morphology dynamics simulated at the Cahn-Hilliard level is established. Analysis of the numerical investigation of phase separation under shear in the diffusive regime revealed no new physical process that occur at the time scale which is slow compared to the Cahn- Hilliard level and fast from the perspective of the Doi-Ohta level. Therefore, there are no new dissipative effects arising in the performed coarse-graining step through the Green- Kubo type formula, i.e., no new physics appear. That leads us to the conclusion that all the dissipation has been introduced into the model by coarse graining from the reversible atomistic level to the Cahn-Hilliard level. This coarse-graining step has been done in the derivation of the hydrodynamic equations of a phase separating fluid mixture from the underlying microscopic dynamics [46]. Furthermore, Espa´ nol and V´ azquez showed in [50] that all dissipation is introduced already at the very first coarse-graining step, going from the atomistic to the Fokker-Planck level. From there then follows the conclusion that the Cahn-Hilliard level is not so fundamental considering that the same dissipation mechanisms occur even at the finer levels, and are afterwards transmitted to an even more coarse-grained Doi-Ohta level, with no new dissipative processes arising. 

The presented results could be used to deal with several interesting open problems. First, possible extension of the procedure to more complicated phase separating systems with the addition of surfactants or block copolymers would be very interesting for wider use. Second, the nature of the dissipative processes in a phase separating binary mixture is still far from clear. While during the diffusive regime, in the absence of thermal noise, we do not recognize any further fast processes that could be considered as fluctuations at the Doi- Ohta level, this becomes far from evident during the late stages of coarsening when inertia becomes important. The nature of the domain coarsening in the inertial hydrodynamics regime is still unknown, as well as the asymptotic behavior of a phase separating system under shear. Furthermore, recent studies showed that a nonequilibrium steady state can be reached at high Reynolds numbers (low shear rates) only in the mixed viscous-inertial regime, i.e., with any non-zero amount of inertia [21–24]. The fact that no steady state could be formed without presence of inertia, no matter how small, suggests that inertia plays the role of a singular perturbation in this problem [24]. The formed steady state, arising from a cyclic occurrence of elongation of domains followed by their break up, coagulation and shape relaxation, is characterized by irregular fluctuations around the attained steady state values. It would be interesting to understand the nonequilibrium steady state and the origin of these 

