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A Multi-Fluid Compressible System as the Limit of Weak Solutions of the Isentropic Compressible Navier–Stokes Equations

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Abstract

This paper mainly concerns the mathematical justification of a viscous compressible multi-fluid model linked to the Baer-Nunziato model used by engineers, see for instance Ishii (Thermo-fluid dynamic theory of two-phase flow, Eyrolles, Paris, 1975), under a “stratification” assumption. More precisely, we show that some approximate finite-energy weak solutions of the isentropic compressible Navier–Stokes equations converge, on a short time interval, to the strong solution of this viscous compressible multi-fluid model, provided the initial density sequence is uniformly bounded with corresponding Young measures which are linear convex combinations of m Dirac measures. To the authors’ knowledge, this provides, in the multidimensional in space case, a first positive answer to an open question, see Hillairet (J Math Fluid Mech 9:343–376, 2007), with a stratification assumption. The proof is based on the weak solutions constructed by Desjardins (Commun Partial Differ Equ 22(5–6):977–1008, 1997) and on the existence and uniqueness of a local strong solution for the multi-fluid model established by Hillairet assuming initial density to be far from vacuum. In a first step, adapting the ideas from Hoff and Santos (Arch Ration Mech Anal 188:509–543, 2008), we prove that the sequence of weak solutions built by Desjardins has extra regularity linked to the divergence of the velocity without any relation assumption between λ and μ. Coupled with the uniform bound of the density property, this allows us to use appropriate defect measures and their nice properties introduced and proved by Hillairet (Aspects interactifs de la m’ecanique des fluides, PhD Thesis, ENS Lyon, 2005) in order to prove that the Young measure associated to the weak limit is the convex combination of m Dirac measures. Finally, under a non-degeneracy assumption of this combination (“stratification” assumption), this provides a multi-fluid system. Using a weak–strong uniqueness argument, we prove that this convex combination is the one corresponding to the strong solution of the multi-fluid model built by Hillairet, if initial data are equal. We will briefly discuss this assumption. To complete the paper, we also present a blow-up criterion for this multi-fluid system following (Huang et al. in Serrin type criterion for the three-dimensional viscous compressible flows, arXiv, 2010).

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References

  1. Beale J.T., Kato T., Majda A.: Remarks on the breakdown of smooth solutions for the 3D euler equations. Commun. Math. Phys. 94, 61–66 (1984)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  2. Brenier, Y.: Homogénéisation variationnelle des équations d’Euler. Séminaire Équations aux dérivées partielles (Polytechnique), p. 17. Exp. No. 10 (1996–1997)

  3. Brenier, Y., De Lellis, C., Scékelyhidi, L. Jr: Weak–strong uniqueness for measure-valued solutions. Submitted, (2009)

  4. Bresch D., Desjardins B., Ghidaglia J.-M., Grenier E.: Global weak solutions to a generic two-fluid model. Arch. Rational Mech. Anal. 196, 599–629 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  5. Chen, G.Q., Perepelitsa, M.: Vanishing viscosity limit of the Navier–Stokes equations to the Euler equations for compressible flow. Submitted (2009)

  6. Desjardins B.: Regularity of weak solutions of the compressible isentropic Navier–Stokes equations. Commun. Partial Differ. Equ. 22(5–6), 977–1008 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Desjardins B., Esteban M.: On weak solutions for fluid-rigid structure interaction: compressible and incompressible models. Commun. Partial Differ. Equ. 25(7–8), 1399–1413 (2000)

    MATH  MathSciNet  Google Scholar 

  8. DiPerna R.J., Majda A.J.: Concentrations in regularizations for 2D incompressible flow. Commun. Pure Appl. Math. 40(3), 301–345 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  9. DiPerna R.J., Majda A.J.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Commun. Math. Phys. 108(4), 667–689 (1987)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Duchon J., Robert R.: Relaxation of Euler equations and hydrodynamics instabilities. Q. Appl. Math. 1(2), 235–255 (1992)

    MathSciNet  Google Scholar 

  11. W.E.: Propagation of oscillations in the solutions of 1d compressible fluid equations. Commun. Partial Differ. Equ. 17, 347–370 (1992)

  12. Embid P., Hunter J., Majda A.: Simplified asymptotic equations for the transition to detonation in reactive granular materials. SIAM J. Appl. Math. 52, 1199–1237 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  13. Evans, L.C.: Weak convergence methods for nonlinear partial differential equations. CBMS, vol. 74. American Mathematical Society, Providence, RI (1990)

  14. Feireisl E.: Dynamics of Viscous Compressible Fluids. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  15. Feireisl E.: On compactness of solutions to the compressible isentropic Navier–Stokes equations when the density is not square integrable. Comment. Math. Univ. Carolin. 42(1), 83–98 (2001)

    MATH  MathSciNet  Google Scholar 

  16. Gallay, TH.: Interaction of vortices in weakly viscous planar flows. Arch. Ration. Mech Anal. (2011, to appear)

  17. Germain, P.: Weak–strong uniqueness for the isentropic compressible Navier–Stokes system. J. Math. Fluid Mech. (2011, to appear)

  18. Hillairet M.: Propagation of density-oscillations in solutions to the barotropic compressible Navier–Stokes system. J. Math. Fluid Mech. 9, 343–376 (2007)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  19. Hillairet, M.: Aspects interactifs de la mécanique des fluides. PhD Thesis, ENS Lyon, 2005

  20. Hoff D.: Global existence of the Navier–Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differ. Equ. 120, 215–254 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hoff D.: Discontinous solutions of the Navier–Stokes equations for multi-dimensional flows of heat-conducting fluids. Arch. Ration. Mech Anal. 139, 303–354 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  22. Hoff, D.: Dynamics of singularly surface for compressible, viscous flows in two space dimensions. Commun. Pure Appl. Math. 1365–1407 (2002)

  23. Hoff D.: Uniqueness of weak solutions of the Navier–Stokes equations of multidimensional, compressible flow. SIAM J. Math. Anal. 37(6), 1742–1760 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hoff D., Santos M.M.: Lagrangean structure and propagation of singularities in multidimensional flow. Arch. Ration. Mech. Anal. 188, 509–543 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Huang, X., Li, J., Xin, Z.P.: Serrin type criterion for the three-dimensional viscous compressible flows, 2010, arXiv:1004.4748

  26. Ishii M.: Thermo-Fluid Dynamic Theory of Two-Phase Flow. Eyrolles, Paris (1975)

    MATH  Google Scholar 

  27. Lions, P.-L.:Mathematical topics in fluid mechanics, vol. 2. The Clarendon Press, Oxford University Press, 1998

  28. Ponce, G.: Remarks on a paper: “Remarks on the breakdown of smooth solutions for the 3D Euler equations” by J.T. Beale, T. Kato and A. Majda. Commun. Math. Phys., 98, 349–353 (1985)

  29. Serre D.: Variations de grande amplitude pour la densité d’un fluide visqueux compressible. Phys. D 48, 113–128 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  30. Sun, Y., Wang, C., Zhang, Z.: A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier—Stokes equations. J. Math. Pure Appl. (2010)

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Authors and Affiliations

  1. Laboratoire de Mathématiques, UMR 5127 CNRS, Université of Savoie, 73376, Le Bourget Du Lac, France

    D. Bresch

  2. Department of Mathematics, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    X. Huang

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  1. D. Bresch
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  2. X. Huang
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Correspondence to D. Bresch.

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Communicated by W. E

This research is supported by the ANR-08-BLAN-0301-01 project. This work has been done during the CNRS post-doctoral position of Xiangdi Huang in the Laboratoire de Mathématiques de l’Université de Savoie.

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Bresch, D., Huang, X. A Multi-Fluid Compressible System as the Limit of Weak Solutions of the Isentropic Compressible Navier–Stokes Equations. Arch Rational Mech Anal 201, 647–680 (2011). https://doi.org/10.1007/s00205-011-0400-8

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