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• W2024482949 abstract "The purpose of this work is to investigate the problem of global in time existence of sequences of weak solutions to the Navier–Stokes equations for viscous compressible and heat conducting fluids. A class of density and temperature dependent viscosity and conductivity coefficients is considered. This result extends P.-L. Lions' work in 1993 [P.-L. Lions, Compacité des solutions des équations de Navier–Stokes compressibles isentropiques, C. R. Acad. Sci. Paris, Sér. I 317 (1993) 115–120] restricted to barotropic flows, and provides weak solutions “à la Leray” to the full compressible model that includes internal energy evolution equation with thermal conduction effects. A partial answer is therefore given to this currently widely open problem, described for instance in P.-L. Lions' book [P.-L. Lions, Mathematical Topics in Fluid Dynamics, vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998]. The proof uses the generalization to the temperature dependent case, of a new mathematical entropy equality derived by the authors in [D. Bresch, B. Desjardins, Some diffusive capillary models of Korteweg type, C. R. Acad. Sci., Paris, Section Mécanique 332 (11) (2004) 881–886]. The construction scheme of approximate solutions, using on additional regularizing effects such as capillarity, is provided in [D. Bresch, B. Desjardins, On the construction of approximate solutions for 2D viscous shallow water model and for compressible Navier–Stokes models, J. Math. Pures Appl. 86 (4) (2006) 362–368], and allows to use the stability arguments of this paper. Le but de ce travail est d'étudier le problème de l'existence globale en temps de solutions faibles pour les équations de Navier–Stokes régissant un fluide visqueux compressible et conducteur de chaleur. Une certaine classe de viscosités et conductivités dépendant de la densité est considérée. Ce résultat étend le travail de P.-L. Lions de 1993 [P.-L. Lions, Compacité des solutions des équations de Navier–Stokes compressibles isentropiques, C. R. Acad. Sci. Paris, Sér. I 317 (1993) 115–120] concernant les fluides barotropes, et procure des solutions faibles « à la Leray » du modèle compressible complet incluant l'équation d'évolution de l'énergie interne avec effets de conductivité thermique. Une réponse partielle est alors donnée à ce problème largement ouvert décrit par exemple dans le livre de P.-L. Lions [P.-L. Lions, Mathematical Topics in Fluid Dynamics, vol. 2, Compressible Models, Oxford Science Publication, Oxford, 1998]. La démonstration utilise une généralisation, au cas dépendant de la température, d'une nouvelle égalité mathématique d'entropie obtenue par les auteurs dans [D. Bresch, B. Desjardins, Some diffusive capillary models of Korteweg type, C. R. Acad. Sci., Paris, Section Mécanique 332 (11) (2004) 881–886]. La construction d'un schéma de solutions approchées, utilisant des effets de régularisation tels que ceux provenant de la capillarité, est donnée dans [D. Bresch, B. Desjardins, On the construction of approximate solutions for 2D viscous shallow water model and for compressible Navier–Stokes models, J. Math. Pures Appl. 86 (4) (2006) 362–368], et elle permet d'utiliser les arguments de stabilité de cet article pour obtenir le résultat d'existence." @default.
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• W2024482949 date "2007-01-01" @default.
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• W2024482949 title "On the existence of global weak solutions to the Navier–Stokes equations for viscous compressible and heat conducting fluids" @default.
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• W2024482949 doi "https://doi.org/10.1016/j.matpur.2006.11.001" @default.
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