Global Newtonian limit for the Relativistic Boltzmann Equation near Vacuum

Abstract

We study the Cauchy Problem for the relativistic Boltzmann equation with near Vacuum initial data. Unique global in time "mild" solutions are obtained uniformly in the speed of light parameter $c \ge 1$. We furthermore prove that solutions to the relativistic Boltzmann equation converge to solutions of the Newtonian Boltzmann equation in the limit as $c\to\infty$ on arbitrary time intervals $[0,T]$, with convergence rate $1/c^{2-\epsilon}$ for any $\epsilon \in(0,2)$. This may be the first proof of unique global in time validity of the Newtonian limit for a Kinetic equation.