙ j uh kp p ,
∆∆
−
λ
λ
λ
λ L
Rd
whence a similar argument as that in [9] gives rise to
Z t
h ∞
˙ j uh k 1 p
˙ j uh (t′ )kLp dt′ + c̄µ22j k∆
˙
f (t′ )k∆
k∆j uλ kLt (Lp ) + λ
λ Lt (L )
λ
(5.15)
0
˙ j (u · ∇uh )k 1 p + k∆
˙ j ((1 + a)∇h Πλ )k 1 p + µk∆
˙ j (a∆uh )k 1 p .
≤k∆
λ Lt (L )
λ Lt (L )
Lt (L )
For s = 0 and ε, applying Lemma 5.2 gives
˙ j (uh · ∇h uh )k
˙ j (u · ∇uh )k 1 p ≤k∆
k∆
λ
Lt (L )
λ
L1t (Lp )
−j(−1+ dp −s)
≤Ccj,r 2
+
˙ j (ud ∂d uh )k 1 p
+ k∆
λ Lt (L )
kuh k
−1+ d
p
e ∞ (Ḃp,r
L
t
)
kuhλ k
1
1− 1
.
kuhλ k 2r 1+ d −s kuhλ k 2r
d −s
−1+
e 1 (Ḃp,r p )
e 1 (Ḃp,r p )
L
L
t
t,f
1+ d −s
e 1 (Ḃp,r p
L
t
)
And applying Lemma 5.3 and (5.14) yields
d
˙ j (a∆uh )k 1 p ≤ Ccj,r 2−j(−1+ p −s) ka0 kL∞ + kak
k∆
λ Lt (L )
d
q
e ∞ (Bq,∞
)
L
t
kuhλ k
1+ d −s
e 1 (Ḃp,r p
L
t
,
)
and
−j(−1+ dp −s)
˙ j ((1 + a)∇h Πλ )k 1 p ≤ Ccj,r 2
k∆
Lt (L )
1 + ka0 kL∞ + kak
d
q
e ∞ (Bq,∞
)
L
t
k∇h Πλ k
d −s
−1+ p
e 1 (Bp,r
L
t
.
)
34
J. HUANG, M. PAICU, AND P. ZHANG
Now let c1 be a small enough positive constant, which will be determined later on, we define T by
n
def
h
∞
T = max t ∈ [0, T ∗ ) : kuh k
−1+ d + µ ka0 kL
−1+ d −ε + ku k
e ∞ (Ḃp,r p )
e ∞ (Ḃp,r p )
L
L
t
t
(5.16)
o
h
h
+
ku
k
+
ku
k
≤
c
µ
.
+ kak
d
d
d
1
1+ p
1+ p −ε
q
e 1 (Ḃp,r
L
t
e ∞ (Bq,∞ )
L
t
In particular, (5.16) implies that ka0 kL∞ + kak
≤ c1 for t ≤ T. Taking c1 so small that
d
q
e ∞ (Bq,∞
L
)
t
Cc1 ≤ 12 , (5.6) and (5.14) ensures that
e 1 (Ḃp,r )
L
t
)
n
1
1
d
˙ j ((1 + a)∇h Πλ )k 1 p ≤Ccj,r 2−j(−1+ p −s) kuh k1− 2r d kuh k 2r
k∆
λ
λ
Lt (L )
1+ −s
e 1 (Ḃp,r p
L
t
+ µ(ka0 kL∞ + kak
h
d
q
e ∞ (Bq,∞
)
L
t
+ µ(ka0 kL∞ + kak
−1+ d
p −s
e 1 (Ḃ
L
p,1
t,f
)
) + ku k
d
d
q
e ∞ (Bq,∞
L
)
t
d
−1+ p
e ∞ (Ḃp,r
L
t
)ku k
1+ d −s
e 1 (Ḃp,r p
L
t
)
o
)
)
kuhλ k
1+ d −s
e 1 (Ḃp,r p
L
t
)
for t ≤ T.
Substituting the above estimates into (5.15), we obtain for s = 0 and ε that
kuhλ k
d −s
−1+ p
e ∞ (Ḃp,r
L
t
)
+ λkuhλ k
−1+ d
p −s
e 1 (Ḃp,r
L
t,f
)
+ c̄µkuhλ k
1+ d −s
e 1 (Ḃp,r p
L
t
n
)
c̄µ h
1
+ C 2r−1 kuhλ k
ku k
1+ d
−1+ d −s
p −s
e 1 (Ḃp,r p )
4 λ Le 1t (Ḃp,r
µ
)
L
t,f
h
h
)
+
ku
k
kuλ k
+ µ(ka0 kL∞ + kak
d
d
1+ d −s
−1+
q
e ∞ (Ḃp,r p )
e 1 (Ḃp,r p )
e ∞ (Bq,∞
)
L
L
L
t
t
t
o
d
+ µ(ka0 kL∞ + kak
for t ≤ T.
)ku
k
d
d
1+ p −s
q
≤kuh0 k
−1+ d −s
Ḃp,r p
+
e ∞ (Bq,∞ )
L
t
Taking λ =
(5.17)
2C
µ2r−1
e 1 (Ḃp,r
L
t
)
in the above inequality and thanks to (5.16), we get
kuhλ k
−1+ d −s
e ∞ (Ḃp,r p )
L
t
≤kuh0 k
−1+ d −s
Ḃp,r p
c̄
+ µkuhλ k
1+ d −s
e 1 (Ḃp,r p )
2
L
t
+ Cµ ka0 kL∞ + kak
d
q
e ∞ (Bq,∞
)
L
t
d
ku k
1+ d −s
e 1 (Ḃp,r p
L
t
for
)
t ≤ T,
provided that c1 in (5.16) is so small that Cc1 ≤ 4c̄ .
On the other hand, it is easy to observe from (5.5) and Definition A.3 that
o
n Z t
c̄
′
′
h
h
λf
(t
)
dt
+
exp
−
ku k
µku
k
d −s
−s
−1+ d
1+
e ∞ (Ḃp,r p )
e 1 (Ḃp,r p )
2
L
L
0
t
t
c̄
≤ kuhλ k
µkuhλ k
d −s +
−1+ p
1+ d −s ,
e ∞ (Ḃp,r
e 1 (Ḃp,r p )
2
L
)
L
t
t
from which, and (5.3), (5.16), (5.17), we infer that for s = 0, ε, and t ≤ T,
µ
h
+
kuh k
kak
d −s
d + ε + ku k
−s
−1+ d
1+
q 2
e ∞ (Ḃp,r p )
e 1 (Ḃp,r p )
e ∞ (Bq,∞
2
L
)
L
L
t
t
t
≤ C µ(ka0 kL∞ + ka0 k dq +ε ) + kuh0 k −1+ dp −s
(5.18)
Bq,∞
Ḃp,r
Z t
n
o
1
′
d
d
d ′ 2r
.
dt
+
ku
k
+
× exp Cε ku k
ku
(t
)k
d
d
1+ −ε
1+
−1+ d + 1
2r−1
e 1 (Ḃp,r p )
e 1 (Ḃp,r p )
L
L
0 µ
Ḃp,r p r
t
t
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
35
5.4. The estimate of ud . By virtue of (1.2), we get, by a similar derivation of (5.15), that
˙ j ud k 1 p ≤ k∆
˙ j ud kLp
˙ j ud kL∞ (Lp ) + c̄µ22j k∆
k∆
0
Lt (L )
t
(5.19)
˙ j (u · ∇ud )k 1 p + k∆
˙ j ((1 + a)∂d Π)k 1 p + µk∆
˙ j (a∆ud )k 1 p .
+ C k∆
Lt (L )
Lt (L )
Lt (L )
Applying Lemma 5.1 and Lemma 5.2 gives for s = 0 and ε that
k∆j (u · ∇ud )kL1t (Lp ) . 2j k∆j (uh ud )kL1t (Lp ) + k∆j (ud divh uh )kL1t (Lp )
−j(−1+ dp −s)
kuh k
. cj,r 2
−1+ d
e ∞ (Ḃp,r p )
L
t
kud k
+ kuh k
1+ d −s
e 1 (Ḃp,r p )
L
t
1+ d
e 1 (Ḃp,r p )
L
t
kud k
−1+ d −s
e ∞ (Ḃp,r p )
L
t
.
Whereas thanks to (5.16) and (5.14), we get, by applying Lemma 5.3 and Proposition 5.2, that
−j(−1+ dp −s)
˙ j ((1 + a)∂d Π)k 1 p ≤ Ccj,r 2
k∆
Lt (L )
−j(−1+ pd −s)
≤ Ccj,r 2
n
(1 + ka0 kL∞ + kak
d
q
e ∞ (Bq,∞
L
)
t
kuh k
kud k
d −s
−1+ p
) + kuh k
1+ d
e 1 (Ḃp,r p )
L
t
+ µ(ka0 kL∞ + kak
d
q
e ∞ (Bq,∞
)
L
t
e ∞ (Ḃp,r
L
t
−1+ d
e ∞ (Ḃp,r p )
L
t
)k∂d Πk
−1+ d
p −s
e 1 (Ḃp,r
L
t
)
kuh k
1+ d −s
e 1 (Ḃp,r p )
L
t
+ kud k
1+ d −s
e 1 (Ḃp,r p )
L
t
)
o
,
for t ≤ T. Substituting the above estimates into (5.19) leads to
(5.20)
kud k
−1+ d −s
e ∞ (Ḃp,r p )
L
t
+ c̄µkud k
1+ d −s
e 1 (Ḃp,r p )
L
t
+ µ(ka0 kL∞ + kak
d
q
e ∞ (Bq,∞
)
L
t
≤ kud0 k
−1+ d −s
Ḃp,r p
) + kuh k
d
−1+ p
e ∞ (Ḃp,r
L
t
)
n
+ C kuh k
1+ d
e 1 (Ḃp,r p )
L
t
kuh k
1+ d −s
e 1 (Ḃp,r p
L
t
)
kud k
−1+ d
p −s
e ∞ (Ḃp,r
L
t
+ kud k
1+ d −s
e 1 (Ḃp,r p
L
t
)
o
,
)
for t ≤ T and s = 0, ε.
1 c̄ , 4C in (5.16), we deduce from (5.20) that
In particular, if we take c1 ≤ min 2C
kud k
(5.21)
d −s
−1+ p
e ∞ (Ḃp,r
L
t
)
+ µc̄kud k
1+ d −s
e 1 (Ḃp,r p
L
t
)
≤ 2kud0 k
−1+ d
p −s
Ḃp,r
+ c2 µ
for t ≤ T and s = 0, ε.
5.5. The proof of Theorem 1.2. According to the arguments at the beginning of this section,
we only need to prove that T = ∞ under the assumption of (1.13). Otherwise, if T < T ∗ < ∞, we
first deduce from (5.21) that
kud k
1
−1+ d
p+r
L2r
)
t (Ḃp,r
(5.22)
1
≤Ckud k
≤Cµ
1
− 2r
(kud0 k −1+ pd
Ḃp,r
1
1− 2r
≤ Ckud k 2r
−1+ d + 1
e 2r (Ḃp,r p r )
L
t
1+ d
e 1 (Ḃp,r p )
L
t
+ c2 µ) for
kud k
−1+ d
p
e ∞ (Ḃp,r
L
t
)
t ≤ T.
Substituting (5.21) and (5.22) into (5.18) gives rise to
kuh k
−1+ d −ε
e ∞ (Ḃp,r p )
L
t
≤ C1 µka0 k
+ kuh k
−1+ d
e ∞ (Ḃp,r p )
L
t
d +ε
q
L∞ ∩Bq,∞
+ µ kak
d+ε
q 2
e ∞ (Bq,∞
L
)
t
+ kuh0 k
−1+ d −ε
−1+ d
Ḃp,r p ∩Ḃp,r p
exp
nC
+ kuh k
2
2r
µ
1+ d −ε
e 1 (Ḃp,r p )
L
t
kud0 k2r−1+ d −ε
Ḃp,r
p
+ kuh k
−1+ d
∩Ḃp,r p
o
1+ d
e 1 (Ḃp,r p )
L
t
for t ≤ T and some positive constants C1 , C2 which depends on c̄, c1 and ε. In particular, if we take
Cr,ε large enough and c0 sufficiently small in (1.13), the above inequality implies that for δ given
by (1.13)
kuh k
−1+ d
p −ε
e ∞ (Ḃp,r
L
t
+ kuh k
)
+ kuh k
1+ d −ε
e 1 (Ḃp,r p )
L
t
−1+ d
p
e ∞ (Ḃp,r
L
t
+ kuh k
)
+ µ ka0 kL∞ + kak
1+ d
e 1 (Ḃp,r p )
L
t
≤ Cδ ≤
c1
µ
2
d+ε
q 2
e ∞ (Bq,∞
L
)
t
for
t ≤ T,
36
J. HUANG, M. PAICU, AND P. ZHANG
which contradicts with (5.16). Whence we conclude that T = T ∗ = ∞, and there holds (1.14).
This completes the proof of Theorem 1.2.
Appendix A. Littlewood-Paley analysis
The proof of Theorem 1.2 requires Littlewood-Paley decomposition. Let us briefly explain how
it may be built in the case x ∈ Rd (see e.g. [6]). Let ϕ be a smooth function supported in the ring
def
def
C = {ξ ∈ Rd , 43 ≤ |ξ| ≤ 38 } and χ be a smooth function supported in the ball B = {ξ ∈ Rd , |ξ| ≤ 43 }
such that
X
X
χ(ξ) +
ϕ(2−q ξ) = 1 for all ξ ∈ Rd and
ϕ(2−j ξ) = 1 for ξ =
6 0.
q≥0
j∈Z
Then for u ∈ S ′ (Rd ), we set
(A.1)
∀ j ∈ Z,
∀ q ∈ N,
˙ j u def
∆
= ϕ(2−j D)u
and
∆q u = ϕ(2−q D)u
and
def
Ṡj u = χ(2−j D)u,
X
Sq u =
∆ℓ u, ∆−1 u = χ(D)u.
ℓ≤q−1
We have the formal decomposition
X
X
˙ j u, ∀ u ∈ S ′ (Rd )/P[Rd ] and u =
(A.2)
u=
∆
∆q u,
q≥−1
j∈Z
∀ u ∈ S ′ (Rd ),
where P[Rd ] is the set of polynomials. Moreover, the Littlewood-Paley decomposition satisfies the
property of almost orthogonality:
(A.3)
˙ j∆
˙ ℓu ≡ 0
∆
if |j − ℓ| ≥ 2 and
˙ j (Ṡℓ−1 u∆
˙ ℓ u) ≡ 0
∆
if
|j − q| ≥ 5.
We recall now the definition of homogeneous Besov spaces and Bernstein type inequalities from
[6].
Definition A.1. Let (p, r) ∈ [1, +∞]2 , s ∈ R and u ∈ Sh′ (R3 ), which means that u ∈ S ′ (R3 ) and
limj→−∞ kṠj ukL∞ = 0, we define
def
def s
s
kukBp,r
= 2qs k∆q ukLp r and Ḃp,r
(Rd ) = u ∈ Sh′ (Rd ) kukḂ s < ∞ .
p,r
ℓ
s (Rd ) can be defined in a similar way so that
Inhomogeneous Besov spaces Bp,r
def
s
= 2qs k∆q ukLp r .
kukBp,r
ℓ (N)
Lemma A.1. Let B be a ball and C a ring of Rd . A constant C exists so that for any positive real
number δ, any non negative integer k, any smooth homogeneous function σ of degree m, and any
couple of real numbers (a, b) with b ≥ a ≥ 1, there hold
1
1
Supp û ⊂ δB ⇒ sup k∂ α ukLb ≤ C k+1 δk+3( a − b ) kukLa ,
|α|=k
(A.4)
Supp û ⊂ δC ⇒ C −1−k δk kukLa ≤ sup k∂ α ukLa ≤ C 1+k δk kukLa ,
|α|=k
1
1
Supp û ⊂ δC ⇒ kσ(D)ukLb ≤ Cσ,m δm+3( a − b ) kukLa .
We shall frequently use Bony’s decomposition from [7] in the homogeneous context:
(A.5)
uv = Ṫu v + Ṙ(u, v) = Ṫu v + Ṫv u + Ṙ(u, v),
WELLPOSEDNESS OF INCOMPRESSIBLE INHOMOGENEOUS NS EQUATIONS
where
def
Ṫu v =
X
def
˙ j v,
Ṡj−1 u∆
Ṙ(u, v) =
j∈Z
def
Ṙ(u, v) =
X
j∈Z
ė v
˙ j u∆
∆
j
with
X
37
˙ j uṠj+2 v,
∆
j∈Z
ė v =
∆
j
X
def
˙ j ′ v.
∆
|j ′ −j|≤1
In order to obtain a better description of the regularizing effect of the transport-diffusion equae λ (Ḃ s (Rd )).
tion, we need to use Chemin-Lerner type spaces L
p,r
T
e λ (Ḃ s (Rd )) as the comDefinition A.2. Let (r, λ, p) ∈ [1, +∞]3 and T ∈ (0, +∞]. We define L
pr
T
pletion of C([0, T ]; S(Rd )) by the norm
Z T
r 1
def X jrs
˙ j u(t)kλLp dt λ r < ∞.
k∆
kukLe λ (Ḃ s ) =
2
T
p,r
0
j∈Z
s ).
eλ (Ḃp,r
with the usual change if r = ∞. For short, we just denote this space by L
T
As one can not use Gronwall’s inequality in the Chemin-Lerner type spaces, we [26, 27] introduced
weighted Chemin-Lerner norm in the context of anisotropic Besov spaces. To prove 1.2, we need
the following version of weighted Chemin-Lerner in the context of isentropic Besov spaces:
Definition A.3. Let (r, p) ∈ [1, +∞]2 and T ∈ (0, +∞]. Let 0 ≤ f (t) ∈ L1 (0, T ). We define the
e 1 (Ḃ s ) as
norm L
p,r
T,f
1
def X jrs
˙ j u(t)kr 1 p r .
kukLe 1 (Ḃ s ) =
2 kf (t)∆
L (L )
T,f
p,r
j∈Z
T
Acknowledgments. Part of this work was done when M. Paicu was visiting Morningside Center
of the Chinese Academy of Sciences in the Spring of 2012. We appreciate the hospitality of MCM
and the financial support from the Chinese Academy of Sciences. P. Zhang is partially supported
by NSF of China under Grant 10421101 and 10931007, the one hundred talents’ plan from Chinese
Academy of Sciences under Grant GJHZ200829 and innovation grant from National Center for
Mathematics and Interdisciplinary Sciences.
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(J. HUANG) Academy of Mathematics & Systems Science, Chinese Academy of Sciences, Beijing
100190, P. R. CHINA
E-mail address: [email protected]
(M. PAICU) Université Bordeaux 1, Institut de Mathématiques de Bordeaux, F-33405 Talence Cedex,
France
E-mail address: [email protected]
(P. ZHANG) Academy of Mathematics & Systems Science and Hua Loo-Keng Key Laboratory of
Mathematics, The Chinese Academy of Sciences, Beijing 100190, CHINA
E-mail address: [email protected]