Stability and modeling error for the Boltzmann equation

El Miloud Zaoui; Marc Laforest

doi:10.3934/krm.2014.7.401

Article Contents

2014, Volume 7, Issue 2: 401-414. Doi: 10.3934/krm.2014.7.401

This issue Previous Article Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation Next Article

Stability and modeling error for the Boltzmann equation

El Miloud Zaoui^1, and
Marc Laforest^2,

1.
École Nationale de l'Industrie Minérale, Laboratoire de Mécanique, Thermique et Matériaux, Avenue Hadj Ahmed Cherkaoui - BP 753, Agdal, Rabat
2.
Département de mathématiques et de génie industriel, École Polytechnique de Montréal, C.P. 6079, succursale centre-ville, Montréal, Québec, Canada, H3C 3A7

Received: July 31, 2012

Revised: October 31, 2013

Published: May 2014

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Abstract

We show that the residual measures the difference in $L^1$ between the solutions to two different Boltzmann models of rarefied gases. This work is an extension of earlier work by Ha on the stability of Boltzmann's model, and more specifically on a nonlinear interaction functional that controls the growth of waves. The two kinetic models that are compared in this research are given by (possibly different) inverse power laws, such as the hard spheres and pseudo-Maxwell models. The main point of the estimate is that the modeling error is measured a posteriori, that is to say, the difference between the solutions to the first and second model can be bounded by a term that depends on only one of the two solutions. This work allows the stability estimate to be used to assess uncertainty, modeling or numerical, present in the solution of the first model without solving the second model.

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Mathematics Subject Classification: Primary: 82B40, 35B35; Secondary: 65M15.

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