# On a rough perturbation of the Navier-Stokes system and its vorticity formulation

## TL;DR

This paper introduces a rough perturbation of the Navier-Stokes equations, analyzing its well-posedness and properties in both two and three dimensions using rough path theory, with implications for stochastic fluid dynamics.

## Contribution

It develops a new framework for analyzing rough perturbations of Navier-Stokes equations, including an intrinsic solution concept and results on well-posedness and dynamical systems.

## Key findings

- Well-posedness and enstrophy balance in 2D
- Rough path continuity and Wong-Zakai approximation
- Existence of local solutions in 3D

## Abstract

We introduce a rough perturbation of the Navier-Stokes system and justify its physical relevance from balance of momentum and conservation of circulation in the inviscid limit. We present a framework for a well-posedness analysis of the system. In particular, we define an intrinsic notion of solution based on ideas from the rough path theory and study the system in an equivalent vorticity formulation. In two space dimensions, we prove that well-posedness and enstrophy balance holds. Moreover, we derive rough path continuity of the equation, which yields a Wong-Zakai result for Brownian driving paths, and show that for a large class of driving signals, the system generates a continuous random dynamical system. In dimension three, the noise is not enstrophy balanced, and we establish the existence of local in time solutions.