  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.
