We consider a diffuse interface model for an incompressible binary viscoelastic fluid flow. The model consists of the Navier-Stokes-Voigt equations where the instantaneous kinematic viscosity has been replaced by a memory term. These equations are coupled with a > Cahn–Hilliard equation with constant mobility and Flory-Huggins (i.e. logarithmic) type potential. The resulting system is subject to no-slip condition for the (volume averaged) fluid velocity and no-flux boundary conditions for the order parameter as well as for the chemical potential. The presence of a memory term entails hyperbolic features (i.e. the fluid velocity does not regularize in finite time). We show that the corresponding initial and boundary value problem is well posed. Moreover, by adding an Eckman-type damping, we can define a dissipative dynamical system in a suitable phase space. This system has the global attractor, while the existence of exponential attractors can be proven in two spatial dimensions. We also discuss some open issues.
