A convex integration scheme for the continuity equation past the Sobolev embedding threshold

2025/04/04 by Colombo, Maria, Colombo, Roberto, Kumar, Anuj · 2 citations
#35A02 #35F10 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2504.03578

Abstract

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 C_tW^1,p_x and nonunique solutions in C_t L^q_x 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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