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

TL;DR

This paper demonstrates that for a dense set of bounded vector fields, smooth regularization does not serve as a selection criterion for the continuity equation, indicating the generic nature of previous counterexamples.

Contribution

It constructs a dense set of vector fields where regularization fails as a selection criterion, showing the phenomenon is generic among such fields.

Findings

Regularization does not select solutions for a dense set of vector fields.

Counterexamples are shown to be generic within the constructed set.

The set of vector fields is dense in a specific function space.

Abstract

We construct a set of bounded vector fields dense in $L^p((0,2);W^{s,p}_{loc}(\mathbb{R}^2;\mathbb{R}^2))$ for $1\leq p <+\infty$ and $0\leq s<1$ 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. Qu\'e. 46 (2022), De Lellis and Giri] are generic.