On some typicality and density results for nonsmooth vector fields and the associated ODE and continuity equation

TL;DR

This paper studies the typicality and density of solution existence and uniqueness for continuity equations and ODEs with nonsmooth vector fields, revealing that uniqueness is generic while existence is rare, yet counterexamples are densely present.

Contribution

It demonstrates that in certain function spaces, solution existence is rare but uniqueness is generic, and constructs dense sets of counterexamples to non-uniqueness for both equations.

Findings

Existence of solutions is meager in certain regimes.

Uniqueness of solutions is a generic property.

Counterexamples to uniqueness form a dense subset.

Abstract

These notes address two problems. First, we investigate the question of ``how many'' are (in Baire sense) vector fields in $L^1_t L^q_x$, $q \in [1, \infty)$, for which existence and/or uniqueness of local, distributional solutions to the associated continuity equation holds. We show that, in certain regimes, existence of solutions (even locally in time, for at least one nonzero initial datum) is a meager property, whereas, on the contrary, uniqueness of solutions is a generic property. Secondly, despite the fact that non-uniqueness is a meager property, we prove that (Sobolev) counterexamples to uniqueness, both for the continuity equation and for the ODE, in the spirit of [Bru\`e, Colombo, Kumar 2024] and [Kumar 2024] respectively, form a dense subset of the natural ambient space they live in.