Low Mach number limit for viscous compressible flows

Raphaël Danchin

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2005)

  • Volume: 39, Issue: 3, page 459-475
  • ISSN: 0764-583X

Abstract

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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 ϵ goes to 0 . Besides, it is shown that the global existence for the limit equations entails the global existence for the compressible model with small ϵ . 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.

How to cite

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Danchin, Raphaël. "Low Mach number limit for viscous compressible flows." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 39.3 (2005): 459-475. <http://eudml.org/doc/245249>.

@article{Danchin2005,
abstract = {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 $\epsilon $ goes to $0$. Besides, it is shown that the global existence for the limit equations entails the global existence for the compressible model with small $\epsilon $. 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.},
author = {Danchin, Raphaël},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {low Mach number limit; compressible Navier-Stokes; slightly compressible; incompressible; well-prepared data; ill-prepared data; convergence to the incompressible Navier-Stokes equations; global existence},
language = {eng},
number = {3},
pages = {459-475},
publisher = {EDP-Sciences},
title = {Low Mach number limit for viscous compressible flows},
url = {http://eudml.org/doc/245249},
volume = {39},
year = {2005},
}

TY - JOUR
AU - Danchin, Raphaël
TI - Low Mach number limit for viscous compressible flows
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2005
PB - EDP-Sciences
VL - 39
IS - 3
SP - 459
EP - 475
AB - 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 $\epsilon $ goes to $0$. Besides, it is shown that the global existence for the limit equations entails the global existence for the compressible model with small $\epsilon $. 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.
LA - eng
KW - low Mach number limit; compressible Navier-Stokes; slightly compressible; incompressible; well-prepared data; ill-prepared data; convergence to the incompressible Navier-Stokes equations; global existence
UR - http://eudml.org/doc/245249
ER -

References

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