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 f(0) is spherically symmetric, compactly supported in momentum space, vanishes on character is tics with vanishing angular momentum, and its L-beta norm is below a critical constant C-beta > 0 whenever beta >= 3/2. It is also shown that, if the bound C on the L-beta norm of f(0) is replaced by a bound C > C-beta, any beta is an element of (1, infinity), then classical initial data exist which lead to a blow-up in finite time. The sharp value of C-beta is computed for all beta is an element of (1, 3/2], with the results C-beta = 0 for beta is an element of (1,3/2) and C-3/2 = 3/8(15/16)(1/3) (when parallel to f(0)parallel to(L1) = 1), while for all beta > 3/2 upper and lower bounds on C-beta are given which coincide as beta down arrow 3/2. Thus, the L-3/2 bound is optimal in the sense that it cannot be weakened to an L-beta bound with beta < 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.