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 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]: where is the velocity, is the pressure, and 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 . The last equation (1.3) is the well-known wave maps when .
It is a simple matter to show that the system (1.1)–(1.4) has a unique local-in-time smooth solution when with in . 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 with in and let be a smooth solution of (1.1)–(1.4) on some interval with . Assume that Then the solution can be extended beyond .
is the homogeneous Besov space. We have ; see Triebel [2].
In the proof of Theorem 1.1, we will use the logarithmic Sobolev inequalities [3–6]: for , and the Gagliardo-Nirenberg inequalities: with , and , and the product estimate due to Kato-Ponce [7]: with and .
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.
Theorem 1.2. Let with in . Then there exists at least a global-in-time smooth solution: for any .
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 , using (1.2), we see that
Testing (1.3) by , using and , we find that
Summing up (2.1) and (2.2), we get from which we get
Applying to (1.1), testing by , using (1.2) and (1.10), we derive where
Taking to (1.3), testing by , we have
By using (1.10), (2.4), and (1.9), can be bounded as follows:
By using (1.10), can be bounded as
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
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 , using , we have
Summing up (2.1) and (3.1), we get
which yields
Applying to (1.11), testing by , using (1.2) and (1.10), we deduce that where
Applying to (1.13), we have
Since we easily see that
Testing (3.6) by , using (3.8), we obtain
By the same calculations as those in [8], we have
By using (1.10), can be bounded as
Combining (3.4), (3.9), (3.10), and (3.11) and using (1.6) and the Gronwall lemma, we conclude that
This completes the proof.
Acknowledgment
This paper is supported by NSFC (no. 11171154).