Accéder directement au contenu Accéder directement à la navigation
Pré-publication, Document de travail

On Koopman Operator for Burgers' Equation

Abstract : We consider the flow of Burgers' equation on an open set of (small) functions in $L^2([0,1])$. We derive explicitly the Koopman decomposition of the Burgers' flow. We identify the frequencies and the coefficients of this decomposition as eigenvalues and eigenfunctionals of the Koopman operator. We prove the convergence of the Koopman decomposition for $t>0$ for small Cauchy data, and up to $t=0$ for regular Cauchy data. The convergence up to $t=0$} leads to a `completeness' property for the basis of Koopman modes. We construct all modes and eigenfunctionals, including the eigenspaces involved in geometric multiplicity. This goes beyond the summation formulas provided by (Page & Kerswell, 2018), where only one term per eigenvalue was given. A numeric illustration of the Koopman decomposition is given and the Koopman eigenvalues compared to the eigenvalues of a Dynamic Mode Decomposition (DMD).
Type de document :
Pré-publication, Document de travail
Liste complète des métadonnées

https://hal-univ-paris13.archives-ouvertes.fr/hal-03225356
Contributeur : Karim Boualem <>
Soumis le : mercredi 12 mai 2021 - 14:19:23
Dernière modification le : mercredi 2 juin 2021 - 03:28:29

Lien texte intégral

Identifiants

  • HAL Id : hal-03225356, version 1
  • ARXIV : 2007.01218

Citation

Mikhael Balabane, Miguel A Mendez, Sara Najem. On Koopman Operator for Burgers' Equation. 2021. ⟨hal-03225356⟩

Partager

Métriques

Consultations de la notice

11