On measure solutions of the Boltzmann equation, Part II: Rate of convergence to equilibrium

Lu Xuguang (1) , Clément Mouhot (2)

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

The paper considers the convergence to equilibrium for measure solutions of the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. We prove the exponential sharp rate of strong convergence to equilibrium for conservative measure solutions having finite mass and energy. The proof is based on the regularizing property of the iterated collision operators, exponential moment production estimates, and some previous results on the exponential rate of strong convergence to equilibrium for square integrable initial data. We also obtain a lower bound of the convergence rate and deduce that no eternal solutions exist apart from the trivial stationary solutions given by the Maxwellian equilibrium. We finally use these convergence rates in order to deduce global-in-time strong stability of measure solutions.

Mots clés: Boltzmann equation; spatially homogeneous; hard potentials; measure solutions; equilibrium; exponential rate of convergence; eternal solution; global-in-time stability

Domaines: Equations aux dérivées partielles [math.AP]; Mathématiques [math]

Lu Xuguang, Clément Mouhot. On measure solutions of the Boltzmann equation, Part II: Rate of convergence to equilibrium. 2015. ⟨hal-00829882v2⟩