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, pp. 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
@incollection{SB_2003-2004__46__29_0,
     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},
     note = {talk:926},
     pages = {29--57},
     publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France},
     address = {Paris},
     number = {299},
     year = {2005},
     mrnumber = {2167201},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2003-2004__46__29_0/}
}
TY  - CHAP
AU  - Gallagher, Isabelle
TI  - Résultats récents sur la limite incompressible
BT  - Séminaire Bourbaki : volume 2003/2004, exposés 924-937
AU  - Collectif
T3  - Astérisque
N1  - talk:926
PY  - 2005
DA  - 2005///
SP  - 29
EP  - 57
IS  - 299
PB  - Association des amis de Nicolas Bourbaki, Société mathématique de France
PP  - Paris
UR  - http://www.numdam.org/item/SB_2003-2004__46__29_0/
UR  - https://www.ams.org/mathscinet-getitem?mr=2167201
LA  - fr
ID  - SB_2003-2004__46__29_0
ER  - 
%0 Book Section
%A Gallagher, Isabelle
%T Résultats récents sur la limite incompressible
%B Séminaire Bourbaki : volume 2003/2004, exposés 924-937
%A Collectif
%S Astérisque
%Z talk:926
%D 2005
%P 29-57
%N 299
%I Association des amis de Nicolas Bourbaki, Société mathématique de France
%C Paris
%G fr
%F SB_2003-2004__46__29_0
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

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[17] W | MR | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

[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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

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

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