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Comptes Rendus Mathématique
Volume 343, n° 3
pages 219-224 (août 2006)
Doi : 10.1016/j.crma.2006.05.016
Received : 1 December 2005 ;  accepted : 15 May 2006
Stabilité de solutions faibles globales pour les équations de Navier-Stokes compressible avec température
Stability of global weak solutions for the Navier-Stokes equations modelling compressible and heat conducting fluids
 

Didier Bresch a , Benoît Desjardins b, c
a LMC-IMAG UMR5223, 51, rue des mathématiques, B.P. 53, 38041 Grenoble, France 
b CEA/DIF, B.P. 12, 91680 Bruyères le Châtel, France 
c DMA E.N.S. Ulm, 45, rue dʼUlm, 75230 Paris cedex 05, France 

Résumé

Nous présentons un résultat de stablité globale en temps de suites de solutions faibles « à la Leray » des équations de Navier-Stokes compressibles modélisant un fluide visqueux conducteur de chaleur dans le cas de lʼespace entier   (ou dans un domaine   avec conditions aux limites périodiques) pour des données initiales arbitrairement grandes. Des hypothèses sont faites sur la dépendance en densité et température de la condutivité thermique et des coefficients de viscosité et μ , qui assurent des propriétés importantes de conservation déjà mises en évidence par les auteurs. Lʼéquation dʼétat est supposée celle dʼun gaz parfait polytropique, à laquelle on ajoute une composante de pression et dʼénergie interne à température nulle, qui ne joue un rôle que pour les faibles densités. Notre résultat complète celui de P.-L. Lions, restreint aux écoulements barotropes. Pour citer cet article : D. Bresch, B. Desjardins, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

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Abstract

We present a global in time stability result for sequences of weak solutions à la Leray' to the Navier-Stokes equations modelling viscous compressible heat conducting fluids in the whole space   (or in the box   with periodic boundary conditions) with arbitrary large initial data. Specific assumptions are made on the density and temperature dependence of the thermal conduction and the viscosity coefficients and μ in order to preserve a particular conservation property discovered by the authors. The underlying mathematical structure is the key ingredient to get additional information on the density which allows to define weak solutions and get strong compactness results needed on the temperature. The equation of state is assumed to be the perfect polytropic gas law with an additional zero isothermal component that plays a role only for small density. Our result extends the work of P.-L. Lions restricted to barotropic flows obtained in 1993. Note that approximate solutions construction process, i.e. sequences of suitable smooth approximate solutions, is explained elsewhere. To cite this article: D. Bresch, B. Desjardins, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

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