The method shows nonunique solutions in L^q and W^{1,p} for vector fields, indicating improved conditions under convex integration schemes.
We introduce a convex integration scheme for the continuity equation in the context of the Di Perna-Lions theory that allows to build incompressible vector fields in CₜW1,pₓ and nonunique solutions in Cₜ Lqₓ for any $p,q$ with 1/p + 1/q > 1 + 1/d- δ for some δ>0. This improves the previous bound, corresponding to δ=0, or equivalently q' > p^*, obtained with convex integration so far, and critical for those schemes in view of the Sobolev embedding that guarantees that solutions are distributional in the opposite range.
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Colombo et al. (2025) studied this question.