The existence and blow-up criterion of liquid crystals system in critical Besov space

We consider the existence of strong solution to liquid crystals system in critical Besov space,then give a criterion which is similar to Serrin's criterion on regularity of weak solution to Navier-Stokes equations.

From the viewpoint of partial differential equations, system (1.1) is a strongly coupled system between the incompressible Navier-Stokes equations and the transported heat flow of harmonic maps. It is difficult to get the global existence for general initial data, even the existence of weak solutions in three-dimensional space. However, Lin et al [14] and Hong [6] obtained the global weak solutions in two-dimensional space. In this paper, a solution is called to be a strong solution if the uniqueness holds. The well-posedness of system (1.1)-(1.3) has been studied by Li and Wang [9], Lin and Ding [15], Wang [16]. We are interesting in the local and global existence of strong solution to system (1.1)-(1.3) in this paper. We will also consider a criterion for system (1.1), which is similar to serrin's criterion on regularity of weak solution to Navier-Stokes equations. A natural way of dealing with uniqueness is to find a function space as large as possible, where the existence and uniqueness of solution hold. In a word, we need to find a "critical" space. This approach has been initiated by Fujita and Kato [5] on Navier-Stokes equations. We give the scaling of (u, p, d) as follows So it is natural to give the following definition: a space X is called critical space if ∀ u ∈ X, u λ X = u X , f or all λ > 0.
In [5], Fujita and Kato usedḢ 1 2 as the critical space on Navier-Stokes equations in threedimensional space. In this paper, we consider the homogeneous Besov spacesḂ s p,r . Inspired by [2], we find where N denotes the spacial dimension. In addition, we suppose |d| = 1 in physics and note thatḢ So it is natural to chooseḂ  2,1 (for d) as the critical spaces in which we study liquid crystals system (1.1). We are able to prove the following theorems.
I. There exists a constant T > 0, such that system (1.1)-(1.3) has a unique strong In addition, for any ρ 1 , In addition, for any ρ 1 , holds.
Space L ρ T (Ḃ s p,r ) and L ρ (Ḃ s p,r ) will be introduced in section 2. We want to point out that if (u, d) is smooth, then we can proof |d| = |d 0 | = 1 by maximal principle. Wang [16], Lin and Ding [15] proved |d| = |d 0 | = 1 by taking |∇d| 2 d as a second fundamental form. The following result is inspired by Kozono and Shimada [7].
It is well-known that using Besov space, one can get the same results in a lower regularity. And the following Remark is very interested.

Remark 1.1 By the embedding theory
we have which is the Serrin's criterion on endpoint.
The rest of our paper is organized as follows. In section 2, we give a short introduction on Besov space. In section 3, we prove Theorem 1.1 and Theorem 1.2.

Littlewood-Paley theory and Besov space
In this section, we present some well-known facts on Littlewood-paley theory, more details see [1,3]. Let S(R N ) be the Schwartz space and S ′ (R N ) be its dual space. Let (χ, ϕ) be a couple of smooth functions such that Then we have suppϕ ⊂ C(0, 3 4 as an open ball with radius R centered at zero, and C(0, For any u ∈ S(R N ), the Fourier transform of u denotes by u or F u. The inverse Fourier transform denotes by F −1 . Let h = F −1 ϕ. We define homogeneous dyadic blocks as follows Then for any u ∈ S ′ (R N ), the following decomposition , s be a real number, and 1 ≤ p, r ≤ ∞. We set The besov spaceḂ s p,r is defined as follows: and define the corresponding spaces Remark 2.1 According to Minkowski inequality, we have Next let's present some results of paradifferential calculus in homogeneous space, which is useful for dealing with nonlinear equations. For u, v ∈ S ′ (R N ), we have the following formal decomposition: Define the operators T , R as follows, Then we have the following so-called homogeneous Bony decomposition: In order to estimate the above terms, let's recall some lemmas (see [1,3]).
The initial problem of heat equation reads The following lemma can be found in [1,3].

Proof of main results
Let P denote the Leray projector on solenoidal vector fields which is defined by By operator P, we can project the second equation of (1.1) onto the divergence free vector field. Then the pressure p can be eliminated. It is easy to notice that P is a homogeneous multiplier of degree zero. Denote τ = d−d 0 . We only need to consider the following equations with initial conditions and far field behaviors u → 0, τ → 0, as |x| → ∞.
Next let us introduce the heat semi-group operator e at∆ . Let v = e at∆ v 0 , then v solves problem (2.7) with G replaced by zero.
Then we have where k is a constant. Case I. By using (3.16), we obtain It is easy to find N > 0, such that For above fixed N , let T be small enough such that Combining (3.17) and (3.18), we obtain (3.14). Case II. By using (3.16), we have By choosing δ 0 ≤ ε 0 C , we finish the proof.
Proof of Theorem 3.1.
In this part, we will use the iterative method to prove Theorem 3.1, which is separated into three steps. Firstly we point out that the case I in the following proof corresponds to I in Theorem 3.1 and case II corresponds to II in Theorem 3.1. Also we should keep in mind thatḂ s 2,1 , s > N 2 is not a Banach space. Then let us consider the following linear equations, with initial conditions Let us set u 1 = e at∆ u 0 , τ 1 = e at∆ τ 0 and begin our proof. First step: Uniform boundedness Case I. We claim that the following estimates hold for some T > 0,  In addition, we can get from (3.20), (3.21) and Lemma 2.3 that and more precisely, Case II. For the small initial data, it is simple to prove providing δ 0 is small enough. Here we omit the details. Second step: Convergence Case I. We will prove {(u n , τ n )|n = 1, 2 · · · } is a Cauchy sequence. Firstly, let's consider .
According to the proof of Lemma 2.3, we need to estimate the following terms: By using (3.22), Lemma 2.1 and Lemma 2.2, we can obtain ), ), , , .

Proof of Theorem 3.2.
We note that (u, τ ) ∈ C([0, T ];Ḃ 2,1 ). By theorem 3.1, we can find T 1 such that T 1 > T and (u, τ ) is a solution on [0, T 1 ] × R N . Since (3.13) holds, for any small ε 0 > 0, we can find δ > 0 which depends only on T ′ , such that , where 0 < θ < 1, and providing ε 0 is small enough. Then we get where the constant C is independent of T 2 . Let's repeat above procedure, we claim that T 1 ≥ T ′ . Indeed, if it is not true, we can find T ∈ [T 1 , T ′ ], such that( T is the first blow-up time) is bounded. But we find that (3.24) with T 2 replaced by T − δ 2 is contradicting with (3.25). So we get T 1 ≥ T ′ , and finish the proof of Theorem 3.2.