Boltzmann equation and hydrodynamics at the Burnett level
Alexander Bobylev Åsa Windfäll
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.
keywords: Hydrodynamics regularized Burnett equations Stability sound propagation.
Transport coefficients in the $2$-dimensional Boltzmann equation
Alexander Bobylev Raffaele Esposito
We show that a rarefied system of hard disks in a plane, described in the Boltzmann-Grad limit by the $2$-dimensional Boltzmann equation, has bounded transport coefficients. This is proved by showing opportune compactness properties of the gain part of the linearized Boltzmann operator.
keywords: transport coefficients. Boltzmann equation
Kinetic modeling of economic games with large number of participants
Alexander Bobylev Åsa Windfäll
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 $\{\gamma, q ,s\}$ where $0<\gamma\leq 1$ 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.
keywords: Maxwell models distribution of wealth market economy. self-similar solutions
Discrete velocity models of the Boltzmann equation and conservation laws
Alexander Bobylev Mirela Vinerean Åsa Windfäll
We consider in this paper the general problem of construction and classification of normal, i.e. without spurious invariants, discrete velocity models (DVMs) of the classical (elastic) Boltzmann equation. We explain in detail how this problem can be solved and present a complete classification of (i.e. we present all distinct) normal plane DVMs with relatively small number $n$ of velocities ($n\leq 10$). Some results for models with larger number of velocities are also presented.
keywords: Boltzmann equation discrete velocity models collision invariants. conservation laws
Upper Maxwellian bounds for the Boltzmann equation with pseudo-Maxwell molecules
Alexander V. Bobylev Irene M. Gamba

We consider solutions to the initial value problem for the spatially homogeneous Boltzmann equation for pseudo-Maxwell molecules and show uniform in time propagation of upper Maxwellians bounds if the initial distribution function is bounded by a given Maxwellian. First we prove the corresponding integral estimate and then transform it to the desired local estimate. We remark that propagation of such upper Maxwellian bounds were obtained by Gamba, Panferov and Villani for the case of hard spheres and hard potentials with angular cut-off. That manuscript introduced the main ideas and tools needed to prove such local estimates on the basis of similar integral estimates. The case of pseudo-Maxwell molecules needs, however, a special consideration performed in the present paper.

keywords: Boltzmann kinetic equation pseudo-Maxwell molecules bounded solutions

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