A strong solution of the Navier-Stokes equations?

[isoaker]

A strong solution to the Navier-Stokes equations?

Ben and perhaps Drenchenator would be able to comment more on this, but if this turns out to be true, a more accurate way to model fluid dynamics may be on the horizon.

Here's a link to the original paper - in Russian

[Drenchenator]

The Navier-Stokes existence and smoothness problem is a huge unsolved problem in mathematics. The Navier-Stokes equations are a set of nonlinear differential equations that describe the motion of fluids, so that's why it's important for water guns. The problem is that there is has been no mathemtical proof in 3D that solutions to the equations exist (the existence problem) and there has been no proof that these solutions are then "smooth solutions" (forgive the scare quotes; these are mathematical definitions). This has been solved in 2D for decades now, but not in 3D.

Now, you may say, well of course 3D solutions exist! I see water flowing all the time. Well, that's reality; this still needs to be proved mathematically, which is a challenge. The Navier-Stokes equations are notoriously difficult to solve, given that fluid flow can easily have turbulence, chaos and a host of other physically difficult phenomena.

Many people have claimed to have solved this in the past and then had to retract their work. I remember a few years ago reading about someone who said he solved this problem and then quickly retracted his solution when an issue arose. Given that this is in Russian, a hundred pages long, and is most likely far more mathematical that I can handle, I can't really comment as to the validity of this particle person's solution (I am very interested in the problem, though). However, I will say that if proved true it will offer a lot of insight into the nature of the Navier-Stokes equations. The big thing people are looking for is some insight into the nature of turbulence, which a solution could provide.

[SSCBen]

Thanks for bringing this to my attention, isoaker. This is a very interesting issue, and I'll be sure to follow these developments closely. Unfortunately, I can't really understand the paper as it's in Russian, but it seems to start from the Navier-Stokes equations and then moves into proving that some energy is bounded. That's the approach I think is most likely to succeed for this problem. Either that or something starting from the Boltzmann equation, which is more general than N-S, but may prove to be more tractable.

a more accurate way to model fluid dynamics may be on the horizon.

The media might suggest that and I've read some laymen suggest the same. Unfortunately, solving the Millennium Prize problem offers probably no help to modeling fluid dynamics. The prize concerns the most basic form of the Navier-Stokes equations. For the more general case, everyone assumes that the solutions exist and are unique. It'd be unprecedented for solutions to not exist and be unique (for "good" starting conditions) for equations in mathematical physics. I think the problem is of more academic interest than practical interest.

Here's another way to think of it: A proof of the existence, etc., of solutions to N-S is like saying "There's more than enough rocket fuel for us to go to Saturn and back!" That doesn't make manned spaceflight to Saturn easy. Though, it at least lets you know if you are wasting your time. If N-S turns out to be a bad model because solutions don't necessarily exist when they should, etc., then perhaps I should start looking for a new line of work, or look to fix these issues.