KdV limit for the Vlasov-Poisson-Landau system

Abstract

Two fundamental models in plasma physics are given by the Vlasov-Poisson-Landau system and the compressible Euler-Poisson system which both capture the complex dynamics of plasmas under the self-consistent electric field interactions at the kinetic and fluid levels, respectively. Although there have been extensive studies on the long wave limit of the Euler-Poisson system towards Korteweg-de Vries equations, few results on this topic are known for the Vlasov-Poisson-Landau system due to the complexity of the system and its underlying multiscale feature. In this article, we derive and justify the Korteweg-de Vries equations from the Vlasov-Poisson-Landau system modelling the motion of ions under the Maxwell-Boltzmann relation. Specifically, under the Gardner-Morikawa transformation $$(t,x,v)\to ({\delta }^{\frac{3}{2}}t,{\delta }^{\frac{1}{2}}(x-\sqrt{\frac{8}{3}}t),v)$$ with ${\epsilon }^{\frac{2}{3}}\le \delta \le {\epsilon }^{\frac{2}{5}}$ and $\epsilon >0$ being the Knudsen number, we construct smooth solutions of the rescaled Vlasov-Poisson-Landau system over an arbitrary finite time interval that can converge uniformly to smooth solutions of the Korteweg-de Vries equations as $\delta \to 0$. Moreover, the explicit rate of convergence in $\delta$ is also obtained. The proof is obtained by an appropriately chosen scaling and the intricate weighted energy method through the micro-macro decomposition around local Maxwellians.

68 pages. All comments are welcome. arXiv admin note: text overlap with arXiv:2212.07654