Remarks on unsolved basic problems of the Navier-Stokes equations

TL;DR

This paper discusses fundamental unresolved issues of the Navier-Stokes equations, including the existence of singularities, energy conservation, and the mathematical consistency of the model in describing fluid dynamics.

Contribution

It reviews key physical assumptions, explores the implications of weak solutions, and highlights open problems related to singularities and the continuum hypothesis.

Findings

Discussion of weak solutions and energy conservation

Analysis of finite time singularities and their implications

Identification of unresolved fundamental problems in NSE

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.