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A semigroup approach to the justification of kinetic theory


Reference:

Matthies, K. and Theil, F., 2012. A semigroup approach to the justification of kinetic theory. SIAM Journal on Mathematical Analysis (SIMA), 44 (6), 4345–4379.

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    Official URL:

    http://dx.doi.org/10.1137/120865598

    Abstract

    This paper develops a method to rigorously show the validity of continuum description for the deterministic dynamics of many interacting particles with random initial data. We consider a hard sphere flow where particles are removed after the first collision. A fixed number of particles is drawn randomly according to an initial density f0(u; v) depending on d-dimensional position u and velocity v. In the Boltzmann Grad scaling, we derive the validity of a Boltzmann equation without gain term for arbitrary long times, when we assume finiteness of moments up to order two and initial data that are L∞ in space. We characterize the many particle flow by collision trees which encode possible collisions. The convergence of the many-particle dynamics to the Boltzmann dynamics is achieved via the convergence of associated probability measures on collision trees. These probability measures satisfy nonlinear Kolmogorov equations, which are shown to be well-posed by semigroup methods.

    Details

    Item Type Articles
    CreatorsMatthies, K.and Theil, F.
    DOI10.1137/120865598
    DepartmentsFaculty of Science > Mathematical Sciences
    Publisher StatementMT_SIMA2012.pdf: ©SIAM
    RefereedYes
    StatusPublished
    ID Code32273

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