Kinetic and Related Models (KRM)

Stability and modeling error for the Boltzmann equation

Pages: 401 - 414, Volume 7, Issue 2, June 2014      doi:10.3934/krm.2014.7.401

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El Miloud Zaoui - École Nationale de l'Industrie Minérale, Laboratoire de Mécanique, Thermique et Matériaux, Avenue Hadj Ahmed Cherkaoui - BP 753, Agdal, Rabat, Morocco (email)
Marc Laforest - Département de mathématiques et de génie industriel, École Polytechnique de Montréal, C.P. 6079, succursale centre-ville, Montréal, Québec, Canada, H3C 3A7, Canada (email)

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.

Keywords:  Error estimate, modeling error, a posteriori, Boltzmann equation, stability.
Mathematics Subject Classification:  Primary: 82B40, 35B35; Secondary: 65M15.

Received: August 2012;      Revised: November 2013;      Available Online: March 2014.