Entropy dissipation estimates for the linear Boltzmann operator

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.

References in corpus (2)

• Integral representation of the linear Boltzmann operator for granular gas dynamics with applications
• Equilibrium Solution to the Inelastic Boltzmann Equation Driven by a Particles Thermal Bath

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