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.
From: Antonio Agresti [view email]
[v1]
Thu, 13 Jun 2024 16:06:42 UTC (114 KB)
[v2]
Tue, 1 Oct 2024 21:32:38 UTC (115 KB)
[v3]
Fri, 31 Jul 2026 09:52:19 UTC (44 KB)
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