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

TL;DR

This paper demonstrates that adding a specific transport noise to 3D hyperviscous Navier-Stokes equations with small hyperviscosity ensures the existence of global smooth solutions with high probability, improving deterministic results.

Contribution

It introduces a transport noise that guarantees global smooth solutions for hyperviscous NSEs with small hyperviscosity, extending well-posedness results beyond deterministic cases.

Findings

Transport noise leads to global solutions with high probability.

Improves known well-posedness results for hyperviscous NSEs.

Applicable for all hyperviscosity exponents greater than one.

Abstract

The existence of global smooth solutions to the Navier-Stokes equations (NSEs) with hyperviscosity $(-\Delta)^{\gamma}$ 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.