High order approximation for the Boltzmann equation without angular cutoff under moderately soft potentials

Abstract

The Boltzmann equation has an angular singularity inherent in the long range interaction between molecules. Angular singularity produces many difficulties in both theoretical and numerical study of Boltzmann equation. As a result, many rely on angular cutoff models to approximate the Boltzmann equation. Cutoff models have no angular singularity and can be solved by existent numerical methods. However, as the singularity goes stronger, the proportion of grazing collision becomes larger, which renders ineffectiveness of the cutoff Boltzmann equation. Based on the theoretical result that the limit of grazing collision is Landau operator, we propose to add a suitably scaled Landau operator to the cutoff equation to form a new approximate equation. This new approximate equation was studied in [27] in the case of hard potentials and its approximation accuracy is proved to be one order higher than that of angular cutoff models, which is a significant improvement in numerical computing. In this work, under moderately soft potentials, we establish the well-posedness theory of the new approximate equation, prove regularity propagation of its solution, check the high order accuracy. The new approximate equation can be solved by existing numerical methods, and this work may provide a theoretical foundation and a new direction to high order numerical methods for solving the Boltzmann equation.