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arXiv · 1011.2265

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

Brent Young
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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 $β\ge 3/2$ has $\mathfrak{L}^β$-norm strictly below a positive, critical value $\mathcal{C}_β$. Everything else being equal, data leading to finite time blow-up can be found with $\mathfrak{L}^β$-norm surpassing $\mathcal{C}_β$ for any $β>1$, with $\mathcal{C}_β>0$ if and only if $β\geq 3/2$. 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 $\mathcal{C}_β$ is calculated in terms of the famous Lane-Emden functions. Numerical computations of the $\mathcal{C}_β$ are presented along with some elementary asymptotics near the critical exponent ${3}/{2}$.

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Brent Young. 2010-11-10. Optimal $\mathfrak{L}^β$-Control for the Global Cauchy Problem of the Relativistic Vlasov-Poisson System. https://doi.org/10.1080/00411450.2011.651032

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