Millennium Problem: Navier-Stokes existence and smoothness

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Navier-Stokes equations are about motion of fluids. Example of fluids being air and water. Mathematicians and physicists believe that an explanation for both the breeze and the turbulence of fluids can be found through an understanding of solutions to the Navier-Stokes equations.  A fluid is something that you can assume to be a continuum- i.e. not made of discrete particles. To this end, we can already see that the Navier-stokes equations must be approximations, because fluids are made of atoms!


The Navier-Stokes equations are, in essence, just Newton’s 2nd law written in a form that is applicable to continuum bodies, rather than discrete objects.
We know that Newton’s second law as F=ma, which can be written as

Where v is the velocity. If we are dealing with a fluid, we don’t care about mass, we care about density- which for now we assume is constant-i.e., the fluid is incompressible.

Therefore, just by taking the continuum form of Newton’s Second Law, we get:


Where f is the “body force” – the infinitesimal force on an infinitesimal chunk of the fluid.

Since the velocity of the fluid is both a function of space and of time – we have:


We note that dx/dt are just components of the velocity, and hence we can put this into a vector notation:



Which means:

So now we need to think about what these forces could be.
There are two types of forces that can act on a fluid:
  • Internal forces, due to interactions between the components
  • External forces such as gravity

There are two major internal forces that we need to consider: pressure forces, and the viscosity.
If we write this out, we have:
f=fviscous + fext - ∇P

Where P is the internal pressure, and ∇ is just a vector derivative.
Substituting we get


 The Navier-Stokes equation has one final step which is to write down the form of the viscous force.
The viscous force is the internal friction of all the particles rubbing up against each other, and takes the form:


 So we can see that just by applying Newton’s 2nd law to a small chunk of a fluid, we have derived:


 This really is the incompressible Navier-Stokes equation!
So that’s what it is, in relatively simple terms: it is just “force is the product of mass times acceleration” adjusted for pressure and viscosity.

The mathematics behind this, however, is where things get interesting and challenging.
This is an incredibly hard equation to solve. We can solve it exactly for certain specific cases, and you can make approximations to make it easier to study. It’s so difficult that it has not even been proved that solutions always exist.

The prize problem can be broken into two parts. The first focuses on the existence of solutions to the equation. The second focuses on whether these solutions are bounded (remain finite).  If you are able to show that you will win $1,000,000.

 Reference
http://www.claymath.org/sites/default/files/navierstokes.pdf

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