Vlasov-Maxwell equation: Difference between revisions

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:<math>B_t = - \nabla \times E</math>
:<math>B_t = - \nabla \times E</math>


where <math>f(t,x,v)</math> is the particle density (and is non-negative), <math>j(t,x) = \int v f(t,x,v) dv</math> is the current density, <math>\rho(t,x) = \int f(t,x,v) dv</math> is the charge density.  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.  
where <math>f(t,x,v)</math> is the particle density (and is non-negative), <math>j(t,x) = \int \underline{v} f(t,x,v) dv</math> is the current density, <math>\rho(t,x) = \int f(t,x,v) dv</math> is the charge density, and <math>\underline{v} = v / (1 + |v|^2)^{1/2}</math> 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].  
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].  

Revision as of 13:20, 11 July 2007

The Vlasov-Maxwell equations equations are given by

where is the particle density (and is non-negative), is the current density, is the charge density, and 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 is replaced by . Considerably more is known for the existence theory of this equation.