Skip to main content
SpringerLink
Log in

Singular Limits for Models of Compressible, Viscous, Heat Conducting, and/or Rotating Fluids

  • Reference work entry
  • First Online:
Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Abstract

The complete set of equations describing the motion of a general compressible, viscous, heat-conducting, and possibly rotating fluid arises as a mathematical model in a large variety of real world applications. The scale analysis aims at two different objectives: Rigorous derivation of a simplified asymptotic set of equations and understanding the passage from the original primitive system to the simplified target system. These issues are discussed in the context of compressible, viscous, heat conducting, and/or rotating fluids.

The research of E.F. leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ ERC Grant Agreement 320078. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 1,799.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 2,499.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. T. Alazard, Low Mach number flows and combustion. SIAM J. Math. Anal. 38(4), 1186–1213 (electronic) (2006)

    Article  MathSciNet  Google Scholar 

  2. T. Alazard, Low Mach number limit of the full Navier-Stokes equations. Arch. Ration. Mech. Anal. 180, 1–73 (2006)

    Article  MathSciNet  Google Scholar 

  3. A.S. Almgren, J.B. Bell, C.A. Rendleman, M. Zingale, Low Mach number modeling of type Ia supernovae. I. hydrodynamics. Astrophys. J. 637, 922–936 (2006)

    Article  Google Scholar 

  4. S.N. Antontsev, A.V. Kazhikhov, V.N. Monakhov, Krajevyje zadaci mechaniki neodnorodnych zidkostej. Novosibirsk (1983)

    Google Scholar 

  5. A. Babin, A. Mahalov, B. Nicolaenko, Global regularity of 3D rotating Navier-Stokes equations for resonant domains. Indiana Univ. Math. J. 48, 1133–1176 (1999)

    MathSciNet  MATH  Google Scholar 

  6. A. Babin, A. Mahalov, B. Nicolaenko, 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity. Indiana Univ. Math. J. 50(Special Issue), 1–35 (2001)

    Article  MathSciNet  Google Scholar 

  7. C. Bardos, T.T. Nguyen, Remarks on the inviscid liit for the compressible flows. Arxive Preprint Series (2014). arXiv 1410.4952v1

    Google Scholar 

  8. S.E. Bechtel, F.J. Rooney, M.G. Forest, Connection between stability, convexity of internal energy, and the second law for compressible Newtonian fuids. J. Appl. Mech. 72, 299–300 (2005)

    Article  Google Scholar 

  9. 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 Appl. 87, 57–90 (2007)

    Article  MathSciNet  Google Scholar 

  10. M. Bulíček, J. Málek, K.R. Rajagopal, Navier’s slip and evolutionary Navier-Stokes-like systems with pressure and shear- rate dependent viscosity. Indiana Univ. Math. J. 56, 51–86 (2007)

    Article  MathSciNet  Google Scholar 

  11. L. Caffarelli, R.V. Kohn, L. Nirenberg, On the regularity of the solutions of the Navier-Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)

    Article  Google Scholar 

  12. T. Chatelain, A. Henrot, Some results about Schiffer’s conjectures. Inverse Probl. 15(3), 647–658 (1999)

    Article  MathSciNet  Google Scholar 

  13. J.-Y. Chemin, B. Desjardins, I. Gallagher, E. Grenier, Mathematical Geophysics. Volume 32 of Oxford Lecture Series in Mathematics and Its Applications (The Clarendon Press/Oxford University Press, Oxford, 2006)

    Google Scholar 

  14. C. Cheverry, I. Gallagher, T. Paul, L. Saint-Raymond, Semiclassical and spectral analysis of oceanic waves. Duke Math. J. 161(5), 845–892 (2012)

    Article  MathSciNet  Google Scholar 

  15. A.J. Chorin, J.E. Marsden, A Mathematical Introduction to Fluid Mechanics (Springer, New York, 1979)

    Book  Google Scholar 

  16. C.M. Dafermos, The second law of thermodynamics and stability. Arch. Ration. Mech. Anal. 70, 167–179 (1979)

    Article  MathSciNet  Google Scholar 

  17. A.-L. Dalibard, L. Saint-Raymond, Mathematical study of the β-plane model for rotating fluids in a thin layer. J. Math. Pures Appl. (9) 94(2), 131–169 (2010)

    Google Scholar 

  18. R. Danchin, Low Mach number limit for viscous compressible flows. M2AN Math. Model Numer. Anal. 39, 459–475 (2005)

    Article  MathSciNet  Google Scholar 

  19. R. Danchin, X. Liao, On the well-posedness of the full low Mach number limit system in general critical Besov spaces Commun. Contemp. Math. 14(3), 1250022-1–1250022-47 (2012)

    Article  MathSciNet  Google Scholar 

  20. B. Desjardins, E. Grenier, Low Mach number limit of viscous compressible flows in the whole space. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455(1986), 2271–2279 (1999)

    Article  MathSciNet  Google Scholar 

  21. B. Desjardins, E. Grenier, P.-L. Lions, N. Masmoudi, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. 78, 461–471 (1999)

    Article  MathSciNet  Google Scholar 

  22. D.B. Ebin, The motion of slightly compressible fluids viewed as a motion with strong constraining force. Ann. Math. 105, 141–200 (1977)

    Article  MathSciNet  Google Scholar 

  23. R. Farwig, H. Kozono, H. Sohr, An Lq-approach to Stokes and Navier-Stokes equations in general domains. Acta Math. 195, 21–53 (2005)

    Article  MathSciNet  Google Scholar 

  24. C.L. Fefferman, Existence and smoothness of the Navier-Stokes equation, in The Millennium Prize Problems (Clay Mathematics Institute, Cambridge, 2006), pp. 57–67

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  26. E. Feireisl, Vanishing dissipation limit for the Navier-Stokes-Fourier system. Commun. Math. Sci., 14(6), pp. 1535–1551 (2015)

    Article  MathSciNet  Google Scholar 

  27. E. Feireisl, A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids (Birkhäuser-Verlag, Basel, 2009)

    Book  Google Scholar 

  28. E. Feireisl, A. Novotný, Weak-strong uniqueness property for the full Navier-Stokes-Fourier system. Arch. Ration. Mech. Anal. 204, 683–706 (2012)

    Article  MathSciNet  Google Scholar 

  29. E. Feireisl, A. Novotný, Inviscid incompressible limits of the full Navier-Stokes-Fourier system. Commun. Math. Phys. 321, 605–628 (2013)

    Article  MathSciNet  Google Scholar 

  30. E. Feireisl, A. Novotný, Inviscid incompressible limits under mild stratification: a rigorous derivation of the EulerBoussinesq system. Appl. Math. Optim. 70, 279–307 (2014)

    Article  MathSciNet  Google Scholar 

  31. E. Feireisl, A. Novotný, Multiple scales and singular limits for compressible rotating fluids with general initial data. Commun. Partial Differ. Equ. 39, 1104–1127 (2014)

    Article  MathSciNet  Google Scholar 

  32. E. Feireisl, M.E. Schonbek, On the Oberbeck-Boussinesq approximation on unbounded domains, in Abel Symposium Lecture Notes (Springer, Berlin, 2011)

    Google Scholar 

  33. E. Feireisl, A. Novotný, H. Petzeltová, On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids. J. Math. Fluid Mech. 3, 358–392 (2001)

    Article  MathSciNet  Google Scholar 

  34. E. Feireisl, A. Novotný, Y. Sun, Suitable weak solutions to the Navier–Stokes equations of compressible viscous fluids. Indiana Univ. Math. J. 60, 611–631 (2011)

    Article  MathSciNet  Google Scholar 

  35. E. Feireisl, I. Gallagher, D. Gerard-Varet, A. Novotný Multi-scale analysis of compressible viscous and rotating fluids. Commun. Math. Phys. 314, 641–670 (2012)

    Article  MathSciNet  Google Scholar 

  36. E. Feireisl, B.J. Jin, A. Novotný, Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system. J. Math. Fluid Mech. 14, 712–730 (2012)

    MathSciNet  MATH  Google Scholar 

  37. E. Feireisl, O. Kreml, Š Nečasová, J. Neustupa, J. Stebel, Weak solutions to the barotropic NavierStokes system with slip boundary conditions in time dependent domains. J. Differ. Equ. 254, 125–140 (2013)

    Article  Google Scholar 

  38. E. Feireisl, Š Nečasová, Y. Sun, Inviscid incompressible limits on expanding domains. Nonlinearity 27(10), 2465–2478 (2014)

    Article  MathSciNet  Google Scholar 

  39. E. Feireisl, R. Klein, A. Novotný, E. Kaminska, On singular limits arising in the scale analysis of stratified fluid flows. Math. Models Methods Appl. Sci. 26(3) 419–443 (2016)

    Article  MathSciNet  Google Scholar 

  40. I. Gallagher, Résultats récents sur la limite incompressible. Astérisque (299). Exp. No. 926, vii, 29–57, 2005. Séminaire Bourbaki, vol. 2003/2004

    Google Scholar 

  41. I. Gallagher, L. Saint-Raymond, Weak convergence results for inhomogeneous rotating fluid equations. C. R. Math. Acad. Sci. Paris 336(5), 401–406 (2003)

    Article  MathSciNet  Google Scholar 

  42. I. Gallagher, L. Saint-Raymond, Mathematical study of the betaplane model: equatorial waves and convergence results. Mém. Soc. Math. Fr. (N.S.) (107), v+116 (2006/2007)

    Article  Google Scholar 

  43. G. Gallavotti, Statistical Mechanics: A Short Treatise (Springer, Heidelberg, 1999)

    Book  Google Scholar 

  44. G. Gallavotti, Foundations of Fluid Dynamics (Springer, New York, 2002)

    Book  Google Scholar 

  45. P.A. Gilman, G.A. Glatzmaier, Compressible convection in a rotating spherical shell. I. Anelastic equations. Astrophys. J. Suppl. 45(2), 335–349 (1981)

    Article  MathSciNet  Google Scholar 

  46. G.A. Glatzmaier, P.A. Gilman, Compressible convection in a rotating spherical shell. II. A linear anelastic model. Astrophys. J. Suppl. 45(2), 351–380 (1981)

    MathSciNet  Google Scholar 

  47. D. Gough, The anelastic approximation for thermal convection. J. Atmos. Sci. 26, 448–456 (1969)

    Article  Google Scholar 

  48. E. Grenier, Y. Guo, T.T. Nguyen, Spectral stability of Prandtl boundary layers: an overview. Analysis (Berlin) 35(4), 343–355 (2015)

    Google Scholar 

  49. D. Hoff, The zero-Mach limit of compressible flows. Commun. Math. Phys. 192(3), 543–554 (1998)

    Article  MathSciNet  Google Scholar 

  50. D. Hoff, T.-P. Liu, The inviscid limit for the Navier-Stokes equations of compressible, isentropic flow with shock data. Indiana Univ. Math. J. 38(4), 861–915 (1989)

    Article  MathSciNet  Google Scholar 

  51. E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213–231 (1951)

    Article  MathSciNet  Google Scholar 

  52. T. Kato, On classical solutions of the two-dimensional nonstationary Euler equation. Arch. Ration. Mech. Anal. 25, 188–200 (1967)

    Article  MathSciNet  Google Scholar 

  53. T. Kato, Nonstationary flows of viscous and ideal fluids in r3. J. Funct. Anal. 9, 296–305 (1972)

    Article  Google Scholar 

  54. T. Kato, Remarks on the zero viscosity limit for nonstationary Navier–Stokes flows with boundary, in Seminar on PDE’s, ed. by S.S. Chern (Springer, New York, 1984)

    Google Scholar 

  55. T. Kato, Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary, in Seminar on Nonlinear Partial Differential Equations, Berkeley, 1983. Volume 2 of Mathematical Sciences Research Institute Publications (Springer, New York, 1984), pp. 85–98

    Google Scholar 

  56. T. Kato, C.Y. Lai, Nonlinear evolution equations and the Euler flow. J. Funct. Anal. 56, 15–28 (1984)

    Article  MathSciNet  Google Scholar 

  57. S. Klainerman, A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34, 481–524 (1981)

    Article  MathSciNet  Google Scholar 

  58. R. Klein, Scale-dependent models for atmospheric flows, in Annual Review of Fluid Mechanics. Annual Review of Fluid Mechanics, vol. 42 (Annual Reviews, Palo Alto, 2010), pp. 249–274

    Article  MathSciNet  Google Scholar 

  59. R. Klein, N. Botta, T. Schneider, C.D. Munz, S. Roller, A. Meister, L. Hoffmann, T. Sonar, Asymptotic adaptive methods for multi-scale problems in fluid mechanics. J. Eng. Math. 39, 261–343 (2001)

    Article  MathSciNet  Google Scholar 

  60. J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)

    Article  MathSciNet  Google Scholar 

  61. J. Lighthill, On sound generated aerodynamically I. General theory. Proc. R. Soc. Lond. A 211, 564–587 (1952)

    Article  MathSciNet  Google Scholar 

  62. J. Lighthill, On sound generated aerodynamically II. General theory. Proc. R. Soc. Lond. A 222, 1–32 (1954)

    Google Scholar 

  63. P.-L. Lions, Mathematical Topics in Fluid Dynamics. Incompressible Models, vol. 1 (Oxford Science Publication, Oxford, 1996)

    Google Scholar 

  64. P.-L. Lions, Mathematical Topics in Fluid Dynamics. Compressible Models, vol. 2 (Oxford Science Publication, Oxford, 1998)

    Google Scholar 

  65. P.-L. Lions, N. Masmoudi, Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998)

    Article  MathSciNet  Google Scholar 

  66. P.-L. Lions, N. Masmoudi, Une approche locale de la limite incompressible. C. R. Acad. Sci. Paris Sér. I Math. 329(5), 387–392 (1999)

    Article  MathSciNet  Google Scholar 

  67. F.B. Lipps, R.S. Hemler, A scale analysis of deep moist convection and some related numerical calculations. J. Atmos. Sci. 39, 2192–2210 (1982)

    Article  Google Scholar 

  68. A. Majda, High Mach number combustion. Lect. Notes Appl. Math. 24, 109–184 (1986)

    MathSciNet  Google Scholar 

  69. A. Majda, Introduction to PDE’s and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematics, vol. 9 (Courant Institute, New York, 2003)

    Google Scholar 

  70. N. Masmoudi, The Euler limit of the Navier-Stokes equations, and rotating fluids with boundary. Arch. Ration. Mech. Anal. 142(4), 375–394 (1998)

    Article  MathSciNet  Google Scholar 

  71. N. Masmoudi, Ekman layers of rotating fluids: the case of general initial data. Commun. Pure Appl. Math.. 53, 432–483 (2000)

    Article  MathSciNet  Google Scholar 

  72. N. Masmoudi, Incompressible, inviscid limit of the compressible Navier-Stokes system. Ann. Inst. Henri Poincaré, Anal. non linéaire 18, 199–224 (2001)

    Article  MathSciNet  Google Scholar 

  73. N. Masmoudi, Examples of singular limits in hydrodynamics, in Handbook of Differential Equations, III, ed. by C. Dafermos, E. Feireisl (Elsevier, Amsterdam, 2006)

    Google Scholar 

  74. N. Masmoudi, Rigorous derivation of the anelastic approximation. J. Math. Pures Appl. 88, 230–240 (2007)

    Article  MathSciNet  Google Scholar 

  75. G. Métivier, S. Schochet, The incompressible limit of the non-isentropic Euler equations. Arch. Ration. Mech. Anal. 158, 61–90 (2001)

    Article  MathSciNet  Google Scholar 

  76. G. Métivier, S. Schochet, Averaging theorems for conservative systems and the weakly compressible Euler equations. J. Differ. Equ. 187, 106–183 (2003)

    Article  MathSciNet  Google Scholar 

  77. M. Michálek, Stability result for Navier-stokes equations with entropy transport. J. Math. Fluid Mech. 17(2), 279–285 (2015)

    Article  MathSciNet  Google Scholar 

  78. Y. Ogura, M. Phillips, Scale analysis for deep and shallow convection in the atmosphere. J. Atmos. Sci. 19, 173–179 (1962)

    Article  Google Scholar 

  79. M. Oliver, Classical solutions for a generalized Euler equation in two dimensions. J. Math. Anal. Appl. 215, 471–484 (1997)

    Article  MathSciNet  Google Scholar 

  80. J. Pedlosky, Geophysical Fluid Dynamics (Springer, New York, 1987)

    Book  Google Scholar 

  81. L. Saint-Raymond, Hydrodynamic limits: some improvements of the relative entropy method. Annal. I.H.Poincaré AN 26, 705–744 (2009)

    Article  MathSciNet  Google Scholar 

  82. S. Schochet, The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit. Commun. Math. Phys. 104, 49–75 (1986)

    Article  MathSciNet  Google Scholar 

  83. S. Schochet, Fast singular limits of hyperbolic PDE’s. J. Differ. Equ. 114, 476–512 (1994)

    Article  MathSciNet  Google Scholar 

  84. S. Schochet, The mathematical theory of low Mach number flows. M2ANMath. Model Numer. Anal. 39, 441–458 (2005)

    Article  MathSciNet  Google Scholar 

  85. J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer, New York, 1967)

    MATH  Google Scholar 

  86. F. Sueur, On the inviscid limit for the compressible Navier-Stokes system in an impermeable bounded domain. J. Math. Fluid Mech. 16(1), 163–178 (2014)

    Article  MathSciNet  Google Scholar 

  87. V.A. Vaigant, A.V. Kazhikhov, On the existence of global solutions to two-dimensional Navier-Stokes equations of a compressible viscous fluid (in Russian). Sibirskij Mat. Z. 36(6), 1283–1316 (1995)

    MATH  Google Scholar 

  88. G.K. Vallis, Atmospheric and Oceanic Fluid Dynamics (Cambridge University Press, Cambridge, 2006)

    Book  Google Scholar 

  89. S. Wang, S. Jiang, The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations. Commun. Partial Differ. Equ. 31(4–6), 571–591 (2006)

    Article  MathSciNet  Google Scholar 

  90. E. Weinan, Boundary layer theory and the zero-viscosity limit of the Navier-Stokes equation. Acta Math. Sin. Engl. Ser. 16, 207–218 (2000)

    Article  MathSciNet  Google Scholar 

  91. R.Kh. Zeytounian. Asymptotic Modeling of Atmospheric Flows (Springer, Berlin, 1990)

    Book  Google Scholar 

  92. R.Kh. Zeytounian, Asymptotic Modelling of Fluid Flow Phenomena. Volume 64 of Fluid Mechanics and Its Applications (Kluwer Academic, Dordrecht, 2002)

    Google Scholar 

  93. R.Kh. Zeytounian, Joseph Boussinesq and his approximation: a contemporary view. C. R. Mecanique 331, 575–586 (2003)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Evolution Differential Equations (EDE), Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Prague, Czech Republic

    Eduard Feireisl

Authors
  1. Eduard Feireisl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eduard Feireisl .

Editor information

Editors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Tokyo, Japan

    Yoshikazu Giga

  2. Université de Toulon IMATH, Toulon, France

    Antonín Novotný

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this entry

Check for updates. Verify currency and authenticity via CrossMark

Cite this entry

Feireisl, E. (2018). Singular Limits for Models of Compressible, Viscous, Heat Conducting, and/or Rotating Fluids. In: Giga, Y., Novotný, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-13344-7_70

Download citation

Publish with us

Policies and ethics

Search

Navigation