On the relativistic Vlasov-Poisson system

The Cauchy problem is revisited for the so-called relativistic Vlasov-Poisson system in the attractive case, originally studied by Glassey and Schaeffer in 1985. It is proved that a unique global classical solution exists whenever the positive, integrable initial datum f₀ is spherically symmetric, compactly supported in momentum space, vanishes on characteristics with vanishing angular momentum, and its £^β norm is below a critical constant C_β > 0 whenever β ≥ 3/2. It is also shown that, if the bound C_β on the £^β norm of₀ is replaced by a bound C > Cβ, any β ∈ (1, ∞), then classical initial data exist which lead to a blow-up in finite time. The sharp value of Cβ is computed for all β ∈ (1, 3/2], with the results Cβ = 0 for β e (1, 3/2 ) and c_3/2 = 3/8(15/16)^1/3 (when ||f₀o||£ = 1) while for all β > 3/2 upper and lower bounds on Cβ are given which coincide as β ↓ 3/2. Thus, the £^3/2 bound is optimal in the sense that it cannot be weakened to an £^β bound with β 3/2, whatever that bound. A new, non-gravitational physical vindication of the model which (unlike the gravitational one) is not restricted to weak fields, is also given.