Non-uniqueness of smooth solutions of the Navier-Stokes equations from almost the same initial conditions

TL;DR

This paper presents numerical evidence suggesting that the Navier-Stokes equations can have multiple distinct solutions originating from nearly identical initial conditions, challenging the traditional notion of uniqueness.

Contribution

It provides the first numerical evidence of non-uniqueness of Navier-Stokes solutions from almost identical initial data using high-precision simulations.

Findings

Different solutions from initial conditions differing by as little as 10^{-40}

Implications for the Navier-Stokes existence and uniqueness problem

Potential impact on the Millennium Prize Problem

Abstract

Using clean numerical simulation (CNS) which can give very accurate spatiotemporal trajectory of Navier-Stokes turbulence in a finite but long enough interval of time, we give some numerical evidences that the Navier-Stokes equations admit distinct global solutions from almost the same initial conditions whose difference is very small, i.e. even at the order $10^{-40}$ of magnitude. Hopefully these examples could provide some enlightenments for the uniqueness and existence of Navier-Stokes equations, which are related to one Millennium Prize Problem of Clay Institute.