# Global Strong Solution to the Density-Dependent 2-D Liquid Crystal Flows

Authors: Yong Zhou; Jishan Fan; Gen Nakamura
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The initial-boundary value problem for the density-dependent flow of nematic crystals is studied in a 2-D bounded smooth domain. For the initial density away from vacuum, the existence and uniqueness is proved for the global strong solution with the large initial velocityu0 and small ∇d0. We also give a regularity criterion ∇d∈Lp(0,T;Lq(Ω))(2/q)+(2/p)=1,2<q≤∞ of the problem with the Dirichlet boundary condition u=0, d=d0 on  ∂Ω.


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## Body

## 1. Introduction and Main Results

LetΩ⊆ℝ2 be a bounded domain with smooth boundary ∂Ω, and ν is the unit outward normal vector on ∂Ω. We consider the global strong solution to the density-dependent incompressible liquid crystal flow [1–4] as follows:
(1)divu=0,(2)∂tρ+div(ρu)=0,(3)∂t(ρu)+div(ρu⊗u)+∇π-Δu=-∇·(∇d⊙∇d),(4)∂td+u·∇d-Δd=|∇d|2d,
in (0,∞)×Ω with initial and boundary conditions
(5)(ρ,u,d)(·,0)=(ρ0,u0,d0)inΩ,(6)u=0,∂νd=0on∂Ω,
where ρ denotes the density, u the velocity, d the unit vector field that represents the macroscopic molecular orientations, and π the pressure. The symbol ∇d⊙∇d denotes a matrix whose (i,j)th entry is ∂id∂jd, and it is easy to find that ∇d⊙∇d=∇dT∇d.Whend is a given constant unit vector, then (1), (2), and (3) represent the well-known density-dependent Navier-Stokes system, which has received many studies; see [5–7] and references therein.Whenρ≡1 and Ω∶=ℝ2, Xu and Zhang [8] proved global existence of weak solutions to the problem if u0∈L2,∇d0∈L2,|d0|=1, and
(7)exp(216(∥u0∥L22+116)2)∥∇d0∥L22<116.Whenρ≡1 and (6) is replaced by
(8)u=0,d=d0on∂Ω.Lin et al. [9] proved the global existence of weak solutions to the system (1)–(5) and (8), which are smooth away from at most finitely many singular times, and they also prove a regularity criterion
(9)d∈L2(0,T;H2(Ω)).Whenρ=1 and the term |∇d|2 in (4) is replaced by (1-|d|2)d, then the problem has been studied in [10–15].Very recently, Wen and Ding [16] proved the global existence and uniqueness of strong solutions to the problem (1)–(6) with small u0 and ∇d0 and the local strong solutions with large initial data when Ω⊆ℝ2 is a smooth bounded domain.Fan et al. [17] studied the regularity criterion of the Cauchy problem (1)–(5) when Ω∶=ℝ2.We will prove the following.Theorem 1.
Let0<m≤ρ0≤M<∞, ρ0∈W1,r for some r∈(2,∞), u0∈H01∩H2, and d0∈H3 with divu0=0, and |d0|=1 in Ω. If
(10)∥∇d0∥L22exp[216C02m(∥ρ0u0∥L22+18C02)2]≤18C02,
with an absolute constant C0 in (22), then the problem (1)–(6) has a unique global-in-time strong solution (ρ,u,d) satisfying
(11)∥ρ∥L∞(0,T;W1,r)≤C,∥ρt∥L∞(0,T;Lr)≤C,∥u∥L∞(0,T;H2)∩L2(0,T;W2,s)≤C,forsomes>2,∥d∥L∞(0,T;H3)≤C.Remark 2.
WhenΩ∶=ℝ2, Theorem 1 is also correct, thus improving the result in [18], where u0 and ∇d0 are assumed to be small.Next, we consider (1)–(4) with ρ≡1 as follows:
(12)divu=0,(13)∂tu+u·∇u+∇π-Δu=-∇·(∇d⊙∇d),(14)∂td+u·∇d-Δd=|∇d|2d,(15)u=0,d=d0on∂Ω,(16)(u,d)(·,0)=(u0,d0)inΩ.We will prove the following.Theorem 3.
Letu0∈L2 and d0∈H1 with divu0=0 and |d0|=1 in Ω and d0∈C2,β(∂Ω) for some β∈(0,1). If d satisfies
(17)∇d∈Lq(0,T;Lp),2q+2p=1,2<p≤∞,
then the strong solution (u,d) can be extended beyond T>0.Remark 4.
In [9], the authors prove the regularity criterion (9) for the problem (12)–(16), and our condition (17) is weaker than (9). Moreover, (17) is scaling invariant for (12)–(14).


## 2. Proof of Theorem1

This section is devoted to the proof of Theorem1. Since the local-in-time well-posedness has been proved in [16], we only need to establish a priori estimates. Also, by the local well-posedness result in [16], we note that ∇d is absolutely continuous on [0,T] for any given T>0.By the maximum principle, it follows from (1) and (2) that
(18)0<m≤ρ≤M<∞.Testing (3) by u and using (1) and (2), we see that
(19)12ddt∫‍ρu2dx+∫‍|∇u|2dx=-∫‍(u·∇)d·Δddx.Testing (4) by -Δd-|∇d|2d, using |d|=1, we find that
(20)12ddt∫‍|∇d|2dx+∫‍|Δd+|∇d|2d|2dx=∫‍(u·∇)d·Δddx.Summing up (19) and (20) and integrating over (0,T), we get
(21)∫(ρu2+|∇d|2)dx+2∫0T‍∫‍(|∇u|2+|Δd+|∇d|2d|)dxdt≤∫(ρ0u02+|∇d0|2)dx.Since∂νd=0 on (0,∞)×∂Ω, we have the following Gagliardo-Nirenberg inequality:
(22)∥∇d∥L42≤C0∥∇d∥L2∥Δd∥L2.By (20) and the Ladyzhenskaya inequality in 2D, we derive
(23)12ddt∫‍|∇d|2dx+∫‍|Δd+|∇d|2d|2dx≤∥u∥L4∥∇d∥L4∥Δd∥L2≤2∥u∥L21/2∥∇u∥L21/2·C0∥∇d∥L21/2∥Δd∥L23/2≤∥Δd∥L228+216C02∥u∥L22∥∇u∥L22∥∇d∥L22≤∥Δd∥L228+216C02m(∥ρ0u0∥L22+∥∇d0∥L22)∥∇u∥L22∥∇d∥L22.On the other hand, since(a+b)2≥(a2/2)-b2, we have
(24)∫|Δd+|∇d|2d|2dx≥∥Δd∥L222-∥∇d∥L44≥∥Δd∥L222-C02∥∇d∥L22∥Δd∥L22.If the initial data∥∇d0∥L22<(1/C02)(1/8), then there exists T1>0 such that for any t∈[0,T1],
(25)∥∇d(t)∥L22≤1C02·14.We denote byT1* the maximal time such that (25) holds on [0,T1*]. Therefore, by (23), (24), and (25), it follows that for any t∈[0,T1*],
(26)ddt∫‍|∇d|2dx+14∥Δd∥L22≤432C02m(∥ρ0u0∥L22+∥∇d0∥L22)∥∇u∥L22∥∇d∥L22≤432C02m(∥ρ0u0∥L22+18C02)∥∇u∥L22∥∇d∥L22,
which gives
(27)∥∇d(t)∥L22+14∫0t‍∥Δd(τ)∥L22dτ≤∥∇d0∥L22exp[432C02m(∥ρ0u0∥L22+18C02)≤∥∇d0∥L22exph4g×∫0T1*‍∥∇u∥L22dτ(∥ρ0u0∥L22+18C02)]≤∥∇d0∥L22exp[216C02m(∥ρ0u0∥L22+18C02)2]≤18C02,
which implies that T1*=T if the initial data satisfies
(28)∥∇d0∥L22exp[216C02m(∥ρ0u0∥L22+18C02)2]≤18C02.LetT* be a maximal existence time for the solution (ρ,u,d). Then, (18), (21), and (27) ensure that T*=∞ by continuity argument.Testing (3) by ut, using (1), (18), (21), (22), |d|=1, and the Gagliardo-Nirenberg inequalities, we obtain
(29)12ddt∫‍|∇u|2dx+∫‍ρut2dx=-∫‍ρu·∇u·utdx-∫‍ut·∇d·Δddx≤C∥ρut∥L2(∥u∥L4∥∇u∥L4+∥∇d∥L4∥Δd∥L4)≤C∥ρut∥L2[∥u∥L21/2∥∇u∥L2(∥Δu∥L21/2+∥u∥L21/2)≤C∥ρut∥L2g+∥∇d∥L21/2∥Δd∥L2(∥∇Δd∥L21/2+∥d∥L21/2)]≤C∥ρut∥L2(∥∇u∥L2∥Δu∥L21/2+∥∇u∥L2+∥Δd∥L2≤C∥ρut∥L2f×∥∇Δd∥L21/2+∥Δd∥L2).On the other hand, (3) can be rewritten as
(30)-Δu+∇π=f∶=-ρut-ρu·∇u-∇·(∇d⊙∇d).By theH2-theory of Stokes system, we have
(31)∥Δu∥L2≤C∥f∥L2≤C∥ρut∥L2+C∥u∥L4∥∇u∥L4+C∥∇d∥L4∥Δd∥L4≤C∥ρut∥L2+C∥∇u∥L2∥Δu∥L21/2+C∥∇u∥L2ffjgf+C∥Δd∥L2∥∇Δd∥L21/2+C∥Δd∥L2,
which yields
(32)∥Δu∥L2≤C∥ρut∥L2+C∥∇u∥L22+C∥Δu∥L2+C∥Δd∥L2∥∇Δd∥L21/2+C∥Δd∥L2.Inserting (32) into (29), we deduce that
(33)12ddt∫‍|∇u|2dx+∫‍ρut2dx≤C∥ρut∥L23/2∥∇u∥L2+C∥ρut∥L2(∥∇u∥L22+∥∇u∥L2)+C∥ρut∥L2∥Δd∥L2∥∇Δd∥L21/2+C∥ρut∥L2∥Δd∥L2≤18∥ρut∥L22+C∥∇u∥L24+C+18∥∇Δd∥L22+C∥Δd∥L24.ApplyingΔ to (4), testing by Δd, using |d|=1, (21) and (22), and the Gagliardo-Nirenberg inequalities, we have
(34)12ddt∫‍|Δd|2dx+∫‍|∇Δd|2dx≤∫‍|∇(|∇d|2d)||∇Δd|dx+∫‍|∇(u·∇d)||∇Δd|dx≤C(∥∇d∥L63+∥∇d∥L4∥Δd∥L4+∥u∥L4∥Δd∥L4ffffff+∥∇u∥L2∥∇d∥L∞)∥∇Δd∥L2≤C(∥∇d∥L2∥Δd∥L22+∥Δd∥L2∥∇Δd∥L21/2+∥Δd∥L2fffffffffff+∥∇u∥L21/2∥Δd∥L21/2∥∇Δd∥L21/2fffffffffff+∥∇u∥L21/2∥Δd∥L21/2+∥∇u∥L2fffffffffj×∥∇d∥L21/2∥∇Δd∥L21/2)∥∇Δd∥L2≤18∥∇Δd∥L22+C∥Δd∥L24+C+C∥∇u∥L24.Here, we have used the Gagliardo-Nirenberg inequalities(35)∥∇d∥L63≤C∥∇d∥L2∥Δd∥L22,∥∇d∥L∞2≤∥∇d∥L2∥∇Δd∥L2,∥Δd∥L42≤C∥Δd∥L2∥∇Δd∥L2+C∥Δd∥L2.Combining (33) and (34) and using the Gronwall inequality, we have
(36)∥u∥L∞(0,T;H1)+∥u∥L2(0,T;H2)≤C,(37)∥ρut∥L2(0,T;L2)≤C,(38)∥d∥L∞(0,T;H2)+∥d∥L2(0,T;H3)≤C.Now, by the similar calculations as those in [17], we arrive at
(39)∥(ut,∇dt)∥L∞(0,T;L2)∩L2(0,T;H1)≤C,∥(u,∇d)∥L∞(0,T;H2)≤C,∥u∥L2(0,T;W2,s)≤Cforsomes>2,∥ρ∥L∞(0,T;W1,r)≤C,∥ρt∥L∞(0,T;Lr)≤C.This completes the proof.


## 3. Proof of Theorem3

This section is devoted to the proof of Theorem3. By the results in [9], we only need to prove (9).Similar to (21), we still have
(40)∫(u2+|∇d|2)dx+2∫0T∫(|∇u|2+|Δd+|∇d|2d|)dxdt‍≤∫‍(u02+|∇d0|2)dx.We will use the following Gagliardo-Nirenberg inequalities:(41)∥u∥L2p/(p-2)≤C∥u∥L21-(2/p)∥∇u∥L22/p,(42)∥∇d∥L2p/(p-2)≤C∥∇d∥L21-(2/p)∥Δd∥L22/p+C∥∇d∥L2.Testing (14) by -Δd, using |d|=1, (40), (41), and (42), we have
(43)12ddt∫‍|∇d|2dx+∫‍|Δd|2dx=∫‍(u·∇d-|∇d|2d)Δddx≤(∥u∥L2p/(p-2)∥∇d∥Lp+∥∇d∥Lp∥∇d∥L2p/(p-2))∥Δd∥L2≤C∥∇d∥Lp(∥u∥L21-(2/p)∥∇u∥L22/p+∥∇d∥L2≤C∥∇d∥Lpc+∥∇d∥L21-(2/p)∥Δd∥L22/p)∥Δd∥L2≤C∥∇d∥Lp(∥∇u∥L22/p+1+∥Δd∥L22/p)∥Δd∥L2≤14∥Δd∥L22+C∥∇d∥Lp2(∥∇u∥L24/p+1+∥Δd∥L24/p)≤12∥Δd∥L22+∥∇u∥L22+C∥∇d∥Lp2p/(p-2)+C,
which gives (9).This completes the proof.



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