Cercignani's Conjecture is Sometimes True and Always Almost True

By Cédric Villani. 2003

183 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/0cc8f883f9277d3cebcbbd2e0df20d7ac8e597ad

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In Applied Mathematics

Citations in the corpus

Kac’s program in kinetic theory (2011, background): “Moreover the linear inequality between the entropy and the entropy production is conjectured to be wrong for any physical collision kernels in [75], see for instance [23, 24] for partial confirmations to this conjecture.”
On the convergence to equilibrium of Kac’s random walk on matrices (2007, background): “See [4, 6, 16] and the references therein for more works in similar directions.”
Exponential decay toward equilibrium via entropy methods for reaction–diffusion equations (2006, background): “), integral equations (such as the spatially homogeneous Boltzmann equation [24, 25, 26]), or kinetic equations ([4], [10, 11], [13]).”
Mathematical tools for kinetic equations (2004, background): “For large time behavior, we refer to [87] (part I), Desvillettes and Villani [44] for bounded domains, and to [101] for the whole space.”
Entropy methods for reaction-diffusion equations: slowly growing a-priori bounds (2008, background): “), integral equations (such as the spatially homogeneous Boltzmann equation [33, 34, 35]), or kinetic equations ([3], [12, 13], [14]).”
Hypocoercivity (2006, background): “Known entropy production estimates from [52] make it possible to estimate D(f) from below by Kε [ H(f)−H(Mρu T ) ]1+ε .”
Contractive Probability Metrics and Asymptotic Behavior of Dissipative Kinetic Equations (2007, result): “The same result was conjectured to hold by Cercignani [48] for the spatially homogeneous Boltzmann equation in any dimension and with any kernel, provided a suitable lower bound on the entropy production holds true; for an exhaustive discussion on this conjecture see [105].”
Almost Exponential Decay Near Maxwellian (2006, methodology): “This surprising new decay result relies crucially on recent energy estimates in Guo (2002, 2003a) as well as other extensive and delicate work (Desvillettes and Villani, 2000, 2001, 2002, 2005; Mouhot, 2005; Toscani and Villani, 1999; Villani, 2003).”
On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation (2005, methodology): “Theorem 4 (quantitative H Theorem from [60]).”
Hypocoercive Diffusion Operators (2006, background): “Among a large literature, I shall only quote some of my own works: entropy production estimates for the spatially homogeneous Boltzmann equation, in collaboration with Giuseppe Toscani [25], [26], [28]; and for certain nonlinear diffusion equations with a convex mean-field interaction, in collaborat”
Rate of Convergence to Equilibrium for the Spatially Homogeneous Boltzmann Equation with Hard Potentials (2006, background): “In Section 2 we introduce suitable approximations of the non-local parts of Lm and L, and state and prove various technical estimates on these linearized operators useful for the sequel.”
Factorization for non-symmetric operators and exponential H-theorem (2010, background): “many cases in the spatially homogeneous cases in [30] (in the sense that polynomial inequalities relating the relative entropy and the entropy production hold for a power as close to 1 as wanted), and was an important inspiration for the work [12] in the inhomogeneous case.”