Advanced Search
Article Contents
Article Contents

The Vlasov-Navier-Stokes equations as a mean field limit

Abstract Full Text(HTML) Related Papers Cited by
  • Convergence of particle systems to the Vlasov-Navier-Stokes equations is a difficult topic with only fragmentary results. Under a suitable modification of the classical Stokes drag force interaction, here a partial result in this direction is proven. A particle system is introduced, its interaction with the fluid is modelled and tightness is proved, in a suitable topology, for the family of laws of the pair composed by solution of Navier-Stokes equations and empirical measure of the particles. Moreover, it is proved that every limit law is supported on weak solutions of the Vlasov-Navier-Stokes system. Open problems, like weak-strong uniqueness for this system and its relevance for the convergence of the particle system, are outlined.

    Mathematics Subject Classification: Primary: 35J05, 76T20, 60K35.


    \begin{equation} \\ \end{equation}
  • 加载中
  •   G. Allaire , Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes, Arch. Ration. Mech. Anal, 113 (1990) , 209-259.  doi: 10.1007/BF00375065.
      E. Bernard, L. Desvillettes, F. Golse and V. Ricci, A derivation of the Vlasov-Navier-Stokes model for aerosol flows from kinetic theory, Commun. Math. Sci., 15 (2017), 1703–1741, arXiv: 1608.00422. doi: 10.4310/CMS.2017.v15.n6.a11.
      L. Boudin , L. Desvillettes , C. Grandmont  and  A. Moussa , Global existence of solutions for the coupled vlasov and Navier-Stokes equations, Diff. Int. Eq., 22 (2009) , 1247-1271. 
      B. Desjardins  and  M., J. Esteban , Existence of weak solutions for the motion of rigid bodies in a viscous fluid, Arch. Ration. Mech. Anal., 146 (1999) , 59-71.  doi: 10.1007/s002050050136.
      L. Desvillettes , F. Golse  and  V. Ricci , The mean-field limit for solid particles in a Navier-Stokes flow, J. Stat. Phys., 131 (2008) , 941-967.  doi: 10.1007/s10955-008-9521-3.
      L. Desvillettes  and  J. Mathiaud , Some aspects of the asymptotics leading from gas-particles equations towards multiphase flows equations, J. Stat. Phys., 141 (2010) , 120-141.  doi: 10.1007/s10955-010-0044-3.
      E. Feireisl , Y. Namlyeyeva  and  Š. Nečasová , Homogenization of the evolutionary Navier-Stokes system, Manuscr. Math., 149 (2016) , 251-274.  doi: 10.1007/s00229-015-0778-y.
      F. Flandoli, A fluid-particle system related to Vlasov-Navier-Stokes equations, to appear in Lecture Notes RIMS Kyoto, Ed. Y. Maekawa.
      D. Gérard-Varet  and  M. Hillairet , Regularity issues in the problem of fluid structure interaction, Arch. Ration. Mech. Anal., 195 (2010) , 375-407.  doi: 10.1007/s00205-008-0202-9.
      O. Glass, A. Munnier and F. Sueur, Point vortex dynamics as zero-radius limit of the motion of a rigid body in an irrotational fluid, preprint hal.inria.fr 2016.
      T. Goudon , P.-E. Jabin  and  A. Vasseur , Hydrodynamic limit for the Vlasov-Navier-Stokes equations. Ⅰ. Light particles regime, Indiana Univ. Math. J., 53 (2004) , 1495-1515.  doi: 10.1512/iumj.2004.53.2508.
      T. Goudon , P.-E. Jabin  and  A. Vasseur , Hydrodynamic limit for the Vlasov-Navier-Stokes equations.Ⅱ. Fine particles regime, Indiana Univ. Math. J., 53 (2004) , 1517-1536.  doi: 10.1512/iumj.2004.53.2509.
      P.-E. Jabin  and  F. Otto , Identification of the dilute regime in particle sedimentation, Comm. Math. Phys., 250 (2004) , 415-432.  doi: 10.1007/s00220-004-1126-3.
      C. Yu , Global weak solutions to the incompressible Navier-Stokes-Vlasov equations, J. Math. Pures Appl., 100 (2013) , 275-293.  doi: 10.1016/j.matpur.2013.01.001.
  • 加载中

Article Metrics

HTML views(903) PDF downloads(262) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint