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

2 MATHRISK - Mathematical Risk Handling
UPEM - Université Paris-Est Marne-la-Vallée, ENPC - École des Ponts ParisTech, Inria de Paris
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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https://hal-upec-upem.archives-ouvertes.fr/hal-01593131
Submitted on : Wednesday, May 2, 2018 - 12:39:40 PM
Last modification on : Thursday, July 18, 2019 - 3:00:05 PM
Long-term archiving on : Tuesday, September 25, 2018 - 12:04:10 AM

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### Identifiers

• HAL Id : hal-01593131, version 2
• ARXIV : 1710.00695

### Citation

Vlad Bally. Upper bounds for the function solution of the homogenuous 2D Boltzmann equation with hard potential. 2018. ⟨hal-01593131v2⟩

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