Entropy dissipation estimates for the linear Boltzmann operator

TL;DR

This paper establishes an entropy-entropy dissipation inequality for the linear Boltzmann operator, confirming a linear analogue of Cercignani's conjecture and connecting to known inequalities in the grazing collision limit.

Contribution

It proves a new entropy dissipation estimate for the linear Boltzmann operator applicable to physically relevant kernels, including Maxwellian and hard-spheres interactions.

Findings

Proves a linear inequality between entropy and entropy dissipation for the linear Boltzmann operator.

Validates the linear Cercignani's conjecture for this operator.

Connects the results to logarithmic Sobolev inequalities in the grazing collision limit.

Abstract

We prove a linear inequality between the entropy and entropy dissipation functionals for the linear Boltzmann operator (with a Maxwellian equilibrium background). This provides a positive answer to the analogue of Cercignani's conjecture for this linear collision operator. Our result covers the physically relevant case of hard-spheres interactions as well as Maxwellian kernels, both with and without a cut-off assumption. For Maxwellian kernels, the proof of the inequality is surprisingly simple and relies on a general estimate of the entropy of the gain operator due to Matthes and Toscani (2012) and Villani (1998). For more general kernels, the proof relies on a comparison principle. Finally, we also show that in the grazing collision limit our results allow to recover known logarithmic Sobolev inequalities.