Archive for Rational Mechanics and Analysis

, Volume 230, Issue 2, pp 593–639 | Cite as

The Vlasov–Navier–Stokes System in a 2D Pipe: Existence and Stability of Regular Equilibria

  • Olivier Glass
  • Daniel Han-Kwan
  • Ayman MoussaEmail author


In this paper, we study the Vlasov–Navier–Stokes system in a 2D pipe with partially absorbing boundary conditions. We show the existence of stationary states for this system near small Poiseuille flows for the fluid phase, for which the kinetic phase is not trivial. We prove the asymptotic stability of these states with respect to appropriately compactly supported perturbations. The analysis relies on geometric control conditions which help to avoid any concentration phenomenon for the kinetic phase.


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  1. 1.
    Allaire, G.: Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes. Arch. Ration. Mech. Anal. 113(3), 209–259 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Anoshchenko, O., Boutet de Monvel-Berthier, A.: The existence of the global generalized solution of the system of equations describing suspension motion. Math. Methods Appl. Sci. 20(6), 495–519 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bardos, C.: Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels; théorèmes d'approximation; application à l'équation de transport. Ann. Sci. École Norm. Sup. 4(3), 185–233 (1970)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30(5), 1024–1065 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Benjelloun, S., Desvillettes, L., Moussa, A.: Existence theory for the kinetic-fluid coupling when small droplets are treated as part of the fluid. J. Hyperbolic Differ. Equ. 11(1), 109–133 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bernard, É., Desvillettes, L., Golse, F., Ricci, V.: A derivation of the Vlasov-Stokes system for aerosol flows from the kinetic theory of binary gas mixtures. Kinet. Relat. Models 11(1), 43–49 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bernard, É., Salvarani, F.: On the exponential decay to equilibrium of the degenerate linear Boltzmann equation. J. Funct. Anal. 265(9), 1934–1954 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Boudin, L., Desvillettes, L., Grandmont, C., Moussa, A.: Global existence of solutions for the coupled Vlasov and Navier-Stokes equations. Differ. Integral Equ. 22(11–12), 1247–1271 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Boudin, L., Grandmont, C., Lorz, A., Moussa, A.: Modelling and numerics for respiratory aerosols. Commun. Comput. Phys. 18(3), 723–756 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Boudin, L., Grandmont, C., Moussa, A.: Global existence of solutions to the incompressible Navier–Stokes–Vlasov equations in a time-dependent domain. J. Differ. Equ. 262(3), 1317–1340 (2017)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Boyer, F., Fabrie, P.: Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, vol. 183. Applied Mathematical Sciences Springer, New York (2013)zbMATHGoogle Scholar
  12. 12.
    Brézis, H., Gallouet, T.: Nonlinear Schrödinger evolution equations. Nonlinear Anal. 4(4), 677–681 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Carrillo, J., Duan, R., Moussa, A.: Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system. Kinet. Relat. Models 4(1), 227–258 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chae, M., Kang, K., Lee, J.: Global classical solutions for a compressible fluid-particle interaction model. J. Hyperbolic Differ. Equ. 10(3), 537–562 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Choi, Y.-P.: Finite-time blow-up phenomena of Vlasov/Navier–Stokes equations and related systems. J. de Mathématiques Pures et Appliquées 108(6), 991–1021 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Choi, Y.-P., Kwon, B.: Global well-posedness and large-time behavior for the inhomogeneous Vlasov-Navier-Stokes equations. Nonlinearity 28(9), 3309 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Cioranescu, D., Murat, F.: Un terme étrange venu d'ailleurs. In: Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. II (Paris, 1979/1980), volume 60 of Res. Notes in Math., pp. 98–138, 389–390. Pitman, Boston (1982)Google Scholar
  18. 18.
    Desvillettes, L.: Some aspects of the modeling at different scales of multiphase flows. Comput. Methods Appl. Mech. Eng. 199(21–22), 1265–1267 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Desvillettes, L., Golse, F., Ricci, V.: The mean-field limit for solid particles in a Navier-Stokes flow. J. Stat. Phys. 131(5), 941–967 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dufour, G.: Modélisation Multi-fluide Eulérienne Pour les écoulements Diphasiques à Inclusions Dispersées. Ph.D. thesis, Université Paul-Sabatier Toulouse-III, France (2005)Google Scholar
  22. 22.
    Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001) (reprint of the 1998 edition)Google Scholar
  23. 23.
    Glass, O., Han-Kwan, D.: On the controllability of the relativistic Vlasov–Maxwell system. J. Math. Pures Appl. (9) 103(3), 695–740 (2015)Google Scholar
  24. 24.
    Goudon, T., He, L., Moussa, A., Zhang, P.: The Navier-Stokes-Vlasov-Fokker-Planck system near equilibrium. SIAM J. Math. Anal. 42(5), 2177–2202 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Goudon, T., Jabin, P.-E., Vasseur, A.: Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime. Indiana Univ. Math. J. 53(6), 1495–1515 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hamdache, K.: Global existence and large time behaviour of solutions for the Vlasov-Stokes equations. Jpn. J. Ind. Appl. Math. 15(1), 51–74 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Han-Kwan, D., Léautaud, M.: Geometric analysis of the linear Boltzmann equation I. Trend to equilibrium. Ann. PDE, 1(1):Art. 3, 84, (2015)Google Scholar
  28. 28.
    Hillairet, M.: On the homogenization of the Stokes problem in a perforated domain. ArXiv e-prints, April (2016)Google Scholar
  29. 29.
    Jabin, P.-E.: Large time concentrations for solutions to kinetic equations with energy dissipation. Commun. Partial Differ. Equ. 25(3–4), 541–557 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kellogg, R.B., Osborn, J.E.: A regularity result for the Stokes problem in a convex polygon. J. Funct. Anal. 21(4), 397–431 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Li, F., Mu, Y., Wang, D.: Global well-posedness and large time behavior of strong solution to a kinetic-fluid model. ArXiv e-prints, Aug (2015)Google Scholar
  32. 32.
    Mellet, A., Vasseur, A.: Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations. Math. Models Methods Appl. Sci. 17(7), 1039–1063 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Moyano, I.: On the controllability of the 2-D Vlasov–Stokes system. Comm. Math. Sci. 15(3), 711–743 (2017) MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Moyano, I.: Local null-controllability of the 2-D Vlasov–Navier–Stokes system. arXiv preprint arXiv:1607.05578 (2016)
  35. 35.
    O'Rourke, P. J.: Collective Drop Effects on Vaporizing Liquid Sprays. Ph.D. thesis, Los Alamos National Laboratory (1981)Google Scholar
  36. 36.
    Williams, F. A.: Combustion Theory, 2nd edn. Benjamin Cummings (1985)Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CEREMADE, Université Paris-Dauphine, CNRS UMR 7534, PSL Research UniversityParis Cedex 16France
  2. 2.CMLS - École polytechnique, CNRSPalaiseau CedexFrance
  3. 3.Sorbonne Université, CNRS, UMR 7598, LJLLParisFrance

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