Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 182, pp. 1–16.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
WEAK-STRONG UNIQUENESS OF HYDRODYNAMIC FLOW OF
NEMATIC LIQUID CRYSTALS
JI-HONG ZHAO, QIAO LIU
Abstract. This article concerns a simplified model for a hydrodynamic system of incompressible nematic liquid crystal materials. It is shown that the
weak-strong uniqueness holds for the class of weak solutions provided that
−1+3/p+2/q
either (u, ∇d) ∈ C([0, T ), L3 (R3 )); or (u, ∇d) ∈ Lq (0, T ; Ḃp,q
(R3 ))
2
3
with 2 ≤ p < ∞, 2 < q < ∞ and p + q > 1.
1. Introduction
In this article, we study uniqueness criteria for solutions of a hydrodynamical
system modeling the flow of nematic liquid crystals in the whole space R3 , namely
the Cauchy problem
∂t u − ν∆u + u · ∇u + ∇π = −λ div(∇d ∇d),
∂t d + u · ∇d = γ(∆d − g(d)),
div u = 0,
(1.1)
(u, d)|t=0 = (u0 , d0 ).
This system describes the time evolution of nematic liquid crystal materials (cf.
[18]), where u ∈ R3 and π ∈ R denote, respectively, the velocity field and the
pressure of the fluid, and d ∈ R3 denotes the director field of the nematic liquid
4
crystals; ν, λ, γ are positive constants, and g(d) = ∇G(d) with G(d) = |d|4 −
|d|2
2 is a Ginzburg-Landau approximation function; the unusual term ∇d
(h∂xi d, ∂xj di)1≤i,j≤3 is the stress tensor induced by the director field d,
3
∇d =
and the
notation h·, ·i denotes the inner product in R . Since the sizes of the viscosity
constants ν, λ and γ do not play important roles in the proof of our main result,
we shall assume that ν = λ = γ = 1 throughout this paper.
As the authors pointed out in [20], although system (1.1) is a simplified version
of the liquid crystal model proposed by Ericksen [3] and Leslie [16], but it still
retains most of the interesting mathematical properties. We refer the reader to
see [4, 10, 17, 18] and the references therein for more discussions of the physical
background of this problem. In [20], using the modified Galerkin method and the
2000 Mathematics Subject Classification. 35A02, 35B35, 76A15.
Key words and phrases. Nematic liquid crystal flow; weak solutions; stability;
weak-strong uniqueness.
c
2012
Texas State University - San Marcos.
Submitted July 12, 2012. Published October 19, 2012.
1
2
J. ZHAO, Q. LIU
EJDE-2012/182
compactness argument, Lin and Liu proved global existence of weak solutions of
(1.1) with g(d) = ∇G(d) for some smooth and bounded function G : R3 → R.
Moreover, when g(d) = 0, they established global existence of strong solutions
if the initial data is sufficiently small (or if the viscosity ν is sufficiently large).
The same as for the Navier-Stokes equations (which are equations obtained by
putting d = 0 in (1.1)), it is well known that weak solution of (1.1) is unique
and regular in R2 . However, the question of regularity and uniqueness of weak
solution is an outstanding open problem in R3 . Hence, it is meaningful to find
sufficient conditions on a strong solution of (1.1) such that all weak solutions sharing
the same initial data must coincide with the one which additionally satisfies these
sufficient conditions, and we say then weak-strong uniqueness holds. For the three
dimensional Navier-Stokes equations, Prodi [25] and Serrin [27] proved that weakstrong uniqueness holds in the class
P = Lq (0, T ; Lp (R3 ))
with
3 2
+ = 1, 3 < p ≤ ∞.
p q
Von Wahl [28] and Giga [9] improved this result in the class
P = C([0, T ], L3 (R3 )).
Moreover, this last result was extended in the limit case by Kozono and Sohr [13],
and Escauriaza, Seregin and S̆verák [5], who proved that weak strong uniqueness
holds for
P = L∞ (0, T ; L3 (R3 )).
For uniqueness criteria related to the Sobolev spaces, we refer the reader to [1, 26].
Recently, many researches have refined the above results. Kozono and Taniuchi
[14] proved that weak-strong uniqueness holds in the class
P = L2 (0, T ; BM O).
Gallagher and Planchon [6] proved that weak-strong uniqueness holds for
−1+3/p+2/q
P = Lq (0, T ; Ḃp,q
(R3 ))
with 2 ≤ p < ∞, 2 < q < ∞ and
3 2
+ > 1.
p q
Lemarié-Rieusset [15] and Germain [8] proved that weak-strong uniqueness holds
for
(0)
P = C([0, T ], X1 ) or P = L2/(1−r) (0, T ; Xr ) with r ∈ [−1, 1).
Chen, Miao and Zhang [2] improved the above results by showing weak-strong
uniqueness for
r
P = Lq (0, T ; Ḃp,∞
(R3 ))
with
3
< p ≤ ∞, r ∈ (0, 1] and (p, r) 6= (∞, 1).
1+r
We refer the reader to see [8] and [15] for definitions of these function spaces.
In this article, we are interested in finding uniqueness criteria for weak solutions
of (1.1). For the two n × n matrixes A = (aij )ni,j=1 and B = (bij )ni,j=1 , we define
Pn
A : B =
i,j=1 aij bij , and denote by ⊗ the tensor product. Let us recall the
definition of weak solutions.
Definition 1.1. The vector-valued function (u, d) is called a weak solution of (1.1)
on R3 × (0, T ) if it satisfies the following conditions:
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WEAK-STRONG UNIQUENESS OF HYDRODYNAMIC FLOW
3
(1) (u, ∇d) ∈ L∞ (0, T ; L2 (R3 )) ∩ L2 (0, T ; Ḣ 1 (R3 )) := (LS), where Ḣ 1 (R3 ) is
the usual homogeneous Sobolev space; i.e., the space of functions whose
gradient belongs to L2 (R3 ).
(2) (u, d) satisfies (1.1) in the sense of distributions; i.e., div u = 0 in the
distributional sense and for all v ∈ C0∞ (R3 ×(0, T )) and e ∈ C0∞ (R3 ×(0, T ))
with div v = 0, we have
Z TZ
Z TZ
Z TZ
u · ∂t v dx dt −
∇u : ∇v dx dt +
u ⊗ u : ∇v dx dt
R3
T
0
∇d ∇d : ∇v dx dt
R3
0
and
T Z
T
Z
Z
d · ∂t e dx dt −
0
Z
R3
0
Z
=−
Z
R3
0
Z
R3
T Z
Z
T
Z
u ⊗ d : ∇e dx dt
∇d : ∇e dx dt +
0
R3
0
R3
g(d) · e dx dt.
=
0
R3
(3) The following energy inequality holds (see (3.6) in the appendix):
Z t
ku(t)k2L2 + k∇d(t)k2L2 + 2
(k∇u(τ )k2L2 + k∆d(τ )k2L2 )dτ
0
Z t
+6
kd · ∇dk2L2 (τ )dτ
0
Z t
2
2
≤ ku0 kL2 + k∇d0 kL2 +
k∇d(τ )k2L2 dτ for all t ≥ 0.
0
Before presenting the exact statement of our result, let us first recall the definition of the homogeneous Besov spaces. Let S(R3 ) be the Schwartz space. We
denote by {∆j , Sj }j∈Z the Littlewood-Paley decomposition. Let Z(R3 ) = f ∈
S(R3 ) : ∂ α fb(0) = 0, ∀α ∈ (N ∪ {0})3 , and denote its dual by Z 0 (R3 ). Recall
s
that for s ∈ R and (p, q) ∈ [1, ∞] × [1, ∞], the homogeneous Besov space Ḃp,q
(R3 )
is defined by
s
Ḃp,q
(R3 ) = f ∈ Z 0 (R3 ) : kf kḂ s < ∞ ,
p,q
where
( P
kf kḂ s =
p,q
2jsq k∆j f kqLp
supj∈Z 2js k∆j f kLp
j∈Z
1/q
for 1 ≤ q < ∞,
for q = ∞.
s
(R3 ), k · kḂ s )
It is well-known that if either s < p3 or s = p3 and q = 1, then (Ḃp,q
p,q
is a Banach space. For more details about the homogeneous Besov spaces, we refer
the reader to see [15]. Next we introduce some notations. Given 0 < T < ∞ and a
Banach space X, we denote by C([0, T ], X) the Banach space of all bounded and
continuous mappings from [0, T ] to X, and for p ≥ 1, we denote by Lp (0, T ; X)
the set of Bochner measurable X-valued time dependent functions f such that
t → kf kX belongs to Lp (0, T ). The product of Banach spaces X × Y will be
equipped with the usual norm k(f, g)kX ×Y = kf kX + kgkY , and if X = Y, we use
k(f, g)kX to denote k(f, g)kX ×X .
The main result of this paper is as follows.
4
J. ZHAO, Q. LIU
EJDE-2012/182
Theorem 1.2. Assume that (u, d) and (ũ, d̃) are two weak solutions of (1.1) for
a given initial data (u0 , ∇d0 ) ∈ L2 (R3 ). Assume furthermore that for some T > 0,
either
(u, ∇d) ∈ C([0, T ], L3 (R3 ))
(1.2)
or
−1+3/p+2/q
(u, ∇d) ∈ Lq (0, T ; Ḃp,q
(R3 ))
(1.3)
with 2 ≤ p < ∞, 2 < q < ∞ and
interval [0, T ].
3
p
+
2
q
Remark 1.3. Theorem 1.2 holds with
q
−1+3/p+2/q
(0, T ; Ḃp,q
(R3 ))
L
assume that
> 1. Then u = ũ and d = d̃ on the time
3
p
q
+
2
q
= 1 in (1.3) as well, with the space
replaced by L (0, T ; Lp (R3 )), when p > 3, namely, if we
(u, ∇d) ∈ Lq (0, T ; , Lp (R3 ))
with 3 < p ≤ ∞, 2 ≤ q < ∞ and
3 2
+ = 1,
p q
then u = ũ and d = d̃ on the time interval [0, T ]. This can be seen as a consequence
of Prodi-Serrin’s uniqueness criterion.
Remark 1.4. We extend, in Theorem 1.2, the uniqueness criteria of weak solutions
of [28] and [6] for the system (1.1).
Let us sketch an idea leading to the proof of Theorem 1.2. We introduce the
function
F = ∇d.
T
Let F be the transpose of F . Then, taking the gradient of second equation of
(1.1), noticing the facts that F F = F T F and
n
n
n
∂ X ∂di X ∂uj ∂di X
∂ ∂di uj
=
+
uj
= (F ∇u + u · ∇F )ik
∂xk j=1 ∂xj
∂xk ∂xj j=1 ∂xj ∂xk
j=1
for all i, k = 1, 2, . . . , n, system (1.1) reads
∂t u − ∆u = −∇π − u · ∇u − div(F T F ),
∂t F − ∆F = −u · ∇F − F ∇u − (3|d|2 − 1)F,
div u = 0,
(1.4)
(u, F )|t=0 = (u0 , F0 ),
where F0 = ∇d0 . System (1.4) is more related to the viscoelastic fluids, which
had attracted much attention recently; see for instance [21]. Using the technical
matrixes analysis, the energy inequality and the similar argument in the studying
of the incompressible Navier-Stokes equations in [28] and [6], we can obtain some
important estimates which yield the proof of Theorem 1.2.
Before ending this section, we mention some well-posedness results of the system
(1.1). Recently, when g(d) = 0, by using the maximal regularity of Stokes equations
and the parabolic equations, Hu and Wang [12] proved global existence of strong
solutions to the system (1.1) for small initial data belonging to Besov spaces of
positive-order. They also proved that when the strong solution exists, all global
weak solutions constructed by [20] must be equal to the unique strong solution.
In [19] and [22], the authors studied the system (1.1) with g(d) = |∇d|2 d in two
dimensions. They established the global existence, uniqueness and partial regularity
EJDE-2012/182
WEAK-STRONG UNIQUENESS OF HYDRODYNAMIC FLOW
5
of weak solutions and performed the blow-up analysis at each singular time. Hong
[11] proved independently the global existence of weak solutions of the system (1.1)
in two dimensions. In [29], Wang established global well-posedness of (1.1) with
g(d) = |∇d|2 d for small initial data in BM O−1 × BM O. Some regularity criteria
for weak solutions of the system (1.1) were also established, see [7], [23] and [24].
The rest of this paper is organized as follows. In Section 2, we present the proof
of Theorem 1.2. In appendix, we shall establish the basic energy inequality of the
system (1.1), which gives global existence of weak solutions of (1.1).
2. The proof of Theorem 1.2
Throughout this section, we assume that (u0 , ∇d0 ), (ũ0 , ∇d̃0 ) ∈ L2 (R3 ), and
denote by (u, d) and (ũ, d̃), respectively, be two weak solutions associated with
initial conditions (u0 , ∇d0 ) and (ũ0 , ∇d̃0 ), respectively.
Let us define F = ∇d, F̃ = ∇d̃, F0 = ∇d0 and F̃0 = ∇d̃0 . Obviously, by
Definition 1.1, (u, F ) and (ũ, F̃ ) verify equations (1.4) and satisfy
Z t
ku(t)k2L2 + kF (t)k2L2 + 2
(k∇u(τ )k2L2 + k∇F (τ )k2L2 )dτ
0
Z t
+6
k|d|F k2L2 (τ )dτ
0
Z t
2
2
≤ ku0 kL2 + kF0 kL2 + 2
kF (τ )k2L2 dτ,
(2.1)
0
Z t
kũ(t)k2L2 + kF̃ (t)k2L2 + 2
(k∇ũ(τ )k2L2 + k∇F̃ (τ )k2L2 )dτ
0
Z t
+6
k|d̃|F̃ k2L2 (τ )dτ
0
Z t
≤ kũ0 k2L2 + kF̃0 k2L2 + 2
kF̃ (τ )k2L2 dτ.
(2.2)
0
Setting w = u − ũ, E = F − F̃ , w0 = u0 − ũ0 and E0 = F0 − F̃0 , we divide the
proof of Theorem 1.2 into the following two cases.
Case 1. (u, ∇d) ∈ C([0, T ], L3 (R3 )). We shall prove the following stability
result.
Proposition 2.1. Assume that (u, ∇d) ∈ C([0, T ], L3 (R3 )). Then
k(w(t), E(t))k2L2
≤
k(w0 , E0 )k2L2
Z
+2
t
k(∇w(τ ), ∇E(τ ))k2L2 dτ
0
exp Ct k(u, F )k2C([0,T ],L3 (R3 )) + 1 ,
(2.3)
where C is a constant depending on k(d, d̃)kL∞ (0,T ;Ḣ 1 ) and k(d, d̃)kL2 (0,T ;Ḣ 2 ) .
It is clear that, under the condition (1.2), Theorem 1.2 is an immediate consequence of Proposition 2.1. Note that, by (2.1) and (2.2), the left hand side of (2.3)
6
J. ZHAO, Q. LIU
EJDE-2012/182
satisfies
k(w(t), E(t))k2L2
Z
t
k(∇w(τ ), ∇E(τ ))k2L2 dτ
Z t
+ k(ũ(t), F̃ (t))k2L2 + 2
k(∇u(τ ), ∇F (τ ))k2L2 dτ
+2
0
= k(u(t), F (t))k2L2
0
Z t
+2
k(∇ũ(τ ), ∇F̃ (τ ))k2L2 dτ − 2 u(t)|ũ(t)
0
Z t
Z t
(2.4)
− 2 F (t)|F̃ (t) − 4
∇u(τ )|∇ũ(τ ) dτ − 4
∇F (τ )|∇F̃ (τ ) dτ
0
0
Z t
Z t
≤ k(u0 , F0 )k2L2 + 2
kF (τ )k2L2 dτ − 6
k|d|F k2L2 (τ )dτ + k(ũ0 , F̃0 )k2L2
0
0
Z t
Z t
2
2
+2
kF̃ (τ )kL2 dτ − 6
k|d̃|F̃ kL2 (τ )dτ − 2 u(t)|ũ(t) − 2 F (t)|F̃ (t)
0
0
Z t
Z t
−4
∇u(τ )|∇ũ(τ ) dτ − 4
∇F (τ )|∇F̃ (τ ) dτ,
0
0
where we denote by (·|·) the scalar product in L2 (R3 ). Hence, we aim at proving
the following lemma.
Lemma 2.2. Under the assumptions of Proposition 2.1, the following equality holds
for all t ≤ T ,
Z t
Z t
u(t)|ũ(t) + F (t)|F̃ (t) + 2
∇u(τ )|∇ũ(τ ) dτ + 2
∇F (τ )|∇F̃ (τ ) dτ
0
0
Z tZ
Z tZ
= (u0 |ũ0 ) + (F0 |F̃0 ) −
w · ∇w · u dx dτ +
F̃ T F̃ : ∇u dx dτ
0
R3
0
R3
Z tZ
Z tZ
+
F T F : ∇u dx dτ −
∇u : (F̃ T F + F T F̃ ) dx dτ
3
3
0
R
0
R
Z tZ
Z tZ
−
w · ∇F : F̃ dx dτ +
F : F̃ ∇w dx dτ
0
R3
0
R3
Z tZ
Z tZ
−
(3|d|2 − 1)F : F̃ dx dτ −
(3|d̃|2 − 1)F̃ : F dx dτ.
R3
0
0
R3
(2.5)
Proof. Let us choose two smooth sequences {(ũn , F̃n )} (div ũn = 0) and {(un , Fn )}
(div un = 0) such that
lim (ũn , F̃n ) = (ũ, F̃ )
n→∞
lim (ũn , F̃n ) = (ũ, F̃ )
n→∞
in L2 (0, T ; Ḣ 1 (R3 )),
weakly-star in L∞ (0, T ; L2 (R3 ))
(2.6)
and
lim (un , Fn ) = (u, F )
n→∞
in L2 (0, T ; Ḣ 1 (R3 )) ∩ C([0, T ], L3 (R3 )),
lim (un , Fn ) = (u, F )
n→∞
weakly-star in L∞ (0, T ; L2 (R3 )).
We split the proof into the following two steps.
(2.7)
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WEAK-STRONG UNIQUENESS OF HYDRODYNAMIC FLOW
7
Step 1. Taking the scalar product with ũn and un of the equation (1.4) on u
and ũ respectively, after integration in time and integration by parts in the space
variables, we obtain
Z t
(∂τ u|ũn ) + (∇u|∇ũn ) + (u · ∇u|ũn ) + (div(F T F )|ũn ) dτ = 0
(2.8)
0
and
Z t
(∂τ ũ|un ) + (∇ũ|∇un ) + (ũ · ∇ũ|un ) + (div(F̃ T F̃ )|un ) dτ = 0.
(2.9)
0
By (2.6) and (2.7), it is obvious that
Z t
Z t
Z t
lim
(∇u|∇ũn )dτ +
(∇ũ|∇un )dτ = 2
(∇u|∇ũ)dτ.
n→∞
0
0
(2.10)
0
Applying the Hölder inequality and the Sobolev embedding inequality, it follows
that
Z t
Z t
(ũ · ∇ũ|un )dτ ≤ C
kũkL6 k∇ũkL2 kun kL3 dτ
(2.11)
0
0
≤ Ckũk2L2 (0,T ;Ḣ 1 ) kun kC([0,T ],L3 ) .
Since un converges to u in C([0, T ], L3 (R3 )), (2.11) implies that
Z t
Z t
lim
(ũ · ∇ũ|un )dτ =
(ũ · ∇ũ|u)dτ.
n→∞
0
(2.12)
0
Similarly, by applying (2.6), (2.7) and (2.11), we obtain the following three equalities:
Z t
Z t
lim
(u · ∇u|ũn )dτ = − lim
(u · ∇ũn |u)dτ
n→∞ 0
n→∞ 0
(2.13)
Z t
Z t
=−
(u · ∇ũ|u)dτ =
(u · ∇u|ũ)dτ,
0
t
0
Z t
(div(F T F )|ũn )dτ = − lim
(F T F |∇ũn )dτ
n→∞ 0
n→∞ 0
Z t
Z t
T
=−
(F F |∇ũ)dτ =
(div(F T F )|ũ)dτ,
Z
lim
0
(2.14)
0
and
Z
lim
n→∞
=
t
T
(div(F̃ F̃ )un )dτ = lim
n→∞
0
Z tX
3
0
Z tX
3
0
(∂xi F̃ T F̃ + F̃ T ∂xi F̃ )|un dτ
i=1
Z t
(∂xi F̃ T F̃ + F̃ T ∂xi F̃ )u dτ =
(div(F̃ T F̃ )|u)dτ.
(2.15)
0
i=1
Since ∂t ũ = ∆ũ − ũ · ∇ũ − div(F̃ T F̃ ) − ∇π holds in the sense of distribution, the
estimates (2.10), (2.12)–(2.15) and div un = 0 imply in particular that
Z t
Z t
(∂τ ũ|un )dτ = − lim
(∇ũ|∇un ) + (ũ · ∇ũ|un ) + (div(F̃ T F̃ )|un ) dτ
lim
n→∞ 0
n→∞ 0
Z t
=−
(∇ũ|∇u) + (ũ · ∇ũ|u) + (div(F̃ T F̃ )|u) dτ
0
8
J. ZHAO, Q. LIU
Z
EJDE-2012/182
t
=
(∂τ ũ|u)dτ.
0
It can be proved analogously that
Z t
Z t
lim
(∂τ u|ũn )dτ =
(∂τ u|ũ)dτ.
n→∞
0
0
Putting these estimates together, and noticing that
Z t
(∂τ ũ|u) + (∂τ u|ũ)dτ = (u(t)|ũ(t)),
0
Z t
Z t
(ũ · ∇ũ|u) − (u · ∇ũ|u) dτ =
(w · ∇w|u)dτ,
0
(2.16)
(2.17)
0
we obtain
Z t
(u(t)|ũ(t)) + 2
(∇u|∇ũ)dτ
0
Z t
Z tZ
= (u0 |ũ0 ) −
(w · ∇w|u)dτ +
F̃ T F̃ : ∇u dx dτ
3
0
0
R
Z tZ
+
F T F : ∇u dx dτ.
(2.18)
R3
0
Step 2. Proceeding in the same way as (2.8) and (2.9), we obtain
Z t
(∂τ F |F̃n ) + (∇F |∇F̃n ) + (u · ∇F |F̃n )
0
+ (F ∇u|F̃n ) + ((3|d|2 − 1)F |F̃n ) dτ = 0
(2.19)
and
Z t
(∂τ F̃ |Fn ) + (∇F̃ |∇Fn ) + (ũ · ∇F̃ |Fn )
+ (F̃ ∇ũ|Fn ) + ((3|d̃|2 − 1)F̃ |Fn ) dτ = 0.
0
(2.20)
By using assumptions (2.6)–(2.7) and similar argument in the proof of (2.11), we
obtain
Z t
Z t
Z t
lim
(∇F |∇F̃n )dτ +
(∇F̃ |∇Fn )dτ = 2
(∇F |∇F̃ )dτ,
(2.21)
n→∞
0
0
0
Z t
Z t
lim
(ũ · ∇F̃ |Fn )dτ =
(ũ · ∇F̃ |F )dτ,
(2.22)
n→∞ 0
0
Z t
Z t
lim
(u · ∇F |F̃n )dτ =
(u · ∇F |F̃ )dτ,
(2.23)
n→∞ 0
0
Z t
Z t
lim
(F̃ ∇ũ|Fn )dτ =
(F̃ ∇ũ|F )dτ,
(2.24)
n→∞ 0
0
Z t
Z t
lim
(F ∇u|F̃n )dτ =
(F ∇u|F̃ )dτ.
(2.25)
n→∞
0
0
EJDE-2012/182
WEAK-STRONG UNIQUENESS OF HYDRODYNAMIC FLOW
9
To estimate the remaining terms, the Hölder inequality and the Sobolev embedding
theorem yield
Z t
(F (τ )|F̃n (τ ))dτ ≤ kF kL2 (0,T ;L2 ) kF̃n kL2 (0,T ;L2 )
0
and
t
Z
Z
2
(3|d| F |F̃n )(τ )dτ ≤ C
0
t
kd(τ )kL6 kF (τ )kL2 kF̃n (τ )kL6 dτ
0
≤ CkdkL∞ (0,T ;Ḣ 1 ) kF kL2 (0,T ;L2 ) kF̃n kL2 (0,T ;Ḣ 1 ) .
Hence, by (2.6)–(2.7), we can easily see that
Z t
Z t
2
lim
((3|d| − 1)F |F̃n )dτ =
((3|d|2 − 1)F |F̃ )dτ.
n→∞
0
(2.26)
0
Similarly,
Z
lim
n→∞
t
((3|d̃|2 − 1)F̃ |Fn )dτ =
0
Z
t
((3|d̃|2 − 1)F̃ |F )dτ.
(2.27)
0
As in the derivations of estimates (2.16) and (2.17), the above estimates (2.21)–
(2.27) imply
Z t
Z t
lim
((∂τ F |F̃n )dτ =
(∂τ F |F̃ )dτ,
n→∞ 0
0
Z t
Z t
lim
(∂τ F̃ |Fn )dτ =
(∂τ F̃ |F )dτ.
n→∞
0
0
Since
Z
t
(∂τ F |F̃ ) + (∂τ F̃ |F )dτ = (F (t)|F̃ (t)) − (F0 |F̃0 ),
Z tZ
ũ · ∇F̃ : F + ũ · ∇F : F̃ dx dτ =
ũ · ∇(F̃ : F ) dx dτ = 0,
0
Z tZ
R3
0
R3
0
T
F̃ ∇u : F + F̃ : F ∇u = ∇u : (F̃ F + F T F̃ ),
we have
Z tZ
0
u · ∇F : F̃ + ũ · ∇F̃ : F + F ∇u : F̃ + F̃ ∇ũ : F dx dτ
R3
Z tZ
=
0
∇u : (F̃ T F + F T F̃ ) + (u − ũ) · ∇F : F̃ − F : F̃ ∇(u − ũ) dx dτ.
R3
Here we have used the facts div ũ = 0 and AB : C = A : CB T = B : AT C for any
three n × n matrixes A, B and C. Finally, putting all above estimates together, we
obtain
Z t
(F (t)|F̃ (t)) + 2
(∇F |∇F̃ )dτ
0
Z tZ = (F0 |F̃0 ) −
∇u : (F̃ T F + F T F̃ ) + w · ∇F : F̃ − F : F̃ ∇w dx dτ
0
R3
Z tZ −
(3|d|2 − 1)F : F̃ + (3|d̃|2 − 1)F̃ : F dx dτ.
0
R3
(2.28)
10
J. ZHAO, Q. LIU
EJDE-2012/182
Now it is easy see that (2.5) follows from (2.18) and (2.28). This proves Lemma
2.3.
The following result plays a very important role in the proof of Proposition 2.1.
Lemma 2.3 ([28]). Let u be a measurable function in (LS) ∩ C([0, T ], L3 (R3 )).
Then for each ε > 0 we can split u on [0, T ] in u = m + l with m ∈ L∞ ([0, T ] × R3 )
and klkL∞ (0,T ;L3 ) < ε.
Proof. The proof of this lemma is due to [28], but we give it for completeness. Since
u ∈ C([0, T ], L3 (R3 )), by the uniform continuity, we can choose N large enough such
that
N
−1
X
k ε
u(x, t) −
χ[ k T, k+1 T ] (t)u(x, T )L∞ (0,T ;L3 ) < ,
N
N
N
2
k=0
where χ[a,b] denotes the characteristic function on the interval [a, b]. Now we may
k
approximate each u(·, N
T ) by a function mk,N ∈ L∞ (R3 ) with an error controlled
k
3
in L -norm by ku(·, N T ) − mk,N (·)kL3 < ε/2. Now we define m as m(x, t) =
P
0≤k≤N −1 χ[ k T, k+1 T ] (t)mk,N (x), and l = u − m. This proves Lemma.
N
N
Proof of Proposition 2.1. Since div w = 0, we obtain
By (2.4) and Lemma 2.2, it follows immediately that
k(w(t), E(t))k2L2
Z
+2
RtR
0
R3
w·∇F : F dx dτ = 0.
t
k(∇w(τ ), ∇E(τ ))k2L2 dτ
0
≤ k(u0 , F0 )k2L2 + k(ũ0 , F̃0 )k2L2 − 2(u0 |ũ0 ) − 2(F0 |F̃0 )
Z tZ
Z tZ
w · ∇w · u dx dτ − 2
F̃ T F̃ : ∇u dx dτ
+2
0
R3
0
R3
Z tZ
Z tZ
−2
F T F : ∇ũ dx dτ + 2
∇u : (F̃ T F + F T F̃ ) dx dτ
0
R3
0
R3
Z t
Z tZ
Z tZ
w · ∇F : F̃ dx dτ − 2
F : F̃ ∇w dx dτ − 2
kEk2L2 dτ
+2
0
R3
0
R3
0
Z tZ
|d|2 E : F + |d̃|2 E : F̃ dx dτ
−6
0
R3
Z tZ
Z tZ
≤ k(w0 , E0 )k2L2 + 2
w · ∇w · u dx dτ − 2
E T E : ∇u dx dτ
3
3
0
R
0
R
Z t
Z tZ
Z tZ
T
kEk2L2 dτ
−2
w · ∇F : E dx dτ + 2
E F : ∇w dx dτ − 2
3
3
0
R
0
R
0
Z tZ
|d|2 E : F + |d̃|2 E : F̃ dx dτ.
−6
0
R3
(2.29)
Since we have assumed that (u, F ) ∈ C([0, T ], L3 (R3 )), by Lemma 2.3, we can
split u = u1 + u2 and F = F1 + F2 such that (u1 , F1 ) ∈ L∞ ([0, T ] × R3 ) and
k(u2 , F2 )kL∞ (0,T ;L3 ) < ε, respectively, where ε > 0 is a constant to be determined
EJDE-2012/182
WEAK-STRONG UNIQUENESS OF HYDRODYNAMIC FLOW
later. Then we see that
Z Z
t
w · ∇w · u dx dτ 0
R3
Z t
≤ Cku2 kC([0,T ],L3 )
k∇wk2L2 dτ
0
Z t
1/2 Z t
1/2
2
∞
3
+ ku1 kL ([0,T ]×R )
k∇wkL2 dτ
kwk2L2 dτ
0
0
Z t
Z t
4
ku1 k2L∞ ([0,T ]×R3 )
kwk2L2 dτ.
≤ 2Cε
k∇wk2L2 dτ +
Cε
0
0
Similarly, we obtain
Z tZ
E T E : ∇u dx dτ 0
R3
Z tZ
=−
div(E T E) · u dx dτ 0
R3
Z t
Z t
4
≤ 2Cε
k∇Ek2L2 dτ +
ku1 k2L∞ ([0,T ]×R3 )
kEk2L2 dτ ;
Cε
0
0
Z tZ
w · ∇F : E dx dτ 0
R3
Z tZ
(w ⊗ F ) · ∇E dx dτ =
0
R3
Z t
Z t
4
kF1 k2L∞ ([0,T ]×R3 )
kwk2L2 dτ ;
≤ 2Cε
k(∇w, ∇E)k2L2 dτ +
Cε
0
0
Z t
Z tZ
E T F : ∇w dx dτ ≤ 2Cε
k(∇w, ∇E)k2L2 dτ
3
0
R
0
Z t
4
+
kF1 k2L∞ ([0,T ]×R3 )
kEk2L2 dτ ;
Cε
0
Z tZ
|d|2 E : F + |d̃|2 E : F̃ dx dτ 0
R3
Z t
≤C
(kdk2L6 kF kL6 + kd̃k2L6 kF̃ kL6 )kEkL2 dτ
0
Z t
2
≤C
(kdk2Ḣ 1 kF kḢ 1 + kd̃kḢ
1 kF̃ kḢ 1 )kEkL2 dτ
0
Z t
kEk2L2 dτ,
≤C
11
(2.30)
(2.31)
(2.32)
(2.33)
(2.34)
0
where C is a constant depending on k(d, d̃)kL∞ (0,T ;Ḣ 1 ) and k(d, d̃)kL2 (0,T ;Ḣ 2 ) . Returning back to the estimate (2.29), putting (2.30)–(2.34) together, and choosing ε
sufficiently small such that 16Cε < 1, we obtain
Z t
2
k(w(t), E(t))kL2 +
k(∇w(τ ), ∇E(τ ))k2L2 dτ
0
(2.35)
Z t
2
2
2
2
≤ k(w0 , E0 )kL2 + C k(u2 , F2 )kL∞ ((0,T )×R3 ) + 1
(kwkL2 + kEkL2 )dτ.
0
12
J. ZHAO, Q. LIU
EJDE-2012/182
The estimate above together with the Gronwall inequality yield the desired estimate
(2.3) immediately. We complete the proof of Proposition 2.1.
−1+3/p+2/q
Case 2. (u, ∇d) ∈ Lq (0, T ; Ḃp,q
(R3 )). It suffices to establish the following stability result.
−1+3/p+2/q
Proposition 2.4. Assume that (u, ∇d) ∈ Lq (0, T ; Ḃp,q
(R3 )) with 2 ≤ p <
2
3
∞, 2 < q < ∞ and p + q > 1. Then
Z t
k(w(t), E(t))k2L2 + 2
k(∇w(τ ), ∇E(τ ))k2L2 dτ
0
(2.36)
Z t
q
2
≤ k(w0 , E0 )kL2 × exp Ct + C
k(u(τ ), F (τ ))k −1+3/p+2/q dτ .
Ḃp,q
0
where C is a constant depending on k(d, d̃)kL∞ (0,T ;Ḣ 1 ) and k(d, d̃)kL2 (0,T ;Ḣ 2 ) .
To prove Proposition 2.4, the key tool we shall use is the following Lemma whose
proof can be found in [6].
Lemma 2.5 ([6]). Let 2 ≤ p < ∞ and 2 < q < ∞ such that 2q + p3 > 1. Then for
every T > 0, the trilinear form
Z TZ
−1+3/p+2/q
(u, v, w) ∈ (LS) × (LS) × Lq (0, T ; Ḃp,q
(R3 )) 7→
u · ∇v · w dx dt
0
R3
is continuous. In particular, the following estimate holds:
Z T Z
u · ∇v · w dx dt
R3
0
1−2/q
2/q
≤ CkukL∞ (0,T ;L2 ) k∇ukL2 (0,T ;L2 ) k∇vkL2 (0,T ;L2 ) kwkLq (0,T ;Ḃ −1+3/p+2/q )
p,q
+
1−2/q
2/q
k∇ukL2 (0,T ;L2 ) kvkL∞ (0,T ;L2 ) k∇vkL2 (0,T ;L2 ) kwkLq (0,T ;Ḃ −1+3/p+2/q )
p,q
1−1/q
1/q
1/q
(2.37)
1−1/q
+ kukL∞ (0,T ;L2 ) k∇ukL2 (0,T ;L2 ) kvkL∞ (0,T ;L2 ) k∇vkL2 (0,T ;L2 )
× kwkLq (0,T ;Ḃ −1+3/p+2/q ) .
p,q
Note that (2.37) holds in both scalar and vector cases.
Note that for the Navier-Stokes equations, Gallagher and Planchon [6] proved
that weak-strong uniqueness holds in the class
3 2
−1+3/p+2/q
P = Lq (0, T ; Ḃp,q
(R3 )) with 2 ≤ p < ∞, 2 < q < ∞ and + > 1.
p q
Hence, we need only to deal with the remaining terms div(F T F ) and F ∇u (the
term u·∇F can be treated as the term u·∇u). Similarly as we have done before, we
choose two smooth sequences of {(ũn , F̃n )} (div ũn = 0) and {(un , Fn )} (div un =
0) such that
lim (ũn , F̃n ) = (ũ, F̃ )
n→∞
lim (ũn , F̃n ) = (ũ, F̃ )
n→∞
in L2 (0, T ; Ḣ 1 (R3 )),
weakly-star in L∞ (0, T ; L2 (R3 ))
and
lim (un , Fn ) = (u, F )
n→∞
−1+n/p+2/q
in L2 (0, T ; Ḣ 1 (Rn )) ∩ Lq (0, T ; Ḃp,q
(R3 )),
lim (un , Fn ) = (u, F )
n→∞
weakly-star in L∞ (0, T ; L2 (R3 )).
EJDE-2012/182
WEAK-STRONG UNIQUENESS OF HYDRODYNAMIC FLOW
Applying the above assumptions and Lemma 2.5, we obtain
Z t
Z t
lim
(div(F T F )|ũn )dτ = − lim
(F T F |∇ũn )dτ
n→∞ 0
n→∞ 0
Z t
Z t
=−
(F T F |∇ũ)dτ =
(div(F T F )|ũ)dτ
0
13
(2.38)
0
and
Z
lim
n→∞
=
t
(div(F̃ T F̃ )un )dτ = lim
n→∞
0
Z tX
n
0
Z tX
n
0
(∂xi F̃ T F̃ + F̃ T ∂xi F̃ )|un dτ
i=1
Z t
T
T
(∂xi F̃ F̃ + F̃ ∂xi F̃ ) u dτ =
(div(F̃ T F̃ )|u)dτ.
(2.39)
0
i=1
Hence, (2.18) still holds under the assumption of Proposition 2.4.
It is clear that by Lemma 2.5,
Z t
Z t
lim
(F̃ ∇ũ|Fn )dτ =
(F̃ ∇ũ|F )dτ.
n→∞
0
(2.40)
0
Since ∇F̃n converges to ∇F̃ in L2 (0, T ; L2 (R3 )), and {F̃n } is bounded in he space
L∞ (0, T ; L2 (R3 )) which was ensured by the Banach-Steinhaus theorem due to F̃n
weakly-star converge to F̃ in L∞ (0, T ; L2 (R3 )), by Lemma 2.5, we obtain
Z t
Z t
lim
(F ∇u|F̃n )dτ =
(F ∇u|F̃ )dτ.
(2.41)
n→∞
0
0
The two estimates (2.40)–(2.41) imply that the equality (2.28) still holds under the
assumption of Proposition 2.4.
Now we finish the proof of Proposition 2.4. Using the similar argument as in the
proof of Lemma 2.5 (see [6]), we obtain
Z tZ
w · ∇w · u dx dτ 0
R3
Z t
2/q
2−2/q
(2.42)
≤C
kwkL2 k∇wkL2 kukḂ −1+3/p+2/q dτ
≤
1
2
Z
p,q
0
t
0
k∇wk2L2 dτ + C
Z
0
t
kwk2L2 kukq −1+3/p+2/q dτ ;
Ḃp,q
Z tZ
E T E : ∇u dx dτ 3
0
R
Z tZ
=−
div(E T E) · u dx dτ 3
0
R
Z t
2/q
2−2/q
≤C
kEkL2 k∇EkL2 kukḂ −1+3/p+2/q dτ
p,q
0
Z t
Z t
1
≤
k∇Ek2L2 dτ + C
kEk2L2 kukq −1+3/p+2/q dτ ;
Ḃp,q
2 0
0
(2.43)
14
J. ZHAO, Q. LIU
EJDE-2012/182
Z tZ
w · ∇F : E dx dτ 3
0
R
Z tZ
=
(w ⊗ F ) · ∇E dx dτ 3
0
R
Z
Z t
1 t
2
k(∇w, ∇E)kL2 dτ + C
k(w, E)k2L2 kF kq −1+3/p+2/q dτ
≤
Ḃp,q
2 0
0
(2.44)
and
1Z t
k(∇w, ∇E)k2L2 dτ
E F : ∇w dx dτ ≤
2 0
3
R
Z t
+C
k(w, E)k2L2 kF kq −1+3/p+2/q dτ.
Z tZ
0
T
0
(2.45)
Ḃp,q
Returning back to the estimate (2.29) and putting the above estimates (2.42)–(2.45)
and (2.34) together, we obtain
Z t
k(w(t), E(t))k2L2 + 2
k(∇w(τ ), ∇E(τ ))k2L2 dτ
0
Z t
(2.46)
≤ k(w0 , E0 )k2L2 + C
(kwk2L2 + kEk2L2 )(1 + kukq −1+3/p+2/q
+
Ḃp,q
0
kF kq −1+3/p+2/q )dτ.
Ḃp,q
Applying the Gronwall inequality, we obtain (2.36) immediately. The proof of
Proposition 2.4 is complete.
3. Appendix
In this section we shall establish the basic energy inequality (see Definition 1.1)
governing the system (1.1). In order to do so, let us consider a classical solution
(u, d) of the problem (1.1). We first multiply the first equation of (1.1) by u,
2
integrate over R3 , and use the fact ∇ · (∇d ∇d) = ∇( |∇d|
2 ) + ∆d · ∇d, we see
that
1 d
kuk2L2 + k∇uk2L2 + (∆d · ∇d, u) = 0.
(3.1)
2 dt
Next, we multiply the second equation of (1.1) by −∆d + g(d), integrate over R3 ,
and use the fact that (u · ∇d, g(d)) = (u, ∇G(d)) = 0, we see that
Z
1 d
d
k∇dk2L2 +
G(d)dx + k∆d − g(d)k2L2 − (u · ∇d, ∆d) = 0.
(3.2)
2 dt
dt R3
Equations (3.1) and (3.2) together imply
Z
1 d
2
2
kukL2 + k∇dkL2 + 2
G(d)dx + k∇uk2L2 + k∆d − g(d)k2L2 = 0. (3.3)
2 dt
R3
R
1
Note that R3 G(d)dx = 4 kd(t)k4L4 − 21 kd(t)k2L2 . Hence, in order to calculate the
R
d
term dt
G(d)dx, we multiply the second equation of (1.1) by d to yield that
R3
Z
1 d
2
2
kdkL2 + k∇dkL2 +
g(d) · ddx = 0;
2 dt
R3
i.e.,
1 d
kdk2L2 + k∇dk2L2 + kdk4L4 = kdk2L2 .
(3.4)
2 dt
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WEAK-STRONG UNIQUENESS OF HYDRODYNAMIC FLOW
Similarly, multiplying the second equation of (1.1) by |d|2 d, we obtain
1 d
kdk4L4 + 3kd · ∇dk2L2 + kdk6L6 = kdk4L4 .
4 dt
On the other hand, it is obvious that
15
(3.5)
k∆d − g(d)k2L2 = (∆d − |d|2 d + d, ∆d − |d|2 d + d)
= k∆dk2L2 − 2(∆d, |d|2 d) + 2(∆d, d) − 2(|d|2 d, d) + (|d|2 d, |d|2 d) + (d, d)
= k∆dk2L2 + 6k|d|∇dk2L2 − 2k∇dk2L2 − 2kdk4L4 + kdk6L6 + kdk2L2 .
Putting the estimates (3.3)–(3.5) together, we obtain
1 d
kuk2L2 + k∇dk2L2 + k∇uk2L2 + k∆dk2L2 + 3k|d|∇dk2L2 = k∇dk2L2 . (3.6)
2 dt
This yields immediately the energy inequality in Definition 1.1. Finally, by applying
the Gronwall inequality, we obtain the following basic energy inequality:
Z t
2
2
ku(t)kL2 + k∇d(t)kL2 +
k∇u(τ )k2L2 + k∆d(τ )k2L2 dτ
(3.7)
0
≤ C(ku0 k2L2 , k∇d0 k2L2 )e2t .
Combining the above energy estimate, the Galerkin approximate procedure and the
compactness argument give global existence of weak solutions of (1.1).
Acknowledgments. This research was supported by the National Natural Science
Foundation of China (11171357) and by the Doctoral Fund of Northwest A&F
University (Z109021118).
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Ji-hong Zhao
College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China
E-mail address: zhaojihong2007@yahoo.com.cn
Qiao Liu
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China
E-mail address: liuqao2005@163.com