Nonuniqueness in law of stochastic 3d navierstokes equations with general multiplicative noise

TL;DR

This paper demonstrates the non-uniqueness in law for stochastic 3D Navier-Stokes equations with general multiplicative noise by constructing infinitely many solutions using stochastic convex integration and Ito calculus.

Contribution

It introduces a stochastic convex integration method to establish non-uniqueness and existence of multiple ergodic stationary solutions for these equations.

Findings

Existence of infinitely many global-in-time solutions

Non-uniqueness in law for the stochastic equations

Presence of infinitely many ergodic stationary solutions

Abstract

We are concerned with the three dimensional navier-stokes equations driven by a general multiplicative noise. For every divergence free and mean free initial condition in L2, we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, which implies non-uniqueness in law. Moreover, we prove the existence of infinitely many ergodic stationary solutions. Our results are based on a stochastic version of the convex integration and the Ito calculus.