Non-Leray-Hopf solutions to 3D stochastic hyper-viscous Navier-Stokes equations: Beyond the lions exponent

Wenping Cao, Zirong Zeng and Deng Zhang

Stochastic Processes and their Applications, 2026, vol. 193, issue C

Abstract: We consider the 3D stochastic Navier-Stokes equations where the viscosity exponent can be larger than the Lions exponent 5/4. Though it is well-known that the Leray-Hopf solutions are unique in this high viscous regime, we prove that the uniqueness would fail in two scaling-supercritical regimes with respect to the Ladyžhenskaya-Prodi-Serrin criteria. The constructed solutions can be non-Leray-Hopf and very close to the Leray-Hopf solutions. Furthermore, we prove the vanishing noise limit result, which relates together the stochastic solutions and the deterministic convex integration solutions constructed by Buckmaster-Vicol [1] and the recent work [2].

Keywords: Convex integration; Hyper-viscous Navier-Stokes equations; Non-uniqueness; Ladyžhenskaya-Prodi-Serrin criteria; Probabilistically strong solutions (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1016/j.spa.2025.104836

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