Steady states of the Vlasov-Maxwell system

The Vlasov-Maxwell system models collisionless plasma. Solutions are considered that depend on one spatial variable, $x$, and two velocity variables, $v_1$ and $v_2$. As $x\rightarrow - \infty$ it is required that the phase space densities of particles approach a prescribed function, $F\left (v_1,v_2\right )$, and all field components approach zero. It is assumed that $F\left (v_1,v_2\right ) = 0$ if $v_1 \leq W_1$, where $W_1$ is a positive constant. An external magnetic field is prescribed and taken small enough so that no particle is reflected ($v_1$ remains positive). The main issue is to identify the large-time behavior; is a steady state approached and, if so, can it be identified from the time independent Vlasov-Maxwell system? The time-dependent problem is solved numerically using a particle method, and it is observed that a steady state is approached (on a bounded $x$ interval) for large time. For this steady state, one component of the electric field is zero at all points, the other oscillates without decay for $x$ large; in contrast the magnetic field tends to zero for large $x$. Then it is proven analytically that if the external magnetic field is sufficiently small, then (a reformulation of) the steady problem has a unique solution with $B \rightarrow 0$ as $x \rightarrow +\infty$. Thus the “downstream” condition, $B \rightarrow 0$ as $x\rightarrow + \infty$, is used to identify the large time limit of the system.