Diffusion limit for a kinetic equation with a thermostatted interface

Giada Basile; Tomasz Komorowski; Stefano Olla

doi:10.3934/krm.2019045

Article Contents

2019, Volume 12, Issue 5: 1185-1196. Doi: 10.3934/krm.2019045

This issue Previous Article Uniform estimates on the Fisher information for solutions to Boltzmann and Landau equations Next Article

Diffusion limit for a kinetic equation with a thermostatted interface

1.
Dipartimento di Matematica, Università di Roma La Sapienza, Roma, Italy
2.
IMPAN, Polish Academy of Sciences, Warsaw, Poland
3.
CEREMADE, UMR CNRS, Université Paris-Dauphine, PSL Research University, 75016 Paris, France

^* Corresponding author: Stefano Olla
^* Corresponding author: Stefano Olla

Received: March 2019

Published online: July 13, 2019

T.K. acknowledges the support of the National Science Centre: NCN grant 2016/23/B/ST1/00492. SO's research is supported by ANR-15-CE40-0020-01 grant LSD. Both T.K. and S.O. were partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.

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Abstract

We consider a linear phonon Boltzmann equation with a reflecting/transmitting/absorbing interface. This equation appears as the Boltzmann-Grad limit for the energy density function of a harmonic chain of oscillators with inter-particle stochastic scattering in the presence of a heat bath at temperature $ T $ in contact with one oscillator at the origin. We prove that under the diffusive scaling the solutions of the phonon equation tend to the solution $ \rho(t, y) $ of a heat equation with the boundary condition $ \rho(t, 0)\equiv T $.

Keywords:

Diffusion limit,
thermostatted boundary conditions.

Mathematics Subject Classification: Primary: 60F17; Secondary: 82C70.

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References

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