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

Hengyan Li; Xin Zhao; Weiping Yan

doi:10.1515/anona-2020-0052

Article Open Access

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

Hengyan Li , Xin Zhao and Weiping Yan

Published/Copyright: December 31, 2019

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From the journal Advances in Nonlinear Analysis Volume 9 Issue 1

Abstract

This paper is devoted to the study of the dynamical behavior for the 3D incompressible flow of liquid crystals. We prove that this system under smooth external forces possesses time dependent periodic solutions, bifurcating from a steady solution.

Keywords: Nematic liquid crystals; periodic solution; bifurcation

MSC 2010: 37N10; 35B10; 70K50

1 Introduction and Main Results

We consider the 3D incompressible flow of liquid crystals under external time-independent force

∂U∂t+U⋅∇U−μ△U+∇P=−λ∇⋅(∇d⊙∇d)+fα, (1)

∂d∂t+U⋅∇d=y(△d−f(d))+hα, (2)

∇⋅U=0, (3)

where U ∈ R³ denotes the velocity, d ∈ R³ the director field 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, y stand 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; f_α and h_α 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,

∇⋅(∇d⊙∇d)=∇(|∇d|22)+(∇d)T△d, (4)

where (∇ d)^T denotes the transpose of the 3 × 3 matrix ∇ d. In (2), f(d) is the penalty function which will be assumed to be

f(d)=|∇d|2d. (5)

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 flow, including the velocity vector u and direction vector d of the fluid. Since the general Ericksen-Leslie system is very complicated, we only consider a simplified 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 simplified one (1)-(3) describe the time evolution of liquid crystal materials under the influence of both the velocity field u and the director field d. Hence, a natural question of the existence of time-periodic solution arises when (1)-(3) under the effect of the external forces.

Since the Ericksen-Leslie system (1)-(3) with |u| = 1 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| = 1 for the functional ∫|∇ u|²dx, Lin and Liu [18, 19] considered Ginzburg-Landau functionals

∫Ω[|∇d|2+12ϵ2(1−|d|2)2]dx,

for any function d ∈ H¹(Ω;R³) with a parameter ϵ > 0. They obtained the global existence of weak solutions with large initial data and the global existence of classical solutions was also obtained if the coefficient 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 R² with initial data except for at a finite 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 R³. 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 diffusion model in R^a. We mention that Crandall and Rabinowitz [5] gave an abstract infinite-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 flow of liquid crystals. But we can not directly use the method of dealing with Navier-Stokes equation to three-dimensional incompressible flow of liquid crystals because the presence of the velocity field and its interaction with the director field in the liquid crystals flow of large oscillation. A weighted Young theorem (see Lemma 6) is derived to deal with strong coupled between the velocity field and the director field.

We assume that f_α and h_α depend smoothly on some parameter α, which can be chosen suitably so that (u_α(x) + u_c, d_α(x) + d_c, p_α(x)) (the steady solution has certain smoothness property) is the solution of the three-dimensional steady incompressible flow of liquid crystals

U⋅∇U−μ△U+∇P=−λ∇⋅(∇d⊙∇d)+fα, (6)

U⋅∇d=y(△d−f(d))+hα, (7)

∇⋅U=0, (8)

with u_c = (c₁, 0, 0)^T, d_c = (c₁, 0, 0)^T and

lim|x|→∞uα(x)=0, lim|x|→∞dα(x)=0,

where 0 = (0, 0, 0)^T.

To seek the periodic solution, we linearize system (1)-(2) about the steady state (u_α, d_α, p_α) by writing

U(x,t) = u(x,t)+uα(x),d(x,t) =z(x,t)+dα(x),P =p+pα.

Then, the deviation (u, z, p) from the stationary (u_α, d_α, p_α) satisfies

∂u∂t−μ△u+c1∂x1u+uα⋅∇u+u⋅∇uα + u⋅∇u+∇p=−λ∇(|∇z|22)−λ∇(|∇z||∇dα|) −λ(∇z)T△(z+dα)−λ(∇dα)T△z, (9)

∂z∂t−y△z+c1y∂x1z + uα⋅∇z+u⋅∇dα+u⋅∇z=−y|∇z|2z−y|∇z|2dα−y|∇dα|2z−2y|∇z||∇dα|(z+dα), (10)

Here, for general matrices u = (u_ij)_i,j=1,2,3,

∇⋅u=(∑j=13∂x1u1j,∑j=13∂x1u2j,∑j=13∂x1u3j)T.

We introduce a 3 × 3 matrix

v=∇z, vα=∇zα, (11)

and take the gradient of (10) and notice (4)-(5) to rewrite (9)-(10) as

∂u∂t−μ△u+c1∂x1u+uα⋅∇u+u⋅∇uα+u⋅∇u+∇p=−λ∇(|v|22)−λ∇(|v||∇dα|)+λvT∇(v+vα)+λvαT∇v, (12)

∂v∂t−y△v+c1y∂x1v+uα⋅∇v+v∇uα+u⋅∇vα+vα∇u+u⋅∇v+v∇u=−y∇(|v|2z+|v|2dα)−y∇(|∇dα|2z+2|v||∇dα|(z+dα)), (13)

with incompressible condition

∇⋅u=0, (14)

where we used, for all i, j, k = 1, 2, 3,

∂∂xk(uj∂di∂xj)=∂uj∂xk∂di∂xj+uj∂∂xj(∂di∂xk)=(v∇u+u⋅∇v)ik.

In fact, by the incompressible condition (14), it follows that

∇⋅(uvT)=u⋅∇u+u∇⋅u=u⋅∇u. (15)

Thus using (14) and (15) to (12)-(13), we obtain

∂u∂t−μ△u+c1∂x1u+∇⋅(uαuT)+∇⋅(uuαT)+∇⋅(uuT)+∇p=−λ∇(|v|22)−λ∇(|v||∇dα|)+λvT∇(v+vα)+λvαT∇v, (16)

∂v∂t−y△v+c1y∂x1v+uα⋅∇v+v∇uα+u⋅∇vα+vα∇u+u⋅∇v+v∇u=−y∇(|v|2z+|v|2dα)−y∇(|∇dα|2z+2|v||∇dα|(z+dα)). (17)

The vorticity associated with velocity field u of the fluid is defined by ω = ∇ × u. Then, using

∇×∇⋅(uuT)=∇⋅(ωuT−uωT),

we can rewrite system (16) as

∂ω∂t−μ△ω+c1∂x1ω+∇⋅(ωαuT−uαωT)+∇⋅(ωuαT−uωαT)+∇⋅(ωuT−uωT)=−λ∇(∇×|v|22)−λ∇(∇×(|v||∇dα|))+λ∇×(vT∇(v+vα))+λ∇×(vαT∇v). (18)

Note that the space of divergence free vector fields is invariant under the evolution (18). We can assume that

∇⋅ω=0.

Moreover, we can reconstruct the velocity u from the vorticity ω by solving the equation

∇×u=ω, ∇⋅ω=0.

The velocity field u is defined in terms of the vorticity via the Biot-Savart law

u(x)=−14π∫R3(x−y)⊥×ω(y)|x−y|3dy, x∈R3. (19)

Denote φ = (ω, v)^T. Then, we can write system (17)-(18) as the evolution equation form

dφdt+Nφ+G(φ)=F(φ), (20)

where

N=−μ△+c1∂x100−y△+c1∂x1,

and

G(φ)=g1g2, F(φ)=g3g4

with

g1=∇⋅(ωαuT−uαωT)+∇⋅(ωuαT−uωαT)+λ∇(∇×(|v||∇dα|))−λ∇×(vT∇vα)−λ∇×(vαT∇v),g2=uα⋅∇v+v∇uα+u⋅∇vα+vα∇u+y∇(|∇dα|2z+2|v||∇dα|dα),g3=−∇⋅(ωuT−uωT)−λ∇(∇×|v|22)+λ∇×(vT∇v),g4=−u⋅∇v−v∇u−y∇(|v|2z+|v|2dα)−2y∇(|v||∇dα|z).

For convenience, we denote the Fourier coefficient of operators 𝓝 and 𝓖 by 𝓝̂ and 𝓖̂, respective. To overcome the essential spectrum of operator −(𝓝̂ + Ĝ) up to the imaginary axis, we need the following assumption:

For any α ∈ [α_c − α₀, α_c + α₀], 0 is not an eigenvalue of 𝓝̂+Ĝ.
For α = α_c, the operator −(𝓝̂ + Ĝ) has two pair eigenvalues (λ0+,μ0+) and (λ0−,μ0−) satisfying

λ0±(αc)=μ0±(αc)=±iξ0≠0, for ξ0>0,ddαRe(λ0±(α))∣α=αc,ddαRe(μ0±(α))∣α=αc>0.
The rest eigenvalue of −(𝓝̂ + Ĝ) is strictly bounded away from the imaginary axis in the left half plane for all α ∈ [α_c − α₀, α_c + α₀].

Under the generic assumption the cubic coefficient terms a₁, a₂ ≠ 0 in (64)-(65), Hopf-bifurcation result about 3D incompressible flow of liquid crystals is stated:

Theorem 1

Assume that (H1)-(H3) hold. Then system (1)-(3) admits a one dimensional family of small time-periodic solutions, i.e.

U(x,t)=U(x,t+2π/ξ1), d(x,t)=d(x,t+2π/ξ2)

with α = α_c + ϵ, ϵ ∈ (0, α₀), and positive frequencies ξ₁ and ξ₂. Moreover,

ξ1=ξ0+O(ϵ), ξ2=ξ0+O(ϵ),

and

∥U(x,t)∥Cb0(R3×[0,2π/ξ1])=O(ϵ), ∥d(x,t)∥Cb1(R3×[0,2π/ξ2])=O(ϵ).

Above result also holds in a three dimensional torus T³ 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.

2 Preliminary and Some notations

We start this section by introducing some notations. Consider the following standard Sobolev space, spatially weighted Lebesgue space

Wκq:={u:∥u∥κq:=∑|α|≤κ∥Dαu∥Lqq<∞},Lsp:={u:∥u∥sp:=∫R3ρs(x)up(x)dx<∞},

where weighted function ρ(x) = 1+|x|2 . The Fourier transform is a continuous mapping from Lsp into Wκq . Especially, when p = 2, the Fourier transform is an isomorphism between H^p and Lp2 with ∥u∥Lp2=∥ρpu∥L2.

To investigate periodic solutions of system (1)-(2), we also introduce the space

X:={u=(un)n∈Z:∥u∥X<∞}

and weighted space

Lsp=Lsp×Lsp, Hm=Hm×Hm, X=X×X,

with norms

∥u∥X=∑n∈Z∥un∥Hp, ∥φ∥X:=∥u∥X+∥v∥X,∥φ^∥Lsp:=∥u^∥Lsp+∥v^∥Lsp, ∥φ^∥Hm:=∥u∥Hm+∥v∥Hm.

for φ = (u, v)^T ∈ Lsp or 𝓧, respectively.

In this paper, we consider the following form of time-periodic solution

ω=ω(x,t/ξ1), v=v(x,t/ξ2),

where ξ₁, ξ₂ ∈ R⁺ denote the corresponding frequencies.

Thus we need to find 2π time periodic solutions of

Ξdφdt+Nφ+G(φ)=F(φ), (21)

where

Ξ=ξ100ξ2, N=−μ△+c1∂x100−y△+c1∂x1,

and

G(φ)=g1g2, F(φ)=g3g4

with

g1=∇⋅(ωαuT−uαωT)+∇⋅(ωuαT−uωαT)+λ∇(∇×(|v||∇dα|))−λ∇×(vT∇vα)−λ∇×(vαT∇v),

g2=uα⋅∇v+v∇uα+u⋅∇vα+vα∇u+y∇(|∇dα|2z+2|v||∇dα|dα),

g3=−∇⋅(ωuT−uωT)−λ∇(∇×|v|22)+λ∇×(vT∇v), (22)

g4=−u⋅∇v−v∇u−y∇(|v|2z+|v|2dα)−2y∇(|v||∇dα|z). (23)

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

essspec(N^)={λ∈C2:λ=(−μ|y|2+icy1,−y|y|2+icy1), y∈R3}.

Moreover, the spectra of 𝓝 + G and 𝓝 only differ by isolated eigenvalues of finite multiplicity. Above spectrum properties are critical to prove our main result.

For convenience, we can rewrite (21) as

ξ1ωt=M1ω+g3(ω,u,v), (24)

ξ2vt=M2v+g4(ω,u,v), (25)

where g³ and g⁴ defined in (22)-(23),

M1=M1¯+g1=μ△+c1∂x1+g1,M2=M2¯+g2=y△+c1∂x1+g2.

We make the ansatz

ω(x,t)=∑n∈Zωn(x)eint, v(x,t)=∑n∈Zvn(x)eint

to (24)-(25), we obtain

(inξ1−M1)ωn=gn3(ω,u,v), (26)

(inξ2−M2)vn=gn4(ω,u,v), (27)

where

g3(ω,u,v)(x,t)=∑n∈Zgn3(ω,u,v)eint,g4(ω,u,v)(x,t)=∑n∈Zgn4(ω,u,v)eint.

Note that we are interested in real valued solution only. We will always suppose that (ω_n, v_n) = (ω_−n, v_−n) for n ∈ Z. These series are uniformly convergent on R³ × [0, 2π] in the spaces which we have chosen. More precisely, we have the following results:

Lemma 1

A linear operator J : 𝓧 ⟶ Cb0 (R³ × [0, π], C²) is defined by

(Ju)(x,t)=u~(x,t):=∑n∈Zun(x)eint, u=(un)n∈Z∈X.

Then J is bounded.

The counterpart to multiplication uv in physical space is given by the convolution (∑k∈Zun−kvk)n∈Z, since

uv=∑l∈Zul(x)eilt∑j∈Zvj(x)eijt=∑n∈Z∑k∈Zun−k(x)vk(x)eint.

Lemma 2

For u = (u_n)_n∈Z, v = (v_n)_n∈Z ∈ X, the convolution u ∗ v ∈ X is defined by

(u∗v)n=∑k∈Zun−kvk, n∈Z.

Then there exists C > 0 such that

∥u∗v∥X≤C∥u∥X∥v∥X.

Lemma 3

Let a linear operator M_i : X ⟶ X be defined component-wise as (M_iu)_n = M_in u_n for u = (u_n)_n∈Z. Then

∥Miu∥X=(∥Mi0∥Hm⟶Hm+supn∈Z∖{0}∥Mi∥Hm⟶Hm)∥u∥X, for i=1,2.

The proof of above three Lemmas are standard, so we omit it.

For any bounded analytic semigroup Aσϑ , the following result holds.

Lemma 4

[28] For every 0 < ϑ < 1 and p > 1 there exists a constant M > 0 such that for all t > 0 one has

∥AσϑeAσt∥Lp⟶Lp≤Mtϑ.

The proof of following result can be found in [8] for bounded domain and [28] for Rⁿ.

Lemma 5

For every 12 < ϑ < 1 and p > 1 there exists a constant C > 0 such that

∥Aσ−ϑf∥Lp≤C∥f∥W−2ϑ,p.

The following result shows a weighted Young theorem.

Lemma 6

There exists a positive constant C such that

∥ω∗u∥Lm2≤C∥ω∥Lm2∥u∥Lm2.

Proof

It is easy to check that

ρ(x)≤ρ(x−y)ρ(y),∀x,y∈R3, (28)

where we take the weighted function as

ρ(x)=(1+|x|2)12.

Then, There exist positive constants s₁, s₂, s such that s₁ + s₂ = m + s, with s₁, s₂, s < m. using Young inequality and (28), we derive

∥ω∗u∥Lm22=∫R3ρ2m(ω∗u)2(x)dx=∫R3∫R3ω(x−y)u(y)ρm(x)dy2dx=∫R3ρ−2s(x)∫R3ρs1(x−y)ω(x−y)ρs2(y)u(y)dy2dx≤∫R3ρ−2s(x)∫R3ρ2s1(z)ω2(z)dz∫R3ρ2s2(y)u2(y)dydx≤C∥ω∥Ls122∥u∥Ls222≤C∥ω∥Lm22∥u∥Lm22.

This completes the proof. □

3 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 λ0± (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_±1,c by

P1,cω=(ψ+,∗,ω)L2ψ+, P−1,cω=(ψ−,∗,ω)L2ψ−, (29)

P1,cv=(ψ+,∗,v)L2ψ+, P−1,cv=(ψ−,∗,v)L2ψ−, (30)

where ψ^± denotes the associated normalized eigenfunctions, ψ^±1,* denotes the associated normalized eigenfunctions of the adjoint operator Mi∗ . The bounded “stable” part of the projection is P_±1,s = I − P_±1,c, we also know that P_±,c M_i = M_i P_±,c and P_±,s M_i = M_iP_±,s. Thus we can split ω_±1 and v_±1 as

ω1=ω1,c+ω1,s, ω−1=ω−1,c+ω−1,s,v1=v1,c+v1,s, v−1=v−1,c+v−1,s

with

ω±1,c=P±1,cω1, ω±1,s=P±1,sω1,v±1,c=P±1,cv1, v±1,s=P±1,sv1.

Using above decompositions to (26)-(27), we have

(inξ1−M1)ωn=gn3(ω,u,v), n=±2,±3,…, (31)

(inξ2−M2)vn=gn4(u,v), n=±2,±3,…, (32)

M1ω0=g03(ω,u,v), n=0, (33)

M2v0=g04(u,v), n=0, (34)

(±iξ1−M1)ω±1,s=P±1,sg±13(ω,u,v), (35)

(±iξ2−M2)v±1,s=P±1,sg±14(u,v), (36)

(±iξ1−M1)ω±1,c=P±1,cg±13(ω,u,v), (37)

(±iξ2−M2)v±1,c=P±1,cg±14(u,v). (38)

The organization of proof of Theorem 1 is that we first solve the equations (33)-(34). Then using the fixed point theorem to solve equations (31)-(32) and (35)-(36) which is nontrivial due to the nonlinear term gn3 (ω, u, v) and gn4 (u, v). At last, we employ the implicit function theorem to solve equation (37)-(38). The process of solving equation (37)-(38) is inspired by the classical Hopf-Bifurcation result [25].

Rewrite (31)-(38) as

(inΞ+N+G)φn=Fn(φ,u), n=±2,±3,…, (39)

(N+G)φ0=F0(φ,u), n=0, (40)

(±iΞ+N+G)φ±1,s=P±1,sF±1(φ,u), (41)

(±iΞ+N+G)φ±1,c=P±1,cF±1(φ,u). (42)

Now we first 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.

Lemma 7

For j = 1, 2 and f = (f¹, f²)^T ∈ (𝓗^m−1 ∩ 𝓛¹), the equation

Nφ=∂jf

admits a unique solution φ = 𝓝⁻¹ ∂_jf ∈ 𝓗^m. Moreover,

∥φ∥Hm≤C∥f∥Hm−1∩L1.

Proof

Define a smooth cut-off function χ taking its value in [0, 1] as

χ(y):=1, |y|≤1,0, |y|≥2.

We denote

(f^11,f^12)=(f^1χ,f^2χ) and (f^21,f^22)=(f^1(1−χ),f^2(1−χ))

with f^=(f1,f2)=(f^11+f^21,f^12+f^22). Then

ω^1(y)=iyjf^11inξ1−μ|y|2−ic1y1 and ω^2(y)=iyjf^21inξ1−μ|y|2−ic1y1,v^1(y)=iyjf^12inξ2−y|y|2−ic1y1 and v^2(y)=iyjf^22inξ2−y|y|2−ic1y1.

Note that (ω, v) = (ω₁ + ω₂, v₁ + v₂). Moreover, it holds

∥ω1∥Hm2=∥ω^1∥Lm22=∫R2|yj|2|f^χ(y)|2|inξ1−μ|y|2−ic1y1|2ρ2m(y)dy≤C∥f∥L12∫|y|≤2|yj|2r4+c2y12dy≤C∥f∥L12,

and

∥ω2∥Hm2=∥ω^2∥Lm22=∫R2|yj|2|f^(1−χ(y))|2|inξ1−μ|y|2−ic1y1|2ρ2m(y)dy≤C∫R2|f^(y)|2ρ(2m−1)(y)dy≤C∥f∥Hm−12.

By the same process, we can obtain

∥v1∥Hm2=∥v^1∥Lm22≤C∥f∥L12,∥v2∥Hm2=∥v^2∥Lm22≤C∥f∥Hm−12.

This completes the proof. □

This Lemma tells us that 𝓝̂(iy_i, iy_i)^T ⋅ is bounded compact operator in from Lm2×Lm2 to itself. Furthermore, the spectra of 𝓝̂ + Ĝ and 𝓝̂ only differ by isolated eigenvalues of finite 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

φ0=(N+G)−1F0(φ,u). (43)

Moreover,

∥φ0∥Hm≤C∥yj−1I2×2F0(φ,u)^∥Lm2.

Proof

Since the operator 𝓝̂⁻¹Ĝ : Lm2⟶Lm2 is compact, the operator I + 𝓝̂⁻¹Ĝ is Fredholm with index 0. If (I + 𝓝̂⁻¹ Ĝ)φ̂ = 0 had a nontrivial solution, then (𝓝̂ + Ĝ)φ̂ = 𝓝̂(I + 𝓝̂⁻¹Ĝ)φ̂ = 0 would also have a nontrivial solution. This would contradict (H1). Hence the Fredholm property implies that the existence of (I + 𝓝̂⁻¹Ĝ)⁻¹ : Lm2⟶Lm2 . Then we have

N^(I+N^−1G^)φ^=iyjI2×2f^,

where I_2×2 is the unit matrix.

Thus, by Lemma 7, we obtain

∥φ∥Hm=∥φ^∥Lm2≤∥(I+N^−1G^)−1∥Lm2⟶Lm2∥N^−1iyjI2×2f^∥Lm2≤C∥f^∥Lm2.

This completes the proof. □

Lemma 9

There exist a constant C > 0 such that

∥u∥Hm≤C∥ω∥Hm, ∥∂xiu∥Hm≤C∥ω∥Hm. (44)

Proof

The related equation of the velocity u and the vorticity ω is

∇×u = ω,∇⋅u = 0, ∇⋅ω=0.

This leads in Fourier space to

0−iy3iy2iy30−iy1−iy2iy10iy1iy2iy3u^1u^2u^3=ω^1ω^2ω^30.

We can get

N^ω^=−1|y|20iy3−iy2iy1−iy30iy1iy2iy2−iy10iy3ω^1ω^2ω^3=u^1u^2u^3=u^.

Using Hölder’s inequality, for 1p1+1p2 = 1, p₁, p₂ > 1, s₁ + s₂ = 2m and s₁, s₂ > 0, we have

∥u∥Hm2=∥u^∥Lm22 ≤ C(∥χ|y|≤1N^∥Ls122p12∥ω^∥Ls222p22+∥χ|y|≥1N^∥L∞2∥ω^∥Lm22)≤ C(∥χ|y|≤1N^∥Ls122p12+∥χ|y|≥1N^∥L∞2)∥ω^∥Lm22≤ C∥ω^∥Lm22=C∥ω∥Hm2,

where we use the weighted function ρ(y)=|y|(1+|y|)12, the boundedness of ∥χ|y|≥1iyi|y|2∥L∞2 and

∥χ|y|≤1iyi|y|2∥Ls122p12 = ∫|y|≤1|iyi|y|2|2p1ρp1sdy= ∫|y|≤1|iyi|y|2|2p1|y|p1s(1+|y|)p1s2dy≤C∫01ϱ2p1ϱ4p1ϱp1s(1+ϱ)p1sϱ2dϱ= C∫01ϱp1s−2p1+2(1+ϱ)p1sϱ2dϱ≤∞, for p1s−2p1+2>0.

The second estimate in (44) is followed by

∥∂xiu∥Hm=∥iyiu^ρm∥L2≤∥iyiN^∥L∞∥ω^∥Lm2≤C∥ω∥Hm.

This completes the proof. □

From the form of the nonlinear terms g³ and g⁴, it is critical to estimate the term as uv and u². For convenience, we derive some estimates about the nonlinear term N¹(φ) = φ² and N²(φ, ψ) = φψ. This proof is similar with Lemma 4 in [1], so we omit it.

Lemma 10

Define N¹ : 𝓧 ⟶ 𝓧 by N¹(φ)_n = Nn1 (Jφ) and N² : 𝓧 × 𝓧 ⟶ 𝓧 by N²(φ)_n = Nn2 (Jφ, Jψ) for φ, ψ ∈ 𝓧. Then there exists C > 0 such that

∥N1(φ)∥X≤C∥φ∥X2, ∥N2(φ,ψ)∥X≤C∥ψ∥X∥φ∥X (45)

for φ, ψ ∈ 𝓧 with ∥φ∥_𝓧 ≤ 1 and ∥ψ∥_𝓧 ≤ 1. Moreover, there exists C > 0 such that

∥N1(φ1)−N1(φ2)∥X≤C(∥φ1∥X+∥φ2∥X)∥φ1−φ2∥X, (46)

∥N2(φ1,ψ1)−N2(φ2,ψ2)∥X ≤ C∥φ1∥X+∥φ2∥X+∥ψ1∥X+∥ψ2∥X× ∥φ1−φ2∥X+∥ψ1−ψ2∥X, (47)

for φ¹, φ², ψ¹, ψ² ∈ 𝓧 with ∥φ¹∥_𝓧, ∥φ²∥_𝓧, ∥ψ¹∥_𝓧, ∥ψ²∥_𝓧 ≤ 1.

Then we have the following result.

Lemma 11

Assume that ξ_i close enough to ξ₀ for i = 1, 2. Then there exists a constant C > 0 such that

∥(inΞ+N)−1∥X⟶X≤C,∥(inΞ+N−G)−1∥X⟶X≤C,∥(inΞ+N−G)−1P±1,s∥X⟶X≤C,

for n ≠ 0.

Proof

We observe that the solution φ of the equation (inΞ + 𝓝)φ = f is given by

φ^(y)=inξ1+μ|y|2−ic1y100inξ2+y|y|2−ic1y1−1f^(y), y∈R3.

For δ₁ = μ2ξ2ξ12+4c12 and δ₂ = y2ξ22ξ22+4c12 , we have

|inξ1+ν|y|2−ic1y1|2=μ2|y|4+(c1y1+nξ1)2≥ω24c12χ|y|≤ω2c1+δ12(1+|y|2)χ|y|≥ω2c1,|inξ2+y|y|2−ic1y1|2=y2|y|4+(c1y1+nξ2)2≥ω24c12χ|y|≤ω2c1+δ22(1+|y|2)χ|y|≥ω2c1.

It follows for f ∈ 𝓗^m that φ̂ ∈ Lm+22 , thus φ ∈ 𝓗^m+2.

Let f̂ ∈ Lm+22 ⊂ Lm2 , ω̄̂ = ρ(y, ϵ)ω̂ and v̄̂ = ρ(y, ϵ)v̂ with ρ(x, ϵ) = 1+ϵ|x|2 . Note that φ is a solution of the equation (in Ξ + 𝓝)φ = f. By a direct computation, we have

(inΞ+N^φ¯^)+ϵL(y,ϵ)φ¯^=g^,

where φ̄ = (ω̂, v̂)^T, ĝ = ρ(y, ϵ)f̂ and

ϵL(y,ϵ)=(inξ1+μ|y|2−icy1)(1−ρ−1(y,ϵ))00(inξ2+y|y|2−icy1)(1−ρ−1(y,ϵ))

Here we use the fact that 𝓝 is elliptic of order of 2. Hence it derives from the form of ρ(y, ϵ) = 1+ϵ|y|2 that

L(y,ϵ)∼ϵ|y|41+ϵ|y|2+1+ϵ|y|2μ00y.

Using a Neumann series, it derives from the boundness of the operator L : Lm+22 ⟶ Lm2 that

(inΞ+N^)+ϵL:Lm+22⟶Lm2

is invertible with a bounded inverse, for sufficient small ϵ > 0. This implies that φ̄ ∈ Lm+22 , i.e., φ ∈ 𝓗^m+2. Moreover, we have

∥φ∥Hm+2=∥φ^∥Lm+22=∥φ¯^∥Lm+22≤C∥g^∥Lm2=C∥f∥Hm+2.

Above result shows that (inΞ + 𝓝)^–1 : 𝓗^m ⟶ 𝓗^m+2 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 ≠ 0 and ϵ > 0 sufficient small. In a similar manner to prove the first 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 ξ₀ for i = 1, 2. Then there exists a constant C > 0 such that

∥(inξi−Mi¯)−1∥Hm⟶Hm≤C, ∥(inξi−Mi¯)−1∇j⋅∥Hm⟶Hm≤C,∥(inξi−Mi)−1∥Hm⟶Hm≤C, ∥(inξi−Mi)−1∇j⋅∥Hm⟶Hm≤C,∥(inξi−Mi)−1P±1,s∥Hm⟶Hm≤C, ∥(inξi−Mi)−1∇j⋅P±1,s∥Hm⟶Hm≤C,

for n ≠ 0 and j = 1, 2.

Thus by Lemmas 11-12, we can obtain the solution of equations (39) and (41) as

φn=(inΞ+N)−1Fn(φ,u), n=±2,±3,…,φ±1,s=(±iΞ+N)−1P±1,sF±1(φ,u),

i.e.

ωn=(inξ1−M1)−1gn3(ω,u,v), n=±2,±3,…, (48)

vn=(inξ2−M2)−1gn4(ω,u,v), n=±2,±3,…, (49)

ω±1,s=(±iξ1−M1)−1P±1,sg±13(ω,u,v), (50)

v±1,s=(±iξ2−M2)−1P±1,sg±14(ω,u,v). (51)

The following Lemma shows the solvable of equations (48)-(51).

Lemma 13

Assume that there exist σ₁, σ₂ > 0 such that for all ξ₁, ξ₂ > 0 with |ξ₁ – ξ₀|, |ξ₂ – ξ₀| ≤ σ₁ and all ω_±1,c, v_±1,c ∈ H^m with ∥ω_±1,c∥_H^m, ∥v_±1,c∥_H^m ≤ σ₂. Then equations (48)-(51) has a unique solution (ω̃, ṽ) = Φ(ω_c, v_c) ∈ 𝓧, where

ωc=(ω−1,c,ω1,c), vc=(v−1,c,v1,c),ω~=(…,ω−2,ω−1,c+ω−1,s,ω0,ω1,c+ω1,s,ω2,…),v~=(…,v−2,v−1,c+v−1,s,v0,v1,c+v1,s,v2,…).

Moreover, there exits C > 0 such that

Φ(0,0)=(0,0), ∥ω~−ωc∥X≤C(∥ω−1,c∥Hm2+∥ω1,c∥Hm2), (52)

∥v~−vc∥X≤C(∥v−1,c∥Hm2+∥v1,c∥Hm2), (53)

with

ω~−ωc:=(…,,0,ω−1,c,0,ω1,c,0,…),v~−vc:=(…,,0,v−1,c,0,v1,c,0,…).

Proof

For fixed ξ₁, ξ₂ > 0 so close to ξ₀ and given ω_±1,c, v_±1,c ∈ H^m with

∥ω±1,c∥Hm,∥v±1,c∥Hm≤σ2.

Define the operator

Γ:(ω~∗,v~∗) ⟼ (ω~,v~) =(ω~∗+(…,0,ω−1,c,0,ω1,c,0,…),v~∗+(…,0,v−1,c,0,v1,c,0,…))⟼ (ω,v)⟼(ω~∗∗,v~∗∗)=right hand side of (48)−(51),

where (ω, v) = (Jω̃, Jṽ) are defined in Lemma 1 and

(ω~∗,v~∗) = ((…,ω−2,ω−1,s,ω0,ω1,s,ω2,…),(…,v−2,v−1,s,v0,v1,s,v2,…)),(ω~,v~) = (ω~∗+ωc,v~∗+vc) = (ω~∗+(…,0,ω−1,c,0,ω1,c,0,…),v~∗+(…,0,v−1,c,0,v1,c,0,…)).

By Lemma 2, Young inequality, (11) and the form of nonlinear terms g³ and g⁴ in (22)-(23), we derive

∥g3∥Hm−2 ≤ C(∥∇⋅(ωuT−uωT)∥Hm−2+∥∇(∇×|v|22)∥Hm−2+∥∇×(vT∇v)∥Hm−2)≤ C(∥ω∥Hm−1∥u∥Hm−1+∥v∥Hm2)≤ C(∥ω∥Hm−12+∥u∥Hm−12+∥v∥Hm2),

∥g4∥Hm−2 ≤ C∥u⋅∇v∥Hm−2+∥v∇u∥Hm−2+∥∇(|v|2z+|v|2dα)∥Hm−2+∥∇(|v||∇dα|z)∥Hm−2≤ C∥u∥Hm−1∥v∥Hm−1+∥v∥Hm−12∥z∥Hm−1+∥v∥Hm∥z∥Hm≤ C∥u∥Hm−12+∥v∥Hm−12+∥z∥Hm−12+∥v∥Hm−14+∥v∥Hm2+∥z∥Hm2≤ C∥u∥Hm−12+∥v∥Hm−12+∥v∥Hm−14.

Now we prove the operator Γ is a self-map of a sufficiently small ball in 𝓧. Using Lemma 9 and Lemma 12, we have

∥ω~∗∗∥X ≤ Csup{∥(inξ1−M1)−1∥Hm⟶Hm,∥(±iξ1−M1)−1P±1,s∥Hm⟶Hm, ∥(inξ1−M1)−1∇j∥Hm⟶Hm,∥(±iξ1−M1)−1∇jP±1,s∥Hm⟶Hm :n∈Z∖{−1,1}}×(∥ω~∥X2+∥u∥X2+∥v~∥X2)≤ C(∥ω~∗∥X2+∥ω−1,c∥Hm2+∥ω1,c∥Hm2+∥v~∗∥X2+∥v−1,c∥Hm2+∥v1,c∥Hm2)≤ C(∥ω~∗∥X2+∥v~∗∥X2+σ22), (54)

∥v~∗∗∥X ≤ Csup{∥(inξ2−M2)−1∥Hm⟶Hm,∥(±iξ2−M2)−1P±1,s∥Hm⟶Hm, ∥(inξ2−M2)−1∇j∥Hm⟶Hm,∥(±iξ2−M2)−1∇jP±1,s∥Hm⟶Hm :n∈Z∖{−1,1}}×(∥u~∥X2+∥v~∥X2+∥v~∥X4)≤ C(∥ω~∗∥X2+∥ω−1,c∥Hm2+∥ω1,c∥Hm2+∥v~∗∥X2+∥v−1,c∥Hm2+∥v1,c∥Hm2)≤ C(∥ω~∗∥X2+∥v~∗∥X2+σ22). (55)

Thus, for σ₂ ≤ 12C and (ω̃^*, ṽ^*) ∈ 𝓧 with ∥(ω̃^*, ṽ^*)∥_𝓧 ≤ 12C , we have

∥Γ(ω~∗,v~∗)∥X=∥ω~∗∗∥X+∥v~∗∗∥X≤C(∥ω~∗∥X+∥v~∗∥X)2+σ22≤1,

which implies that for sufficient small σ₂ > 0, Γ maps the ∥⋅∥_𝓧 ball of radius r = 1. Hence, we obtain a unique fixed point (ω̃^*, ṽ^*) ∈ 𝓧 of Γ, which means that equations (48)-(51) has solution of (ω̃, ṽ) = (ω̃^* + ω_c, ṽ^* + v_c). Moreover, if (ω_±1,c, v_±1,c) = (0, 0), then Φ(0, 0) = (0, 0). Next we prove the second inequality in (52). Note that

(ω~∗,v~∗)=Γ(ω~∗,v~∗)=(ω~∗∗,v~∗∗),

which combine with (54)-(55), we derive

∥ω~−ωc∥X=∥ω~∗∥X=∥ω~∗∗∥X≤C(∥ω~∗∥X2+∥ω−1,c∥Hm2+∥ω1,c∥Hm2),∥v~−vc∥X=∥v~∗∥X=∥v~∗∗∥X≤C(∥v~∗∥X2+∥v−1,c∥Hm2+∥v−1,c∥Hm2).

Thus we deduce that for sufficient small ball B_r(0) ⊂ B₁(0),

∥ω~−ωc∥X≤C(∥ω−1,c∥Hm2+∥ω1,c∥Hm2), (56)

∥v~−vc∥X≤C(∥v−1,c∥Hm2+∥v1,c∥Hm2), (57)

where

ω~−ωc:=(…,,0,ω−1,c,0,ω1,c,0,…), v~−vc:=(…,,0,v−1,c,0,v1,c,0,…).

This completes the proof.□

Proof of Theorem 1

To prove Theorem 1, the rest remains to analyze equations (37)-(38). We restate equations:

(±iξ1−M1)ω±1,c=P±1,cg±13(ω,u,v),(±iξ2−M2)v±1,c=P±1,cg±14(u,v).

It follows from (ω_–1, v_–1) = (ω₁, v₁) and (g±13,g±14)=(g±13¯,g±14¯) that the ”–” equation is the complex conjugate of the ”+” equation. By Lemma 1, we can denote (ω, v) = (Jω̃, Jṽ) by means of

(ω~,v~)=Φ(ωc,vc)=Φ((ω1,c¯,ω1,c),(v1,c¯,v1,c)).

Our target is to find (ξ₁, β) and (ξ₂, β) close to (ξ₀, β_c) and a nontrivial solution (ω_1,c, v_1,c) = (ω_1,c, v_1,c)(x) of

−iξ1ω1,c+M1ω1,c+P1,cg13(JΦ(ω1,c¯,ω1,c,v1,c¯,v1,c))=0, (58)

−iξ2v1,c+M2v1,c+P1,cg14(JΦ(ω1,c¯,ω1,c,v1,c¯,v1,c))=0. (59)

Due to ω_1,c, v_1,c ∈ Cψ⁺ and (M₁ψ⁺, M₂ψ⁺) = (λ0+(β)ψ+,μ0+(β)ψ+), we can write

ω1,c=ηψ+, v1,c=δψ+.

Then by (58)-(59), we obtain

−iξ1ηψ++λ0+(β)ηψ++P1,cg13(JΦ(ηψ+¯,ηψ+,δψ+¯,δψ+))=0, (60)

−iξ2δψ++μ0+(β)δψ++P1,cg14(JΦ(ηψ+¯,ηψ+,δψ+¯,δψ+))=0, (61)

for some η, δ ∈ C ∖ {0}.

To be simple, we introduce (p_1,c, θ_1,c) by

(P1,cω,P1,cv)=(p1,c(ω)ψ+,θ1,c(v)ψ+).

Then equations (60)-(61) can be written as

−iξ1η+λ0+(β)η+g3(β,η,δ)=0, for some η∈C, (62)

−iξ2δ+μ0+(β)δ+g4(β,η,δ)=0, for some δ∈C, (63)

where the cubic coefficient μ ≠ 0 in

g3(β,η,δ):=p1,cg13(JΦ(ηψ+¯,ηψ+,δψ+¯,δψ+), (64)

g4(β,η,δ):=θ1,cg14(JΦ(ηψ+¯,ηψ+,δψ+¯,δψ+). (65)

Note that

|p1,c(ω)|≤C∥P1,cω∥Hm≤C∥ω∥Hm, (66)

|p1,c(v)|≤C∥P1,cv∥Hm≤C∥v∥Hm. (67)

So by (64)-(65), (66)-(67), we derive

|p1,cg13(JΦ(ηψ+¯,ηψ+,δψ+¯,δψ+)|≤C∥g13(JΦ(ηψ+¯,ηψ+,δψ+¯,δψ+)∥Hm≤C∥Φ(ηψ+¯,ηψ+,δψ+¯,δψ+)∥X≤C(∥ω1,c¯∥Hm2+∥ω1,c∥Hm2+∥v1,c¯∥Hm2+∥v1,c∥Hm2)≤C(∥ηψ+∥Hm2+∥δψ+∥Hm2)≤C(|η|2+|δ|2),

|θ1,cg14(JΦ(ηψ+¯,ηψ+,δψ+¯,δψ+)|≤C∥g14(JΦ(ηψ+¯,ηψ+,δψ+¯,δψ+)∥Hm≤C∥Φ(ηψ+¯,ηψ+,δψ+¯,δψ+)∥X≤C(∥ω1,c¯∥Hm2+∥ω1,c∥Hm2+∥v1,c¯∥Hm2+∥v1,c∥Hm2+∥v1,c¯∥Hm4+∥v1,c∥Hm4)≤C(∥ηψ+∥Hm2+∥δψ+∥Hm2+∥δψ+∥Hm4)≤C(|η|2+|δ|2+|δ|4),

where we use the notation

(ω~,v~)=Φ(ωc,vc)=Φ(ηψ+¯,ηψ+,δψ+¯,δψ+).

Inspired by the classical Hopf-Bifurcation result [25], if we exclude the zero solution, we can employ the implicit function theorem to find real value solutions (i.e. find (y₁, y₂) = (η, δ) ∈ R²) of equations (62)-(63). Hence, we define the complex-valued smooth function

Υ1(y1,y2;ϱ,β):=−i(ξ0+ϱ)+λ0+(βc+ϵ)+y1−1g3(βc+ϵ,y1,y2), y1≠0,−i(ξ0+ϱ)+λ0+(βc+ϵ), y1=0,

Υ2(y1,y2;ϱ,β):=−i(ξ0+ϱ)+μ0+(βc+ϵ)+y2−1g4(βc+ϵ,y1,y2), y2≠0,−i(ξ0+ϱ)+μ0+(βc+ϵ), y2=0.

It follows from (λ0+(βc),μ0+(βc))=(iξ0,iξ0) that (Υ¹(0, 0, 0, 0), Υ²(0, 0, 0, 0)) = (0, 0). Moreover, by assumption (H2) the Jacobi Matrix

Dρ,ϵΥ1(y1,y2;ϱ,ϵ)|y1=y2=ϱ=ϵ=0=0ddβReλ0+(β)|β=βc−1ddβImλ0+(β)|β=βc,

Dρ,ϵΥ2(y1,y2;ϱ,ϵ)|y1=y2=ϱ=ϵ=0=0ddβReμ0+(β)|β=βc−1ddβImμ0+(β)|β=βc

with respect to ρ, ϵ has

detDρ,ϵΥ1(y1,y2;ϱ,ϵ)|y1=y2=ϱ=ϵ=0=ddβReλ0+(β)|β=βc>0,detDρ,ϵΥ2(y,y2;ϱ,ϵ)|y1=y2=ϱ=ϵ=0=ddβReμ0+(β)|β=βc>0.

Thus, for sufficient small y₁, y₂ > 0, we find a function y₁ ↦ (ϱ(y₁), ϵ(y₁)) and y₂ ↦ (ϱ(y₂), ϵ(y₂)) with ϱ(0) = ϵ(0) = 0 such that

−i(ξ0+ϱ(y1))+λ0+(βc+ϵ(y1))−y1−1g3(βc+ϵ(y1),y1,βc+ϵ(y2),y2)=0,−i(ξ0+ϱ(y2))+μ0+(βc+ϵ(y2))−y2−1g4(βc+ϵ(y1),y1,βc+ϵ(y2),y2)=0

Note that the degree of nonlinearity. Then it follows from differentiating this equation that ϵⁱ ≠ 0 for some first i. Hence, the function y₁ ↦ ϵ(y₁) and y₁ ↦ ϵ(y₂) are locally invertible, and have ϵ ↦ y₁(ϵ) and ϵ ↦ y₂(ϵ). It implies that the following equation holds

− i(ξ0+ϱ(y1(ϵ)))y1(ϵ)+λ0+(βc+ϵ)y1(ϵ)−g3(βc+ϵ,y1(ϵ),y2(ϵ))=0,− i(ξ0+ϱ(y2(ϵ)))y2(ϵ)+μ0+(βc+ϵ)y2(ϵ)−g4(βc+ϵ,y1(ϵ),y2(ϵ))=0,

for sufficient small ϵ > 0.

Therefore we obtain the desired solutions of (58)-(59) by setting (ξ₁, ξ₂) = (ξ₀ + ϱ(y₁(ϵ)), ξ₀ + ϱ(y₂(ϵ))), β = β_c + ϵ and (ω1,c,v1,c)=(y1(ϵ)ψβc+ϵ+,y2(ϵ)ψβc+ϵ+)(x). This result combining with Lemma 8, Lemma 13 and (19) give the proof of Theorem 1.

Acknowledgement

This work is supported by NSFC No 11771359, and the Fundamental Research Funds for the Central Universities (Grant No. 20720190070, No.201709000061 and No. 20720180009). The second author is supported by Huizhou University Professor Doctor Launch Project Grant, No. 20187B037.

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Received: 2019-09-24

Accepted: 2019-10-14

Published Online: 2019-12-31

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https://doi.org/10.1515/anona-2020-0052

Keywords for this article

Nematic liquid crystals; periodic solution; bifurcation

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