Boltzmann Equation with Cutoff Rutherford Scattering Cross Section Near Maxwellian

The well-known Rutherford differential cross section, denoted by $ d\Omega/d\sigma$, corresponds to a two body interaction with Coulomb potential. It leads the logarithmically divergence of momentum transfer (or transport section) which is described $$\int_{{\mathbb S}^2} (1-\cos\theta) \frac{d\Omega}{d\sigma} d\sigma\sim \int_0^{\pi} \theta^{-1}d\theta. $$ Here $\theta$ deviation angle in scattering event. Due screening effect, physically one can assume that $\theta_{\min}$ order magnitude smallest angles for still be regarded as scattering. Under ad hoc cutoff $\theta \geq \theta_{\min}$ on angle, L. D. Landau derived new equation \cite{landau1936transport} weakly interacting gas now referred Fokker-Planck-Landau or equation. In present work, we establish unified framework justify Landau's formal derivation and so-called approximation problem proposed \cite{alexandre2004landau} close-to-equilibrium regime. Precisely, (i). prove global well-posedness Boltzmann section perhaps most singular kernel both relative velocity angle. (ii). global-in-time error estimate between solutions equations logarithm accuracy, consistent famous logarithm. Key ingredients into proofs these results include complete coercivity linearized collision operator, uniform spectral gap novel linear-quasilinear method.