Existence and properties of the Navier-Stokes equations

TL;DR

This paper investigates the existence and properties of solutions to the Navier-Stokes equations, revealing issues with viscous solutions in three dimensions and proposing modifications to better model fluid flow phenomena.

Contribution

It demonstrates the non-existence of viscous solutions in 3D Navier-Stokes equations and introduces modified equations to achieve desirable flow solutions.

Findings

No viscous solutions in 3D Navier-Stokes equations due to divergence-free velocity limitations.

Modified equations enable modeling of flow phenomena like blowup and laminar-turbulent transition.

Insights into fluid detachment and magnetic dynamo problem implications.

Abstract

A proof of existence, uniqueness and smoothness of the Navier-Stokes equations is an actual problem, which solution is important for different branches of science. The subject of this study is obtaining the smooth and unique solutions of the three-dimension Stokes-Navier equations for the initial and boundary value problem. The analysis shows that there exist no viscous solutions of the Navier-Stokes equations in three dimensions. The reason is the insufficient capability of the divergence-free velocity field. It is necessary to modify the Navier-Stokes equations for obtaining the desirable solutions. The modified equations describe a three-dimension flow of incompressible fluid which sticks to a body surface. The equation solutions show the resonant blowup of the laminar flow, laminar-turbulent transition, the fluid detachment that opens the way to solve the magnetic dynamo problem.