Global Existence of Solutions to the 2D Incompressible Generalized Liquid Crystal Flow

Abstract

We consider the global existence of solutions to the 2D incompressible generalized liquid crystal flow. It is proved that the local solution exists globally with β = 0 , α ...5; 2 .

(ProQuest: ... denotes non-US-ASCII text omitted.)

Academic Editor:Yonghui Xia

School of Mathematical Sciences, Xiamen University, Xiamen, China

Received 11 April 2014; Accepted 21 April 2014; 6 May 2014

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.

1. Introduction

In this paper, we consider the following 2D liquid crystal flow: [figure omitted; refer to PDF] where α ...5; 0 , β ...5; 0 are real parameters and u is the velocity, d is a vectorial function modeling the orientation of the crystal molecules, and p is the scalar pressure. Here f ( d ) [: =] ( ^{| d | 2} - 1 ) d and Λ = ( - Δ ^{) 1 / 2} is defined in terms of Fourier transform by [figure omitted; refer to PDF]

When α = β = 1 , it has been shown that (1)-(4) has unique global weak and smooth solutions [1-3]. In [4], global regularity for this system with mixed partial viscosity is proved. Some regularity criteria are established for the system with zero dissipation in [5].

The aim of this paper is to establish the following global regularity for the 2D liquid crystal model with fractional diffusion.

Theorem 1.

Assume ( _{u 0} , _{d 0} ) ∈ ^{H 3} ( ^{... 2} ) × ^{H 4} ( ^{... 2} ) . Let ( u , d ) be the local strong solution to the problem (1)-(4). If α and β satisfy β = 0 , α ...5; 2 , then the 2D liquid crystal model has a unique global classical solution ( u , d ) satisfying [figure omitted; refer to PDF]

Remark 2.

This work is partially motivated by the recent progress on the 2D incompressible MHD system with fractional diffusion; we refer to [6-10] and references therein. In [7], Tran et al. obtained the global regularity of 2D GMHD equations for the following three cases: (1) α ...5; 1 , β ...5; 1 ; (2) 0 ...4; α < 1 / 2 , 2 α + β > 2 ; (3) α ...5; 2 , β = 0 . Combining them with the result in [10], we know that if α + β ...5; 2 , 2D incompressible MHD system with fractional diffusion possesses a global smooth solution. Fan et al. [8] proved the global existence of smooth solutions with α > 0 , β = 1 . Global regularity for the case α = 0 , β > 1 was established by Jiu and Zhao [9] which improves the result in [6]. Very recently, the authors improved the case α = 0 , β > 1 for the 2D liquid crystal model in [11].

2. Proof of Theorem 1

It is sufficient to prove Theorem 1 with α = 2 , β = 0 .

We will prove Theorem 1 if we can demonstrate the boundedness of ^{|| u || H 3 2} + ^{|| d || H 4 2} . In order to reach our purpose, we will show this by contradiction: assume [figure omitted; refer to PDF] for some finite time T > 0 . Our thought is that when _{T 0} is close enough to T , ^{|| u || H 3 2} + ^{|| d || H 4 2} remains uniformly bounded for _{T 0} < t < T under such assumption, thus reaching a contradiction.

First, we do ^{L 2} estimate for d . Multiplying (2) by d and using (3), after integration by parts, we see that [figure omitted; refer to PDF] By using the Gronwall inequality, we have [figure omitted; refer to PDF] Then, we will show the ^{L 2} estimate for u and ∇ d . Multiplying (1) and (2) by u and - Δ d , respectively, we find that [figure omitted; refer to PDF] Thanks to Gronwall's inequality and (9), we have [figure omitted; refer to PDF] which means ∇ u ∈ ^{L 2} ( 0 , T ; BMO ) .

The ^{H 1} estimate for u and ^{H 2} estimate for d will be shown as follows. Multiplying (1) by Δ u , applying Δ to (2), multiplying by Δ d , and then summing them up, we obtain [figure omitted; refer to PDF] Let us introduce the following commutator and bilinear estimates established in [12, 13]: [figure omitted; refer to PDF] with s > 0 and 1 / p = 1 / _{p 1} + 1 / _{q 1} = 1 / _{p 2} + 1 / _{q 2} .

Now, we do the ^{H 2} estimate for u and ^{H 3} estimate for d . Applying ^{Λ 2} to (1), multiplying by ^{Λ 2} u , and dealing with (2) in the same way by ^{Λ 3} and ^{Λ 3} d , after summing them up, we have [figure omitted; refer to PDF] Using Hölder's inequality, Gagliardo-Nirenberg inequality, Young's inequality, and (13), we have the following estimates: [figure omitted; refer to PDF] Now we estimate I _{I 1} , I _{I 2} , and I _{I 3} one by one: [figure omitted; refer to PDF] _{K 1} and _{K 2} can be estimated as follows: [figure omitted; refer to PDF] Combining _{K 1} and _{K 2} , we have [figure omitted; refer to PDF] Summing all the above estimates to (14), we obtain [figure omitted; refer to PDF] Now, we will show the ^{H 3} estimate for u and ^{H 4} estimate for d . Applying ^{Λ 3} to (1), multiplying by ^{Λ 3} u , and dealing with (2) in the same way by ^{Λ 4} and ^{Λ 4} d , after summing them up, we have [figure omitted; refer to PDF] Using Hölder's inequality, Gagliardo-Nirenberg inequality, Young's inequality, and (13), we have the following estimates: [figure omitted; refer to PDF] Now we estimate _{J 31} , _{J 32} , _{J 33} , and _{J 34} one by one: [figure omitted; refer to PDF] The estimate for _{J 4} is as follows: [figure omitted; refer to PDF] We calculate _{J 41} , _{J 42} , and _{J 43} : [figure omitted; refer to PDF] Combining _{J 41} , _{J 42} , and _{J 43} , we get [figure omitted; refer to PDF] Combining the above estimates to (20), we get [figure omitted; refer to PDF] Now we estimate the term ^{∫ _{T 0} t} ... _{|| ^{Λ 3} u || ^{L 2}} by applying the Gronwall inequality to (12): [figure omitted; refer to PDF] Here _{T 0} ∈ ( 0 , T ) will be fixed later and we denote ∇ _{u 0} [: =] ∇ u ( · , _{T 0} ) , Δ _{d 0} [: =] Δ d ( · , _{T 0} ) . Set A ( t ) [: =] _{max ... τ ∈ ( T 0 , t )} ( ^{|| u || H 3 2} + ^{|| d || H 4 2} ) ( τ ) . Now applying the logarithmic inequality [14] [figure omitted; refer to PDF] we get [figure omitted; refer to PDF] Since _{|| ∇ u || BMO} ∈ ^{L 1} ( _{T 0} , T ) , we can take _{T 0} close enough to T , so that [figure omitted; refer to PDF] for some small positive number δ to be fixed later. With such choice of _{T 0} we have [figure omitted; refer to PDF] Hölder's inequality gives [figure omitted; refer to PDF] Fix _{T 0} satisfying [figure omitted; refer to PDF]

Combining the above estimates together, we get [figure omitted; refer to PDF] Integrating the above inequality, we have [figure omitted; refer to PDF] where _{A 0} [: =] ^{|| u || H 3 2} ( _{T 0} ) + ^{|| d || H 4 2} ( _{T 0} ) .

Taking δ = 1 / 24 , we have [figure omitted; refer to PDF] Thus (35) tells us that [figure omitted; refer to PDF] This in turn gives [figure omitted; refer to PDF] We set B ( t ) [: =] ( 1 + A ( t ) ^{) 1 / 24} , _{B 0} : = ( 1 + _{A 0} ^{) 1 / 24} and divide the above inequality by ( 1 + A ( t ) ^{) 23 / 24} ; using the monotonicity of A ( t ) we reach [figure omitted; refer to PDF] The standard Gronwall's inequality now gives [figure omitted; refer to PDF] which leads to [figure omitted; refer to PDF] As ^{∫ _{T 0} t} ... _{|| ∇ u || BMO} + _{|| ^{Λ 2} u || ^{L 2}} d τ remains bounded as t ... T , the above inequality contradicts that A ( t ) ... ∞ as t ... T , so we complete our proof of Theorem 1.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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