On the well-posedness of (nonlinear) rough continuity equations

TL;DR

This paper establishes well-posedness and flow representations for rough differential equations and rough PDEs with non-Lipschitz drifts, applying to fluid dynamics models like the 2D Euler equations with rough noise.

Contribution

It introduces new well-posedness results for RDEs and RPDEs with non-Lipschitz drifts, extending classical theories to rough settings relevant for fluid dynamics.

Findings

Proves existence and uniqueness of solutions for rough continuity equations.

Provides flow representation formulas for linear rough PDEs.

Applies results to 2D Euler equations with rough transport noise.

Abstract

Motivated by applications to fluid dynamics, we study rough differential equations (RDEs) and rough partial differential equations (RPDEs) with non-Lipschitz drifts. We prove well-posedness and existence of a flow for RDEs with Osgood drifts, as well as well-posedness of weak $L^p$-valued solutions to linear rough continuity and transport equations on $\mathbb{R}^d$ under DiPerna--Lions regularity conditions; a combination of the two then yields flow representation formula for linear RPDEs. We apply these results to obtain existence, uniqueness and continuous dependence for $L^1\cap L^\infty$-valued solutions to a general class of nonlinear continuity equations. In particular, our framework covers the $2$D Euler equations in vorticity form with rough transport noise, providing a rough analogue of Yudovich's theorem. As a consequence, we construct an associated continuous random dynamical system, when the driving noise is a fractional Brownian motion with Hurst parameter $H \in (1/3,1)$. We further prove weak existence of solutions for initial vorticities in $L^1\cap L^p$, for any $p\in [1,\infty)$.