On the lack of selection for the transport equation over a dense set of vector fields

The purpose of this work is to demonstrate that the lack of selection by smooth regularisation for the continuity equation with a bounded, divergence-free vector field as demonstrated in DeLellisGiri22 by De Lellis and Giri takes place over a dense set of vector fields. More precisely, we construct a set of bounded vector fields D dense in Lᵖ₋₎₂ (0, 2 ²;²) such that for each vector field D, there are two smooth regularisations of, for which the unique solution of the Cauchy problem for the continuity equation along each regularisation converges to two distinct solutions of the Cauchy problem along.