Global smooth solutions by transport noise of 3D Navier-Stokes equations with small hyperviscosity

The existence of global smooth solutions to the Navier-Stokes equations (NSEs) with hyperviscosity $(-\mathrm{\Delta }{)}^{\gamma }$ is open unless $\gamma$ is close to the J.-L. Lions exponent $\frac{5}{4}$ at which the energy balance is strong enough to prevent singularity formation. If $1<\gamma \ll \frac{5}{4}$, then the global well-posedness of the hyperviscous NSEs is widely open as for the usual NSEs. In this paper, for all $\gamma >1$, we show the existence of a transport noise for which global smooth solutions to the stochastic hyperviscous NSEs on the three-dimensional torus exist with high probability. In particular, a suitable transport noise considerably improves the known well-posedness results in the deterministic setting.