Asymptotic Analysis of the Spatially Homogeneous Boltzmann Equation: Grazing Collisions Limit

Abstract

In the present work, we consider the asymptotic problem of the spatially homogeneous Boltzmann equation when almost all collisions are grazing, that is, the deviation angle \(\theta \) of the collision is limited near zero (i.e., \(\theta \le \epsilon \)). We show that by taking the proper scaling to the cross-section which was used in [37], that is, assuming $$\begin{aligned} B^\epsilon ( v-v_{*},\sigma )=2(1-s)|v-v_*|^{\gamma }\epsilon ^{-3}\sin ^{-1}\theta \left( \frac{\theta }{\epsilon }\right) ^{-1-2s}\mathrm {1}_{\theta \le \epsilon }, \end{aligned}$$ where \(\theta = \langle \theta ={\frac{\upsilon -\upsilon _*}{|\upsilon -\upsilon _*|}}.\sigma \rangle , \) the solution \(f^\epsilon \) of the Boltzmann equation with initial data \(f_0\) can be globally or locally expanded in some weighted Sobolev space as $$\begin{aligned} f^\epsilon = f+ O(\epsilon ), \end{aligned}$$ where the function \(f\) is the solution of Landau equation, which is associated with the grazing collisions limit of Boltzmann equation, with the same initial data \(f_0\). This gives the rigorous justification of the Landau approximation in the spatially homogeneous case. In particular, if taking \(\gamma =-3\) and \(s=1-\epsilon \) in the cross-section \(B^\epsilon \), we show that the above asymptotic formula still holds and in this case \(f\) is the solution of Landau equation with the Coulomb potential. Going further, we revisit the well-posedness problem of the Boltzmann equation in the limiting process. We show there exists a common lifespan such that the uniform estimates of high regularities hold for each solution \(f^\epsilon \). Thanks to the weak convergence results on the grazing collisions limit in [37], in other words, we establish a unified framework to establish the well-posedness results for both Boltzmann and Landau equations.