Exponential stability of the solutions to the Boltzmann equation for the Benard problem
Leif Arkeryd Raffaele Esposito Rossana Marra Anne Nouri
Kinetic & Related Models 2012, 5(4): 673-695 doi: 10.3934/krm.2012.5.673
We complete the result in [2] by showing the exponential decay of the perturbation of the laminar solution below the critical Rayleigh number and of the convective solutions above the critical Rayleigh number, in the kinetic framework.
keywords: Boltzmann equation hydrodynamical limit Benard problem.
Ghost effect by curvature in planar Couette flow
Leif Arkeryd Raffaele Esposito Rossana Marra Anne Nouri
Kinetic & Related Models 2011, 4(1): 109-138 doi: 10.3934/krm.2011.4.109
We study a rarefied gas, described by the Boltzmann equation, between two coaxial rotating cylinders in the small Knudsen number regime. When the radius of the inner cylinder is suitably sent to infinity, the limiting evolution is expected to converge to a modified Couette flow which keeps memory of the vanishing curvature of the cylinders ( ghost effect [18]). In the $1$-d stationary case we prove the existence of a positive isolated $L_2$-solution to the Boltzmann equation and its convergence. This is obtained by means of a truncated bulk-boundary layer expansion which requires the study of a new Milne problem, and an estimate of the remainder based on a generalized spectral inequality.
keywords: Couette flow. Boltzmann equation Ghost effect Hydrodynamical limit
Erratum to: Ghost effect by curvature in planar Couette flow [1]
Leif Arkeryd Raffaele Esposito Rossana Marra Anne Nouri
Kinetic & Related Models 2012, 5(3): 669-672 doi: 10.3934/krm.2012.5.669
keywords: Couette flow. Boltzmann equation hydrodynamical limit ghost effect
A kinetic equation for spin polarized Fermi systems
Leif Arkeryd
Kinetic & Related Models 2014, 7(1): 1-8 doi: 10.3934/krm.2014.7.1
This paper considers a kinetic Boltzmann equation, having a general type of collision kernel and modelling spin-dependent Fermi gases at low temperatures. The distribution functions have values in the space of positive hermitean $2\times2$ complex matrices. Global existence of weak solutions is proved in $L^1\cap L^{\infty}$ for the initial value problem of this Boltzmann equation in a periodic box.
keywords: low temperature kinetics. Fermi system with spin Boltzmann spin kinetics
A Milne problem from a Bose condensate with excitations
Leif Arkeryd Anne Nouri
Kinetic & Related Models 2013, 6(4): 671-686 doi: 10.3934/krm.2013.6.671
This paper deals with a half-space linearized problem for the distribution function of the excitations in a Bose gas close to equilibrium. Existence and uniqueness of the solution, as well as its asymptotic properties are proven for a given energy flow. The problem differs from the ones for the classical Boltzmann and related equations, where the hydrodynamic mass flow along the half-line is constant. Here it is no more constant. Instead we use the energy flow which is constant, but no more hydrodynamic.
keywords: Milne problem. Low temperature kinetics Bose condensate two component model
On a Boltzmann equation for Haldane statistics
Leif Arkeryd Anne Nouri
Kinetic & Related Models 2019, 12(2): 323-346 doi: 10.3934/krm.2019014

The study of quantum quasi-particles at low temperatures including their statistics, is a frontier area in modern physics. In a seminal paper Haldane [10] proposed a definition based on a generalization of the Pauli exclusion principle for fractional quantum statistics. The present paper is a study of quantum quasi-particles obeying Haldane statistics in a fully non-linear kinetic Boltzmann equation model with large initial data on a torus. Strong $L^1$ solutions are obtained for the Cauchy problem. The main results concern existence, uniqueness and stabililty. Depending on the space dimension and the collision kernel, the results obtained are local or global in time.

keywords: Anyon Haldane statistics low temperature kinetic theory quantum Boltzmann equation

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