The BGK model with external confining potential: existence, long-time behaviour and time-periodic Maxwellian equilibria
type: Item
instance of: scholarly article
title: The BGK model with external confining potential: existence, long-time behaviour and time-periodic Maxwellian equilibria
zbMATH Open document ID: 1188.82060
author: Roberta Bosi
author: Maria José Cáceres
published in: Journal of Statistical Physics
publication date: 2009-09-22T00:00:00Z
review text: The existence and the long time behavior solution of the Boltzmann equation with the external potential force \({\mathbf F}=\nabla_x\Phi\) are analyzed: \[ \partial_t f+{\mathbf v}\cdot\nabla_x f+\nabla_x \Phi\nabla_v f= J \] The BGK approximation of the collision integral \((J= J_{\text{BGK}})\) and the collision integral \((J= J_B(f))\) with three collision invariants \((\int J(f)I_m d{\mathbf v}= 0,\;I_m= n,\;v,|v^2|)\) are investigated. The Cauchy problem for the Boltzmann equation is studied analytically. Some mathematical aspects of the Boltzmann equation solution including the relaxation problem of rarefied gases are investigated. Strong convergence on \(L^1\) to a Maxwellian equilibrium state are proved if the initial datum \(f(r,v, t=0)\geq 0\) and mass, entropy and total energy are bounded. It is shown that in the case of a quadratic potential \(\Phi({\mathbf x})= \sum_j a_i x_j+{\mathbf b}\cdot{\mathbf x}\), an infinite family of time dependent local Maxwellian solutions exists.
reviewed by: Oleg A. Sinkevich
Mathematics Subject Classification ID: 82C40
Mathematics Subject Classification ID: 76P05
Mathematics Subject Classification ID: 35Q20
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 82B40
Mathematics Subject Classification ID: 82D05
zbMATH DE Number: 5606287
zbMATH Keywords: Boltzmann equation
zbMATH Keywords: BGK approximation
zbMATH Keywords: external force
zbMATH Keywords: solution
zbMATH Keywords: equilibrium
zbMATH Keywords: convergence
zbMATH Keywords: mathematical aspects
zbMATH Keywords: analytical results
MaRDI profile type: Publication
full work available at URL: s10955-009-9782-5
OpenAlex ID: W2106537309
cites work: A consistent BGK-type model for gas mixtures
cites work: A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems
cites work: Construction of BGK models with a family of kinetic entropies for a given system of conservation laws
cites work: Entropy satisfying flux vector splittings and kinetic BGK models
cites work: On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with Coulombic and Newtonian potentials
cites work: A BGK model for small Prandtl number in the Navier-Stokes approximation
cites work: Equilibration Rate for the Linear Inhomogeneous Relaxation-Time Boltzmann Equation for Charged Particles
cites work: The Boltzmann equation and its applications
cites work: A hybrid kinetic--fluid model for solving the Vlasov--BGK equation
cites work: Convergence to equilibrium in large time for Boltzmann and B.G.K. equations
cites work: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation
cites work: On the Cauchy problem for Boltzmann equations: Global existence and weak stability
cites work: Global weak solutions of Vlasov‐Maxwell systems
cites work: Global solutions of Boltzmann's equation and the entropy inequality
cites work: \(L^ p\) regularity of velocity averages
cites work: On a quantum Boltzmann equation for a gas of photons.
cites work: Dispersion and moment lemmas revisited
cites work: Regularity of the moments of the solution of a transport equation
cites work: Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II
cites work: Compactness in Boltzmann's equation via Fourier integral operators and applications. III
cites work: Q4022295
cites work: Uniqueness for the BGK-equation in \(\mathbb{R}^ N\) and rate of convergence for a semi-discrete scheme
cites work: Global existence to the BGK model of Boltzmann equation
cites work: Q4792090
cites work: Boltzmann Type Schemes for Gas Dynamics and the Entropy Property
cites work: Weighted \(L^ \infty\) bounds and uniqueness for the Boltzmann BGK model
cites work: Du modèle BGK de l'équation de Boltzmann aux équations d'Euler des fluides incompressibles. (From the BGK Boltzmann model to the Euler equations of incompressible fluids)
cites work: From the BGK model to the Navier–Stokes equations
DOI: 10.1007/S10955-009-9782-5
description: scientific article; zbMATH DE number 5606287