Non-uniqueness of Leray-Hopf Solutions to Forced Stochastic Hyperdissipative Navier-Stokes Equations up to Lions Index

2025/03/23 by Chen, Weiquan, Dong, Zhao, Zheng, Yang
#Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2503.18041

Abstract

We show non-uniqueness of local strong solutions to stochastic fractional Navier-Stokes equations with linear multiplicative noise and some certain deterministic force. Such non-uniqueness holds true even if we perturb such deterministic force in appropriate sense.This is closely related to a critical condition on force under which Leray-Hopf solution to the stochastic equations is locally unique. Meanwhile, by a new idea, we show that for some stochastic force the system admits two different global Leray-Hopf solutions smooth on any compact subset of (0,∞) × ℝ^d.

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