This paper deals with a half-space linearized problem for the distribution function of the excitations in a Bose gas close to equilibrium. Existence and uniqueness of the solution, as well as its asymptotic properties are proven for a given energy flow. The problem differs from the ones for the classical Boltzmann and related equations, where the hydrodynamic mass flow along the half-line is constant. Here it is no more constant. Instead we use the energy flow which is constant, but no more hydrodynamic.
The electric potential plays a key role in the
confinement properties of tokamak plasmas, with the subsequent
impact on the performances of fusion reactors. Understanding its
structure in the peripheral plasma -- interacting with solid
materials -- is of crucial importance, since it governs the
boundary conditions for the burning core plasma. This paper aims
at highlighting the dedicated impact of the plasma-wall boundary
layer on this peripheral region. Especially, the physics of
plasma-wall interactions leads to non-linear constraints along the
magnetic field. In this framework, the existence and uniqueness of
the electric potential profile are mathematically investigated.
The working model is two-dimensional in space and time evolving.
We complete the result in  by showing the exponential decay of the perturbation of the laminar solution below the critical Rayleigh number and of the convective solutions above the critical Rayleigh number, in the kinetic framework.
We study a rarefied gas, described by the Boltzmann equation, between two coaxial rotating cylinders in the small Knudsen number regime. When the radius of the inner cylinder is suitably sent to infinity, the limiting evolution is expected to converge to a modified Couette flow which keeps memory of the vanishing curvature of the cylinders ( ghost effect ). In the $1$-d stationary case we prove the existence of a positive isolated $L_2$-solution to the Boltzmann equation and its convergence.
This is obtained by means of a truncated bulk-boundary layer expansion which requires the study of a new Milne problem, and an estimate of the remainder based on a generalized spectral inequality.
A finite Larmor radius approximation is rigourously derived from the Vlasov equation, in the limit of large (and uniform) external magnetic field. Existence and uniqueness of a solution is proven in the stationary frame.
A kinetic chemotaxis model with attractive interaction by quasistationary chemical signalling is considered. The special choice of the turning operator, with velocity jumps biased towards the chemical concentration gradient, permits closed ODE systems for moments of the distribution function of arbitrary order. The system for second order moments exhibits a critical mass phenomeneon. The main result is existence of an aggregated steady state for supercritical mass.