# American Institute of Mathematical Sciences

June  2018, 11(3): 521-545. doi: 10.3934/krm.2018023

## Landau damping in the multiscale Vlasov theory

 1 École Polytechnique de Montréal, C.P.6079 suc. Centre-ville, Montréal, H3C 3A7, Québec, Canada 2 Mathematical Institute, Faculty of Mathematics, Charles University, Sokolovská 83, 18675 Prague, Czech Republic

* Corresponding author: Miroslav Grmela

Received  March 2017 Revised  August 2017 Published  March 2018

Fund Project: This research was supported by the Natural Sciences and Engineering Research Council of Canada and by the Czech Science Foundation, project no. 17-15498Y.

Vlasov kinetic theory is extended by adopting an extra one particle distribution function as an additional state variable characterizing the micro-turbulence internal structure. The extended Vlasov equation keeps the reversibility, the Hamiltonian structure, and the entropy conservation of the original Vlasov equation. In the setting of the extended Vlasov theory we then argue that the Fokker-Planck type damping in the velocity dependence of the extra distribution function induces the Landau damping. The same type of extension is made also in the setting of fluid mechanics.

Citation: Miroslav Grmela, Michal Pavelka. Landau damping in the multiscale Vlasov theory. Kinetic & Related Models, 2018, 11 (3) : 521-545. doi: 10.3934/krm.2018023
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