Upper bounds for the function solution of the homogenuous 2D Boltzmann equation with hard potential

Vlad Bally (1, 2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 MATHRISK - Mathematical Risk Handling

We deal with $f_{t}(dv),$ the solution of the homogeneous $2D$ Boltzmann equation without cutoff. The initial condition $f_{0}(dv)$ may be any probability distribution (except a Dirac mass). However, for sufficiently hard potentials, the semigroup has a regularization property (see \cite{[BF]}): $f_{t}(dv)=f_{t}(v)dv$ for every $t>0.$ The aim of this paper is to give upper bounds for $f_{t}(v),$ the most significant one being of type $f_{t}(v)\leq Ct^{-\eta}e^{-\left\vert v\right\vert ^{\lambda}}$ for some $\eta,\lambda>0.$

Mots clés: Boltzmann equation without cutoff; Interpolation criterion; Integration-by-parts; Hard potentials

Domaines: Probabilités [math.PR]

Vlad Bally. Upper bounds for the function solution of the homogenuous 2D Boltzmann equation with hard potential. The Annals of Applied Probability, 2019. ⟨hal-02429468v3⟩