Global renormalized solutions to Boltzmann systems modeling mixture gases of monatomic and polyatomic species

Inspired by DiPerna-Lions’ work \cite{Diperna-Lions}, we study the renormalized solutions to the large-data Cauchy problem of the Boltzmann systems modeling mixture gases of monatomic and polyatomic species, in which the distribution functions $f_{\alpha }$ characterized the polyatomic species contain the continuous internal energy variable $I\in {\mathbb{R}}_{+}$. We first construct the smooth approximated problem and establish the corresponding uniform and physically natural bounds. Then, by employing the averaged velocity (-internal energy) lemma, we can show that the weak $L^1$ limit of the approximated solution is exactly a renormalized solution what we required. Moreover, we also justify that the constructed renormalized solution subjects to the entropy inequality.