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

Lucio Galeati, James-Michael Leahy, Torstein Nilssen

TL;DR

This work develops a rigorous framework for well-posedness of rough differential equations and rough partial differential equations with non-Lipschitz drifts, combining rough path theory with DiPerna–Lions techniques. It establishes flow representations, stability, and renormalizability for linear RPDEs and nonlinear rough continuity equations, culminating in a Yudovich-type theory for rough 2D Euler and the construction of continuous random dynamical systems under rough noise such as fractional Brownian motion with $H o (1/3,1)$. The results apply to vorticity formulations, provide weak existence for broader initial data, and open avenues toward viscous limits, large deviations, and domain-general RPDEs. Overall, the paper significantly extends rough PDE analysis to low-regularity drifts and nonlinear couplings relevant to fluid dynamics.

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 $p\in [1,\infty)$.

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

TL;DR

This work develops a rigorous framework for well-posedness of rough differential equations and rough partial differential equations with non-Lipschitz drifts, combining rough path theory with DiPerna–Lions techniques. It establishes flow representations, stability, and renormalizability for linear RPDEs and nonlinear rough continuity equations, culminating in a Yudovich-type theory for rough 2D Euler and the construction of continuous random dynamical systems under rough noise such as fractional Brownian motion with . The results apply to vorticity formulations, provide weak existence for broader initial data, and open avenues toward viscous limits, large deviations, and domain-general RPDEs. Overall, the paper significantly extends rough PDE analysis to low-regularity drifts and nonlinear couplings relevant to fluid dynamics.

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 -valued solutions to linear rough continuity and transport equations on 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 -valued solutions to a general class of nonlinear continuity equations. In particular, our framework covers the 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 . We further prove weak existence of solutions for initial vorticities in , for any .

Paper Structure

This paper contains 26 sections, 55 theorems, 391 equations.

Table of Contents

  1. Introduction
  2. Main results and discussion
  3. Relations to the existing literature
  4. Future perspectives
  5. Notation and preliminaries
  6. Notation
  7. Function spaces
  8. Spaces of $\mathfrak{p}$-variation and rough paths
  9. Osgood moduli of continuity
  10. Uniform convergence in weak topologies
  11. RDEs with Osgood drifts
  12. Forced RDEs and a priori estimates
  13. Construction of flows and their properties
  14. Linear rough continuity and transport equations
  15. A primer on unbounded rough drivers
  16. ...and 11 more sections

Key Result

Theorem 1.1

Assume that $Z$ lifts to a rough path $\bZ \in \clC^{\mathfrak{p}}$ for $\mathfrak{p} \in [2,3)$, $\xi \in C_b^3$ and $b$ is a bounded Osgood vector field. Then the rough differential equation is well-posed and induces a flow $(t,x) \mapsto \Phi_t(x)$ of homeomorphisms on $\mathbb{R}^d$, which satisfies for some modulus of continuity $F$. The same estimate holds with $\Phi$ replaced by its inver

Theorems & Definitions (144)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • ...and 134 more