OAI.RVM.global_classical_solution

The theorem states that for every initial datum d=(f₀,E₀,B₀) of a relativistic Vlasov–Maxwell system in three space dimensions that is Admissible, there is a solution (f,E,B) with three properties. Admissible means: f₀ is a C^∞, compactly supported, nonnegative function of position and momentum (x,v) in ℝ³×ℝ³; E₀ and B₀ are C^∞ vector fields on ℝ³ all of whose iterated derivatives are bounded, and both lie in L²; div E₀ equals the charge density ρ₀(x)=∫f₀(x,v)dv; and div B₀=0. The solution, with f a real function of (t,x,v) and E,B vector fields of (t,x), is Classical for d: f, E and B are C¹ on t≥0, f is nonnegative, E and B are continuous in time into L² on t≥0, for each finite time horizon T the support of f(t,·) for t in [0,T] lies in a single compact set of phase space, the initial conditions f(0)=f₀, E(0)=E₀, B(0)=B₀ hold, and for every t≥0 the equations hold pointwise (with one-sided time derivatives at t=0). These are the Vlasov equation ∂ₜf+v̂·∇ₓf+(E+v̂×B)·∇ᵥf=0 with relativistic velocity v̂=v/√(1+|v|²), the equations ∂ₜE−curl B=−j and ∂ₜB+curl E=0 where j(x)=∫f(x,v)v̂ dv, and the constraints div E=ρ and div B=0 with ρ=∫f dv. Second, the solution is SmoothOnFiniteHorizons: f, E and B are C^∞ on each slab 0≤t≤T. Third, it is unique among classical solutions: any other Classical solution for d agrees with it, for f, E and B, at every time t≥0.