Celebrating Cercignani's conjecture for the Boltzmann equation

TL;DR

This paper reviews the progress on Cercignani's conjecture for the Boltzmann equation, discussing both positive and negative results in the context of entropy inequalities and convergence to equilibrium.

Contribution

It provides a comprehensive survey of the developments related to Cercignani's conjecture since its proposal in the 1980s.

Findings

Summary of positive results confirming the conjecture in certain cases

Overview of negative results disproving the conjecture in some contexts

Discussion of implications for convergence to thermodynamical equilibrium

Abstract

Cercignani's conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann's nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at the core of the renewal of the mathematical theory of convergence to thermodynamical equilibrium for rarefied gases over the past decade. In this review paper, we survey the various positive and negative results which were obtained since the conjecture was proposed in the 1980s.