## Section: New Results

### Lagrangian averaged gyrokinetic-waterbag continuum

Participant : Nicolas Besse [correspondent] .

In this paper [26] , we first present the derivation of the anisotropic Lagrangian averaged gyrowaterbag continuum (LAGWBC-$\alpha $) equations. The gyrowaterbag (nickname for gyrokinetic-waterbag) continuum can be viewed as a special class of exact weak solution of the gyrokinetic-Vlasov equation, allowing to reduce this latter into an infinite dimensional set of hydrodynamic equations while keeping its kinetic features such as Landau damping. In order to obtain the LAGWBC-$\alpha $ equations from the gyrowaterbag continuum we use an Eulerian variational principle and Lagrangian averaging techniques introduced by Holm, Marsden, Ratiu and Shkoller for the mean motion of ideal incompressible flows, extended to barotropic compressible flows by Bhat and co-workers and some supplementary approximations for the electrical potential fluctuations. Regarding to the original gyrowaterbag continuum, the LAGWBC-$\alpha $ equations show some additional properties and several advantages from the mathematical and physical viewpoints, which make this model a good candidate for describing accurately gyrokinetic turbulence in magnetically confined plasma. In the second part of this paper we prove local-in-time well-posedness of an approximate version of the anisotropic LAGWBC-$\alpha $ equations, that we call the “isotropic” LAGWBC-$\alpha $ equations, by using quasilinear PDE type methods and elliptic regularity estimates for several operators.