On Measure Solutions of the Boltzmann Equation, part I: Moment Production and Stability Estimates

Xuguang Lu (1) , Clément Mouhot (2)

1 Department of Mathematical Sciences
2 DPMMS - Department of Pure Mathematics and Mathematical Statistics

The spatially homogeneous Boltzmann equation with hard potentials is considered for measure valued initial data having finite mass and energy. We prove the existence of \emph{weak measure solutions}, with and without angular cutoff on the collision kernel; the proof in particular makes use of an approximation argument based on the Mehler transform. Moment production estimates in the usual form and in the exponential form are obtained for these solutions. Finally for the Grad angular cutoff, we also establish uniqueness and strong stability estimate on these solutions.

Mots clés: Boltzmann equation; spatially homogeneous; hard potentials; hard spheres; long-range interactions; measure solution; moment estimate; moment production; exponential moment; stability estimate; Mehler transform

Domaines: Equations aux dérivées partielles [math.AP]

Xuguang Lu, Clément Mouhot. On Measure Solutions of the Boltzmann Equation, part I: Moment Production and Stability Estimates. Journal of Differential Equations, 2012, 252 (4), pp.3305-3363. ⟨10.1016/j.jde.2011.10.021⟩. ⟨hal-00561793v2⟩