Some theoretical results concerning diphasic viscoelastic flows of the Oldroyd kind

Abstract

We present several existence and uniqueness results for the equations satisfied by the three-dimensional non-Newtonian mixture of the Oldroyd kind. We study the coupling of the constitutive law, the Navier-Stokes equations and the Cahn-Hilliard equation which stands for a model of a multiphase non-Newtonian fluid. We prove that a strong local solution exists, but also that a global solution exists if the data are small enough. This last result is established even if the fluids are strongly elastic.