Solutions to the non-cutoff Boltzmann equation in the grazing limit

Abstract

It is known that in the parameter range $-2\le \gamma <-2s$, a spectral gap does not exist for the linearized Boltzmann operator without cutoff, but it does for the linearized Landau operator. This paper is devoted to the understanding of the formation of a spectral gap in this range through the grazing limit. Precisely, we study the Cauchy problems of these two classical collisional kinetic equations around global Maxwellians in a torus and establish the following results which are uniform in the vanishing grazing parameter $\epsilon$: (i) spectral-gap-type estimates for the collision operators; (ii) global existence of small-amplitude solutions for initial data with low regularity; (iii) propagation of regularity in both space and velocity variables, as well as velocity moments without smallness; (iv) global-in-time asymptotics of the Boltzmann solution toward the Landau solution at the rate $O(\epsilon )$; (v) continuous transition of decay structure of the Boltzmann operator to the Landau operator. In particular, the result in part (v) captures the uniform-in-$\epsilon$ transition of intrinsic optimal time-decay structures of solutions and reveals how the spectrum of the linearized non-cutoff Boltzmann equation in the mentioned parameter range changes continuously under the grazing limit.