Continuation Criterion for the 2D Liquid Crystal Flows

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 ∇𝑑,𝜕_𝑡𝑑∈𝐿¹(0,𝑇;𝐵̇^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]:𝜕𝜕_𝑡𝑢+𝑢⋅∇𝑢+∇d𝜋iv−𝑢Δ=𝑢0=,_𝑘𝜕_𝑡𝑑^𝑘⋅∇𝑑^𝑘,((11..12))_2𝑡𝑢,𝑑,𝜕_𝑡𝑑+𝑢⋅∇𝑑−Δ𝑑=𝑑||∇𝑑||²−||𝜕_𝑡𝑑||²,||𝑑||=1,(1.3)𝑑(𝑥,0)=𝑢₀,𝑑₀,𝑑₁(𝑥),𝑥∈ℝ²,||𝑑₀||=1,𝑑₀⋅𝑑₁=0,(1.4) where 𝑢 is the velocity, 𝜋 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 ∑_𝑘𝜕_𝑡𝑑^𝑘⋅∇𝑑^𝑘. The last equation (1.3) is the well-known wave maps when 𝑢=0.

It is a simple matter to show that the system (1.1)–(1.4) has a unique local-in-time smooth solution when 𝑢₀,∇𝑑₀,𝑑₁∈𝐻^1+𝑠(ℝ²) with 𝑠>0,div𝑢₀=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 𝑢₀,∇𝑑₀,𝑑₁∈𝐻^1+𝑠(ℝ²) with 𝑠>0,div𝑢₀=0,|𝑑₀|=1,𝑑₀⋅𝑑₁=0 in ℝ² and let (𝑢,𝑑) be a smooth solution of (1.1)–(1.4) on some interval [0,𝑇] with 0<𝑇<∞. Assume that ∇𝑑,𝜕_𝑡𝑑∈𝐿¹0,𝑇;𝐵̇^0∞,∞ℝ².(1.5) Then the solution (𝑢,𝑑) can be extended beyond 𝑇>0.

𝐵̇^0∞,∞ is the homogeneous Besov space. We have 𝐿^∞⊂𝐵𝑀𝑂⊂𝐵̇^0∞,∞; see Triebel [2].

In the proof of Theorem 1.1, we will use the logarithmic Sobolev inequalities [3–6]:‖∇‖𝑑𝑢‖‖_𝐿^∞_𝐿∞≤𝐶‖𝑢‖_𝐻¹log^1/2𝑒+‖𝑢‖_𝐻^1+𝑠,(1.6)‖‖𝜕≤𝐶1+‖∇𝑑‖_̇𝐵^0∞,∞log𝑒+‖∇𝑑‖_𝐻^1+𝑠,(1.7)_𝑡𝑑‖‖_𝐿∞≤𝐶1+‖‖𝜕_𝑡𝑑‖‖_{̇𝐵0∞,∞}log𝑒+‖‖𝜕_𝑡𝑑‖‖_𝐻1+𝑠,(1.8) for 𝑠>0, and the Gagliardo-Nirenberg inequalities:‖Λ‖_𝑠𝑤‖_𝐿⁴≤𝐶‖𝑤‖^𝛼𝐿₂‖‖Λ^1+𝑠𝑤‖‖^𝐿1−𝛼₂,𝑤‖_𝐿⁴≤𝐶‖𝑤‖^𝐿1−𝛼₂‖‖Λ^1+𝑠𝑤‖‖^𝛼𝐿₂,(1.9) with Λ∶=(−Δ)^1/2,𝛼∶=1−(1/2)⋅(1/(1+𝑠)), and 𝑠>0, and the product estimate due to Kato-Ponce [7]:‖Λ^𝑠(𝑓𝑔)‖_𝐿^𝑝≤𝐶‖𝑓‖_𝐿^𝑝1‖Λ^𝑠𝑔‖_𝐿^𝑞1+‖𝑔‖_𝐿^𝑝2‖Λ^𝑠𝑓‖_𝐿^𝑞2,(1.10) with 𝑠>0 and 1/𝑝=1/𝑝₁+1/𝑞₁=1/𝑝₂+1/𝑞₂.

Motivated by the problem (1.1)–(1.4), we consider the following liquid crystal flows:𝜕𝜕_𝑡𝑢+𝑢⋅∇𝑢+∇d𝜋iv−𝑢Δ=𝑢0=,_𝑘𝜕_𝑡𝑑^𝑘⋅∇𝑑^𝑘,((11..1112))_2𝑡𝜆∶=𝑑||𝜕_𝑡+𝑢⋅∇𝑑+(−Δ)²𝑑=−𝜆𝑑,||𝑑||=1,(1.13)𝑢,𝑑,𝜕_𝑡𝑑||²+||Δ𝑑||²+Δ||∇𝑑||²+2_𝑘𝜕_𝑘𝑑⋅Δ𝜕_𝑘𝑑,(1.14)𝑑(𝑥,0)=𝑢₀,𝑑₀,𝑑₁(𝑥,0),𝑥∈ℝ²,||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 𝑢₀∈𝐻²,(∇𝑑₀,𝑑₁)∈𝐻³×𝐻² with div𝑢₀=0,|𝑑₀|=1,𝑑₀⋅𝑑₁=0 in ℝ². Then there exists at least a global-in-time smooth solution: 𝑢,∇𝑑,𝜕_𝑡𝑑∈𝐿^∞0,𝑇;𝐻²×𝐿^∞0,𝑇;𝐻³×𝐿^∞0,𝑇;𝐻²(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 𝑢, using (1.2), we see that12𝑑𝑑𝑡𝑢²𝑑𝑥+||∇𝑢||²𝑑𝑥=(𝑢⋅∇)𝑑⋅𝜕_𝑡𝑑𝑑𝑥.(2.1)

Testing (1.3) by 𝜕_𝑡𝑑, using |𝑑|=1 and 𝑑⋅𝜕_𝑡𝑑=0, we find that12𝑑𝑑𝑡||∇𝑑||²+||𝜕_𝑡𝑑||²𝑑𝑥=−(𝑢⋅∇)𝑑⋅𝜕_𝑡𝑑𝑑𝑥.(2.2)

Summing up (2.1) and (2.2), we get12𝑑𝑑𝑡𝑢²+||∇𝑑||²+||𝜕_𝑡𝑑||²𝑑𝑥+||∇𝑢||²𝑑𝑥=0,(2.3) from which we get𝑢²+||∇𝑑||²+||𝜕_𝑡𝑑||²𝑑𝑥+^𝑇0||∇𝑢||²𝑑𝑥𝑑𝑡≤𝐶.(2.4)

Applying Λ^1+𝑠 to (1.1), testing by Λ^1+𝑠𝑢, using (1.2) and (1.10), we derive12𝑑𝑑𝑡||Λ^1+𝑠=−Λ𝑢||²𝑑_1+𝑠𝑥+||Λ^2+𝑠𝑢||²𝑑𝑥≤𝐶‖𝑢‖_𝐿∞div(𝑢⊗𝑢)⋅Λ^1+𝑠𝑢𝑑𝑥+Λ^1+𝑠𝜕_𝑡𝑑⋅∇𝑑⋅Λ^1+𝑠𝑢𝑑𝑥‖‖Λ^2+𝑠𝑢‖‖_𝐿2‖‖Λ^1+𝑠𝑢‖‖_𝐿2+𝐶‖‖𝜕_𝑡𝑑‖‖_𝐿∞‖‖Λ^2+𝑠𝑑‖‖_𝐿2+‖∇𝑑‖_𝐿^∞‖‖Λ^1+𝑠𝜕_𝑡𝑑‖‖_𝐿2‖‖Λ^1+𝑠𝑢‖‖_𝐿2≤12‖‖Λ^2+𝑠𝑢‖‖^2𝐿₂+𝐶‖𝑢‖^2𝐿_∞‖‖Λ^1+𝑠𝑢‖‖^2𝐿₂+𝐶‖‖𝜕_𝑡𝑑,∇𝑑‖‖_𝐿∞𝑦²+‖‖Λ^1+𝑠𝑢‖‖^2𝐿₂,(2.5) where𝑦²∶=‖‖Λ^1+𝑠𝜕_𝑡𝑑‖‖^2𝐿₂+‖‖Λ^2+𝑠𝑑‖‖^2𝐿₂.(2.6)

Taking Λ^1+𝑠 to (1.3), testing by Λ^1+𝑠𝜕_𝑡𝑑, we have12𝑑𝑑𝑡𝑦²=−ΛΛ^1+𝑠𝑑||∇𝑑||²−||𝜕_𝑡𝑑||²⋅Λ^1+𝑠𝜕_𝑡𝑑𝑑𝑥_1+𝑠(𝑢⋅∇𝑑)⋅Λ^1+𝑠𝜕_𝑡𝑑𝑑𝑥=∶𝐼₁+𝐼₂.(2.7)

By using (1.10), (2.4), and (1.9), 𝐼₁ can be bounded as follows:𝐼₁≤𝐶‖𝑑‖_𝐿^∞‖‖Λ^1+𝑠||∇𝑑||²−||𝜕_𝑡𝑑||²‖‖_𝐿2+‖‖Λ^1+𝑠𝑑‖‖_𝐿4‖∇𝑑‖_𝐿^∞‖∇𝑑‖_𝐿⁴+‖‖𝜕_𝑡𝑑‖‖_𝐿∞‖‖𝜕_𝑡𝑑‖‖_𝐿4‖‖Λ^1+𝑠𝜕_𝑡𝑑‖‖_𝐿2≤𝐶+𝑐‖𝑦∇_𝛼𝑑‖_𝐿^∞‖‖Λ^2+𝑠𝑑‖‖_𝐿2+‖‖𝜕_𝑡𝑑‖‖_𝐿∞‖‖Λ^1+𝑠𝜕_𝑡𝑑‖‖_𝐿2‖∇𝑑‖_𝐿^∞𝑦^1−𝛼+‖‖𝜕_𝑡𝑑‖‖_𝐿∞𝑦^1−𝛼‖‖Λ^1+𝑠𝜕_𝑡𝑑‖‖_𝐿2≤𝐶‖‖𝜕_𝑡𝑑,∇𝑑‖‖_𝐿∞𝑦².(2.8)

By using (1.10), 𝐼₂ can be bounded as𝐼₂≤𝐶‖𝑢‖_𝐿^∞‖‖Λ^2+𝑠𝑑‖‖_𝐿2+‖∇𝑑‖_𝐿^∞‖‖Λ^1+𝑠𝑢‖‖_𝐿2‖‖Λ^1+𝑠𝜕_𝑡𝑑‖‖_𝐿2≤𝐶‖𝑢‖_𝐿^∞𝑦²+𝐶‖∇𝑑‖_𝐿^∞𝑦²+‖‖Λ^1+𝑠𝑢‖‖^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 at‖𝑢‖‖‖∇𝑑,_𝐿^∞_(0,𝑇;𝐻^1+𝑠₎𝜕+_𝑡‖𝑢‖_𝐿²_(0,𝑇;𝐻^2+𝑠₎≤𝐶,𝑑‖‖_{𝐿∞(0,𝑇;𝐻1+𝑠)}≤𝐶.(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 𝜕_𝑡𝑑, using 𝑑⋅𝜕_𝑡𝑑=0, we have12𝑑𝑑𝑡||Δ𝑑||²+||𝜕_𝑡𝑑||²𝑑𝑥=−(𝑢⋅∇)𝑑⋅𝜕_𝑡𝑑𝑑𝑥.(3.1)

Summing up (2.1) and (3.1), we get𝑢²+||Δ𝑑||²+||𝜕_𝑡𝑑||²𝑑𝑥+^𝑇0||∇𝑢||²1𝑑𝑥𝑑𝑡≤𝐶,2𝑑𝑑𝑡||∇𝑑||²𝑑𝑥=−Δ𝑑⋅𝜕_𝑡𝑑𝑑𝑥≤‖Δ𝑑‖_𝐿²‖‖𝜕_𝑡𝑑‖‖_𝐿2≤𝐶,(3.2)

which yields||∇𝑑||²𝑑𝑥≤𝐶.(3.3)

Applying Δ to (1.11), testing by Δ𝑢, using (1.2) and (1.10), we deduce that12𝑑𝑑𝑡||Δ𝑢||²=−Δd𝑑i𝑥v+(𝑢⊗||∇𝑢Δ)𝑢⋅||Δ²𝑑𝑢𝑥𝑑𝑥+Δ𝜕_𝑡≤𝐶‖𝑢‖_𝐿∞𝑑,∇𝑑⋅Δ𝑢𝑑𝑥‖∇Δ𝑢‖_𝐿²‖Δ𝑢‖_𝐿²+𝐶‖‖𝜕_𝑡𝑑‖‖_𝐿∞‖∇Δ𝑑‖_𝐿²+‖∇𝑑‖_𝐿^∞‖‖Δ𝜕_𝑡𝑑‖‖_𝐿2‖Δ𝑢‖_𝐿²≤12‖∇Δ𝑢‖^2𝐿2+𝐶‖𝑢‖^2𝐿_∞‖Δ𝑢‖^2𝐿₂+𝐶‖‖𝜕_𝑡𝑑‖‖^𝐿1/2₂‖‖Δ𝜕_𝑡𝑑‖‖^𝐿1/2₂‖Δ𝑑‖^𝐿1/2₂‖‖Δ²𝑑‖‖^𝐿1/2₂+‖∇𝑑‖_𝐿^∞‖‖Δ𝜕_𝑡𝑑‖‖_𝐿2‖Δ𝑢‖^𝐿2≤12‖∇Δ𝑢‖^2𝐿2+𝐶‖𝑢‖^2𝐿_∞‖Δ𝑢‖^2𝐿₂+𝐶𝑦²+𝐶‖Δ𝑢‖^2𝐿₂+𝐶‖∇𝑑‖_𝐿^∞𝑦²+‖Δ𝑢‖^2𝐿₂,(3.4) where𝑦²∶=‖‖Δ𝜕_𝑡𝑑‖‖^2𝐿₂+‖‖Δ²𝑑‖‖^2𝐿₂.(3.5)

Applying Δ to (1.13), we haveΔ𝜕^2𝑡𝑑+Δ³𝑑=−(𝜆Δ𝑑+2∇𝜆⋅∇𝑑+𝑑Δ𝜆)−Δ(𝑢⋅∇𝑑).(3.6)

Since0=Δ𝑑𝜕_𝑡𝑑=𝑑Δ𝜕_𝑡𝑑+𝜕_𝑡𝑑Δ𝑑+2_𝑘𝜕_𝑘𝑑𝜕_𝑘𝜕_𝑡𝑑,(3.7) we easily see that−𝑑Δ𝜕_𝑡𝑑=𝜕_𝑡𝑑Δ𝑑+𝜕_𝑡||∇𝑑||².(3.8)

Testing (3.6) by Δ𝜕_𝑡𝑑, using (3.8), we obtain12𝑑𝑑𝑡𝑦²=−−Δ𝜆Δ(𝑢𝑑⋅⋅∇Δ𝑑𝜕)_𝑡𝑑⋅Δ+𝜕∇_𝑡𝜆2∇𝑑⋅Δ𝜕_𝑡𝑑+∇𝜕_𝑡𝑑Δ𝑑+2∇𝑑⋅∇𝜕_𝑡𝑑𝑑𝑥𝑑𝑑𝑥=∶𝐽₁+J₂.(3.9)

By the same calculations as those in [8], we have𝐽₁≤𝐶1+‖∇𝑑‖_𝐿^∞𝑦²≤𝐶1+‖∇𝑑‖_𝐻¹log(𝑒+𝑦)𝑦².(3.10)

By using (1.10), 𝐽₂ can be bounded as𝐽₂≤𝐶‖Δ𝑢‖_𝐿²‖∇𝑑‖_𝐿^∞+‖𝑢‖_𝐿^∞‖∇Δ𝑑‖_𝐿²‖‖Δ𝜕_𝑡𝑑‖‖_𝐿2≤𝐶‖∇𝑑‖_𝐿^∞𝑦²+‖Δ𝑢‖^2𝐿₂+𝐶‖𝑢‖^𝐿1/22‖Δ𝑢‖^𝐿1/2₂‖Δ𝑑‖^𝐿1/2₂‖‖Δ²𝑑‖‖^𝐿1/2₂‖‖Δ𝜕_𝑡𝑑‖‖_𝐿2≤𝐶‖∇𝑑‖_𝐿^∞𝑦²+‖Δ𝑢‖^2𝐿₂+𝐶𝑦²+𝐶‖Δ𝑢‖^2𝐿₂.(3.11)

Combining (3.4), (3.9), (3.10), and (3.11) and using (1.6) and the Gronwall lemma, we conclude that‖∇‖𝑑𝑢‖‖_𝐿^∞_(0,𝑇;𝐻²_{)𝐿∞(0,𝑇;𝐻3)}+‖𝑢‖_𝐿²_(0,𝑇;𝐻³₎≤𝐶,+‖‖𝜕_𝑡𝑑‖‖_{𝐿∞(0,𝑇;𝐻2)}≤𝐶.(3.12)

This completes the proof.

Acknowledgment

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