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

Yongqian Han

TL;DR

The paper investigates non-uniqueness of regular solutions to the incompressible static Euler equations under given boundary conditions and introduces turbulent global solutions for the incompressible Navier–Stokes equations. It develops a symplectic-inspired representation $u = A(\nabla)\phi + \{A(\nabla)\times\nabla\}\psi$ with a divergence-free constraint, and shows that, when $\phi = \lambda\psi$ and $-\Delta\psi = \lambda^2\psi$, the resulting $u_e$ solves the Euler system while a corresponding viscous flow $u_{ns} = e^{-\nu\lambda^2 t}(\lambda A(\nabla)\psi + \{A(\nabla)\times\nabla\}\psi)$ solves NS. By constructing eigenfunction-based families and analyzing zero-viscosity limits, the work demonstrates infinite nonunique Euler solutions under periodic or Dirichlet boundaries and reveals random path limits (turbulent solutions) that do not admit a well-defined double limit in $(\nu,t)$. The results argue for intrinsic randomness and turbulence in incompressible fluids, without invoking Prandtl boundary layers, and provide explicit examples and corollaries (e.g., periodic pipeline flows) to illustrate the phenomena. Overall, the paper connects boundary-value problems, nonuniqueness, and turbulence through explicit analytical constructions in the NS–Euler framework.

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.

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

The paper investigates non-uniqueness of regular solutions to the incompressible static Euler equations under given boundary conditions and introduces turbulent global solutions for the incompressible Navier–Stokes equations. It develops a symplectic-inspired representation with a divergence-free constraint, and shows that, when and , the resulting solves the Euler system while a corresponding viscous flow solves NS. By constructing eigenfunction-based families and analyzing zero-viscosity limits, the work demonstrates infinite nonunique Euler solutions under periodic or Dirichlet boundaries and reveals random path limits (turbulent solutions) that do not admit a well-defined double limit in . The results argue for intrinsic randomness and turbulence in incompressible fluids, without invoking Prandtl boundary layers, and provide explicit examples and corollaries (e.g., periodic pipeline flows) to illustrate the phenomena. Overall, the paper connects boundary-value problems, nonuniqueness, and turbulence through explicit analytical constructions in the NS–Euler framework.

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 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.

Paper Structure

This paper contains 2 sections, 9 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Nonu-SEE-ppDb']}

Key Result

Theorem 1.2

Assume that $A(\nabla)$ is any linear operator defined by Definition AOp-def, $\lambda$ is constant, $\psi(x)$ is regular enough and satisfies the following equation and $\phi(x)=\lambda\psi(x)$. Let then $u=u_e(x)$ is the solution of static incompressible Euler equations SEE1SEE2. Moreover let then $u=u_{ns}$ is the solution of incompressible Navier-Stokes equations NS1NS2.

Theorems & Definitions (11)

  • Definition 1.1
  • Theorem 1.2: Global Turbulent Solution
  • Theorem 1.3: Global Turbulent Periodic Solution
  • Corollary 1.4: Non-unique Periodic Solutions of Euler Equations
  • Corollary 1.5: Global Well-posedness of NS Periodic Solution
  • Theorem 1.6: Global Turbulent Dirichlet Solution
  • Theorem 1.7: Non-unique Solutions of Periodic Pipeline Euler Flow
  • Theorem 1.8: Global Solution of Periodic Pipeline NS Flow
  • Corollary 1.9: Typification without Prandtl Layer
  • Theorem 1.10: Non-unique Solutions of 2d Euler Equations
  • ...and 1 more