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 >= 3. For the power index q below the compactness threshold, i.e. q ∈ (1, 2d/d+2), we show ill-posedness of Leray–Hopf solutions. For a wider class of indices q ∈ (1, 3d+2/d+2) we show ill-posedness of distributional (non-Leray–Hopf) solutions, extending the seminal paper of Buckmaster & Vicol [10]. In this wider class we also construct non-unique solutions for every datum in L²