doi: 10.3934/dcdsb.2018313

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

1. 

Scuola Normale Superiore of Pisa - Italy

2. 

University of Pisa - Italy

3. 

University of Florence - Italy

Received  April 2018 Revised  July 2018 Published  October 2018

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.

Citation: Franco Flandoli, Marta Leocata, Cristiano Ricci. The Vlasov-Navier-Stokes equations as a mean field limit. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018313
References:
[1]

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.

[2]

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.

[3]

L. BoudinL. DesvillettesC. Grandmont and A. Moussa, Global existence of solutions for the coupled vlasov and Navier-Stokes equations, Diff. Int. Eq., 22 (2009), 1247-1271.

[4]

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.

[5]

L. DesvillettesF. 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.

[6]

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.

[7]

E. FeireislY. Namlyeyeva and Š. Nečasová, Homogenization of the evolutionary Navier-Stokes system, Manuscr. Math., 149 (2016), 251-274. doi: 10.1007/s00229-015-0778-y.

[8]

F. Flandoli, A fluid-particle system related to Vlasov-Navier-Stokes equations, to appear in Lecture Notes RIMS Kyoto, Ed. Y. Maekawa.

[9]

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.

[10]

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.

[11]

T. GoudonP.-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.

[12]

T. GoudonP.-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.

[13]

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.

[14]

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.

show all references

References:
[1]

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.

[2]

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.

[3]

L. BoudinL. DesvillettesC. Grandmont and A. Moussa, Global existence of solutions for the coupled vlasov and Navier-Stokes equations, Diff. Int. Eq., 22 (2009), 1247-1271.

[4]

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.

[5]

L. DesvillettesF. 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.

[6]

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.

[7]

E. FeireislY. Namlyeyeva and Š. Nečasová, Homogenization of the evolutionary Navier-Stokes system, Manuscr. Math., 149 (2016), 251-274. doi: 10.1007/s00229-015-0778-y.

[8]

F. Flandoli, A fluid-particle system related to Vlasov-Navier-Stokes equations, to appear in Lecture Notes RIMS Kyoto, Ed. Y. Maekawa.

[9]

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.

[10]

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.

[11]

T. GoudonP.-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.

[12]

T. GoudonP.-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.

[13]

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.

[14]

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.

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