Well-Posedness and Asymptotics of Grazing Collisions Limit of Boltzmann Equation with Coulomb Interaction

Abstract

In the present work, we consider the well-posedness and asymptotics of grazing collisions limit of the spatially inhomogeneous Boltzmann equation with Coulomb interaction. Under the screening hypothesis on the cross-section $B^\epsilon(z,\sigma)$ of the Boltzmann collision operator, that is, $ B^\epsilon(z, \sigma)=|\log \epsilon|^{-1}|z|^{-3}\theta^{-3}(\sin\theta)^{-1}{1}_{\{\theta\ge \epsilon\}}$, where $\cos\theta= \frac{z}{|z|}\cdot\sigma$, we prove that there exists a common lifespan $T$ such that for any $\epsilon$, the Boltzmann equation admits a unique nonnegative solution $f^\epsilon(t,x,v)$ in the function space $C([0,T]; L^2_x L^2_{2N(4N+5/2)+l}\cap \mathcal{H}^N_l)$ when the initial data belongs to the weighted Sobolev space $L^2_xL^2_{2N(4N+5/2)+l}\cap \mathcal{H}^N_l $ with $N\in \mathbb{N}$ and $l\in \mathbb{R}^+$. Moreover, it is proved that the solution $f^\epsilon$ is uniformly bounded with respect to the parameter $\epsilon$ in the above function spaces. As a consequence, by using the equation and taking the grazing collisions limit $\epsilon\rightarrow0$, we derive the local existence and uniqueness of the nonnegative solution $f$ of the Landau equation with a Coulomb potential corresponding to the same initial data as the one to the Boltzmann equation. In other words, we establish the well-posedness for both Boltzmann and Landau equations with Coulomb interaction in a unified framework. Finally, we show that the limiting process ($\epsilon\rightarrow0$) can be explicitly described by the asymptotic formula $f^\epsilon= f+ O(|\log \epsilon|^{-1})$, which holds in some weighted Sobolev spaces. It gives the first rigorous result on the justification of the Landau approximation for the Coulomb potential in the general spatially inhomogeneous setting.