The global solution of Navier-Stokes equations is one of the biggest mysteries in theoretical physics, and it belongs to list of Millenium problems by Clay Institute.
But in a recent paper, Dr. Algirdas Maknickas provides a solution to Navier-Stokes equations by transforming these equations into parabolic differential equations, and then to Helmholtz equation. The solution is in the form of sinusoidal wave. See http://vixra.org/abs/1311.0164
So, do you think that the global solution of Navier-Stokes equations has been found? Your comments are welcome.
http://vixra.org/abs/1311.0164
Several years ago I managed to reduce Navier-Stokes equation for planar and axially symmetric gas flows to a solvable equation whose solutions can in general be composed geometrically. Now, to finalize construction of solutions I need to develop some geometry, but I am busy with other work.
I have not looked at this paper, but I would not hold my breath for this one. Numerous claims have surfaced (most of them amateurish, but some high-profile like Penny Smith from Lehigh in 2006 and Mukhtarbay Otelbaev of the Eurasian National University in 2013), but in every case catastrophic flaws have been identified. The problem is still open and is likely to stay that way for some time (Nature has a nice synopsis at http://www.nature.com/news/fiendish-million-dollar-proof-eludes-mathematicians-1.15659 ). There is a general lack of appreciation of just how difficult this problem is, especially since there is no general theory of non-linear PDEs (as there is for linear PDEs). Terrence Tao recently submitted a paper describing some of the attributes a yet-to-be-developed solution would have, including resolution of the fine structure of the non-linear operator (something that is tied intimately to the cascade of turbulence interactions down to their smallest physical length and time scales).
As far as I remember, the problem in Clay Inst formulation is to prove existence of smooth solutions, not to solve the equation itself. My construction shows that shock waves almost always appear.
Yes, the Clay problem involves demonstrating properties of the equations, specifically in 3-D. "Solving the equations" is something more associated with applied problems, for example by invoking specific boundary and initial conditions. There is an enormous body of literature and know-how for doing this task in an approximate sense (CFD, perturbation solutions, etc). I would point out that the formal statement for the Clay problem (available here: http://www.claymath.org/millenium-problems/navier%E2%80%93stokes-equation ) specifies the so-called incompressible form of the N-S equations, which precludes shock waves.
I have some unknown incompressible solutions too, but gas with thermodynamics is much simpler.
Thank you, Alejandro, Zafar and Michael, for your answers. Best wishes
The Global? May be. But certainly not general:
the Q(x,t) in Eq.(10) has special form of sum of sin-functions. Some physical functions can not be expressed as sum of sin-functions. Same problem in Eq.(13).
Number of modas of N-S Equation was found by Dr. Oscar Manley and co-author, published in JFM, around 1982 or 1983.
@Alexander, P.K. Mohanty, and Rui. Thanks for your answers. Best wishes
I think it is interesting to read this article concerning the question http://terrytao.wordpress.com/2014/02/04/finite-time-blowup-for-an-averaged-three-dimensional-navier-stokes-equation/
The article above posted by Olena discusses Terrence Tao's paper I cited earlier. Since it has not yet been mentioned, I will point out that the 2-D version of the problem has already been answered in the affirmative: there *are* smooth global solutions for the 2-D equations (see Ladyzhenskaya's 1969 text: The Mathematical Theory of Viscous Incompressible Flows). The Clay problem is specifically for the (much more difficult) case of the 3-D equations. There has been a rumor floating around for a few years that Grigori Perelman is also working on this problem (he solved the Clay problem on the Poincare conjecture). I would look for any progress on the N-S problem to come from mathematicians of the caliber of Tao and Perelman.
@Michael. Thanks, yes 3D N-S problem is much more difficult. Best wishes
In general, not yet, only for sime particular cases. Even the problem about the existence of unique solution is still not solved. the last try for this was recently done, now is under provement. There are no general methods for solution of the non-linear PDE.
Not all is published. I said, I have unpublished materials on quite general solutions for compressible flows with thermodynamics.
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