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

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

2010-11-10

arxiv.org/abs/1011.2265v1

Physics

Mathematical Physics

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

Scientific paper

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 ${3}/{2}$.

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