Mathematical problems of dynamics of generalized Newtonial fluids

The talk will be devoted to boundary value and initial-boundary value problems for equations describing the flow of generalized Newtonial fluids, which include the Navier–Stokes equations as a particular case. Problems we will concern in the lecture are quite typical for the theory of PDE's. Among of them are existense, uniqueness and differentiability properties for solutions of corresponding boundary value and initial-boundary value problems. Although we have already answered to the part of questions, many problems remain to be open.
Another theme of the talk will be connected with investigation of semigroups generated by equations of the motion of genralized Newtonial fluids. In particular, it is interesting to know whether they possess attractors, what properties semigroups do have attractors, whether attractors are finite dimensial, what majorants for the Hausdorff and fractal dimensions and etc. To answer these questions we have to involve various methods from other fields of mathematics and, in particular, from non-linear fractional analysis.