Non-uniqueness of Leray-Hopf solutions for stochastic forced Navier-Stokes equations

TL;DR

This paper demonstrates that for stochastic forced Navier-Stokes equations in three dimensions, non-uniqueness of Leray-Hopf solutions can occur both locally in time and in law, with the set of such forces being dense in certain function spaces.

Contribution

It extends the understanding of non-uniqueness in Navier-Stokes solutions to stochastic settings, showing density of non-uniqueness forces and joint non-uniqueness in law.

Findings

Non-uniqueness of solutions occurs locally in time.

Non-uniqueness in law is established for solutions.

The set of forces causing non-uniqueness is dense in $L^{1}_{t}L^{2}_{x}$.

Abstract

We consider stochastic forced Navier--Stokes equations on $\mathbb{R}^{3}$ starting from zero initial condition. The noise is linear multiplicative and the equations are perturbed by an additional body force. Based on the ideas of Albritton, Bru\'e and Colombo \cite{ABC22}, we prove non-uniqueness of local-in-time Leray--Hopf solutions as well as joint non-uniqueness in law for solutions on $\mathbb{R}^{+}$. In the deterministic setting, we show that the set of forces, for which Leray--Hopf solutions are non-unique, is dense in $L^{1}_{t}L^{2}_{x}$. In addition, by a simple controllability argument we show that for every divergence-free initial condition in $L^{2}_{x}$ there is a force so that non-uniqueness of Leray--Hopf solutions holds.