Non-uniqueness of Leray-Hopf Solutions to Forced Stochastic Hyperdissipative Navier-Stokes Equations up to Lions Index
Weiquan Chen, Zhao Dong, Yang Zheng
TL;DR
The paper proves non-uniqueness of Leray–Hopf solutions for stochastic hyperdissipative Navier–Stokes with fractional dissipation $\Lambda^\alpha$, by transforming the stochastic problem into a random PDE, employing similarity-variable analysis, and constructing unstable background flows to generate two distinct solution branches. A key mechanism relies on spectral analysis of the perturbed linear operator $\mathbf{L}_{\bar{U}}$, establishing the existence of eigenvalues with positive real parts and corresponding semigroup growth, which, together with a fixed-point framework, yields non-uniqueness of local solutions and, in a stochastic setting, two global Leray–Hopf solutions smooth on compact subsets of $(0,\infty)\times\mathbb{R}^d$. The results delineate a stability-uniqueness boundary at the Lions exponent $\alpha=1+\frac{d}{2}$ and demonstrate robust non-uniqueness under perturbations of the forcing term, with extensions to 2D Euler/fractional NS. This work advances understanding of ill-posedness in stochastic fluid models near critical dissipation levels and highlights the role of forcing structure in solution multiplicity.
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 $(0,\infty) \times \mathbb{R}^d$.