Long Wave Asymptotics for the Vlasov–Poisson–Landau Kinetic Equation

The work is devoted to some mathematical problems of dynamics of collisional plasma. The difficulty is that in plasma case we have at least three different length scales: Debye radius r D , mean free path 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 > > 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. An approximate asymptotic formula for the oscillating electric field near this limit is derived from VLPE. Still the limiting equations, which are used in many publications by physicists, can lead in some cases to correct results for the distribution function. Both these conclusions are confirmed by more explicit analysis of the linearized Vlasov–Poisson equation. We also study the well-posedness of limiting kinetic equations and the corresponding criterion in the class of weakly inhomogeneous initial data. It is shown that the collisional effects do not play an important role in this problem. In particular the equations are well-posed in the case of small deviations from equilibrium, as it was already known for related collisionless models.