Random dynamical system generated by the 3D Navier-Stokes equation with rough transport noise

TL;DR

This paper constructs a measurable random dynamical system for the 3D Navier-Stokes equations with rough transport noise, addressing the challenge of non-uniqueness by selecting solutions with the semigroup property and energy dissipation criteria.

Contribution

It introduces a novel method to define a single-valued measurable dynamical system for the 3D Navier-Stokes equations with rough noise, despite the lack of uniqueness.

Findings

Established existence of a measurable random dynamical system.

Selected solutions satisfy the semigroup property with shifted rough paths.

Solutions adhere to energy dissipation maximization.

Abstract

We consider the Navier-Stokes system in three dimensions perturbed by a transport noise which is sufficiently smooth in space and rough in time. The existence of a weak solution was proved recently, however, as in the deterministic setting the question of uniqueness remains a major open problem. An important feature of systems with uniqueness is the semigroup property satisfied by their solutions. Without uniqueness, this property cannot hold generally. We select a system of solutions satisfying the semigroup property with appropriately shifted rough path. In addition, the selected solutions respect the well accepted admissibility criterium for physical solutions, namely, maximization of the energy dissipation. Finally, under suitable assumptions on the driving rough path, we show that the Navier-Stokes system generates a measurable random dynamical system. To the best of our knowledge, this is the first construction of a measurable single-valued random dynamical system in the state space for an SPDE without uniqueness.