Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules

Milana Pavić-Čolić; Maja Tasković

doi:10.3934/krm.2018025

Article Contents

2018, Volume 11, Issue 3: 597-613. Doi: 10.3934/krm.2018025

This issue Previous Article High order approximation for the Boltzmann equation without angular cutoff Next Article Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities

Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules

Milana Pavić-Čolić^1, , and
Maja Tasković^2,

1.
Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia
2.
Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab., 209 South 33rd Street, Philadelphia, PA 19104, USA

* Corresponding author: Milana Pavić-Čolić
* Corresponding author: Milana Pavić-Čolić

Received: March 31, 2017

Revised: August 31, 2017

Published: May 2018

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Abstract

We study the spatially homogeneous Boltzmann equation for Maxwell molecules, and its 1-dimensional model, the Kac equation. We prove propagation in time of stretched exponential moments of their weak solutions, both for the angular cutoff and the angular non-cutoff case. The order of the stretched exponential moments in question depends on the singularity rate of the angular kernel of the Boltzmann and the Kac equation. One of the main tools we use are Mittag-Leffler moments, which generalize the exponential ones.

Keywords:

Streched exponential moments,
Kac equation,
Boltzmann equation with Maxwell molecules,
cutoff and non-cutoff case,
Mittag-Leffler moments.

Mathematics Subject Classification: Primary: 35Q20, 35Q82; Secondary: 82C40.

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References

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