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Some Problems in Kinetic Theory and Applications
Karlstad University, Faculty of Technology and Science, Department of Mathematics.
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
Mathematics
Identifiers
ISBN: 978-91-7063-388-1OAI: oai:DiVA.org:kau-8498DiVA: diva2:446956
Public defence
2011-12-01, 21A 342, Karlstads universitet, Karlstad, 13:15 (English)
Supervisors
Available from: 2011-11-07 Created: 2011-10-10 Last updated: 2011-11-07Bibliographically approved
List of papers
1. Discrete velocity models of the Boltzmann equation and conservation laws
Open this publication in new window or tab >>Discrete velocity models of the Boltzmann equation and conservation laws
2010 (English)In: Kinetic and Related Models, ISSN 1937-5093, E-ISSN 1937-5077, Vol. 3, no 1, 35-58 p.Article in journal (Refereed) Published
Place, publisher, year, edition, pages
Springfield, MO: American Institute of Mathematical Sciences, 2010
Mathematics
Mathematics
Identifiers
urn:nbn:se:kau:diva-8709 (URN)10.3934/krm.2010.3.35 (DOI)000273989700004 ()
Available from: 2011-11-03 Created: 2011-11-03 Last updated: 2015-12-29Bibliographically approved
2. Boltzmann equation and hydrodynamics at the Burnett level
Open this publication in new window or tab >>Boltzmann equation and hydrodynamics at the Burnett level
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
Keyword
Hydrodynamics, regularized Burnett equations, Stability, sound propagation.
Mathematics
Mathematics
Identifiers
urn:nbn:se:kau:diva-8710 (URN)10.3934/krm.2012.5.237 (DOI)000302962700002 ()
Available from: 2011-11-03 Created: 2011-11-03 Last updated: 2012-12-04Bibliographically approved
3. Kinetic modeling of economic games with large number of participants
Open this publication in new window or tab >>Kinetic modeling of economic games with large number of participants
2011 (English)In: Kinetic and Related Models, ISSN 1937-5093, Vol. 4, no 1, 169-185 p.Article in journal (Refereed) Published
Abstract [en]

We study a Maxwell kinetic model of socio-economic behavior introduced in the paper A. V. Bobylev, C. Cercignani and I. M. Gamba, Commun. Math. Phys., 291 (2009), 599-644. The model depends on three non-negative parameters $\displaystyle{\left\lbrace\gamma,{q},{s}\right\rbrace}$ where $\displaystyle{0}<\gamma\leq{1}$ is the control parameter. Two other parameters are fixed by market conditions. Self-similar solution of the corresponding kinetic equation for distribution of wealth is studied in detail for various sets of parameters. In particular, we investigate the efficiency of control. Some exact solutions and numerical examples are presented. Existence and uniqueness of solutions are also discussed.

Place, publisher, year, edition, pages
American Institute of Mathematical Sciences, 2011
Keyword
Maxwell models, self-similar solutions, distribution of wealth, market economy
Mathematics
Mathematics
Identifiers
urn:nbn:se:kau:diva-8711 (URN)10.3934/krm.2011.4.169 (DOI)000286926200010 ()
Available from: 2011-11-03 Created: 2011-11-03 Last updated: 2012-12-04Bibliographically approved
4. On a special function related to the Riemann zeta-function
Open this publication in new window or tab >>On a special function related to the Riemann zeta-function
(English)Manuscript (preprint) (Other academic)
Mathematics
Mathematics
Identifiers
urn:nbn:se:kau:diva-8712 (URN)
Available from: 2011-11-03 Created: 2011-11-03 Last updated: 2011-11-07Bibliographically approved

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