A Lagrangian approach for the compressible Navier-Stokes equations

Raphaël Danchin[1]

  • [1] Université Paris-Est LAMA, UMR 8050 & Institut Universitaire de France 61 avenue du Général de Gaulle 94010 Créteil Cedex (France)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 2, page 753-791
  • ISSN: 0373-0956

Abstract

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Here we investigate the Cauchy problem for the barotropic Navier-Stokes equations in ℝ n , in the critical Besov spaces setting. We improve recent results as regards the uniqueness condition: initial velocities in critical Besov spaces with (not too) negative indices generate a unique local solution. Apart from (critical) regularity, the initial density just has to be bounded away from 0 and to tend to some positive constant at infinity. Density-dependent viscosity coefficients may be considered. Using Lagrangian coordinates is the key to our statements as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence, and Lipschitz continuity of the flow map (in Lagrangian coordinates) is established.

How to cite

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Danchin, Raphaël. "A Lagrangian approach for the compressible Navier-Stokes equations." Annales de l’institut Fourier 64.2 (2014): 753-791. <http://eudml.org/doc/275445>.

@article{Danchin2014,
abstract = {Here we investigate the Cauchy problem for the barotropic Navier-Stokes equations in $\mathbb\{R\}^n$, in the critical Besov spaces setting. We improve recent results as regards the uniqueness condition: initial velocities in critical Besov spaces with (not too) negative indices generate a unique local solution. Apart from (critical) regularity, the initial density just has to be bounded away from $0$ and to tend to some positive constant at infinity. Density-dependent viscosity coefficients may be considered. Using Lagrangian coordinates is the key to our statements as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence, and Lipschitz continuity of the flow map (in Lagrangian coordinates) is established.},
affiliation = {Université Paris-Est LAMA, UMR 8050 & Institut Universitaire de France 61 avenue du Général de Gaulle 94010 Créteil Cedex (France)},
author = {Danchin, Raphaël},
journal = {Annales de l’institut Fourier},
keywords = {Compressible fluids; uniqueness; critical regularity; Lagrangian coordinates; compressible fluids},
language = {eng},
number = {2},
pages = {753-791},
publisher = {Association des Annales de l’institut Fourier},
title = {A Lagrangian approach for the compressible Navier-Stokes equations},
url = {http://eudml.org/doc/275445},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Danchin, Raphaël
TI - A Lagrangian approach for the compressible Navier-Stokes equations
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 2
SP - 753
EP - 791
AB - Here we investigate the Cauchy problem for the barotropic Navier-Stokes equations in $\mathbb{R}^n$, in the critical Besov spaces setting. We improve recent results as regards the uniqueness condition: initial velocities in critical Besov spaces with (not too) negative indices generate a unique local solution. Apart from (critical) regularity, the initial density just has to be bounded away from $0$ and to tend to some positive constant at infinity. Density-dependent viscosity coefficients may be considered. Using Lagrangian coordinates is the key to our statements as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence, and Lipschitz continuity of the flow map (in Lagrangian coordinates) is established.
LA - eng
KW - Compressible fluids; uniqueness; critical regularity; Lagrangian coordinates; compressible fluids
UR - http://eudml.org/doc/275445
ER -

References

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