Optimal ${\mathfrak{L}}^{\beta }$-Control for the Global Cauchy Problem of the Relativistic Vlasov-Poisson System

Abstract

Recently, M.K.-H. Kiessling and A.S. Tahvildar-Zadeh proved that a unique global classical solution to the relativistic Vlasov-Poisson system exists whenever the positive, integrable initial datum is spherically symmetric, compactly supported in momentum space, vanishes on characteristics with vanishing angular momentum, and for $\beta \ge 3/2$ has ${\mathfrak{L}}^{\beta }$-norm strictly below a positive, critical value ${\mathcal{C}}_{\beta }$. Everything else being equal, data leading to finite time blow-up can be found with ${\mathfrak{L}}^{\beta }$-norm surpassing ${\mathcal{C}}_{\beta }$ for any $\beta >1$, with ${\mathcal{C}}_{\beta }>0$ if and only if $\beta \ge 3/2$. In their paper, the critical value for $\beta =3/2$ is calculated explicitly while the value for all other $\beta$ is merely characterized as the infimum of a functional over an appropriate function space. In this work, the existence of minimizers is established, and the exact expression of ${\mathcal{C}}_{\beta }$ is calculated in terms of the famous Lane-Emden functions. Numerical computations of the ${\mathcal{C}}_{\beta }$ are presented along with some elementary asymptotics near the critical exponent $3/2$.

24 pages, 2 figures Refereed and accepted for publication in Transport Theory and Statistical Physics