Finite Larmor radius regime: Collisional setting and fluid models

The subject matter of this paper concerns the derivation of fluid limits for gyro-kinetic models. The arguments apply for any collision kernel satisfying the usual conservations (mass, momentum, kinetic energy) and possessing a production entropy sign. We describe the set of equilibria in terms of several moments, we determine the average collision invariants, and we write the associated macroscopic equations and the entropy inequality.

Reduced fluid models for self-propelled particles interacting through alignment

The asymptotic analysis of kinetic models describing the behavior of particles interacting through alignment is performed. We will analyze the asymptotic regime corresponding to large alignment frequency where the alignment effects are dominated by the self-propulsion and friction forces. The former hypothesis leads to a macroscopic fluid model due to the fast averaging in velocity, while the second one imposes a fixed speed in the limit, and thus a reduction of the dynamics to a sphere in the velocity space. The analysis relies on averaging techniques successfully used in the magnetic confinement of charged particles. The limiting particle distribution is supported on a sphere, and therefore we are forced to work with measures in velocity. As for the Euler-type equations, the fluid model comes by integrating the kinetic equation against the collision invariants and its generalizations in the velocity space. The main difficulty is their identification for the averaged alignment kernel in our functional setting of measures in velocity.

A DISCRETE BOLTZMANN EQUATION BASED ON HEXAGONS

We develop the theory of a Boltzmann equation which is based on a hexagonal discretization of the velocity space. We prove that such a model contains all the basic features of classical kinetic theory, like collision invariants, H-theorem, equilibrium solutions, features of the linearized problem etc. This theory includes the infinite as well as finite hexagonal grids which may be used for numerical purposes.

GENERALIZED DISCRETE VELOCITY MODELS

Starting from a mesoscopic principle of moment conservation, discrete Boltzmann collision operators Jh are constructed, which both converge to bounded collision operators JΩ and have the same collision invariants as the original Boltzmann collision operator J. The crucial point of this construction is the application of a weak formulation of the gain operator to remove the post-collision velocities from it as well as the development of moment conserving integration formulas for the approximation of surface integrals over the unit sphere. Finally two applications for the discrete operators are presented.

A NEW DISCRETIZED MODEL IN NONLINEAR KINETIC THEORY—THE SEMICONTINUOUS BOLTZMANN EQUATION

This paper deals with the mathematical modeling and analysis of a new model of the Boltzmann equation with a finite number of velocity moduli, but with a continuous dependence on the velocity directions. The mathematical model is derived in the first part of the paper. Then the analysis of the equilibrium Maxwellian state is dealt with in the second part of the paper with the purpose of showing that the space of collision invariants is the correct one.