Future global stability of Maxwell-Jüttner equilibria and vacuum for the massless Boltzmann equation on FLRW spacetimes

Abstract

In this work we study the general relativistic massless Boltzmann equation on Friedmann-Lemaître-Robertson-Walker spacetimes with spatial topology ${\mathbb{T}}^3$ in the linear and decelerated expanding regimes, where the scale factor is $t^{\mathfrak{q}}$ with $\mathfrak{q}\in [0,1]$. The massless Boltzmann equation on these backgrounds admits non-stationary Maxwell-Jüttner equilibria of the form $\exp (-|t^{2\mathfrak{q}}p|)$. For $0\le \mathfrak{q}\le 1$, we prove future global-in-time existence and uniqueness of small perturbations of these equilibria in the case of hard ball interaction without symmetry assumptions. For $0\le \mathfrak{q}<1/3$, we prove that the perturbation -- measured in a suitable $L_p^2$ based energy norm -- decays at the superpolynomial time-decay rate of $t^{-3\mathfrak{q}}\exp (-t^{1-3\mathfrak{q}})$, whereas for $1/3<\mathfrak{q}\le 1$ we obtain the polynomial time-decay rate of $t^{-3\mathfrak{q}}$. In the borderline case $\mathfrak{q}=1/3$, we show the time-decay of $t^{-3\mathfrak{q}-c}$ with a uniform constant $c>0$. Finally, for $\frac{1}{3}<\mathfrak{q}\le 1$, we prove future global-in-time existence and uniqueness of small perturbations of the vacuum solution on ${\mathbb{T}}^3$.

71 pages