Earlier posts have given evidence that the Clay Navier-Stokes Millennium Problem, as formulated by Charles Fefferman, is ill-posed and as such has no physically meaningful resolution.
Here is how the scope, meaning and scientific relevance of the problem is presented by the Clay Institute:
- Mathematicians and physicists believe that an explanation for and the prediction of both the breeze and the turbulence can be found through an understanding of solutions to the Navier-Stokes equations.
- Although these equations were written down in the 19th Century, our understanding of them remains minimal.
- The challenge is to make substantial progress toward a mathematical theory which will unlock the secrets hidden in the Navier-Stokes equations.
We see that the Clay problem is motivated by the flight of an airplane as being described by the Navier-Stokes equations, but that the understanding remains minimal.
It seems therefore natural to ask that the present ill-posed (physically meaningless) Clay Navier-Stokes Problem will be reformulated into the problem of developing a well-posed (physically meaningful) Theory of Flight based on understanding (incompressible) Navier-Stokes solutions unlocking secrets of the Navier-Stokes equations.
To motivate such a reformulation, recall that the existing theory of flight was formulated by the German mathematician Martin Wilhelm Kutta in 1904 as as circulation theory, which however has shown to be an unphysical theory. The challenge was then to understand flight by understanding the mathematics of flight and the challenge is the same today with the additional requirement that the mathematical understanding should have a physical meaning. A real challenge because of the requirement of real meaning!
PS1 An example of mathematics without real meaning is Cantor's theory about transfinite numbers. Mathematics with real meaning is close to constructive mathematics. For Navier-Stokes equations this means that construction of solutions by computational mathematics is fundamental, and that analytical analytical mathematics is used to understand properties and qualities of the computed solutions as reflections of real phenomena.
PS2 In all modesty, our new theory of flight revealing The Secret of Flight would be a candidate for the prize.
PS3 The progress towards a solution of the problem in its original formulation by Fefferman, has been nil since 2000, or since Leray 1934, despite major efforts by the brightest mathematicians and there is not even a conjecture or possibility to explore. In such a situation Mr Clay may be expected to demand a reformulation into problem for which there is some hope of resolution, in order for his prize to compete with the Milner prize in mathematics coming up. A prize for a solving a meaningless unsolvable problem, has little meaning.
Inlägg
It seems therefore natural to ask that the present ill-posed (physically meaningless)
SvaraRaderaIsn't that a quite old view on the properties of ill-posed problems? Hadamard was a bit to quick to dismiss them.
See Petrov, Sizcov, Well-posed, Ill-posed, and Intermediate Problems with Applications, (VSP, 2005) p. 132 for instance:
Hadamard advanced a statement that ill-posed problems are physically
meaningless problems; in other words, if an equation that describes some
applied problem is an ill-posed equation, then the problem presents an arti-
ficial (unreal) problem or this problem is described in an inadequate manner
(for instance, some constraints on the solution making the problem correct
are not taken into account). Hadamard gave several examples of ill-posed
problems, for instance, the Cauchy problem for the Laplace equation (see
Section 4.3). This problem, however, has many applications in astronomy,
geophysics, cosmonautics, etc., i. e., presents a physically meaningful prob-
lem.
Moreover, many applied problems (in signal and image processing, to-
mography, spectroscopy, control theory, etc.) are ill-posed problems, which
fact was widely recognized in several last decades. It should therefore be
concluded that the statement advanced by Hadamard was invalid and has
slowed down the development of many fields of research in pure and applied
mathematics.
It is true that there are so-called ill-posed problems (such as inverse problems) of importance, but in order to make an ill-posed problem physically meaningful and computable, it has to be regularized by smoothing which eliminates unphysical solutions. The effect of the regularization is to turn an ill-posed problem into a well-posed problem. Hadamard was not off the point, but misunderstanding is always possible.
SvaraRadera