The Navier-Stokes equations with transport noise in critical $H^{1/2}$ space

TL;DR

This paper investigates the Navier-Stokes equations with transport noise in the critical $H^{1/2}$ space, establishing local existence, uniqueness, and conditions for near-certain global solutions, independent of spatial domain compactness.

Contribution

It introduces a probabilistic framework for Navier-Stokes with transport noise in critical spaces, proving local well-posedness and near-certain global existence for small initial data.

Findings

Existence and uniqueness of local solutions in $H^{1/2}$ space.

Probability of global existence approaches 1 with small initial data.

Results hold on both torus and whole space, independent of domain compactness.

Abstract

We study the Navier-Stokes equations with transport noise in critical function spaces. Assuming the initial data belongs to $H^{1/2}$ almost surely, we establish the existence and uniqueness of a local-in-time probabilistically strong solution. Moreover, we show that the probability of global existence can be made arbitrarily close to $1$ by choosing the initial data norm sufficiently small, and that the solution norm remains small for all time. Our analysis is independent of the compactness of the spatial domain, and consequently, the results apply both to the three-dimensional torus and to the whole space.