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On solutions of the Vlasov-Poisson-Landau equations for slowly varying in space initial data

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FKTW03 - Frontiers in kinetic equations for plasmas and collective behaviour

The talk 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 ε = rD/L << 1, where  rD  stands for the Debye radius. It is also assumed that the Knudsen number Kn = 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 ε > 0. It is known that the formal limit of VPLE at ε = 0 does not describe a rapidly oscillating part of the electrical field [1]. Our aim is to study the behavior of the “true” electrical 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 electrical field is derived and then confirmed numerically by using a simplified BGK -type model of VPLE . Another class of initial conditions that leads to strong oscillations having the amplitude of order O(1/ε) is also considered. A formal asymptotic expansion of solution in powers of ε is constructed. Numerical solutions of that class are studied for different values of parameters ε and Kn. The work is based on papers [1], [2]. [1]  Bobylev A.V.,  Potapenko I.F.,  Long  wave asymptotics for Vlasov-Poisson-Landau kinetic equation, J.Statist. Phys., 175 (2019), 1-18.[2]  Bobylev A.V.,  Potapenko I.F.,  On solutions of the Vlasov-Poisson-Landau equations for slowly varying in space initial data (submitted to Kinet. Relat. Models in Jan. 2022).

This talk is part of the Isaac Newton Institute Seminar Series series.

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