Rigorous derivation of the anelastic approximation

Journal des Mathematiques Pures et Appliquees

Volumn 88, Issue 3, 2007, Pages 230-240

  Masmoudi, Nader ^a  

^a NEW YORK UNIVERSITY   (United States)

Author keywords

Anelastic approximation; Compressible incompressible limit; Froude number; Mach number; Singular limit

Indexed keywords

EID: 34548425680     PISSN: 00217824     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.matpur.2007.06.001     Document Type: Article

References (33)

1. Alazard T. Low Mach number limit of the full Navier-Stokes equations. Arch. Ration. Mech. Anal. (2005)
2. Babin A., Mahalov A., and Nicolaenko B. Regularity and integrability of 3D Euler and Navier-Stokes equations for rotating fluids. Asymptot. Anal. 15 2 (1997) 103-150
3. Bois P.-A. A unified asymptotic theory of the anelastic approximation in geophysical gases and liquids. Mech. Res. Comm. 33 5 (2006) 628-635
4. Bresch D., Desjardins B., Grenier E., and Lin C.-K. Low Mach number limit of viscous polytropic flows: formal asymptotics in the periodic case. Stud. Appl. Math. 109 2 (2002) 125-149
5. Chemin J.-Y., Desjardins B., Gallagher I., and Grenier E. Mathematical Geophysics. Oxford Lecture Series in Mathematics and its Applications vol. 32 (2006), The Clarendon Press, Oxford University Press, Oxford An introduction to rotating fluids and the Navier-Stokes equations
6. Danchin R. Low Mach number limit for viscous compressible flows. M2AN Math. Model. Numer. Anal. 39 3 (2005) 459-475
7. Desjardins B., Grenier E., Lions P.-L., and Masmoudi N. Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. (9) 78 5 (1999) 461-471
8. Durran D.R. Numerical Methods for Wave Equations in Geophysical Fluid Dynamics (1999), Springer-Verlag, New York
9. Ebin D.G. The motion of slightly compressible fluids viewed as a motion with strong constraining force. Ann. of Math. (2) 105 1 (1977) 141-200
10. 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 (2001) 83-98
11. Feireisl E. Dynamics of Viscous Compressible Fluids. Oxford Lecture Series in Mathematics and its Applications vol. 26 (2004), Oxford University Press, Oxford
12. E. Feireisl, J. Málek, A. Novotný, I. Straskraba, Anelastic approximation as a singular limit of the compressible Navier-Stokes system, Preprint, 2006
13. Feireisl E., and Petzeltová H. On integrability up to the boundary of the weak solutions of the Navier-Stokes equations of compressible flow. Comm. Partial Differential Equations 25 3-4 (2000) 755-767
14. I. Gallagher, Résultats récents sur la limite incompressible, Séminaire Bourbaki (926), 2004
15. Grenier E. Oscillatory perturbations of the Navier-Stokes equations. J. Math. Pures Appl. (9) 76 6 (1997) 477-498
16. Klainerman S., and Majda A. Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math. 34 4 (1981) 481-524
17. Klein R. Multiple spatial scales in engineering and atmospheric low Mach number flows. M2AN Math. Model. Numer. Anal. 39 3 (2005) 537-559
18. Leray J. Essai sur les mouvements plans d'un liquide visqueux emplissant l'espace. Acta Math. 63 (1934) 193-248
19. Leray J. Essai sur les mouvements plans d'un liquide visqueux qui limitent des parois. J. Math. Pures Appl. 13 (1934) 331-418
20. Lions P.-L. Mathematical Topics in Fluid Mechanics, vol. 1. Incompressible Models. Oxford Science Publications (1996), The Clarendon Press, Oxford University Press, New York
21. Lions P.-L. Mathematical Topics in Fluid Mechanics, vol. 2. Compressible Models. Oxford Science Publications (1998), The Clarendon Press, Oxford University Press, New York
22. Lions P.-L. Bornes sur la densité pour les équations de Navier-Stokes compressibles isentropiques avec conditions aux limites de Dirichlet. C. R. Acad. Sci. Paris Sér. I Math. 328 8 (1999) 659-662
23. Lions P.-L., and Masmoudi N. Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. (9) 77 6 (1998) 585-627
24. Lions P.-L., and Masmoudi N. Une approche locale de la limite incompressible. C. R. Acad. Sci. Paris Sér. I Math. 329 5 (1999) 387-392
25. Lipps F., and Hemler R. A scale analysis of deep moist convection and some related numerical calculations. J. Atmos. Sci. 29 (1982) 2192-2210
26. Majda A. Introduction to PDEs and Waves for the Atmosphere and Ocean. Courant Lecture Notes in Mathematics vol. 9 (2003), New York University Courant Institute of Mathematical Sciences, New York
27. Masmoudi N. Asymptotic problems and compressible-incompressible limit. Advances in Mathematical Fluid Mechanics. Paseky, 1999 (2000), Springer, Berlin 119-158
28. Masmoudi N. Ekman layers of rotating fluids: the case of general initial data. Comm. Pure Appl. Math. 53 4 (2000) 432-483
29. Masmoudi N. Incompressible, inviscid limit of the compressible Navier-Stokes system. Ann. Inst. H. Poincaré Anal. Non Linéaire 18 2 (2001) 199-224
30. Masmoudi N. Examples of singular limits in hydrodynamics. Evolutionary Equations, vol. III, Handbook of Differential Equations (2007), Elsevier/North-Holland, Amsterdam 195-276
31. Métivier G., and Schochet S. Averaging theorems for conservative systems and the weakly compressible Euler equations. J. Differential Equations 187 1 (2003) 106-183
32. Ogura Y., and Phillips N. Scale analysis for deep and shallow convection in the atmosphere. J. Atmos. Sci. 19 (1962) 173-179
33. Schochet S. The mathematical theory of low Mach number flows. M2AN Math. Model. Numer. Anal. 39 3 (2005) 441-458