Boltzmann equation with cutoff Rutherford scattering cross section near Maxwellian

Abstract

The well-known Rutherford differential cross section, denoted by $ d\Omega/d\sigma$, corresponds to a two body interaction with Coulomb potential. It leads to the logarithmically divergence of the momentum transfer (or the transport cross section) which is described by $$\int_{{\mathbb S}^2} (1-\cos\theta) \frac{d\Omega}{d\sigma} d\sigma\sim \int_0^{\pi} \theta^{-1}d\theta. $$ Here $\theta$ is the deviation angle in the scattering event. Due to screening effect, physically one can assume that $\theta_{\min}$ is the order of magnitude of the smallest angles for which the scattering can still be regarded as Coulomb scattering. Under ad hoc cutoff $\theta \geq \theta_{\min}$ on the deviation angle, L. D. Landau derived a new equation in \cite{landau1936transport} 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 \cite{landau1936transport} and the so-called Landau approximation problem proposed in \cite{alexandre2004landau} 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 Boltzmann and Landau equations with logarithm accuracy, which is consistent with the famous Coulomb logarithm. Key ingredients into 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 method.