The inviscid limit for density-dependent incompressible fluids

Danchin, Raphaël. "The inviscid limit for density-dependent incompressible fluids." Annales de la faculté des sciences de Toulouse Mathématiques 15.4 (2006): 637-688. <http://eudml.org/doc/10018>.

@article{Danchin2006,
abstract = {This paper is devoted to the study of smooth flows of density-dependent fluids in $\{\mathbb\{R\}\}^N$ or in the torus $\{\mathbb\{T\}\}^N$. We aim at extending several classical results for the standard Euler or Navier-Stokes equations, to this new framework.Existence and uniqueness is stated on a time interval independent of the viscosity $\mu $ when $\mu $ goes to $0$. A blow-up criterion involving the norm of vorticity in $L^1(0,T;L^\infty )$ is also proved. Besides, we show that if the density-dependent Euler equations have a smooth solution on a given time interval $[0,T_0]$, then the density-dependent Navier-Stokes equations with the same data and small viscosity have a smooth solution on $[0,T_0]$. The viscous solution tends to the Euler solution when the viscosity $\mu $ goes to $0$. The rate of convergence in $L^2$ is of order $\mu $.An appendix is devoted to the proof of elliptic estimates in Sobolev spaces with positive or negative regularity indices, interesting for their own sake.},
affiliation = {Centre de Mathématiques, Univ. Paris 12, 61 av. du Général de Gaulle, 94010 Créteil Cedex, France},
author = {Danchin, Raphaël},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {incompressible nonhomogeneous fluids},
language = {eng},
number = {4},
pages = {637-688},
publisher = {Université Paul Sabatier, Toulouse},
title = {The inviscid limit for density-dependent incompressible fluids},
url = {http://eudml.org/doc/10018},
volume = {15},
year = {2006},
}

TY - JOUR
AU - Danchin, Raphaël
TI - The inviscid limit for density-dependent incompressible fluids
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2006
PB - Université Paul Sabatier, Toulouse
VL - 15
IS - 4
SP - 637
EP - 688
AB - This paper is devoted to the study of smooth flows of density-dependent fluids in ${\mathbb{R}}^N$ or in the torus ${\mathbb{T}}^N$. We aim at extending several classical results for the standard Euler or Navier-Stokes equations, to this new framework.Existence and uniqueness is stated on a time interval independent of the viscosity $\mu $ when $\mu $ goes to $0$. A blow-up criterion involving the norm of vorticity in $L^1(0,T;L^\infty )$ is also proved. Besides, we show that if the density-dependent Euler equations have a smooth solution on a given time interval $[0,T_0]$, then the density-dependent Navier-Stokes equations with the same data and small viscosity have a smooth solution on $[0,T_0]$. The viscous solution tends to the Euler solution when the viscosity $\mu $ goes to $0$. The rate of convergence in $L^2$ is of order $\mu $.An appendix is devoted to the proof of elliptic estimates in Sobolev spaces with positive or negative regularity indices, interesting for their own sake.
LA - eng
KW - incompressible nonhomogeneous fluids
UR - http://eudml.org/doc/10018
ER -