Annals of Applied Probability

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

Vlad Bally

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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 Probab. Theory Related Fields 151 (2011) 659–704): $f_{t}(dv)=f_{t}(v)\,dv$ 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 $f_{t}(v)\leq Ct^{-\eta}e^{-\vert v\vert^{\lambda}}$ for some $\eta,\lambda>0$.

Article information

Ann. Appl. Probab., Volume 29, Number 3 (2019), 1929-1961.

Received: May 2018
Revised: August 2018
First available in Project Euclid: 19 February 2019

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Zentralblatt MATH identifier

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60J75: Jump processes 82C40: Kinetic theory of gases

Boltzmann equation without cutoff Hard potentials interpolation criterion integration by parts


Bally, Vlad. Upper bounds for the function solution of the homogeneous $2D$ Boltzmann equation with hard potential. Ann. Appl. Probab. 29 (2019), no. 3, 1929--1961. doi:10.1214/18-AAP1451.

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