Vlasov-Maxwell equation

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The Vlasov-Maxwell equations equations are given by

$f_t + \underline{v} \cdot \nabla_x f + (E + \underline{v} \times B) \cdot \nabla_v f = 0$

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

$\nabla \cdot B = 0$

$E_t = \nabla \times B - 4 \pi j$

$B_t = - \nabla \times E$

where $f (t, x, v)$ is the particle density (and is non-negative), $j(t,x) = \int \underline{v} f(t,x,v) dv$ is the current density, $\rho(t,x) = \int f(t,x,v) dv$ 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.