Non Uniqueness of Power-Law Flows

We apply the technique of convex integration to obtain non-uniqueness and existence results for power-law fluids, in dimension $d\geq 3$. For the power index $q$ below the compactness threshold, i.e. $q \in (1, \frac{2d}{d+2})$, we show ill-posedness of Leray-Hopf solutions. For a wider class of indices $q \in (1, \frac{3d+2}{d+2})$ we show ill-posedness of distributional (non-Leray-Hopf) solutions, extending the seminal paper of Buckmaster and Vicol (Nonuniqueness of weak solutions to the Navier-Stokes equation). In this wider class we also construct non-unique solutions for every datum in $L^2$.