The Boltzmann equation for a multi-species mixture close to global equilibrium

We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the $3$-dimensional torus. The ultimate aim of this work is to obtain existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in $L^1_vL^\infty_x(m)$, where $m\sim (1+|v|^k)$ is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann operator. Then we develop an $L^2-L^\infty$ theory \textit{\`a la Guo} for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (\textit{e.g.} Carleman representation, Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the $L^1_vL^\infty_x$ framework is dealt with for any $k>k_0$, recovering the optimal physical threshold of finite energy $k_0=2$ in the particular case of a multi-species hard spheres mixture with same masses.