Non-uniqueness of Regular Solutions for Incompressible Static Euler Equations with Given Boundary Conditions and Turbulent Global Solutions of Incompressible Navier-Stokes Equations

TL;DR

This paper demonstrates the non-uniqueness of regular solutions for incompressible static Euler equations with boundary conditions and explores turbulent solutions of Navier-Stokes equations, revealing inherent fluid randomness and turbulence.

Contribution

It introduces the existence of infinite non-trivial and random solutions for static Euler equations and links these to turbulent behaviors in Navier-Stokes flows at high Reynolds numbers.

Findings

Existence of infinite regular solutions for static Euler equations.

Presence of random solutions and turbulence phenomena.

Non-existence of double limits in Navier-Stokes flows.

Abstract

The incompressible Navier-Stokes equations and static Euler equations are considered. We find that there exist infinite non-trivial regular solutions of incompressible static Euler equations with given boundary conditions. Moreover there exist random solutions of incompressible static Euler equations. Provided Reynolds number is large enough and time variable $t$ goes to infinity, these random solutions of static Euler equations are the path limits of corresponding Navier-Stokes flows. But the double limits of these Navier-Stokes flows do not exist. These phenomena reveal randomness and turbulence of incompressible fluids. Therefore these solutions are called turbulent solutions. Here some typing models without Prandtl layer are given.