Comptes Rendus
no. 5
Mathematical Physics
Geometric dissipation in kinetic equations
[Dissipation géométrique dans les équations cinétiques]

Présenté par : Philippe G. Ciarlet

Darryl D. Holm1, 2 ; Vakhtang Putkaradze3, 4 ; Cesare Tronci1, 5
1 Department of Mathematics, Imperial College London, London SW7 2AZ, UK
2 Computer and Computational Science Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
3 Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA
4 Institute for Theoretical Physics, Universität Köln, Zuplicher Str. 77, 50968 Köln, Germany
5 TERA Foundation for Oncological Hadrontherapy, 11 V. Puccini, Novara 28100, Italy
Comptes Rendus. Mathématique, Volume 345 (2007) no. 5, pp. 297-302.

Une nouvelle approche est proposée pour modeliser la dissipation dans les équations cinétiques. Cette approche produit une structure à double crochet dans l'espace des phases qui aboutit aux équations cinétiques d'une dynamique coadjointe après transformations canoniques. L'exemple de Vlasov admet alors des solutions pour une seule particule. Ces solutions sont réversibles ; l'entropie totale est un Casimir et elle est donc préservée.

A new symplectic variational approach is developed for modeling dissipation in kinetic equations. This approach yields a double bracket structure in phase space which generates kinetic equations representing coadjoint motion under canonical transformations. The Vlasov example admits measure-valued single-particle solutions. Such solutions are reversible. The total entropy is a Casimir, and thus it is preserved.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.07.001
Darryl D. Holm 1, 2 ; Vakhtang Putkaradze 3, 4 ; Cesare Tronci 1, 5

1 Department of Mathematics, Imperial College London, London SW7 2AZ, UK
2 Computer and Computational Science Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
3 Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA
4 Institute for Theoretical Physics, Universität Köln, Zuplicher Str. 77, 50968 Köln, Germany
5 TERA Foundation for Oncological Hadrontherapy, 11 V. Puccini, Novara 28100, Italy 
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Darryl D. Holm; Vakhtang Putkaradze; Cesare Tronci. Geometric dissipation in kinetic equations. Comptes Rendus. Mathématique, Volume 345 (2007) no. 5, pp. 297-302. doi : 10.1016/j.crma.2007.07.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.07.001/

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