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Dynamical bifurcation for the Kuramoto–Sivashinsky equation
Nonlinear Analysis 74 (2011) 1155–1163
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Dynamical bifurcation for the Kuramoto–Sivashinsky equation✩ Yindi Zhang ∗ , Lingyu Song, Wang Axia School of Science, Chang’an University, Xi’an 710064, PR China College of Science, Xi’an Jiaotong University, Xi’an 710049, PR China
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Article history: Received 9 April 2010 Accepted 27 September 2010 Keywords: Kuramoto–Sivashinsky equation Attractor bifurcation Eigenvalue analysis Center manifold
abstract In this paper, by using the center manifold reduction method, together with the eigenvalue analysis, we made bifurcation analysis for the Kuramoto–Sivashinsky equation, and proved that the Kuramoto–Sivashinsky equation with constraint condition bifurcates an attractor Aλ as λ crossed the first critical value λ0 = 1 under the two cases. Our analysis was based on a new and mature attractor bifurcation theory developed by Ma and Wang (2005) [17,18]. Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved.
1. Introduction The Kuramoto–Sivashinsky equation is a canonical nonlinear evolution equation arising in a variety of physical contexts, e.g. long wave on thin film, long wave on the interface between two viscous fluid [1], unstable drift waves in plasmas, reaction–diffusion systems [2] and flame front instability [3,4], it represents models of pattern formation on unstable flame fronts and thin hydrodynamic films [3,4], the Kuramoto–Sivashinsky equation has thus been studied extensively; see [5–11] for details. Also, for more studies on dynamical properties of the nonlinear equations see [12–16]. We are interested in the following Kuramoto–Sivashinsky equation in one-dimensional space with periodic condition:
∂ 2u ∂u ∂ u ∂ 4u + 4 + λ 2 + λu = 0, ∂t ∂x ∂x ∂x ∂ ju ∂ ju (−π , t ) = j (π , t ), j = 0, 1, 2, 3 j ∂ x ∂x u(0, x) = ϕ(x),
(1.1)
supplemented with the following constraint
∫
π
u(x, t )dx = 0,
(1.2)
−π
where x ∈ Ω = (−π , π ), u : (−π , π ) × [0, ∞) → R1 is a real scalar function. Refs. [17–19] recently developed a new bifurcation theory based on a notion of bifurcation called attractor bifurcation. The main objective of this paper is to conduct bifurcation and stability analysis for the Kuramoto–Sivashinsky equation by using this new attractor bifurcation theory. In the first case, we look for solutions to problem (1.1) with constraint (1.2), which are odd functions with respect to x, i.e., u(−x, t ) = −u(x, t ). We prove that, for λ ≤ λ0 = 1, u = 0 is an asymptotically stable equilibrium point of the Kuramoto–Sivashinsky equation, and for 4 > λ > λ0 = 1, the Kuramoto–Sivashinsky equation bifurcates from (u, λ) = (0, λ0 ) to an attractor Aλ , homologic to S 0 . In the second case, we look for solutions without the oddness assumption, and ✩ Supported by the NSF of China 10971165 and 10971166; and CHD2010JC094.
∗
Corresponding author at: School of Science, Chang’an University, Xi’an 710064, PR China. E-mail address: [email protected] (Y. Zhang).
0362-546X/$ – see front matter Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.09.052
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we show that, for λ ≤ λ0 = 1, u = 0 is an asymptotically stable equilibrium point of the Kuramoto–Sivashinsky equation, and for 4 > λ > λ0 = 1, the Kuramoto–Sivashinsky equation bifurcates from (u, λ) = (0, λ0 ) to an attractor Aλ , homologic to S 2−1 = S 1 . Moreover, in both cases, we have found that the attractor Aλ consists of the steady state solutions of the bifurcation equation attained by using the center manifold reduction [17]. 2. Preliminaries In this section, we shall recall some results of the dynamic bifurcation of abstract nonlinear evolution equations developed in Ref. [17]. Let H and H1 be two Hilbert spaces, H1 ⊂ H a compact and dense inclusion. Consider the following abstract nonlinear evolution equations given by
du
= Lλ u + G(u, λ), dt u(0) = ϕ,
(2.1)
where u(t ) : [0, ∞) → H is the unknown scalar function, λ ∈ R1 is a system parameter, and Lλ : H1 −→ H a parameterized linear completely continuous field continuously depending on λ ∈ R1 , which satisfies Lλ = −A + Bλ , A : H1 → H , Bλ : H1 → H ,
a sectorial operator, a linear homeomorphism, the parameterized linear compact operator.
(2.2)
We can 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 (see [20,21]). We now assume that the nonlinear terms G(·, λ) : Hα → H for some 0 ≤ α < 1 are a family of parameterized C r (r ≥ 1) bounded operators which continuously depend on the parameter λ ∈ R1 , such that G(u, λ) = o(‖u‖α ),
∀λ ∈ R1 .
(2.3)
Definition 2.1 ([17]). A set Σ ⊂ H is called an invariant set, Σ ⊂ H of (2.1) is said to be an attractor if Σ is compact, and there exists a neighborhood Σ ⊂ H of Σ such that for any ϕ ∈ U we have lim distH (u(t , ϕ), Σ ) = 0.
(2.4)
t →∞
The largest open set U satisfying (2.4) is called the basin of attraction of Σ . Definition 2.2 ([17]). (1) We say that Eq. (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 an attractor bifurcation. (3) If Ωλ are attractors, which are homotopic equivalent to an m-dimensional sphere S m , then the bifurcation is called S m -attractor bifurcation at (0, λ0 ). Let the eigenvalues (counting multiplicity) of Lλ be given by β1 (λ), β2 , (λ), . . . , βk (λ) ∈ C (k ≥ 1), where C is the complex space. Suppose that
< 0, Reβi (λ) = = 0, > 0, Reβi (λ0 ) < 0,
if λ < λ0 , if λ = λ0 , if λ > λ0 .
∀1 ≤ i ≤ m
(2.5)
∀m + 1 ≤ j.
(2.6)
Let the eigenspace of Lλ at λ0 be E0 = 1≤i≤m k {u ∈ H1 (Lλ0 − βi (λ0 )) u = 0, k = 1, 2, . . .}. By (2.5), we know that dim E0 = m. The following attractor bifurcation theorem for the infinite-dimensional system (2.1) was found in [17].
k
Theorem 2.1. Assume that conditions (2.2)–(2.3) and (2.5)–(2.6) hold true, and u = 0 is a locally asymptotically stable equilibrium point of (2.1) at λ = λ0 . Then the following assertions hold true. (1) Eq. (2.1) bifurcates from (u, λ) = (0, λ0 ) an attractor Ωλ for λ > λ0 , with m − 1 ≤ dim Ωλ ≤ m, which is connected as m > 1.
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(2) The attractor Ωλ is a sequence of m-dimensional annulus Ak with Ak+1 ⊂ Ak ; especially, if Ωλ is a finite simplistic complex, then Ωλ has the homotopy type of the m − 1-dimensional sphere S m−1 . (3) For any uλ ∈ Ωλ , uλ can be expressed as uλ = vλ + o(‖vλ ‖H1 ),
∀vλ ∈ E0 .
(4) If the number of equilibrium points of (2.1) in Ωλ is finite, then we have the index formula:
−
ind[−(Lλ + G), ui ] =
ui ∈Ωλ
2, 0,
if m = odd if m = even,
where ind(f , x0 ) = deg(f , U , 0), and U is a small neighborhood of x0 . (5) If u = 0 is globally asymptotically stable for (2.1) at λ = λ0 , then for any bounded open set U ⊂ H with Ωλ ⊂ U, there is an ε > 0, such that λ0 < λ < λ0 + ε , the attractor Ωλ attracts U \ Γ , where Γ is the stable manifold of u = 0 with codimension m. In particular, if (2.1) has a global attractor for all λ near λ0 , then the ε here can be chosen independently of U. Consider the equation given by du dt
= Lu + G(u),
(2.7)
where L is symmetric, then all eigenvalues of L are real. Let the eigenvalue βk of L satisfy
βi = 0, βj < 0,
if 1 ≤ i ≤ m if m + 1 ≤ j < ∞.
(2.8)
Set E0 = {u ∈ H1 |Lu = 0} E0⊥ = {u ∈ H1 |⟨u, v⟩ = 0, ∀v ∈ E0 }. The following theorem for (2.7) was proved by Ma and Wang [17], too Theorem 2.2. Let L : H1 → H be symmetric with a spectrum given by (2.8), and G : H1 → H satisfy the following orthogonal condition:
⟨G(u), u⟩H = 0,
∀u ∈ H1 .
(2.9)
Then one and only one of the following two assertions holds true: (1) there exists a sequence of invariant sets Γn ⊂ E0 of (2.7) such that 0 ̸∈ Γn ,
lim dist(Γn , 0) = 0.
n→∞
(2) The trivial equilibrium point u = 0 of (2.7) is locally asymptotically stable under the H-norm. Furthermore, if (2.7) has no invariant sets in E0 except the trivial one {0}, then u = 0 is globally asymptotically stable. The basic idea of Theorem 2.1 can be demonstrated as follows [17,18]. Near λ = λ0 , the flows of (2.1) in a small neighborhood of u = 0 is squeezed into an m-dimensional center manifold which can be expressed by a function h = h(x, λ), where x ∈ Rm . The flow in the center manifold has the same topological structure as the flow in Rm of the equations: dx dt
= Lλ x + PG(x, h(x, λ)),
(2.10)
where Lλ = Lλ |E0 , E0 is the subspace of Rm on which all eigenvalues of Lλ possess nonnegative real parts at λ = λ0 and P is the projection. It suffices then to consider the dynamic bifurcation for (2.10). When λ < λ0 , near x = 0 the linear term Lλ x determines the dynamical behavior of (2.10). Therefore the flow converges to x = 0, when λ = λ0 , assuming that x = 0 is locally asymptotically stable for (2.10). Then when λ − λ0 > 0 is sufficiently small, (2.10) can be rewritten as dx dt
= PG(x, h(x, 0)) + Lλ x + K (x, λ),
(2.11)
where K (x, λ) = PG(x, h(x, λ)) − PG(x, h(x, 0)) → 0, as λ → 0. Hence the flow of (2.11) is a superposition of the flow of dx dt
= PG(x, h(x, 0))
(2.12)
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and the flow of dx
= Lλ x + K (x, λ). (2.13) dt The flow of (2.13) is outward from x = 0 and the flow of (2.12) is inward to x = 0. Near x = 0, the linear term dominates the flow and for x far away from x = 0, the nonlinear term PG(x, h(x, 0)) dominates the flow structure. Thus the inward flow and the out flow squeeze an attractor near x = 0. 3. Mathematical setting We shall discuss the attractor bifurcation of (1.1) and (1.2) in the following two cases: Case 1. Case with odd solutions. In this case, we look for solutions of (1.1) and (1.2), which are odd functions with respect to x. Let
∫
π
v dx = 0 , (−π , π )|v(−x) = −v(x), −π ∫ π v dx = 0 , H = v ∈ L2 (−π , π )|v(−x) = −v(x), H1 =
v∈
4 Hper
−π 4 Hper
here is the Sobolev space with periodic boundary conditions as given in (1.1). In this case, let Lλ = −A + λB : H1 → H and G(u, λ) : H1 → H be two maps defined by
∂ 4u , ∂ x4
Au =
Bu = −
∂ 2u , ∂ x2
G(u, λ) = −λu
∂u , ∂x
(3.1)
where u ∈ H1 . Case 2. General case. In this case, we look for solution without the oddness assumption. Let
∫ π 4 v ∈ Hper (−π , π )| v dx = 0 , −π ∫ π 2 ˜ H = v ∈ L (−π , π )| v dx = 0 .
˜1 = H
−π
˜ 1 → H˜ and G(u, λ) : H˜ 1 → H˜ be two maps defined by In this case, let Lλ = −A + λB : H ∂ 4u , ∂ x4
Au =
Bu = −
∂ 2u , ∂ x2
G(u, λ) = −λu
∂u , ∂x
(3.2)
˜ 1. where u ∈ H ˜ Thus, in both cases, problem (1.1) can be rewritten into the following abstract form in H (or H) du
= Lλ u + G(u, λ), dt u(x, 0) = ϕ(x).
(3.3)
4. Main results In this section, we first give an attractor bifurcation theorem for the Kuramoto–Sivashinsky equation in the space H consisting of odd periodic functions, i.e. under Case 1 in Section 3. Theorem 4.1. For the Kuramoto–Sivashinsky equation (1.1) with constraint (1.2), we have the following assertions: (1) For λ ≤ λ0 = 1, u = 0 is a globally asymptotically stable equilibrium point of the equation. (2) For 4 > λ > λ0 = 1, the Kuramoto–Sivashinsky equation bifurcates from (u, λ) = (0, λ0 ) to an attractor Aλ , and Aλ has the homotopy type of the zero-dimensional sphere S 0 . (3) The bifurcation attractor in (2) is
Aλ = {u± 1 sin x + h.o.t.},
for 4 > λ > λ0 = 1, and near λ0 = 1,
√ −1 where u± (λ − 1)(4 − λ), and h.o.t. is a high order term of |u1 |. 1 = ±4λ Proof. We shall apply Theorems 2.1 and 2.2 to prove this problem, together with the center manifold reduction. Consider the equation: Lλ u = β(λ)u, where Lλ is defined by (3.1).
(4.1)
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It is easy to see that the eigenvalues and eigenvectors of the linear operator Lλ : H1 → H are given by βn (λ) = n2 (λ− n2 ), en = sin nx, for n = 1, 2, . . . , that is
β1 = λ − 1, β2 = 4(λ − 4), −∞ ← · · · < β2 < β1 = λ − 1.
β3 = 9(λ − 9), . . . ;
It is obviously
< 0, if λ < λ0 = 1 Reβ1 (λ) = = 0, if λ = λ0 = 1 > 0, if λ > λ0 = 1 βj (λ0 ) < 0, ∀j ≥ 2.
(4.2)
So βn (λ) satisfy conditions (2.5) and (2.6) with m = 1 at λ0 = 1. Obviously, the operator G(u, λ) : H1 → H defined by (3.1) satisfies G(u, λ) = −λuux = o(‖u‖α ),
for some 0 ≤ α < 1,
so condition (2.4) is satisfied. By the periodic boundary condition in (2.7) we have
(G(u), u) = −λ
∫
π
1 u2 ux dx = − u3 |π−π = 0, 3 −π
where (·, ·) is an inner product in H. So G(u, λ) satisfies the orthogonal condition (2.9) in Theorem 2.2. At λ0 = 1 Eq. (3.3) has no invariant sets in E0 = span{sin x}. In fact, for any δ ̸= 0, take any δ sin x ∈ E0 , we obtain G(δ sin x, λ0 ) = G(δ sin x) = −λ0 δ 2 sin x cos x 1 = − λ0 δ 2 sin 2x ̸∈ E0 2
for any δ ̸= 0,
then by Theorem 2.2, we know that u = 0 is globally asymptotically stable for du dt
= Lλ0 u + G(u, λ0 ).
So assertion (1) and assertion (2) in Theorem 4.1 follow from Theorem 2.1. We reduce the equation ∑∞ to its center manifold near λ0 . Let u ∈ H, and u = j=1 uj sin x, so H = E0 ⊕ E0⊥ ,
u = u1 sin x +
∞ −
uj sin jx,
j =2
where u1 sin x ∈ E0 , Set
∑∞
j =2
uj sin jx ∈ E0⊥ .
u = u1 sin x + Φ ,
Φ : E0 → E0⊥ near u = 0,
where Φ is a center manifold function. Let P1 : H → E0 , P2 : H → E0⊥ , so we have P1 ut = P1 Lλ u + P1 G(u),
P2 ut = P2 Lλ u + P2 G(u)
and
(ut , sin x) = (Lλ u, sin x) + (G(u), sin x), then by computing we have
π
∂u = π (λ − 1)u1 + ∂t
π
∫
G(u) sin xdx,
−π
where Lλ sin x = (λ − 1) sin x. So we may obtain the following bifurcation equation:
∂ u1 1 = (λ − 1)u1 + ∂t π Let us compute Φ .
∫
π −π
G(u) sin xdx.
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Set Lλ2 = Lλ |E0 : E0⊥ → E0⊥ , it follows from (3.73) in Ref. [17] that
Φ = −(Lλ2 )−1 P2 G(u1 sin x) + h.o.t. This implies that Lλ2 Φ = −G2 G(u1 sin x) + h.o.t.
(4.3)
and
−P2 G(u1 sin x) = P2 {(u1 sin x)(u1 cos x)} λ λ = P2 u21 sin 2x = u21 sin 2x. 2
2
The center manifold function is
Φ (x) = Φ2 sin 2x + Φ3 sin 3x + · · · . By (4.3), it follows that
λ
Lλ2 Φ =
2
sin 2x + h.o.t..
It is easy to know that Lλ sin 2x = β2 sin 2x,
(Lλ )−1 sin 2x =
1
β2
sin 2x,
thus we infer that
Φ=
λ 2
u21 (Lλ2 )−1 sin 2x + h.o.t. =
λu21 sin 2x + h.o.t.. 2β2
The direct calculation shows that π
λu21 G u1 sin x + sin 2x + h.o.t. sin xdx π −π 2β2 ∫ π λ λu21 λu21 =− u1 sin x + sin 2x + h.o.t. u1 cos x + cos 2x sin xdx + h.o.t. π −π 2β2 β2 ∫ λu31 λu31 λ2 u41 λ π u21 sin x cos x + sin 2x cos x + sin x cos 2x + =− sin 2x cos 2x sin xdx + h.o.t. π −π 2β2 β2 2β22 ∫ λu31 λ π λu31 2 2 sin x cos 2x + sin 2x dx + h.o.t. =− π −π β2 4β2 ∫ λ2 u31 π 1 2 2 =− 2 sin x cos 2x + sin 2x dx + h.o.t. 2π β2 −π 2 2 3 λ u1 =− + h.o.t. 4β2 λ2 u31 = + h.o.t. 4 · 4(λ − 4) 1
∫
hence, by the discussion above, we have the following conclusion:
∂ u1 λ2 u31 = (λ − 1)u1 + + h.o.t.. ∂t 4 · 4(λ − 4)
(4.4)
It is obvious that the stead state solutions of (4.4) are solutions of the following equation:
(λ − 1)u1 +
λ2 u31 = 0, 4 · 4(λ − 4)
which implies that u1 = 0,
or u1 = ±4λ−1
(λ − 1)(4 − λ) + h.o.t.,
for 4 > λ > λ0 = 1
are steady state solutions of Eq. (4.4). Thus, (4.2) in Theorem 4.1 is verified. So the proof of Theorem 4.1 is complete.
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Second, we consider the attractor bifurcation for the Kuramoto–Sivashinsky equation in the general case without the ˜ consisting of oddness assumption, i.e. under Case 2 in Section 3. Namely, we study the dynamics in the phase spaces H, periodic functions. Theorem 4.2. For the Kuramoto–Sivashinsky equation (1.1) with constraint (1.2), we have the following assertions: (1) For λ ≤ λ0 = 1, u = 0 is a globally asymptotically stable equilibrium point of the equation; (2) For 4 > λ > λ0 = 1, the Kuramoto–Sivashinsky equation bifurcates from (u, λ) = (0, λ0 ) to an attractor Aλ , and Aλ has the homotopy type of the one-dimensional sphere S 2−1 = S 1 ; (3) The bifurcation attractor in (2) is
± ± 2 ± 2 u± 1 sin x + v1 cos x + h.o.t. |(u1 ) + (v1 ) =
Aλ =
16(λ − 1)(4 − λ)
λ2
for 4 > λ > λ0 = 1, and near λ0 = 1 ,
(4.5)
i.e. Aλ consists of only steady states of (1.1) with (1.2). Proof. We shall apply Theorems 2.1 and 2.2 to prove this theorem. ∑ ˜ then u = ∞ Let u ∈ H, j=1 uj (sin jx + cos jx). Let e2n−1 = sin nx, e2n = cos nx, n = 1, 2, 3, . . . , then we consider the
˜ 1 → H˜ , Lλ u = β(λ)u, Lλ e2n−1 = β2n−1 (λ)e2n−1 , Lλ e2n = β2n (λ)e2n . eigenvalues βk (λ) and eigenvectors ek (x) of Lλ : H The eigenvalues are given by β2n−1 (λ) = β2n (λ) = n2 (λ − n2 ),
(4.6)
for any n ≥ 1, and
β1 = β2 = λ − 1, β3 = β4 = 4(λ − 4), −∞ ← · · · < β4 = β3 < β2 = β1 = λ − 1.
β5 = β6 = 9(λ − 9), . . .
It is obvious that
< 0, Reβ1 (λ) = Reβ2 (λ) = 0, > 0, βj (λ0 ) < 0,
if λ < λ0 = 1, if λ = λ0 = 1, if λ > λ0 = 1;
(4.7)
∀j ≥ 3.
(4.8)
So βn (λ) satisfy conditions (2.5) and (2.6) with m = 2 at λ0 = 1, and G(u, λ) = −λuux = o(‖u‖Hα ),
for some 0 ≤ α < 1.
˜ 1 → H˜ defined by (3.2) satisfies the orthogonal condition (2.9) in Theorem 2.2. Obviously, the operator G(u, λ) : H And at λ0 = 1 Eq. (3.3) has no invariant sets in E0 = span{sin x, cos x}. In fact, for any δ ̸= 0, take any δ sin x ∈ E0 , we have G(δ sin x, λ0 ) = G(δ sin x) = −λδ sin x(δ sin x)′
λ = λδ 2 sin x cos x = − δ 2 sin 2x ̸∈ E0 , 2
then by Theorem 2.2, we get that u = 0 is globally asymptotically stable for (3.3), assertion (2) in Theorem 4.2 follow from Theorem 2.1. We reduce the equation to its center manifold near λ0 . ˜ = E0 ⊕ E0⊥ , Let H u = u1 sin x + v1 cos x +
∞ − {uj sin jx + vj cos jx}, j =2
˜ where u1 sin x, v1 cos x ∈ E0 , u ∈ H. ˜ → E0 , P2 : H˜ → E0⊥ , so we have Let P1 : H P1 ut = P1 Lλ u + P1 G(u),
P2 ut = P2 Lλ u + P2 G(u),
du dt
= Lλ0 u + G(u, λ0 ). So assertion (1) and
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and
(ut , sin x) = (Lλ u, sin x) + (G(u), sin x), (ut , cos x) = (Lλ u, cos x) + (G(u), cos x), ˜ Then by computing it follows that here (·, ·) is an inner product in H. ∫ π ∂ u1 G(u) sin xdx, = π (λ − 1)u1 + ∂t −π ∫ π ∂v1 π G(u) cos xdx, = π (λ − 1)v1 + ∂t −π π
where Lλ sin x = (λ − 1) sin x. So we may obtain the following bifurcation equation:
∂ u1 1 = (λ − 1)u1 + ∂t π ∂v1 1 = (λ − 1)v1 + ∂t π
π
∫
−π ∫ π
G(u) sin xdx,
(4.9)
G(u) cos xdx.
(4.10)
−π
Let u = u1 sin x + v1 cos x + Φ , Φ : E0 → E0⊥ near u = 0, where Φ is the center manifold function. Let us compute Φ . Set Lλ2 = Lλ |E ⊥ : E0⊥ → E0⊥ , it follows from (3.73) in Ref. [17] that 0
λ −1
Φ = −(L2 ) P2 G(u1 sin x + v1 cos x) + h.o.t.. Thus Lλ2 Φ = −P2 G(u1 sin x + v1 cos x) + h.o.t.
= λP2 (u1 sin x + v1 cos x)(u1 sin x + v1 cos x)′ + h.o.t. = λP2 (u1 sin x + v1 cos x)(u1 cos x + v1 sin x) + h.o.t. 2 u1 − v12 = λP2 sin 2x + u1 v1 cos 2x + h.o.t.. 2
Since Lλ sin 2x = β3 sin 2x, (Lλ )−1 sin 2x = β1 sin 2x, (Lλ )−1 cos 2x = β1 cos 2x, which implies that 3 3
Φ=λ
u21 − v12 2β3
sin 2x + λ
u1 v 1
β3
cos 2x + h.o.t..
The direct calculation in (4.9) shows that 1
π
∫
π
∫ λ π u2 − v12 u1 v 1 u1 sin x + v1 cos x + λ 1 sin 2x + λ cos 2x π −π 2β3 β3 ′ 2 2 u1 − v 1 u1 v 1 × u1 sin x + v1 cos x + λ sin 2x + λ cos 2x sin xdx + h.o.t. 2β3 β3 ∫ λ π u1 v 1 u21 − v12 − u1 sin x + v1 cos x + λ sin 2x + λ cos 2x π −π 2β3 β3 u2 − v12 u1 v 1 cos 2x − 2λ sin 2x sin xdx + h.o.t. × u1 cos x − v1 sin x + λ 1 β3 β3 ∫ λ2 u1 π u21 − v12 2v12 − sin x cos 2x − cos x sin 2x π −π β3 β3 u2 − v12 v2 + 1 cos x sin 2x − 1 cos 2x sin x sin xdx + h.o.t. 2β3 β3 2 λ u1 (u21 + v12 ) + h.o.t. 4β3 λ2 u1 (u21 + v12 ) + h.o.t.. 4 · 4(λ − 4)
G(u, λ) sin xdx = −
−π
=
=
= =
(4.11)
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Hence, by the discussion above, it follows that
∂ u1 1 = (λ − 1)u1 + u1 (u21 + v12 ) + h.o.t. ∂t 16(λ − 4)
(4.12)
By the same methods, it is easy to obtain from (4.10) that
∂v1 1 = (λ − 1)v1 + v1 (u21 + v12 ) + h.o.t.. ∂t 16(λ − 4)
(4.13)
It is obvious that the steady state solutions of (4.12) and (4.13) are solutions of the following equations:
(λ − 1)u1 + (λ − 1)v1 +
1 16(λ − 4) 1 16(λ − 4)
u1 (u21 + v12 ) = 0,
v1 (u21 + v12 ) = 0,
which are
(u1 , v1 ) = (0, 0),
or
(u1 , v1 ) | u21 + v12 =
16(λ − 1)(4 − λ)
λ2
,
for 4 > λ > λ0 = 1 and near λ0 = 1, so (4.5) in Theorem 4.2 is proved. Thus, the three assertions in Theorem 4.2 are verified. So the proof of Theorem 4.2 is complete. Acknowledgement We would like to express our gratitude and admiration to Professor T. Ma, whose mathematical talent and illuminating guidance drew our interest to the study of attractor bifurcation for various evolution equations. References [1] A.P. Hoope, R. Grimshaw, Nonlinear instability at the interface between two viscous fluids, Phys. Fluids 28 (1985) 37–45. [2] Y. Kuramoto, T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Progr. Theoret. Phys. 55 (2) (1976) 356–369. [3] G.I. Sivashinsky, On flame propagation under condition of stoichiometry, SIAM J. Appl. Math. 39 (1980) 67–82. [4] G.I. Sivashinsky, Instabilities, Pattern-formation and turbulence in flames, Annu. Rev. Fluid Mech. 15 (1983) 179–199. [5] R. Grimshaw, A.P. Hooper, The non-existence of a certain class of travelling wave solutions of the Kuramoto–Sivashinsky equation, Physica D 50 (1991) 231–238. [6] C. Foias, I. Kukavica, Determing Nodes for the Kuramoto–Sivashinsky equation, J. Dyn. Differ. Equ. 7 (2) (1995) 365–373. [7] X. Liu, Gevrey class regularity and approximate inertial manifolds for the Kuramoto–Sivashinsky equation, Physica D 50 (1991) 135–151. [8] A.H. Khater, R.s. Temsah, Numerical solutions of the generalized Kuramoto–Sivashinsky equation by Chebyshev spectral collocation methods, Comput. Math. Appl. 56 (2008) 1465–1472. [9] B. Nicolaenko, B. Scheurer, R. Temam, Some global dynamical properties of the Kuramoto–Sivashinsky equation: nonlinear stability and attractors, Physica D 16 (1985) 155–183. [10] E. Tadmor, The well-posedness of the Kuramoto–Sivashinsky equation, SIAM J. Math. Anal. 17 (4) (1986) 884–893. [11] R. Temam, X.M. Wang, Estimates on the lowest dimension of inertial manifolds for the Kuramoto–Sivashinsky equation in the general case, Differential Integral Equations 7 (3–4) (1994) 1095–1108. [12] J.W. Cholewa, T. Dlotko, Global attractor for the Cahn–Hilliard system, Bull. Austral. Math. Soc. 49 (1994) 277–292. [13] T. Dlotko, Global attractor for the Cahn–Hilliard equation in H 2 and H 3 , J. Differential Equations 113 (1994) 381–393. [14] J.K. Hale, Asymptotic Behaviour of Dissipative Systems, AMS, Providence RI, 1988. [15] B. Nicolaenko, B. Scheurer, R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. Partial Differential Equations 14 (1989) 245–297. [16] A. Novick-Cohen, L.A. Segel, Nonlinear aspects of the Cahn–Hilliard equation, Physica 10D (1984) 277–298. [17] T. Ma, S.H. Wang, Dynamic bifurcation of nonlinear evolution equations, Chin. Ann. Math. 26 (2) (2005) 185–206. [18] T. Ma, S.H. Wang, Bifurcation Theory and Applications, World Scientific Publishing, London, 2005. [19] Z.P. Wang, C.K. Zhong, Dynamic bifurcation for the generalized Burgers equations, J. Lanzhou Univ. Nat. Sci. 45 (4) (2009) 133–139. [20] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, in: Appl. Math. Sci., vol. 44, Springer-Verlag, 1983. [21] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., in: Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997.
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Dynamical bifurcation for the Kuramoto–Sivashinsky equation✩ Yindi Zhang ∗ , Lingyu Song, Wang Axia School of Science, Chang’an University, Xi’an 710064, PR China College of Science, Xi’an Jiaotong University, Xi’an 710049, PR China
article
info
Article history: Received 9 April 2010 Accepted 27 September 2010 Keywords: Kuramoto–Sivashinsky equation Attractor bifurcation Eigenvalue analysis Center manifold
abstract In this paper, by using the center manifold reduction method, together with the eigenvalue analysis, we made bifurcation analysis for the Kuramoto–Sivashinsky equation, and proved that the Kuramoto–Sivashinsky equation with constraint condition bifurcates an attractor Aλ as λ crossed the first critical value λ0 = 1 under the two cases. Our analysis was based on a new and mature attractor bifurcation theory developed by Ma and Wang (2005) [17,18]. Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved.
1. Introduction The Kuramoto–Sivashinsky equation is a canonical nonlinear evolution equation arising in a variety of physical contexts, e.g. long wave on thin film, long wave on the interface between two viscous fluid [1], unstable drift waves in plasmas, reaction–diffusion systems [2] and flame front instability [3,4], it represents models of pattern formation on unstable flame fronts and thin hydrodynamic films [3,4], the Kuramoto–Sivashinsky equation has thus been studied extensively; see [5–11] for details. Also, for more studies on dynamical properties of the nonlinear equations see [12–16]. We are interested in the following Kuramoto–Sivashinsky equation in one-dimensional space with periodic condition:
∂ 2u ∂u ∂ u ∂ 4u + 4 + λ 2 + λu = 0, ∂t ∂x ∂x ∂x ∂ ju ∂ ju (−π , t ) = j (π , t ), j = 0, 1, 2, 3 j ∂ x ∂x u(0, x) = ϕ(x),
(1.1)
supplemented with the following constraint
∫
π
u(x, t )dx = 0,
(1.2)
−π
where x ∈ Ω = (−π , π ), u : (−π , π ) × [0, ∞) → R1 is a real scalar function. Refs. [17–19] recently developed a new bifurcation theory based on a notion of bifurcation called attractor bifurcation. The main objective of this paper is to conduct bifurcation and stability analysis for the Kuramoto–Sivashinsky equation by using this new attractor bifurcation theory. In the first case, we look for solutions to problem (1.1) with constraint (1.2), which are odd functions with respect to x, i.e., u(−x, t ) = −u(x, t ). We prove that, for λ ≤ λ0 = 1, u = 0 is an asymptotically stable equilibrium point of the Kuramoto–Sivashinsky equation, and for 4 > λ > λ0 = 1, the Kuramoto–Sivashinsky equation bifurcates from (u, λ) = (0, λ0 ) to an attractor Aλ , homologic to S 0 . In the second case, we look for solutions without the oddness assumption, and ✩ Supported by the NSF of China 10971165 and 10971166; and CHD2010JC094.
∗
Corresponding author at: School of Science, Chang’an University, Xi’an 710064, PR China. E-mail address: [email protected] (Y. Zhang).
0362-546X/$ – see front matter Crown Copyright © 2010 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.09.052
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Y. Zhang et al. / Nonlinear Analysis 74 (2011) 1155–1163
we show that, for λ ≤ λ0 = 1, u = 0 is an asymptotically stable equilibrium point of the Kuramoto–Sivashinsky equation, and for 4 > λ > λ0 = 1, the Kuramoto–Sivashinsky equation bifurcates from (u, λ) = (0, λ0 ) to an attractor Aλ , homologic to S 2−1 = S 1 . Moreover, in both cases, we have found that the attractor Aλ consists of the steady state solutions of the bifurcation equation attained by using the center manifold reduction [17]. 2. Preliminaries In this section, we shall recall some results of the dynamic bifurcation of abstract nonlinear evolution equations developed in Ref. [17]. Let H and H1 be two Hilbert spaces, H1 ⊂ H a compact and dense inclusion. Consider the following abstract nonlinear evolution equations given by
du
= Lλ u + G(u, λ), dt u(0) = ϕ,
(2.1)
where u(t ) : [0, ∞) → H is the unknown scalar function, λ ∈ R1 is a system parameter, and Lλ : H1 −→ H a parameterized linear completely continuous field continuously depending on λ ∈ R1 , which satisfies Lλ = −A + Bλ , A : H1 → H , Bλ : H1 → H ,
a sectorial operator, a linear homeomorphism, the parameterized linear compact operator.
(2.2)
We can 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 (see [20,21]). We now assume that the nonlinear terms G(·, λ) : Hα → H for some 0 ≤ α < 1 are a family of parameterized C r (r ≥ 1) bounded operators which continuously depend on the parameter λ ∈ R1 , such that G(u, λ) = o(‖u‖α ),
∀λ ∈ R1 .
(2.3)
Definition 2.1 ([17]). A set Σ ⊂ H is called an invariant set, Σ ⊂ H of (2.1) is said to be an attractor if Σ is compact, and there exists a neighborhood Σ ⊂ H of Σ such that for any ϕ ∈ U we have lim distH (u(t , ϕ), Σ ) = 0.
(2.4)
t →∞
The largest open set U satisfying (2.4) is called the basin of attraction of Σ . Definition 2.2 ([17]). (1) We say that Eq. (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 an attractor bifurcation. (3) If Ωλ are attractors, which are homotopic equivalent to an m-dimensional sphere S m , then the bifurcation is called S m -attractor bifurcation at (0, λ0 ). Let the eigenvalues (counting multiplicity) of Lλ be given by β1 (λ), β2 , (λ), . . . , βk (λ) ∈ C (k ≥ 1), where C is the complex space. Suppose that
< 0, Reβi (λ) = = 0, > 0, Reβi (λ0 ) < 0,
if λ < λ0 , if λ = λ0 , if λ > λ0 .
∀1 ≤ i ≤ m
(2.5)
∀m + 1 ≤ j.
(2.6)
Let the eigenspace of Lλ at λ0 be E0 = 1≤i≤m k {u ∈ H1 (Lλ0 − βi (λ0 )) u = 0, k = 1, 2, . . .}. By (2.5), we know that dim E0 = m. The following attractor bifurcation theorem for the infinite-dimensional system (2.1) was found in [17].
k
Theorem 2.1. Assume that conditions (2.2)–(2.3) and (2.5)–(2.6) hold true, and u = 0 is a locally asymptotically stable equilibrium point of (2.1) at λ = λ0 . Then the following assertions hold true. (1) Eq. (2.1) bifurcates from (u, λ) = (0, λ0 ) an attractor Ωλ for λ > λ0 , with m − 1 ≤ dim Ωλ ≤ m, which is connected as m > 1.
Y. Zhang et al. / Nonlinear Analysis 74 (2011) 1155–1163
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(2) The attractor Ωλ is a sequence of m-dimensional annulus Ak with Ak+1 ⊂ Ak ; especially, if Ωλ is a finite simplistic complex, then Ωλ has the homotopy type of the m − 1-dimensional sphere S m−1 . (3) For any uλ ∈ Ωλ , uλ can be expressed as uλ = vλ + o(‖vλ ‖H1 ),
∀vλ ∈ E0 .
(4) If the number of equilibrium points of (2.1) in Ωλ is finite, then we have the index formula:
−
ind[−(Lλ + G), ui ] =
ui ∈Ωλ
2, 0,
if m = odd if m = even,
where ind(f , x0 ) = deg(f , U , 0), and U is a small neighborhood of x0 . (5) If u = 0 is globally asymptotically stable for (2.1) at λ = λ0 , then for any bounded open set U ⊂ H with Ωλ ⊂ U, there is an ε > 0, such that λ0 < λ < λ0 + ε , the attractor Ωλ attracts U \ Γ , where Γ is the stable manifold of u = 0 with codimension m. In particular, if (2.1) has a global attractor for all λ near λ0 , then the ε here can be chosen independently of U. Consider the equation given by du dt
= Lu + G(u),
(2.7)
where L is symmetric, then all eigenvalues of L are real. Let the eigenvalue βk of L satisfy
βi = 0, βj < 0,
if 1 ≤ i ≤ m if m + 1 ≤ j < ∞.
(2.8)
Set E0 = {u ∈ H1 |Lu = 0} E0⊥ = {u ∈ H1 |⟨u, v⟩ = 0, ∀v ∈ E0 }. The following theorem for (2.7) was proved by Ma and Wang [17], too Theorem 2.2. Let L : H1 → H be symmetric with a spectrum given by (2.8), and G : H1 → H satisfy the following orthogonal condition:
⟨G(u), u⟩H = 0,
∀u ∈ H1 .
(2.9)
Then one and only one of the following two assertions holds true: (1) there exists a sequence of invariant sets Γn ⊂ E0 of (2.7) such that 0 ̸∈ Γn ,
lim dist(Γn , 0) = 0.
n→∞
(2) The trivial equilibrium point u = 0 of (2.7) is locally asymptotically stable under the H-norm. Furthermore, if (2.7) has no invariant sets in E0 except the trivial one {0}, then u = 0 is globally asymptotically stable. The basic idea of Theorem 2.1 can be demonstrated as follows [17,18]. Near λ = λ0 , the flows of (2.1) in a small neighborhood of u = 0 is squeezed into an m-dimensional center manifold which can be expressed by a function h = h(x, λ), where x ∈ Rm . The flow in the center manifold has the same topological structure as the flow in Rm of the equations: dx dt
= Lλ x + PG(x, h(x, λ)),
(2.10)
where Lλ = Lλ |E0 , E0 is the subspace of Rm on which all eigenvalues of Lλ possess nonnegative real parts at λ = λ0 and P is the projection. It suffices then to consider the dynamic bifurcation for (2.10). When λ < λ0 , near x = 0 the linear term Lλ x determines the dynamical behavior of (2.10). Therefore the flow converges to x = 0, when λ = λ0 , assuming that x = 0 is locally asymptotically stable for (2.10). Then when λ − λ0 > 0 is sufficiently small, (2.10) can be rewritten as dx dt
= PG(x, h(x, 0)) + Lλ x + K (x, λ),
(2.11)
where K (x, λ) = PG(x, h(x, λ)) − PG(x, h(x, 0)) → 0, as λ → 0. Hence the flow of (2.11) is a superposition of the flow of dx dt
= PG(x, h(x, 0))
(2.12)
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Y. Zhang et al. / Nonlinear Analysis 74 (2011) 1155–1163
and the flow of dx
= Lλ x + K (x, λ). (2.13) dt The flow of (2.13) is outward from x = 0 and the flow of (2.12) is inward to x = 0. Near x = 0, the linear term dominates the flow and for x far away from x = 0, the nonlinear term PG(x, h(x, 0)) dominates the flow structure. Thus the inward flow and the out flow squeeze an attractor near x = 0. 3. Mathematical setting We shall discuss the attractor bifurcation of (1.1) and (1.2) in the following two cases: Case 1. Case with odd solutions. In this case, we look for solutions of (1.1) and (1.2), which are odd functions with respect to x. Let
∫
π
v dx = 0 , (−π , π )|v(−x) = −v(x), −π ∫ π v dx = 0 , H = v ∈ L2 (−π , π )|v(−x) = −v(x), H1 =
v∈
4 Hper
−π 4 Hper
here is the Sobolev space with periodic boundary conditions as given in (1.1). In this case, let Lλ = −A + λB : H1 → H and G(u, λ) : H1 → H be two maps defined by
∂ 4u , ∂ x4
Au =
Bu = −
∂ 2u , ∂ x2
G(u, λ) = −λu
∂u , ∂x
(3.1)
where u ∈ H1 . Case 2. General case. In this case, we look for solution without the oddness assumption. Let
∫ π 4 v ∈ Hper (−π , π )| v dx = 0 , −π ∫ π 2 ˜ H = v ∈ L (−π , π )| v dx = 0 .
˜1 = H
−π
˜ 1 → H˜ and G(u, λ) : H˜ 1 → H˜ be two maps defined by In this case, let Lλ = −A + λB : H ∂ 4u , ∂ x4
Au =
Bu = −
∂ 2u , ∂ x2
G(u, λ) = −λu
∂u , ∂x
(3.2)
˜ 1. where u ∈ H ˜ Thus, in both cases, problem (1.1) can be rewritten into the following abstract form in H (or H) du
= Lλ u + G(u, λ), dt u(x, 0) = ϕ(x).
(3.3)
4. Main results In this section, we first give an attractor bifurcation theorem for the Kuramoto–Sivashinsky equation in the space H consisting of odd periodic functions, i.e. under Case 1 in Section 3. Theorem 4.1. For the Kuramoto–Sivashinsky equation (1.1) with constraint (1.2), we have the following assertions: (1) For λ ≤ λ0 = 1, u = 0 is a globally asymptotically stable equilibrium point of the equation. (2) For 4 > λ > λ0 = 1, the Kuramoto–Sivashinsky equation bifurcates from (u, λ) = (0, λ0 ) to an attractor Aλ , and Aλ has the homotopy type of the zero-dimensional sphere S 0 . (3) The bifurcation attractor in (2) is
Aλ = {u± 1 sin x + h.o.t.},
for 4 > λ > λ0 = 1, and near λ0 = 1,
√ −1 where u± (λ − 1)(4 − λ), and h.o.t. is a high order term of |u1 |. 1 = ±4λ Proof. We shall apply Theorems 2.1 and 2.2 to prove this problem, together with the center manifold reduction. Consider the equation: Lλ u = β(λ)u, where Lλ is defined by (3.1).
(4.1)
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It is easy to see that the eigenvalues and eigenvectors of the linear operator Lλ : H1 → H are given by βn (λ) = n2 (λ− n2 ), en = sin nx, for n = 1, 2, . . . , that is
β1 = λ − 1, β2 = 4(λ − 4), −∞ ← · · · < β2 < β1 = λ − 1.
β3 = 9(λ − 9), . . . ;
It is obviously
< 0, if λ < λ0 = 1 Reβ1 (λ) = = 0, if λ = λ0 = 1 > 0, if λ > λ0 = 1 βj (λ0 ) < 0, ∀j ≥ 2.
(4.2)
So βn (λ) satisfy conditions (2.5) and (2.6) with m = 1 at λ0 = 1. Obviously, the operator G(u, λ) : H1 → H defined by (3.1) satisfies G(u, λ) = −λuux = o(‖u‖α ),
for some 0 ≤ α < 1,
so condition (2.4) is satisfied. By the periodic boundary condition in (2.7) we have
(G(u), u) = −λ
∫
π
1 u2 ux dx = − u3 |π−π = 0, 3 −π
where (·, ·) is an inner product in H. So G(u, λ) satisfies the orthogonal condition (2.9) in Theorem 2.2. At λ0 = 1 Eq. (3.3) has no invariant sets in E0 = span{sin x}. In fact, for any δ ̸= 0, take any δ sin x ∈ E0 , we obtain G(δ sin x, λ0 ) = G(δ sin x) = −λ0 δ 2 sin x cos x 1 = − λ0 δ 2 sin 2x ̸∈ E0 2
for any δ ̸= 0,
then by Theorem 2.2, we know that u = 0 is globally asymptotically stable for du dt
= Lλ0 u + G(u, λ0 ).
So assertion (1) and assertion (2) in Theorem 4.1 follow from Theorem 2.1. We reduce the equation ∑∞ to its center manifold near λ0 . Let u ∈ H, and u = j=1 uj sin x, so H = E0 ⊕ E0⊥ ,
u = u1 sin x +
∞ −
uj sin jx,
j =2
where u1 sin x ∈ E0 , Set
∑∞
j =2
uj sin jx ∈ E0⊥ .
u = u1 sin x + Φ ,
Φ : E0 → E0⊥ near u = 0,
where Φ is a center manifold function. Let P1 : H → E0 , P2 : H → E0⊥ , so we have P1 ut = P1 Lλ u + P1 G(u),
P2 ut = P2 Lλ u + P2 G(u)
and
(ut , sin x) = (Lλ u, sin x) + (G(u), sin x), then by computing we have
π
∂u = π (λ − 1)u1 + ∂t
π
∫
G(u) sin xdx,
−π
where Lλ sin x = (λ − 1) sin x. So we may obtain the following bifurcation equation:
∂ u1 1 = (λ − 1)u1 + ∂t π Let us compute Φ .
∫
π −π
G(u) sin xdx.
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Set Lλ2 = Lλ |E0 : E0⊥ → E0⊥ , it follows from (3.73) in Ref. [17] that
Φ = −(Lλ2 )−1 P2 G(u1 sin x) + h.o.t. This implies that Lλ2 Φ = −G2 G(u1 sin x) + h.o.t.
(4.3)
and
−P2 G(u1 sin x) = P2 {(u1 sin x)(u1 cos x)} λ λ = P2 u21 sin 2x = u21 sin 2x. 2
2
The center manifold function is
Φ (x) = Φ2 sin 2x + Φ3 sin 3x + · · · . By (4.3), it follows that
λ
Lλ2 Φ =
2
sin 2x + h.o.t..
It is easy to know that Lλ sin 2x = β2 sin 2x,
(Lλ )−1 sin 2x =
1
β2
sin 2x,
thus we infer that
Φ=
λ 2
u21 (Lλ2 )−1 sin 2x + h.o.t. =
λu21 sin 2x + h.o.t.. 2β2
The direct calculation shows that π
λu21 G u1 sin x + sin 2x + h.o.t. sin xdx π −π 2β2 ∫ π λ λu21 λu21 =− u1 sin x + sin 2x + h.o.t. u1 cos x + cos 2x sin xdx + h.o.t. π −π 2β2 β2 ∫ λu31 λu31 λ2 u41 λ π u21 sin x cos x + sin 2x cos x + sin x cos 2x + =− sin 2x cos 2x sin xdx + h.o.t. π −π 2β2 β2 2β22 ∫ λu31 λ π λu31 2 2 sin x cos 2x + sin 2x dx + h.o.t. =− π −π β2 4β2 ∫ λ2 u31 π 1 2 2 =− 2 sin x cos 2x + sin 2x dx + h.o.t. 2π β2 −π 2 2 3 λ u1 =− + h.o.t. 4β2 λ2 u31 = + h.o.t. 4 · 4(λ − 4) 1
∫
hence, by the discussion above, we have the following conclusion:
∂ u1 λ2 u31 = (λ − 1)u1 + + h.o.t.. ∂t 4 · 4(λ − 4)
(4.4)
It is obvious that the stead state solutions of (4.4) are solutions of the following equation:
(λ − 1)u1 +
λ2 u31 = 0, 4 · 4(λ − 4)
which implies that u1 = 0,
or u1 = ±4λ−1
(λ − 1)(4 − λ) + h.o.t.,
for 4 > λ > λ0 = 1
are steady state solutions of Eq. (4.4). Thus, (4.2) in Theorem 4.1 is verified. So the proof of Theorem 4.1 is complete.
Y. Zhang et al. / Nonlinear Analysis 74 (2011) 1155–1163
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Second, we consider the attractor bifurcation for the Kuramoto–Sivashinsky equation in the general case without the ˜ consisting of oddness assumption, i.e. under Case 2 in Section 3. Namely, we study the dynamics in the phase spaces H, periodic functions. Theorem 4.2. For the Kuramoto–Sivashinsky equation (1.1) with constraint (1.2), we have the following assertions: (1) For λ ≤ λ0 = 1, u = 0 is a globally asymptotically stable equilibrium point of the equation; (2) For 4 > λ > λ0 = 1, the Kuramoto–Sivashinsky equation bifurcates from (u, λ) = (0, λ0 ) to an attractor Aλ , and Aλ has the homotopy type of the one-dimensional sphere S 2−1 = S 1 ; (3) The bifurcation attractor in (2) is
± ± 2 ± 2 u± 1 sin x + v1 cos x + h.o.t. |(u1 ) + (v1 ) =
Aλ =
16(λ − 1)(4 − λ)
λ2
for 4 > λ > λ0 = 1, and near λ0 = 1 ,
(4.5)
i.e. Aλ consists of only steady states of (1.1) with (1.2). Proof. We shall apply Theorems 2.1 and 2.2 to prove this theorem. ∑ ˜ then u = ∞ Let u ∈ H, j=1 uj (sin jx + cos jx). Let e2n−1 = sin nx, e2n = cos nx, n = 1, 2, 3, . . . , then we consider the
˜ 1 → H˜ , Lλ u = β(λ)u, Lλ e2n−1 = β2n−1 (λ)e2n−1 , Lλ e2n = β2n (λ)e2n . eigenvalues βk (λ) and eigenvectors ek (x) of Lλ : H The eigenvalues are given by β2n−1 (λ) = β2n (λ) = n2 (λ − n2 ),
(4.6)
for any n ≥ 1, and
β1 = β2 = λ − 1, β3 = β4 = 4(λ − 4), −∞ ← · · · < β4 = β3 < β2 = β1 = λ − 1.
β5 = β6 = 9(λ − 9), . . .
It is obvious that
< 0, Reβ1 (λ) = Reβ2 (λ) = 0, > 0, βj (λ0 ) < 0,
if λ < λ0 = 1, if λ = λ0 = 1, if λ > λ0 = 1;
(4.7)
∀j ≥ 3.
(4.8)
So βn (λ) satisfy conditions (2.5) and (2.6) with m = 2 at λ0 = 1, and G(u, λ) = −λuux = o(‖u‖Hα ),
for some 0 ≤ α < 1.
˜ 1 → H˜ defined by (3.2) satisfies the orthogonal condition (2.9) in Theorem 2.2. Obviously, the operator G(u, λ) : H And at λ0 = 1 Eq. (3.3) has no invariant sets in E0 = span{sin x, cos x}. In fact, for any δ ̸= 0, take any δ sin x ∈ E0 , we have G(δ sin x, λ0 ) = G(δ sin x) = −λδ sin x(δ sin x)′
λ = λδ 2 sin x cos x = − δ 2 sin 2x ̸∈ E0 , 2
then by Theorem 2.2, we get that u = 0 is globally asymptotically stable for (3.3), assertion (2) in Theorem 4.2 follow from Theorem 2.1. We reduce the equation to its center manifold near λ0 . ˜ = E0 ⊕ E0⊥ , Let H u = u1 sin x + v1 cos x +
∞ − {uj sin jx + vj cos jx}, j =2
˜ where u1 sin x, v1 cos x ∈ E0 , u ∈ H. ˜ → E0 , P2 : H˜ → E0⊥ , so we have Let P1 : H P1 ut = P1 Lλ u + P1 G(u),
P2 ut = P2 Lλ u + P2 G(u),
du dt
= Lλ0 u + G(u, λ0 ). So assertion (1) and
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Y. Zhang et al. / Nonlinear Analysis 74 (2011) 1155–1163
and
(ut , sin x) = (Lλ u, sin x) + (G(u), sin x), (ut , cos x) = (Lλ u, cos x) + (G(u), cos x), ˜ Then by computing it follows that here (·, ·) is an inner product in H. ∫ π ∂ u1 G(u) sin xdx, = π (λ − 1)u1 + ∂t −π ∫ π ∂v1 π G(u) cos xdx, = π (λ − 1)v1 + ∂t −π π
where Lλ sin x = (λ − 1) sin x. So we may obtain the following bifurcation equation:
∂ u1 1 = (λ − 1)u1 + ∂t π ∂v1 1 = (λ − 1)v1 + ∂t π
π
∫
−π ∫ π
G(u) sin xdx,
(4.9)
G(u) cos xdx.
(4.10)
−π
Let u = u1 sin x + v1 cos x + Φ , Φ : E0 → E0⊥ near u = 0, where Φ is the center manifold function. Let us compute Φ . Set Lλ2 = Lλ |E ⊥ : E0⊥ → E0⊥ , it follows from (3.73) in Ref. [17] that 0
λ −1
Φ = −(L2 ) P2 G(u1 sin x + v1 cos x) + h.o.t.. Thus Lλ2 Φ = −P2 G(u1 sin x + v1 cos x) + h.o.t.
= λP2 (u1 sin x + v1 cos x)(u1 sin x + v1 cos x)′ + h.o.t. = λP2 (u1 sin x + v1 cos x)(u1 cos x + v1 sin x) + h.o.t. 2 u1 − v12 = λP2 sin 2x + u1 v1 cos 2x + h.o.t.. 2
Since Lλ sin 2x = β3 sin 2x, (Lλ )−1 sin 2x = β1 sin 2x, (Lλ )−1 cos 2x = β1 cos 2x, which implies that 3 3
Φ=λ
u21 − v12 2β3
sin 2x + λ
u1 v 1
β3
cos 2x + h.o.t..
The direct calculation in (4.9) shows that 1
π
∫
π
∫ λ π u2 − v12 u1 v 1 u1 sin x + v1 cos x + λ 1 sin 2x + λ cos 2x π −π 2β3 β3 ′ 2 2 u1 − v 1 u1 v 1 × u1 sin x + v1 cos x + λ sin 2x + λ cos 2x sin xdx + h.o.t. 2β3 β3 ∫ λ π u1 v 1 u21 − v12 − u1 sin x + v1 cos x + λ sin 2x + λ cos 2x π −π 2β3 β3 u2 − v12 u1 v 1 cos 2x − 2λ sin 2x sin xdx + h.o.t. × u1 cos x − v1 sin x + λ 1 β3 β3 ∫ λ2 u1 π u21 − v12 2v12 − sin x cos 2x − cos x sin 2x π −π β3 β3 u2 − v12 v2 + 1 cos x sin 2x − 1 cos 2x sin x sin xdx + h.o.t. 2β3 β3 2 λ u1 (u21 + v12 ) + h.o.t. 4β3 λ2 u1 (u21 + v12 ) + h.o.t.. 4 · 4(λ − 4)
G(u, λ) sin xdx = −
−π
=
=
= =
(4.11)
Y. Zhang et al. / Nonlinear Analysis 74 (2011) 1155–1163
1163
Hence, by the discussion above, it follows that
∂ u1 1 = (λ − 1)u1 + u1 (u21 + v12 ) + h.o.t. ∂t 16(λ − 4)
(4.12)
By the same methods, it is easy to obtain from (4.10) that
∂v1 1 = (λ − 1)v1 + v1 (u21 + v12 ) + h.o.t.. ∂t 16(λ − 4)
(4.13)
It is obvious that the steady state solutions of (4.12) and (4.13) are solutions of the following equations:
(λ − 1)u1 + (λ − 1)v1 +
1 16(λ − 4) 1 16(λ − 4)
u1 (u21 + v12 ) = 0,
v1 (u21 + v12 ) = 0,
which are
(u1 , v1 ) = (0, 0),
or
(u1 , v1 ) | u21 + v12 =
16(λ − 1)(4 − λ)
λ2
,
for 4 > λ > λ0 = 1 and near λ0 = 1, so (4.5) in Theorem 4.2 is proved. Thus, the three assertions in Theorem 4.2 are verified. So the proof of Theorem 4.2 is complete. Acknowledgement We would like to express our gratitude and admiration to Professor T. Ma, whose mathematical talent and illuminating guidance drew our interest to the study of attractor bifurcation for various evolution equations. References [1] A.P. Hoope, R. Grimshaw, Nonlinear instability at the interface between two viscous fluids, Phys. Fluids 28 (1985) 37–45. [2] Y. Kuramoto, T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Progr. Theoret. Phys. 55 (2) (1976) 356–369. [3] G.I. Sivashinsky, On flame propagation under condition of stoichiometry, SIAM J. Appl. Math. 39 (1980) 67–82. [4] G.I. Sivashinsky, Instabilities, Pattern-formation and turbulence in flames, Annu. Rev. Fluid Mech. 15 (1983) 179–199. [5] R. Grimshaw, A.P. Hooper, The non-existence of a certain class of travelling wave solutions of the Kuramoto–Sivashinsky equation, Physica D 50 (1991) 231–238. [6] C. Foias, I. Kukavica, Determing Nodes for the Kuramoto–Sivashinsky equation, J. Dyn. Differ. Equ. 7 (2) (1995) 365–373. [7] X. Liu, Gevrey class regularity and approximate inertial manifolds for the Kuramoto–Sivashinsky equation, Physica D 50 (1991) 135–151. [8] A.H. Khater, R.s. Temsah, Numerical solutions of the generalized Kuramoto–Sivashinsky equation by Chebyshev spectral collocation methods, Comput. Math. Appl. 56 (2008) 1465–1472. [9] B. Nicolaenko, B. Scheurer, R. Temam, Some global dynamical properties of the Kuramoto–Sivashinsky equation: nonlinear stability and attractors, Physica D 16 (1985) 155–183. [10] E. Tadmor, The well-posedness of the Kuramoto–Sivashinsky equation, SIAM J. Math. Anal. 17 (4) (1986) 884–893. [11] R. Temam, X.M. Wang, Estimates on the lowest dimension of inertial manifolds for the Kuramoto–Sivashinsky equation in the general case, Differential Integral Equations 7 (3–4) (1994) 1095–1108. [12] J.W. Cholewa, T. Dlotko, Global attractor for the Cahn–Hilliard system, Bull. Austral. Math. Soc. 49 (1994) 277–292. [13] T. Dlotko, Global attractor for the Cahn–Hilliard equation in H 2 and H 3 , J. Differential Equations 113 (1994) 381–393. [14] J.K. Hale, Asymptotic Behaviour of Dissipative Systems, AMS, Providence RI, 1988. [15] B. Nicolaenko, B. Scheurer, R. Temam, Some global dynamical properties of a class of pattern formation equations, Comm. Partial Differential Equations 14 (1989) 245–297. [16] A. Novick-Cohen, L.A. Segel, Nonlinear aspects of the Cahn–Hilliard equation, Physica 10D (1984) 277–298. [17] T. Ma, S.H. Wang, Dynamic bifurcation of nonlinear evolution equations, Chin. Ann. Math. 26 (2) (2005) 185–206. [18] T. Ma, S.H. Wang, Bifurcation Theory and Applications, World Scientific Publishing, London, 2005. [19] Z.P. Wang, C.K. Zhong, Dynamic bifurcation for the generalized Burgers equations, J. Lanzhou Univ. Nat. Sci. 45 (4) (2009) 133–139. [20] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, in: Appl. Math. Sci., vol. 44, Springer-Verlag, 1983. [21] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd ed., in: Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997.