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

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Abstract

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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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  9. Boudin, L., Grandmont, C., Lorz, A., Moussa, A.: Modelling and numerics for respiratory aerosols. Commun. Comput. Phys. 18(3), 723–756 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  12. Brézis, H., Gallouet, T.: Nonlinear Schrödinger evolution equations. Nonlinear Anal. 4(4), 677–681 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. Dufour, G.: Modélisation Multi-fluide Eulérienne Pour les écoulements Diphasiques à Inclusions Dispersées. Ph.D. thesis, Université Paul-Sabatier Toulouse-III, France (2005)

  22. Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001) (reprint of the 1998 edition)

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  28. Hillairet, M.: On the homogenization of the Stokes problem in a perforated domain. ArXiv e-prints, April (2016)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  33. Moyano, I.: On the controllability of the 2-D Vlasov–Stokes system. Comm. Math. Sci. 15(3), 711–743 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  34. Moyano, I.: Local null-controllability of the 2-D Vlasov–Navier–Stokes system. arXiv preprint arXiv:1607.05578 (2016)

  35. O'Rourke, P. J.: Collective Drop Effects on Vaporizing Liquid Sprays. Ph.D. thesis, Los Alamos National Laboratory (1981)

  36. Williams, F. A.: Combustion Theory, 2nd edn. Benjamin Cummings (1985)

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Authors and Affiliations

  1. CEREMADE, Université Paris-Dauphine, CNRS UMR 7534, PSL Research University, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Olivier Glass

  2. CMLS - École polytechnique, CNRS, 91128, Palaiseau Cedex, France

    Daniel Han-Kwan

  3. Sorbonne Université, CNRS, UMR 7598, LJLL, 75005, Paris, France

    Ayman Moussa

Authors
  1. Olivier Glass
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  2. Daniel Han-Kwan
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  3. Ayman Moussa
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Correspondence to Ayman Moussa.

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Communicated by A. Bressan

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Glass, O., Han-Kwan, D. & Moussa, A. The Vlasov–Navier–Stokes System in a 2D Pipe: Existence and Stability of Regular Equilibria. Arch Rational Mech Anal 230, 593–639 (2018). https://doi.org/10.1007/s00205-018-1253-1

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  • DOI: https://doi.org/10.1007/s00205-018-1253-1

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