The existence of global smooth solutions to the Navier-Stokes equations (NSEs) with hyperviscosity (-Δ)γ is open unless γ is close to the J.-L. Lions exponent 5/4 at which the energy balance is strong enough to prevent singularity formation. If 1<γ 5/4, then the global well-posedness of the hyperviscous NSEs is widely open as for the usual NSEs. In this paper, for all γ>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.
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Antonio Agresti (2024) studied this question.
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