In this seminar we shall consider a mathematical problem derived from the Ellis model used in Fluid Mechanics to describe the response of a great variety of generalized fluid flows.
For pseudoplastic fluids, it is well-known that the weak solutions to that problem extinct in a finite time.
In order to obtain the same property for Newtonian and dilatant fluids, we modify the problem by introducing an absorption term in the momentum equation.
The proof of this property relies on a suitable energy method, Sobolev type interpolation inequalities and also on a generalized Korn's inequality.
For the modified problem with different exponents on the absorption term, we prove also that, under different conditions on the forces field, we obtain decays of power-time rates or exponential decays.
Then we extend our results for several cases: slip boundary conditions, anisotropic absorption and non-homogeneous fluid flows.
We also discuss existence and uniqueness of weak solutions for the
modified problem.
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