The current state of Navier-Stokes existence and smoothness

[AVFistula]

Can anyone point me to some publications or archives which feature developments in solving the N-S existence and smoothness problem? Basically, I'd like to read up about how far people have gone towards solving the problem, e.g. a new method to analyze the equations.

Also, what fields of mathematics and physics do the Navier-Stokes equations employ? I've taken differential equations (the class had barely any content on partial diffeq) and just physics I+II. Can I even begin to understand the equations in-depth without any other math or fluid mechanics?

 

[The_Duck]

I don't know anything about status of the millenium problem. But the Navier-Stokes equations are just a partial differential equation. If and when you have a good handle on PDE's the Navier-Stokes equations are not too hard to understand. The physics they encode is quite simple--it's just the equivalent of F = ma except it describes a continuous medium instead of a point particle.

 

[torquil]

Also, what fields of mathematics and physics do the Navier-Stokes equations employ?

Regarding the math, I would guess that the theory of Sobolev spaces would be very relevant.

 

[AVFistula]

Okay, thanks for the heads up. I've since read a lot of explanations and derivations of the N-S equations, so I think the math isn't too bad. I'll read up about Sobolev spaces, though.

If anyone knows where I can read general science/math publications so that I can search for developments in this particular subject, I am still looking and would greatly appreciate a heads-up.

 

[sardar]

The Navier-Stokes equations are a set of partial differential equations that describe the motion of fluids. They are widely used in many fields of physics and engineering, including fluid mechanics, aerodynamics, and weather forecasting. The equations are notoriously difficult to solve, and the question of their existence and smoothness has been a topic of much research for over a century.

There are many publications and archives that feature developments in solving the Navier-Stokes existence and smoothness problem. One of the most notable papers on this topic is the Clay Mathematics Institute's Millennium Prize Problem, which offers a $1 million prize to anyone who can prove the existence and smoothness of solutions to the Navier-Stokes equations. Other notable publications include the work of mathematicians such as Terence Tao and Cédric Villani, who have made significant contributions to understanding the equations.

The Navier-Stokes equations employ a wide range of mathematical and physical concepts, including vector calculus, differential geometry, and fluid mechanics. A deep understanding of these topics is necessary to fully comprehend the equations and their solutions. However, with a strong foundation in differential equations and basic physics, you can certainly begin to understand the basics of the Navier-Stokes equations. It is a complex topic, but with dedication and further study, you can build a deeper understanding of these important equations in fluid dynamics.