[Paper Review] Remarks on unsolved basic problems of the Navier-Stokes equations

Alexander Rauh|ArXiv.org|Jan 4, 2001

Fluid Dynamics and Turbulent Flows4 references3 citations

TL;DR

This paper examines the self-consistency of the Navier-Stokes equations (NSE) by analyzing whether finite-time singularities in weak solutions violate the continuum hypothesis. It argues that if velocity gradients blow up in finite time, molecular-scale physics must intervene, undermining the deterministic NSE framework and raising fundamental questions about turbulence's nature as purely deterministic chaos.

ABSTRACT

There is renewed interest in the question of whether the Navier-Stokes equations (NSE), one of the fundamental models of classical physics and widely used in engineering applications, are actually self-consistent. After recalling the essential physical assumptions inherent in the NSE, the notion of weak solutions, possible implications for the energy conservation law, as well as existence and uniqueness in the incompressible case are discussed. Emphasis will be placed on the possibility of finite time singularities and their consequences for length scales which should be consistent with the continuum hypothesis.

Motivation & Objective

• To assess the self-consistency of the Navier-Stokes equations (NSE) under the assumption of the continuum hypothesis.
• To investigate whether finite-time singularities in weak solutions of the NSE imply a breakdown of the continuum description at small length scales.
• To examine the implications of such singularities for energy conservation and the validity of deterministic fluid dynamics.
• To clarify the role of weak solutions and the integral I(τ) in proving uniqueness and existence of solutions.
• To evaluate whether hydrodynamic turbulence could be a manifestation of deterministic chaos alone or requires stochastic effects due to singularities.

Proposed method

• Derives the incompressible NSE from conservation laws (mass and momentum) using the Reynolds transport theorem and Newtonian fluid assumptions.
• Introduces the concept of weak solutions via the L2 norm of velocity and its time derivative, ensuring integrability conditions for physical consistency.
• Defines the integral I(τ) as the L∞ norm of the velocity gradient tensor Dv over time, crucial for uniqueness proofs.
• Applies the Poincaré inequality and Schwarz inequality to bound the viscous and nonlinear terms in the energy estimate for the difference of two solutions.
• Uses Gronwall’s lemma to derive an inequality showing that if I(τ) exists and initial conditions are identical, solutions must remain equal, proving uniqueness.
• Analyzes the implications of I(τ) blowing up at finite time τ*, indicating a potential finite-time singularity in velocity gradients.

Experimental results

Research questions

• RQ1 Can finite-time singularities in the Navier-Stokes equations violate the continuum hypothesis by introducing length scales smaller than molecular scales?
• RQ2 To what extent does the existence of weak solutions with unbounded velocity gradients challenge the self-consistency of the NSE?
• RQ3 Is the phenomenon of hydrodynamic turbulence purely deterministic, or could it require stochastic forces due to singularities?
• RQ4 How does the integral I(τ), related to the L∞ norm of velocity gradients, determine the uniqueness and existence of solutions?
• RQ5 What are the consequences of finite-time singularities for energy conservation and the validity of the momentum balance in the NSE?

Key findings

• Finite-time singularities in the Navier-Stokes equations would imply that velocity gradients ∂vi/∂xk diverge as (τ*−t)−γ with γ≥1, leading to arbitrarily small length scales.
• Such singularities would violate the continuum hypothesis, as length scales would fall below the molecular scale, requiring the inclusion of stochastic forces.
• The integral I(τ), defined as the time integral of the L∞ norm of the velocity gradient tensor, is both a sufficient and potentially necessary condition for solution uniqueness.
• If I(τ) exists up to time τ*, then solutions remain unique; if I(τ) blows up at τ*, a singularity occurs, breaking uniqueness and physical consistency.
• The proof of uniqueness via Gronwall’s lemma shows that identical initial conditions and finite I(τ) imply u(t)=0, so solutions remain equal.
• In the inviscid limit (ν₀→0), the uniqueness argument still holds if I(τ) remains finite, suggesting that singularities could emerge even without viscosity.