Asymptotic solutions of the Vlasov–Poisson–Landau kinetic equations

The paper is devoted to analytical and numerical study of solutions to the Vlasov–Poisson–Landau kinetic equations (VPLE) for distribution functions with typical length L such that $ \varepsilon = r_D/L \ll 1 $, where $ r_D $ stands for the Debye radius. It is also assumed that the Knudsen number $ {\rm K n} = l/L = O(1)$, where $ l $ denotes the mean free pass of electrons. We use the standard model of plasma of electrons with a spatially homogeneous neutralizing background of infinitely heavy ions. The initial data is always assumed to be close to neutral. We study an asymptotic behavior of the system for small $ \varepsilon > 0$. It is known that the formal limit of VPLE at $ \varepsilon = 0$ does not describe a rapidly oscillating part of the electric field. Our aim is to study the behavior of the “true” electric field near this limit. We consider the problem with standard isotropic in velocities Maxwellian initial conditions, and show that there is almost no damping of these oscillations in the collisionless case. An approximate formula for the electric field is derived and then confirmed numerically by using a simplified Bathnagar–Gross–Krook (BGK-type) model of Vlasov–Poisson–Landau equation (VPLE). Another class of initial conditions that leads to strong oscillations having the amplitude of order $ O(1/\varepsilon) $ is also considered. Numerical solutions of that class are studied for different values of parameters $ \varepsilon $ and $ {\rm K n} $.