Optimal -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 β⩾3/2 has -norm strictly below a positive, critical value . Everything else being equal, data leading to finite time blow-up can be found with -norm surpassing for any β>1, with if and only if β⩾3/2. In their paper, the critical value for β=3/2 is calculated explicitly while the value for all other β 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 is calculated in terms of the famous Lane-Emden functions. Numerical computations of the are presented along with some elementary asymptotics near the critical exponent 3/2.