Sharpness of the Osgood Criterion for the Continuity Equation with Divergence-free Vector Fields

TL;DR

This paper demonstrates that for certain non-Osgood moduli of continuity, divergence-free velocity fields can lead to non-uniqueness of solutions in the continuity equation and flow maps, challenging classical uniqueness results.

Contribution

The paper introduces a novel 'parallelization' technique and a fixed-point framework to construct non-unique solutions for the continuity equation with non-Osgood moduli.

Findings

Constructed divergence-free velocity fields with non-unique flow maps.

Established existence of multiple solutions to the continuity equation starting from the same initial data.

Showed solutions can be absolutely continuous in space and time despite non-uniqueness.

Abstract

For any modulus of continuity $\omega$ that fails the Osgood condition, we construct a divergence-free velocity field $v \in C_t C^\omega_x$ for which the associated ODE admits at least two distinct flow maps. In other words, non-uniqueness does not occur merely for a single or even finitely many trajectories, but instead on a set of initial conditions $E$ of positive Lebesgue measure. In fact, the set $E$ has full measure inside a cube where the construction is supported. Moreover, we also construct a divergence-free velocity field $v \in C_{t}C^\omega_x$ for which the associated continuity equation admits two distinct solutions $\mu^1$ and $\mu^2$ which are absolutely continuous with respect to Lebesgue measure for almost every time, and start from the same initial datum $\bar \mu \ll \mathscr{L}^{d}$. Our construction introduces two novel ideas: (i) We introduce the notion of "parallelization", where at each time, the velocity field consists of simultaneous motion across multiple nested spatial scales. This differs from most explicit constructions in the literature on mixing or anomalous dissipation, where the velocity on different scales acts at separate times. This is crucial to cover the whole class of non-Osgood moduli of continuity. (ii) Inspired by a recent work of Bru\`e, Colombo and Kumar, we develop a new fixed-point framework that naturally incorporates the parallelization mechanism. This framework allows us to construct anomalous solutions of the continuity equation that belong to $L^1(\mathbb{R}^d)$ a.e. in time.