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Nonlinear Analysis: Real World Applications 40 (2018) 290–306
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications
www.elsevier.com/locate/nonrwa
The 3D nematic liquid crystal equations with blow-up criteria in
terms of pressure✩
Qiao Liu*, Pei Wang
Key Laboratory of High Performance Computing and Stochastic Information Processing (HPCSIP),
(Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal
University, Changsha, Hunan 410081, PR China
article
info
Article history:
Received 12 April 2017
Received in revised form 6 August
2017
Accepted 16 August 2017
abstract
In this paper, we concern the 3D nematic liquid crystal equations and prove three
almost Serrin-type blow-up criteria for the breakdown of local in time smooth
solutions in terms of pressure and gradient of the orientation field. More precisely,
let T∗ be the maximal time of the local smooth solution, then T∗ < +∞ if and only
if
∫
Keywords:
Nematic liquid crystal flows
Local in time smooth solution
Blow-up criteria
Anisotropic Lebesgue spaces
T∗
0
with



∥P (·, t)∥Lp 
 q

x1
L
x2
β



+ ∥∇d(·, t)∥8L4 dt = ∞,
Lr
x3
2
1
1
1
1
1
1
+ + + = 2 and 2 ≤ p, q, r ≤ ∞, 1 − ( + + ) ≥ 0,
β
p
q
r
p
q
r
and
T∗
∫
0
with



∥∇P (·, t)∥Lp 
 q

x1
L
x2
β



+ ∥∇d(·, t)∥8L4 dt = ∞,
Lr
x3
1
1
1
1
1
1
2
+ + + = 3 and 1 ≤ p, q, r ≤ ∞, 1 − (
+
+
) ≥ 0,
β
p
q
r
2p
2q
2r
and
∫
0
with
T∗


∥∂ P (·, t)∥ γ β
 3
Lx  α
3
Lx x
1 2
+ ∥∇d(·, t)∥8L4 dt = ∞,
2
1
2
3
1
+ +
= k ∈ [2, 3) and
≤γ≤α<
.
β
γ
α
k
k−2
© 2017 Published by Elsevier Ltd.
✩
*
This work is partially supported by the National Natural Science Foundation of China (11401202).
Corresponding author.
E-mail addresses: liuqao2005@163.com (Q. Liu), 315627178@qq.com (P. Wang).
https://doi.org/10.1016/j.nonrwa.2017.08.008
1468-1218/© 2017 Published by Elsevier Ltd.
Q. Liu, P. Wang / Nonlinear Analysis: Real World Applications 40 (2018) 290–306
291
1. Introduction
Liquid crystals exhibit a phase of matter that has properties between those of a conventional liquid
and those of a solid crystal, hence, it is commonly considered as the fourth state of matter, different
from gases, liquid, and solid. To the present state of knowledge, three main types of liquid crystals are
distinguished, nematic, termed smectic and cholesteric. The nematic phase appears to be the most common,
and the hydrodynamic theory of liquid crystals in the nematic case has been established by Ericksen [1]
and Leslie [2] (see also [3]) in 1960s, which has been widely and successful used both for theoretical and
experimental research (see [4]).
In 1989, Lin [3] proposed a simplified version which still retains most of the interesting mathematical
properties (without destroying the basic nonlinear structure) of the original Ericksen–Leslie model for the
hydrodynamics of nematic liquid crystals, and the simplified version reads as follows:
⎧
in R3 × R+ ,
⎨ut − ν∆u + (u · ∇)u + ∇P = −λ div(∇d ⊙ ∇d)
2
(1.1)
d + (u · ∇)d = µ(∆d + |∇d| d)
in R3 × R+ ,
⎩ t
∇ · u = 0, |d| = 1
in R3 × R+ .
Here, u : R3 × R+ → R3 is the unknown velocity field of the flow, P : R3 × R+ → R is the scalar pressure
and d : R3 × R+ → S2 , the unit sphere in R3 , is the unknown (averaged) macroscopic/continuum molecule
orientation of the nematic liquid crystal flow, and ν, µ and λ are three positive viscosity constants. The
notation ∇d ⊙ ∇d denotes the 3 × 3 matrix whose (i, j)th entry is given by ∂i d · ∂j d (1 ≤ i, j ≤ 3).
∇ · u = 0 represents the incompressible condition. We will consider the Cauchy problem (1.1) with the initial
conditions
u|t=0 = u0 (x), and d|t=0 = d0 (x),
|d0 (x)| = 1
in R3 ,
(1.2)
and far field behaviors
u → 0,
d → d0
as
|x| → ∞.
(1.3)
Here, u0 is a given initial velocity with ∇ · u0 = 0 in distribution sense, d0 : R3 → S2 is a given initial liquid
crystal orientation field, and d0 is a constant vector with |d0 | = 1. Since the concrete values of ν, µ and λ
play no role in our discussion, for simplicity, we shall assume them to be all equal to one throughout this
paper.
Mathematically, (1.1)–(1.3) is a strongly coupled system between the incompressible Navier–Stokes
equations (i.e., d ≡ d¯0 ) and the transported heat flows of harmonic map (i.e., u ≡ 0). Hence, it is full
2
of challenges to study its mathematical analysis. If |∇d| d in the second equation of (1.1) is replaced by
(1−|d|2 )d
f (d) =
(ε > 0), the corresponding system has been studied in a series of papers by Lin and Liu [5,6].
ε
In two independent papers [7] and [8], the authors established global existence of Leray–Hopf type weak
solutions to (1.1)–(1.3) on bounded domain in R2 under suitable initial and boundary value conditions. The
uniqueness of such weak solutions was subsequently obtained by Lin and Wang [9] and Xu and Zhang [10].
When the space dimension is three, Lin–Wang [11] obtained the existence of global weak solutions recently
when the initial data (u0 , n0 ) ∈ L2 × H 1 with the initial director field d0 in the upper hemisphere S2+ .
The global existence of weak solutions to (1.1) with general initial data in dimensions three is still an open
problem (see [4]). The global existence and uniqueness of strong solutions to system (1.1)–(1.3) with small
initial data were also considered by many authors, we refer the readers to see [12–16] and the references
cited there.
Notice that the strong solutions of the heat flow of harmonic maps (i.e., u ≡ 0) must be blowing up at
finite time (see [17]), one cannot expect that (1.1)–(1.3) has a global smooth solution with general initial
Q. Liu, P. Wang / Nonlinear Analysis: Real World Applications 40 (2018) 290–306
292
data. Hence, the development of blow-up/nonblow-up theory is of major importance for both theoretical
and practical purpose. Local existence of smooth solutions of (1.1)–(1.3) has been announced in [15,18].
That is, for any u0 ∈ H s (R3 , R3 ) with ∇ · u0 = 0 and d0 ∈ H s+1 (R3 , S2 ) for s ≥ 3, then there exists
0 < T∗ < +∞ depending only on the initial value such that (1.1)–(1.3) has a unique local smooth solution
(u, d) ∈ R3 × [0, T∗ ) satisfying
(u, d) ∈ C([0, T ]; H s (R3 , R3 ) × H s+1 (R3 , S2 )) ∩ C 1 ([0, T ]; H s−2 (R3 , R3 ) × H s−1 (R3 , S2 ))
(1.4)
for all 0 < T < T∗ . However, whether this smooth solution of (1.1)–(1.3) on [0, T∗ ) will lead to a singularity
at t = T∗ is an outstanding open problem. For the well-known Navier–Stokes equations with dimension
three, the Serrin conditions (see [19,20]) state that if 0 < T∗ < ∞ is the first finite singular time of the
smooth solutions u, then u does not belong to the class Lα (0, T∗ ; Lβ (R3 )) for all α2 + β3 ≤ 1, 2 < α < ∞,
3 < β < ∞. Some other blow-up criteria results including assumptions on the pressure have been given by
Chae and Lee [21], Berselli and Galdi [22], and Zhou [23,24]. More precisely, it is shown that if the solution
u blows up at time T∗ , then
∫ T∗
2
3
3
α
∥P (·, t)∥Lβ dt = ∞ with + ≤ 2 and < β ≤ ∞.
α β
2
0
Moreover, they also established that if u blows up at time T∗ , then
∫ T∗
3
2
α
∥∇P (·, t)∥Lβ dt = ∞ with + ≤ 3 and 1 < β ≤ +∞.
α
β
0
We also refer [25–30] and references therein for related works.
On the other hand, as for the heat flow of harmonic maps into S2 , Wang in [31] established a Serrin-type
condition, which implies that if the solution d blows up at time T∗ , that is
sup ∥∇d(·, t)∥Ln = ∞.
0≤t<T∗
For system (1.1)–(1.3), when space dimension n = 2, Lin, Lin and Wang obtained that the local smooth
solution (u, d) to (1.1)–(1.3) can be continued past any time T > 0 provided that there holds
∫ T
4
∥∇d(·, t)∥L4 dt < +∞.
0
Huang and Wang [18] established a blow-up criterion for classical solutions to (1.1)–(1.3) in two and three
dimensions. More precisely, 0 < T∗ < +∞ is the maximal existence time if
∫ T∗
2
∥∇ × u(·, t)∥L∞ + ∥∇d(·, t)∥L∞ dx = ∞ when dimension n = 3;
0
∫ T∗
2
∥∇d(·, t)∥L∞ dx = ∞ when dimension n = 2.
0
Recently, Liu [32] established that if (0, T∗ ] with T∗ < ∞ is the maximal time interval, then
∫ T∗
2
3
3
β
8
∥P (·, t)∥Lα + ∥∇d(·, t)∥L4 dt = ∞, with + ≤ 2 and < β ≤ ∞,
α
β
2
0
and
∫
0
T∗
β
8
∥∇P (·, t)∥Lα + ∥∇d(·, t)∥L4 dt = ∞, with
We also refer [4,33] and references therein for related works.
2
3
+ ≤ 3 and 1 ≤ β ≤ ∞.
α β
Q. Liu, P. Wang / Nonlinear Analysis: Real World Applications 40 (2018) 290–306
293
Inspired by [21–24,28,30] on the Navier–Stokes equations and [8,18,31–33] on nematic liquid crystal flows,
the purpose of this paper is to establish blow-up criteria for local smooth solutions to system (1.1)–(1.3) in
terms of the pressure and the orientation field on the anisotropic Lebesgue spaces. Firstly, let us recall the
following definition of the anisotropic Lebesgue spaces:
Definition 1.1. Let 1 ≤ p, q, r ≤ ∞. We say that a function f belongs to Lp (Rx1 ; Lq (Rx2 ; Lr (Rx3 ))) if f
is measurable on R3 and the following norm is finite:



∥f ∥ p 
Lx  q

1 L
x2
⎛




∫ (∫ (∫
p
|f (x1 , x2 , x3 )| dx1
:≜ ⎝
Lrx
3
R
R
) rq
) pq
dx2
⎞ r1
dx3 ⎠ .
R
Now, we state our main result as follows:
Theorem 1.2. For u0 ∈ H 3 (R3 , R3 ) with ∇ · u0 = 0 and d0 ∈ H 4 (R3 , S2 ), let T∗ > 0 be the maximum
value such that the nematic liquid crystal flow (1.1)–(1.3) has a unique solution (u, d) satisfying (1.4). If
T∗ < +∞, then
∫ T∗ 
 
β

8

∥P (·, t)∥ p 
+ ∥∇d(·, t)∥L4 dt = ∞,

Lx
q 

1
0
Lx
2
Lrx
3
2
1 1 1
1 1 1
with + + + = 2 and 2 ≤ p, q, r ≤ ∞, 1 − ( + + ) ≥ 0,
β
p q
r
p q
r
(1.5)
and
T∗
∫
0
with



∥∇P (·, t)∥ p 
Lx  q

1 L
x2
β



8
+ ∥∇d(·, t)∥L4 dt = ∞,
Lrx
3
2
1 1 1
1
1
1
+ + + = 3 and 1 ≤ p, q, r ≤ ∞, 1 − ( +
+ ) ≥ 0,
β
p q
r
2p 2q 2r
(1.6)
and
∫
0
with
T∗

β


∥∂3 P (·, t)∥Lγx  α
3
Lx x
1 2
8
+ ∥∇d(·, t)∥L4 dt = ∞,
1
2
3
1
2
+ + = k ∈ [2, 3) and ≤ γ ≤ α <
.
β
γ
α
k
k−2
(1.7)
Remark 1.3. 1. Our main observation is that if (1.5) (or (1.6) or (1.7)) is not true, then it holds that (see
Lemma 2.4)
∫ T∗
4
4
(∥u(·, t)∥L12 + ∥∇d(·, t)∥L12 )dt ≤ C̃,
0
for some positive constant C̃.
2. When d ≡ d0 , system (1.1) becomes the Navier–Stokes equations, and the criteria (1.5)–(1.7) are still
hold and new for the Navier–Stokes equations. Moreover, Theorem 1.2 can be understood as a generalization
of the results from [21,23,24,28].
The remaining of the paper is devoted to proving Theorem 1.2. Throughout the paper, the norms of the
usual Lebesgue spaces Lp (R3 ) are denoted by ∥ · ∥Lp , while the directional derivatives of a function ϕ are
∂ϕ
denoted by ∂i ϕ = ∂x
(i = 1, 2, 3). C is the generic positive constant which may depend on the norms of the
i
initial data and may change from line to line.
Q. Liu, P. Wang / Nonlinear Analysis: Real World Applications 40 (2018) 290–306
294
2. The proof of Theorem 1.2
In this section, we shall give the proof of Theorem 1.2. In order to do it, let us first give the following two
lemmas, which can be viewed as the generalization of the Sobolev and Ladyzhenskaya inequality.
Lemma 2.1. Let 2 ≤ p, q, r ≤ ∞ and 1 − ( p1 + 1q + 1r ) ≥ 0. Then there exists a positive constant C such
that for every f ∈ C0∞ (R3 )








1
1
1
1−( 1 + 1 + 1 )



(2.1)
≤ C∥∂1 f ∥Lp 2 ∥∂2 f ∥Lq 2 ∥∂3 f ∥Lr 2 ∥f ∥L2 p q r .
∥f ∥ 2p  2q 

 2r
p−2 
Lx
q−2 

1
Lx
r−2
2
Lx
3
2p
1
1
1
= ∞ if p = 2. Let 1 ≤ p, q, r ≤ ∞ and 1 − ( 2p
+ 2q
+ 2r
) ≥ 0. Then there exists a
Here, we define p−2
∞
3
positive constant C such that for every f ∈ C0 (R )








1
1
1
1−( 1 + 1 + 1 )



≤ C∥∂1 f ∥L2p2 ∥∂2 f ∥L2q2 ∥∂3 f ∥L2r2 ∥f ∥L2 2p 2q 2r .
(2.2)
∥f ∥ 2p  2q 
 2r

p−1 
Lx
q−1 

1
Lx
r−1
2
2p
p−1
Here, we define
Lx
3
= ∞ if p = 1.
Proof . We only give the proof of (2.1), and (2.2) can be obtained in a similar way. Let Λp1 be the Fourier
multiplier defined as
∫
p
F1 (Λp1 f )(ξ1 , x2 , x3 ) = |ξ1 | F1 f (ξ1 , x2 , x3 ) with F1 f (ξ1 , x2 , x3 ) =
e−iξ1 x1 f (x1 , x2 , x3 )dx1 ,
R
and analogically Λq2 and Λr3 . Then according to the Sobolev embedding theorem, the Minkowski inequality
and the Hölder inequality, one can deduce that












 





 1 






1 


 p 




 
Λ p f  2q 
≤ 
Λ
f
≤





∥f ∥ 2p  2q 
 1  2  2q 
 1  q−2  





 2r

p−2 
Lx
Lx
Lx
2r
2r
q−2 
q−2 




1 Lx
1
2
Lx
L2
x1 L r−2
2
2
Lxr−2
Lxr−2
x3
3
3







 1 1 



1
1
1
1
 1 q p 
 q p 
 q p 


r

≤ 
≤ 
Λ2 Λ1 f 
Λ2 Λ1 f 
≤

2r 




Λ3 Λ2 Λ1 f  2 .
2r



 r−2
L2
L r−2
L
x1 x2
Lx
L2
x1 x2
x3
3
By using the Plancherel Theorem and the Hölder inequality, it follows that


 1 1q p1 
Λ r Λ Λ f 
 3 2 1 
(∫
2
2
2
) 12
2
|ξ1 | p |ξ2 | q |ξ3 | r |Ff (ξ1 , ξ2 , ξ3 )| dξ1 dξ2 dξ3
≤C
R3
L2
(∫
2
2
2
2
2
2
|ξ1 | p |Ff (ξ)| p |ξ2 | q |Ff (ξ)| q |ξ3 | r |Ff (ξ)| r |Ff (ξ)|
=C
2+2+2)
2−( p
q r
) 12
dξ1 dξ2 dξ3
R3
1−1−1
1− p
q r
≤ C∥Ff ∥L2
1
1− 1 − 1
q−r
≤ C∥f ∥L2 p
Here, Ff (ξ1 , ξ2 , ξ3 ) =
∫
R3
(∫
2
2
1 (∫
) 2p
|ξ1 | |Ff | dξ
2
1 (∫
) 2q
|ξ1 | |Ff | dξ
R3
1
2
R3
1
∥∂1 f ∥Lp 2 ∥∂2 f ∥Lq 2 ∥∂3 f ∥Lr 2 .
e−iξ·x f (x)dx. This completes the proof of Lemma 2.1.
2
|ξ1 | |Ff | dξ
R3
1
2
□
1
) 2r
Q. Liu, P. Wang / Nonlinear Analysis: Real World Applications 40 (2018) 290–306
295
Lemma 2.2. Let 1 ≤ γ, α, ξ, a, t ≤ ∞, 1 < s ≤ ∞, and 0 ≤ θ ≤ 1. Then there exists a positive constant C
such that1
⏐
 (1−θ)(s−1)
⏐∫

 1s

 θ(s−1)

s
⏐

⏐

s





⏐
f ghdx⏐⏐ ≤ C ∥∂1 f ∥Lγx  α ∥∂1 f ∥Lγx  θ(s−1)t 
⏐
∥f ∥Lξ 
1
R3
s−1
s
L2
Lx x
2 3
1
∥(∂2 , ∂3 )g∥Ls 2 ∥h∥
× ∥g∥
x1
Lx x
2 3
1
s−1
s
L2
(1−θ)(s−1)a
Lx x
2 3
1
∥(∂2 , ∂3 )h∥Ls 2 ,
(2.3)
for f, g, h ∈ C0∞ (R3 ), where γ, α, ξ, s and θ satisfying
1 1
α−1
+ =
,
a
t
α
(2.4)
θ
1−θ
1
+ =
.
(s − 1)γ
γ
ξ(γ − 1)
(2.5)
and
Proof . The proof is essentially due to [25], for the readers convenience, we give a simple proof. By using
of the Hölder inequality and Gagliardo–Nirenberg inequalities, one can deduce that
⏐
⏐ ⏐∫
⏐∫
⏐
⏐ ⏐
⏐
⏐
⏐=⏐
⏐
f
ghdx
dx
dx
f
ghdx
1
2
3⏐
⏐ ⏐ 3
⏐ 3
R
R
(∫
) 21 (∫
) 12 }
∫ {
2
2
≤C
max |f |
|g| dx1
|h| dx1
dx2 dx3
R2
x1 ∈R
{∫
≤C
R
R
} 1s {∫
s
(∫
(max |f |) dx2 dx3
|g| dx1
R2 x1 ∈R
{∫
≤C
s−1
|f |
R2
(∫
R2
|g|
|∂1 f |dx1 dx2 dx3
2s
s−1
|h|
dx1
dx2 dx3
R2
R
dx2 dx3
R
} 12 {∫ (∫
) s−1
s
} s−1
2s
s
) s−1
2
|h| dx1
dx2 dx3
R
} 1s {∫ (∫
R3
} s−1
{∫
2s
s
) s−1
2
R
2s
s−1
} 21
) s−1
s
dx2 dx3
dx1
R2
 1s



s−1
s−1
1
1



∥f ∥s−1
∥g∥L2s ∥(∂2 , ∂3 )g∥Ls 2 ∥h∥L2s ∥(∂2 , ∂3 )h∥Ls 2
γ(s−1) 

Lx x 
α
γ−1 
2 3
Lx
α−1

 1s


≤C ∥∂1 f ∥Lγx  α
1
Lx ,x
2 3
1
 1s



≤C ∥∂1 f ∥Lγx  α
1
Lx x
2 3

 1s


≤C ∥∂1 f ∥Lγx  α
1
Lx x
2 3
 1s


(1−θ)(s−1) 

∥∂1 f ∥θ(s−1)
∥f
∥
γ
ξ


L
Lx
x1
1




∥∂1 f ∥Lγx 
1
θ(s−1)
s


∥f ∥ ξ
θ(s−1)t 
L
s−1
s
α
2
L
α−1
Lx ,x
2 3
(1−θ)(s−1)
s
∥g∥



x1 
Lx x
2 3
(1−θ)(s−1)a
Lx x
2 3
1
s−1
1
∥(∂2 , ∂3 )g∥Ls 2 ∥h∥L2s ∥(∂2 , ∂3 )h∥Ls 2
s−1
1
s−1
1
∥g∥L2s ∥(∂2 , ∂3 )g∥Ls 2 ∥h∥L2s ∥(∂2 , ∂3 )h∥Ls 2 ,
where we have used (2.4) and (2.5) in the above inequality. This completes the proof of Lemma 2.1.
□
We also need to recall the following commutator and product estimates (see [19,34]), which can be used
later in the proof of Theorem 1.2.
Lemma 2.3. For α > 0, it holds that
∥Λα (f g) − f Λα g∥Lp ≤ C(∥∇f ∥Lp1 ∥Λα−1 g∥Lq1 + ∥Λα f ∥Lp2 ∥g∥Lq2 );
∥Λα (f g)∥Lp ≤ C(∥f ∥Lp1 ∥Λα g∥Lq2 + ∥Λα f ∥Lp2 ∥g∥Lq2 )
with 1 < p, p1 , p2 , q1 , q2 < ∞ such that
1
p
=
1
p1
+
1
q1
=
1
p2
+
1
q2 .
1
Here, we denote by Λ = (−∆) 2 .
1
We notice that our result (2.3), which can be viewed as a generalization of (1.10) of Cao and Titi [25], is different to Lemmas
2.1 and 2.2 of Qian [27].
296
Q. Liu, P. Wang / Nonlinear Analysis: Real World Applications 40 (2018) 290–306
By using Lemmas 2.1–2.3, we now turn to give the proof of our main result Theorem 1.2. Since we deal
with the local smooth solutions, and notice that [0, T∗ ) is the maximal existence interval of local smooth
solution associated with initial value (u0 , d0 ) ∈ H 3 (R3 , R3 )×H 4 (R3 , S2 ). We shall prove Theorem 1.2 arguing
by contradiction. Suppose, that (1.5) or (1.6) or (1.7) is not true. Then, there is 0 < M < ∞ such that
T∗
∫
0



∥P (·, t)∥ p 
Lx  q

1 L
x2
β



8
+ ∥∇d(·, t)∥L4 dt ≤ M,
Lrx
3
2
1 1 1
1 1 1
with + + + = 2 and 2 ≤ p, q, r ≤ ∞, 1 − ( + + ) ≥ 0,
β
p q
r
p q
r
(2.6)
or
T∗
∫
0



∥∇P (·, t)∥ p 

L

x1 L q
x2
β



Lrx
8
+ ∥∇d(·, t)∥L4 dt ≤ M,
3
2
1 1 1
1
1
1
with + + + = 3 and 1 ≤ p, q, r ≤ ∞, 1 − ( +
+ ) ≥ 0,
β
p q
r
2p 2q 2r
(2.7)
or
∫
T∗
0
with

β


∥∂x3 P (·, t)∥Lγx  α
3
8
Lx x
1 2
+ ∥∇d(·, t)∥L4 dt ≤ M,
2
1
2
3
1
+ + = k ∈ [2, 3) and ≤ γ ≤ α ≤
.
β
γ
α
k
k−2
(2.8)
We shall prove that if assumption (2.6) (or (2.7) or (2.8)) holds, then there holds
lim (∥u(·, t)∥H 3 + ∥d(·, t)∥H 4 ) ≤ C,
(2.9)
t→T∗−
for some positive constant C depends only on u0 , d0 , T∗ and M . The estimate (2.9) is enough to ensure the
smooth solution (u, d) that can be extended beyond time T∗ . That is to say, [0, T∗ ) is not a maximal interval
of existence, which leads to the contradiction.
Before going to derive the inequality (2.9), we need to prove the following lemma.
Lemma 2.4. For u0 ∈ H 3 (R3 , R3 ) with ∇ · u0 = 0 and d0 ∈ H 4 (R3 , S2 ), 0 < T∗ < ∞, let (u, d) be a
solution to system (1.1)–(1.3) satisfying (1.4), and suppose that there holds (2.6) (or (2.7) or (2.8)). Then
we have
sup
0<t≤T∗
(
∫
)
4
4
∥u(·, t)∥L4 + ∥∇d(·, t)∥L4 + 2
0
T∗
(
)
2 2
2 2
∥∇|u| ∥L2 + ∥∇|∇d| ∥L2 dt ≤ C̃,
(2.10)
where C̃ is a positive constant depending only on u0 , d0 and M .
Proof . We firstly differentiate (1.1)2 with respect to x, it follows that
2
∇dt − ∆∇d + ∇u · ∇d + u · ∇(∇d) = ∇(|∇d| d).
(2.11)
Q. Liu, P. Wang / Nonlinear Analysis: Real World Applications 40 (2018) 290–306
2
297
2
Multiplying both sides of Eqs. (1.1)1 and (2.11) by 4|u| u and 4|∇d| ∇d, respectively, and then integrating
the resulting equations with respect to x over R3 . After suitable integration by parts, it follows that
)
d (
2
4
4
2
2 2
2 2
∥u(·, t)∥L4 + ∥∇d(·, t)∥L4 + 4(∥|∇u| |u|∥L2 + ∥|∇2 d| |∇d|∥L2 ) + 2(∥∇|u| ∥L2 + ∥∇|∇d| ∥L2 )
dt ∫
∫
∫
2
=−4
2
∇P · |u| udx − 4
3
∫R
2
2
(div(∇d ⊙ ∇d)) · |u| udx − 4
R3
(∇u · ∇d) · |∇d| ∇ddx
R3
2
+4
∇(|∇d| d) · |∇d| ∇ddx
∫ R3
∫
∫
2
2
2
=4
P div(|u| u)dx + 4
(∇d ⊙ ∇d) · ∇(|u| u)dx + 4
(u · ∇d)∇(|∇d| ∇d)dx
R3∫
R3
R3
2
2
+4
∇(|∇d| d) · |∇d| ∇ddx
3
∫ R
∫
∫
∫
2
2
2
2
2
2
2
≤4
P u · ∇|u| dx + 4
|∇d| |∇|u| | |u|dx + 4
|∇d| |u| |∇u|dx + 4
|u||∇d| |∇|∇d| |dx
3
3
3
3
R ∫
R
R ∫
R
∫
3
2
3
6
2
+4
|u| |∇d| |∇ d|dx + 4
|∇|∇d| ||∇d| dx + 4
|∇d| dx
R3
R3
R3
:≜I1 + I2 + I3 + I4 + I5 + I6 + I7 ,
(2.12)
where we have used the fact that |d| = 1 and the following identities due to the divergence-free condition:
∫
∫
1
4
u · ∇|u| dx = 0;
2
3
3
R
∫
∫R
1
2
4
(u · ∇∇d) · |∇d| ∇ddx =
u · ∇|∇d| dx = 0;
2 R3
R3
∫
∫
∫
2
2
2
2
2
(∆u) · |u| udx = −
|∇u| |u| dx − 2
|∇|u| | |u| dx
3
3
3
R
∫R
∫R
1
2
2
2
=−
|∇u| |u| dx −
|∇|u| |dx.
2
3
3
R
R
2
(u · ∇u) · |u| udx =
In what follows, we will estimate the term Ii (i = 1, 2, . . . , 7) term by term to make sure that (2.10) holds
under the assumption (2.6) (or (2.7) or (2.8)).
Case I: If (2.6) holds:
Let us first estimate the integral I1 , before going to do it, we take div on both sides of (1.1)1 to get
∆P = − div div(u ⊗ u + ∇d ⊙ ∇d).
Therefore the Calderon–Zygmund inequality ensures
2
2
2
2
∥P ∥Lq ≤ C(∥|u| ∥Lq + ∥|∇d| ∥Lq ) = C(∥u∥L2q + ∥∇d∥L2q );
2
∥∇P ∥Lq ≤ C1 (∥|u| |∇u|∥Lq + ∥|∇ d| |∇d|∥Lq ),
(2.13)
(2.14)
for all 1 < q < ∞. Then, using the Hölder’s inequality and the Young’s inequality implies that
∫
2
P u · ∇|u| dx ≤
I1 = 4
R3
1
2 2
∥∇|u| ∥L2 + C
4
∫
2
2
|P | |u| dx.
R3
(2.15)
Q. Liu, P. Wang / Nonlinear Analysis: Real World Applications 40 (2018) 290–306
298
To estimate the second term on the right side of (2.15), by using the Hölder’s inequality and (2.1) of Lemma
2.1, one can deduce that for 2 ≤ p, q, r ≤ ∞ and 1 − ( p1 + 1q + 1r ) ≥ 0







∫
∫












 2


2
2
2

p 
∥P
∥
∥
|P | |u| dx =
|P | |P | |u| dx ≤ C 
2p


 2
∥P
|u|


Lx
q 

2q 

1 Lx
p−2 
L
R3
R3
L
2r
2 Lr
q−2
x


x3
1
Lx
2
Lxr−2
3

 
1
1
1+1+1)
1


1−( p

2
p
q
q
r
r

∥∂1 P ∥L2 ∥∂2 P ∥L2 ∥∂3 P ∥L2 ∥P ∥L2
≤C 
∥u∥L4
∥P ∥Lpx  q 
1
Lx
2
Lrx
3




p 
∥
≤C 
∥P
Lx
q

1 L




∥∇P ∥Lp 2




≤C 
∥P ∥Lpx1 Lq




(
∥|u| |∇u|∥L2 + ∥|∇2 d| |∇d|∥L2
) p1 + 1q + r1 (




p 
∥
≤C 
∥P
Lx
q

1 L




(
∥|u| |∇u|∥L2 + ∥|∇2 d| |∇d|∥L2
) p1 + 1q + r1 (
x2
x2
x2
1+1+1
q r
Lrx
3
Lrx
Lrx
1+1+1)
1−( p
q r
∥P ∥L2
2
∥u∥L4
2
2
)1−( p1 + 1q + r1 )
2
2
)2−( p1 + 1q + r1 )
∥u∥L4 + ∥∇d∥L4
2
∥u∥L4
3
∥u∥L4 + ∥∇d∥L4
3
2

 
)
)
 2−( p1 + 1q + r1 ) (

1(
2

2
4
4
2


≤
∥|u| |∇u|∥L2 + ∥|∇ d| |∇d|∥L2 + C ∥P ∥Lpx  q 
∥u∥L4 + ∥∇d∥L4 .
1 Lx
4
2 Lr
x3
Hence
2

 
)
)
 2−( p1 + 1q + r1 ) (

1(
2

4
4
2 2
2
2


∥u∥L4 + ∥∇d∥L4 .
I1 ≤
∥∇|u| ∥L2 + ∥|u| |∇u|∥L2 + ∥|∇ d| |∇d|∥L2 + C ∥P ∥Lpx  q 
1 Lx
4
2 Lr
x3
Applying the Hölder’s inequality, the Young’s inequality and the interpolation inequality, it follows that
2
2
2
I2 + I3 ≤ C(∥∇d∥L6 ∥∇|u| ∥L2 ∥u∥L6 + ∥∇d∥L6 ∥u∥L6 ∥|∇u| |u|∥L2 )
1
1
2 2
2
2
4
≤ ∥∇|u| ∥L2 + ∥|∇u||u|∥L2 + C∥u∥L6 ∥∇d∥L6
4
2
1
1
2
2
2 2
2
≤ ∥∇|u| ∥L2 + ∥|∇u||u|∥L2 + C∥u∥L4 ∥u∥L12 ∥∇d∥L4 ∥∇d∥L12
4
2
1
1
1
2 2
2
2
2
2
≤ ∥∇|u| ∥L2 + ∥|∇u||u|∥L2 + C∥u∥L4 ∥∇|u| ∥L2 2 ∥∇d∥L4 ∥∇|∇d| ∥L2
4
2
1
1
1
2 2
2
2 2
8
4
≤ ∥∇|u| ∥L2 + ∥|∇u||u|∥L2 + ∥∇|∇d| ∥L2 + C∥∇d∥L4 ∥u∥L4 .
2
2
2
Similarly,
2
2
2
I4 + I5 ≤ C(∥u∥L6 ∥∇d∥L6 ∥∇|∇d| ∥L2 + ∥u∥L6 ∥∇d∥L6 ∥|∇2 d||∇d|∥L2 )
1
1
2
2 2
2
4
≤ ∥∇|∇d| ∥L2 + ∥|∇2 d||∇d|∥L2 + C∥u∥L6 ∥∇d∥L6
4
2
1
1
1
2
2 2
2 2
8
4
≤ ∥∇|∇d| ∥L2 + ∥|∇2 d||∇d|∥L2 + ∥∇|u| ∥L2 + C∥∇d∥L4 ∥u∥L4 .
2
2
2
For the rest terms I6 and I7 , one has
3
2
6
1
2 2
6
∥∇|∇d| ∥L2 + C∥∇d∥L6
4
3
1
3
2
2 2
≤ ∥∇|∇d| ∥L2 + C∥∇d∥L4 ∥∇|∇d| ∥L2 2
4
I6 + I7 ≤ C(∥∇d∥L6 ∥∇|∇d| ∥L2 + ∥∇d∥L6 ) ≤
1
3
3
2 2
∥∇|∇d| ∥L2 + C∥∇d∥L4 ∥∇d∥L12
4
1
2 2
8
4
≤ ∥∇|∇d| ∥L2 + C∥∇d∥L4 ∥∇d∥L4 .
2
≤
Q. Liu, P. Wang / Nonlinear Analysis: Real World Applications 40 (2018) 290–306
299
Inserting all estimates of Ii (i = 1, 2, ·, 7) above into (2.12), it follows that
d
4
(∥u(·, t)∥L4
dt
⎛


⎝
≤C 
∥P ∥Lpx
1
3
3
2
4
2 2
2
2 2
+ ∥∇d(·, t)∥L4 ) + 2∥∇|u| ∥L2 + ∥|∇u| |u|∥L2 + 2∥∇|∇d| ∥L2 + ∥|∇2 d| |∇d|∥L2
2
2
⎞
2
 
1+1+1)
2−( p
 
8
4
4
q
r
+ ∥∇d∥L4 ⎠ (∥u∥L4 + ∥∇d∥L4 ).
 q 

Lx
Lrx
2
3
By using the Gronwall’s inequality to the above inequality in the interval [0, T∗ ), one obtains from (2.6)
that
4
∫
4
sup (∥u(·, t)∥L4 + ∥∇d(·, t)∥L4 ) + 2
0≤t<T∗
4
≤(∥u0 ∥L4
T∗
2
2
2
2
∥∇|u| (·, t)∥L2 + ∥∇|∇d| (·, t)∥L2 dt
0
⎧
⎞ ⎫
⎛
2
⎨ ∫ T∗ 
 

⎬
1+1+1)


2−(


4
⎝∥P ∥ p   p q r + ∥∇d∥8 4 ⎠ dt
+ ∥∇d0 ∥L4 ) exp C
L
Lx
q 

⎭
⎩
1 Lx
0
2 Lr
x3
4
≤(∥u0 ∥L4
+
4
∥∇d0 ∥L4 )eCM
≤ C̃,
where C̃ only depending on u0 , d0 and M . Thus we obtain (2.10).
Case II: If (2.7) holds:
By using the Hölder inequality and Lemma 2.1 again, one can estimate I1 for 1 ≤ p, q, r ≤ ∞ and
1
1
1
+ 2q
+ 2r
) ≤ 0 as
1 − ( 2p
∫
I1 = −4
∫
2
3
∇P · |u| udx ≤ C
R3
R3




 

 
1
|∇P | 2   
2p 


L4 
Lx
2q
1 Lx 
2




p 
≤ C∥∇P ∥L2 
∥
∥∇P
Lx
q

1 L
1
2
x2
(
≤ C ∥|u| |∇u|∥L2
≤
1
4
1
2
L2r
x3









 2 


|u|  2p  2q 


p−1 
Lx
q−1 

1
Lx
2
∥u∥L4
2r
Lxr−1
3
1 
1
1
1
1−( 1 + 1 + 1 )
2 
2p 
2p 2q 2r
2  2q 
2  2r 
2
 ∂1 |u|2 
|u|
|u|
∥u∥L4

∂

∂

|u|
 2
2
3

2
2
2
L
L
L
L
Lrx
3


) 21 


+ ∥|∇d| |∇ d|∥L2 
∥∇P ∥Lpx 
2
1
2
2
1
|∇P | 2 |∇P | 2 |u| |u|dx
R3


1

≤ C |∇P | 2 
(
∫
|∇P ||u| dx = C
∥|u| |∇u|∥L2 + ∥|∇d| |∇2 d|∥L2
q
Lx
2
1 
1+1+1
2 
3−( 1 + 1 + 1 )
2p 2q 2r
 ∇|u|2 
∥u∥L4 p q r
 2

L
Lrx
3


2 )




2
p 
+ ∇|u| 
+ C
∥
∥∇P
Lx

2
L
1
q
Lx
2
2

 3−( p1 + 1q + r1 )
4

∥u∥L4 .

Lrx
3
The estimates of Ii (i = 2, 3, . . . , 7) are similar as that in Case I. Then, inserting all estimates of
Ii (i = 1, 2, . . . , 7) into (2.12), one has
d
5
5
2
4
4
2 2
2
2 2
(∥u(·, t)∥L4 + ∥∇d(·, t)∥L4 ) + ∥∇|u| ∥L2 + ∥|∇u| |u|∥L2 + ∥∇|∇d| ∥L2 + ∥|∇2 d| |∇d|∥L2
dt
2
2
⎛
⎞
2

 
1
1
1


  3−( p + q + r )
8
4
4
≤C ⎝
+ ∥∇d∥L4 ⎠ (∥u∥L4 + ∥∇d∥L4 ).
∥∇P ∥Lpx  q 
1
Lx
2
Lrx
3
Q. Liu, P. Wang / Nonlinear Analysis: Real World Applications 40 (2018) 290–306
300
By using the Gronwall’s inequality to the above inequality in the interval [0, T∗ ), it follows from (2.7)
that
∫ T∗
2
2
2
2
4
4
∥∇|u| (·, t)∥L2 + ∥∇|∇d| (·, t)∥L2 dt
sup (∥u(·, t)∥L4 + ∥∇d(·, t)∥L4 ) + 2
0≤t<T∗
0
⎧
⎞ ⎫
⎛
2
⎬
 

⎨ ∫ T∗ 
1+1+1)


3−(


4
4
⎝∥∇P ∥ p   p q r + ∥∇d∥8 4 ⎠ dt
≤(∥u0 ∥L4 + ∥∇d0 ∥L4 ) exp C
L
Lx
q 

⎭
⎩
1 L
0
4
x2
Lrx
3
4
≤(∥u0 ∥L4 + ∥∇d0 ∥L4 )eCM ≤ C̃,
where C̃ only depending on u0 , d0 and M . Thus we get (2.10).
Case III: If (2.8) holds:
We notice that the estimates for Ii (i = 2, 3, . . . , 7) do not change. For the term I1 , we need to estimate
the second term on the right side of (2.15), notice that for 1 ≤ γ ≤ α ≤ ∞ and α > 2, by set
ξ = (1 − θ)(s − 1)a,
s=
αγ + αξ(γ − 1)
γ + αγ − α
then ξ, s and θ satisfying (2.5), and s − 1 =
a=
αξ(γ−1)−γ+α
γ+αγ−α
and
θ=
α−γ
,
αξ(γ − 1) − γ + α
(2.16)
> 0. By select that
γ + γα − α
α(γα + γ − α)
and t =
,
α(γ − 1)
α−γ
(2.17)
then a and t satisfying (2.4). Furthermore, it is easy to check that the selected γ, α, ξ, s, θ, a and t satisfying
the assumptions of Lemma 2.2, and it holds that
∫
∫
2
2
2
|P | |u| dx =
|P | |P | |u| dx
R3
R3
αγ
(s−1)(γ+αγ−α)+γ−α
1


s−1 
s−1
1


 s(γ+αγ−α)
2 s
2
∥P ∥Lξ s(γ+αγ−α)
≤C ∥∂3 P ∥Lγx  α
∥P ∥L2s ∥(∂1 , ∂2 )P ∥Ls 2 ∥|u| ∥L2s (∂1 , ∂2 )|u|  2
3
Lx x
1 2
L
αγ
(s−1)(γ+αγ−α)+γ−α


1
s−1 
s−1
1
 s(γ+αγ−α)


2 s
2
≤C ∥∂3 P ∥Lγx  α
∥P ∥Lξ s(γ+αγ−α)
∥P ∥L2s ∥∇P ∥Ls 2 ∥|u| ∥L2s ∇|u| 
2
3
Lx x
1 2
L
αγ
(s−1)(γ+αγ−α)+γ−α


s−1
s−1

 s(γ+αγ−α)
2
2
2
(∥|u| ∥L2 + ∥|∇d| ∥L2 ) s ∥|u| ∥L2s
≤C ∥∂3 P ∥Lγx  α
∥P ∥Lξ s(γ+αγ−α)
3
Lx x
1 2
1

 )1 
(

2 s
× ∥|u| |∇u|∥L2 + |∇d| |∇2 d|L2 s ∇|u| 
L2
αγ
(s−1)(γ+αγ−α)+γ−α


s−1
s−1

 s(γ+αγ−α)
2
2
2
≤C ∥∂3 P ∥Lγx  α
∥P ∥Lξ s(γ+αγ−α)
(∥|u| ∥L2 + ∥|∇d| ∥L2 ) s ∥|u| ∥L2s
3
(
×
Lx x
1 2

2 ) 1s

2

2
2
∥|u| |∇u|∥L2 + |∇d| |∇2 d|L2 + ∇|u|  2
L
≤
1
4
(
2
∥|u| |∇u|∥L2
2
(∥u∥L4
αγ
(s−1)(γ+αγ−α)+γ−α

2 )






 (s−1)(γ+αγ−α)
2
2 2

γ
+ ∇|∇d| |∇ d| L2 + ∇|u|  2 + C ∥∂3 P ∥Lx  α
∥P ∥Lξ (s−1)(γ+αγ−α)
L
2
2
∥∇d∥L4 )∥u∥L4
3
Lx x
1 2
×
+
(
αγ

2 )




1


 (s−1)(γ+αγ−α)
2
2
2 2

≤
∥|u| |∇u|∥L2 + |∇d| |∇ d| L2 + ∇|u|  2 + C ∥∂3 P ∥Lγx  α
3 Lx x
4
L
1 2
(s−1)(γ+αγ−α)+γ−α
)
(
(s−1)(γ+αγ−α)
2
2
4
4
× ∥u∥L2ξ + ∥∇d∥L2ξ
(∥u∥L4 + ∥∇d∥L4 ).
For k ∈ [2, 3), by selecting
s=
(4 − k)αγ
,
γ + αγ − α
Q. Liu, P. Wang / Nonlinear Analysis: Real World Applications 40 (2018) 290–306
301
then we have from (2.16) that
ξ=
(3 − k)γ
.
γ−1
Utilizing the Hölder’s inequality with
kαγ − α − 2γ
2αγ(3 − k) − α(γk − 3)
+
= 1,
2[(3 − k)αγ − γ + α]
2[(3 − k)αγ − γ + α]
kαγ − α − 2γ
∈ [0, 1) (by(2.8)),
2[(3 − k)αγ − γ + α]
∫
2
|P | |P | |u| dx can be further estimated as
(
2 )


2
1

2
2
2
∥|u| |∇u|∥L2 + |∇d| |∇2 d|L2 + ∇|u|  2
|P | |P | |u| dx ≤
4
L
R3
 1
(
)

 3−k
2
2
4
4
+ C ∥u∥L6 + ∥∇d∥L6 ∥∂3 P ∥Lγx  α (∥u∥L4 + ∥∇d∥L4 )
3 Lx x
1 2
(

2 )

2
1

2
2
2
∥|u| |∇u|∥L2 + |∇d| |∇ d|L2 + ∇|u|  2
≤
4
L

 1
(
)
2

 3−k
2
4
4
+ C ∥∇u∥L2 + ∥∇2 d∥L2 ∥∂3 P ∥Lγx  α (∥u∥L4 + ∥∇d∥L4 ),
then the term
∫
R3
3
when
Lx x
1 2
kαγ−α−2γ
2[(3−k)αγ−γ+α]
= 0 (i.e., α = γ = k3 ), or
(
∫

2 )

2
1

2
2
2
|P | |P | |u| dx ≤
∥|u| |∇u|∥L2 + |∇d| |∇2 d|L2 + ∇|u| 
4
L2
R3
(
2γ(3−k)
2γ(3−k)
+C
2· 2γ(3−k)−3(γ−1)
∥u∥
L
4
2
(3−k)γ
γ−1
+ ∥∇d∥
2· 2γ(3−k)−3(γ−1)
L
2
(3−k)γ
γ−1

 2αγ

 αγk−α−2γ
+ ∥∂3 P ∥Lγx  α
3
)
Lx x
1 2
4
× (∥u∥L4 + ∥∇d∥L4 ),
kαγ−α−2γ
1
< 1 (i.e., k3 < γ ≤ α < k−2
).
when 0 < 2[(3−k)αγ−γ+α]
Thus, we can estimate I1 as
(

2 )


1

2
2
2 2

I1 ≤
∥|u| |∇u|∥L2 + |∇d| |∇ d| L2 + ∇|u|  2
4
L
⎧(

 1
)
3
2
⎪

 3−k
2
4
4
2
⎪
if α = γ = ,
⎪
⎪ ∥∇u∥L2 + ∥∇ d∥L2 ∥∂3 P ∥Lγx3 Lα (∥u∥L4 + ∥∇d∥L4 )
⎪
k
x1 x2
⎪
⎪
)
⎨(
2γ(3−k)
2γ(3−k)

 2αγ
2· 2γ(3−k)−3(γ−1)
2· 2γ(3−k)−3(γ−1)
αγk−α−2γ


4
4
+C
∥u∥ (3−k)γ
+ ∥∇d∥ (3−k)γ
+ ∥∂3 P ∥Lγx  α
(∥u∥L4 + ∥∇d∥L4 )
⎪
⎪
2
2
3
L
⎪
x1 x2
L γ−1
L γ−1
⎪
⎪
⎪
3
1
⎪
⎩
if < γ ≤ α <
.
k
k−2
Then, inserting all estimates of Ii (i = 1, 2, . . . , 7) into (2.12), one has
5
5
d
2
4
4
2 2
2
2 2
(∥u(·, t)∥L4 + ∥∇d(·, t)∥L4 ) + ∥∇|u| ∥L2 + ∥|∇u| |u|∥L2 + ∥∇|∇d| ∥L2 + ∥|∇2 d| |∇d|∥L2
dt
2
2
⎧(
 1
)
⎪
3
2

 3−k
2
4
4
2
⎪
⎪
γ
∥∇u∥
+
∥∇
d∥
P
∥
(∥u∥L4 + ∥∇d∥L4 )
if α = γ = ,
∥∂
2
2
3
⎪
Lx  α
L
L
⎨
3 Lx x
k
1 2
)
≤C (
2αγ
2γ(3−k)


2· 2γ(3−k)−3(γ−1)
⎪
3
1

 αγk−α−2γ
⎪
4
4
⎪
+ ∥∂3 P ∥Lγx  α
(∥u∥L4 + ∥∇d∥L4 ) if < γ ≤ α <
.
⎪
⎩ ∥(u, ∇d)∥ 2 (3−k)γ
3
k
k−2
Lx x
1 2
L γ−1
(2.18)
Q. Liu, P. Wang / Nonlinear Analysis: Real World Applications 40 (2018) 290–306
302
∫T
8
Notice that if 0 ∗ ∥∇d∥L4 dt ≤ M , by multiplying both sides of (1.1)1 with u, and (1.1)2 with −∆d, and
then adding the two results and integrating it over R3 , it is easy to get that
d
2
2
2
2
(∥u∥L2 + ∥∇d∥L2 ) + (∥∇u∥L2 + ∥∇2 d∥L2 )
dt
∫
∫
∫
2
=−
∇ · (∇d ⊙ ∇d) · udx +
u · ∇d∆ddx −
|∇d| d · ∆ddx
3
3
3
R
R
∫ R
1
2
8
2
2
2
≤
|∇d| |d| |∆d|dx ≤ C∥∇ d∥L2 ∥∇d∥L4 ∥d∥L∞ ≤ ∥∇2 d∥L2 + C(1 + ∥∇d∥L4 ).
2
R3
Thus we have for 0 ≤ t ≤ T∗
2
2
∥u(·, t)∥L2 +∥∇d(·, t)∥L2
2
≤(∥u0 ∥L2
∫
T∗
2
2
(∥∇u(·, τ )∥L2 + ∥∇2 d(·, τ )∥L2 )dτ
0
∫ T∗
C
(1+∥∇d∥8 4 )dt
2
L
,
+ ∥∇d0 ∥L2 )e 0
+
which yields that
u, ∇d ∈ L∞ (0, T∗ ; L2 (R2 )) ∩ L2 (0, T∗ ; H 1 (R3 )).
Moreover, the above fact together with the standard interpolation inequality yields that
2γ(3−k)
2· 2γ(3−k)−3(γ−1)
u, ∇d ∈ L
(0, T∗ ; L2
(3−k)γ
γ−1
(R3 )),
(2.19)
1
if k3 ≤ γ ≤ α ≤ k−2
and k ∈ [2, 3). Then, by applying the Gronwall’s inequality to (2.19) in the interval
[0, T∗ ), and using (2.8) and (2.19), one can deduce that
sup
0≤t<T∗
4
(∥u(·, t)∥L4
4
≤(∥u0 ∥L4
4
≤(∥u0 ∥L4
+
4
∥∇d(·, t)∥L4 )
∫
T∗
2
2
2
2
∥∇|u| (·, t)∥L2 + ∥∇|∇d| (·, t)∥L2 dt
+2
0
⎧
{ ∫
}
 1
T
(
)
∗
⎪
3
2
3−k
⎪


2
⎪
exp C
∥∇u∥L2 + ∥∇2 d∥L2 ∥∂3 P ∥Lγx  α dt
if α = γ = ,
⎪
⎪
3
⎪
k
L
x1 x2
0
⎪
⎪
⎨
{ ∫
(
) }
2γ(3−k)

 2αγ
4
T∗
2· 2γ(3−k)−3(γ−1)
+ ∥∇d0 ∥L4 ) ×
αγk−α−2γ


exp C
∥(u, ∇d)∥ (3−k)γ
+ ∥∂3 P ∥Lγx  α
dt
⎪
⎪
2
3 Lx x
⎪
0
⎪
1 2
L γ−1
⎪
⎪
⎪
3
1
⎪
⎩
if < γ ≤ α <
,
k
k−2
4
+ ∥∇d0 ∥L4 )eCM ≤ C̃,
where C̃ only depending on u0 , d0 and M . Thus, (2.10) is established, and the proof of Lemma 2.4 is
completed. □
By using Lemma 2.4, we are now in position to prove Theorem 1.2 under the assumption (2.6) or (2.7)
or (2.8).
Proof of Theorem 1.2. It is sufficient to prove (2.9). To do it, let us first recall that for all local smooth
solutions to system (1.1)–(1.3), the assumption (2.6) or (2.7) or (2.8) ensures that (2.10) holds, and then
yields that
∫
0
T∗
4
4
∥u(·, t)∥L12 + ∥∇d(·, t)∥L12 dt ≤ C̃ < +∞.
(2.20)
Q. Liu, P. Wang / Nonlinear Analysis: Real World Applications 40 (2018) 290–306
303
Now, taking ∇ on (1.1)1 and multiplying the resulting equation by ∇u, taking ∇i ∇j on (1.1)2 and multiplying
the resulting equation by ∇i ∇j d, then adding the two resulting equations and integrating with respect to x
over R3 , and using the fact ∇ · u = 0, one can deduce that
1 d
2
2
2
2
(∥∇u(·, t)∥L2 + ∥∇2 d(·, t)∥L2 ) + (∥∇2 u(·, t)∥L2 + ∥∇3 d(·, t)∥L2 )
2 dt
∫
2
=−
[∇u · ∇u∇u + ∇(∇ · (∇d ⊙ ∇d))∇u + ∇2 u · ∇d∇2 d + 2∇u · ∇∇d∇2 d − ∇2 (|∇d| d)∇2 d]dx
R3
≜J1 + J2 + J3 + J4 + J5 .
(2.21)
By using the Hölder’s inequality, the interpolation inequality and the Young’s inequality, we obtain
]
))
(
(
∫
∫ [
2
|∇d|
∆u dx = −
(∇u · ∇u∇u − ∇d∆d∆u) dx
J1 + J2 = −
∇u · ∇u∇u − ∇d∆d + ∇
2
R3
R3
1
2
3
2
2
3
∥∇2 u∥L2 + C(∥∇u∥L3 + ∥∆d∥L3 ∥∇d∥L6 )
≤ C(∥∇u∥L3 + ∥∇2 u∥L2 ∥∆d∥L3 ∥∇d∥L6 ) ≤
16
(
(
)2 (
)2 )
1
1
2
2
1
1
1
2
≤
∥∇2 u∥L2 + C (∥u∥L3 12 ∥∇2 u∥L3 2 )3 + ∥∇3 d∥L3 2 ∥∇d∥L3 12
∥d∥L2 ∞ ∥∇d∥L2 12
16
(
)
7
1 2 2
1
2
4
3
2
≤ ∥∇ u∥L2 + ∥∇ d∥L2 + C ∥u∥L12 + ∥∇d∥L12
8
16
(
)
1
1
2
2
4
4
≤ ∥∇2 u∥L2 + ∥∇3 d∥L2 + C ∥u∥L12 + ∥∇d∥L12 + 1 ,
8
16
where we have used the fact that |d| = 1. Similarly,
∫
∫
J3 + J4 = −
∇2 u · ∇d∇2 ddx − 2
∇u · ∇∇d∇2 ddx
Rn
R3
(
)
2
≤ C ∥∇2 u∥L2 ∥∇2 d∥L3 ∥∇d∥L6 + ∥∇u∥L3 ∥∇2 d∥L3
(
)
1
1
2
2
1
1
2
4
2
3
2
3
3
3
3
3
2
2
3
3
≤ C ∥∇ u∥L2 ∥∇ d∥L2 ∥∇d∥L12 ∥d∥L∞ ∥∇d∥L12 + ∥∇ u∥L2 ∥u∥L12 ∥∇ d∥L2 ∥∇d∥L2
≤
1
1
2
2
4
4
∥∇2 u∥L2 + ∥∇3 d∥L2 + C(1 + ∥u∥L12 + ∥∇d∥L12 ).
16
16
2
By using facts |d| = 1 and ∆d · d = −|∇d| , we see that
∫
∫
2
2
2
2
J5 =
∇ (|∇d| d)∇ ddx =
∇(2∇2 d∇dd + |∇d| ∇d)∇2 ddx
R3
∫
=
R3
2
(2∇3 d∇dd + 2|∇2 d| d + 5∇2 d∆dd)∇2 ddx
R3
3
2
≤ C(∥∇3 d∥L2 ∥∇d∥L6 ∥∇2 d∥L3 + ∥∇2 d∥L3 + ∥∇2 d∥L3 ∥∆d∥L3 )
)
(
1
1
2
1
≤ C ∥∇3 d∥L2 (∥d∥L2 ∞ ∥∇d∥L2 12 )(∥∇∆d∥L3 2 ∥∇d∥L3 12 ) + ∥∇∆d∥L2 ∥∇d∥L12
≤
7
1
1
2
2
4
4
∥∇3 d∥L2 + C(∥∇d∥L2 12 + ∥∇d∥L12 ) ≤
∥∇3 d∥L2 + C(1 + ∥∇d∥L12 ).
16
16
Inserting the estimates of Ji (i = 1, 2, . . . , 5) into (2.21), one gets
d
2
2
2
2
4
4
(∥∇u(·, t)∥L2 + ∥∇2 d(·, t)∥L2 ) + (∥∇2 u(·, t)∥L2 + ∥∇3 d(·, t)∥L2 ) ≤ C(1 + ∥u∥L12 + ∥∇d∥L12 ),
dt
304
Q. Liu, P. Wang / Nonlinear Analysis: Real World Applications 40 (2018) 290–306
which implies that
sup
0<t≤T∗
2
2
(∥∇u(·, t)∥L2 +∥∇2 d(·, t)∥L2 )
∫
≤C
0
T∗
T∗
∫
+
0
2
2
(∥∇2 u(·, t)∥L2 + ∥∇2 d(·, t)∥L2 )dt
4
4
(1 + ∥u∥L12 + ∥∇d∥L12 )dt ≤ C C̃,
(2.22)
where C̃ is the bounded positive constant in (2.10) (or (2.20)).
We now derive the higher energy estimates, i.e., the a priori estimates of Λ3 u and Λ4 d. Taking Λ3 on (1.1)1
and multiplying Λ3 u, Taking Λ4 on (1.1)2 and multiplying Λ4 d, then adding the two resulting equations and
integrating with respect to x over R3 , and using integration by parts and Lemma 2.3, one obtains
)
1 d ( 3
2
2
2
2
∥Λ u(·, t)∥L2 + ∥Λ4 d(·, t)∥L2 + ∥Λ4 u(·, t)∥L2 + ∥Λ5 d(·, t)∥L2
2 dt
∫
∫
∫
∫
2
3
3
3
3
4
4
=−
Λ (u · ∇u) · Λ udx −
Λ (∆d · ∇d) · Λ udx −
Λ (u · ∇d) · Λ ddx +
Λ4 (|∇d| d) · Λ4 ddx
3
3
3
3
R
R
R
R
∫
∫
∫
=−
[Λ3 (u · ∇u) − u · ∇Λ3 u] · Λ3 udx +
Λ3 (∆d · ∇d) · Λ3 udx −
[Λ4 (u · ∇d) − u · ∇Λ4 d] · Λ4 ddx
3
3
3
R
R
R
∫
]
[ 3
2
2
2
2
−
Λ (|∇d| )d · Λ5 d + 3Λ2 (|∇d| )Λd · Λ5 d + 3Λ(|∇d| )Λ2 d · Λ5 d + |∇d| Λ3 d · Λ5 d dx
3
(R
≤C ∥[Λ3 (u · ∇u) − u · ∇Λ3 u]∥ 3 ∥Λ3 u∥L3 + ∥Λ3 (∆d · ∇d)∥ 3 ∥Λ3 u∥L3
L2
+ ∥Λ4 (u · ∇d) − u · ∇Λ4 d∥
3
L2
L2
∥Λ4 d∥L3
)
2
2
+ ∥Λ5 d∥L2 (∥∇d∥L6 ∥Λ4 d∥L3 + ∥Λ2 d∥L4 ∥Λ3 d∥L4 + ∥∇d∥L6 ∥Λ3 d∥L6 + ∥∇d∥L6 ∥Λ2 d∥L6 )
(
2
≤C ∥∇u∥L3 ∥Λ3 u∥L3 + (∥∆d∥L2 ∥Λ4 d∥L6 + ∥Λ5 d∥L2 ∥∇d∥L6 )∥Λ3 u∥L3
+ (∥∇d∥L6 ∥Λ4 u∥L2 + ∥∇u∥L2 ∥Λ4 d∥L2 )∥Λ4 d∥L3
)
2
2
+ ∥Λ5 d∥L2 (∥∇d∥L6 ∥Λ4 d∥L3 + ∥Λ2 d∥L4 ∥Λ3 d∥L4 + ∥∇d∥L6 ∥Λ3 d∥L6 + ∥∇d∥L6 ∥Λ2 d∥L6 )
(
11
5
7
1
≤C ∥∇u∥L6 2 ∥Λ4 u∥L62 + ∥∆d∥L2 ∥Λ5 d∥L2 ∥∇u∥L6 2 ∥Λ4 u∥L6 2 + (∥∇u∥L2 ∥Λ5 d∥L2
)
5
5
2
7
7
1
+ ∥Λ4 u∥L2 ∥∆d∥L2 )∥∆d∥L6 2 ∥Λ5 d∥L6 2 + ∥Λ5 d∥L2 (∥∆d∥L6 2 ∥Λ5 d∥L6 2 + ∥∆d∥L3 2 ∥Λ5 d∥L3 2 )
)
(
)
38
1( 4 2
2
24
4
14
14
5
7
≤
∥Λ u∥L2 + ∥Λ d∥L2 + C ∥∇u∥L2 + ∥∆d∥L2 + ∥∆d∥L2 + ∥∇u∥L2 + ∥∆d∥L2
2
)
(
(
)
19
1
2
2
∥Λ4 u∥L2 + ∥Λ5 d∥L2 + C C̃ 7 + C̃ 7 + C̃ 12 + C̃ 2 ,
(2.23)
≤
2
∫
∫
where we have used the fact that div u = 0 implies R3 (u · ∇Λ3 u) · Λ3 udx = R3 (u · ∇Λ4 d) · Λ4 ddx = 0, and
have used (2.20) and the following interpolation inequalities:
1
5
∥∇u∥L3 ≤ C∥∇u∥L6 2 ∥Λ4 u∥L6 2 ;
1
3
∥Λ2 d∥L4 ≤ C∥∆d∥L4 2 ∥Λ5 d∥L4 2 ;
2
5
1
∥Λ3 u∥L3 ≤ C∥∇u∥L6 2 ∥Λ4 u∥L6 2 ;
5
1
∥Λ3 d∥L4 ≤ C∥∆d∥L6 2 ∥Λ5 d∥L6 2 ;
5
1
∥Λ4 d∥L3 ≤ C∥∆d∥L6 2 ∥Λ5 d∥L6 2 ;
1
2
∥Λ3 d∥L6 ≤ C∥∆d∥L3 2 ∥Λ5 d∥L3 2 ;
1
and ∥Λ2 d∥L6 ≤ C∥∆d∥L3 2 ∥Λ5 d∥L3 2 .
By letting C̃ > 1, one can deduce (2.23) that
d
1
2
2
2
2
(∥Λ3 u(·, t)∥L2 + ∥Λ4 d(·, t)∥L2 ) + (∥Λ4 u(·, t)∥L2 + ∥Λ5 d(·, t)∥L2 ) ≤ C C̃ 12 .
dt
2
Integrating the above inequality with respect to t over [0, T∗ ], it follows that
∫
)
1 T∗ ( 4
2
2
2
2
sup (∥Λ3 u(·, t)∥L2 + ∥Λ4 d(·, t)∥L2 ) +
∥Λ u(·, τ )∥L2 + ∥Λ5 d(·, τ )∥L2 dτ ≤ C < ∞,
2 0
0≤t≤T∗
Q. Liu, P. Wang / Nonlinear Analysis: Real World Applications 40 (2018) 290–306
305
where C only depends on the initial data (u0 , d0 ), C̃ and T∗ . Therefore, the above inequality together with
the standard energy inequality (see e.g. [18]) ensures that
∥(u, d)∥L∞ (0,T∗ ;H 3 ×H 4 ) + ∥(u, d)∥L2 (0,T∗ ;H 4 ×H 5 ) ≤ C < +∞,
from which, one obtains (2.9). This completes the proof of Theorem 1.2.
□
Acknowledgments
The authors would like to acknowledge their sincere thanks to the referees for their careful reading of the
work and their many valuable comments and suggestions.
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