Non-Newtonian fluids and convex integration

The viscosity of a fluid is usually a constant, independent of the stress. There are however in nature several examples of fluids (ice, molten lava, blood, certain polymers, some salt solutions) where viscosity changes under applied forces. Such fluids are called non-Newtonian. I will focus on a simple model for such fluids, the “power law” model (Ladyzhenskaya, 1966): it is known that such model is well-posed in the “subcritical" regime and it has energy solutions above the "compactness threshold”. In a recent joint work with  J. Burczak and L. Székelyhidi, we show that a picture dual to the above one holds: the power-law model is ill posed below the "compactness threshold" and it has many (very) weak solutions in the "supercritical regime".