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 $L^p((0,2);W_{loc}^{s,p}({\mathbb{R}}^2;{\mathbb{R}}^2))$ 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 [Calc. Var. Partial Differ. Equ. 59 (2019), Ciampa, Crippa and Spirito] and [Ann. Math. Qué. 46 (2022), De Lellis and Giri] are generic.