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

Abstract

We construct a set of bounded vector fields dense in $L^p\left(\right(0,2);W_{loc}^{s,p}(R^d;R^d\left)\right)$ for $1\le p<+\infty$ and $0\le s<1$ with $p<1/s$ for which smooth regularisation of the vector field does not give a selection criterion for the continuity equation, thereby showing that the two examples constructed in [10], [12] are generic.