# NS Millennium Problem: Uniqueness & Solutions Explained

In summary, if solutions to the NS equations are found to be non-unique, then the equations would need to be corrected. If a solution is found to be unique, then it would be proof that the solution exists and is stable.

I recently came across the NS millennium problem and I read that uniqueness for the NS equations is unknown. I have two questions.

First question, if solutions are found to be non-unique, would the NS equations have to be corrected?

Second question, since uniqueness is unknown, if someone finds a blow-up solution would he or she have to prove that it's unique to answer the problem? (e.g. turbulent and smooth solution possible for exactly the same IC, BC, etc..? kinda strange.)

I'm curious what uniqueness means in relation to smoothness and turbulence. I'm not an expert in math so sorry if the answers are obvious. haha. Thanks in advance.

There are equations that represent the dynamics of systems which are much simpler than NS, and are known to have non-unique solutions. A simple example is an oscillating mass on a spring, where the stiffness of the spring is nonlinear (Google for Duffing's equation). There are situations where the same applied force can produce different stable oscillations with completely different amplitudes. This is not just a theoretical curiosity. For example there was a lot of rather urgent work done in the aerospace industry in the 1990s, when it was discovered (by chance) that something similar to this could occur in existing designs of aircraft jet engines, with fairly obvious consequences for safety.

Non unique solutions to NS would certainly be "interesting" (i.e. worrying!) for computer simulations. A good way to find steady-state solutions to the NS equations in "real life" situations (for example modelling supersonic aircraft performance) is to start from a solution with the "wrong" flow conditions, and compute how the flow would change over time until it reaches a steady state. But if there several possible steady states, that approach would only find one of them, and there might not be a good way to "guess" what different starting conditions would find the others - especially if you don't even know how many different solutions you are looknig for.

I'm curious what uniqueness means in relation to smoothness and turbulence.

... Yup, you and a whole lot of other people - which is why answering the question is worth a Millenium Prize!

That would be very strange indeed. So if non uniqueness is proven it would give rise to a new strange and interesting field.

Btw, I tried googling Duffing's equation, but I couldn't find something that talked about non-uniqueness. I've never heard of non-uniqueness in a spring mass system. Can you please point me to a link that talks about it? Just in a general/basic sense, if possible,...I'm too lazy to think very hard. haha. Thanks.

## 1. What is the NS Millennium Problem?

The NS Millennium Problem is a mathematical problem that was proposed by the Clay Mathematics Institute in 2000. It consists of seven unsolved mathematical problems, each with a prize of \$1 million for anyone who can provide a solution. The problem is considered one of the most challenging and important problems in mathematics.

## 2. What is the specific problem of Uniqueness and Solutions in the NS Millennium Problem?

The Uniqueness and Solutions problem in the NS Millennium Problem is one of the seven unsolved problems. It deals with the question of whether there is a unique solution for certain mathematical equations, and if so, how to prove it. The problem is considered important because it has applications in many fields, including physics, engineering, and computer science.

## 3. How is the Uniqueness and Solutions problem related to other problems in the NS Millennium Problem?

The Uniqueness and Solutions problem is closely related to the other problems in the NS Millennium Problem, as it deals with the question of whether certain mathematical equations have a unique solution. This question is also relevant to other problems in the NS Millennium Problem, such as the Navier-Stokes Equations and the Riemann Hypothesis.

## 4. Why is it important to find a solution to the Uniqueness and Solutions problem in the NS Millennium Problem?

Finding a solution to the Uniqueness and Solutions problem in the NS Millennium Problem is important because it has significant implications in various fields, including physics, engineering, and computer science. It could also lead to a better understanding of the fundamental principles of mathematics and potentially lead to the development of new mathematical techniques and theories.

## 5. How close are we to finding a solution to the Uniqueness and Solutions problem in the NS Millennium Problem?

As of now, there is no known solution to the Uniqueness and Solutions problem in the NS Millennium Problem. However, progress has been made by mathematicians in understanding the problem and developing new techniques to approach it. It is difficult to predict when a solution will be found, but the Clay Mathematics Institute is actively encouraging and supporting research in this area.