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

Clément Mouhot (1) , Lukas Neumann (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 RICAM - Johann Radon Institute for Computational and Applied Mathematics

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 existence of classical solutions near equilibrium for the full non-linear 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 are 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.

Mots clés: collisional kinetic models; Boltzmann equation; Landau equation; relaxation; semi-classical relaxation; bosons; fermions; Fokker-Planck; weak external field; Poisson self-consistent potential; rate of convergence to equilibrium; explicit; energy method

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Clément Mouhot, Lukas Neumann. Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus. Nonlinearity, 2006, 19, pp.969-998. ⟨10.1088/0951-7715/19/4/011⟩. ⟨hal-00087173⟩