Strong solutions to Cauchy problem of 2D compressible nematic liquid crystal flows

This paper studies the local existence of strong solutions to the Cauchy problem of the 2D simplified Ericksen-Leslie system modeling compressible nematic liquid crystal flows, coupled via $ρ$ (the density of the fluid), $u$ (the velocity of the field), and $d$ (the macroscopic/continuum molecular orientations). Notice that the technique used for the corresponding 3D local well-posedness of strong solutions fails treating the 2D case, because the $L^p$-norm ($p>2$) of the velocity $u$ cannot be controlled in terms only of $ρ^{\frac{1}{2}}u$ and $\nabla u$ here. In the present paper, under the framework of weighted approximation estimates introduced in [J. Li, Z. Liang, On classical solutions to the Cauchy problem of the two-dimensional barotropic compressible Navier-Stokes equations with vacuum, J. Math. Pures Appl. (2014) 640-671] for Navier-Stokes equations, we obtain the local existence of strong solutions to the 2D compressible nematic liquid crystal flows.

Li-Liu-Zhong [13] got the same result under small initial data without the additional geometric condition (5). Extended to the more complicated compressible case, the simplified Ericksen-Leslie system is strongly coupled via the compressible Navier-Stokes equation and the transported harmonic map heat flow to S 2 , with significant progresses made during past years. Among them, Ding-Lin-Wang-Wen [1,2] obtained the global existence for weak and strong solutions of the 1D problem. For the 2D case, under the condition with the image of d 0 contained in the upper hemisphere S 2 + , Jiang-Jiang-Wang [9] established the existence of global weak solutions. In the 3D case, Huang-Wang-Wen [6] studied the local existence of strong solutions of (1). Moreover, Huang-Wang-Wen [6,7] and Huang-Wang [8] obtained some blow-up criteria. The local strong solution has been shown to be global for small initial energy in [11]. Very recently, Lin-Lai-Wang [19] established the existence of finite energy weak solutions with large initial data, provided the initial orientational director field d 0 lies in S 2 + . Recently, Li-Liang [12] established the local existence of classical solutions to the Cauchy problem of Navier-Stokes equation via weighted estimates instead of the general ones. It can be found that the systems (1) and (6) share the same continuity equation. However, (1) is more complicated than (6) because of the super critical nonlinearity |∇d| 2 d in the transported heat flow of harmonic map equation (1) 3 and the strong coupling nonlinear term ∆d · ∇d in the momentum equation (1) 2 involved here. The aim of this paper is to establish the local existence of strong solutions to the 2D Cauchy problem (1). Notice that the local well-posedess of strong solutions for the 3D case obtained by Huang-Wang-Wen [6] is not admitted for the 2D case. This is mainly due to that in the 2D case the L p -norm (p > 2) of the velocity u cannot be controlled in terms only of ρ 1 2 u and ∇u. Moreover, the coupling of u and d, and the presence of |∇d| 2 d bring additional difficulties. So, some new ideas and careful estimates are necessary to deal with the 2D case. In the present paper, we will use the framework of weighted approximation estimates introduced in [12] for Navier-Stokes equations to overcome these difficulties. It would be interesting to consider the global existence of the strong solutions to the problem (1), for which one should study the blow-up mechanism with the structure of possible singularities to the strong solutions of (1). Along this direction, very recently, Liu-Wang [22] and Wang [25] obtained for the 2D isentropic compressible nematic liquid crystal flows (1) that if T * ∈ (0, ∞) is the maximal time of existence for strong solutions (ρ, u, d), then Throughout the paper, we use simplified notations with p ∈ [1, ∞], k ≥ 0, and Ω = R 2 or Ω = B R := {x ∈ R 2 | |x| < R}.
Denotex := (e + |x| 2 ) 1 2 log 1+η0 (e + |x| 2 ), The main result of this paper is stated as the following theorem: with q > 2 and 1 < a < 2. Then there exist T 0 , N > 0 such that the problem (1)-(3) has a unique strong solution (ρ, u, d) on and Notice that in the framework of the paper, some weighted assumptions on ρ 0 and ∇d 0 , differently from the case of R 3 , are required in Theorem 1.1.
The rest of the paper is organized as follows. In Section 2, we recall some elementary facts and inequalities used in the sequel. Sections 3 deals with an approximation problem on B R to derive uniform estimates for the unique strong solution with respect to R. Finally, we give the proof of Theorem 1.1 in Section 4.

2.
Preliminaries. In this section, we recall some known results as preliminaries.
Consider an approximation problem for (1) By an argument similar to that in [6], it is easy to establish the local existence and uniqueness of classical solutions to (9). We give this result as a lemma below without proof. Then we will prove in the next two sections that the classical solutions (ρ R , u R , d R ) of (9)-(10) converge to the strong solution of the original Cauchy problem (1) by letting R → ∞.
Then there exist T R > 0 and a unique classical solution (ρ R , u R , d R ) to (9) on In addition, we cite a lemma involving estimates on weighted bounds for functions inD 1,2 (Ω).
3. Uniform estimates for approximation problem. In this section, we will derive some uniform estimates for the solution (ρ R , u R , d R ) to the approximation problem (9)-(10) ensured by Lemma 2.1, independent of the lower bound of the initial density and the size of the domain B R , which are crucial to prove the local existence of strong solutions to the Cauchy problem (1) with the initial vacuum permitted. For simplicity, denote (ρ R , u R , d R ) by (ρ, u, d).
Without loss of generality, assume that there exists N 0 > 0 such that with R > 4N 0 ≥ 4, due to the assumption (11). Now we deal with the required estimates to the approximation solution (ρ, u, d). For simplicity, denote (11) and (15). Then there exist T 0 , M > 0, both depending only on µ, γ, q, a, η 0 , N 0 , and E 0 , such that Proposition 1 will be proved via the next three lemmas.
where and throughout the paper, denote by C generic positive constants independent of R.
Proof. Notice that there is the common continuity equation between the nematic liquid crystal flows (1) and the Navier-Stokes equations (6), with different velocity field u involved. So, observing the framework of [12,Lemma 3.4] for proving an estimate similar to (58), it suffices to verify the following estimate: It follows from (35), (18), and (38) that Choosing p = q in (34), with notationu := u t + u · ∇u, we have where by (21) and (28). Combine (60)-(62) with (38) and (18) to get that and that On the other hand, we have by (18), (19) and the Gagliardo-Nirenberg inequality that The desired (59) follows from (60)-(65). Now, we can deal with the proof of Proposition 1.
Proof of Proposition 1. We have from (17) 4. Proof of Theorems 1.1. Now we make the approximation procedure to prove Theorem 1.1.