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Some Recent Asymptotic Results in Fluid Mechanics

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Analysis and Simulation of Fluid Dynamics

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

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Abstract

The general equations of fluid mechanics are the law of mass conservation, the Navier-Stokes equation, the law of energy conservation and the laws of thermodynamics. These equations are merely written in this generality. Instead, one often prefers simplified forms. To obtain reduced systems, the easiest route is to introduce dimensionless numbers which quantify the importance of various physical processes. Many recent works are devoted to the study of the classical solutions when such a dimensionless number goes to zero. A few results in this field are here reviewed.

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References

  1. T. Alazard — “Alentours de la limite incompressible”, in Séminaire: Équations aux Dérivées Partielles, 2004–2005, École Polytech., 2005, Exp. No. XXIV.

    Google Scholar 

  2. T. Alazard — “Incompressible limit of the nonisentropic Euler equations with solid wall boundary conditions”, Adv. in Differential Equations 10 (2005), p. 19–44.

    MathSciNet  MATH  Google Scholar 

  3. T. Alazard — “Low Mach number of the full Navier-Stokes equations”, Arch. Ration. Mech. Anal. (2005), in press.

    Google Scholar 

  4. T. Alazard — “Low Mach number flows, and combustion”, preprint.

    Google Scholar 

  5. S. Alinhac — “Temps de vie des solutions régulières des équations d’Euler compressibles axisymétriques en dimension deux”, Invent. Math. 111 (1993), no. 3, p. 627–670.

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Bresch, D. Gérard-Varet & E. Grenier — “Derivation of the planetary geostrophic equations”, preprint ENS, 2005.

    Google Scholar 

  7. F. Charve — “Etude de phénomènes dispersifs en mécanique des fluides géophysiques”, Thèse de doctorat de l’école Polytechnique, 2004.

    Google Scholar 

  8. F. Charve — “Global well-posedness and asymptotics for a geophysical fluid system”, Comm. Partial Differential Equations 29 (2004), no. 11–12, p. 1919–1940.

    MathSciNet  MATH  Google Scholar 

  9. F. Charve — “Convergence of weak solutions for the primitive system of the quasigeostrophic equations”, Asymptot. Anal. 42 (2005), no. 3–4, p. 173–209.

    MathSciNet  MATH  Google Scholar 

  10. J.-Y. Chemin, B. Desjardins, I. Gallagher & E. Grenier — “Basis of mathematical geophysics”, to appear, 2005.

    Google Scholar 

  11. C. Cheverry — “Propagation of oscillations in real vanishing viscosity limit”, Comm. Math. Phys. 247 (2004), no. 3, p. 655–695.

    Article  MathSciNet  MATH  Google Scholar 

  12. C. Cheverry, O. Guès & G. Métivier — “Oscillations fortes sur un champ linéairement dégénéré”, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 5, p. 691–745.

    MATH  Google Scholar 

  13. C. Cheverry, O. Guès & G. Métivier — “Large-amplitude high-frequency waves for quasilinear hyperbolic systems”, Adv. Differential Equations 9 (2004), no. 7–8, p. 829–890.

    MathSciNet  MATH  Google Scholar 

  14. J.-F. Coulombel & T. Goudon — “The strong relaxation limit of multidimensional isothermal euler equations”, to appear in Trans. Am. Soc., 2005.

    Google Scholar 

  15. R. Danchin — “Global existence in critical spaces for flows of compressible viscous and heat-conductive gases”, Arch. Ration. Mech. Anal. 160 (2001), no. 1, p. 1–39.

    Article  MathSciNet  MATH  Google Scholar 

  16. R. Danchin — “Zero Mach number limit for compressible flows with periodic boundary conditions”, Amer. J. Math. 124 (2002), no. 6, p. 1153–1219.

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Danchin — “Zero Mach number limit in critical spaces for compressible Navier-Stokes equations”, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 1, p. 27–75.

    MathSciNet  MATH  Google Scholar 

  18. R. Danchin — “Low mach number limit for viscous compressible flows”, in special issue vol. 39 No. 3 (May–June 2005), M2AN Math. Model. Numer. Anal., 2005.

    Google Scholar 

  19. 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 (1999), no. 1986, p. 2271–2279.

    Article  MathSciNet  MATH  Google Scholar 

  20. 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. (9) 78 (1999), no. 5, p. 461–471.

    Article  MathSciNet  Google Scholar 

  21. A. Dutrifoy & T. Hmidi — “The incompressible limit of solutions of the two-dimensional compressible Euler system with degenerating initial data”, Comm. Pure Appl. Math. 57 (2004), no. 9, p. 1159–1177.

    Article  MathSciNet  MATH  Google Scholar 

  22. I. Gallagher — “Résultats récents sur la limite incompressible”, Séminaire Bourbaki 2003–2004 Exp. No. 926.

    Google Scholar 

  23. I. Gallagher & L. Saint-Raymond — “On pressureless gases driven by a strong inhomogeneous magnetic field”, SIAM J. Math. Anal. 36 (2005), no. 4, p. 1159–1176.

    Article  MathSciNet  MATH  Google Scholar 

  24. I. GALLAGHER & L. SAINT-RAYMOND — “Résultats asymptotiques pour des fluides en rotation inhomogène, in Séminaire: Équations aux Dérivées Partielles, 2003–2004, École Polytech., 2004, Exp. No. III.

    Google Scholar 

  25. P. Godin — “The lifespan of a class of smooth spherically symmetric solutions of the compressible euler equations with variable entropy in three space dimensions several spaces variables”, Arch. Ration. Mech. Anal. 177 (2005), no. 3, p. 479–511.

    Article  MathSciNet  MATH  Google Scholar 

  26. B. Hanouzet & R. Natalini — “Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy”, Arch. Ration. Mech. Anal. 169 (2003), no. 2, p. 89–117.

    Article  MathSciNet  MATH  Google Scholar 

  27. D. Hoff — “The zero-Mach limit of compressible flows”, Comm. Math. Phys. 192 (1998), no. 3, p. 543–554.

    Article  MathSciNet  MATH  Google Scholar 

  28. P.-L. Lions & N. Masmoudi — “Incompressible limit for a viscous compressible fluid”, J. Math. Pures Appl. (9) 77 (1998), no. 6, p. 585–627.

    MathSciNet  MATH  Google Scholar 

  29. A. Majda — Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984.

    MATH  Google Scholar 

  30. M. Majdoub & M. Paicu — “Uniform local existence for inhomogeneous rotating fluid equations”, in preparation, 2005.

    Google Scholar 

  31. G. Métivier & S. Schochet — “The incompressible limit of the non-isentropic Euler equations”, Arch. Ration. Mech. Anal. 158 (2001), no. 1, p. 61–90.

    Article  MathSciNet  MATH  Google Scholar 

  32. G. Métivier & S. Schochet — “Averaging theorems for conservative systems and the weakly compressible Euler equations”, J. Differential Equations 187 (2003), no. 1, p. 106–183.

    Article  MathSciNet  MATH  Google Scholar 

  33. S. Schochet — “The mathematical theory of low mach numbers flows”, in special issue vol. 39 No. 3 (May–June 2005), M2AN Math. Model. Numer. Anal., 2005.

    Google Scholar 

  34. T.C. Sideris, B. Thomases & D. Wang — “Long time behavior of solutions to the 3D compressible Euler equations with damping”, Comm. Partial Differential Equations 28 (2003), no. 3–4, p. 795–816.

    Article  MathSciNet  MATH  Google Scholar 

  35. W.-A. Yong — “Entropy and global existence for hyperbolic balance laws”, Arch. Ration. Mech. Anal. 172 (2004), no. 2, p. 247–266.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. MAB Université de Bordeaux I, F-33405, Talence Cedex, France

    Thomas Alazard

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  1. Thomas Alazard
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Editors and Affiliations

  1. Laboratoire Paul Painleve, UMR 8524, CNRS-Université des Sciences et Techno., F-59655, Villeneuve d’Ascq cedex, France

    Caterina Calgaro , Jean-François Coulombel  & Thierry Goudon ,  & 

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© 2006 Birkhäuser Verlag Basel/Switzerland

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Alazard, T. (2006). Some Recent Asymptotic Results in Fluid Mechanics. In: Calgaro, C., Coulombel, JF., Goudon, T. (eds) Analysis and Simulation of Fluid Dynamics. Advances in Mathematical Fluid Mechanics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7742-7_1

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