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

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.