  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. {In
particular, we fully exploit the fact that the collision operator is preserving
both momentum and kinetic energy.} 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.
