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 Cβ > 0 whenever β ≥ 3/2. It is also shown that, if the bound Cβ 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 c3/2 = 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.
All Science Journal Classification (ASJC) codes
- Global existence
- Optimal bounds
- Spherical symmetry