## Abstract

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 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 of_{0} 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 ||f_{0}o||£ = 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.

Original language | English (US) |
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Pages (from-to) | 3177-3207 |

Number of pages | 31 |

Journal | Indiana University Mathematics Journal |

Volume | 57 |

Issue number | 7 |

DOIs | |

State | Published - 2008 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Keywords

- Global existence
- Optimal bounds
- Spherical symmetry