For the inverse power law potential \(U(r)=r^{-p}\) , the Boltzmann kernel has the asymptotic behavior \(B(v-v_{*}, \sigma ) \sim \theta ^{-2-2s} |v-v_{*}|^{\gamma }\) 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 \(\gamma >-3, 0<s<1\) 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 \(\gamma >-2s -3, 0<s<1\) . This range is optimal by recalling that the dominant part of the Boltzmann operator behaves like the factional Laplace operator \((-\Delta )^{s}\) 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 \(s \rightarrow 1^{-}\) from a new perspective for any \(\gamma >-5\) that includes the Coulomb potential \(\gamma =-3\) . As a byproduct, the Landau equation is globally well-posed for any \(\gamma >-5\) .
