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H 1 Linear Boltzmann equation and fractional diffusion

Auteurs : Golse, François (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : (Work in collaboration with C. Bardos and I. Moyano). Consider the linear Boltzmann equation of radiative transfer in a half-space, with constant scattering coefficient $\sigma$. Assume that, on the boundary of the half-space, the radiation intensity satisfies the Lambert (i.e. diffuse) reflection law with albedo coefficient $\alpha$. Moreover, assume that there is a temperature gradient on the boundary of the half-space, which radiates energy in the half-space according to the Stefan-Boltzmann law. In the asymptotic regime where $\sigma \to +\infty$ and $1 − \alpha ∼ C/\sigma$, we prove that the radiation pressure exerted on the boundary of the half-space is governed by a fractional diffusion equation. This result provides an example of fractional diffusion asymptotic limit of
    a kinetic model which is based on the harmonic extension definition of $\sqrt{−\Delta}$. This fractional diffusion limit therefore differs from most of other such limits for kinetic models reported in the literature, which are based on specific properties of the equilibrium distributions (“heavy tails”) or of the scattering coefficient as in [U. Frisch-H. Frisch: Mon. Not. R. Astr. Not. 181 (1977), 273-280].

    Keywords : linear Boltzmann equation; radiative transfer equation; diffusion approximation; fractional diffusion

    Codes MSC :
    35Q20 - Boltzmann equations
    45K05 - Integro-partial differential equations
    45M05 - Asymptotics
    82C70 - Transport processes
    85A25 - Radiative transfer (astronomy and astrophysics)
    35R11 - Fractional partial differential equations

    Ressources complémentaires :

    Informations sur la rencontre

    Nom de la rencontre : Non standard diffusions in fluids, kinetic equations and probability / Diffusions non standards en mécanique des fluides, équations cinétiques et probabilités
    Organisateurs de la rencontre : Imbert, Cyril ; Mouhot, Clément ; Tristani, Isabelle
    Dates : 10/12/2018 - 14/12/2018
    Année de la rencontre : 2018
    URL Congrès : https://conferences.cirm-math.fr/1862.html

    Citation Data

    DOI : 10.24350/CIRM.V.19483603
    Cite this video as: Golse, François (2018). Linear Boltzmann equation and fractional diffusion. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19483603
    URI : http://dx.doi.org/10.24350/CIRM.V.19483603

    Voir aussi


    1. Bardos, C., Golse, F., & Moyano, I. (2018). Linear Boltzmann equation and fractional diffusion. Kinetic & Related Models, 2018, 11(4), 1011-1036 - http://dx.doi.org/10.3934/krm.2018039