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

Vlad Bally 1, 2
2 MATHRISK - Mathematical Risk Handling
Inria de Paris, ENPC - École des Ponts ParisTech, UPEM - Université Paris-Est Marne-la-Vallée
Abstract : 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 $\eta,\lambda>0.$
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Submitted on : Wednesday, May 2, 2018 - 12:39:40 PM
Last modification on : Tuesday, October 19, 2021 - 4:07:28 PM
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  • HAL Id : hal-02429468, version 3
  • ARXIV : 1710.00695
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Vlad Bally. Upper bounds for the function solution of the homogenuous 2D Boltzmann equation with hard potential. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2019. ⟨hal-02429468v3⟩

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