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

Abstract: 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 \cite{DeLellis_Giri22} 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 $L^p_{loc}([0,2]\times \R^2;\R^2)$ such that for each vector field $\bb\in D$, there are two smooth regularisations of $\bb$, 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 $\bb$.