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

TL;DR

This work investigates whether smooth solutions to the Navier–Stokes equations can be non-unique when starting from almost identical initial data. By applying Clean Numerical Simulation (CNS) to a 2D Kolmogorov flow with $n_K=16$ and $Re=2000$, the authors compare four runs whose initial conditions differ by a tiny amount up to $10^{-40}$. They observe divergent spatiotemporal trajectories, symmetry properties, and statistics (e.g., the dissipation measure $ig\\langle D \\rangle_A$ and PDFs) among Flow CNS and Flow CNS'$_{1,2,3}$, indicating non-uniqueness of global smooth NS solutions within a finite time window. The results, supported by negligible numerical noise in CNS over $t  [0,300]$, offer numerical counterpoints to the uniqueness question and illustrate CNS as a powerful tool for probing fundamental Navier–Stokes questions, potentially informing 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.

Figures (3)

• Figure 1: Comparison of time histories of the spatially averaged kinetic energy dissipation rate $\langle D \rangle_A$ of the 2D turbulent Kolmogorov flow, governed by Eqs. (\ref{['eq_psi']}) and (\ref{['boundary_condition']}) for $n_K=16$ and $Re=2000$, given by Flow CNS (red solid line), Flow CNS$'_1$ (black dash line), Flow CNS$'_2$ (green dash-dot line), and Flow CNS$'_3$ (blue dash-dot-dot line).
• Figure 2: Vorticity fields $\omega(x,y)$ at $t=200$ of the 2D turbulent Kolmogorov flow governed by (\ref{['eq_psi']}) and (\ref{['boundary_condition']}) for $n_K=16$ and $Re=2000$, given by CNS subject to the initial conditions (\ref{['initial_condition1']}) (marked by Flow CNS), (\ref{['initial_condition2-1']}) (marked by Flow CNS$'_1$), (\ref{['initial_condition2-2']}) (marked by Flow CNS$'_2$), and (\ref{['initial_condition2-3']}) (marked by Flow CNS$'_3$).
• Figure 3: Comparison of (a) probability density functions (PDFs) of kinetic energy dissipation rate $D(x,y,t)$ and (b) spatio-temporal averaged kinetic energy dissipation rate $\langle D\rangle_{x,t}(y)$ of the 2D turbulent Kolmogorov flow, governed by Eqs. (\ref{['eq_psi']}) and (\ref{['boundary_condition']}) for $n_K=16$ and $Re=2000$, given by Flow CNS (red line), Flow CNS$'_1$ (black line), Flow CNS$'_2$ (green line or triangle), and Flow CNS$'_3$ (blue line or cirlce).