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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.06133 |
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Table of Contents:
- We consider the 3D stochastic Navier-Stokes equations (NSE) on torus where the viscosity exponent can be larger than the Lions exponent 5/4. For arbitrarily prescribed divergence-free initial data in $L^{2}_x$, we construct infinitely many probabilistically strong and analytically weak solutions in the class $L^{r}_ΩL_{t}^γW_{x}^{s,p}$, where $r\geq1$ and $(s, γ, p)$ lie in two supercritical regimes with respect to the Ladyžhenskaya-Prodi-Serrin (LPS) criteria.It shows that even in the high viscosity regime beyond the Lions exponent, though solutions are unique in the Leray-Hopf class, the uniqueness fails in the mixed Lebesgue spaces and, actually, there exist infinitely manly non-Leray-Hopf solutions which can be very close to the Leray-Hopf solutions. Furthermore, we prove the vanishing noise limit result, which relates together the stochastic solutions and the deterministic solutions constructed by Buckmaster-Vicol [4] and the recent work [23].