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

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arxiv 2406.09267 v3 pith:C7LTVBQM submitted 2024-06-13 math.AP math.PR

Antonio Agresti This is my paper
classification math.APmath.PR
keywords gammaglobalnsesnoisesmoothsolutionstransportequations
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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.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Absence of blow-up in the 3D Navier-Stokes equations with transport noise

    math.PR 2026-07 conditional novelty 8.0 of 10

    3D Navier–Stokes equations with a strong, carefully chosen transport noise have global smooth solutions with probability arbitrarily close to 1, for arbitrarily large subcritical initial data.

  2. Closed Estimates of Leray Projected Transport Noise and Strong Solutions of the Stochastic Euler Equations

    math.AP 2025-07 conditional novelty 8.0 of 10

    Closed Sobolev estimates for Leray-projected transport noise yield unique local strong solutions of the 3D stochastic Euler equations, with blow-up characterized by L1([0,T]; W^{1,∞}).

  3. Zero-noise selection and Large Deviations in $L^\infty_t L^p_x$ for the stochastic transport equation beyond DiPerna-Lions

    math.PR 2025-06 conditional novelty 8.0 of 10

    Rough transport noise selects the unique DiPerna-Lions solution in the zero-noise limit and yields a large deviations principle in the non-separable space L^∞_t L^p_x.

  4. Fractal dimension of singular times for SPDEs: Energy bounds, criticality, and weak-strong uniqueness

    math.PR 2026-02 conditional novelty 7.0 of 10

    Singular times of weak SPDE solutions have Hausdorff/Minkowski dimension at most 1−ℓ·Exc, giving stochastic 3D Navier-Stokes with multiplicative noise the classical 1/2-dimensional bound.

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