On the Navier-Stokes equation perturbed by rough transport noise

TL;DR

This paper investigates the Navier-Stokes equations with space-time dependent rough transport noise using rough path theory, establishing existence, uniqueness, and stability of solutions in two dimensions.

Contribution

It introduces a novel framework for weak solutions of Navier-Stokes with rough noise, applying unbounded rough drivers to prove existence and uniqueness in 2D.

Findings

Existence of weak solutions under rough transport noise

Uniqueness and stability of solutions in two dimensions

Application of rough path theory to fluid dynamics

Abstract

We consider the Navier-Stokes system in two and three space dimensions perturbed by transport noise and subject to periodic boundary conditions. The noise arises from perturbing the advecting velocity field by space-time dependent noise that is smooth in space and rough in time. We study the system within the framework of rough path theory and, in particular, the recently developed theory of unbounded rough drivers. We introduce an intrinsic notion of a weak solution of the Navier-Stokes system, establish suitable a priori estimates and prove existence. In two dimensions, we prove that the solution is unique and stable with respect to the driving noise.