A Regularity Criterion for the Nematic Liquid Crystal Flows

Volume 2010, Article ID 589697,9pages doi:10.1155/2010/589697

Research Article

A Regularity Criterion for the Nematic Liquid Crystal Flows

Yong Zhou

and Jishan Fan

1Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang 321004, China

2Department of Applied Mathematics, Nanjing Forestry University, Nanjing, Jiangsu 210037, China

3Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

Correspondence should be addressed to Yong Zhou,[email protected] Received 25 September 2009; Accepted 16 April 2010

Academic Editor: Michel C. Chipot

Copyrightq2010 Y. Zhou and J. Fan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A logarithmically improved regularity criterion for the 3D nematic liquid crystal flows is established.

1. Introduction

We consider the following hydrodynamical systems modeling the flow of nematic liquid crystal materials1,2:

utu· ∇u∇π−μΔu−λ∇ ·

∇d ∇d

Δd−fd

⊗d

, 1.1

d_tu· ∇d−d· ∇uγ

Δd−fd

, 1.2

divu0, 1.3

v, d|t0 v₀, d₀ inR³. 1.4 ux, t ∈ R³ is the velocity field of the flow. dx, t ∈ R³ is the averagedmacroscopic/

continuum molecular orientations vector inR³.πx, tis a scalar function representing the pressureincluding both the hydrostatic part and the induced elastic part from the orientation field.μis a positive viscosity constant. The constantλrepresents the competition between kinetic energy and potential energy. The constantγis the microscopic elastic relaxation time Deborah numberfor the molecular orientation field.fd 1/²|d|²−1d. For simplicity,

we will takeμλγ1. The 3×3 matrix is defined by∇ ∇d_ij ∂id·∂jd.⊗is the usual Kronecker multiplication, for example,a⊗b_ij aibjfora, b∈R³.

Very recently, results for the local existence of classical solutions for the problems1.1–

1.4were presented in3. The aim of this paper is to establish a regularity criterion for it.

We will prove the following.

Theorem 1.1. Letu0, d₀∈H²×H³with divu₀ 0 inR³. Suppose that a local smooth solution u, dsatisfies

_T

0

∇ut^r_Lp

1lne∇utL^pdt <∞, with 2 r 3

p 2, 2≤p≤3. 1.5

Thenu, dcan be extended beyondT.

Remark 1.2. Equation1.5can be regarded as a logarithmically improved regularity criterion of the form∇u∈ L^r0, T;L^pR³with2/r 3/p 2. Condition1.5only involves the velocity field u, which plays a dominant role in regularity theorem. Similar phenomenon already appeared in the studies of MHD equationssee4–6for details.

Remark 1.3. When λ 0 in 1.1, then 1.1 and 1.2 are the well-known Navier-Stokes equations. Similar conditions to1.5have been established in7–10. But previous methods can not be used here.

Remark 1.4. A natural region for p in1.5should be 3/2≤ p≤ ∞, but we only can prove it for 2≤p≤3 here. We are unable to establish any other regularity criterion in terms ofuorπ.

2. Proof of Theorem 1.1

Since we deal with the regularity conditions of the local smooth solutions, we only need to establish the needed a priori estimates. We mainly will follow the method introduced in9.

First, it has been proved in3that

1 2

d dt

R³

|u|²x, t |∇d|²x, t |d|²−1²x, t dx

R³

|∇u|²x, t |Δd−fd|²x, t dx0.

2.1

Hence

u_L∞0,T;L²u_L20,T;H¹≤C. 2.2

Multiplying1.3byd, integration by parts yields

1 2

d dt

R³|d|²x, tdx

R³

|∇d|²x, t |d|⁴x, t dx

R³

|d|²x, t d· ∇u·dx, t dx

≤ 1 2

R³|d|⁴x, tdx

R³

|d|²x, t 1

2|∇u|²x, t

dx.

2.3

Thanks to2.1,2.2, and the Gronwall inequality, we get

d_L^∞_0,T;H¹d_L²_0,T;H²≤C. 2.4

Letu u1, u2, u3^Tandd d1, d2, d3^T, then theithi1,2,3component ofusatisfies

∂tuiu· ∇ui∂iπ−Δui− ³

j1

∂j

k

∂idk∂jdk Δdi−

|d|²−1 di

dj

. 2.5

Multiplying2.5 by−Δui, after integration by parts, summing overi, and using1.2, we find that

1 2

d dt

R³|∇u|²x, tdx

R³|Δu|²x, tdx −

i,j,k

R³∂_ku_j·∂_ju_i·∂_ku_idx−

i,k

R³Δdk·∂_i∇dk· ∇uidx

−

i,k

R³∂_id_k· ∇Δdk· ∇uidx

i,j

R³∂_j d_jΔdi

·Δuidx

−

i,j

R³∂_j

|d|²−1 d_id_j

·Δuidx

:I1I2I3I4I5.

2.6

ApplyingΔon1.3, multiplying it byΔd, and using1.2, we have 1

2 d dt

R³|Δd|²x, tdx

R³

|∇Δd|²x, t Δfd·Δdx, t dx

i,k

R³∂idk· ∇Δdk· ∇uidx−

i,j,k

R³∂i∂jdk·∂j∇dk· ∇uidx

i,j

R³

djΔdi

·∂jΔuidx−

i,j

R³ΔdjΔdi·∂juidx

−2

i,j

R³∇dj·∂_ju_i· ∇Δdidx :I₆I₇I₈I₉I₁₀.

2.7

Combining2.6and2.7together, noting thatI3I60,I4I80, we deduce that 1

2 d dt

R³

|∇u|²x, t |Δd|²x, t dx

R³|Δu|²x, tdx

R³

|∇Δd|²x, t Δfd·Δdx, t

dxI1I2I5I7I9I10.

2.8

We do estimates forI_ii1,2,5,7,9,10as follows:

I₁ ≤C∇u_L^p∇u²_L2p/p−1

≤C∇u_L^p∇u^21−3/2p_L2 Δu^3/p_L2

≤Δu²_L2C∇u^2p/2p−3_Lp ∇u²_L2, for any >0.

2.9

Here we have used the following Gagliardo-Nirenberg inequality:

∇u_L2p/p−1 ≤C∇u^1−3/2p_L2 Δu^3/2p_L2 . 2.10

Similarly, by using2.10, we have

I₂I₇I₉≤C∇u_L^pΔd²_L2p/p−1

≤C∇u_L^pΔd^21−3/2p_L2 ∇Δd^3/p_L2

≤∇Δd²_L2C∇u^2p/2p−3_Lp Δd²_L2, for any >0.

2.11

I5is simply bounded as follows:

I₅≤C

R³

|d||d|³

|∇d| · |Δu|dx

≤C

d_L⁶∇d_L³d³_L6∇d_L∞

Δu_L²

≤C∇d_L³∇d_L^∞Δu_L²

≤C

∇d^1/2_L2 Δd^1/2_L2 ∇d^1/4_L2 ∇Δd^3/4_L2

Δu_L2

≤Δu²_L2CΔd_L2C∇Δd^3/2_L2

≤Δu²_L2CΔd²_L2∇Δd²_L2C,

2.12

for any >0.

Whenp2 or 3,I10can be estimated easily and hence omitted here. If 2 < p <3, we do estimates as follows:

I₁₀≤C∇u_L^p∇d_L2p/p−2∇Δd_L2

≤C∇u_L^p· Δd^2−3/p_L2 · ∇Δd^3/p_L2

≤∇Δd²_L2C∇u^2p/2p−3_Lp · Δd²_L2,

2.13

for any >0. Here we have used the Gagliardo-Nirenberg inequality:

∇d_L2p/p−2 ≤CΔd^2−3/p_L2 ∇Δd^3/p−1_L2 . 2.14

Finally, we omit the trivial term

R³Δfd·Δd dx−

i

R³∂_ifd·∂_iΔd dx. 2.15

Now, putting the above estimates forIis into2.8and takingsmall enough, we obtain d

dt

R³

|∇u|²|Δd|² dx

R³

|Δu|²|∇Δd|² dx

≤C∇u^2p/2p−3_Lp

∇u²_L2Δd²_L2

CΔd²_L2C

≤C

1∇u^2p/2p−3_Lp

1∇u²_L2Δd²_L2

.

2.16

Due to the integrability of1.5, we conclude that for any small constant >0, there exists a timeT_∗< T such that

_T

T∗

1∇ut^2p/2p−3_Lp

1lne∇ut_Lpdt≤. 2.17

Easily, from2.16and2.17it follows that d

dt

1∇u²_L2Δd²_L2

≤C 1∇u^2p/2p−3_Lp

1lne∇u_L^plneΔu_L2∇Δd_L2

1∇u²_L2Δd²_L2

,

2.18

which implies that fort∈T_∗, T,

∇ut²_L2Δdt²_L2≤C

1sup

T∗,tΔu·_L²sup

T∗,t∇Δd·_L² _C

. 2.19

We are going to do the estimate forΔuand∇Δd. To this end, we introduce the following commutator estimates due to the work of Kato and Ponce11:

Λ^α fg

−fΛ^αg

L^p≤CΛ^α−1g

L^q1∇f

L^p1Λ^αf

L^p2g

L^q2

, 2.20

Λ^αfg

L^p≤C

f_Lp1Λ^αg_Lq1Λ^αf_Lp2g_Lq2

, 2.21

whereΛ^α −Δ^α/2, forα >1, and 1/p 1/p1 1/q1 1/p2 1/q2.

ApplyingΔto2.5and multiplying it byΔui, after integration by parts, and summing overiyield

1 2

d dt

R³|Δu|²x, tdx

R³|∇Δu|²x, tdx

≤

R³Δu· ∇u−u· ∇·Δu·Δu dx

i,j

R³∂jΔ

∂id·∂jd

·Δuidx

i,j

R³∂_jΔ

|d|²−1 d_id_j

·Δuidx

i,j

R³d_jΔ²d_i·∂_jΔuidx

i,j

R³Δdi·Δdj·∂_jΔuidx 2

i,j

R³

∇dj· ∇Δdi·∂_jΔuidx :J1J2J3J4J5J6.

2.22

ApplyingΛ³to1.3, multiplying it byΛ³d, we deduce that 1

2 d dt

R³|Λ³d|²x, tdx

R³|Λ⁴d|²x, tdx

≤

R³

Λ³u· ∇d−u· ∇Λ³d

·Λ³d dx

R³Λ³fd·Λ³d dx −

i,j

R³djΔ²di·∂jΔuidx

−

i,j

R³∂juiΔdj·Δ²didx−2

i,j

R³∇dj· ∇∂jui·Δ²didx :J₇J₈J₉J₁₀J₁₁.

2.23

Summing up2.22and2.23, usingJ₄J₉0, we have 1

2 d dt

R³

|Δu|²x, t |Λ³d|²x, t dx

R³

|∇Δu|²x, t |Λ⁴d|²x, t dx

≤J₁J₂J₃J₅J₆J₇J₈J₁₀J₁₁.

2.24

Now we estimate each termJ_ias follows.

By using2.20, we estimateJ1as

J₁≤C∇u_L³Δu²_L3 ≤C∇u^3/4_L2 ∇Δu^1/4· ∇u^1/2_L2 ∇Δu^3/2_L2

≤∇Δu²_L2C∇u¹⁰_L2, for any >0;

2.25

here we used the following Gagliardo-Nirenberg inequalities:

∇u_L³ ≤C∇u^3/4_L2 ∇Δu^1/4_L2 , Δu_L³ ≤C∇u^1/4_L2 ∇Δu^3/4_L2 . 2.26

Using2.21, we estimateJ2as

J₂≤C∇d_L∞Λ⁴d_L²Δu_L²

≤CΔd^3/4_L2 Λ⁴d^5/4_L² · ∇u^1/2_L2 ∇Δu^1/2_L2

≤∇Δu²_L2Λ⁴d²_L²C∇u⁴_L2Δd⁶_L2,

2.27

for any >0. Here we have used the following Gagliardo-Nirenberg inequalities:

∇d_L^∞ ≤CΔd^3/4_L2 Λ⁴d^1/4_L² , Δu_L² ≤C∇u^1/2_L2 ∇Δu^1/2_L2 . 2.28

J3only involves lower derivatives ofdand is easy to handle, so we omit it here:

J5≤CΔd²_L4∇Δu_L2

≤CΔd^5/4_L2 Λ⁴d^3/4_L2 ∇Δu_L2

≤∇Δu²_L2Λ⁴d²_L2CΔd¹⁰_L2,

2.29

for any >0. Here we have used

Δd_L4≤CΔd^5/8_L2 Λ⁴d^3/8_L2 , J₆≤C∇d_L⁶∇Δd_L³∇Δu_L²

≤CΔd_L²· Δd^1/4_L2 Λ⁴d^3/4_L² ∇Δu_L²

≤∇Δu²_L2Λ⁴d²_L²CΔd¹⁰_L2,

2.30

for any >0. Where we have used the following inequality

∇Δd_L³ ≤CΔd^1/4_L2 Λ⁴d^3/4_L² . 2.31

By using2.20, we estimateJ₇as follows:

J₇≤C∇u_L2Λ³d²_L4CΛ³u_L2∇d_L4Λ³d_L4

≤C∇u_L2Δd^1/4_L2 Λ⁴d^7/4_L2 CΛ³u_L2∇d_L4Δd^1/8_L2 Λ⁴d^7/8_L2

≤Λ³u²_L2Λ⁴d²_L2CΔd²_L2∇u⁸_L2CΔd²_L2∇d¹⁶_L4,

2.32

for any >0. Here we have used

Λ³d_L4 ≤CΔd^1/8_L2 Λ⁴d^7/8_L2 . 2.33

The termJ8is trivial, and we omit it here:

J10≤CΔd_L∞∇u_L2Λ⁴d_L2

≤C∇u_L²· Δd^1/4_L2 · Λ⁴d^7/4_L²

≤Λ⁴d²_L2C∇u⁸_L2Δd²_L2,

2.34

for any >0. Where we have used the following inequality:

Δd_L^∞ ≤CΔd^1/4_L2 Λ⁴d^3/4_L² . 2.35

Finally, using2.26,J11can be bounded as follows:

J₁₁≤C∇d_L6Δu_L3Λ⁴d_L2

≤CΔd_L2· ∇u^1/4_L2 · Λ³u^3/4_L2 Λ⁴d_L2

≤Λ³u²_L²Λ⁴d²_L²C∇u²_L2Δd⁸_L2,

2.36

for any >0. Now, inserting the above estimates forJ_is into2.24, using2.19, and taking be small enough, we get

u_L∞0,T;H²u_L²_0,T;H³≤C, dL^∞0,T;H³dL²0,T;H⁴ ≤C.

2.37

This completes the proof.

Acknowledgments

The authors thank the referee for his/her careful reading and helpful suggestions. This work is partially supported by Zhejiang Innovation ProjectGrant no. T200905, NSF of Zhejiang Grant no. R6090109, and NSF of ChinaGrant no. 10971197.

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