article

Title: On the relativistic Vlasov-Poisson system

Authors: M. K.-H. Kiessling and A. S. Tahvildar-Zadeh

Issue: Volume 57 (2008), Issue 7, 3177-3208

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 $\mathfrak{L}^{\beta}$ norm is below a critical constant $C_{\beta} > 0$ whenever $\beta \geq \tfrac{3}{2}$. It is also shown that, if the bound $C_{\beta}$ on the $\mathfrak{L}^{\beta}$ norm of $f_0$ is replaced by a bound $C > C_{\beta}$, any $\beta \in (1,\infty)$, then classical initial data exist which lead to a blow-up in finite time. The sharp value of $C_{\beta}$ is computed for all $\beta \in (1,\tfrac{3}{2}]$, with the results $C_{\beta} = 0$ for $\beta \in (1,\tfrac{3}{2})$ and $C_{3/2} = \tfrac{3}{8}(\tfrac{15}{16})^{1/3}$ (when $\|f_0\|_{\mathfrak{L}^1} = 1$), while for all $\beta > \tfrac{3}{2}$ upper and lower bounds on $C_{\beta}$ are given which coincide as $\beta \downarrow \tfrac{3}{2}$. Thus, the $\mathfrak{L}^{3/2}$ bound is optimal in the sense that it cannot be weakened to an $\mathfrak{L}^{\beta}$ bound with $\beta < \tfrac{3}{2}$, 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.$