The inviscid limit for density-dependent incompressible fluids

Cet article est consacré à l’étude des fluides incompressibles à densité variable dans ${ℝ}^N$ ou ${𝕋}^N$. 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 $\mu$ du fluide ainsi qu’un critère d’explosion faisant intervenir la norme du tourbillon dans $L^1(0,T;L^{\infty })$. On montre en outre que si les équations d’Euler ont une solution régulière sur un intervalle de temps $[0,T_0]$ 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 $L^2$ est de l’ordre de $\mu$.

En appendice, on démontre des estimations a priori de type elliptique dans des espaces de Sobolev à indice positif ou négatif.

This paper is devoted to the study of smooth flows of density-dependent fluids in ${ℝ}^N$ or in the torus ${𝕋}^N$. 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.

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