Mathematical analysis of a nonlinear parabolic equation arising in the modelling of non-newtonian flows

The mathematical properties of a nonlinear parabolic equation arising in the modelling of concentrated suspension flows are investigated. The peculiarity of this equation is that it may degenerate into a hyperbolic equation (in fact, a linear advection equation). Depending on the initial data, at least two situations can be encountered: the equation may have a unique solution in a convenient class, or it may have infinitely many solutions. The present article is the theoretical side of a joint project with rheologists, aiming at better understanding the flows of complex fluids.