Talk:Navier–Stokes existence and smoothness

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[edit] Possible links to examples of simpler solutions?

It says the analog for R2 is solved but doesn't point to the article or reference stating examples, solutions, or proofs.

JWhiteheadcc 09:24, 14 May 2007 (UTC)

[edit] Notation clue?

What if, where it identifies Δ, we add parenthetically the notation I am more used to seeing, e.g:

(also written \nabla^2)

One of the biggest barriers I have in reading math is that there are so many different notations for the same concept, and so many different usages for the same symbols. Δ is used in quite a few different ways, for example. In this case, the change in notation really had me scratching my head, since this is a familiar equation. For the casual reader, it could be a bigger stumbling block.

Note that Δ is the form in the original problem statement, so I don't think we should rewrite it, just provide the conceptual link. But many of the linked references use the \nabla^2 notation.

Bob Kerns (talk) 21:39, 23 December 2007 (UTC)

Δ is the notation most used by mathematicians (especially PDE theorists), so it makes sense to use it here. I see your point, though. In fact, this article needs a lot of rewriting, and more background and history. As it is, it is based too closely on the Clay Institute problem description. The Clay description is ultraprecise and narrow for a good reason: there should be room no dispute on whether a solution warrants the million dollar prize. However, that's not necessarily right for Wikipedia. The Riemann hypothesis and Hodge conjecture articles, for example, are not mere writeups of the official Clay Institute problem descriptions. What this article should be, I think, is an overview of rigorous, mathematical approaches to the Navier-Stokes equations. The Clay Institute problem should be there, but in context. It should be pointed out that there are other mathematical problems under the same heading which are also very important even if there is no million dollar prize, one example being the compressible Navier-Stokes equations. Perturbationist (talk) 03:04, 31 December 2007 (UTC)
Bob, reading the above comment it looks like I went on a rant about this article and sort of ignored your suggestion. I didn't mean to do that. My point was, I think we should do what you said, but it wouldn't really flow into the text the way it's written now, so we should rewrite the article first. Perturbationist (talk) 02:37, 1 January 2008 (UTC)
Not to worry, I didn't take it as a rant at all! You raise a good point -- should this be about the Clay Institute problem, or about the mathematical problem? While the former may be more exciting to read about, the latter is probably more useful.
I also like that you identify that this notation is favored by PDE theorists. I think that while the math world has all these notational camps, Wikipedia should try to cross-reference them. I don't care which notation is used, just that notation differences should be minimized as a stumbling block for readers familiar with one or the other. Placing it in context as "Δ is the notation most used by mathematicians (especially PDE theorists)" is really helpful to learn when to expect which notation. Despite my ancient background with Macsyma and PDE theorists, I've had more to do with engineers and physicists overall, so your phrase lights a little LED over my head...
It almost seems like a "Math notation" Wikipedia project would be helpful, to catalog the various systems of notation and variants, and which articles using a particular notation could simply reference.
I find keeping the notation systems straight for, say, category theory, to be harder than the theory itself! (At the level I grasp it, anyway!) Bob Kerns (talk) 06:18, 4 January 2008 (UTC)
It sounds like we have similar ideas about the direction this article should go in. I want to run by you the changes I have in mind and see what you think. The Clay problem should be mentioned, but alongside more general problems. For example, perhaps the most natural problem, and one to which a lot of effort has been devoted, is to solve the Navier-Stokes equations on a compact region Ω in \mathbb{R}^3 with smooth boundary, with the boundary condition u = 0 on \partial\Omega. The formulation of the Clay problem avoids boundary conditions -- this is meant to make the problem easier -- but that doesn't mean we should. As for the notation, the text is way too dense. The quantifier symbols (\forall and \exists) should be replaced by English, and there is no need to repeat expressions like f \mathcal{2} (C^\infty(\mathbb{R}^3))^3 all over the place. It would be better to say something like "let f be a smooth time-dependent vector field on \mathbb{R}^3".
The math notation project certainly sounds like a good idea, though it may be a lot of work to get off the ground. Perturbationist (talk) 04:09, 6 January 2008 (UTC)

[edit] Advices for person who want to find this exact analytical solution of Navier-Stokes equation

1. This problem is extremely difficult.

If you will work fast or for money (for 1 million dollars) you will destroy your brain yourself. It is very dangerous problem for health. Our brain cannot to solve it fast.

2. There were 12 attempts to solve it in USA and Europe till 2008 year. But they had faults. One person wrote that he lost 8 hours during 6 years every day. And he made mistakes. He could not solve it.

3. Mathematicians created very complex notation, which stop them. I think, it interferes in of you to find this solution.

4. This problems useful not for mathematicians, but for all scientists from much science. The Navier-Stokes equation - it is very complex mathematical model. But mathematicians think today, that it is simple mathematical model. Look please Lecture by Luis Cafarelli. http://claymath.msri.org/navierstokes.mov He thinks so and many others.

5. If your organization needs for this solution, I can tell you for 4 minutes. But when I try to explain it some professors in the Universities, they don't understand me, because they didn't know the theory of differential equation or they don't know physics.

Without mathematics you cannot create a mathematical model, for example for radioactive decay. If you don't know radioactivity, you don't understand the solution of this equation, if you are mathematician only. And I think, it is very interesting for us to know, when people can rediscovery solution of this very difficult problem again. May be after 5 or 8 years?

6. I don't want to publish my results, because it will be use for design of weapon. When I try to send my e-mail for scientists, my letters were blocked or disappear. It is amazing, but some people to try hampered me. I discovered in Montreal some numbers of strange people. One person worked with a garbage in McGill University. If he wants, you cannot read the book. They use special machine, may be electromagnetic field (do you know what it is means 8 terahertz - you can see the shade,shadow of beam before it will starts heat you) to stop any person, if they want. They use chemical microinjection of drugs with rings.

7. Today, Canadian research consul (he are working for Government) don't want to support your work with using exact solution of Navier-Stokes equation anywhere, for application. It is not interesting for any Canadian organizations. It was strange for me. I was upset. But when people developed math theory for radio, they had the same problems many years ago. But I think, they has not money. For people today, there is no difference, who are you.They have enough official scientists. And they have to solve any problems only. It is funny.

8. Exact solution of Navier-Stokes equation for turbulent flow is nonlinear waves. Navier-Stokes equation can describe two physical and one biological process, for example. Turbulence, mitosis and other. I think, that you see this solution in the future.I try to use solution for describing finance process with the stocks on tsx.com It is interesting to know, there is a chaos in the stock exchange, likes in the fluid or not? Is it possible to describe it with Navier-Stokes? I think my comments will help you.