High order approximation for the Boltzmann equation without angular cutoff

Abstract

In order to solve the Boltzmann equation numerically, in the present work, we propose a new model equation to approximate the Boltzmann equation without angular cutoff. Here the approximate equation incorporates Boltzmann collision operator with angular cutoff and the Landau collision operator. As a first step, we prove the well-posedness theory for our approximate equation. Then in the next step we show the error estimate between the solutions to the approximate equation and the original equation. Compared to the standard angular cutoff approximation method, our method results in higher order of accuracy.

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

“…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.…”

“…In this subsection, we specify some challenges and innovations of the current work and compare it with the paper [27]. In [27], equation ( 14) was studied in the case of hard potentials. In the current paper, we study equation ( 14) with soft potentials.…”

“…That is, we do not need L 2 when do L 1 moment estimate. Therefore in the case of hard potential, [27] follows…”

“…Global existence. In the hard potential case in [27], coercivity estimate (16) gains some weight thanks to γ > 0. As a consequence, the dissipation functional D(f (t)) is stronger than the energy functional E(f (t)), and we have…”

“…which gives the uniform-in-time estimate of energy E(f (t)) E(f 0 ). By a standard continuity argument, [27] utilizes the uniform-in-time a priori estimate and localin-time well-posedness to get global-in-time well-posedness.…”