Periodic solutions of the 1D Vlasov–Maxwell system with boundary conditions

We study the 1D Vlasov–Maxwell system with time‐periodic boundary conditions in its classical and relativistic form. We are mainly concerned with existence of periodic weak solutions. We shall begin with the definitions of weak and mild solutions in the periodic case. The main mathematical difficulty in dealing with the Vlasov–Maxwell system consist of establishing L ∞ estimates for the charge and current densities. In order to obtain this kind of estimates, we impose non‐vanishing conditions for the incoming velocities, which assure a finite lifetime of all particles in the computational domain ]0, L [. The definition of the mild solution requires Lipschitz regularity for the electro‐magnetic field. It would be enough to have a generalized flow but the result of DiPerna Lions ( Invent. Math . 1989; 98 : 511–547) does not hold for our problems because of boundary conditions. Thus, in the first time, the Vlasov equation has to be regularized. This procedure leads to the study of a sequence of approximate solutions. In the same time, an absorption term is introduced in the Vlasov equation, which guarantees the uniqueness of the mild solution of the regularized problem. In order to preserve the periodicity of the solution, a time‐averaging vanishing condition of the incoming current is imposed: 1 \def\d{{\rm d}}\def\incdist#1#2{\int_{0}^{T}\d t\int_{v_{x}#10}\int_{v_{y}}v_xg_{#2}(t,v_x,v_y)\,\d v}$$\incdist{>}{0}+\incdist{<}{L}=0$$\nopagenumbers\end where g 0 , g L are incoming distributors 2\documentclass{article}\usepackage{amsfonts}\begin{document}\pagestyle{empty}$$f(t,0,v_{x},v_{y})=g_{0}(t,v_{x},v_{y}),\quad t \in \mathbb{R}_{t},\quad v_{x}>0,\quad v_{y} \in \mathbb{R}_{v}$$\end{document}3\documentclass{article}\usepackage{amsfonts}\begin{document}\pagestyle{empty}$$f(t,L,v_{x},v_{y})=g_{L}(t,v_{x},v_{y}),\quad t \in \mathbb{R}_{t},\quad v_{x}<0,\quad v_{y} \in \mathbb{R}_{v}$$\end{document} The existence proof uses the Schauder fixed point theorem and also the velocity averaging lemma of DiPerna and Lions ( Comm. Pure Appl. Math. 1989; XVII : 729–757). In the last section we treat the relativistic case. Copyright © 2000 John Wiley & Sons, Ltd.