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 , .

1. Introduction

In this paper, we consider the following 2D liquid crystal flow: where , are real parameters and is the velocity, is a vectorial function modeling the orientation of the crystal molecules, and is the scalar pressure. Here and is defined in terms of Fourier transform by

When , 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 . Let be the local strong solution to the problem (1)–(4). If and satisfy , , then the 2D liquid crystal model has a unique global classical solution satisfying

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) , ; (2) , ; (3) , . Combining them with the result in [10], we know that if , 2D incompressible MHD system with fractional diffusion possesses a global smooth solution. Fan et al. [8] proved the global existence of smooth solutions with , . Global regularity for the case , was established by Jiu and Zhao [9] which improves the result in [6]. Very recently, the authors improved the case , for the 2D liquid crystal model in [11].

2. Proof of Theorem 1

It is sufficient to prove Theorem 1 with , .

We will prove Theorem 1 if we can demonstrate the boundedness of . In order to reach our purpose, we will show this by contradiction: assume for some finite time . Our thought is that when is close enough to , remains uniformly bounded for under such assumption, thus reaching a contradiction.

First, we do estimate for . Multiplying (2) by and using (3), after integration by parts, we see that By using the Gronwall inequality, we have Then, we will show the estimate for and . Multiplying (1) and (2) by and , respectively, we find that Thanks to Gronwall’s inequality and (9), we have which means .

The estimate for and estimate for will be shown as follows. Multiplying (1) by , applying to (2), multiplying by , and then summing them up, we obtain Let us introduce the following commutator and bilinear estimates established in [12, 13]: with and .

Now, we do the estimate for and estimate for . Applying to (1), multiplying by , and dealing with (2) in the same way by and , after summing them up, we have Using Hölder’s inequality, Gagliardo-Nirenberg inequality, Young’s inequality, and (13), we have the following estimates: Now we estimate , , and one by one: and can be estimated as follows: Combining and , we have Summing all the above estimates to (14), we obtain Now, we will show the estimate for and estimate for . Applying to (1), multiplying by , and dealing with (2) in the same way by and , after summing them up, we have Using Hölder’s inequality, Gagliardo-Nirenberg inequality, Young’s inequality, and (13), we have the following estimates: Now we estimate , , , and one by one: The estimate for is as follows: We calculate , , and : Combining , , and , we get Combining the above estimates to (20), we get Now we estimate the term by applying the Gronwall inequality to (12): Here will be fixed later and we denote , . Set . Now applying the logarithmic inequality [14] we get Since , we can take close enough to , so that for some small positive number to be fixed later. With such choice of we have Hölder’s inequality gives Fix satisfying

Combining the above estimates together, we get Integrating the above inequality, we have where .