Transport equations for Osgood velocity fields

TL;DR

This paper develops a new well-posedness theory for the transport equation with Osgood velocity fields, especially in cases where traditional Lebesgue-based methods fail, by employing Riemann--Stieltjes integration.

Contribution

It introduces a novel interpretation of the weak formulation for Osgood velocity fields using Riemann--Stieltjes integration, extending the theory beyond existing approaches.

Findings

Well-posedness established for 1D Osgood velocity fields

Extension to multi-dimensional log-Lipschitz velocities

Addresses cases with unbounded divergence of velocity

Abstract

We consider the transport equation with a velocity field satisfying the Osgood condition. The weak formulation is not meaningful in the usual Lebesgue sense, meaning that the usual DiPerna--Lions treatment of the problem is not applicable {(in particular, the divergence of the velocity might be unbounded)}. Instead, we use Riemann--Stieltjes integration to interpret the weak formulation, leading to a well-posedness theory in regimes not covered by existing works. The most general results are for the one-dimensional problem, with generalisations to multiple dimensions in the particular case of log-Lipschitz velocities.