Boltzmann Equation with Cutoff Rutherford Scattering Cross Section Near Maxwellian

The well-known Rutherford differential cross section, denoted by dΩ/dσ, corresponds to a two body interaction with Coulomb potential. This leads to the logarithmic divergence of the momentum transfer (or the transport cross section), which is described by ∫S2(1-cosθ)dΩdσdσ∼∫0πθ-1dθ.Here θ is the deviation angle in the scattering event. Due to a screening effect, physically one can assume that θmin is the order of magnitude of the smallest angles for which the scattering can still be regarded as Coulomb scattering. Under ad hoc cutoff θ≧θmin on the deviation angle, L. D. Landau derived a new equation in [17] for the weakly interacting gas which is now referred to as the Fokker-Planck-Landau or Landau equation. In the present work, we establish a unified framework to justify Landau's formal derivation in [17] and the so-called Landau approximation problem proposed in [5] in the close-to-equilibrium regime. Precisely, (i) we prove global well-posedness of the Boltzmann equation with cutoff Rutherford cross section which is perhaps the most singular kernel both in relative velocity and deviation angle; (ii) we prove a global-in-time error estimate between solutions to the Boltzmann and Landau equations with logarithm accuracy, which is consistent with the famous Coulomb logarithm. Key ingredients in the proofs of these results include a complete coercivity estimate of the linearized Boltzmann collision operator, a uniform spectral gap estimate and a novel linear-quasilinear energy method.