  For a general class of linear collisional kinetic models in the torus,
including in particular the linearized Boltzmann equation for hard spheres, the
linearized Landau equation with hard and moderately soft potentials and the
semi-classical linearized fermionic and bosonic relaxation models, we prove
explicit coercivity estimates on the associated integro-differential operator
for some modified Sobolev norms. We deduce existence of classical solutions
near equilibrium for the full non-linear models associated, with explicit
regularity bounds, and we obtain explicit estimates on the rate of exponential
convergence towards equilibrium in this perturbative setting. The proof are
based on a linear energy method which combines the coercivity property of the
collision operator in the velocity space with transport effects, in order to
deduce coercivity estimates in the whole phase space.
