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Celebrating Cercignani’s conjecture for the Boltzmann equation
 Kinet. Relat. Models
"... 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 ..."
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Cited by 7 (1 self)
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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. This paper is dedicated to the memory of the late Carlo Cercignani, powerful mind and great scientist, one of the founders of the modern theory of the Boltzmann equation.
Asymptotic of grazing collisions and particle approximation for the Kac equation without cutoff
 COMM. MATH. PHYS
, 2011
"... The subject of this article is the Kac equation without cutoff. We first show that in the asymptotic of grazing collisions, the Kac equation can be approximated by a FokkerPlanck equation. The convergence is uniform in time and we give an explicit rate of convergence. Next, we replace the small c ..."
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Cited by 5 (2 self)
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The subject of this article is the Kac equation without cutoff. We first show that in the asymptotic of grazing collisions, the Kac equation can be approximated by a FokkerPlanck equation. The convergence is uniform in time and we give an explicit rate of convergence. Next, we replace the small collisions by a small diffusion term in order to approximate the solution of the Kac equation and study the resulting error. We finally build a system of stochastic particles undergoing collisions and diffusion, that we can easily simulate, which approximates the solution of the Kac equation without cutoff. We give some estimates on the rate of convergence.
Flow on sweeping networks
, 2013
"... We introduce a cellular automaton model coupled with a transport equation for flows on graphs. The direction of the flow is described by a switching process where the switching probability dynamically changes according to the value of the transported quantity in the neighboring cells. A motivation i ..."
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We introduce a cellular automaton model coupled with a transport equation for flows on graphs. The direction of the flow is described by a switching process where the switching probability dynamically changes according to the value of the transported quantity in the neighboring cells. A motivation is pedestrian dynamics in a small corridor where the propagation of people in a part of the corridor can be either left or rightgoing. Under the assumptions of propagation of chaos and meanfield limit, we derive a master equation and the corresponding meanfield kinetic and macroscopic models. Steady–states are computed and analyzed analytically and exhibit the possibility of multiple metastable states and hysteresis.
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"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.