Global attractor for a time discretization of damped forced KdV - - PowerPoint PPT Presentation

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor

Global attractor for a time discretization of damped forced KdV equation

Mostafa ABOUNOUH abounouh@fstg-marrakech.ac.ma

Univ´ ersit´ e Cadi Ayyad Marrakech-Maroc

CANUM , 26-30 Mai 2008

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor

The Korteweg-de Vries equation (KdV equation) models one directional long water waves of small amplitude, propagating in a channel.

• D. J. Korteweg and G. de Vries, On the change of form of

long waves advancing in a rectangular canal, and on new type

• f long stationary waves, Phil. Mag. (5), 39, 422-443, (1895).

The continuous model of the damped forced KdV equation reads as : ∂tu + αu + u∂xu + ∂xxxu = f . (1) u : R+

t × Tx

− → R (t, x) − → u(t, x) periodic with respect to x. α > 0 : the damping parameter, f : the externel force, not depending on t.

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor

We recall that the semigroup associated to (1) possesses a global attractor. J-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time, J. Diff. Eq., 74, pp 369-390, (1988). J-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations, J. Diff. Eq. 110, 356-359, (1994).

• O. Goubet, Asymptotic smoothing effect for weakly damped

forced Korteweg-de Vries equations, Discrete Contin. Dynam. Systems 6 (2000), no. 3, 625–644.

• O. Goubet, R. Rosa Asymptotic smoothing and the global

attractor of a weakly damped KdV equation on the real line ,

• J. Differential Equations 185 (2002), no. 1, 25–53.

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor

To proceed to numerical computations one discretizes in time and space. Into the framwork of infinite dimensional dynamical systems, a challenging issue is to study the system provided by a time discretization, keeping the space variable continuous. This might provide some new insight for large time computations associated to this scheme. We aim to perform such analysis for discrete KdV equation as already done in

• O. Goubet and A. Zahrouni On a time discretization of a

weakly damped forced nolinear Schr¨

• dinger equation,to

appear.

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor Introduction of the scheme Well posedness of the scheme

When f = 0 and α = 0, the first three conserved quantities are : I0 =

• T

u(t, x)dx =

• T

u0(x)dx I1 =

• T

u2(t, x)dx =

• T

u2

0(x)dx

and I2 =

• T

u2

x(t, x)dx − 1

3

• T

u3(t, x)dx =

• T

(u0)2

x(x)dx − 1

3

• T

u3

0(x)dx

We’ll propose a scheme satisfying the properties I0 and I1 in the conservative case.

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor Introduction of the scheme Well posedness of the scheme

(1) ⇐ ⇒ ∂t(eαtu) + 1 2∂x(e

α 2 tu)2 + ∂3

x(eαtu) = eαtf

(2) We discretize (2) as : eα(n+1)∆tun+1 − eαn∆tun ∆t +1 2∂x e

α 2 (n+1)∆tun+1 + e α 2 n∆tun

2 2 + ∂3

x

eα(n+1)∆tun+1 + eαn∆tun 2

• = eα(n+1)∆t + eαn∆t

2 f (3) After multiplication of (3) by e−α(n+1)∆t, we obtain : un+1 − δun ∆t + 1

2∂x

• un+1 + δ

1 2un

2 2 + ∂3

x

un+1 + δun 2

• = 1 + δ

2 f , where δ = e−α∆t.

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor Introduction of the scheme Well posedness of the scheme

As α∆t ≪ 1, we replace δ

1 2 by δ in ordre to have the same

continuous case estimations. To simplify calculus, we take f instead of 1 + δ 2 f . So the scheme considered is : un+1 − δun ∆t +1 2∂x un+1 + δun 2 2 +∂3

x

un+1 + δun 2

• = f

(4) This scheme is one ordre. u(t + ∆t) − δu(t) ∆t + 1 2∂x u(t + ∆t) + δu(t) 2 2 + ∂3

x

u(t + ∆t) + δu(t) 2

• −f = 1

4

• ∂xu2 − 2f
• (α.∆t) + o(∆t2)

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor Introduction of the scheme Well posedness of the scheme

We prove that this scheme is stable in L2(T), uniformly in ∆t, assuming that α∆t ≪ 1. Proposition 1 Assume that α∆t is small enough. Then un2

L2 ≤ δnu02 L2 + (1 − δn) 8

α2f 2

L2.

(5) Proof : The scalar product of (4) with un+1 + δun in L2(T) = ⇒ un+12

L2 ≤ δ2un2 L2 + ∆t| < f , un+1 + δun > |

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor Introduction of the scheme Well posedness of the scheme

Young inequality = ⇒

• 1 − α∆t

4

• un+12

L2 ≤

• 1 + α∆t

4

• δ2un2

L2 + 2∆t

α f 2

L2

α∆t small enough = ⇒ δ

• 1 − α∆t

4 −1 1 + α∆t 4

• ≤ 1. So

un+12

L2 ≤ δun2 L2 + 4∆t

α f 2

L2.

The discrete Gronwall’s lemma gives then un2

L2 ≤ δnu02 L2 + (1 − δn)

4α∆t α2(1 − δ)f 2

L2.

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor Introduction of the scheme Well posedness of the scheme

We introduce the set E = {v ∈ L2(T) ; √ ∆tv2

L2 <

c}. Remark 1 It follows from Proposition 1 that if u0 ∈ E and f ∈ L2(T) satisfying 8 √ ∆tf 2

L2 ≤

cα2, then un ∈ E for all n ∈ I N∗. We can write (4) as un+1 + δun 2 − δun ∆t 2 + ∂3

x

un+1 + δun 2

• + 1

2∂x un+1 + δun 2 2 = f (6)

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor Introduction of the scheme Well posedness of the scheme

If u0, u1, ..., un are obtained in E, instead of seeking un+1, we seek the quantity un+1 + δun 2 as fixed point of the problem : F(w) = δRun + ∆t 2 Rf − ∆t 4 R∂xw 2 (7) where R =

• 1 + ∆t

2 ∂3

x

−1 satisfies the following estimates :

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor Introduction of the scheme Well posedness of the scheme

Lemma 1 There exists a constant c positive such as one has : (i) RL(L2,Hs) ≤ 2 ∆t s

3 ,

∀ 0 ≤ s ≤ 3, (ii) ∂xRL(L1,L2) ≤ c √ ∆t . un+1 + δun 2 is more regular than the solution un. Lemma 2 There exists a positive constant K = K(α−1, u0L2, f L2) such that :

• un+1 + δun

2

• H3(T)

≤ K ∆t .

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor Introduction of the scheme Well posedness of the scheme

Proof of Lemma 2 : (6), Proposition 1 and lemma 1 = ⇒

• un+1 + δun

2

• H3

≤ 1 ∆t unL2 + f L2 +

• ∂x

un+1 + δun 2 2

• L2

≤ 1 ∆t unL2 + f L2 +

• un+1 + δun

2

• L∞
• un+1 + δun

2

• x
• L2

≤ 1 ∆t unL2 + f L2 +

• un+1 + δun

2

• 1

2

L2

• un+1 + δun

2

• 3

2

H1

≤ 1 ∆t unL2 + f L2 + c

• un+1 + δun

2

• 3

2

L2

• un+1 + δun

2

• 1

2

H3

≤ C (f L2, u0L2, α−1)

• 1

∆t +

• un+1 + δun

2

• 1

2

H3

• ≤

1 ∆t C (f L2, u0L2, α−1) + 1 2

• un+1 + δun

2

• H3

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor Introduction of the scheme Well posedness of the scheme

Proposition 2 For f ∈ L2(T) satisfying 8 √ ∆tf 2

L2 ≤

cα2, the equation (4) has a discrete semi-group Sn , n ≥ 1, satisfying : ∀u0 ∈ E , un+1 = Sun = Sn+1u0 is solution of (4). Proof : We establish that if un exists then un+1 is obtained in the following way : If w is a fixed point of F then un+1 = 2w − δun. We perform a fixed point argument for F on F, where F(w) = δRun + ∆t 2 Rf − ∆t 4 R∂xw 2 F =

• v ∈ L2(T) : vL2 ≤ R = 2unL2 + f L2 ≤ 3

√

• c

∆t

1 4

• .

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor Introduction of the scheme Well posedness of the scheme

Using Lemma 1, we have F(w)L2 ≤ δunL2 + ∆t 2 f L2 + c(∆t)

1 2

4 w2

L2

≤ unL2 + 1 2f L2 + c(∆t)

1 2

4 R2 ≤ R 2 + 3c √

• c(∆t)

1 4

4 R. If 3c √

• c(∆t)

1 4

4 < 1 2, then F sends F into F. For the contraction, we have F(w1) − F(w2) = ∆t 4 R∂x [(w1 − w2)(w1 + w2)] F(w1) − F(w2)L2 ≤ c(∆t)

1 2Rw1 − w2L2

≤ 3c √

• c(∆t)

1 4w1 − w2L2.

If 3c √

• c(∆t)

1 4 < 1, then F is a contraction. Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor

Theorem 1 The discrete semi-group S : E − → E defined by Sun = un+1 possesses a global attractor that is a compact subset of H3(T). Proof : Any solution of (4) splits as follows un = v n + w n , ∀n ∈ I N (8) v n+1 − δv n ∆t + ∂3

x

v n+1 + δv n 2

• =

v 0 = u0 (9)        w n+1 − δw n ∆t + ∂3

x

w n+1 + δw n 2

• +

1 2 ∂x un+1 + δun 2 2 = f w 0 = 0 (10)

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor

Proposition 3 The problems (9) and (10) are well posed and the solutions v n and w n satisfy the following results : lim

n− →∞ v nL2 = 0,

(11) and there exists a positive constant c = c (α−1, f L2, u0L2) such that ∂3

xw nL2 ≤

c ∆t . (12)

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor

It follows from the splitting Snu0 = v n + w n that the semi-group splits into a compact mapping and another one that converges towards 0, uniformly on bounded sets, when n goes to the infinity. Applying Theorem I.1.1 in

• R. Temam, Infinite Dimensional Dynamical Systems in

Mechanics and Physics, Springer-Verlag, Second Edition, 1997. we then prove the existence of a global attractor A∆t that is bounded subset of H3(T). To prove the compactness of the attractor, we use the J. Ball argument

• J. Ball, Global attractors for damped semilinear wave
• equations. Partial differential equations and applications,

Discrete Contin. Dyn. Syst. 10 (2004), no. 1-2, 31–52.

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor

We consider the discrete semi-group S defined in section 3. S is differentiable on any point u ∈ A∆t. We consider v = Su ∈ A∆t. Then w = DS(u)h is solution of w − δh ∆t +∂3

x

w + δh 2

• +∂x

w + δh 2 v + δu 2

• = 0

(13) We then have : Proposition 4 A∆t has a finite Hausdorff dimension in H1.

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor

Proof : We deduce from (13) w + ∆t 2 R∂x[w(v + δu)] = δ

• R
• 1 − ∆t

2 ∂3

x

• h − ∆t

2 R∂x[h(v + δu)]

• .

(14) Set Gξ = 1 2∂x[ξ(v + δu)] and U = R

• 1 − ∆t

2 ∂3

x

• .

The equality (14) becomes w + ∆t.RGw = δ (U − ∆t.RG) h. (15)

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor

We consider now an orthonormal family of functions h1, ..., hm in H1(T) and the Gram determinant of w1 = DS(u)h1 , ... , wm = DS(u)hm that we will estimate. We denote by Qm the H1(T) orthogonal projector onto the space spanned by h1, ..., hm. We then have : Gram(Lh1, ... , Lhm) = det(QmLtLQm) Gram(h1 , ... , hm), (16)

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor

To estimate det(QmLtLQm) we use the fact that Lh = w if and only if the equation (15) is valid and we use the following Lemma : Lemma 3 Set Lh = δ (U − ∆t.RG) h, we have : Gram( Lh1, ..., Lhm) ≤ δ2me2∆tTr|RG|. (17) If ∆tRG ≤ 1 2, we have : Gram(Lh1, ... , Lhm) ≤ Gram( Lh1, ..., Lhm)e4∆tTr|RG|. (18)

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor

It follow from inequalities (17) and (18) that Gram(Lh1, ..., Lhm) ≤ e2∆t(−αm+3Tr|RG|) (19) and then that the discrete semi-group Sn contract the m-dimensional infinitesimal volumes if 3Tr|RG| ≤ α.m (20) To bound the trace, we use : Lemma 4 For all Φ ∈ L2(T) we have : RGΦH1 ≤ c (∆t)

5 6 ΦL2.

(21)

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor

We recall that Tr|RG| = max(

m

• k=1

RGΦk), where the maximum is taken on the set of families of m vectors, that are

• rthonormal in H1(T).

Using formula (21), we bound by above the trace as follows : 3Tr|RG| ≤ (22) c (∆t)

5 6 max

m

• k=1

ΦkL2(T) ΦkH1(T)

• ≤

c (∆t)

5 6

m

• k=1

1 k ≤ c (∆t)

5 6 ln(m) Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Existence and regularity of the global attractor Estimation of the Hausdorff dimension of the global attractor

where the maximum is computed over the orthogonal families in H1(T) of cardinal m and where the eigenvalues for the Laplacian operator are ∼ ck2. Hence if m is large enough such that αm ≥ c (∆t)

5 6 ln(m),

then the dimension of A∆t is less than m.

Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV