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

TL;DR

This paper determines the exact critical $ ext{L}^eta$-norm thresholds for global existence of solutions to the relativistic Vlasov-Poisson system, using variational methods and Lane-Emden functions, with numerical and asymptotic analysis.

Contribution

It establishes the existence of minimizers and explicitly computes the critical $ ext{L}^eta$-norm values in terms of Lane-Emden functions, advancing understanding of blow-up conditions.

Findings

Exact critical $ ext{L}^eta$-norms are derived for all $eta eq 3/2$.

Numerical computations of the critical values are provided.

Asymptotic behavior near the critical exponent $3/2$ is analyzed.

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 $\beta\geq 3/2$. 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 ${3}/{2}$.