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ISRN Mathematical Analysis
VolumeΒ 2012Β (2012), Article IDΒ 248473, 7 pages
http://dx.doi.org/10.5402/2012/248473
Research Article

Continuation Criterion for the 2D Liquid Crystal Flows

1Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, China
2Department of Applied Physics, Waseda University, Tokyo 169-8555, Japan

Received 16 December 2011; Accepted 17 January 2012

Academic Editors: A.Β Montes-Rodriguez and T.-P.Β Tsai

Copyright Β© 2012 Jishan Fan and Tohru Ozawa. 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.

Abstract

We consider the 2D liquid crystal systems, which consists of Navier-Stokes system coupled with wave maps or biharmonic wave maps, respectively. By logarithmic Sobolev inequalities, we obtain a blow-up criterion βˆ‡π‘‘,πœ•π‘‘π‘‘βˆˆπΏ1̇𝐡(0,𝑇;0∞,∞(ℝ2)) for the case with wave maps, and we prove the existence of a global-in-time strong solutions for the case with biharmonic wave maps.

1. Introduction

First, we consider the following simplified liquid crystal flows in two space dimensions [1]:πœ•π‘‘ξ“π‘’+π‘’β‹…βˆ‡π‘’+βˆ‡πœ‹βˆ’Ξ”π‘’=π‘˜πœ•π‘‘π‘‘π‘˜β‹…βˆ‡π‘‘π‘˜,πœ•(1.1)div𝑒=0,(1.2)2𝑑||||𝑑+π‘’β‹…βˆ‡π‘‘βˆ’Ξ”π‘‘=π‘‘βˆ‡π‘‘2βˆ’||πœ•π‘‘π‘‘||2,||𝑑||ξ€·=1,(1.3)𝑒,𝑑,πœ•π‘‘π‘‘ξ€Έξ€·π‘’(π‘₯,0)=0,𝑑0,𝑑1ξ€Έ(π‘₯),π‘₯βˆˆβ„2,||𝑑0||=1,𝑑0⋅𝑑1=0,(1.4) where 𝑒 is the velocity, πœ‹ is the pressure, and 𝑑 represents the macroscopic average of the liquid crystal orientation field with values in the unit circle.

The first two equations (1.1) and (1.2) are the well-known Navier-Stokes system with the Lorentz force βˆ‘π‘˜πœ•π‘‘π‘‘π‘˜β‹…βˆ‡π‘‘π‘˜. The last equation (1.3) is the well-known wave maps when 𝑒=0.

It is a simple matter to show that the system (1.1)–(1.4) has a unique local-in-time smooth solution when 𝑒0,βˆ‡π‘‘0,𝑑1∈𝐻1+𝑠(ℝ2) with 𝑠>0,div𝑒0=0,|𝑑0|=1,𝑑0⋅𝑑1=0 in ℝ2. The aim of this paper is to study the regularity criterion of smooth solutions to the problem (1.1)–(1.4). We will prove the following.

Theorem 1.1. Let 𝑒0,βˆ‡π‘‘0,𝑑1∈𝐻1+𝑠(ℝ2) with 𝑠>0,div𝑒0=0,|𝑑0|=1,𝑑0⋅𝑑1=0 in ℝ2 and let (𝑒,𝑑) be a smooth solution of (1.1)–(1.4) on some interval [0,𝑇] with 0<𝑇<∞. Assume that βˆ‡π‘‘,πœ•π‘‘π‘‘βˆˆπΏ1̇𝐡0,𝑇;0∞,βˆžξ€·β„2.ξ€Έξ€Έ(1.5) Then the solution (𝑒,𝑑) can be extended beyond 𝑇>0.

̇𝐡0∞,∞ is the homogeneous Besov space. We have πΏβˆžΜ‡π΅βŠ‚π΅π‘€π‘‚βŠ‚0∞,∞; see Triebel [2].

In the proof of Theorem 1.1, we will use the logarithmic Sobolev inequalities [3–6]:β€–π‘’β€–πΏβˆžβ‰€πΆβ€–π‘’β€–π»1log1/2𝑒+‖𝑒‖𝐻1+𝑠,(1.6)β€–βˆ‡π‘‘β€–πΏβˆžξ‚€β‰€πΆ1+β€–βˆ‡π‘‘β€–Μ‡π΅0∞,βˆžξ€·log𝑒+β€–βˆ‡π‘‘β€–π»1+π‘ ξ€Έξ‚β€–β€–πœ•,(1.7)π‘‘π‘‘β€–β€–πΏβˆžξ‚€β€–β€–πœ•β‰€πΆ1+𝑑𝑑‖‖̇𝐡0∞,βˆžξ€·β€–β€–πœ•log𝑒+𝑑𝑑‖‖𝐻1+𝑠,(1.8) for 𝑠>0, and the Gagliardo-Nirenberg inequalities:‖𝑀‖𝐿4≀𝐢‖𝑀‖𝛼𝐿2β€–β€–Ξ›1+𝑠𝑀‖‖𝐿1βˆ’π›Ό2,‖Λ𝑠𝑀‖𝐿4≀𝐢‖𝑀‖𝐿1βˆ’π›Ό2β€–β€–Ξ›1+𝑠𝑀‖‖𝛼𝐿2,(1.9) with Ξ›βˆΆ=(βˆ’Ξ”)1/2,π›ΌβˆΆ=1βˆ’(1/2)β‹…(1/(1+𝑠)), and 𝑠>0, and the product estimate due to Kato-Ponce [7]:‖Λ𝑠‖(𝑓𝑔)𝐿𝑝≀𝐢‖𝑓‖𝐿𝑝1β€–Ξ›π‘ π‘”β€–πΏπ‘ž1+‖𝑔‖𝐿𝑝2β€–Ξ›π‘ π‘“β€–πΏπ‘ž2ξ€Έ,(1.10) with 𝑠>0 and 1/𝑝=1/𝑝1+1/π‘ž1=1/𝑝2+1/π‘ž2.

Motivated by the problem (1.1)–(1.4), we consider the following liquid crystal flows:πœ•π‘‘ξ“π‘’+π‘’β‹…βˆ‡π‘’+βˆ‡πœ‹βˆ’Ξ”π‘’=π‘˜πœ•π‘‘π‘‘π‘˜β‹…βˆ‡π‘‘π‘˜,πœ•(1.11)div𝑒=0,(1.12)2𝑑𝑑+π‘’β‹…βˆ‡π‘‘+(βˆ’Ξ”)2||𝑑||||πœ•π‘‘=βˆ’πœ†π‘‘,=1,(1.13)πœ†βˆΆ=𝑑𝑑||2+||||Δ𝑑2||||+Ξ”βˆ‡π‘‘2+2π‘˜πœ•π‘˜π‘‘β‹…Ξ”πœ•π‘˜ξ€·π‘‘,(1.14)𝑒,𝑑,πœ•π‘‘π‘‘ξ€Έξ€·π‘’(π‘₯,0)=0,𝑑0,𝑑1ξ€Έ(π‘₯,0),π‘₯βˆˆβ„2,||𝑑0||=1.(1.15) The last two equations (1.13) and (1.14) are the biharmonic wave maps. It is also a simple matter to show that the problem (1.11)–(1.15) has at least one local-in-time strong solution. The aim of this paper is to prove the global-in-time regularity. We obtain the following.

Theorem 1.2. Let 𝑒0∈𝐻2,(βˆ‡π‘‘0,𝑑1)∈𝐻3×𝐻2 with div𝑒0=0,|𝑑0|=1,𝑑0⋅𝑑1=0 in ℝ2. Then there exists at least a global-in-time smooth solution: 𝑒,βˆ‡π‘‘,πœ•π‘‘π‘‘ξ€ΈβˆˆπΏβˆžξ€·0,𝑇;𝐻2ξ€ΈΓ—πΏβˆžξ€·0,𝑇;𝐻3ξ€ΈΓ—πΏβˆžξ€·0,𝑇;𝐻2ξ€Έ(1.16) for any 𝑇>0.

Remark 1.3. We are unable to prove the uniqueness of strong solutions in Theorem 1.2.

2. Proof of Theorem 1.1

We only need to prove a priori estimates.

Testing (1.1) by 𝑒, using (1.2), we see that12π‘‘ξ€œπ‘’π‘‘π‘‘2ξ€œ||||𝑑π‘₯+βˆ‡π‘’2ξ€œ(𝑑π‘₯=π‘’β‹…βˆ‡)π‘‘β‹…πœ•π‘‘π‘‘π‘‘π‘₯.(2.1)

Testing (1.3) by πœ•π‘‘π‘‘, using |𝑑|=1 and π‘‘β‹…πœ•π‘‘π‘‘=0, we find that12π‘‘ξ€œ||||π‘‘π‘‘βˆ‡π‘‘2+||πœ•π‘‘π‘‘||2ξ€œ(𝑑π‘₯=βˆ’π‘’β‹…βˆ‡)π‘‘β‹…πœ•π‘‘π‘‘π‘‘π‘₯.(2.2)

Summing up (2.1) and (2.2), we get12π‘‘ξ€œπ‘’π‘‘π‘‘2+||||βˆ‡π‘‘2+||πœ•π‘‘π‘‘||2ξ€œ||||𝑑π‘₯+βˆ‡π‘’2𝑑π‘₯=0,(2.3) from which we getξ€œπ‘’2+||||βˆ‡π‘‘2+||πœ•π‘‘π‘‘||2ξ€œπ‘‘π‘₯+𝑇0ξ€œ||||βˆ‡π‘’2𝑑π‘₯𝑑𝑑≀𝐢.(2.4)

Applying Ξ›1+𝑠 to (1.1), testing by Ξ›1+𝑠𝑒, using (1.2) and (1.10), we derive12π‘‘ξ€œ||Λ𝑑𝑑1+𝑠𝑒||2ξ€œ||Λ𝑑π‘₯+2+𝑠𝑒||2ξ€œΞ›π‘‘π‘₯=βˆ’1+𝑠div(π‘’βŠ—π‘’)β‹…Ξ›1+π‘ ξ€œΞ›π‘’π‘‘π‘₯+1+π‘ ξ€·πœ•π‘‘ξ€Έπ‘‘β‹…βˆ‡π‘‘β‹…Ξ›1+𝑠𝑒𝑑π‘₯β‰€πΆβ€–π‘’β€–πΏβˆžβ€–β€–Ξ›2+𝑠𝑒‖‖𝐿2β€–β€–Ξ›1+𝑠𝑒‖‖𝐿2ξ€·β€–β€–πœ•+πΆπ‘‘π‘‘β€–β€–πΏβˆžβ€–β€–Ξ›2+𝑠𝑑‖‖𝐿2+β€–βˆ‡π‘‘β€–πΏβˆžβ€–β€–Ξ›1+π‘ πœ•π‘‘π‘‘β€–β€–πΏ2ξ€Έβ€–β€–Ξ›1+𝑠𝑒‖‖𝐿2≀12β€–β€–Ξ›2+𝑠𝑒‖‖2𝐿2+𝐢‖𝑒‖2πΏβˆžβ€–β€–Ξ›1+𝑠𝑒‖‖2𝐿2β€–β€–ξ€·πœ•+𝐢𝑑‖‖𝑑,βˆ‡π‘‘πΏβˆžξ‚€π‘¦2+β€–β€–Ξ›1+𝑠𝑒‖‖2𝐿2,(2.5) where𝑦2β€–β€–Ξ›βˆΆ=1+π‘ πœ•π‘‘π‘‘β€–β€–2𝐿2+β€–β€–Ξ›2+𝑠𝑑‖‖2𝐿2.(2.6)

Taking Ξ›1+𝑠 to (1.3), testing by Ξ›1+π‘ πœ•π‘‘π‘‘, we have12𝑑𝑦𝑑𝑑2=ξ€œΞ›1+𝑠𝑑||||βˆ‡π‘‘2βˆ’||πœ•π‘‘π‘‘||2⋅Λ1+π‘ πœ•π‘‘βˆ’ξ€œΞ›π‘‘π‘‘π‘₯1+𝑠(π‘’β‹…βˆ‡π‘‘)β‹…Ξ›1+π‘ πœ•π‘‘π‘‘π‘‘π‘₯=∢𝐼1+𝐼2.(2.7)

By using (1.10), (2.4), and (1.9), 𝐼1 can be bounded as follows:𝐼1ξ‚ƒβ€–β‰€πΆπ‘‘β€–πΏβˆžβ€–β€–Ξ›1+𝑠||||βˆ‡π‘‘2βˆ’||πœ•π‘‘π‘‘||2‖‖𝐿2+β€–β€–Ξ›1+𝑠𝑑‖‖𝐿4ξ€·β€–βˆ‡π‘‘β€–πΏβˆžβ€–βˆ‡π‘‘β€–πΏ4+β€–β€–πœ•π‘‘π‘‘β€–β€–πΏβˆžβ€–β€–πœ•π‘‘π‘‘β€–β€–πΏ4β€–β€–Ξ›ξ€Έξ€»1+π‘ πœ•π‘‘π‘‘β€–β€–πΏ2β‰€πΆξ€Ίξ€·β€–βˆ‡π‘‘β€–πΏβˆžβ€–β€–Ξ›2+𝑠𝑑‖‖𝐿2+β€–β€–πœ•π‘‘π‘‘β€–β€–πΏβˆžβ€–β€–Ξ›1+π‘ πœ•π‘‘π‘‘β€–β€–πΏ2ξ€Έ+π‘π‘¦π›Όξ€·β€–βˆ‡π‘‘β€–πΏβˆžπ‘¦1βˆ’π›Ό+β€–β€–πœ•π‘‘π‘‘β€–β€–πΏβˆžπ‘¦1βˆ’π›Όβ€–β€–Ξ›ξ€Έξ€»1+π‘ πœ•π‘‘π‘‘β€–β€–πΏ2β€–β€–ξ€·πœ•β‰€πΆπ‘‘ξ€Έβ€–β€–π‘‘,βˆ‡π‘‘πΏβˆžπ‘¦2.(2.8)

By using (1.10), 𝐼2 can be bounded as𝐼2ξ€·β‰€πΆβ€–π‘’β€–πΏβˆžβ€–β€–Ξ›2+𝑠𝑑‖‖𝐿2+β€–βˆ‡π‘‘β€–πΏβˆžβ€–β€–Ξ›1+𝑠𝑒‖‖𝐿2ξ€Έβ€–β€–Ξ›1+π‘ πœ•π‘‘π‘‘β€–β€–πΏ2β‰€πΆβ€–π‘’β€–πΏβˆžπ‘¦2+πΆβ€–βˆ‡π‘‘β€–πΏβˆžξ‚€π‘¦2+β€–β€–Ξ›1+𝑠𝑒‖‖2𝐿2.(2.9)

Combining (2.5), (2.7), (2.8), and (2.9) and using (1.6), (1.7), (1.8), and the Gronwall lemma, we arrive atβ€–π‘’β€–πΏβˆž(0,𝑇;𝐻1+𝑠)+‖𝑒‖𝐿2(0,𝑇;𝐻2+𝑠)‖‖≀𝐢,βˆ‡π‘‘,πœ•π‘‘π‘‘ξ€Έβ€–β€–πΏβˆž(0,𝑇;𝐻1+𝑠)≀𝐢.(2.10)

This completes the proof.

3. Proof of Theorem 1.2

For simplicity, we only present a priori estimates.

First, we still have (2.1).

Testing (1.13) by πœ•π‘‘π‘‘, using π‘‘β‹…πœ•π‘‘π‘‘=0, we have12π‘‘ξ€œ||||𝑑𝑑Δ𝑑2+||πœ•π‘‘π‘‘||2ξ€œ(𝑑π‘₯=βˆ’π‘’β‹…βˆ‡)π‘‘β‹…πœ•π‘‘π‘‘π‘‘π‘₯.(3.1)

Summing up (2.1) and (3.1), we getξ€œπ‘’2+||||Δ𝑑2+||πœ•π‘‘π‘‘||2ξ€œπ‘‘π‘₯+𝑇0ξ€œ||||βˆ‡π‘’21𝑑π‘₯𝑑𝑑≀𝐢,2π‘‘ξ€œ||||π‘‘π‘‘βˆ‡π‘‘2ξ€œπ‘‘π‘₯=βˆ’Ξ”π‘‘β‹…πœ•π‘‘π‘‘π‘‘π‘₯≀‖Δ𝑑‖𝐿2β€–β€–πœ•π‘‘π‘‘β€–β€–πΏ2≀𝐢,(3.2)

which yieldsξ€œ||||βˆ‡π‘‘2𝑑π‘₯≀𝐢.(3.3)

Applying Ξ” to (1.11), testing by Δ𝑒, using (1.2) and (1.10), we deduce that12π‘‘ξ€œ||||𝑑𝑑Δ𝑒2ξ€œ||||𝑑π‘₯+βˆ‡Ξ”π‘’2ξ€œξ€œΞ”ξ€·πœ•π‘‘π‘₯=βˆ’Ξ”div(π‘’βŠ—π‘’)⋅Δ𝑒𝑑π‘₯+𝑑𝑑,βˆ‡π‘‘β‹…Ξ”π‘’π‘‘π‘₯β‰€πΆβ€–π‘’β€–πΏβˆžβ€–βˆ‡Ξ”π‘’β€–πΏ2‖Δ𝑒‖𝐿2ξ€·β€–β€–πœ•+πΆπ‘‘π‘‘β€–β€–πΏβˆžβ€–βˆ‡Ξ”π‘‘β€–πΏ2+β€–βˆ‡π‘‘β€–πΏβˆžβ€–β€–Ξ”πœ•π‘‘π‘‘β€–β€–πΏ2‖Δ𝑒‖𝐿2≀12β€–βˆ‡Ξ”π‘’β€–2𝐿2+𝐢‖𝑒‖2πΏβˆžβ€–Ξ”π‘’β€–2𝐿2ξ‚€β€–β€–πœ•+𝐢𝑑𝑑‖‖𝐿1/22β€–β€–Ξ”πœ•π‘‘π‘‘β€–β€–πΏ1/22‖Δ𝑑‖𝐿1/22β€–β€–Ξ”2𝑑‖‖𝐿1/22+β€–βˆ‡π‘‘β€–πΏβˆžβ€–β€–Ξ”πœ•π‘‘π‘‘β€–β€–πΏ2‖Δ𝑒‖𝐿2≀12β€–βˆ‡Ξ”π‘’β€–2𝐿2+𝐢‖𝑒‖2πΏβˆžβ€–Ξ”π‘’β€–2𝐿2+𝐢𝑦2+𝐢‖Δ𝑒‖2𝐿2+πΆβ€–βˆ‡π‘‘β€–πΏβˆžξ‚€π‘¦2+‖Δ𝑒‖2𝐿2,(3.4) where𝑦2β€–β€–βˆΆ=Ξ”πœ•π‘‘π‘‘β€–β€–2𝐿2+β€–β€–Ξ”2𝑑‖‖2𝐿2.(3.5)

Applying Ξ” to (1.13), we haveΞ”πœ•2𝑑𝑑+Ξ”3𝑑=βˆ’(πœ†Ξ”π‘‘+2βˆ‡πœ†β‹…βˆ‡π‘‘+π‘‘Ξ”πœ†)βˆ’Ξ”(π‘’β‹…βˆ‡π‘‘).(3.6)

Sinceξ€·0=Ξ”π‘‘πœ•π‘‘π‘‘ξ€Έ=π‘‘Ξ”πœ•π‘‘π‘‘+πœ•π‘‘ξ“π‘‘Ξ”π‘‘+2π‘˜πœ•π‘˜π‘‘πœ•π‘˜πœ•π‘‘π‘‘,(3.7) we easily see thatβˆ’π‘‘Ξ”πœ•π‘‘π‘‘=πœ•π‘‘π‘‘Ξ”π‘‘+πœ•π‘‘||||βˆ‡π‘‘2.(3.8)

Testing (3.6) by Ξ”πœ•π‘‘π‘‘, using (3.8), we obtain12𝑑𝑦𝑑𝑑2ξ€œ=βˆ’πœ†Ξ”π‘‘β‹…Ξ”πœ•π‘‘ξ€Ίπ‘‘+βˆ‡πœ†2βˆ‡π‘‘β‹…Ξ”πœ•π‘‘ξ€·πœ•π‘‘+βˆ‡π‘‘π‘‘Ξ”π‘‘+2βˆ‡π‘‘β‹…βˆ‡πœ•π‘‘π‘‘βˆ’ξ€œξ€Έξ€»π‘‘π‘₯Ξ”(π‘’β‹…βˆ‡π‘‘)β‹…Ξ”πœ•π‘‘π‘‘π‘‘π‘₯=∢𝐽1+J2.(3.9)

By the same calculations as those in [8], we have𝐽1≀𝐢1+β€–βˆ‡π‘‘β€–πΏβˆžξ€Έπ‘¦2≀𝐢1+β€–βˆ‡π‘‘β€–π»1𝑦log(𝑒+𝑦)2.(3.10)

By using (1.10), 𝐽2 can be bounded as𝐽2≀𝐢‖Δ𝑒‖𝐿2β€–βˆ‡π‘‘β€–πΏβˆž+β€–π‘’β€–πΏβˆžβ€–βˆ‡Ξ”π‘‘β€–πΏ2ξ€Έβ€–β€–Ξ”πœ•π‘‘π‘‘β€–β€–πΏ2β‰€πΆβ€–βˆ‡π‘‘β€–πΏβˆžξ‚€π‘¦2+‖Δ𝑒‖2𝐿2+𝐢‖𝑒‖𝐿1/22‖Δ𝑒‖𝐿1/22‖Δ𝑑‖𝐿1/22β€–β€–Ξ”2𝑑‖‖𝐿1/22β€–β€–Ξ”πœ•π‘‘π‘‘β€–β€–πΏ2β‰€πΆβ€–βˆ‡π‘‘β€–πΏβˆžξ‚€π‘¦2+‖Δ𝑒‖2𝐿2+𝐢𝑦2+𝐢‖Δ𝑒‖2𝐿2.(3.11)

Combining (3.4), (3.9), (3.10), and (3.11) and using (1.6) and the Gronwall lemma, we conclude thatβ€–π‘’β€–πΏβˆž(0,𝑇;𝐻2)+‖𝑒‖𝐿2(0,𝑇;𝐻3)≀𝐢,β€–βˆ‡π‘‘β€–πΏβˆž(0,𝑇;𝐻3)+β€–β€–πœ•π‘‘π‘‘β€–β€–πΏβˆž(0,𝑇;𝐻2)≀𝐢.(3.12)

This completes the proof.

Acknowledgment

This paper is supported by NSFC (no. 11171154).

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