SINGULARITY SPHERE IN THE DYNAMIC ... - Semantic Scholar

SINGULARITY SPHERE IN THE DYNAMIC BIFURCATION OF GINZBURG-LANDAU EQUATION TIAN MA, JUNGHO PARK AND SHOUHONG WANG Abstract. We study in this paper the bifurcation and singularities of the Ginzburg–Landau equation. The existence and structure of the bifurcated solutions are obtained by T. Ma, J. Park and S. Wang [8] and now we prove that the bifurcated invariant set Σλ contains at least Cnk = n···(n−k+1) singularity manifolds T k+1 provided λ crosses the second k! eigenvalue α of −α∆. And we achieved the similar results when λ crosses the rest of eigenvalues.

1. Introduction Complex Ginzburg–Landau (GL) equation, which reads ∂u − (α + iβ)∆u + (σ + iρ)|u|2 u − λu = 0, (1.1) ∂t is an important equation in a number of scientific fields. It is directly related to the GL theory of superconductivity. In this context, the unknown function is the order parameter, the constants β and ρ are zero, and the the bifurcation parameter λ is the GL parameter. We consider the unknown function u : Ω × [0, ∞) → C is a complexvalued furntion and Ω ⊂ Rn is an open , bounded and smooth domain in Rn (1 ≤ n ≤ 3). The parameters α, β, ρ, σ and λ are real numbers and (1.2)

α > 0,

σ > 0.

The initial condition for (1.1) is given by (1.3)

u(x, 0) = φ + iψ.

Also, we give the periodic boundary condition, (1.4)

Ω = (0, 2π)n and u is Ω − periodic.

In fluid dynamics the GL equation is found, for example, in the study of Poiseuille flow, the nonlinear growth of convection rolls in the Rayleigh– B´enard problem and Taylor–Couette flow. The equation also arises in the study of chemical systems governed by reaction-diffusion equations. There are extensive studies for GL equation, in particular, Bartuccelli, Constantin, Doering, Gibbon and Gisself¨alt [2] for the turbulence, Doering, Gibbon, Holm and Nicolaenko [3] for low-dimensional behavior, Tang and 1991 Mathematics Subject Classification. 35, 37. Key words and phrases. Ginzburg–Landau equation, bifurcation, singularity sphere. 1

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T. MA, J. PARK AND S. WANG

Wang [12] for the existence, uniqueness and long time behavior of the solutions of a nonstationary GL superconductivity model and Ma, Park and Wang [8] for the dynamic bifurcation. We study in this paper the spheres of singularities in bifurcated S m invariant set for the GL equation. The existence of bifurcation and full classification of structure of bifurcated solutions are achieved in [8] in which the authors used a newly developed notion called attractor bifurcation; see [10]. The result which will be addressed and proved can be summarized as following. For the GL equation with the periodic boundary condition, (1) if the system parameter λ crosses the first eigenvalue α of −α∆ then the GL equation bifurcates from trivial solution (u, λ) = (0, α) an invariant sphereΣ and it contains at least Cnk = n···(n−1+k) (k + 1)– k! k+1 dimensional tori T for every 1 ≤ k ≤ n proveded β = ρ = 0. (2) if λ crosses the rest of eigenvalues αλk which has multiplicity mthen the invariant set Σ consists of at least Csk (m = 2s, k ≤ s) singularity manifolds T nk +1 for some 1 ≤ nk ≤ n. The paper is organized as follows. In Section 2, we will recall the attractor bifurcation theory and the results of the theory for GL equation which were achieved in [8]. In Section 3 we will state the main results about the singularity sphere in the dynamic bifurcation of GL equation and will prove it. 2. Abstract bifurcation theory 2.1. Preliminary. We recall in this section a general theory on attractor bifurcation for nonlinear evolution equations; see [10, 9]. Let H and H1 be two Hilbert spaces and H1 ,→ H be a dense and compact inclusion. We consider the nonlinear evolution equations du (2.1) = Lλ u + G(u, λ), dt (2.2) u(0) = u0 , where u : [0, ∞) → H is the unknown function, λ ∈ R is the system parameter, and Lλ : H1 → H are parameterized linear completely continuous fields depending continuously on λ ∈ R1 , which satisfy    Lλ = −A + Bλ is a sectorial operator, A : H1 → H a linear homeomorphism, (2.3)   B : H → H the parameterized linear compact operators. 1 λ It is easy to see that Lλ generates an analytic semigroup {e−tLλ }t≥0 . Then we can define fractional power operators Lαλ for any 0 ≤ α ≤ 1 with domain Hα = D(Lαλ ) such that Hα1 ⊂ Hα2 if α1 > α2 , and H0 = H. Furthermore, we assume that the nonlinear terms G(·, λ) : Hα → H for some 1 > α ≥ 0 are a family of parameterized C r bounded operators (r ≥ 1)

SINGULARITY SPHERE

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continuously depending on the parameter λ ∈ R1 , such that (2.4)

G(u, λ) = o(kukHα ) ∀ λ ∈ R1 .

Actually, in this paper we need only the following conditions for the operator Lλ = −A + Bλ , which ensure that Lλ is a sectorial operator. Let there be a eigenvalue sequence {ρk } ⊂ C and an eigenvector sequence {ek , hk } ⊂ H1 of A :    Azk = ρk zk , zk = ek + ihk , Reρk → +∞ as k → ∞, (2.5)   |Imρ /(Reρ + a)| ≤ C for some constants a, C > 0, k k such that {ek , hk } is a basis of H. Condition (2.5) implies that A is a sectorial operator. Hence we can define fractional power operator Aα with domain Hα = D(Aα ). Then for the operator Bλ : H1 −→ H, we assume that there is a constant 0 ≤ θ < 1 such that (2.6)

Bλ : Hθ −→ H bounded ∀ λ ∈ R.

Let {Sλ (t)}t≥0 be an operator semigroup generated by the equation (2.1) which enjoys the following properties: (i) For any t ≥ 0, Sλ (t) : H → H is a linear continuous operator. (ii) Sλ (0) = I : H → H is the identity on H and (iii) For any t, s ≥ 0, Sλ (t + s) = Sλ (t) · Sλ (s). Then the solution of (2.1) and (2.2) can be expressed as u(t) = Sλ (t)u0 ,

t ≥ 0.

Definition 2.1. A set Σ ⊂ H is called an invariant set of (2.1) if S(t)Σ = Σ for any t ≥ 0. An invariant set Σ ⊂ H of (2.1) is called an attractor if Σ is compact, and there exists a neighborhood U ⊂ H of Σ such that for any ϕ ∈ U we have (2.7)

lim distH (u(t, ϕ), Σ) = 0.

t→∞

The largest open set U satisfying (2.7) is called the basin of attraction of Σ. Definition 2.2. (1) We say that (2.1) bifurcates from (u, λ) = (0, λ0 ) an invariant set Ωλ if there exists a sequence of invariant sets {Ωλn } of (2.1), 0 ∈ / Ωλn , such that lim λn = λ0 ,

n→∞

lim max |x| = 0.

n→∞ x∈Ωλn

(2) If the invariant sets Ωλ are attractors of (2.1), then the bifurcation is called attractor bifurcation. (3) If Ωλ are attractors and are homotopy equivalent to an m-dimensional sphere S m , then the bifurcation is called an S m -attractor bifurcation.

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T. MA, J. PARK AND S. WANG

2.2. Attractor Bifurcation. A complex number β = α1 +iα2 ∈ C is called an eigenvalue of Lλ : H1 → H if there are x, y ∈ H1 such that Lλ z = βz, z = x + iy, or, equivalently, Lλ x = α1 x − α2 y, Lλ y = α2 x + α1 y. Now let the eigenvalues (counting the multiplicity) of Lλ be given by β1 (λ), β2 (λ), · · · , βk (λ), · · · ∈ C. Suppose that (2.8)

  < 0 Reβi (λ) = = 0   >0

(2.9)

if λ < λ0 , if λ = λ0 if λ > λ0 ,

Reβj (λ0 ) < 0

(1 ≤ i ≤ m),

∀ m + 1 ≤ j.

Let the eigenspace of Lλ at λ0 be m [ ∞ n o [ E0 = u ∈ H1 | (Lλ0 − βi (λ0 ))k u = 0 . i=1 k=1

It is known that dim E0 = m. Let H1 = H = Rn . The following attractor bifurcation theorem was proved in [10]. Theorem 2.3 (Attractor Bifurcation Theorem). Let H1 = H = Rn , the conditions (2.8) and (2.9) hold true, and u = 0 be a locally asymptotically stable equilibrium point of (2.1) at λ = λ0 . Then the following assertions hold true. (1) Equation (2.1) bifurcates from (u, λ) = (0, λ0 ) an attractor Aλ for λ > λ0 , with m − 1 ≤ dim Aλ ≤ m, which is connected as m > 1. (2) The attractor Aλ is a limit of a sequence of m-dimensional annulus Mk with Mk+1 ⊂ Mk . In particular, if Aλ is a finite simplicial complex, then Aλ has the homotopy type of S m . (3) For any uλ ∈ Aλ , uλ can be expressed as uλ = vλ + o(kvλ kH1 ),

vλ ∈ E0 .

(4) If G : H1 → H is compact and the equilibrium points of (2.1) in Aλ are finite, then we have the index formula ( X 2 if m = odd, ind [−(Lλ + G), ui ] = 0 if m = even. u ∈A i

λ

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(5) If u = 0 is globally stable for (2.1) at λ = λ0 , then for any bounded open set U ⊂ H with 0 ∈ U there is an ε > 0 such that as λ0 < λ < λ0 + ε, the attractor Aλ bifurcated from (0, λ0 ) attracts U/Γ in H, where Γ is the stable manifold of u = 0 with codimension m. In particular, if (2.1) has a global attractor in H then U = H. Remark 2.4. As H1 and H are infinite dimensional Hilbert spaces, if (2.1) satisfies the conditions (2.4)–(2.6),(2.8), and (2.9) and u = 0 is a locally (global) asymptotically stable equilibrium point of (2.1) at λ = λ0 , then the assertions (1)–(5) of Theorem 2.3 hold; see [10, 9]. And the bifurcation of the GL equation is achieved in [8] as follows. Theorem 2.5. For the GL equation (1.1) with the periodic boundary condition (1.4), we have the following assertions. (1) (a) As λ > α, the problem (1.1) with (1.4) bifurcates from (u, λ) = (0, α) an invariant set Σλ . Σλ has dimension between 4n − 1 and 4n and is a limit of a sequence of 4n annulus Mk with Mk+1 ⊂ Mk ; i.e., Σλ = ∩∞ k=1 Mk . (b) If |ρ| + |β| = 6 0, then Σλ contains no steady state solutions of (1.1) with (1.4). (2) (a) As λ ≤ 0, u = 0 is globally asymptotically stable. (b) As λ > 0 the problem (1.1) with (1.4) bifurcates from (u, λ) = (0, 0) an attractor Σλ ⊂ L2 (Ω, C). The bifurcated attractor Σλ has dimension between 1 and 2, and is a limit of a sequence of 2D annulus Mk with Mk+1 ⊂ Mk , i.e. Σλ = ∩∞ k=1 Mk . (c) If ρ 6= 0 then Σλ is a periodic orbit. (d) Σλ attracts L2 (Ω, C)/Γ, where Γ is the stable manifold of u = 0 with codimension two in L2 (Ω, C). Theorem 2.6. Let λk be an eigenvalue of −∆ with the periodic boundary condition with multiplicity m ≥ 1. Then as λ > αλk , the problem (1.1) bifurcates from (u, λ) = (0, αλk ) an invariant set Σλ . This invariant set Σλ has dimension between 2m − 1 and 2m and is a limit of a sequence of 2m-dimensional annuli Mk with Mk+1 ⊂ Mk ; i.e., Σλ = ∩∞ k=1 Mk . If |β| + |ρ| 6= 0, then there is no singularity in Σλ . 3. Singularity sphere in S m In this section we study the singularities in the bifurcated invariant set Σλ and it can be summarized as following. Theorem 3.1. If |β| + |ρ| 6= 0 then Σλ contains no steady state solution of GL equation and as λ > α, Σλ contains of at least Cnk = n···(n+1−k) k! (k + 1)–dimensional singularity tori T k+1 for every k (1 ≤ k ≤ n) provided |β| + |ρ| = 0. Proof. Step 1. We first prove in the case of single index.

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T. MA, J. PARK AND S. WANG

We know that the eigenvalue problem ½ − ∆ek = ρk ek , ek (x + 2kπ) = ek (x), has the eigenvalues ρk = k12 + · · · + kn2 ≥ 1,

ki = 0, 1, · · · (1 ≤ i ≤ n),

and the eigenvectors, corresponding to ρk , sin(k1 x1 + · · · + kn xn ),

cos(k1 x1 + · · · + kn xn ).

For the second eigenvalue ρ1 = 1, we have eigenvectors sin xj ,

cos xj

(1 ≤ j ≤ n).

Define Lλ + G : H1 → H the operator by Lλ u + Gu = ∆u + λu − |u|2 u.

(3.1) Note that the space ( Hj =

u ∈ H1 : u =

∞ X

) yk sin kxj , yk = (y1k , y2k ) ∈ R2

k=1

is an invariant subspace of Lλ + G. Then the Lyapunov-Schmidt reduction equations of the stationary equation of (3.1) on Hj are given by ¶ ∞ µ Z 2π X σ 2 (λ − α)yi1 − |u| sin kxj sin xj dxj yik = 0 π 0 k=1 ∞

X 1 yik + 2 k α−λ r=1

µ Z 2π ¶ σ 2 |u| sin rxj sin kxj dxj yik = 0 π 0

for i = 1, 2 and k ≥ 2. Let σ = π then we have for i = 1, 2, fkr

(3.2)

Z 0

2π

|u|2 sin rxj sin kxj dxj

(λ − α)yi1 −

∞ X

f1k yik = 0,

k=1 ∞

(3.3)

yik

X 1 fkr yik =− 2 k α−λ

(k ≥ 2).

r=1

For k ≥ 2, we let [yik ]1 = −

1 f 1. −λ k

k2 α

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Then (3.3) can be rewritten as ¶ ∞ µ X 1 yik = − 2 fkr yir k α−λ r=1 ¶ ∞ µ X 1 1 1 =− 2 f yi1 + − 2 fkr yir k α−λ k k α−λ = [yik ]1 yi1 −

r=2 ∞ X

1 −λ

k2 α

fkr yir .

r=2

By the straightforward calculation, we have [yik ]1 = O(|y|2 ), ∞ X

y = (y11 , y21 ),

fkr yir = O(|y|4 ).

r=2

By induction, we define ∞

[yik ]s+1 = −

X j 1 fk [yir ]s k2 α − λ

(s ≥ 1).

r=2

Then, we have (3.4)     [yik ]s+1 =

∞ X

(−1)s+1 f rs f rs−1 · · · fr11 , 2 α − λ)(r 2 α − λ) · · · (r 2 α − λ) k rs (k s 1 =2

r1 ,··· ,rs    [y ] = O(|x|2s ). ik s

From (3.3), we have (3.5)

yik = yi1

∞ X [yik ]s . s=1

Inserting (3.5) into (3.2), we obtain (λ − α)yi1 −

∞ X

f1k yi1

∞ X [yik ]s = 0, s=1

k=1

or equivalently, (λ − α)yi1 −

yi1 f11

− yi1

∞ X ∞ X

f1k [yik ]s = 0.

k=2 s=1

Hence (3.2) referred to (3.6)

(λ − α) − f11 −

∞ X ∞ X k=2 s=1

f1k [yik ]s = 0

(i = 1, 2).

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T. MA, J. PARK AND S. WANG

Since the expression [yik ]s is independent on i = 1, 2, (3.6) is only one equation. Note that Z σ 2π 2 1 f1 = |u| sin xj sin xj dxj π 0  !2 Ã ∞ ! Z 2π ÃX ∞ X σ  = y1k sin kxj + y2k sin kxj  sin2 xj dxj π 0 k=1

k=1

3 2 2 = σ(y11 + y21 ) + o(|y|2 ), 4 and

∞ X ∞ X

f1k [yik ]s = O(|y|4 ).

k=2 s=1

Therefore, from (3.6), we have the steady state bifurcation equation of (1.1) near λ = α as follows 3 2 2 + y21 + o(|y|2 ) − O(|y|4 )) = 0 (3.7) (λ − α) − σ(y11 4 or equivalently, (3.8)

2 y11 + y212 + o(|y|2 ) =

4 (λ − α) 3σ

For λ > α, sufficiently small, the set of solutions of (3.8) is a cycle S 1 . Therefore the set of singularities of GL equation is also sphere and say it Φ, that is, ½ ¾ 4 2 Φ = β1 sin x1 + β2 sin x2 + (|β|) : |β| = (λ − α) = S 1 . 3σ note that the solutions of GL equation are translation invariant u(x, t) → u(x + θ, t),

θ = (θ1 , · · · , θn ) ∈ Rn .

Hence the functions in the set Φθ := {u(x + θ) : u ∈ Φ} are also singularities of GL equation. If we set Γ=

n [

[

Φθ

i=1 0≤θi ≤2π

then Γ is a manifold homeomorphic to T 2 = S 1 × S 1 , which consists of singularity of GL equation. Step 2. Now we give the same proof in the case of a multiple index. For a multiple index (j1 , · · · k ), 1 ≤ js ≤ n, jl 6= jm if l 6= m, we take e = sin xj1 + · · · sin xjk .

SINGULARITY SPHERE

Then the space

( He =

u ∈ H1 : u =

∞ X

9

) yk e

k

k=1

is invariant for Lλ + G. By the same way, we can get that Lλ + G has a singularity manifold T k+1 in He . The number of index (j1 , · · · jk ) for k ≤ n and so we complete the proof. is Cnk = n(n−1)···(n−k+1) k! ¤ Corollary 3.2. Let λk > 0 be an eigenvalue of −∆ with the multiplicity m. Then as λ > αλk , the bifurcated invariant set Σλ consists of at least Csk = s(s−1)···(s−k+1) (m = 2s, k ≤ s) singularity manifolds Γ = T nk +1 for k! some nk (1 ≤ nk ≤ n) provided |β| + |ρ| = 0. Proof. The proof follows the same steps as in the proof of Theorem 3.1. ¤ References [1] M. Bartuccelli, P. Constantin, C. R. Doering, J. D. Gibbon, and M. Gis¨ lt, Hard turbulence in a finite-dimensional dynamical system?, Phys. Lett. A, selfa 142 (1989), pp. 349–356. [2] , On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Phys. D, 44 (1990), pp. 421–444. [3] C. R. Doering, J. D. Gibbon, D. D. Holm, and B. Nicolaenko, Low-dimensional behaviour in the complex Ginzburg-Landau equation, Nonlinearity, 1 (1988), pp. 279– 309. [4] D. Henry, Geometric theory of semilinear parabolic equations, vol. 840 of Lecture Notes in Math., Springer-Verlag, Berlin, 1981. ´c ˇ, Ginzburg-Landau dynamics with a time-dependent [5] H. G. Kaper and P. Taka magnetic field, Nonlinearity, 11 (1998), pp. 291–305. [6] I. Kukavica, An upper bound for the winding number for solutions of the GinzburgLandau equation, Indiana Univ. Math. J., 41 (1992), pp. 825–836. [7] , Hausdorff length of level sets for solutions of the Ginzburg-Landau equation, Nonlinearity, 8 (1995), pp. 113–129. [8] T. Ma, J. Park and S. Wang, Dynamic bifurcation of the Ginzburg–Landau equation, SIAM J.Appl. dyna. Sys. Vol. 3, No. 4, (2004), pp. 620–635. [9] T. Ma and S. Wang, Attractor bifurcation theory and its applications to RayleighB´enard convection, Commun. Pure Appl. Anal., 2 (2003), pp. 591–599. [10] , Dynamic bifurcation of nonlinear evolution equations, Chinese Ann. Math.Ser. B, 2004. [11] , Bifurcation Theory and Applications, World Scientific, to appear in 2005. [12] Q. Tang and S. Wang, Time dependent Ginzburg-Landau equations of superconductivity, Phys. D, 88 (1995), pp. 139–166. [13] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, vol. 68 of Appl. Math. Sci. 68, Springer-Verlag, New York, second ed., 1997. Institute for Scientific Computing and Applied mathematics & Department of Mathematics, Indiana University, Bloomington, IN 47405 E-mail address: [email protected]