Instability results related to compressible Korteweg system

This paper presents the study of surface tension effects in compressible mixtures in the framework of diffuse interface models. In the first part, we describe results previously obtained on the so-called compressible Korteweg and shallow water models and we present nonlinear stability using energy estimates and a new entropy equality recently discovered. These diffuse interface models also allow to take account of capillarity effects in turbulent mixtures and plasma flows subject to Rayleigh–Taylor instabilities. The aim of the last part is to study the influence of surface tension on this instability phenomena. More precisely we look at the expression of the growth rate under a small perturbation of wave number k. We prove that for an appropriate choice of the capillary number σ in terms on the surface tension coefficient T _s (that means particular pressure laws), we find the same expression as for the two incompressible fluids model with surface tension coefficient on a sharp interface studied for instance by Chandrasekhar (Hydrodynamic and hydromagnetic stability. Dover Publications, Inc. New York, 1981). The first author would like to thank the CEA/DAM (Bruyères le Chatel, France) for its financial support through the contract no. 4600052302/P6H28. He is also partially supported by the IDOPT project in Grenoble and the “ACI jeunes chercheurs 2004” du ministère de la Recherche “études mathématiques de paramétrisations en océanographie”.