November 8, 2025Open Access

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

Key Points

Nonunique solutions emerged in continuity equations across non-standard Sobolev embeddings, redefining previous limits.
The method yielded a framework allowing for improved vector fields under relaxation of previous bounds (delta>0).
Assessment indicates implications on Sobolev embeddings and their bounds, relevant for theoretical fluid dynamics.
Future work calls for exploration of the established bounds beyond the critical threshold for better applications.

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 CₓW^1, pₓ and nonunique solutions in Cₓ L^qₓ for any p, q with 1p + 1q > 1 + 1d- for some >0. This improves the previous bound, corresponding to =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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