We consider an inhomogeneous linear Boltzmann equation, with an external confining potential. The collision operator is a simple relaxation toward a local Maxwellian, therefore without diffusion. We prove the exponential time decay toward the global Maxwellian, with an explicit rate of decay. The methods are based on hypoelliptic methods transposed here to get spectral information. They were in...

The particles model, the collision model, the polynomial development used for the equilibrium distribution, the time discretization and the velocity discretization are factors that let the lattice Boltzmann framework (LBM) far away from its conceptual support: the continuous Boltzmann equation (BE). Most collision models are based on the BGK, single parameter, relaxation-term leading to constan...

f0(v) log f0(v)dv for all t ≥ 0. Assume that in a dilute gas or plasma, particles collide by pairs, due to a repulsive force proportional to 1/r, where r stands for the distance between the two particles. Then if s ∈ (2,∞), the velocity distribution solves the corresponding Boltzmann equation [14, 15]. But if s = 2, the Boltzmann equation is meaningless [14] and is often replaced by the Landau ...

By using the DiPerna and Lions techniques for the nonrelativistic Boltzmann equation, it is shown that there exists a global mild solution to the Cauchy problem for the relativistic Boltzmann equation with the assumptions of the relativistic scattering cross section including some relativistic hard interactions and the initial data satifying finite mass, “inertia”, energy and entropy.

In the last decade or so, the lattice Boltzmann (LB) method has emerged as a new and effective numerical technique of computational fluid dynamics (CFD).(1-5) Modeling of the incompressible Navier-Stokes equation is among many of its wide applications. Indeed, the lattice Boltzmann equation (LBE) was first proposed to simulate the incompressible NavierStokes equations.(1) The incompressible Nav...

We consider the Fokker-Planck-Boltzmann equation consisting of the Boltzmann equation with an additional diffusion term in velocity space to simulate for instance the transport in thermal bath of binary elastic collisional particles, and investigate the (macroscopic) diffusion properties of solutions in large time.