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

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  • Cao, Wenping
  • Zeng, Zirong
  • Zhang, Deng

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].

Suggested Citation

  • Cao, Wenping & Zeng, Zirong & Zhang, Deng, 2026. "Non-Leray-Hopf solutions to 3D stochastic hyper-viscous Navier-Stokes equations: Beyond the lions exponent," Stochastic Processes and their Applications, Elsevier, vol. 193(C).
    • Handle: RePEc:eee:spapps:v:193:y:2026:i:c:s0304414925002807
    DOI: 10.1016/j.spa.2025.104836
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