Long wave asymptotics for the Vlasov–Poisson–Landau equation

The work is devoted to some mathematical problems of dynamics of collisional plasma. These problems are related to the general question of different length and time scales in plasma physics. The difficulty is that in plasma case we have at least three different length scales: Debye radius $r_D$, mean free pass $l$ and macroscopic length $L$. This is true even for the simplest model (plasma of electrons with a neutralizing background of infinitely heavy ions), considered in the paper. We study at the formal level of mathematical rigour solutions of the VLPE, having the typical length of the order $l\gg r_D$, and try to clarify some mathematical questions related to corresponding limit. In particular, we study the existence of the limit for electric field and show that, generally speaking, it does not exist because of rapidly oscillating terms. Still the limiting equations, which are used in many publications by physicists, can lead in some cases to correct results for the distribution function. We also study the well-posedness of these equations and formulate the corresponding criterion for different classes of weakly inhomogeneous initial data. It is shown that the situation with well-posedness in our case is qualitively similar to the same problem for Vlasov–Dirac–Benney equation, which was studied in detail in recent publications of Bardos et al.