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 $$d\ge 3$$ d ≥ 3 . For the power index q below the compactness threshold, i.e. $$q \in (1, \frac{2d}{d+2})$$ q ∈ ( 1 , 2 d d + 2 ) , we show ill-posedness of Leray–Hopf solutions. For a wider class of indices $$q \in (1, \frac{3d+2}{d+2})$$ q ∈ ( 1 , 3 d + 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 $$L^2$$ L 2 .

On the density of “wild” initial data for the compressible Euler system

We consider a class of “wild” initial data to the compressible Euler system that give rise to infinitely many admissible weak solutions via the method of convex integration. We identify the closure of this class in the natural $$L^1$$ L 1 -topology and show that its complement is rather large, specifically it is an open dense set.