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Pré-Publication, Document De Travail Année : 2015

From hard sphere dynamics to the Stokes-Fourier equations: an $L^2$ analysis of the Boltzmann-Grad limit

Résumé

We derive the linear acoustic and Stokes-Fourier equations as the limiting dynamics of a system of $N$ hard spheres of diameter $\varepsilon$ in two space dimensions, when~$N\to \infty$, $\varepsilon \to 0$, $N\varepsilon =\alpha \to \infty$, using the linearized Boltzmann equation as an intermediate step. Our proof is based on Lanford's strategy \cite{lanford}, and on the pruning procedure developed in \cite{BGSR1} to improve the convergence time. The main novelty here is that uniform $L^2$ a priori estimates combined with a subtle symmetry argument provide a useful cumulant expansion describing the asymptotic decorrelation between the particles. A refined geometric analysis of recollisions is also required in order to discard the possibility of multiple recollisions.
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Dates et versions

hal-01226094 , version 1 (08-11-2015)
hal-01226094 , version 2 (14-12-2015)
hal-01226094 , version 3 (20-10-2016)

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Thierry Bodineau, Isabelle Gallagher, Laure Saint-Raymond. From hard sphere dynamics to the Stokes-Fourier equations: an $L^2$ analysis of the Boltzmann-Grad limit. 2015. ⟨hal-01226094v1⟩
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