Incorporate a DiPerna–Lions drift into the Lagrangian rough-flow framework

Determine how to incorporate a DiPerna–Lions-type drift vector field into the Lagrangian flow framework for rough transport equations, despite the ill-posedness of the associated flow equation at the available regularity.

Background

The paper contrasts its purely Eulerian unbounded-rough-driver approach with a Lagrangian approach based on rough flows. For rough transport equations driven by finite-dimensional rough paths, solutions can be represented using a flow and its inverse. However, adding a drift vector field with DiPerna–Lions regularity leads to an associated flow equation that is not well posed in the classical sense. The authors note that regular Lagrangian flows might provide a possible direction, but the corresponding treatment is absent from the cited literature.

References

Moreover, it is not clear how to include a drift term $b$ in this framework since in this case the flow equation is ill-posed. One could imagine constructing regular Lagrangian flows as in to give meaning to these equations by relaxing to almost every initial condition $x \in Rd$, but this point of view seems to be missing from the literature.

— Higher Order Unbounded Rough Drivers  (2609.05353 - Nilssen et al., 4 Sep 2026) in Section 1, subsection “Related works,” subsection “Low temporal regularity,” footnote following equation (\ref{eq:friz flow})