Spectrum Analysis of Some Kinetic Equations

We analyze the spectrum structure of some kinetic equations qualitatively by using semigroup theory and linear operator perturbation theory. The models include the classical Boltzmann equation for hard potentials with or without angular cutoff and the Landau equation with γ≧ - 2. As an application, we show that the solutions to these two fundamental equations are asymptotically equivalent (mod time decay rate t^{- 5 / 4}) as t→ ∞ to that of the compressible Navier–Stokes equations for initial data around an equilibrium state.