Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions

Laurent Desvillettes (1) , Clément Mouhot (2)

1 CMLA - Centre de Mathématiques et de Leurs Applications
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In this paper, we prove some a priori stability estimates (in weighted Sobolev spaces) for the spatially homogeneous Boltzmann equation without angular cutoff (covering every physical collision kernels). These estimates are conditioned to some regularity estimates on the solutions, and therefore reduce the stability and uniqueness issue to the one of proving suitable regularity bounds on the solutions. We then prove such regularity bounds for a class of interactions including the so-called (non cutoff and non mollified) hard potentials and moderately soft potentials. In particular, we obtain the first result of global existence and uniqueness for these long-range interactions.

Mots clés: moderately soft potentials; Boltzmann equation; spatially homogeneous; non-cutoff; long-range interactions; hard potentials; soft potentials; moderately soft potentials.

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

Laurent Desvillettes, Clément Mouhot. Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions. Archive for Rational Mechanics and Analysis, 2009, 193, pp.227-253. ⟨10.1007/s00205-009-0233-x⟩. ⟨hal-00079713⟩