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Boltzmann equation and hydrodynamics at the Burnett level
Karlstad University, Faculty of Technology and Science, Department of Mathematics. (Kinetisk teori)
Karlstad University, Faculty of Technology and Science, Department of Mathematics. (Kinetisk teori)
2012 (English)In: Kinetic and Related Models, ISSN 1937-5093, Vol. 5, no 2, 237-260 p.Article in journal (Refereed) Published
Abstract [en]

The hydrodynamics at the Burnett level is discussed in detail. First we explain the shortest way to derive the classical Burnett equations from the Boltzmann equation. Then we sketch all the computations needed for details of these equations. It is well known that the classical Burnett equations are ill-posed. We therefore explain how to make a regularization of these equations and derive the well-posed generalized Burnett equations (GBEs). We discuss briefly an optimal choice of free parameters in GBEs and consider a specific version of these equations. It is remarkable that this version of GBEs is even simpler than the original Burnett equations, it contains only third derivatives of density. Finally we prove a linear stability for GBEs. We also present some numerical results on the sound propagation based on GBEs and compare them with the Navier-Stokes results and experimental data.

Place, publisher, year, edition, pages
American Institute of Mathematical Sciences, 2012. Vol. 5, no 2, 237-260 p.
Keyword [en]
Hydrodynamics, regularized Burnett equations, Stability, sound propagation.
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kau:diva-8710DOI: 10.3934/krm.2012.5.237ISI: 000302962700002OAI: oai:DiVA.org:kau-8710DiVA: diva2:453603
Available from: 2011-11-03 Created: 2011-11-03 Last updated: 2012-12-04Bibliographically approved
In thesis
1. Some Problems in Kinetic Theory and Applications
Open this publication in new window or tab >>Some Problems in Kinetic Theory and Applications
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers. the first is devoted to discrete velocity models, the second to hydrodynamic equation beyond Navier-Stokes level, the third to a multi-linear Maxwell model for economic or social dynamics and the fourth is devoted to a function related to the Riemann zeta-function.

In Paper 1, we consider the general problem of construction and classification of normal, i.e. without spurious invariants, discrete velocity models (DVM) of the classical Boltzman equation. We explain in detail how this problem can be solved and present a complete classification of normal plane DVMs with relatively small number n of velocities (n≤10). Some results for models with larger number of velocities are also presented.

In Paper 2, we discuss hydrodynamics at the Burnett level. Since the Burnett equations are ill-posed, we describe how to make a regularization of these. We derive the well-posed generalized Burnett equations (GBEs) and discuss briefly an optimal choice of free parameters and consider a specific version of these equations. Finally we prove linear stability for GBE and present some numerical result on the sound propagationbased on GBEs.

In Paper 3, we study a Maxwell kinetic model of socio-economic behavior. The model can predict a time dependent distribution of wealth among the participants in economic games with an arbitrary, but sufficiently large, number of players. The model depends on three different positive parameters {γ,q,s} where s and q are fixed by market conditions and γ is a control parameter. In particular, we investigate the efficiency of control. Some exact solutions and numerical examples are presented.

In Paper 4, we study a special function u(s,x), closely connected to the Riemann zeta-function ζ(s), where s is a complex number. We study in detail the properties of u(s,x) and in particular the location of its zeros s(x), for various x≥0. For x=0 the zeros s(0) coincide with non-trivial zeros of ζ(s). We perform a detailed numerical study of trajectories of various zeros s(x) of u(s,x).

Place, publisher, year, edition, pages
Karlstad: Karlstad University, 2011. 22 p.
Series
Karlstad University Studies, ISSN 1403-8099 ; 2011:52
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-8498 (URN)978-91-7063-388-1 (ISBN)
Public defence
2011-12-01, 21A 342, Karlstads universitet, Karlstad, 13:15 (English)
Opponent
Supervisors
Available from: 2011-11-07 Created: 2011-10-10 Last updated: 2011-11-07Bibliographically approved

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Bobylev, AlexanderWindfäll, Åsa

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