Résultats récents sur la limite incompressible  [ Recent results on the incompressible limit ]
Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 926, p. 29-57

In the last two decades, a great amount of progress has been made in the understanding of the passage from the equations governing compressible fluids, to the incompressible equations. The aim of this talk is to present the evolution of the mathematical methods used to study that limit, from the works of S. Klainerman and A. Majda in the eighties, to the recent studies of G. Métivier and S. Schochet (for the non isentropic equations). The methods followed are different according to the initial conditions as well as the boundary conditions; we will describe methods of geometrical optics as well as others linked to the theory of defect measures, Strichartz estimates, and also small divisor type computations.

La compréhension du passage des équations de la mécanique des fluides compressibles aux équations incompressibles a fait de grands progrès ces vingt dernières années. L'objectif de cet exposé est de présenter l'évolution des méthodes mathématiques mises en œuvre pour étudier ce passage à la limite, depuis les travaux de S. Klainerman et A. Majda dans les années quatre-vingts, jusqu'à ceux récents de G. Métivier et S. Schochet (pour les équations non isentropiques). Suivant les conditions initiales et les conditions aux bords, les méthodes utilisées sont variées, et nous décrirons des résultats de type optique géométrique aussi bien que d'autres liés à la théorie des mesures de défaut, à des estimations de Strichartz ou encore à des calculs de petits diviseurs.

Classification:  76B03,  76N10,  78M35
Keywords: incompressible limit, oscillations, dispersion, defect measures
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     author = {Gallagher, Isabelle},
     title = {R\'esultats r\'ecents sur la limite incompressible},
     booktitle = {S\'eminaire Bourbaki : volume 2003/2004, expos\'es 924-937},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France},
     address = {Paris},
     number = {299},
     year = {2005},
     note = {talk:926},
     pages = {29-57},
     mrnumber = {2167201},
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Gallagher, Isabelle. Résultats récents sur la limite incompressible, in Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Talk no. 926, pp. 29-57. http://www.numdam.org/item/SB_2003-2004__46__29_0/

[1] T. Alazard - “Incompressible limit of the non-isentropic Euler equations with solid wall boundary conditions”, soumis. | Zbl 1101.35050

[2] V. Arnold - “Sur la géométrie différentiable des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits”, Ann. Inst. Fourier (Grenoble) 16 (1966), p. 319-361. | Numdam | MR 202082 | Zbl 0148.45301

[3] H. Bahouri & J.-Y. Chemin - “Équations d'ondes quasilinéaires et estimation de Strichartz”, Amer. J. Math. 121 (1999), p. 1337-1377. | MR 1719798 | Zbl 0952.35073

[4] H. Beirao Da Veiga - “Singular limits in compressible fluid dynamics”, Arch. Rational Mech. Anal. 128 (1994), no. 4, p. 313-327. | MR 1308856 | Zbl 0829.76073

[5] -, “On the sharp singular limit for slightly compressible fluids”, Math. Methods Appl. Sci. 18 (1995), no. 4, p. 295-306. | MR 1320000 | Zbl 0819.35111

[6] D. Bresch, B. Desjardins, E. Grenier & C. K. Lin - “Low Mach number limit of viscous polytropic flows : formal asymptotics in the periodic case”, Stud. Appl. Math. 109 (2002), no. 2, p. 125-149. | MR 1917042 | Zbl 1114.76347

[7] G. Browning & H.-O. Kreiss - “Problems with different time scales for nonlinear partial differential equations”, SIAM J. Appl. Math. 42 (1982), no. 4, p. 704-718. | MR 665380 | Zbl 0506.35006

[8] N. Burq - “Mesures semi-classiques et mesures de défaut”, in Sém. Bourbaki (1996/97), Astérisque, vol. 245, Société Mathématique de France, 1997, Exp. No.826, p. 167-195. | Numdam | MR 1627111 | Zbl 0954.35102

[9] J.-Y. Chemin - Fluides parfaits incompressibles, Astérisque, vol. 230, Société Mathématique de France, 1995. | Numdam | MR 1340046 | Zbl 0829.76003

[10] R. Danchin - “Fluides légèrement compressibles et limite incompressible”, in Sém. Équations aux Dérivées Partielles, 2000-2001, École polytechnique, 2001, Exp. No. III, 19 pages. | Numdam | MR 1860675 | Zbl 1061.35511

[11] -, “Zero Mach number limit for compressible flows with periodic boundary conditions”, Amer. J. Math. 124 (2002), no. 6, p. 1153-1219. | MR 1939784 | Zbl 1048.35075

[12] -, “Zero Mach number limit in critical spaces for compressible Navier-Stokes equations” 35 (2002), no. 1, p. 27-75. | Numdam | MR 1886005 | Zbl 1048.35054

[13] B. Desjardins & E. Grenier - “Low Mach number limit of viscous compressible flows in the whole space”, Proc. Roy. Soc. London Ser. A 455 (1999), no. 1986, p. 2271-2279. | MR 1702718 | Zbl 0934.76080

[14] 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. | MR 1697038 | Zbl 0992.35067

[15] B. Desjardins & C.-K. Lin - “A survey of the compressible Navier-Stokes equations”, Taiwanese J. Math. 3 (1999), no. 2, p. 123-137. | MR 1692781 | Zbl 0930.35121

[16] A. Dutrifoy & T. H'Midi - “The incompressible limit of solutions of the two-dimensional compressible Euler system with degenerating initial data”, C. R. Acad. Sci. Paris Sér. I Math. 336 (2003), no. 6, p. 471-474. | MR 1975081 | Zbl 1044.35044

[17] W | MR 1163429 | Zbl 0760.35007

[18] C. Fermanian & P. Gérard - “Mesures semi-classiques et croisements de mode”, Bull. Soc. math. France 130 (2002), no. 1, p. 123-168. | Numdam | MR 1906196 | Zbl 0996.35004

[19] I. Gallagher - “Asymptotics of the solutions of hyperbolic equations with a skew-symmetric perturbation”, J. Differential Equations 150 (1998), p. 363-384. | MR 1658597 | Zbl 0921.35095

[20] P. Gérard - “Microlocal defect measures”, Comm. Partial Differential Equations 16 (1991), p. 1761-1794. | MR 1135919 | Zbl 0770.35001

[21] J. Ginibre & G. Velo - “Generalized Strichartz inequalities for the wave equation”, J. Funct. Anal. 133 (1995), p. 50-68. | MR 1351643 | Zbl 0849.35064

[22] E. Grenier - “Oscillatory perturbations of the Navier-Stokes equations”, J. Math. Pures Appl. 76 (1997), p. 477-498. | MR 1465607 | Zbl 0885.35090

[23] G. Hagedorn - “Proof of the Landau-Zener formula in an adiabatic limit with small eigenvalue gap”, Comm. Math. Phys. 110 (1987), p. 519-524. | MR 1099690 | Zbl 0723.35068

[24] T. Hagstrom & J. Lorenz - “All-time existence of classical solutions for slightly compressible flows”, SIAM J. Appl. Math. 28 (1998), no. 3, p. 652-672. | MR 1617767 | Zbl 0907.76073

[25] -, “On the stability of approximate solutions of hyperbolic-parabolic systems and the all-time existence of smooth, slightly compressible flows”, Indiana Univ. Math. J. 51 (2002), no. 6, p. 1339-1387. | MR 1948453 | Zbl 1039.35085

[26] T. Iguchi - “The incompressible limit and the initial layer of the compressible Euler equation in + n , Math. Methods Appl. Sci. 20 (1997), no. 11, p. 945-958. | MR 1458527 | Zbl 0884.35127

[27] H. Isozaki - “Singular limits for the compressible Euler equation in an exterior domain”, J. reine angew. Math. 381 (1987), p. 1-36. | MR 918838 | Zbl 0618.76073

[28] -, “Wave operators and the incompressible limit of the compressible Euler equation”, Comm. Math. Phys. 110 (1987), no. 3, p. 519-524. | MR 891951 | Zbl 0627.76081

[29] -, “Singular limits for the compressible Euler equation in an exterior domain. II. Bodies in a uniform flow”, Osaka J. Math. 26 (1989), no. 2, p. 399-410. | MR 1017594 | Zbl 0716.35065

[30] A. Joye - “Proof of the Landau-Zener formula”, Asymptotic Anal. 9 (1994), p. 209-258. | MR 1295294 | Zbl 0814.35109

[31] T. Kato - Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1980. | Zbl 0435.47001

[32] S. Klainerman - “Global existence for nonlinear wave equations”, Comm. Pure Appl. Math. 33 (1980), p. 43-101. | MR 544044 | Zbl 0405.35056

[33] S. Klainerman & M. Machedon - “Remark on Strichartz type inequalites”, Internat. Math. Res. Notices 5 (1996), p. 201-220, with an appendix of J. Bourgain and D. Tataru. | MR 1383755 | Zbl 0853.35062

[34] S. Klainerman & A. Majda - “Singular limits of quasilinear hyperbolic systems with large parameters, and the incompressible limit of compressible fluids”, Comm. Pure Appl. Math. 34 (1981), p. 481-524. | MR 615627 | Zbl 0476.76068

[35] -, “Compressible and incompressible fluids”, Comm. Pure Appl. Math. 35 (1982), p. 629-651. | MR 668409 | Zbl 0478.76091

[36] R. Klein - “Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics, I, One-dimensional flow”, J. Comput. Phys. 121 (1995), no. 2, p. 213-237. | MR 1354300 | Zbl 0842.76053

[37] H.-O. Kreiss, J. Lorenz & M. J. Naughton - “Convergence of the solutions of the compressible to the solutions of the incompressible Navier-Stokes equations”, Adv. in Appl. Mech. 12 (1991), no. 2, p. 187-214. | MR 1101207 | Zbl 0728.76084

[38] P. Lax - “Hyperbolic systems of conservation laws II”, Comm. Pure Appl. Math. 10 (1957), p. 537-566. | MR 93653 | Zbl 0081.08803

[39] P.-L. Lions - Mathematical Topics in Fluid Mechanics, Vol. I, Incompressible Models, Oxford Science Publications, 1997. | Zbl 0866.76002

[40] -, Mathematical Topics in Fluid Mechanics, Vol. II, Compressible Models, Oxford Science Publications, 1997. | Zbl 0908.76004

[41] P.-L. Lions & N. Masmoudi - “Incompressible limit for a viscous compressible fluid”, J. Math. Pures Appl. 77 (1998), no. 6, p. 585-627. | MR 1628173 | Zbl 0909.35101

[42] -, “Une approche locale de la limite incompressible”, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 5, p. 387-392. | MR 1710123 | Zbl 0937.35132

[43] A. Majda - Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. | MR 748308 | Zbl 0537.76001

[44] A. Majda & S. Osher - “Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary”, Comm. Pure Appl. Math. 28 (1975), p. 607-675. | MR 410107 | Zbl 0314.35061

[45] C. Marchioro & M. Pulvirenti - Mathematical Theory of Incompressible Nonviscous Fluids, Applied Mathematical Sciences, vol. 96, Springer-Verlag, New York, 1994. | MR 1245492 | Zbl 0789.76002

[46] N. Masmoudi - “Asymptotic problems and compressible-incompressible limit”, in Advances in mathematical fluid mechanics (Paseky, 1999), Springer, Berlin, 2000, p. 119-158. | MR 1863211 | Zbl 0970.35119

[47] -, “Incompressible, inviscid limit of the compressible Navier-Stokes system”, Ann. Inst. H. Poincaré. Anal. Non Linéaire 18 (2001), no. 2, p. 199-224. | Numdam | MR 1808029 | Zbl 0991.35058

[48] A. Meister - “Asymptotic single and multiple scale expansions in the low Mach number limit”, SIAM J. Appl. Math. 60 (1999), no. 1, p. 256-271. | MR 1740844 | Zbl 0941.35052

[49] G. Métivier & S. Schochet - “Limite incompressible des équations d'Euler non isentropiques”, in Sém. Équations aux Dérivées Partielles, 2000-2001, École polytechnique, 2001, Exp. No. X, 17 pages. | Numdam | MR 1860682 | Zbl 1061.76074

[50] -, “The incompressible limit of the non-isentropic Euler equations”, Arch. Rational Mech. Anal. 158 (2001), no. 1, p. 61-90. | MR 1834114 | Zbl 0974.76072

[51] -, “Averaging theorems for conservative systems and the weakly compressible Euler equations”, J. Differential Equations 187 (2003), no. 1, p. 106-183. | MR 1946548 | Zbl 1029.34035

[52] C. D. Munz - “Computational fluid dynamics and aeroacoustics for low Mach number flow”, in Hyperbolic partial differential equations (Hamburg, 2001), Vieweg, Braunschweig, 2002, p. 269-320. | MR 1913809

[53] T. Schneider, N. Botta, K. J. Geratz & R. Klein - “Extension of finite volume compressible flow solvers to multi-dimensional, variable density zero Mach number flows”, J. Comput. Phys. 155 (1999), no. 2, p. 248-286. | MR 1723333 | Zbl 0968.76054

[54] S. Schochet - “The compressible Euler equations in a bounded domain : existence of solutions and the incompressible limit”, Comm. Math. Phys. 104 (1986), p. 49-75. | MR 834481 | Zbl 0612.76082

[55] -, “Fast singular limits of hyperbolic PDEs”, J. Differential Equations 114 (1994), p. 476-512. | MR 1303036 | Zbl 0838.35071

[56] P. Secchi - “On the incompressible limit of inviscid compressible fluids”, Ann. Univ. Ferrara Sez. VII (N.S.) 46 (2000), p. 21-33, in Navier-Stokes equations and related nonlinear problems (Ferrara, 1999). | MR 1896921 | Zbl 1006.76081

[57] -, “On the singular incompressible limit of inviscid compressible fluids”, J. Fluid Mech. 2 (2000), no. 2, p. 107-125. | MR 1765773 | Zbl 0965.35127

[58] T. Sideris - “Formation of singularities in three-dimensional compressible fluids”, Comm. Math. Phys. 101 (1985), no. 4, p. 475-485. | MR 815196 | Zbl 0606.76088

[59] L. Tartar - “H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations”, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), p. 193-230. | MR 1069518 | Zbl 0774.35008

[60] S. Ukai - “The incompressible limit and the initial layer of the compressible Euler equation”, J. Math. Kyoto Univ. 26 (1986), no. 2, p. 323-331. | MR 849223 | Zbl 0618.76074

[61] V. Yudovitch - “Non stationary flows of an ideal incompressible fluid”, Zhurnal Vych Matematika 3 (1963), p. 1032-1066. | MR 158189

[62] R. Zeytounian - Asymptotic modelling of fluid flow phenomena, Fluid Mechanics and its Applications, vol. 64, Kluwer Academic Publishers, Dordrecht, 2002. | MR 1894273 | Zbl 1045.76001

[63] -, Theory and applications of nonviscous fluid flows, Springer-Verlag, Berlin, 2002. | MR 1885859 | Zbl 0992.76002