A refined estimate of the grazing limit from Boltzmann to Landau operator in Coulomb potential

We derive a refined estimate of the grazing limit from Boltzmann to Landau operator in Coulomb potential in Sobolev space. The estimate is an improvement over the result in He and Yang (2014) in terms of the regularity order. In addition, the resulting order 3 of regularity is reminiscent of and can be seen as a complement of the estimate in Desvillettes (1992). Our result relies on some careful use of Taylor expansion and a delicate analysis of a 7-dimensional change of variable, which is new to our best knowledge. Our result can be employed to relax regularity requirement in well-posedness and stability theory of Landau equation. (C) 2019 Elsevier Ltd. All rights reserved.