Boltzmann equation with cutoff Rutherford scattering cross section near Maxwellian

TL;DR

This paper rigorously justifies the Landau equation as an approximation to the Boltzmann equation with Coulomb-like interactions, establishing global well-posedness and error estimates in the close-to-equilibrium regime.

Contribution

It provides the first comprehensive proof of global solutions and error bounds for the Boltzmann equation with Rutherford scattering cross section near Maxwellian, bridging Boltzmann and Landau models.

Findings

Proved global well-posedness for Boltzmann with Rutherford cross section.

Established a logarithmic error estimate between Boltzmann and Landau solutions.

Developed new coercivity and spectral gap estimates for the collision operator.

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.