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Annals of PDE

, 4:1 | Cite as

Stationary Solutions to the Boltzmann Equation in the Hydrodynamic Limit

  • Raffaele Esposito
  • Yan Guo
  • Chanwoo Kim
  • Rossana Marra
Article

Abstract

Despite its conceptual and practical importance, a rigorous derivation of the steady incompressible Navier–Stokes–Fourier system from the Boltzmann theory has been an outstanding open problem for general domains in 3D. We settle this open question in the affirmative, in the presence of a small external field and a small boundary temperature variation for the diffuse boundary condition. We employ a recent quantitative \(L^{2}\)\(L^{\infty }\) approach with new \(L^{{6}}\) estimates for the hydrodynamic part \(\mathbf {P}f\) of the distribution function. Our results also imply the validity of Fourier law in the hydrodynamical limit, and our method leads to an asymptotical stability of steady Boltzmann solutions as well as the derivation of the unsteady Navier–Stokes–Fourier system.

Keywords

Boltzmann equation Navier–Stokes equation Hydrodynamic limit 

Notes

Acknowledgements

We thank Prof. Fujun Zhou for pointing out some inaccuracies in previous versions of this paper. Y. Guo’s research is supported in part by NSFC Grant 10828103, NSF Grant DMS-0905255, and BICMR. C. Kim’s research is supported in part by NSF DMS-1501031, KAIST-CMC, the Herchel Smith Foundation, and the University of Wisconsin-Madison Graduate School with funding from the Wisconsin Alumni Research Foundation. R. Marra is partially supported by MIUR-Prin.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.International Research Center M&MOCSUniv. dell’AquilaCisterna di LatinaItaly
  2. 2.Division of Applied MathematicsBrown UniversityProvidenceUSA
  3. 3.Department of MathematicsUniversity of WisconsinMadisonUSA
  4. 4.Dipartimento di Fisica and Unità INFNUniversità di Roma Tor VergataRomeItaly

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