Hypocoercivity for kinetic equations with linear relaxation terms

By Jean Dolbeault, Clément Mouhot, Christian Schmeiser. 2008

156 citations recorded by the source corpus.

Abstract This Note is devoted to a simple method for proving the hypocoercivity associated to a kinetic equation involving a linear time relaxation operator. It is based on the construction of an adapted Lyapunov functional satisfying a Gronwall-type inequality. The method clearly distinguishes the coercivity at microscopic level, which directly arises from the properties of the relaxation operator, and a spectral gap inequality at the macroscopic level for the spatial density, which is connected to the diffusion limit. It improves on previously known results. Our approach is illustrated by the linear BGK model and a relaxation operator which corresponds at macroscopic level to the linearized fast diffusion. To cite this article: J. Dolbeault et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

View Paper: https://www.semanticscholar.org/paper/a1a318087a65efe91d50009d83d95b7a9484cb5a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Factorization for non-symmetric operators and exponential H-theorem (2010, background): “Let us first recall a recent result proved in [14, 13] for the kinetic Fokker-Planck equation in the space L(2)(μ).”
HYPOCOERCIVITY FOR LINEAR KINETIC EQUATIONS CONSERVING MASS (2010, background): “n: [15, 14, 16, 17, 18, 29, 30, 31]; on hyperbolic estimates based on micro-macro decompositions: [23, 24, 25, 34]. Some of the results of this paper, namely Theorems 9 and 14, have been announced in [11] without complete proofs. Our purpose is to establish, in a simpliﬁed framework, suﬃcient condit”
Hypercontractivity and applications for stochastic Hamiltonian systems (2017, background): “See [6, 7, 8, 10] and references within for L(2)-exponential convergence of the same type degenerate diffusion semigroups.”
Exponential Rate of Convergence to Equilibrium for a Model Describing Fiber Lay-Down Processes (2012, methodology): “In the present paper we prove the convergence at an exponential rate towards a unique stationary state in a weighted L(2) norm by adapting the method developed in [4, 3] to the setting of non-moving belts.”
Exponential Stability of Slowly Decaying Solutions to the Kinetic-Fokker-Planck Equation (2014, background): “Such a result has been proved in [20], see also [16, 15, 13, 8, 9].”
Spectral analysis of semigroups and growth-fragmentation equations (2013, background): “We refer the interested reader to [104] for a pedagogical introduction as well as to [56, 53, 88, 54, 55, 105, 29, 28] for some of the original articles.”
A Kinetic Flocking Model with Diffusion (2010, background): “16), thank to the existence of Lyapunov functionals, the hypocoercivity with almost exponential rate or exponential rate in time has been extensively studied by Desvillettes-Villani [7], Mouhot-Neumann [18], Dolbeault-Mouhot-Schmeiser [8] and Villani [25] in a general framework.”