Triggered by the fact that, in the hydrodynamic limit, several different
kinetic equations of physical interest all lead to the same
Navier-Stokes-Fourier system, we develop in the paper an abstract framework
which allows to explain this phenomenon. The method we develop can be seen as a
significant improvement of known approaches for which we fully exploit some
structural assumptions on the linear and nonlinear collision operators as well
as a good knowledge of the Cauchy theory for the limiting equation. We adopt a
perturbative framework in a Hilbert space setting and first develop a general
and fine spectral analysis of the linearized operator and its associated
semigroup. Then, we introduce a splitting adapted to the various regimes
(kinetic, acoustic, hydrodynamic) present in the kinetic equation which allows,
by a fixed point argument, to construct a solution to the kinetic equation and
prove the convergence towards suitable solutions to the Navier-Stokes-Fourier
system. Our approach is robust enough to treat, in the same formalism, the case
of the Boltzmann equation with hard and moderately soft potentials, with and
without cut-off assumptions, as well as the Landau equation for hard and
moderately soft potentials in presence of a spectral gap. New well-posedness
and strong convergence results are obtained within this framework. In
particular, for initial data with algebraic decay with respect to the velocity
variable, our approach provides the first result concerning the strong
Navier-Stokes limit from Boltzmann equation without Grad cut-off assumption or
Landau equation. The method developed in the paper is also robust enough to
apply, at least at the linear level, to quantum kinetic equations for
Fermi-Dirac or Bose-Einstein particles