Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus

By Clément Mouhot, Lukas Neumann. 2006

213 citations recorded by the source corpus.

For a general class of linear collisional kinetic models in the torus, including in particular the linearized Boltzmann equation for hard spheres, the linearized Landau equation with hard and moderately soft potentials and the semi-classical linearized fermionic and bosonic relaxation models, we prove explicit coercivity estimates on the associated integro-differential operator for some modified Sobolev norms. We deduce the existence of classical solutions near equilibrium for the full nonlinear models associated with explicit regularity bounds, and we obtain explicit estimates on the rate of exponential convergence towards equilibrium in this perturbative setting. The proof is based on a linear energy method which combines the coercivity property of the collision operator in the velocity space with transport effects, in order to deduce coercivity estimates in the whole phase space.

Subfields: Applied Mathematics, Computational Mechanics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/316645380c42941718d4ef7d4a47baaa47f1aaa8

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In Applied Mathematics

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Citations in the corpus

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.”
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.”
Factorization for non-symmetric operators and exponential H-theorem (2010, methodology): “First we shall recall the decomposition of L devised in [22], then recall an hypocoercivity property in the torus proved in [24], then introduce a diffusive approximation and finally combine these preliminary steps with Corollary 4.”
Short and long time behavior of the Fokker-Planck equation in a confining potential and applications (2005, background): “The systematic use of crossed derivatives in order to get short-time, long-time and derivative estimates follows from [15] and was also developed in [26]-[19].”
Hypocoercivity (2006, background): “6) is not made in [38], but it is satisfied in all the examples discussed therein: linear relaxation, semi-classical relaxation, linear Fokker–Planck equation, Boltzmann and Landau equations for hard potentials.”
HYPOCOERCIVITY FOR LINEAR KINETIC EQUATIONS CONSERVING MASS (2010, methodology): “Following the vocabulary used in [33, 20, 26], such a strategy will be called hypocoercivity.”
Exponential Stability of Slowly Decaying Solutions to the Kinetic-Fokker-Planck Equation (2014, methodology): “Such a result has been proved in [20], see also [16, 15, 13, 8, 9].”
Hypocoercivity for kinetic equations with linear relaxation terms (2008, methodology): “Villani, see [10,11], and has been successfully applied to various models, see for instance [8].”
Hypocoercive Diffusion Operators (2006, methodology): “Instead, one can introduce an adequate auxiliary operator A into the estimates, in such a way that (i) A∗A + [A, B]∗[A, B] is coercive, and (ii) A “almost commutes” with L. Recently, Clément Mouhot and Lukas Neumann [21] have derived such a hypocoercivity theorem in the particular framework of kinet”