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 of long stationary waves, Phil. Mag. (5), 39, 422-443, (1895). The continuous model of the damped forced KdV equation reads as : ∂ t u + α u + u ∂ x u + ∂ xxx u = f . (1) R + u : t × T x − → 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 ¨ odinger equation ,to appear. Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Introduction of the scheme Existence and regularity of the global attractor Well posedness of the scheme Estimation of the Hausdorff dimension of the global attractor When f = 0 and α = 0, the first three conserved quantities are : � � I 0 = u ( t , x ) dx = u 0 ( x ) dx T T � � u 2 ( t , x ) dx = u 2 I 1 = 0 ( x ) dx T T and � � x ( t , x ) dx − 1 u 2 u 3 ( t , x ) dx I 2 = 3 T T � � x ( x ) dx − 1 ( u 0 ) 2 u 3 = 0 ( x ) dx 3 T T We’ll propose a scheme satisfying the properties I 0 and I 1 in the conservative case. Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Introduction of the scheme Existence and regularity of the global attractor Well posedness of the scheme Estimation of the Hausdorff dimension of the global attractor ⇒ ∂ t ( e α t u ) + 1 2 t u ) 2 + ∂ 3 α x ( e α t u ) = e α t f (1) ⇐ 2 ∂ x ( e (2) We discretize (2) as : � e � 2 e α ( n +1)∆ t u n +1 − e α n ∆ t u n 2 ( n +1)∆ t u n +1 + e α α 2 n ∆ t u n +1 + 2 ∂ x ∆ t 2 � e α ( n +1)∆ t u n +1 + e α n ∆ t u n � = e α ( n +1)∆ t + e α n ∆ t ∂ 3 f x 2 2 (3) After multiplication of (3) by e − α ( n +1)∆ t , we obtain : � � 2 � u n +1 + δ u n � u n +1 − δ u n u n +1 + δ 1 2 u n = 1 + δ + 1 + ∂ 3 2 ∂ x f , 2 2 2 ∆ t x where δ = e − α ∆ t . Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Introduction of the scheme Existence and regularity of the global attractor Well posedness of the scheme Estimation of the Hausdorff dimension of the global attractor 1 2 by δ in ordre to have the same As α ∆ t ≪ 1, we replace δ continuous case estimations. To simplify calculus, we take f instead of 1 + δ f . So the scheme considered is : 2 � u n +1 + δ u n � 2 � u n +1 + δ u n � u n +1 − δ u n +1 + ∂ 3 2 ∂ x = f x ∆ t 2 2 (4) This scheme is one ordre. � u ( t + ∆ t ) + δ u ( t ) � 2 u ( t + ∆ t ) − δ u ( t ) + 1 2 ∂ x + ∆ t 2 � u ( t + ∆ t ) + δ u ( t ) � � � − f = 1 ∂ x u 2 − 2 f ∂ 3 ( α. ∆ t ) + o (∆ t 2 ) x 2 4 Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Introduction of the scheme Existence and regularity of the global attractor Well posedness of the scheme Estimation of the Hausdorff dimension of the global attractor We prove that this scheme is stable in L 2 ( T ), uniformly in ∆ t , assuming that α ∆ t ≪ 1. Proposition 1 Assume that α ∆ t is small enough. Then L 2 + (1 − δ n ) 8 � u n � 2 L 2 ≤ δ n � u 0 � 2 α 2 � f � 2 (5) L 2 . Proof : The scalar product of (4) with u n +1 + δ u n in L 2 ( T ) = ⇒ L 2 + ∆ t | < f , u n +1 + δ u n > | � u n +1 � 2 L 2 ≤ δ 2 � u n � 2 Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Introduction of the scheme Existence and regularity of the global attractor Well posedness of the scheme Estimation of the Hausdorff dimension of the global attractor Young inequality = ⇒ � � � � 1 − α ∆ t 1 + α ∆ t L 2 + 2∆ t � u n +1 � 2 δ 2 � u n � 2 α � f � 2 L 2 ≤ L 2 4 4 � � − 1 � � 1 − α ∆ t 1 + α ∆ t α ∆ t small enough = ⇒ δ ≤ 1. So 4 4 L 2 + 4∆ t � u n +1 � 2 L 2 ≤ δ � u n � 2 α � f � 2 L 2 . The discrete Gronwall’s lemma gives then 4 α ∆ t � u n � 2 L 2 ≤ δ n � u 0 � 2 α 2 (1 − δ ) � f � 2 L 2 + (1 − δ n ) L 2 . Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Introduction of the scheme Existence and regularity of the global attractor Well posedness of the scheme Estimation of the Hausdorff dimension of the global attractor We introduce the set √ E = { v ∈ L 2 ( T ) ; ∆ t � v � 2 L 2 < � c } . Remark 1 It follows from Proposition 1 that if u 0 ∈ E and f ∈ L 2 ( T ) √ c α 2 , then u n ∈ E for all n ∈ I N ∗ . ∆ t � f � 2 satisfying 8 L 2 ≤ � We can write (4) as u n +1 + δ u n � u n +1 + δ u n � � u n +1 + δ u n � 2 − δ u n + 1 2 + ∂ 3 2 ∂ x = f 2 2 ∆ t x 2 (6) Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Introduction of the scheme Existence and regularity of the global attractor Well posedness of the scheme Estimation of the Hausdorff dimension of the global attractor If u 0 , u 1 , ..., u n are obtained in E , instead of seeking u n +1 , we seek the quantity u n +1 + δ u n as fixed point of the problem : 2 F ( w ) = δ R u n + ∆ t 2 R f − ∆ t 4 R ∂ x w 2 (7) � � − 1 1 + ∆ t 2 ∂ 3 where R = satisfies the following x estimates : Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV

Introduction Semi-discrete Korteweg de Vries equations Introduction of the scheme Existence and regularity of the global attractor Well posedness of the scheme Estimation of the Hausdorff dimension of the global attractor Lemma 1 There exists a constant c positive such as one has : � 2 � s 3 , ( i ) �R� L ( L 2 , H s ) ≤ ∀ 0 ≤ s ≤ 3 , ∆ t c √ ( ii ) � ∂ x R� L ( L 1 , L 2 ) ≤ ∆ t . u n +1 + δ u n is more regular than the solution u n . 2 Lemma 2 There exists a positive constant K = K ( α − 1 , � u 0 � L 2 , � f � L 2 ) such that : � � u n +1 + δ u n � � K � � ≤ ∆ t . � � 2 H 3 ( T ) Mostafa ABOUNOUH Global attractor for a time discretization of damped forced KdV