The plasma-charge model in a convex domain

Abstract

The aim of this paper is to study the initial-boundary value problems of a Vlasov type system in a convex domain, so called the plasma-charge model, in which there are two kinds of singular sets, one caused by the boundary effect, the other by the heavy point charges. We prove the local existence of classical solutions for the case that the point charges are moving and global existence of classical solutions for the case that the point charges are fixed away from the boundary. The crucial tools are the extended Velocity Lemma for the plasma-charge model and the Pfaffelmoser’s method developed by Hwang and Velázquez (2010 Arch. Ration. Mech. Anal. 195 763–96) and Marchioro et al (2011 Arch. Ration. Mech. Anal. 201 1–26). In the Pfaffelmoser’s argument, a new idea is that the plasma particles can only be close to one of the singular sets during the time interval [ t − δ , t ] with small length δ, which allows us to obtain the global existence for the fixed point charges case by adapting the techniques established by Hwang et al (2013 Discrete Contin. Dyn. Syst. 33 723–37; 2010 Arch. Ration. Mech. Anal. 195 763–96) and Marchioro et al (2011 Arch. Ration. Mech. Anal. 201 1–26) to the corresponding singular sets respectively.