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# 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
Domaine :
Liste complète des métadonnées

https://hal-univ-paris13.archives-ouvertes.fr/hal-03225356
Contributeur : Karim Boualem Connectez-vous pour contacter le contributeur
Soumis le : mercredi 12 mai 2021 - 14:19:23
Dernière modification le : mercredi 27 octobre 2021 - 16:13:03

### 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⟩

### Métriques

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