On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids
Résumé
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.