From: Stephen Montgomery-Smith Subject: Do you want to be a millionaire? Date: Fri, 02 Jun 2000 01:34:51 GMT Newsgroups: sci.math Summary: [missing] I was looking at the recent prize offer of a million dollars for various math problems, and I was wondering about the status of them - in the sense - are these problems that people have largely given up on, or are they the subject of active research? ( http://www.claymath.org ) I know the answer for the Navier-Stokes problem, as I have been thinking about it for about three/four years, and I know many other people have been too. It is a nice problem, because it is not a career destroyer, that is, as a new Ph.D. or postdoc you can work on this problem or related ones, and still produce nice papers and make a career even if you don't solve the problem. As evidence of the large worldwide interest in this problem, look at the web sites: http://wwwlma.univ-bpclermont.fr/NSenet/ http://www.math.missouri.edu/math-fluids/ Now the Riemann Hypthesis - I recall that Louis de Branges recently circulated a paper claiming to solve the problem, but other than that I get the impression that people have largely given up. It is kind of like Fermat's last theorem - before it was solved it was considered that only a crackpot would seriously work on it - so that Wiles worked on it in secret. Is that also the case with the Riemann Hypothesis? Similarly for the P=NP conjecture - do people seriously work on solving it - does anyone have any idea how a solution might look? I know that people do a lot of heuristic or random methods for solving it, but not to work on the real problem. The only other problem I understand is the Poincare conjecture. I remember a false solution, maybe 10 or 15 years ago, by someone in England. It was a joint work between a Professor (O'Rourke I think was his name) and his student. I wonder what happened to the student, and whether he managed to get a good career going in mathematics? Anyone know what people are doing with the these problems? Are my impressions correct? Stephen Montgomery-Smith ============================================================================== From: Dries van Oosten Subject: Re: Understanding Navier-Stokes: what's pressure? Date: Fri, 4 Aug 2000 11:45:47 +0200 Newsgroups: sci.math,sci.physics Summary: [missing] On 3 Aug 2000, Axel Boldt wrote: > Hello, > > I'm trying to understand the right-hand side of the Navier Stokes > equation for an incompressible viscous fluid, which looks like > > - gradient of pressure + eta * laplace of velocity > > This is supposedly the total force per unit volume; eta is a constant, > the first coefficient of viscosity. Pressure is a scalar field > depending on position and time, velocity is a vector field depending > on position and time, and gradient and laplace are computed with > respect to the position variables only. > > My first problem is the definition of pressure. It is defined to be > the force acting perpendicularly on a small area. I assume this area > is floating with the fluid, so relative to the fluid it is at rest. > Because of the internal stresses, it seems to matter how I orient > the area, doesn't it? But then it wouldn't be a scalar field anymore. Equation in hydrodynamics are written down for fluid elements. The pressure would be the total pressure on a fluid element. If you take a cubic box, you add the pressure on each side. If the are all equal (i.e. if the gradient over the size of you box is zero) then there is no pressure force of the element. > My second question has to do with the stress tensor, which supposedly > gives rise to the laplace term. Is pressure the same as the diagonal > entries of the stress tensor? Are those three numbers equal? If not, > what do the diagonal entries signify? I believe I understand the shear > stresses (nondiagonal entries). I don't have the book handy and I never use hydrodynamics, so I can't remember the exact shape of the stress tensor. Your phrase: "which supposedly gives rise to the laplace term" suggest that you have not perform the derivation. I suggest you do, it's not a very difficult derivation if you use for instance the book by Landau and Lifshitz. > Thanks much, > Axel > > -- > Axel Boldt ** axel@uni-paderborn.de ** math-www.uni-paderborn.de/~axel/ > Sponsor free software at the Free Software Bazaar visar.csustan.edu/bazaar/ > Groeten, Dries ============================================================================== From: Bob Terrell Subject: Leray paper on Navier Stokes Date: Tue, 05 Sep 2000 12:41:13 -0400 Newsgroups: sci.mech.fluids,sci.math,sci.physics Summary: [missing] For those who may be interested, I have posted a translation to English of Leray's classic paper on the Navier Stokes equations, from Acta Mathematica 1934, to my web page http://www.math.cornell.edu/~bterrell This great paper proves more than it is commonly quoted for. Right now I am still correcting typos every day or two. Please email corrections or other feedback to bterrell@math.cornell.edu Bob