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# Communications in Mathematical Sciences

## Volume 14 (2016)

### Number 3

### Lagrangian averaged gyrokinetic-waterbag continuum

Pages: 593 – 626

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n3.a1

#### Author

#### Abstract

In this paper, we first present the derivation of the anisotropic Lagrangian averaged gyrowaterbag continuum (LAGWBC-$\alpha$) equations. The gyrowaterbag (short for *gyrokinetic-waterbag*) continuum can be viewed as a special class of exact weak solution of the gyrokinetic-Vlasov equation, allowing us to reduce the 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, and Ratiu [27, 28], Marsden and Shkoller [32, 33] for the mean motion of ideal incompressible flows, extended to barotropic compressible flows by Bhat et al. [13] and some supplementary approximations for the electrical potential fluctuations. Regarding 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 accurately describing 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, which we call the *isotropic* LAGWBC-$\alpha$ equations, by using quasilinear PDE type methods and elliptic regularity estimates for several operators.

#### Keywords

gyrokinetic-waterbag model, gyrowaterbag model, well-posed problem, gyrokinetic turbulence, Lagrangian averaged models, Eulerian and Lagrangian variational principles, gyrokinetic-Vlasov equations, multi-fluids systems, infinite-dimensional hyperbolic system of conservation laws in several space dimension, magnetically confined fusion plasmas

#### 2010 Mathematics Subject Classification

35F55, 35L65, 35Q35, 35Q83, 76B03, 76F02, 76N10

Published 26 February 2016