Bifurcation of time-periodic solutions for the incompressible flow of nematic liquid crystals in three dimension

where U ∈ R3 denotes the velocity, d ∈ R3 the director eld for the averaged macroscopic molecular orientations, P ∈ R the pressure arising from the incompressibility; and they all depend on the spatial variable x = (x1, x2, x3) ∈ R3 and the time variable t > 0. The positive constants μ, λ, γ stand for viscosity, the competition between kinetic energy and potential energy, and microscopic elastic relaxation time or the Deborah number for the molecular orientation eld, respectively; fα and hα are external time independent forces. The symbol∇d ∇d denotes a matrix whose ijth entry is < ∂xid, ∂xjd >, and it is easy to see that ∇d ∇d = (∇d)T∇d,


Introduction and Main Results
We consider the 3D incompressible ow of liquid crystals under external time-independent force where U ∈ R denotes the velocity, d ∈ R the director eld for the averaged macroscopic molecular orientations, P ∈ R the pressure arising from the incompressibility; and they all depend on the spatial variable x = (x , x , x ) ∈ R and the time variable t > . The positive constants µ, λ, γ stand for viscosity, the competition between kinetic energy and potential energy, and microscopic elastic relaxation time or the Deborah number for the molecular orientation eld, respectively; fα and hα are external time independent forces. The symbol ∇d ∇d denotes a matrix whose ijth entry is < ∂x i d, ∂x j d >, and it is easy to see that ∇d ∇d = (∇d) T ∇d, where (∇d) T denotes the transpose of the × matrix ∇d. In (2), f (d) is the penalty function which will be assumed to be One of the most common liquid crystal phases is the nematic, where the molecules have no positional order, but they have long-range orientational order. For more details of physics, we refer the readers to the two books of de Gennes-Prost [7] and Chandrasekhar [2]. Ericksen and Leslie cf. [6,14] established the hydrodynamic theory of liquid crystals in 1960s. The Ericksen-Lislie theory describes the liquid crystal ow, including the velocity vector u and direction vector d of the uid. Since the general Ericksen-Leslie system is very complicated, we only consider a simpli ed model (1)-(3) of the Ericksen-Leslie system, but still retains most of the essential features. One can see [16][17][18]20] for more discussions on the relations of the two models. Both the Ericksen-Leslie system and the simpli ed one (1)-(3) describe the time evolution of liquid crystal materials under the in uence of both the velocity eld u and the director eld d. Hence, a natural question of the existence of time-periodic solution arises when (1)- (3) under the e ect of the external forces.
Since the Ericksen-Leslie system (1)-(3) with |u| = is complicated, Lin and Liu [18,19] proposed to investigate an approximation model of the Ericksen-Leslie system by Ginzburg-Landau functionals. In order to relax the constraint |u| = for the functional |∇u| dx, Lin and Liu [18,19] considered Ginzburg-Landau functionals for any function d ∈ H (Ω; R ) with a parameter ϵ > . They obtained the global existence of weak solutions with large initial data and the global existence of classical solutions was also obtained if the coe cient µ is large enough in three dimensional spaces. Hu and Wang [12] prove the existence and uniqueness of the global strong solution with small initial data are established. Meanwhile, they obtained that when the strong solution exists, all the global weak solutions constructed in [18] must be equal to the unique strong solution.
Hong [11] proved that the global existence of regular solutions to the Ericksen-Leslie system in R with initial data except for at a nite number of singular times. Li and Yan [15] showed this system admits a stable smooth steady solutions by assumption of existence of it.
Since the work of Sattinger [29], Iudovich [24] and Iooss [21] in 1971, the bifurcation of stationary solutions into time periodic solutions (i.e. Hopf-bifurcation) of incompressible Navier-Stokes equation has attracted much attention, see [3,9,13,22,23], etc. When the linearized operator possesses a continuous spectrum up to the imaginary axis and that a pair of imaginary eigenvalues crosses the imaginary axis, Melcher, A, et al. [26] proved Hopf-bifurcation for the vorticity formulation of the incompressible Navier-Stokes equations in R . Their work is mainly motivated by the work of Brand, T, et al. [1] who studied the Hopf-bifurcation problem and its exchange of stability for a coupled reaction di usion model in R a . We mention that Crandall and Rabinowitz [5] gave an abstract in nite-dimensional version of Hopf bifurcation theorem which has found many application. We refer the readers to [4,27,30,[32][33][34][35] corresponding Hopf-bifurcation result (bifurcating from viscous shock waves) has been established in.
In this paper, our aim is to establish the corresponding Hopf-bifurcation result for the three-dimensional incompressible ow of liquid crystals. But we can not directly use the method of dealing with Navier-Stokes equation to three-dimensional incompressible ow of liquid crystals because the presence of the velocity eld and its interaction with the director eld in the liquid crystals ow of large oscillation. A weighted Young theorem (see Lemma 6) is derived to deal with strong coupled between the velocity eld and the director eld.
We assume that fα and hα depend smoothly on some parameter α, which can be chosen suitably so that (uα(x) + uc , dα(x) + dc , pα(x)) (the steady solution has certain smoothness property) is the solution of the three-dimensional steady incompressible ow of liquid crystals To seek the periodic solution, we linearize system (1)-(2) about the steady state (uα , dα , pα) by writing Then, the deviation (u, z, p) from the stationary (uα , dα , pα) satis es Here, for general matrices u = (u ij ) i,j= , , , and take the gradient of (10) and notice (4)-(5) to rewrite (9)-(10) as with incompressible condition where we used, for all i, j, k = , , , In fact, by the incompressible condition (14), it follows that Thus using (14) and (15) to (12)-(13), we obtain The vorticity associated with velocity eld u of the uid is de ned by ω = ∇ × u. Then, using we can rewrite system (16) as Note that the space of divergence free vector elds is invariant under the evolution (18). We can assume that Moreover, we can reconstruct the velocity u from the vorticity ω by solving the equation The velocity eld u is de ned in terms of the vorticity via the Biot-Savart law Denote φ = (ω, v) T . Then, we can write system (17)-(18) as the evolution equation form where For convenience, we denote the Fourier coe cient of operators N and G by N and G, respective. To overcome the essential spectrum of operator −( N+ G) up to the imaginary axis, we need the following assumption: (H1) For any α ∈ [αc − α , αc + α ], is not an eigenvalue of N + G.
Under the generic assumption the cubic coe cient terms a , a ≠ in (64)-(65), Hopf-bifurcation result about 3D incompressible ow of liquid crystals is stated: , and positive frequencies ξ and ξ . Moreover, Above result also holds in a three dimensional torus T and a bounded domain.
This paper is organized as follows. In section 2, we introduce some notations and preliminaries. In section 3, The main proof of Theorem 1 is carried out by using Lyapunov-Schmidt method.

Preliminary and Some notations
We start this section by introducing some notations. Consider the following standard Sobolev space, spatially weighted Lebesgue space where weighted function ρ(x) = + |x| . The Fourier transform is a continuous mapping from L p s into W q κ . Especially, when p = , the Fourier transform is an isomorphism between H p and L p with u L p = ρ p u L .
To investigate periodic solutions of system (1)-(2), we also introduce the space with norms for φ = (u, v) T ∈ L p s or X, respectively. In this paper, we consider the following form of time-periodic solution where ξ , ξ ∈ R + denote the corresponding frequencies.
Thus we need to nd π time periodic solutions of where By the classical result in [10], we know that the essential spectrum of the operator N + G is relatively compact perturbation of N which has the essential spectrum Moreover, the spectra of N + G and N only di er by isolated eigenvalues of nite multiplicity. Above spectrum properties are critical to prove our main result. For convenience, we can rewrite (21) as where g and g de ned in (22)-(23), We make the ansatz (24)- (25), we obtain where Note that we are interested in real valued solution only. We will always suppose that (ωn , vn) = (ω−n , v−n) for n ∈ Z. These series are uniformly convergent on R × [ , π] in the spaces which we have chosen. More precisely, we have the following results: Then J is bounded.
The counterpart to multiplication uv in physical space is given by the convolution (

Lemma 2.
For u = (un) n∈Z , v = (vn) n∈Z ∈ X, the convolution u * v ∈ X is de ned by Then there exists C > such that The proof of following result can be found in [8] for bounded domain and [28] for R n .

Lemma 5.
For every < ϑ < and p > there exists a constant C > such that The following result shows a weighted Young theorem.

Lemma 6.
There exists a positive constant C such that Proof. It is easy to check that where we take the weighted function as Then, There exist positive constants s , s , s such that s + s = m + s, with s , s , s < m. using Young inequality and (28), we derive This completes the proof.

Proof of Theorem 1
In this section, we will give the detail of proof of Theorem 1. By (H2) and (H3), we know that the operator M i has two eigenvalues λ ± (β) and all other eigenvalues of M i are strictly bounded away from the imaginary axis in the left half plane. Thus we construct a M i -invariant projections P ± ,c by where ψ ± denotes the associated normalized eigenfunctions, ψ ± ,* denotes the associated normalized eigenfunctions of the adjoint operator M * i . The bounded "stable" part of the projection is P ± ,s = I − P ± ,c , we also know that P±,c M i = M i P±,c and P±,s M i = M i P±,s. Thus we can split ω ± and v ± as with ω ± ,c = P ± ,c ω , ω ± ,s = P ± ,s ω , v ± ,c = P ± ,c v , v ± ,s = P ± ,s v .
(inξ − M )vn = g n (u, v), n = ± , ± , . . . , M ω = g (ω, u, v), n = , (±iξ − M )ω ± ,s = P ± ,s g ± (ω, u, v), The organization of proof of Theorem 1 is that we rst solve the equations (33)- (34). Then using the xed point theorem to solve equations (31)- (32) and (35) Now we rst solve the equation (40). The linear operator N has essential spectrum up to the imaginary axis, it can be be inverted in the following sense. We denote Then and By the same process, we can obtain This completes the proof.
This Lemma tells us that N(iy i , iy i ) T · is bounded compact operator in from L m × L m to itself. Furthermore, the spectra of N + G and N only di er by isolated eigenvalues of nite multiplicity (see the book of Henry [10] p.136).
The following Lemma gives the solvable of the equation (40).

Lemma 8. Assume that (H1)-(H3) holds. Then the equation (40) has a unique solution
Moreover, where I × is the unit matrix. Thus, by Lemma 7, we obtain This completes the proof.

Lemma 9. There exist a constant C > such that
Proof. The related equation of the velocity u and the vorticity ω is This leads in Fourier space to We can getNω Using Hölder's inequality, for p + p = , p , p > , s + s = m and s , s > , we have where we use the weighted function ρ(y) = |y|( + |y|) , the boundedness of χ |y|≥ The second estimate in (44) is followed by This completes the proof.
From the form of the nonlinear terms g and g , it is critical to estimate the term as uv and u . For convenience, we derive some estimates about the nonlinear term N (φ) = φ and N (φ, ψ) = φψ. This proof is similar with Lemma 4 in [1], so we omit it.

Lemma 10.
De ne N : X −→ X by N (φ)n = N n (Jφ) and N : X × X −→ X by N (φ)n = N n (Jφ, Jψ) for φ, ψ ∈ X. Then there exists C > such that for φ, ψ ∈ X with φ X ≤ and ψ X ≤ . Moreover, there exists C > such that Then we have the following result.
Lemma 11. Assume that ξ i close enough to ξ for i = , . Then there exists a constant C > such that Proof. We observe that the solution φ of the equation (inΞ + N)φ = f is given by For δ = µ ξ ξ + c and δ = γ ξ ξ + c , we have It follows for f ∈ H m that φ ∈ L m+ , thus φ ∈ H m+ . Letf ∈ L m+ ⊂ L m , ω = ρ(y, ϵ)ω and v = ρ(y, ϵ)v with ρ(x, ϵ) = + ϵ|x| . Note that φ is a solution of the equation (inΞ + N)φ = f . By a direct computation, we have Here we use the fact that N is elliptic of order of . Hence it derives from the form of ρ(y, ϵ) = + ϵ|y| that Using a Neumann series, it derives from the boundness of the operator L : L m+ −→ L m that is invertible with a bounded inverse, for su cient small ϵ > . This implies thatφ ∈ L m+ , i.e., φ ∈ H m+ . Moreover, we have Above result shows that (inΞ+N) − : H m −→ H m+ is bounded. But we only need this operator to be bounded X −→ X. This implies that the spectrum of N in X well separated from inΞ for n ≠ and ϵ > su cient small. In a similar manner to prove the rst inequality, the rest two inequalities can be obtained, so we omit it. This completes the proof.
By the same proof in Lemma 11, we obtain the following result.

Lemma 12.
Assume that ξ i close enough to ξ for i = , . Then there exists a constant C > such that for n ≠ and j = , .
Thus we deduce that for su cient small ball Br( ) ⊂ B ( ), This completes the proof.
Note that