The global well-posedness and Newtonian limit for the relativistic Boltzmann equation in a periodic box

In this paper, we study the Newtonian limit for relativistic Boltzmann equation in a periodic box ${\mathbb{T}}^3$. We first establish the global-in-time mild solutions of relativistic Boltzmann equation with uniform-in-$\mathfrak{c}$ estimates and time decay rate. Then we rigorously justify the global-in-time Newtonian limits from the relativistic Boltzmann solutions to the solution of Newtonian Boltzmann equation in $L_p^1{L}_x^{\mathrm{\infty }}$. Moreover, if the initial data of Newtonian Boltzmann equation belong to $W^{1,\mathrm{\infty }}({\mathbb{T}}^3\times {\mathbb{R}}^3)$, based on a decomposition and $L^2-L^{\mathrm{\infty }}$ argument, the global-in-time Newtonian limit is proved in $L_{x,p}^{\mathrm{\infty }}$. The convergence rates of Newtonian limit are obtained both in $L_p^1{L}_x^{\mathrm{\infty }}$ and $L_{x,p}^{\mathrm{\infty }}$.