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Global strong solution to the 2D density-dependent liquid crystal flows with vacuum
Nonlinear Analysis 97 (2014) 185–190
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Global strong solution to the 2D density-dependent liquid crystal flows with vacuum Jishan Fan a , Fucai Li b,∗ , Gen Nakamura c a
Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, PR China
b
Department of Mathematics, Nanjing University, Nanjing 210093, PR China
c
Department of Mathematics, Inha University, Incheon 402-751, Republic of Korea
article
abstract
info
Article history: Received 8 October 2013 Accepted 20 November 2013 Communicated by Enzo Mitidieri
We establish the global existence and uniqueness of strong solutions to the 2D densitydependent liquid crystal flows with vacuum in a bounded smooth domain. © 2013 Elsevier Ltd. All rights reserved.
Dedicated to Professor Yuanming Wang on the occasion of his 80th birthday MSC: 35Q30 76D03 76D09 Keywords: 2D density-dependent liquid crystal flows Vacuum Global strong solution
1. Introduction In this paper we investigate the global existence and uniqueness of strong solutions to the density-dependent incompressible liquid crystal flow [1–4]:
∂t ρ + div (ρ u) = 0, ∂t (ρ u) + div (ρ u ⊗ u) + ∇π − ∆u = −∇ · (∇ d ⊙ ∇ d),
(1.1)
∂t d + u · ∇ d − ∆d = |∇ d| d,
(1.3)
div u = 0,
(1.4)
2
(1.2)
in a bounded domain Ω ⊂ R with smooth boundary ∂ Ω . Here ρ denotes the density, u the velocity, d the unit vector field which represents the macroscopic orientations, and π the pressure. The symbol ∇ d ⊙ ∇ d denotes a matrix whose (i, j)-th entry is ∂i d∂j d. The system (1.1)–(1.4) is supplemented with the following initial and boundary conditions 2
(ρ, ρ u, d)(0, ·) = (ρ0 , ρ0 u0 , d0 ) in Ω , u = 0, ∂ν d = 0 on (0, ∞) × ∂ Ω , where ν is the unit outward normal vector to ∂ Ω . ∗
Corresponding author. Tel.: +86 2583592861. E-mail addresses: [email protected] (J. Fan), [email protected], [email protected] (F. Li), [email protected] (G. Nakamura).
0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.11.024
(1.5) (1.6)
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J. Fan et al. / Nonlinear Analysis 97 (2014) 185–190
When d is a given constant unit vector, (1.1)–(1.4) is reduced to the well-known density-dependent Navier–Stokes system. For some mathematical results on this system, the interested readers can refer [5–9] and the references therein. When ρ ≡ 1 and Ω = R2 , Xu and Zhang [10] obtained the global existence of weak solutions to the system (1.2)–(1.4) if u0 ∈ L2 (R2 ), ∇ d0 ∈ L2 (R2 ), |d0 | = 1, and
1 2 1 ∥∇ d0 ∥2L2 (R2 ) exp 216 ∥u0 ∥2L2 (R2 ) + . < 16
(1.7)
16
When ρ = 1 and the nonlinear term |∇ d|2 d in (1.3) is replaced by (1 −|d|2 )d, we then obtain the so-called Ginzburg–Landau approximation model of liquid crystal flow and many results are available [11–16]. Very recently, Wen and Ding [17] obtained the local strong solutions to the problem (1.1)–(1.6) with vacuum. Zhou, Fan, and Nakamura [18] showed the global strong solutions to the problem (1.1)–(1.6) with the positive initial density. The aim of this paper is to generalize the results in [18] to the case with vacuum. Our results read as follows. Theorem 1.1. Let 0 ≤ ρ0 ≤ M for some M > 0, ρ0 ∈ W 1,r (Ω ) for some r ∈ (2, ∞), u0 ∈ H01 (Ω ) ∩ H 2 (Ω ), and d0 ∈ H 3 (Ω ) with div u0 = 0, |d0 | = 1 in Ω and ∂ν d0 = 0 on ∂ Ω . Assume that there exists (∇π0 , g ) ∈ L2 (Ω ) such that
∇π0 − ∆u0 + ∇ · (∇ d0 ⊙ ∇ d0 ) =
√ ρ0 g in Ω ,
(1.8)
and that
1 ∥∇ d0 ∥2L2 (Ω ) exp 2Λ20 (ρ0 u20 + |∇ d0 |2 )dx ≤ 8Λ20 Ω
(1.9)
for some Λ0 (see Remark 1.1), then the problem (1.1)–(1.6) has a unique global strong solution (ρ, u, d) satisfying
∥ρ∥L∞ (0,T ;W 1,r (Ω )) ≤ C ,
∥ρt ∥L∞ (0,T ;Lr (Ω )) ≤ C ,
√ ρ ut ∞ L
(0,T ;L2 (Ω ))
≤ C,
∥u∥L∞ (0,T ;H 2 (Ω ))∩L2 (0,T ;W 2,s (Ω )) ≤ C for some s > 2, ∥ut ∥L2 (0,T ;H 1 (Ω )) ≤ C , ∥d∥L∞ (0,T ;H 3 (Ω )) ≤ C , ∥dt ∥L∞ (0,T ;H 1 (Ω ))∩L2 (0,T ;H 2 (Ω )) ≤ C .
(1.10)
Remark 1.1. The constant Λ0 in Theorem 1.1 is defined through the Gagliardo–Nirenberg inequality:
∥∇ d∥2L4 (Ω ) ≤ Λ0 ∥∇ d∥L2 ∥∆d∥L2 (Ω ) .
(1.11)
√
Remark 1.2. Roughly speaking, (1.8) is equivalent to the L2 -integrability of ρ ut at t = 0, as can be shown formally by letting t → 0 in (1.2). On the other hand, for any given (ρ0 , d0 ) in Theorem 1.1 and any given g ∈ L2 (Ω ), there exists a unique (u0 , ∇π0 ) ∈ H01 (Ω ) ∩ H 2 (Ω ) × L2 (Ω ) satisfying (1.8) and div u0 = 0 in Ω . Based on these facts, we can easily construct some interesting examples of initial data with vacuum satisfying the compatibility condition (1.8). The remainder of this paper is devoted to proving Theorem 1.1. Below for notation simplicity, we omit the integral domain
Ω . We use C and Ci (i ≥ 1) to denote the constants which may change from line to line. 2. Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1. Since the local strong solutions to the problem (1.1)–(1.6) was established by Wen and Ding [17], the key step in the proof of Theorem 1.1 is to prove a priori estimates. In fact, the existence part follows easily from the a priori estimates (1.10) and the standard Aubin–Lions compactness principle. To prove the uniqueness part of Theorem 1.1, we make a difference of two solutions (ρ1 , u1 , d1 ) and (ρ2 , u2 , d2 ) as (ρ, ˜ u˜ , d˜ ) := (ρ1 −ρ2 , u1 − u2 , d1 − d2 ) which satisfy a differential system similar to (1.1)–(1.6) with additional lower order terms. Because we consider the strong solution case, the uniqueness comes from the standard arguments (taking basic energy estimates and applying Gronwall inequality) and thus we omit the details here. Now we turn to the proof of a priori estimates for the problem (1.1)–(1.6). In the following calculations, we follow the idea developed in [8,9] where the density-dependent Navier–Stokes system was investigated. First, thanks to the maximum principle for the equation satisfied by |d|2 − 1, it is easy to infer that |d| = 1 is kept if |d0 | = 1. It follows from (1.1) and (1.4) that 0 ≤ ρ ≤ M < ∞.
(2.1)
Testing (1.2) by u and using (1.1) and (1.4), we see that 1 d 2 dt
ρ|u|2 dx +
|∇ u|2 dx = −
(u · ∇)d · ∆d dx.
(2.2)
J. Fan et al. / Nonlinear Analysis 97 (2014) 185–190
187
Testing (1.3) by −∆d − |∇ d|2 d and using the facts that |d| = 1 and d · dt = 0, we find that 1 d
2 dt
|∇ d|2 dx +
|∆d + |∇ d |2 d|2 dx =
(u · ∇)d · ∆d dx.
(2.3)
Summing up (2.2) and (2.3) and integrating over (0, T ), we infer that
(ρ|u|2 + |∇ d|2 )dx + 2
T
(|∇ u|2 + |∆d + |∇ d |2 d|2 )dxdt ≤
(ρ0 |u0 |2 + |∇ d0 |2 )dx.
(2.4)
0
It follows from (2.3) and (1.11) that 1 d
2 dt
|∇ d|2 dx +
|∆d + |∇ d |2 d|2 dx =
u∇ di ∂j2 di dx
i ,j
=−
∂j u∇ di ∂j di dx
i ,j
≤ ∥∇ u∥L2 ∥∇ d∥2L4 ≤ Λ0 ∥∇ u∥L2 ∥∇ d∥L2 ∥∆d∥L2 ≤ On the other hand, since (a + b)2 ≥
|∆d + |∇ d |2 d|2 dx ≥ ≥
1 2 1 2
a2 2
1 8
∥∆d∥2L2 + 2Λ20 ∥∇ u∥2L2 ∥∇ d∥2L2 .
(2.5)
− b2 , we get
∥∆d∥2L2 − ∥∇ d∥4L4 ∥∆d∥2L2 − Λ20 ∥∇ d∥2L2 ∥∆d∥2L2 .
(2.6)
Combining (2.5) with (2.6), we deduce that 1 d
2 dt
3
|∇ d|2 dx +
8
|∆d|2 dx ≤ Λ20 ∥∇ d∥2L2 ∥∆d∥2L2 + 2Λ20 ∥∇ u∥2L2 ∥∇ d∥2L2 .
(2.7)
If the initial datum d0 satisfies that
∥∇ d0 ∥2L2 ≤
1 8Λ20
,
then, by continuity, there exists a T1 > 0 such that, for any t ∈ [0, T1 ],
∥∇ d(t )∥2L2 ≤
1 4Λ20
.
(2.8)
We denote by T1∗ the maximal time such that (2.8) holds on [0, T1∗ ]. Therefore, it follows from (2.7) and (2.8) that d
dt
1
|∇ d|2 dx + ∥∆d∥2L2 ≤ 4Λ20 ∥∇ u∥2L2 ∥∇ d∥2L2 , 4
(2.9)
which gives
∥∇ d(t )∥
2 L2
d0 2L2
≤ ∥∇ ∥ exp
4Λ20
T1∗
∥∇ u∥ 0
2 dt L2
≤ ∥∇ d0 ∥2L2 exp 2Λ20 (ρ0 u20 + |∇ d0 |2 )dx ≤
1 8Λ20
.
(2.10)
(2.10) implies T1∗ = T if (1.9) holds true. Combining (2.9) with (2.10) gives
∥d∥L2 (0,T ;H 2 (Ω )) ≤ C .
(2.11)
Let T ∗ be a maximal existence time for the solution (ρ, u, d). Then (2.1), (2.4), (2.10) and (2.11) ensure that T ∗ = ∞ by continuity argument.
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J. Fan et al. / Nonlinear Analysis 97 (2014) 185–190
Testing (1.2) by ut , using (1.1), (1.4) and (2.1), we derive that
d |∇ u|2 dx + ρ|ut |2 dx − ∇ d ⊙ ∇ d : ∇ udx 2 dt dt = − ρ u · ∇ u · ut dx − ∂t (∇ d ⊙ ∇ d) : ∇ udx √ √ ≤ ρ ut L2 ρ L∞ ∥u∥L∞ ∥∇ u∥L2 + 2∥∇ d∥L∞ ∥∇ dt ∥L2 ∥∇ u∥L2 √ 2 ≤ δ ρ ut L2 + δ∥∇ dt ∥2L2 + C (∥u∥2L∞ + ∥∇ d∥2L∞ )∥∇ u∥2L2
1 d
(2.12)
for any 0 < δ < 1. Testing (1.3) by −∆dt , using the interpolation inequality (1.11), we obtain that
1 d
|∆d|2 dx +
2 dt
|∇ dt |2 dx =
∇(|∇ d|2 d − u · ∇ d) · ∇ dt dx
≤ C (∥∇ d∥L∞ ∥∇ d∥2L4 + ∥∇ d∥L∞ ∥∆d∥L2 + ∥u∥L∞ ∥∆d∥L2 + ∥∇ u∥L2 ∥∇ d∥L∞ )∥∇ dt ∥L2 ≤ C (∥∇ d∥2L∞ ∥∆d∥2L2 + ∥u∥2L∞ ∥∆d∥2L2 + ∥∇ u∥2L2 ∥∇ d∥2L∞ ) + δ∥∇ dt ∥2L2
(2.13)
for any 0 < δ < 1. Combining (2.12) with (2.13), taking δ suitably small and C1 suitably large, and using the following logarithmic Sobolev inequality [19]:
∥f ∥L∞ (Ω ) ≤ C {1 + ∥f ∥H 1 (Ω ) log1/2 (e + ∥f ∥W 1,p (Ω ) )}
(2.14)
for any 2 < p < ∞, we have
1 d
|∇ u|2 dx −
2 dt
d
1
d
2
dt
∇ d ⊙ ∇ d : ∇ udx + C1
dt
|∆d|2 dx + C1
|∇ dt |2 dx +
ρ|ut |2 dx
≤ C (∥u∥2L∞ + ∥∇ d∥2L∞ )(∥∇ u∥2L2 + ∥∆d∥2L2 ) ≤ C [1 + (∥u∥2H 1 + ∥d∥2H 2 ) log(e + ∥u∥H 2 + ∥d∥H 3 )](∥∇ u∥2L2 + ∥∆d∥2L2 ), which implies that
(|∇ u|2 + |∆d|2 )dx +
t
(ρ|ut |2 + |∇ dt |2 )dxdτ ≤ C (e + y(t ))C2 ϵ ,
(2.15)
t0
provided that
T t0
(∥u∥2H 1 + ∥d∥2H 2 )dt ≤ ϵ ≪ 1
(2.16)
with y(t ) := sup (∥u∥H 2 + ∥d∥H 3 ) [t0 ,t ]
and C2 is an absolute constant. It follows from (1.2), (1.3), (2.15) and (2.16) that
t
(|∆u|2 + |∇ ∆d|2 )dxdτ ≤ C (e + y(t ))C2 ϵ .
(2.17)
t0
Taking operator ∂t to (1.2), testing by ut , using (1.1) and (1.4), we get 1 d 2 dt
ρ|ut | dx + 2
2
|∇ ut | dx = − −
ρt |ut | dx −
ρt u · ∇ u · ut dx ρ ut · ∇ u · ut dx + ∂t (∇ d ⊙ ∇ d) : ∇ ut dx 2
=: I1 + I2 + I3 + I4 .
(2.18)
J. Fan et al. / Nonlinear Analysis 97 (2014) 185–190
189
We use (2.1), (1.11) and the Hölder inequality to bound Ii (i = 1, 2, 3, 4) as follows:
√ ρ u · ∇|ut |2 dx ≤ C ∥u∥L6 ρ ut L3 ∥∇ ut ∥L2 √ 1 √ 1 C ∥u∥L6 ρ ut L22 ρ ut L26 ∥∇ ut ∥L2 1 √ 1 C ∥u∥L6 ρ ut L22 ∥ut ∥L26 ∥∇ ut ∥L2 3 √ 1 C ∥u∥L6 ρ ut L22 ∥∇ ut ∥L22 √ 2 1 ∥∇ ut ∥2L2 + C ∥u∥4L6 ρ ut L2 , 16 − ρ u · ∇(u · ∇ u · ut )dx √ √ C ρ ut L3 ∥u∥L6 ∥∇ u∥2L4 + C ρ ut L3 ∥∆u∥L2 ∥u∥2L12 + C ∥∇ ut ∥L2 ∥∇ u∥L4 ∥u∥2L8 5 1 √ C ρ ut L3 ∥u∥2H 1 ∥u∥H 2 + C ∥∇ ut ∥L2 ∥u∥H2 1 ∥u∥H2 2 √ 2 1 ∥∇ ut ∥2L2 + C ρ ut L2 + C ∥u∥4H 1 ∥u∥2H 2 + C ∥u∥5H 1 ∥u∥H 2 , 16 √ 2 C ρ ut L4 ∥∇ u∥L2 √ 2 √ 4 C ρ ut L32 ρ ut L38 ∥∇ u∥L2 4 √ 2 C ρ ut L32 ∥ut ∥L38 ∥∇ u∥L2 4 √ 2 C ρ ut L32 ∥∇ ut ∥L32 ∥∇ u∥L2 √ 2 1 ∥∇ ut ∥2L2 + C ∥∇ u∥3L2 ρ ut L2 ,
I1 = −
≤ ≤ ≤ ≤ I2 =
≤ ≤ ≤ I3 ≤
≤ ≤ ≤ ≤
16 I4 ≤ 2∥∇ d∥L4 ∥∇ dt ∥L4 ∥∇ ut ∥L2 1
1
1
≤ C ∥∆d∥L22 ∥∇ dt ∥L22 ∥∆dt ∥L22 ∥∇ ut ∥L2 ≤
1
16
∥∇ ut ∥2L2 +
1
16
∥∆dt ∥2L2 + C ∥∆d∥2L2 ∥∇ dt ∥2L2 .
Plugging the above estimates into (2.18) we arrive at 1 d
2 dt
ρ|ut |2 dx +
1 4
1
|∇ ut |2 dx ≤
16
√ 2 √ 2 ∥∆dt ∥2L2 + C ∥u∥4L6 ρ ut L2 + C ρ ut L2 + C ∥u∥4H 1 ∥u∥2H 2
√ 2 + C ∥u∥5H 1 ∥u∥H 2 + C ∥∇ u∥3L2 ρ ut L2 + C ∥∆d∥2L2 ∥∇ dt ∥2L2 .
(2.19)
Applying operator ∂t to (1.3), testing by −∆dt , and using (1.11) and the Hölder inequality, we have 1 d 2 dt
2
|∇ dt | dx +
(ut · ∇ d + u · ∇ dt − |∇ d|2 dt − d∂t |∇ d|2 )∆dt dx = − ∇(ut · ∇ d) · ∇ dt dx + (u · ∇ dt − |∇ d|2 dt − d∂t |∇ d|2 )∆dt dx
|∆dt | dx = 2
≤ C ∥∇ ut ∥L2 ∥∇ d∥L4 ∥∇ dt ∥L4 + C ∥ut ∥L4 ∥∆d∥L2 ∥∇ dt ∥L4 + C ∥u∥L4 ∥∇ dt ∥L4 ∥∆dt ∥L2 + C ∥∇ d∥2L6 ∥dt ∥L6 ∥∆dt ∥L2 + C ∥∇ d∥L4 ∥∇ dt ∥L4 ∥∆dt ∥L2 1
1
1
3
≤ C ∥∇ ut ∥L2 ∥d∥H 2 ∥∇ dt ∥L22 ∥∆dt ∥L22 + C ∥u∥L4 ∥∇ dt ∥L22 ∥∆dt ∥L22 1
3
+ C ∥∇ d∥2L6 ∥dt ∥L6 ∥∆dt ∥L2 + C ∥∇ d∥L4 ∥∇ dt ∥L22 ∥∆dt ∥L22 ≤
1 16
∥∆dt ∥2L2 +
1 16
∥∇ ut ∥2L2 + C ∥d∥4H 2 ∥∇ dt ∥2L2
+ C ∥u∥4L4 ∥∇ dt ∥2L2 + C ∥∇ d∥4L6 (∥dt ∥2L2 + ∥∇ dt ∥2L2 ) + C ∥∇ d∥4L4 ∥∇ dt ∥2L2 . Since
dt dx = |∇ d|2 d dx ≤ C , Ω
Ω
(2.20)
190
J. Fan et al. / Nonlinear Analysis 97 (2014) 185–190
we have
∥dt ∥L2 ≤ C + ∥∇ dt ∥L2 .
(2.21)
Combining (2.19) and (2.20) with (2.21), using (2.15) and (2.17), and integrating over [t0 , t ], we conclude that
(ρ|ut |2 + |∇ dt |2 )dx +
t
(|∇ ut |2 + |∆dt |2 )dxdτ ≤ C (e + y(t ))C2 ϵ .
(2.22)
t0
It follows from (1.2), (2.1), (2.15), (2.17), (2.22), and the H 2 -theory of Stokes system that
√ ∥u∥H 2 ≤ C ρ ut L2 + C ∥ρ u · ∇ u∥L2 + C ∥∇ d · ∆d∥L2 + C √ ≤ C + C ρ ut 2 + C ∥u∥L6 ∥∇ u∥L3 + C ∥∇ d∥L4 ∥∆d∥L4 , L
which yields 1
∥u∥H 2 ≤ C (e + y(t ))C2 ϵ + ∥d∥H 3 . 2
(2.23)
Similarly, it follows from (1.3), (2.15), (2.17) and (2.22) that
∥d∥H 3 ≤ C + C ∥∇ dt + ∇(u · ∇ d − |∇ d|2 d)∥L2 ≤ C + C ∥∇ dt ∥L2 + C ∥∇ u∥L2 ∥∇ d∥L∞ + C ∥u∥L4 ∥∆d∥L4 + C ∥∇ d∥3L6 + C ∥∇ d∥L4 ∥∆d∥L4 , which implies
∥d∥H 3 ≤ C (e + y(t ))C2 ϵ .
(2.24)
Combining (2.23) with (2.24), we arrive at
∥u∥L∞ (0,T ;H 2 (Ω )) + ∥d∥L∞ (0,T ;H 3 (Ω )) ≤ C . Now it is standard to prove that (1.10) holds true, and thus we omit the details here. This completes the proof.
(2.25)
Acknowledgments The authors are indebted to the referee for many constructive suggestions which considerably improved the presentations of the paper. The first author was supported by NSFC (Grant No. 11171154). The second author was supported by NSFC (Grant No. 11271184), NCET-11-0227, PAPD, and the Fundamental Research Funds for the Central Universities. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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Global strong solution to the 2D density-dependent liquid crystal flows with vacuum Jishan Fan a , Fucai Li b,∗ , Gen Nakamura c a
Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, PR China
b
Department of Mathematics, Nanjing University, Nanjing 210093, PR China
c
Department of Mathematics, Inha University, Incheon 402-751, Republic of Korea
article
abstract
info
Article history: Received 8 October 2013 Accepted 20 November 2013 Communicated by Enzo Mitidieri
We establish the global existence and uniqueness of strong solutions to the 2D densitydependent liquid crystal flows with vacuum in a bounded smooth domain. © 2013 Elsevier Ltd. All rights reserved.
Dedicated to Professor Yuanming Wang on the occasion of his 80th birthday MSC: 35Q30 76D03 76D09 Keywords: 2D density-dependent liquid crystal flows Vacuum Global strong solution
1. Introduction In this paper we investigate the global existence and uniqueness of strong solutions to the density-dependent incompressible liquid crystal flow [1–4]:
∂t ρ + div (ρ u) = 0, ∂t (ρ u) + div (ρ u ⊗ u) + ∇π − ∆u = −∇ · (∇ d ⊙ ∇ d),
(1.1)
∂t d + u · ∇ d − ∆d = |∇ d| d,
(1.3)
div u = 0,
(1.4)
2
(1.2)
in a bounded domain Ω ⊂ R with smooth boundary ∂ Ω . Here ρ denotes the density, u the velocity, d the unit vector field which represents the macroscopic orientations, and π the pressure. The symbol ∇ d ⊙ ∇ d denotes a matrix whose (i, j)-th entry is ∂i d∂j d. The system (1.1)–(1.4) is supplemented with the following initial and boundary conditions 2
(ρ, ρ u, d)(0, ·) = (ρ0 , ρ0 u0 , d0 ) in Ω , u = 0, ∂ν d = 0 on (0, ∞) × ∂ Ω , where ν is the unit outward normal vector to ∂ Ω . ∗
Corresponding author. Tel.: +86 2583592861. E-mail addresses: [email protected] (J. Fan), [email protected], [email protected] (F. Li), [email protected] (G. Nakamura).
0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.11.024
(1.5) (1.6)
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J. Fan et al. / Nonlinear Analysis 97 (2014) 185–190
When d is a given constant unit vector, (1.1)–(1.4) is reduced to the well-known density-dependent Navier–Stokes system. For some mathematical results on this system, the interested readers can refer [5–9] and the references therein. When ρ ≡ 1 and Ω = R2 , Xu and Zhang [10] obtained the global existence of weak solutions to the system (1.2)–(1.4) if u0 ∈ L2 (R2 ), ∇ d0 ∈ L2 (R2 ), |d0 | = 1, and
1 2 1 ∥∇ d0 ∥2L2 (R2 ) exp 216 ∥u0 ∥2L2 (R2 ) + . < 16
(1.7)
16
When ρ = 1 and the nonlinear term |∇ d|2 d in (1.3) is replaced by (1 −|d|2 )d, we then obtain the so-called Ginzburg–Landau approximation model of liquid crystal flow and many results are available [11–16]. Very recently, Wen and Ding [17] obtained the local strong solutions to the problem (1.1)–(1.6) with vacuum. Zhou, Fan, and Nakamura [18] showed the global strong solutions to the problem (1.1)–(1.6) with the positive initial density. The aim of this paper is to generalize the results in [18] to the case with vacuum. Our results read as follows. Theorem 1.1. Let 0 ≤ ρ0 ≤ M for some M > 0, ρ0 ∈ W 1,r (Ω ) for some r ∈ (2, ∞), u0 ∈ H01 (Ω ) ∩ H 2 (Ω ), and d0 ∈ H 3 (Ω ) with div u0 = 0, |d0 | = 1 in Ω and ∂ν d0 = 0 on ∂ Ω . Assume that there exists (∇π0 , g ) ∈ L2 (Ω ) such that
∇π0 − ∆u0 + ∇ · (∇ d0 ⊙ ∇ d0 ) =
√ ρ0 g in Ω ,
(1.8)
and that
1 ∥∇ d0 ∥2L2 (Ω ) exp 2Λ20 (ρ0 u20 + |∇ d0 |2 )dx ≤ 8Λ20 Ω
(1.9)
for some Λ0 (see Remark 1.1), then the problem (1.1)–(1.6) has a unique global strong solution (ρ, u, d) satisfying
∥ρ∥L∞ (0,T ;W 1,r (Ω )) ≤ C ,
∥ρt ∥L∞ (0,T ;Lr (Ω )) ≤ C ,
√ ρ ut ∞ L
(0,T ;L2 (Ω ))
≤ C,
∥u∥L∞ (0,T ;H 2 (Ω ))∩L2 (0,T ;W 2,s (Ω )) ≤ C for some s > 2, ∥ut ∥L2 (0,T ;H 1 (Ω )) ≤ C , ∥d∥L∞ (0,T ;H 3 (Ω )) ≤ C , ∥dt ∥L∞ (0,T ;H 1 (Ω ))∩L2 (0,T ;H 2 (Ω )) ≤ C .
(1.10)
Remark 1.1. The constant Λ0 in Theorem 1.1 is defined through the Gagliardo–Nirenberg inequality:
∥∇ d∥2L4 (Ω ) ≤ Λ0 ∥∇ d∥L2 ∥∆d∥L2 (Ω ) .
(1.11)
√
Remark 1.2. Roughly speaking, (1.8) is equivalent to the L2 -integrability of ρ ut at t = 0, as can be shown formally by letting t → 0 in (1.2). On the other hand, for any given (ρ0 , d0 ) in Theorem 1.1 and any given g ∈ L2 (Ω ), there exists a unique (u0 , ∇π0 ) ∈ H01 (Ω ) ∩ H 2 (Ω ) × L2 (Ω ) satisfying (1.8) and div u0 = 0 in Ω . Based on these facts, we can easily construct some interesting examples of initial data with vacuum satisfying the compatibility condition (1.8). The remainder of this paper is devoted to proving Theorem 1.1. Below for notation simplicity, we omit the integral domain
Ω . We use C and Ci (i ≥ 1) to denote the constants which may change from line to line. 2. Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1.1. Since the local strong solutions to the problem (1.1)–(1.6) was established by Wen and Ding [17], the key step in the proof of Theorem 1.1 is to prove a priori estimates. In fact, the existence part follows easily from the a priori estimates (1.10) and the standard Aubin–Lions compactness principle. To prove the uniqueness part of Theorem 1.1, we make a difference of two solutions (ρ1 , u1 , d1 ) and (ρ2 , u2 , d2 ) as (ρ, ˜ u˜ , d˜ ) := (ρ1 −ρ2 , u1 − u2 , d1 − d2 ) which satisfy a differential system similar to (1.1)–(1.6) with additional lower order terms. Because we consider the strong solution case, the uniqueness comes from the standard arguments (taking basic energy estimates and applying Gronwall inequality) and thus we omit the details here. Now we turn to the proof of a priori estimates for the problem (1.1)–(1.6). In the following calculations, we follow the idea developed in [8,9] where the density-dependent Navier–Stokes system was investigated. First, thanks to the maximum principle for the equation satisfied by |d|2 − 1, it is easy to infer that |d| = 1 is kept if |d0 | = 1. It follows from (1.1) and (1.4) that 0 ≤ ρ ≤ M < ∞.
(2.1)
Testing (1.2) by u and using (1.1) and (1.4), we see that 1 d 2 dt
ρ|u|2 dx +
|∇ u|2 dx = −
(u · ∇)d · ∆d dx.
(2.2)
J. Fan et al. / Nonlinear Analysis 97 (2014) 185–190
187
Testing (1.3) by −∆d − |∇ d|2 d and using the facts that |d| = 1 and d · dt = 0, we find that 1 d
2 dt
|∇ d|2 dx +
|∆d + |∇ d |2 d|2 dx =
(u · ∇)d · ∆d dx.
(2.3)
Summing up (2.2) and (2.3) and integrating over (0, T ), we infer that
(ρ|u|2 + |∇ d|2 )dx + 2
T
(|∇ u|2 + |∆d + |∇ d |2 d|2 )dxdt ≤
(ρ0 |u0 |2 + |∇ d0 |2 )dx.
(2.4)
0
It follows from (2.3) and (1.11) that 1 d
2 dt
|∇ d|2 dx +
|∆d + |∇ d |2 d|2 dx =
u∇ di ∂j2 di dx
i ,j
=−
∂j u∇ di ∂j di dx
i ,j
≤ ∥∇ u∥L2 ∥∇ d∥2L4 ≤ Λ0 ∥∇ u∥L2 ∥∇ d∥L2 ∥∆d∥L2 ≤ On the other hand, since (a + b)2 ≥
|∆d + |∇ d |2 d|2 dx ≥ ≥
1 2 1 2
a2 2
1 8
∥∆d∥2L2 + 2Λ20 ∥∇ u∥2L2 ∥∇ d∥2L2 .
(2.5)
− b2 , we get
∥∆d∥2L2 − ∥∇ d∥4L4 ∥∆d∥2L2 − Λ20 ∥∇ d∥2L2 ∥∆d∥2L2 .
(2.6)
Combining (2.5) with (2.6), we deduce that 1 d
2 dt
3
|∇ d|2 dx +
8
|∆d|2 dx ≤ Λ20 ∥∇ d∥2L2 ∥∆d∥2L2 + 2Λ20 ∥∇ u∥2L2 ∥∇ d∥2L2 .
(2.7)
If the initial datum d0 satisfies that
∥∇ d0 ∥2L2 ≤
1 8Λ20
,
then, by continuity, there exists a T1 > 0 such that, for any t ∈ [0, T1 ],
∥∇ d(t )∥2L2 ≤
1 4Λ20
.
(2.8)
We denote by T1∗ the maximal time such that (2.8) holds on [0, T1∗ ]. Therefore, it follows from (2.7) and (2.8) that d
dt
1
|∇ d|2 dx + ∥∆d∥2L2 ≤ 4Λ20 ∥∇ u∥2L2 ∥∇ d∥2L2 , 4
(2.9)
which gives
∥∇ d(t )∥
2 L2
d0 2L2
≤ ∥∇ ∥ exp
4Λ20
T1∗
∥∇ u∥ 0
2 dt L2
≤ ∥∇ d0 ∥2L2 exp 2Λ20 (ρ0 u20 + |∇ d0 |2 )dx ≤
1 8Λ20
.
(2.10)
(2.10) implies T1∗ = T if (1.9) holds true. Combining (2.9) with (2.10) gives
∥d∥L2 (0,T ;H 2 (Ω )) ≤ C .
(2.11)
Let T ∗ be a maximal existence time for the solution (ρ, u, d). Then (2.1), (2.4), (2.10) and (2.11) ensure that T ∗ = ∞ by continuity argument.
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J. Fan et al. / Nonlinear Analysis 97 (2014) 185–190
Testing (1.2) by ut , using (1.1), (1.4) and (2.1), we derive that
d |∇ u|2 dx + ρ|ut |2 dx − ∇ d ⊙ ∇ d : ∇ udx 2 dt dt = − ρ u · ∇ u · ut dx − ∂t (∇ d ⊙ ∇ d) : ∇ udx √ √ ≤ ρ ut L2 ρ L∞ ∥u∥L∞ ∥∇ u∥L2 + 2∥∇ d∥L∞ ∥∇ dt ∥L2 ∥∇ u∥L2 √ 2 ≤ δ ρ ut L2 + δ∥∇ dt ∥2L2 + C (∥u∥2L∞ + ∥∇ d∥2L∞ )∥∇ u∥2L2
1 d
(2.12)
for any 0 < δ < 1. Testing (1.3) by −∆dt , using the interpolation inequality (1.11), we obtain that
1 d
|∆d|2 dx +
2 dt
|∇ dt |2 dx =
∇(|∇ d|2 d − u · ∇ d) · ∇ dt dx
≤ C (∥∇ d∥L∞ ∥∇ d∥2L4 + ∥∇ d∥L∞ ∥∆d∥L2 + ∥u∥L∞ ∥∆d∥L2 + ∥∇ u∥L2 ∥∇ d∥L∞ )∥∇ dt ∥L2 ≤ C (∥∇ d∥2L∞ ∥∆d∥2L2 + ∥u∥2L∞ ∥∆d∥2L2 + ∥∇ u∥2L2 ∥∇ d∥2L∞ ) + δ∥∇ dt ∥2L2
(2.13)
for any 0 < δ < 1. Combining (2.12) with (2.13), taking δ suitably small and C1 suitably large, and using the following logarithmic Sobolev inequality [19]:
∥f ∥L∞ (Ω ) ≤ C {1 + ∥f ∥H 1 (Ω ) log1/2 (e + ∥f ∥W 1,p (Ω ) )}
(2.14)
for any 2 < p < ∞, we have
1 d
|∇ u|2 dx −
2 dt
d
1
d
2
dt
∇ d ⊙ ∇ d : ∇ udx + C1
dt
|∆d|2 dx + C1
|∇ dt |2 dx +
ρ|ut |2 dx
≤ C (∥u∥2L∞ + ∥∇ d∥2L∞ )(∥∇ u∥2L2 + ∥∆d∥2L2 ) ≤ C [1 + (∥u∥2H 1 + ∥d∥2H 2 ) log(e + ∥u∥H 2 + ∥d∥H 3 )](∥∇ u∥2L2 + ∥∆d∥2L2 ), which implies that
(|∇ u|2 + |∆d|2 )dx +
t
(ρ|ut |2 + |∇ dt |2 )dxdτ ≤ C (e + y(t ))C2 ϵ ,
(2.15)
t0
provided that
T t0
(∥u∥2H 1 + ∥d∥2H 2 )dt ≤ ϵ ≪ 1
(2.16)
with y(t ) := sup (∥u∥H 2 + ∥d∥H 3 ) [t0 ,t ]
and C2 is an absolute constant. It follows from (1.2), (1.3), (2.15) and (2.16) that
t
(|∆u|2 + |∇ ∆d|2 )dxdτ ≤ C (e + y(t ))C2 ϵ .
(2.17)
t0
Taking operator ∂t to (1.2), testing by ut , using (1.1) and (1.4), we get 1 d 2 dt
ρ|ut | dx + 2
2
|∇ ut | dx = − −
ρt |ut | dx −
ρt u · ∇ u · ut dx ρ ut · ∇ u · ut dx + ∂t (∇ d ⊙ ∇ d) : ∇ ut dx 2
=: I1 + I2 + I3 + I4 .
(2.18)
J. Fan et al. / Nonlinear Analysis 97 (2014) 185–190
189
We use (2.1), (1.11) and the Hölder inequality to bound Ii (i = 1, 2, 3, 4) as follows:
√ ρ u · ∇|ut |2 dx ≤ C ∥u∥L6 ρ ut L3 ∥∇ ut ∥L2 √ 1 √ 1 C ∥u∥L6 ρ ut L22 ρ ut L26 ∥∇ ut ∥L2 1 √ 1 C ∥u∥L6 ρ ut L22 ∥ut ∥L26 ∥∇ ut ∥L2 3 √ 1 C ∥u∥L6 ρ ut L22 ∥∇ ut ∥L22 √ 2 1 ∥∇ ut ∥2L2 + C ∥u∥4L6 ρ ut L2 , 16 − ρ u · ∇(u · ∇ u · ut )dx √ √ C ρ ut L3 ∥u∥L6 ∥∇ u∥2L4 + C ρ ut L3 ∥∆u∥L2 ∥u∥2L12 + C ∥∇ ut ∥L2 ∥∇ u∥L4 ∥u∥2L8 5 1 √ C ρ ut L3 ∥u∥2H 1 ∥u∥H 2 + C ∥∇ ut ∥L2 ∥u∥H2 1 ∥u∥H2 2 √ 2 1 ∥∇ ut ∥2L2 + C ρ ut L2 + C ∥u∥4H 1 ∥u∥2H 2 + C ∥u∥5H 1 ∥u∥H 2 , 16 √ 2 C ρ ut L4 ∥∇ u∥L2 √ 2 √ 4 C ρ ut L32 ρ ut L38 ∥∇ u∥L2 4 √ 2 C ρ ut L32 ∥ut ∥L38 ∥∇ u∥L2 4 √ 2 C ρ ut L32 ∥∇ ut ∥L32 ∥∇ u∥L2 √ 2 1 ∥∇ ut ∥2L2 + C ∥∇ u∥3L2 ρ ut L2 ,
I1 = −
≤ ≤ ≤ ≤ I2 =
≤ ≤ ≤ I3 ≤
≤ ≤ ≤ ≤
16 I4 ≤ 2∥∇ d∥L4 ∥∇ dt ∥L4 ∥∇ ut ∥L2 1
1
1
≤ C ∥∆d∥L22 ∥∇ dt ∥L22 ∥∆dt ∥L22 ∥∇ ut ∥L2 ≤
1
16
∥∇ ut ∥2L2 +
1
16
∥∆dt ∥2L2 + C ∥∆d∥2L2 ∥∇ dt ∥2L2 .
Plugging the above estimates into (2.18) we arrive at 1 d
2 dt
ρ|ut |2 dx +
1 4
1
|∇ ut |2 dx ≤
16
√ 2 √ 2 ∥∆dt ∥2L2 + C ∥u∥4L6 ρ ut L2 + C ρ ut L2 + C ∥u∥4H 1 ∥u∥2H 2
√ 2 + C ∥u∥5H 1 ∥u∥H 2 + C ∥∇ u∥3L2 ρ ut L2 + C ∥∆d∥2L2 ∥∇ dt ∥2L2 .
(2.19)
Applying operator ∂t to (1.3), testing by −∆dt , and using (1.11) and the Hölder inequality, we have 1 d 2 dt
2
|∇ dt | dx +
(ut · ∇ d + u · ∇ dt − |∇ d|2 dt − d∂t |∇ d|2 )∆dt dx = − ∇(ut · ∇ d) · ∇ dt dx + (u · ∇ dt − |∇ d|2 dt − d∂t |∇ d|2 )∆dt dx
|∆dt | dx = 2
≤ C ∥∇ ut ∥L2 ∥∇ d∥L4 ∥∇ dt ∥L4 + C ∥ut ∥L4 ∥∆d∥L2 ∥∇ dt ∥L4 + C ∥u∥L4 ∥∇ dt ∥L4 ∥∆dt ∥L2 + C ∥∇ d∥2L6 ∥dt ∥L6 ∥∆dt ∥L2 + C ∥∇ d∥L4 ∥∇ dt ∥L4 ∥∆dt ∥L2 1
1
1
3
≤ C ∥∇ ut ∥L2 ∥d∥H 2 ∥∇ dt ∥L22 ∥∆dt ∥L22 + C ∥u∥L4 ∥∇ dt ∥L22 ∥∆dt ∥L22 1
3
+ C ∥∇ d∥2L6 ∥dt ∥L6 ∥∆dt ∥L2 + C ∥∇ d∥L4 ∥∇ dt ∥L22 ∥∆dt ∥L22 ≤
1 16
∥∆dt ∥2L2 +
1 16
∥∇ ut ∥2L2 + C ∥d∥4H 2 ∥∇ dt ∥2L2
+ C ∥u∥4L4 ∥∇ dt ∥2L2 + C ∥∇ d∥4L6 (∥dt ∥2L2 + ∥∇ dt ∥2L2 ) + C ∥∇ d∥4L4 ∥∇ dt ∥2L2 . Since
dt dx = |∇ d|2 d dx ≤ C , Ω
Ω
(2.20)
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J. Fan et al. / Nonlinear Analysis 97 (2014) 185–190
we have
∥dt ∥L2 ≤ C + ∥∇ dt ∥L2 .
(2.21)
Combining (2.19) and (2.20) with (2.21), using (2.15) and (2.17), and integrating over [t0 , t ], we conclude that
(ρ|ut |2 + |∇ dt |2 )dx +
t
(|∇ ut |2 + |∆dt |2 )dxdτ ≤ C (e + y(t ))C2 ϵ .
(2.22)
t0
It follows from (1.2), (2.1), (2.15), (2.17), (2.22), and the H 2 -theory of Stokes system that
√ ∥u∥H 2 ≤ C ρ ut L2 + C ∥ρ u · ∇ u∥L2 + C ∥∇ d · ∆d∥L2 + C √ ≤ C + C ρ ut 2 + C ∥u∥L6 ∥∇ u∥L3 + C ∥∇ d∥L4 ∥∆d∥L4 , L
which yields 1
∥u∥H 2 ≤ C (e + y(t ))C2 ϵ + ∥d∥H 3 . 2
(2.23)
Similarly, it follows from (1.3), (2.15), (2.17) and (2.22) that
∥d∥H 3 ≤ C + C ∥∇ dt + ∇(u · ∇ d − |∇ d|2 d)∥L2 ≤ C + C ∥∇ dt ∥L2 + C ∥∇ u∥L2 ∥∇ d∥L∞ + C ∥u∥L4 ∥∆d∥L4 + C ∥∇ d∥3L6 + C ∥∇ d∥L4 ∥∆d∥L4 , which implies
∥d∥H 3 ≤ C (e + y(t ))C2 ϵ .
(2.24)
Combining (2.23) with (2.24), we arrive at
∥u∥L∞ (0,T ;H 2 (Ω )) + ∥d∥L∞ (0,T ;H 3 (Ω )) ≤ C . Now it is standard to prove that (1.10) holds true, and thus we omit the details here. This completes the proof.
(2.25)
Acknowledgments The authors are indebted to the referee for many constructive suggestions which considerably improved the presentations of the paper. The first author was supported by NSFC (Grant No. 11171154). The second author was supported by NSFC (Grant No. 11271184), NCET-11-0227, PAPD, and the Fundamental Research Funds for the Central Universities. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
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