KdV limit for the Vlasov–Poisson–Landau system

Abstract

We are concerned with the fluid limit to KdV equations for the one‐dimensional Vlasov–Poisson–Landau system that describes the dynamics of ions in plasma with the electron density determined by the self‐consistent electric potential through the so‐called Boltzmann relation. Formally, it is well known that as the Knudsen number the Vlasov–Poisson–Landau system in the compressible scaling converges to the Euler–Poisson equations which further under the Gardner–Morikawa transformation converge to the KdV equations as the parameter . Our goal of this paper is to construct smooth solutions of the correspondingly rescaled Vlasov–Poisson–Landau system over an arbitrary finite time interval that can converge uniformly to smooth solutions of the KdV equations as and simultaneously under an extra condition . Moreover, the explicit rate of convergence in is also obtained. The proof is established by an appropriately chosen scaling and an intricate weighted energy method through the macro–micro decomposition around local Maxwellians. We design a ‐‐dependent high‐order energy functional to capture the singularity of such fluid limit problem.