Regularity theory for the spatially homogeneous Boltzmann equation with cut-off

Clément Mouhot (1) , Cédric Villani (2)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We develop the regularity theory of the spatially homogeneous Boltzmann equation with cut-off and hard potentials (for instance, hard spheres), by (i) revisiting the Lp-theory to obtain constructive bounds, (ii) establishing propagation of smoothness and singularities, (iii) obtaining estimates about the decay of the sin- gularities of the initial datum. Our proofs are based on a detailed study of the “regularity of the gain operator”. An application to the long-time behavior is presented.

Mots clés: Boltzmann equation; spatially homogeneous; hard spheres; hard potentials; angular cutoff; regularity theory; quantitative; relaxation to equilibrium

Domaines: Equations aux dérivées partielles [math.AP]

Clément Mouhot, Cédric Villani. Regularity theory for the spatially homogeneous Boltzmann equation with cut-off. Archive for Rational Mechanics and Analysis, 2004, 173, pp.169-212. ⟨10.1007/s00205-004-0316-7⟩. ⟨hal-00087274⟩