On the Boltzmann equation with strong kinetic singularity and its grazing limit from a new perspective

Abstract

For the inverse power law potential U(r)=r-p, the Boltzmann kernel has the asymptotic behavior B(v-v∗,σ)∼θ-2-2s|v-v∗|γ as the deviation angle tends to 0. Global well-posedness of the Boltzmann equation with such singular kernels has been built in the parameter range γ>-3,0<s<1 independently by Gressman and Strain (J Am Math Soc 24:771–847, 2011), Alexandre et al. (J Funct Anal 262:915–1010, 2012), triggering many other theoretical developments thereafter. In this work, we consider stronger kinetic singularity and extend the global well-posedness theory to the range γ>-2s-3,0<s<1. This range is optimal by recalling that the dominant part of the Boltzmann operator behaves like the factional Laplace operator (-Δ)s which allows a singularity with exponent -2s-3 in 3-dimensional space. Based on the global well-posedness result, we prove the grazing limit of the Boltzmann equation to the Landau equation as s→1- from a new perspective for any γ>-5 that includes the Coulomb potential γ=-3. As a byproduct, the Landau equation is globally well-posed for any γ>-5.