Vlasov-Maxwell equation

The Vlasov-Maxwell equations equations are given by

$f_{t}+{\underline {v}}\cdot \nabla _{x}f+(E+{\underline {v}}\times B)\cdot \nabla _{\underline {v}}f=0$

$\nabla \cdot E=4\pi \rho$

$\nabla \cdot B=0$

$E_{t}=\nabla \times B-4\pi j$

$B_{t}=-\nabla \times E$

where $f(t,x,{\underline {v}})$ is the particle density (and is non-negative), $j(t,x)=\int {\underline {v}}f(t,x,{\underline {v}})dv$ is the current density, $\rho (t,x)=\int f(t,x,{\underline {v}})dv$ is the charge density, and ${\underline {v}}=v/(1+|v|^{2})^{1/2}$ is the relativistic velocity. The vector fields E(t,x) and B(t,x) represent the electromagnetic field. x and v live in R^3 and t lives in R. This equation is a coupled wave and conservation law system, and models collision-less plasma at relativistic velocities.

Assuming that the particle density remains compactly supported in the velocity domain for all time, GWP in C^1 was proven in GsSr1986b (see also GsSr1986, GsSr1987. An alternate proof of this result is in KlSt2002. A stronger result (which only imposes compact support conditions on the initial data, not on all time) regarding solutions to Vlasov-Maxwell which are purely outgoing (no incoming radiation) is in Cal-p. The velocity demain hypothesis can be removed in the "2.5 dimensional model" where the x_3 dependence is trivial but the v_3 dependence is not [GsScf1990]. Further results are in GsSch1988, Rei1990, Wol1984, Scf1986 The non-relativistic limit of Vlasov-Maxwell is Vlasov-Poisson, in which the electromagnetic field $E+v\times B$ is replaced by $\nabla \Delta ^{-1}4\pi \rho$. Considerably more is known for the existence theory of this equation.