# Regularity Criterion for the 3D Nematic Liquid Crystal Flows

Authors: Jishan Fan; Tohru Ozawa
Journal: ISRN Mathematical Analysis
 (2012)

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## Abstract

We study the hydrodynamic theory of liquid crystals. We prove a logarithmically improved regularity criterion for two simplified Ericksen-Leslie systems.


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## Body

## 1.  Introduction

The hydrodynamic theory of liquid crystals was established by Ericksen and Leslie [1–4]. However, since the equations are too complicated, we consider the first simplified Ericksen-Leslie system:
(1.1)ut+u⋅∇u+∇π-Δu=-∇⋅(∇d⊙∇d),(1.2)dt+u⋅∇d=Δd-f(d),(1.3)div⁡u=0,(1.4)(u,d)(x,0)=(u0,d0)(x)inR3,
which include the velocity vector u:=(u1,u2,u3)t, the scalar pressure π being and the direction vector d:=(d1,d2,d3)t. f(d):=1/η(|d|2-1)d with η, a positive constant. (∇d⊙∇d)i,j:=∑k∂idk∂jdk, and hence ∇·(∇d⊙∇d)=∑kΔdk∇dk+(1/2)∇|∇d|2.Lin-Liu [5] proved that the system (1.1)–(1.4) has a unique smooth solution globally in 2 space dimensions and locally in 3 dimensions. They also proved the global existence of weak solutions. However, the regularity of solutions to the system is still open. Fan-Guo [6] and Fan-Ozawa [7] showed the following regularity criteria:
(1.5)u∈Lr(0,T;Ls(R3))for2r+3s=1,3<s≤∞,∇u∈Lr(0,T;Ls(R3))for2r+3s=2,32<s≤∞,u∈L2(0,T;Ḃ∞,∞0),ω:=curl⁡u∈L1(0,T;Ḃ∞,∞0),
where Ḃ∞,∞0 denotes the homogeneous Besov space.The first aim of this paper is to prove a new regularity criterion as follows.Theorem 1.1.
Letu0∈H3,d0∈H4 with  divu0=0 in  ℝ3. Let  (u,d) be a smooth solution to the problem (1.1)–(1.4) on  [0,T). If  u satisfies
(1.6)∫0T‖u(⋅,t)‖Ḃ∞,∞-s2/(1-s)1+log⁡(e+‖u(⋅,t)‖Ḃ∞,∞-s)dt<∞
for some s with 0<s<1, then the solution (u,d) can be extended beyond T>0.When the penalization parameterη→0, (1.1)–(1.4) reduce to
(1.7)ut+u⋅∇u+∇π-Δu=-∇⋅(∇d⊙∇d),(1.8)div⁡u=0,(1.9)dt+u⋅∇d=Δd+|∇d|2d,|d|=1,(1.10)(u,d)(x,0)=(u0,d0)(x)inR3,|d0|=1.Whenu=0, then (1.9) is the well-known harmonic heat flow equation onto a sphere.Fan-Gao-Guo [8] proved the following blow-up criteria:(1.11)u,∇d∈L2(0,T;Ḃ∞,∞0),ω,Δd∈L1(0,T;Ḃ∞,∞0).We will prove the folowing theoremTheorem 1.2.
Letu0,∇d0∈H3(ℝ3) with  divu0=0,|d0|=1 in  ℝ3. Let  (u,d) be a smooth solution to the problem (1.7)–(1.10) on  [0,T). If the following condition is satisfied:
(1.12)∫0T‖u(⋅,t)‖Ḃ∞,∞-s2/(1-s)+‖∇d(⋅,t)‖Ḃ∞,∞021+log⁡(e+‖u(⋅,t)‖Ḃ∞,∞-s+‖∇d(⋅,t)‖Ḃ∞,∞0)dt<∞,
for some s with 0<s<1, then the solution (u,d) can be extended beyond T>0.


## 2.  Proof of Theorem1.1

This section is devoted to the proof of Theorem1.1. Since it is well-known that there are T0>0 and a unique smooth solution (u,d) to the problem (1.1)–(1.4) in (0,T0], we only need to show a priori estimates.Testing (1.1) by u and using (1.3), we see that
(2.1)12ddt∫u2dx+∫|∇u|2dx=-∫(u⋅∇)d⋅Δddx.Testing (1.2) by Δd-f(d) and using (1.3), we find that
(2.2)ddt∫12|∇d|2+14η(|d|2-1)2dx+∫|Δd-f(d)|2dx=∫(u⋅∇)d⋅Δddx.Summing up (2.1) and (2.2), we infer that
(2.3)∫12u2+12|∇d|2+14η(|d|2-1)2dx+∫0T∫|∇u|2+|Δd-f(d)|2dx≤∫12u02+12|∇d0|2+14η(|d0|2-1)2dx.Testing (1.2) by d and using (1.3), we deduce that
(2.4)12ddt∫d2dx+∫|∇d|2dx+1η∫|d|4dx=1η∫d2dx,
which yields
(2.5)‖d‖L∞(0,T;L2)+‖d‖L2(0,T;H1)≤C.Next, we prove the following estimate:(2.6)‖d‖L∞(0,T;L∞)≤max⁡(‖d0‖L∞,1).Without loss of generality, we assume that1≤∥d0∥L∞. Multiplying (1.2) by d, we get
(2.7)ϕt+u⋅∇ϕ-Δϕ+2|d|2ϕ=-2η(‖d0‖L∞2-1)|d|2-2|∇d|2≤0
with ϕ=|d|2-∥d0∥L∞2 and ϕ|t=0=|d0|2-∥d0∥L∞2≤0. Then (2.6) follows immediately from ϕ≤0 by the maximum principle.Testing (1.1) by -Δu and using (1.3), we see that
(2.8)12ddt∫|∇u|2dx+∫|Δu|2dx=∫(u⋅∇)u⋅Δudx-∑i,k∫Δdk⋅∂i∇dk⋅∇uidx-∑i,k∫∂idk⋅∇Δdk⋅∇uidx.ApplyingΔ to (1.2), testing by Δd, and using (1.3), we find that
(2.9)12ddt∫|Δd|2dx+∫|∇Δd|2dx=∑i,k∫∂idk⋅∇Δdk⋅∇uidx-∑i,k∫∂i∂jdk⋅∂j∇dk⋅∇uidx-∫Δf(d)⋅Δddx.Summing up (2.8) and (2.9), we get
(2.10)12ddt∫|∇u|2+|Δd|2dx+∫|Δu|2+|∇Δd|2dx=∫(u⋅∇)u⋅Δudx-∑i,k∫Δdk⋅∂i∇dk⋅∇uidx-∑i,k∫∂i∂jdk⋅∂j∇dk⋅∇uidx-∫Δf(d)Δddx=:I1+I2+I3+I4.By using (2.6), I4 is simply bounded as
(2.11)I4≤C‖Δd‖L22.By using the inequalities [9]
(2.12)‖u⋅∇u‖L2≤C‖u‖Ḃ∞,∞-s‖∇u‖Ḃ2,2s,‖∇u‖Ḃ2,2s≤C‖∇u‖L21-s‖Δu‖L2s.I1 can be bounded as follows:
(2.13)I1≤‖u⋅∇u‖L2‖Δu‖L2≤C‖u‖Ḃ∞,∞-s‖u‖Ḃ2,21+s‖Δu‖L2≤C‖u‖Ḃ∞,∞-s‖∇u‖L21-s‖Δu‖L21+s≤12‖Δu‖L22+C‖u‖Ḃ∞,∞-s2/(1-s)‖∇u‖L22.We boundI2 and I3 as follows:
(2.14)I2,I3≤C‖∇u‖L2‖Δd‖L42≤C‖∇u‖L2‖∇Δd‖L2≤14‖∇Δd‖L22+C‖∇u‖L22.Here we used the Gagliardo-Nirenberg inequality(2.15)‖Δd‖L42≤C‖d‖L∞‖∇Δd‖L2.
Inserting the above estimates into (2.10), we derive
(2.16)ddt∫|∇u|2+|Δd|2dx+∫|Δu|2+|∇Δd|2dx≤C(‖u‖Ḃ∞,∞-s2/(1-s)+1)(‖∇u‖L22+‖Δd‖L22)≤C1+∥u∥Ḃ∞,∞-s2/(1-s)1+log⁡(e+∥u∥Ḃ∞,∞-s)(1+log⁡(e+‖u‖Ḃ∞,∞-s))(‖∇u‖L22+‖Δd‖L22)≤C1+∥u∥Ḃ∞,∞-s2/(1-s)1+log⁡(e+‖u‖Ḃ∞,∞-s)(1+log⁡(e+y))(‖∇u‖L22+‖Δd‖L22).Due to (1.6), one concludes that for any small constant ϵ>0, there exists T*<T such that
(2.17)∫T*T‖u‖Ḃ∞,∞-s2/(1-s)1+log⁡(e+‖u‖Ḃ∞,∞-s)dt<ϵ.For anyT*<t≤T, we set
(2.18)y(t):=sup⁡T*≤τ≤t‖Λ3(u,∇d)(⋅,τ)‖L2withΛ:=(-Δ)1/2.Applying Gronwall’s inequality to (2.16) in the interval [T*,t], one has
(2.19)‖∇u(⋅,t)‖L22+‖Δd(⋅,t)‖L22≤C(1+y)C0ϵ.Now, we derive a bound ony(t) defined by (2.18). To this end, we will use the following commutator and product estimates due to Kato-Ponce [10]:
(2.20)‖Λα(fg)-fΛαg‖Lp≤C(‖∇f‖Lp1‖Λα-1g‖Lq1+‖Λαf‖Lp2‖g‖Lq2),(2.21)‖Λα(fg)‖Lp≤C(‖f‖Lp1‖Λαg‖Lq1+‖Λαf‖Lp2‖g‖Lq2),
with α>0 and 1/p=1/p1+1/q1=1/p2+1/q2.ApplyingΛ3 to (1.1), testing by Λ3u, and using (1.3), (2.20), (2.21) and (2.19), we obtain
(2.22)12ddt∫|Λ3u|2dx+∫|Λ4u|2dx=-∫(Λ3(u⋅∇u)-u∇Λ3u)Λ3udx+∫Λ3(∇d⊙∇d):Λ3∇udx≤C‖∇u‖L3‖Λ3u‖L32+C‖∇d‖L∞‖Λ4d‖L2‖Λ4u‖L2≤C‖∇u‖L23/4‖Λ3u‖L21/4⋅‖∇u‖L21/3‖Λ4u‖L25/3+116‖Λ4u‖L22+C‖∇d‖L∞2‖Λ4d‖L22≤14‖Λ4u‖L22+C‖∇u‖L213/2‖Λ3u‖L23/2+C‖Δd‖L23/2‖Λ4d‖L21/2⋅‖Δd‖L22/3‖Λ5d‖L24/3≤14‖Λ4u‖L22+14‖Λ5d‖L22+C‖∇u‖L213/2‖Λ3u‖L23/2+C‖Δd‖L213/2‖Λ4d‖L23/2.
Here we have used the following Gagliardo-Nirenberg inequalities:
(2.23)‖∇u‖L3≤C‖∇u‖L23/4‖Λ3u‖L21/4,‖Λ3u‖L3≤C‖∇u‖L21/6‖Λ4u‖L25/6,‖∇d‖L∞≤C‖Δd‖L23/4‖Λ4d‖L21/4,‖Λ4d‖L2≤C‖Δd‖L21/3‖Λ5d‖L22/3.TakingΛ4 to (1.2), testing by Λ4d, and using (1.3), (2.20), (2.23), and (2.6), we have
(2.24)12ddt∫|Λ4d|2dx+∫|Λ5d|2dx=-∫(Λ4(u⋅∇d)-u∇Λ4d)Λ4ddx-∫Λ4f(d)⋅Λ4ddx≤C‖∇u‖L3‖Λ4d‖L6‖Λ4d‖L2+C‖∇d‖L∞‖Λ4u‖L2‖Λ4d‖L2+C‖Λ4d‖L22≤C‖∇u‖L23/4‖Λ3u‖L21/4⋅‖Δd‖L21/3‖Λ5d‖L25/3+14‖Λ4u‖L22+C‖∇d‖L∞2‖Λ4d‖L22+C‖Λ4d‖L22≤14‖Λ4u‖L22+14‖Λ5d‖L22+C‖∇u‖L29/2‖Δd‖L22‖Λ3u‖L23/2+C‖Δd‖L213/2‖Λ4d‖L23/2+C‖Λ4d‖L22.Summing up (2.22) and (2.24) and taking ϵ small enough, we arrive at
(2.25)‖u‖L∞(0,T;H3)+‖u‖L2(0,T;H4)≤C,‖d‖L∞(0,T;H4)+‖d‖L2(0,T;H5)≤C.This completes the proof.


## 3.  Proof of Theorem1.2

In this section, we will prove Theorem1.2. Since it is easy to prove that there are T0>0 and a unique smooth solution (u,p,d) to the problem (1.7)–(1.10) in [0,T0], we only need to prove a priori estimates.First, as in the previous section, we still have (2.1).Testing (1.9) by -Δd, using dΔd=-|∇d|2 and |d|=1, we see that
(3.1)12ddt∫|∇d|2dx+∫|Δd|2dx=∫(u⋅∇d)Δddx+∫(d⋅Δd)2dx≤∫(u⋅∇d)Δddx+∫|Δd|2dx.Summing up (2.1) and (3.1), we find that
(3.2)∫12u2+12|∇d|2dx+∫0T∫|∇u|2dxdt≤∫12u02+|∇d0|2dx.Similarly to (2.10), we have
(3.3)12ddt∫|∇u|2+|Δd|2dx+∫|Δu|2+|∇Δd|2dx=:I1+I2+I3+I5.
Here I1,I2, and I3 are the same as that in (2.10) and can be bounded as in the previous section. The corresponding last term I5 is written and bounded as
(3.4)I5:=-∑k∫∂k(|∇d|2d)⋅∂kΔddx=-∑k∫∂kd⋅|∇d|2⋅∂kΔddx-∑k∫d∂k|∇d|2⋅∂kΔddx=∑k∫∂k(∂kd⋅|∇d|2)⋅Δddx-∑k∫d∂k|∇d|2⋅∂kΔddx≤C‖∇d‖L42‖Δd‖L42+14‖∇Δd‖L22≤C‖Δd‖L2⋅‖∇d‖Ḃ∞,∞0‖∇Δd‖L2+14‖∇Δd‖L22≤12‖∇Δd‖L22+C‖∇d‖Ḃ∞,∞02‖Δd‖L22.
Here we have used the following inequality [11, 12]:
(3.5)‖Δd‖L42≤C‖∇d‖Ḃ∞,∞0‖∇Δd‖L2
and the Gagliardo-Nirenberg inequality
(3.6)‖∇d‖L42≤C‖d‖L∞‖Δd‖L2.Substituting the above estimates into (3.3), we obtain
(3.7)ddt∫|∇u|2+|Δd|2dx≤C(‖u‖Ḃ∞,∞-s2/(1-s)+‖∇d‖Ḃ∞,∞02)(‖∇u‖L22+‖Δd‖L22)≤C‖u‖Ḃ∞,∞-s2/(1-s)+‖∇d‖Ḃ∞,∞021+log⁡(e+‖u‖Ḃ∞,∞-s+‖∇d‖Ḃ∞,∞0)(1+log⁡(e+y))(‖∇u‖L22+‖Δd‖L22).Due to (1.12), one concludes that for any small constant ϵ>0, there exists T*<T such that
(3.8)∫T*T‖u‖Ḃ∞,∞-s2/(1-s)+‖∇d‖Ḃ∞,∞021+log⁡(e+‖u‖Ḃ∞,∞-s+‖∇d‖Ḃ∞,∞0)dt<ϵ.Applying Gronwall’s inequality to (3.7) in the interval [T*,t], one has (2.19).As in the previous section, we still have (2.22).Similarly to (2.24), we obtain
(3.9)12ddt∫|Λ4d|2dx+∫|Λ5d|2dx=-∫(Λ4(u⋅∇d)-u∇Λ4d)Λ4ddx+∫Λ4(d|∇d|2)Λ4ddx=:J1+J2.J1 is bounded as that in (2.24);
(3.10)J2≤C‖∇d‖L∞‖Λ4d‖L22+C‖Λ4(|∇d|2)‖L2‖Λ4d‖L2,≤C‖∇d‖L∞2‖Λ4d‖L22+C‖∇d‖L∞‖Λ5d‖L2‖Λ4d‖L2,≤14‖Λ5d‖L22+C‖∇d‖L∞2‖Λ4d‖L22,
then J2 can be bounded as that in (2.24).Combining (2.22) and (3.9) and taking ϵ small enough, we conclude that
(3.11)‖u‖L∞(0,T;H3)+‖u‖L2(0,T;H4)≤C,‖∇d‖L∞(0,T;H3)+‖∇d‖L2(0,T;H4)≤C.
This completes the proof.



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*Source: 935045-2012-04-11.xml*