This paper is devoted to the study of smooth flows of density-dependent fluids in or in the torus . 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 when goes to . A blow-up criterion involving the norm of vorticity in is also proved. Besides, we show that if the density-dependent Euler equations have a smooth solution on a given time interval , then the density-dependent Navier-Stokes equations with the same data and small viscosity have a smooth solution on . The viscous solution tends to the Euler solution when the viscosity goes to . The rate of convergence in is of order .
An appendix is devoted to the proof of elliptic estimates in Sobolev spaces with positive or negative regularity indices, interesting for their own sake.
Cet article est consacré à l’étude des fluides incompressibles à densité variable dans ou . On cherche à généraliser plusieurs résultats classiques pour les équations d’Euler et de Navier-Stokes incompressibles.
On établit un résultat d’existence et d’unicité sur un intervalle de temps indépendant de la viscosité du fluide ainsi qu’un critère d’explosion faisant intervenir la norme du tourbillon dans . On montre en outre que si les équations d’Euler ont une solution régulière sur un intervalle de temps donné alors les équations de Navier-Stokes avec mêmes données et petite viscosité ont une solution régulière sur le même intervalle de temps. De plus la solution visqueuse tend vers la solution d’Euler quand la viscosité tend vers 0. Le taux de convergence dans est de l’ordre de .
En appendice, on démontre des estimations a priori de type elliptique dans des espaces de Sobolev à indice positif ou négatif.
@article{AFST_2006_6_15_4_637_0, author = {Danchin, Rapha\"el}, title = {The inviscid limit for density-dependent incompressible fluids}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {637--688}, publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques}, address = {Toulouse}, volume = {Ser. 6, 15}, number = {4}, year = {2006}, doi = {10.5802/afst.1133}, mrnumber = {2295208}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/afst.1133/} }
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 SP - 637 EP - 688 VL - 15 IS - 4 PB - Université Paul Sabatier, Institut de mathématiques PP - Toulouse UR - http://archive.numdam.org/articles/10.5802/afst.1133/ DO - 10.5802/afst.1133 LA - en ID - AFST_2006_6_15_4_637_0 ER -
%0 Journal Article %A Danchin, Raphaël %T The inviscid limit for density-dependent incompressible fluids %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2006 %P 637-688 %V 15 %N 4 %I Université Paul Sabatier, Institut de mathématiques %C Toulouse %U http://archive.numdam.org/articles/10.5802/afst.1133/ %R 10.5802/afst.1133 %G en %F AFST_2006_6_15_4_637_0
Danchin, Raphaël. The inviscid limit for density-dependent incompressible fluids. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 15 (2006) no. 4, pp. 637-688. doi : 10.5802/afst.1133. http://archive.numdam.org/articles/10.5802/afst.1133/
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