We construct a large class of examples of non-uniqueness for the linear transport equation and the transport-diffusion equation with divergence-free vector fields in Sobolev spaces .

We construct infinitely many incompressible Sobolev vector fields u ∈ CtW 1,px on the periodic domain T d for which uniqueness of solutions to the transport equation fails in the class of densities ρ ∈ CtL p x , provided

We show that there are continuous, W 1,p (p < d − 1), incompressible vector fields for which uniqueness of solutions to the continuity equation fails.

Abstract. We prove a quadratic interaction estimate for approximate solutions to scalar conservation laws obtained by the wavefront tracking approximation or the Glimm scheme. This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme.The proof is based on the introduction of a quadratic functional Q(t), decreasing at every interaction, and such that its total variation in time is bounded.Differently from other interaction potentials present in the literature, the form of this functional is the natural extension of the original Glimm functional, and coincides with it in the genuinely nonlinear case.

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 .