In this survey paper, we are concerned with the zero Mach number limit for compressible viscous flows. For the sake of (mathematical) simplicity, we restrict ourselves to the case of barotropic fluids and we assume that the flow evolves in the whole space or satisfies periodic boundary conditions. We focus on the case of ill-prepared data. Hence highly oscillating acoustic waves are likely to propagate through the fluid. We nevertheless state the convergence to the incompressible Navier-Stokes equations when the Mach number goes to . Besides, it is shown that the global existence for the limit equations entails the global existence for the compressible model with small . The reader is referred to [R. Danchin, Ann. Sci. Éc. Norm. Sup. (2002)] for the detailed proof in the whole space case, and to [R. Danchin, Am. J. Math. 124 (2002) 1153-1219] for the case of periodic boundary conditions.
Keywords: low Mach number limit, compressible Navier-Stokes
@article{M2AN_2005__39_3_459_0,
author = {Danchin, Rapha\"el},
title = {Low {Mach} number limit for viscous compressible flows},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {459--475},
year = {2005},
publisher = {EDP Sciences},
volume = {39},
number = {3},
doi = {10.1051/m2an:2005019},
mrnumber = {2157145},
zbl = {1080.35067},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an:2005019/}
}
TY - JOUR AU - Danchin, Raphaël TI - Low Mach number limit for viscous compressible flows JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2005 SP - 459 EP - 475 VL - 39 IS - 3 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an:2005019/ DO - 10.1051/m2an:2005019 LA - en ID - M2AN_2005__39_3_459_0 ER -
%0 Journal Article %A Danchin, Raphaël %T Low Mach number limit for viscous compressible flows %J ESAIM: Modélisation mathématique et analyse numérique %D 2005 %P 459-475 %V 39 %N 3 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an:2005019/ %R 10.1051/m2an:2005019 %G en %F M2AN_2005__39_3_459_0
Danchin, Raphaël. Low Mach number limit for viscous compressible flows. ESAIM: Modélisation mathématique et analyse numérique, Special issue on Low Mach Number Flows Conference, Tome 39 (2005) no. 3, pp. 459-475. doi: 10.1051/m2an:2005019
[1] , Work in progress (2004).
[2] , Ondelettes, paraproduits et Navier-Stokes. Diderot Ed., Paris (1995). | Zbl | MR
[3] , Global existence in critical spaces for compressible Navier-Stokes equations. Invent. Math. 141 (2000) 579-614. | Zbl
[4] , Global existence in critical spaces for flows of compressible viscous and heat-conductive gases. Arch. Rational Mech. Anal. 160 (2001) 1-39. | Zbl
[5] , Local theory in critical spaces for compressible viscous and heat-conductive gases. Comm. Partial Differential Equations 26 (2001) 1183-1233. | Zbl
[6] , Zero Mach number limit for compressible flows with periodic boundary conditions. Am. J. Math. 124 (2002) 1153-1219. | Zbl
[7] , Zero Mach number limit in critical spaces for compressible Navier-Stokes equations. Ann. Sci. Éc. Norm. Sup. (2002). | Zbl | MR | Numdam
[8] , On the uniqueness in critical spaces for compressible navier-stokes equations. Nonlinear Differential Equations and Applications, to appear (2002). | Zbl | MR
[9] and, Low Mach number limit of viscous compressible flows in the whole space. Proc. Roy. Soc. London Ser. A, Math. Phys. Eng. Sci. 455 (1999) 2271-2279. | Zbl
[10] ,, and, Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. (2002). | Zbl
[11] and, On the Navier-Stokes initial value problem. I. Arch. Rational Mech. Anal. 16 (1964) 269-315. | Zbl
[12] , A remark on smooth solutions of the weakly compressible periodic Navier-Stokes equations. J. Math. Kyoto Univ. 40 (2000) 525-540. | Zbl
[13] and, Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133 (1995) 50-68. | Zbl
[14] and, All-time existence of classical solutions for slightly compressible flows. SIAM J. Math. Anal. 29 (1998) 652-672. | Zbl
[15] , The zero-Mach limit of compressible flows. Comm. Math. Phys. 192 (1998) 543-554. | Zbl
[16] , Strong -solutions of the Navier-Stokes equation in , with applications to weak solutions. Math. Z. 187 (1984) 471-480. | Zbl
[17] and, Endpoint Strichartz estimates. Am. J. Math. 120 (1998) 955-980. | Zbl
[18] and, Compressible and incompressible fluids. Comm. Pure Appl. Math. 35 (1982) 629-651. | Zbl
[19] , and, Convergence of the solutions of the compressible to the solutions of the incompressible Navier-Stokes equations. Adv. Appl. Math. 12 (1991) 187-214. | Zbl
[20] , Mathematical topics in fluid mechanics. Vol. 1: Incompressible models. Oxford Clarendon Press (1996). | Zbl | MR
[21] , Mathematical topics in fluid mechanics. Vol. 2: Compressible models. Oxford Clarendon Press (1998). | Zbl | MR
[22] and, Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. (9) 77 (1998) 585-627. | Zbl
[23] and, Une approche locale de la limite incompressible. C. R. Acad. Sci. Paris (1999). | Zbl | MR
[24] , Incompressible, inviscid limit of the compressible Navier-Stokes system. Ann. Inst. H. Poincaré Anal. Non Linéaire (2001). | Zbl | MR | Numdam
[25] and, The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20 (1980) 67-104. | Zbl
[26] and, The incompressible limit of the non-isentropic Euler equations. Arch. Rational Mech. Anal. 158 (2001) 61-90. | Zbl
[27] and, Averaging theorems for conservative systems and the weakly compressible Euler equations. J. Differential Equations 187 (2003) 106-183. | Zbl
[28] , Fast singular limits of hyperbolic PDEs. J. Differential Equations 114 (1994) 476-512. | Zbl
[29] , The incompressible limit and the initial layer of the compressible Euler equation. J. Math. Kyoto Univ. 26 (1986) 323-331. | Zbl
Cité par Sources :
