Well-posedness for the three-dimensional compressible liquid crystal flows

This paper is concerned with 
 the initial-boundary value problem for the three-dimensional compressible liquid crystal flows. The system consists of the Navier-Stokes equations describing the evolution of a compressible viscous fluid coupled with various 
kinematic transport equations for the heat flow of harmonic maps into $\mathbb{S}^2$. 
Assuming the initial density has vacuum and the initial data satisfies a natural compatibility condition, the existence and 
uniqueness is established for the local strong solution with 
large initial data and also for the global strong solution with 
 initial data being close to an equilibrium state. The existence result is proved via the local well-posedness and uniform estimates for a proper linearized system with convective terms.

1. Introduction. In this paper, we establish the well-posedness of a simplified hydrodynamic equation, proposed by Ericksen and Leslie, modeling the flow of nematic liquid crystals formulated in [7]- [9] and [15] in the 1960's. A simplified version of the Ericksen-Leslie model was introduced by Lin [17] and analyzed by Lin and Liu [18]- [20] who used a modified Galerkin approach, and by Shkoller [25] who relied on a contraction mapping argument coupled with appropriate energy estimates. When the Ossen-Frank energy configuration functional reduces to the Dirichlet energy functional, the hydrodynamic flow equation of liquid crystals in R 3 can be written as follows (see [17]): ρ t + ∇ · (ρu) = 0, (1.1a) (ρu) t + ∇ · (ρu ⊗ u) + ∇p(ρ) = µ∆u − λ∇ · ∇d ∇d − |∇d| 2 2 where u ∈ R 3 denotes the velocity, d ∈ S 2 (the unit sphere in R 3 ) is the unit-vector field that represents the macroscopic molecular orientations, p(ρ) is the pressure with p = p(·) ∈ C 1 [0, ∞), p(0) = 0; and they all depend on the spatial variable x = (x 1 , x 2 , x 3 ) ∈ R 3 and the time variable t > 0. σ = µ∇u−p I 3 (the 3×3 identity matrix) is the Cauchy stress tensor given by Stokes' law and µ∇u stands for the fluid viscosity part of the stress tensor. The term λ∇ · (∇d ∇d) in the stress tensor represents the anisotropic feature of the system. The parameters µ, λ, θ are positive constants standing for viscosity, the competition between kinetic energy and potential energy, and microscopic elastic relaxation time or the Deborah number for the molecular orientation field, respectively. The symbol ∇d ∇d denotes a matrix whose (i, j)-th entry is ∂ xi d · ∂ xj d for 1 ≤ i, j ≤ 3, and ∇d ∇d = (∇d) ∇d, where (∇d) denotes the transpose of the matrix ∇d.
It is well known liquid crystals are states of matter which are capable of flow and in which the molecular arrangement gives rise to a preferred direction. Roughly speaking, the system (1.1) is a coupling between the compressible Navier-Stokes equations and a transport heat flow of harmonic maps into S 2 . It is a macroscopic continuum description of the evolution for the liquid crystals of nematic type under the influence of both the flow field u, and the macroscopic description of the microscopic orientation configurations d of rod-like liquid crystals. As for the nonlinear constraint d ∈ S 2 , one of the methods used to relax it is to consider a form of penalization, that is, not |∇d| 2 in (1.1c), but the Ginzburg-Landau approximation 1 ε 2 (1 − |d| 2 ) for small ε. There were some similar fundamental results starting from the work in [19], where the density is constant, Lin-Liu proved local existence of the classical solutions the global existence of weak solutions in the two-dimensional and three-dimensional spaces. For the density-dependent case, Liu [22] proved the global existence of weak solutions and classical solutions to the system of incompressible Smectic-A liquid crystals under the general condition of the initial density ρ 0 satisfying 0 < α ≤ ρ 0 ≤ β. The global existence of weak solutions in dimension three was established by Liu-Zhang [24] if ρ 0 ∈ L 2 (Ω). Later Jiang-Tan [14] pointed out that the condition on initial density can be weaken to belong to L γ (Ω) for any γ ≥ 3 2 . As for the compressible case, Liu-Liu-Hao [23] established the global existence of strong solutions under the smallness conditions on the initial data in Sobolev spaces in dimension three.
Compared with the Ginzburg-Landau approximation problem, |∇d| 2 in (1.1c) brings us some new difficulties. Since the strong solutions of a harmonic map must be blowing up at finite time (see Chang-Ding-Ye [1] for the heat flow of harmonic maps), one cannot expect that there exists a global strong solution to system (1.1) with general initial data. In fact, the global existence of weak solutions to (1.1) with large initial data is an outstanding open problem for high dimensions. By so far, only results in one space dimension have been obtained, for instance, we refer to [5,6]. For the homogeneous case of system (1.1), both the regularity and existence of global weak solutions in dimension two were established by Lin-Lin-Wang [21]. More explicitly, they obtained both interior and boundary regularity theorem for such a flow under smallness conditions, and the existence of global weak solutions that are smooth away from at most finitely many singular times in any bounded smooth domain of R 2 . Meanwhile, Hong [10] also showed the global existence of weak solution to this system in two dimensional space. Wang [26] established a global well-posedness theory for the incompressible liquid crystals for rough initial data, provided that u 0 BM O −1 +[d 0 ] BM O ≤ ε 0 for some ε 0 > 0. Assuming that the initial density ρ 0 has a positive bound from below and under smallness conditions on the initial data, Wen-Ding [27] got the global existence and uniqueness of the strong solution to the incompressible density-dependent case in Sobolev spaces in two dimensions, and Li-Wang obtained the result in Sobolev-Besov spaces for three dimensional case in [16]. Concerning the compressible case, Hu-Wu considered the Cauchy problem for the three-dimensional compressible flow of nematic liquid crystals and obtained the existence and uniqueness of the global strong solution in critical Besov spaces provided that the initial data is close to an equilibrium state in a recent work [11]. Local existence of unique strong solutions were proved provided that the initial data ρ 0 , u 0 , d 0 are sufficiently regular and satisfy a natural compatibility condition in [12]. A criterion for possible breakdown of such a local strong solution at finite time was given in terms of blow up of the L ∞ -norms of ρ and ∇d. In [13], an alternative blow-up criteria was derived in terms of the L ∞ -norms of ∇u and ∇d in dimension three.
In this paper we consider the initial-boundary value problem of system (1.1) in a bounded smooth domain Ω ⊂ R 3 , with the following initial-boundary conditions: where ρ 0 ≥ 0, d 0 : Ω → S 2 is given with compatibility. The boundary condition on the velocity implies non-slip on the boundary. We are interested in the strong solutions to the initial-boundary problem (1.1)-(1.3). In order to obtain strong solutions, we make the following assumptions on the initial data: where 3 < q ≤ 6. And, (ρ 0 , u 0 , d 0 ) satisfies a natural compatibility condition with some (p 0 , g) ∈ H 1 × (L 2 ) 2 . We note here that (1.5) is a compensation to the lack of a positive lower bound of the initial density (see [2]). Our main result establishes the local well-posedness of the Ericksen-Leslie problem for any regular enough initial data: Theorem 1.1. Let the initial data satisfies the regularity condition (1.4) and also the compatibility condition (1.5). There exists a time T * such that the initialboundary problem (1.1)-(1.3) has a unique local strong solution (ρ, u, d) satisfying Furthermore, if we suppose that (ρ,ũ,d) is another solution with initial-boundary conditions then for any t ∈ (0, T * ], the quantities Moreover, we shall prove the existence of global strong solution for initial data that is close to an equilibrium state (0, 0, n) with a constant vector n ∈ S 2 . More precisely, Theorem 1.2. Let be a nonnegative constant and n be a constant unit vector in R 3 . Then there exists a suitable positive constant ξ 0 (small) such that if the initial data satisfies further From the viewpoint of partial differential equations, system (1.1) is a highly nonlinear system coupling hyperbolic equations and parabolic equations. It is very challenging to understand and analyze such a system, especially when the density function ρ may vanish or the fluid takes vacuum states (equation (1.1b) becomes a degenerate parabolic-elliptic couples system). Our approach is quite classical. Successive approximation method [3,4] is employed for two variables, which was used in [23]. It consists in deriving energy estimates without loss of derivatives in sufficiently high order Sobolev spaces for a linearized version of (1.1), and then solve the nonlinear problem through an iterative scheme. There are some difficulties in both steps that will be pointed out along the detailed proofs. Moreover, in order to overcome the difficulty that the initial density has vacuum brings, as usual, the technique is to approximate the nonnegative initial density by a positive initial data.
We organize the rest of this paper as follows. In Sect. 2, we introduce a special linear problem of the original system (1.1)-(1.3) and prove local existence of a strong solution to the linear problem with positive initial density. We also derive a series of uniform a priori estimates, which ensure the local strong solution exists when the initial density allows vacuum. In Sect. 3, after constructing a sequence of approximate solutions, a strong solution of (1.1)-(1.3) is obtained. The uniqueness and continuity on the initial data are also proved. Finally, in Sect. 4, our main result on the global existence of a strong solution of (1.1)-(1.3) will be established via iteration and the convergence of the iteration.

2.
A linear problem. In this section, let us consider the following auxiliary linear problem: with v ∈ R 3 and f ∈ R 3 being given vector functions and enjoying the regularities such that v ∈ C([0, T ]; for all T > 0.
Conformally to the initial-boundary conditions of the original problem, we suppose 2.1. Existence of approximate solutions. For each δ > 0, for example, δ ∈ (0, 1), let u δ 0 solve the elliptic boundary value problem: We will prove Theorem 2.1 through a series of lemmas. To begin with, assume there are three constants c 0 , c 1 and c 2 such that Throughout of the whole paper, sometimes, we make use of A B in place of A ≤ C 0 B, where C 0 stands for a "harmless" constant whose exact meaning depends on the context, and A ≈ B means that A B and B A.

XIAOLI LI AND BOLING GUO
Proof. The existence of the solution follows from the method of characteristics. If we define V (t, τ, x) to be the solution of (2.1a) can be rewritten as and the explicit formula for ρ is Applying the gradient operator ∇ to (2.1a), we have where ∇v · ∇ρ = (∇v) ∇ρ. Multiplying (2.11) by |∇ρ| q−2 ∇ρ and integrating over Ω, we obtain Bearing in mind that we find, by Höder's inequality and the imbedding for all t ≤ T . Using (2.10), (2.13) and choosing and, by using (2.12), it follows obviously that for all t ∈ [0, T 1 ], (1 + p(·) + p (·)). (2.14) Proof. Since (2.1b) is a linear parabolic-type system in d, the existence and uniqueness of d to the problem (2.1b),(1.2) and (1.3) can be obtained by the standard Faedo-Galerkin method, and also the regularity of d described in the lemma. Differentiating (2.1b) with respect to t, multiplying the result by d t and then integrating over Ω, we have (2.16) Now we estimate the right-hand side of (2.16) term by term.
where we have used the fact that d t | ∂Ω = 0.

XIAOLI LI AND BOLING GUO
Note here that the positive constant η will be determined later.
Therefore, combining with (2.16), we get Hence, by virtue of Gronwall's inequality, we obtain from (2.17) that From (2.1b), using the elliptic estimates, we get (2.20) Differentiating (2.1b) with respect to time and taking inner product with ∆d t , we have 1 2 (2.21) Here Bearing in mind that the elliptic estimate while substituting the above estimates into (2.21) and taking ε small enough (< θ 4 ),

XIAOLI LI AND BOLING GUO
All the above estimates can deduce that (2.15) holds. The proof of lemma 2.2 is completed.
Applying the Galerkin method again to (2.25) with the initial data u δ 0 and the nonslip boundary condition, we can deduce the existence and regularity of u described in the lemma.
Next, we prove the estimate (2.24). Differentiating (2.1c) with respect to t, multiplying the result by u t and then integrating over Ω, one has (2.26) Here then, combining with the above estimates and (2.26), we have Multiplying (2.1c) by u t , integrating the result over Ω and using the Cauchy-Schwarz inequality, we obtain and hence lim sup Here we have used the compatibility condition (1.5). By virtue of the Gronwall inequality, from (2.27), we have for all t ∈ [0, T 4 ], where T 4 = min{T 3 , c −2 0 c −4 2 }. Next, we need to estimate ∇u 2 H 1 . The classical elliptic estimates applied to (2.1c), using equation (2.1b) and the assumption (2.4)-(2.7), it gives rise to Thus, we deduce The term ∇ 2 u L q can be estimated by the same way as ∇u H 1 above. In fact, Integrating (2.31) over time and using the estimate (2.30), one has for all t ∈ [0, According to the lower semi-continuity of various norms, the estimates (2.8), (2.15) and (2.24) hold also for (ρ, u, d). Therefore, (ρ, u, d) satisfies the system (2.1) almost everywhere on [0, T 5 ] × Ω, which means, (ρ, u, d) is a strong solution to (2.1) with the initial-boundary condition (1.2) (1.3).
We claim the solution (ρ, u, d) is unique. From Lemma 2.1, ρ is the unique solution to the linear equation (2.1a). Once ρ is computed uniquely from (2.1a), and then (2.1b) and (2.1c) are linear parabolic equations in terms of d and u, respectively. Uniqueness is obvious.
Next, we prove the time continuity of the solution (ρ, u, d). Since the solution to the linear equation (2.1a) is unique, then the solution from Lemma 2.1 is the same as from the approximation above, i.e., ρ ∈ C([0, T 5 ]; W 1,q ).
3. Local existence in Theorem 1.1. In this section, we prove the local existence and uniqueness of strong solution in Theorem 1.1. The proof will be divided into several steps, including constructing the approximate solutions by iteration, obtaining the uniform estimate, showing the convergence, consistency and uniqueness.

3.2.
Convergence of the approximate sequence. We claim that {(ρ k , u k , d k )} ∞ k=1 is a Cauchy sequence and thus converges. In fact, For (3.5) 2 , on the one hand, we multiply (3.5) 2 byd k+1 and integrate over Ω to obtain d dt where we have used
We remark here that the time of existence T * * depends (continuously) on the norms of the data, on the bound for the density, on the domain and on the regularity parameters.
Now if we choose small constants ν, ξ 0 such that for all ξ ∈ (0, ξ 0 ], Cν exp C(t + ξ then it is easy to deduce from (4.4) that The proof of Theorem 1.2 is completed.