Viscous Compressible Flows Under Pressure

Part of the Advances in Mathematical Fluid Mechanics book series (AMFM)


This chapter deals with the role of pressure in the theory of viscous compressible flows. The pressure state laws and viscosities are described. Special attention is devoted to non-monotone pressure laws and pressure dependent viscosities. The global existence proofs are discussed for approximate systems. Some relevant physical applications are described, including among others the anelastic Euler equations, shallow water model, granular media, or mixture problems.



D. Bresch is partially supported by SingFlows project, grand ANR-18-CE40-0027. P.–E. Jabin is partially supported by NSF DMS Grants 161453, 1908739 and NSF Grant RNMS (Ki-Net) 1107444.


  1. 1.
    P. Antonelli, S. Spirito. Global existence of weak solutions to the Navier-Stokes-Korteweg equations. ArXiv:1903.02441 (2019).Google Scholar
  2. 2.
    P. Antonelli, S. Spirito. On the compactness of weak solutions to the Navier-Stokes-Korteweg equations for capillary fluids. ArXiv:1808.03495 (2018).Google Scholar
  3. 3.
    P. Antonelli, S. Spirito. Global Existence of Finite Energy Weak Solutions of Quantum Navier-Stokes Equations. Archive of Rational Mechanics and Analysis, 225 (2017), no. 3, 1161–1199.MathSciNetCrossRefGoogle Scholar
  4. 4.
    P. Antonelli, S. Spirito. On the compactness of finite energy weak solutions to the Quantum Navier-Stokes equations. J. of Hyperbolic Differential Equations, 15 (2018), no. 1, 133–147.MathSciNetCrossRefGoogle Scholar
  5. 5.
    F. Ben Belgacem, P.–E. Jabin. Compactness for nonlinear continuity equations. J. Funct. Anal, 264, no. 1, 139–168, (2013).Google Scholar
  6. 6.
    F. Ben Belgacem, P.–E. Jabin. Convergence of numerical approximations to non-linear continuity equations with rough force fields. Submitted (2018).Google Scholar
  7. 7.
    D. Bresch, F. Couderc, P. Noble, J.–P. Vila. A generalization of the quantum Bohm identity: hyperbolic CFL condition for Euler-Korteweg equations. C.R. Acad. Sciences Paris. volume 354, Issue 1, 39–43, (2016).Google Scholar
  8. 8.
    D. Bresch, B. Desjardins. Existence of global weak solutions for 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Comm. Math. Phys., 238 (2003), no.1-3, 211–223.MathSciNetCrossRefGoogle Scholar
  9. 9.
    D. Bresch and B. Desjardins. On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models. J. Math. Pures Appl. (9) 86 (2006), no. 4, 362–368.Google Scholar
  10. 10.
    D. Bresch, B. Desjardins. Quelques modèles diffusifs capillaires de type Korteweg. C. R. Acad. Sci. Paris, section mécanique, 332, no. 11, 881–886, (2004).Google Scholar
  11. 11.
    D. Bresch, B. Desjardins. Weak solutions via the total energy formulation and their quantitative properties - density dependent viscosities. In: Y. Giga, A. Novotný (éds.) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Berlin (2017).Google Scholar
  12. 12.
    D. Bresch, B. Desjardins. On the existence of global weak solutions to the Navier–Stokes equations for viscous compressible and heat-conducting fluids. J. Math. Pures et Appl., 57–90 (2007).Google Scholar
  13. 13.
    D. Bresch, B. Desjardins, Chi-Kun Lin. On some compressible fluid models: Korteweg, lubrication, and shallow water systems. Comm. Partial Differential Equations 28, no. 3-4, 843–868, (2003).Google Scholar
  14. 14.
    D. Bresch, B. Desjardins, D. Gérard-Varet. On compressible Navier-Stokes equations with density dependent viscosities in bounded domains. J. Math. Pures Appl. 87 (9), 227–235 (2007).MathSciNetCrossRefGoogle Scholar
  15. 15.
    D. Bresch, B. Desjardins, J.–M. Ghidaglia, E. Grenier. Global weak solutions to a generic two-fluid model. Arch. Rational Mech. Analysis. 196, Issue 2, 599–629, (2010).Google Scholar
  16. 16.
    D. Bresch, B. Desjardins, E. Zatorska. Two-velocity hydrodynamics in Fluid Mechanics, Part II. Existence of global κ-entropy solutions to compressible Navier-Stokes system with degenerate viscosities. J. Math. Pures Appl. Volume 104, Issue 4, 801–836 (2015).Google Scholar
  17. 17.
    D. Bresch, P.-E. Jabin. Global existence of weak solutions for compressible Navier-Stokes equations: thermodynamically unstable pressure and anisotropic viscous stress tensor. Ann. of Math. (2) 188, no. 2, 577–684 (2018).Google Scholar
  18. 18.
    D. Bresch, P.–E. Jabin. Quantitative regularity estimates for compressible transport equations. New trends and results in mathematical description of fluid flows. Birkhauser 77–113, (2018).Google Scholar
  19. 19.
    D. Bresch, P.–E. Jabin. Quantitative regularity estimates for advective equation with anelastic degenerate constraint. Proc. Int. Cong. of Math, 2161–2186 (2018).Google Scholar
  20. 20.
    D. Bresch, P.–E. Jabin, F. Wang. On heat-conducting Navier-Stokes equations with a truncated virial pressure state law. In preparation (2019).Google Scholar
  21. 21.
    D. Bresch, I. Lacroix-Violet, M. Gisclon. On Navier-Stokes-Korteweg and Euler-Korteweg systems: Application to quantum fluids models. To appear in Arch. Rational Mech. Anal. (2019).Google Scholar
  22. 22.
    D. Bresch, P. Mucha, E. Zatorska. Finite-energy solutions for compressible two-fluid Stokes system. Arch. Rational Mech. Anal., 232, Issue 2, 987–1029, (2019).Google Scholar
  23. 23.
    D. Bresch, A. Vasseur, C. Yu. Global existence of entropy-weak solutions to the compressible Navier-Stokes equations with non-linear density dependent viscosities. Submitted (2019).Google Scholar
  24. 24.
    R. Carles, K. Carrapatoso, M. Hillairet. Rigidity results in generalized isothermal fluids. Annales Henri Lebesgue, 1, (2018), 47–85.MathSciNetCrossRefGoogle Scholar
  25. 25.
    E. Feireisl. Dynamics of viscous compressible fluids. Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004.Google Scholar
  26. 26.
    E. Feireisl, A. Novotný, H. Petzeltová. On the existence of globally defined weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech.3 (2001), 358–392.MathSciNetCrossRefGoogle Scholar
  27. 27.
    E. Feireisl. Compressible Navier–Stokes Equations with a Non-Monotone Pressure Law. J. Diff. Eqs 183, no 1, 97–108, (2002).Google Scholar
  28. 28.
    E. Feireisl, A. Vasseur. New perspectives in fluid dynamics: Mathematical analysis of a model proposed by Howard Brenner. New directions in mathematical fluid mechanics, 153–179, Adv. Math. Fluid Mech., Birkhauser Verlag, Basel, 2010.Google Scholar
  29. 29.
    S. Gavrilyuk, S.M. Shugrin Media with equations of state that depend on derivatives. J. Applied Mech. Physics, vol. 37, No2, (1996).Google Scholar
  30. 30.
    Z. Guo, Q. Jiu, Z. Xin. Spherically symmetric isentropic compressible flows with density-dependent viscosity coefficients. SIAM J. Math. Anal. 39, no. 5, 1402–1427, (2008).Google Scholar
  31. 31.
    S. Jiang, Z. Xin, P. Zhang. Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity. Methods Appl. Anal. 12, no. 3, 239–251, (2005).Google Scholar
  32. 32.
    S. Jiang, P. Zhang. On spherically symmetric solutions of the compressible isentropic Navier-Stokes equations. Comm. Math. Phys. 215, no. 3, 559–581, (2001).Google Scholar
  33. 33.
    A. Jüngel. Global weak solutions to compressible Navier-Stokes equations for quantum fluids. SIAM J. Math. Anal. 42, no. 3, 1025–1045, (2010).Google Scholar
  34. 34.
    A. Jüngel, D. Matthes. The Derrida–Lebowitz-Speer-Spohn equations: Existence, uniqueness, and Decay rates of the solutions. SIAM J. Math. Anal., 39(6), 1996–2015, (2008).MathSciNetCrossRefGoogle Scholar
  35. 35.
    I. Lacroix-Violet, A. Vasseur. Global weak solutions to the compressible quantum Navier-Stokes equation and its semi-classical limit. J. Math. Pures Appl. (9) 114, 191–210, (2018).Google Scholar
  36. 36.
    J. Leray. Sur le mouvement d’un fluide visqueux remplissant l’espace, Acta Math. 63, 193–248, (1934).MathSciNetCrossRefGoogle Scholar
  37. 37.
    J. Li, Z.P. Xin. Global Existence of Weak Solutions to the Barotropic Compressible Navier-Stokes Flows with Degenerate Viscosities. arXiv:1504.06826 (2015).Google Scholar
  38. 38.
    P.-L. Lions.Mathematical topics in fluid mechanics. Vol. 2. Compressible models. Oxford Lecture Series in Mathematics and its Applications, 10. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998.Google Scholar
  39. 39.
    D. Maltese, M. Michalek, P. Mucha, A. Novotny, M. Pokorny, E. Zatorska. Existence of weak solutions for compressible Navier-Stokes with entropy transport. J. Differential Equations, 261, No. 8, 4448–4485 (2016)MathSciNetCrossRefGoogle Scholar
  40. 40.
    A. Mellet, A. Vasseur. On the barotropic compressible Navier-Stokes equations. Comm. Partial Differential Equations 32 (2007), no. 1-3, 431–452.MathSciNetCrossRefGoogle Scholar
  41. 41.
    P.B. Mucha, M. Pokorny, E. Zatorska. Approximate solutions to a model of two-component reactive flow. Discrete Contin. Dyn. Syst. Ser. S, 7, No. 5 , 1079–1099 (2014).Google Scholar
  42. 42.
    A. Novotny. Weak solutions for a bi-fluid model of a mixture of two compressible non interacting fluids. Submitted (2018).Google Scholar
  43. 43.
    A. Novotny, M. Pokorny. Weak solutions for some compressible multi-component fluid models. Submitted (2018).Google Scholar
  44. 44.
    C. Perrin. Pressure Dependent Viscosity Model for Granular Media Obtained from Compressible Navier-Stokes Equations. Appl Math Res Express, vol. 2016, Iss. 2, p. 289–333 (2016).Google Scholar
  45. 45.
    C. Perrin. An overview on congestion phenomena in fluid equations. Proceeding Journées EDP 2018. See hal-01994880Google Scholar
  46. 46.
    P.I. Plotnikov, W. Weigant. Isothermal Navier-Stokes equations and Radon transform. SIAM J. Math. Anal. 47 (2015), no. 1, 626–653.MathSciNetCrossRefGoogle Scholar
  47. 47.
    F. Rousset. Solutions faibles de l’équation de Navier-Stokes des fluides compressible [d’après A. Vasseur et C. Yu]. Séminaire Bourbaki, 69ème année, 2016–2017, no 1135.Google Scholar
  48. 48.
    S.M. Shugrin. Two-velocity hydrodynamics and thermodynamics. J. Applied Mech and Tech. Physics, 39, 522–537, (1994).MathSciNetCrossRefGoogle Scholar
  49. 49.
    E.M. SteinHarmonic Analysis. Princeton Univ. Press 1995 (second edition).Google Scholar
  50. 50.
    V. A. Vaigant, A. V. Kazhikhov. On the existence of global solutions of two-dimensional Navier-Stokes equations of a compressible viscous fluid. (Russian). Sibirsk. Mat. Zh. 36 (1995), no. 6, 1283–1316, ii; translation in Siberian Math. J. 36 (1995), no.6, 1108–1141.Google Scholar
  51. 51.
    A. Vasseur, C. Yu. Global weak solutions to compressible quantum Navier-Stokes equations with damping. SIAM J. Math. Anal. 48 (2016), no. 2, 1489–1511.MathSciNetCrossRefGoogle Scholar
  52. 52.
    A. Vasseur, C. Yu. Existence of Global Weak Solutions for 3D Degenerate Compressible Navier-Stokes Equations. Inventiones mathematicae (2016), 1–40.Google Scholar
  53. 53.
    A. Vasseur, H. Wen, C. Yu. Global weak solution to the viscous two-phase model with finite energy. To appear in J. Math Pures Appl. (2018).Google Scholar

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Authors and Affiliations

  1. 1.University of Savoie Mont BlancLe Bourget du LacFrance
  2. 2.University of MarylandMarylandUSA

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