KRM
Erratum to: Ghost effect by curvature in planar Couette flow [1]
Leif Arkeryd Raffaele Esposito Rossana Marra Anne Nouri
N/A
keywords: Couette flow. Boltzmann equation hydrodynamical limit ghost effect
KRM
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
KRM
Exponential stability of the solutions to the Boltzmann equation for the Benard problem
Leif Arkeryd Raffaele Esposito Rossana Marra Anne Nouri
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.
KRM
Rigorous validity of the Boltzmann equation for a thin layer of a rarefied gas
Raffaele Esposito Mario Pulvirenti
We consider a thin layer of a rarified gas modeled by a large hard-sphere system and show that, as long as the thickness of the layer is much larger than the interaction length, the limiting behavior is described, at least for short times, by a Boltzmann equation with two-dimensional position variable and three-dimensional velocity. By the analysis of the Lorentz gas we argue that, if the thickness of the layer is of the same order of the interaction length, this is not the case.
keywords: Boltzmann Equation Rarified Gases Thin Films.
KRM
Ghost effect by curvature in planar Couette flow
Leif Arkeryd Raffaele Esposito Rossana Marra Anne Nouri
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
KRM
Validity of the Boltzmann equation with an external force
Raffaele Esposito Yan Guo Rossana Marra
We establish local-in-time validity of the Boltzmann equation in the presence of an external force deriving from a $C^2$ potential.
keywords: Rarified gases derivation external force. Boltzmann equation
KRM
Stability of a Vlasov-Boltzmann binary mixture at the phase transition on an interval
Raffaele Esposito Yan Guo Rossana Marra
We consider a kinetic model for a system of two species of particles on a sufficiently large periodic interval, interacting through a long range repulsive potential and by collisions. The model is described by a set of two coupled Vlasov-Boltzmann equations. For temperatures below the critical value and suitably prescribed masses, there is a non homogeneous solution, the double soliton, which is a minimizer of the entropy functional. We prove the stability, up to translations, of the double soliton under small perturbations. The same arguments imply the stability of the pure phases, as well as the stability of the mixed phase above the critical temperature. The mixed phase is proved to be unstable below the critical temperature.
keywords: Vlasov-Boltzmann equation Phase transitions stability.

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