Continuation Criterion for the 2D Liquid Crystal Flows

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 , 𝜕 𝑡 𝑑 ∈ 𝐿 1 𝐵 ( 0 , 𝑇 ; 0 ∞ , ∞ ( ℝ 2 ) ) 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.


Introduction
First, we consider the following simplified liquid crystal flows in two space dimensions 1 : 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 ∂ t d 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 u 0 , ∇d 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.
Then the solution u, d can be extended beyond T > 0.
Ḃ0 ∞,∞ is the homogeneous Besov space.We have L ∞ ⊂ BMO ⊂ Ḃ0 ∞,∞ ; see Triebel 2 .In the proof of Theorem 1.1, we will use the logarithmic Sobolev inequalities 3-6 : for s > 0, and the Gagliardo-Nirenberg inequalities: 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 : with s > 0 and 1/p 1/p 1 1/q 1 1/p 2 1/q 2 .Motivated by the problem 1.1 -1.4 , we consider the following liquid crystal flows: 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.
Then there exists at least a global-in-time smooth solution: for any T > 0.
Remark 1.3.We are unable to prove the uniqueness of strong solutions in Theorem 1.2.

Proof of Theorem 1.1
We only need to prove a priori estimates.Testing 1.1 by u, using 1.2 , we see that

2.7
By using 1.10 , 2.4 , and 1.9 , I 1 can be bounded as follows:

2.10
This completes the proof.

Proof of Theorem 1.2
For simplicity, we only present a priori estimates.First, we still have 2.1 .Testing 1.13 by ∂ t d, using d • ∂ t d 0, we have 1 2 Summing up 2.1 and 3.1 , we get Applying Δ to 1.11 , testing by Δu, using 1.2 and 1.10 , we deduce that 1 2 where 3.5 Applying Δ to 1.13 , we have

3.9
By the same calculations as those in 8 , we have

3.12
This completes the proof.