The physics puzzle that could win you a million dollars

The Navier-Stokes equations explained.

Look at the picture of water below. It flows in very complicated ways. It forms bubbles. It forms intricate swirls and eddies, each of which have many different length scales. So this begs the question: can we model how water flows mathematically? Given how complex, confusing, and unpredictable water flow is, this seems like an impossible problem.

The flow of water is so complicated that it’s the subject of one of the Millennium Prize Problems, the so-called Navier-Stokes existence and smoothness problem. If you can solve this problem, the Clay Math Institute will give you a $1M prize. But what is this problem and why is it so hard?

Enter Euler’s Equation

The first step in modelling fluids was taken in the 18th century, when Euler developed a simple set of equations to describe fluid flow. These are sometimes called the Euler equations:

\(\dfrac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} = - \dfrac{\nabla P}{\rho}.\)

(The exact definitions of each of the terms is beyond the scope of this post. But I’ll provide links down below which explain them thoroughly.)

This equation was quite good at modelling fluids like water and air. But it did not properly model viscous fluids like molasses or ketchup. So in the 1800’s, mathematicians Navier and Stokes modified the equation by adding another term on the right to account for the viscosity of the fluid.

\(\dfrac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u} = - \dfrac{\nabla P}{\rho} + v \nabla^2 \mathbf{u}.\)

This is the famous Navier-Stokes Equation. It is a simultaneous system of partial differential equations that describes everything that flows in the universe: water, air, honey, gas, smoke, and more.

So what does this equation do for us? Say you’re pouring water from a cup and you know the velocity of the water at one point in time. If you plug in all this information into the Navier Stokes’ equation, it could predict how the water will flow for all time.

The million dollar prize is for proving that, given any initial condition (satisfying some mild smoothness conditions), there exists a smooth solution to the Navier-Stokes equations. A word of caution here is needed. The million dollar prize is not for finding the solution given an initial condition, but merely for showing that a solution exists!

Turbulent vs. Smooth Flow

To understand why, this is hard, we only have to look at the two terms on the left-hand side of the Navier-Stokes equation. The first term is the good guy: it’s called the diffusion term. The second term is the bad guy: it’s called the convection term.

\(\underbrace{\dfrac{\partial \mathbf{u}}{\partial t}}_{\text{diffusion term}} + \underbrace{\mathbf{u} \cdot \nabla \mathbf{u}}_{\text{convection term}} = - \dfrac{\nabla P}{\rho} + v \nabla^2 \mathbf{u}.\)

When the diffusion term is much bigger than the convection term, you get nice smooth flow. When the diffusion term is much smaller than the convection term, you get erratic, turbulent flow.

This equation is a constant push and pull between these two extremes. That’s why Navier-Stokes is so complicated. In fact, if we just deleted the bad term, the equation would become a linear equation and we’d be able to solve the equation in minutes.

So what have people done to tame the beast?

As much as we would love to, we cannot just get rid of the nasty convection term. But we can do the next best thing: make the convection term as small as possible. That is, we can look at cases where the diffusion term is much bigger than the convection term.

For example, if the fluid is flowing very slowly, this might happen. This is intuitive because if a fluid flows very slowly, you would not expect the flow to be very turbulent. Roughly speaking, the velocity term v would be very small. So the convection term would be like v², which is very small. In that case, the good guy overwhelms the bad guy, and we win.

This was the result of a ground-breaking paper in 1964 by mathematicians Fujita and Kato from Japan. They proved that if you start off with a slow enough velocity field, then there exists smooth solutions for the Navier-Stokes equations. The definition of “slow enough” is quite technical, but that’s what the paper is for!

Since then, people have tried many other things. In 1934, Jean Leray had a breakthrough and proved existence of so-called weak solutions for the Navier-Stokes equations. In 2016, Terrence Tao showed that a certain average-time Navier Stokes equation satisfies a certain finite-time blow up property. So far, showing existence of strong solutions seems intractably difficult.

But what’s the big picture?

On one hand, we can think of Navier-Stokes as just a math problem. But really, it’s something much more profound. I had a physics professor in my undergrad who told me:

Solving Navier-Stokes is like solving someone’s personality.

Navier-Stokes equation describes everything that flows, ever. Air, water, ketchup, gas, smoke. A solution to Navier Stokes means that there’s one function that describes all of these at once. This is nothing short of solving a personality. It’s taking an incredibly disorderly, complex, random system — and predicting it coldly, with a single equation.

Want to learn more?

Here are some further links to learn more about the Navier-Stokes equations:

1. Edriss S. Titi has a wonderful beginner-friendly talk about the Navier-Stokes equations.

2. More a slightly more technical account of these equations, here is Prof. Gareth McKinley of MITx Videos explaining the various components of the equation in more detail:

3. If you’re feeling ambitious, Terrence Tao has a set of online notes on his blog where he taught a graduate course on fluid equations.

Problem of the Week

As always, here is a challenge problem for this week:

For how many positive integers k do the lines with equations 9x + 4y = 600 and kx − 4y = 24 intersect at a point whose coordinates are positive integers?

If you have a solution to this challenge problem, submit it here for a chance to be featured in the next issue of this newsletter.

Solution from last week

See here for the solution to last week’s problem.

Feedback

If you have any learning resources that you think would be valuable for readers, or if you have any general feedback, let me know here (or just reply to this email) and I’d be happy to incorporate it.

Thanks for reading and happy learning! Until next time,

Adithya