Navier–Stokes existence and smoothness: Difference between revisions
→Attempt at solution: Paragraph is incoherent and references original research which is not allowed. The "Universal Journal of Applied Math" doesn't seem like a reputable journal. |
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==Attempt at solution== |
==Attempt at solution== |
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'''Classical solutions''' |
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In 2013, [[Mukhtarbay Otelbaev]] of the [[L.N.Gumilyov Eurasian National University|Eurasian National University]] in [[Astana]], [[Kazakhstan]], proposed a solution. As an attempt to solve an important open problem, the proof was immediately inspected by others in the field, who found at least one serious flaw.<ref>{{cite journal |url=http://www.nature.com/news/fiendish-million-dollar-proof-eludes-mathematicians-1.15659 |title=Fiendish million-dollar proof eludes mathematicians |first=Katia |last=Moskvitch |journal=Nature |date=5 August 2014|doi=10.1038/nature.2014.15659}}</ref> Otelbaev is attempting to fix the proof, but other mathematicians are skeptical. |
In 2013, [[Mukhtarbay Otelbaev]] of the [[L.N.Gumilyov Eurasian National University|Eurasian National University]] in [[Astana]], [[Kazakhstan]], proposed a solution. As an attempt to solve an important open problem, the proof was immediately inspected by others in the field, who found at least one serious flaw.<ref>{{cite journal |url=http://www.nature.com/news/fiendish-million-dollar-proof-eludes-mathematicians-1.15659 |title=Fiendish million-dollar proof eludes mathematicians |first=Katia |last=Moskvitch |journal=Nature |date=5 August 2014|doi=10.1038/nature.2014.15659}}</ref> Otelbaev is attempting to fix the proof, but other mathematicians are skeptical. |
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'''Alternative solutions''' |
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[http://en.wikipedia.org/wiki/Terence_Tao Terence Tao] in 18 March, 2007[http://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/ announced] '''three possible strategies''' of an alternative solutions if one wants to solve the full Millennium Prize problem for the 3-dimensional Navier-Stokes equation. Strategy 1 “Solve the Navier-Stokes equation exactly and explicitly (or at least '''transform this equation exactly and explicitly to a simpler equation''')” is used in these works: |
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*[http://www.hrpub.org/download/20131107/UJAM1-12600416.pdf Navier –Stokes First Exact Transformation (3 pages)] |
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*[http://www.hrpub.org/download/20140205/UJAM3-12601996.pdf Navier –Stokes Second Exact Transformation (5 pages)] |
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The author of these brief notes [http://continuum-paradoxes.narod.ru/ Alexandr Kozachok] has offered ([https://groups.google.com/forum/#!topic/sci.physics.computational.fluid-dynamics/Ne7CgE7nuG0 in February 2008] – [http://goglee.org/Navier-Stokes-Equations----from-Eric-Weisstein's-World-of- Internet ], in 2008, 2010, 2012 – [http://www.mathkravchuk.narod.ru/ INTERNATIONAL CONFERENCE reports], in November 2013 and February 2014 - [http://www.hrpub.org/journals/jour_archive.php?id=26 INTERNATIONAL journal]) two exact transformations to the simpler equations. These transformations are executed by well-known classical methods of mathematical physics. Therefore [http://vimeo.com/18185364 not only] [http://www.academia.edu/9592018/Temperature_cycles_from_an_exact_reduction_of_the_Navier_Stokes_equation some] [http://www.academia.edu/8480418/A_simple_exact_solution_to_the_Navier-Stokes_equation professionals], but [http://www.indabook.org/preview/Q4KoTvpJMB1-ZXINVIzYVz6ZndQDOhmFC_FVDgag6Qc,/Navier-Stokes-First-Exact-Transformation.html?query=Navier-Stokes-Equation-Examples also] [http://ru.convdocs.org/docs/index-2701.html educational], [http://www.socialscapes.com/search/navier-stokes-existence-and-smoothness-wikipedia-the/ social] and many [http://ebooktake.in/view/ebook/aHR0cDovL3d3dy5ocnB1Yi5vcmcvZG93bmxvYWQvMjAxNDAyMDUvVUpBTTMtMTI2MDE5OTYucGRmW2J3bl1OYXZpZXIg4oCTc3Rva2VzIFNlY29uZCBFeGFjdCBUcmFuc2Zvcm1hdGlvbg other] [http://booksreadr.org/pdf/navier-stokes-millennium-prize-problem-alternative-solution-194170446.html sites] have [http://pdfmanual4.com/navier-stokes-first-exact-transformation/ republished] or [http://www.pdftube.com/read.php?url=aHR0cDovL3d3dy5ocnB1Yi5vcmcvZG93bmxvYWQvMjAxMzExMDcvVUpBTTEtMTI2MDA0MTYucGRm paid attention] to [http://www.pdftube.com/read.php?url=aHR0cDovL3d3dy5ocnB1Yi5vcmcvZG93bmxvYWQvMjAxNDAyMDUvVUpBTTMtMTI2MDE5OTYucGRm these works] . |
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==Notes== |
==Notes== |
Revision as of 20:08, 22 December 2014
Millennium Prize Problems |
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The Navier–Stokes existence and smoothness problem concerns the mathematical properties of solutions to the Navier–Stokes equations, one of the pillars of fluid mechanics (such as with turbulence). These equations describe the motion of a fluid (that is, a liquid or a gas) in space. Solutions to the Navier–Stokes equations are used in many practical applications. However, theoretical understanding of the solutions to these equations is incomplete. In particular, solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolved problems in physics, despite its immense importance in science and engineering.
Even much more basic properties of the solutions to Navier–Stokes have never been proven. For the three-dimensional system of equations, and given some initial conditions, mathematicians have not yet proved that smooth solutions always exist, or that if they do exist, they have bounded energy per unit mass.[citation needed] This is called the Navier–Stokes existence and smoothness problem.
Since understanding the Navier–Stokes equations is considered to be the first step to understanding the elusive phenomenon of turbulence, the Clay Mathematics Institute in May 2000 made this problem one of its seven Millennium Prize problems in mathematics. It offered a US$1,000,000 prize to the first person providing a solution for a specific statement of the problem:[1]
Prove or give a counter-example of the following statement:
In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.
The Navier–Stokes equations
In mathematics, the Navier–Stokes equations are a system of nonlinear partial differential equations for abstract vector fields of any size. In physics and engineering, they are a system of equations that models the motion of liquids or non-rarefied gases (in which the mean free path is short enough so that it can be thought of as a continuum mean instead of a collection of particles) using continuum mechanics. The equations are a statement of Newton's second law, with the forces modeled according to those in a viscous Newtonian fluid—as the sum of contributions by pressure, viscous stress and an external body force. Since the setting of the problem proposed by the Clay Mathematics Institute is in three dimensions, for an incompressible and homogeneous fluid, only that case is considered below.
Let be a 3-dimensional vector field, the velocity of the fluid, and let be the pressure of the fluid.[note 1] The Navier–Stokes equations are:
where is the kinematic viscosity, the external force, is the gradient operator and is the Laplacian operator, which is also denoted by . Note that this is a vector equation, i.e. it has three scalar equations. Writing down the coordinates of the velocity and the external force
then for each there is the corresponding scalar Navier–Stokes equation:
The unknowns are the velocity and the pressure . Since in three dimensions, there are three equations and four unknowns (three scalar velocities and the pressure), then a supplementary equation is needed. This extra equation is the continuity equation describing the incompressibility of the fluid:
Due to this last property, the solutions for the Navier–Stokes equations are searched in the set of "divergence-free" functions. For this flow of a homogeneous medium, density and viscosity are constants.
The pressure p can be eliminated by taking an operator rot (alternative notation curl) of both sides of the Navier–Stokes equations. In this case the Navier–Stokes equations reduce to the vorticity-transport equations.
Two settings: unbounded and periodic space
There are two different settings for the one-million-dollar-prize Navier–Stokes existence and smoothness problem. The original problem is in the whole space , which needs extra conditions on the growth behavior of the initial condition and the solutions. In order to rule out the problems at infinity, the Navier–Stokes equations can be set in a periodic framework, which implies that they are no longer working on the whole space but in the 3-dimensional torus . Each case will be treated separately.
Statement of the problem in the whole space
Hypotheses and growth conditions
The initial condition is assumed to be a smooth and divergence-free function (see smooth function) such that, for every multi-index (see multi-index notation) and any , there exists a constant such that
- for all
The external force is assumed to be a smooth function as well, and satisfies a very analogous inequality (now the multi-index includes time derivatives as well):
- for all
For physically reasonable conditions, the type of solutions expected are smooth functions that do not grow large as . More precisely, the following assumptions are made:
- There exists a constant such that for all
Condition 1 implies that the functions are smooth and globally defined and condition 2 means that the kinetic energy of the solution is globally bounded.
The Millennium Prize conjectures in the whole space
(A) Existence and smoothness of the Navier–Stokes solutions in
Let . For any initial condition satisfying the above hypotheses there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector and a pressure satisfying conditions 1 and 2 above.
(B) Breakdown of the Navier–Stokes solutions in
There exists an initial condition and an external force such that there exists no solutions and satisfying conditions 1 and 2 above.
Statement of the periodic problem
Hypotheses
The functions sought now are periodic in the space variables of period 1. More precisely, let be the unitary vector in the i- direction:
Then is periodic in the space variables if for any , then:
Notice that this is considering the coordinates mod 1. This allows working not on the whole space but on the quotient space , which turns out to be the 3-dimensional torus:
Now the hypotheses can be stated properly. The initial condition is assumed to be a smooth and divergence-free function and the external force is assumed to be a smooth function as well. The type of solutions that are physically relevant are those who satisfy these conditions:
3.
4. There exists a constant such that for all
Just as in the previous case, condition 3 implies that the functions are smooth and globally defined and condition 4 means that the kinetic energy of the solution is globally bounded.
The periodic Millennium Prize theorems
(C) Existence and smoothness of the Navier–Stokes solutions in
Let . For any initial condition satisfying the above hypotheses there exist smooth and globally defined solutions to the Navier–Stokes equations, i.e. there is a velocity vector and a pressure satisfying conditions 3 and 4 above.
(D) Breakdown of the Navier–Stokes solutions in
There exists an initial condition and an external force such that there exists no solutions and satisfying conditions 3 and 4 above.
Partial results
- The Navier–Stokes problem in two dimensions has already been solved positively since the 1960s: there exist smooth and globally defined solutions.[2]
- If the initial velocity is sufficiently small then the statement is true: there are smooth and globally defined solutions to the Navier–Stokes equations.[1]
- Given an initial velocity there exists a finite time T, depending on such that the Navier–Stokes equations on have smooth solutions and . It is not known if the solutions exist beyond that "blowup time" T.[1]
- The mathematician Jean Leray in 1934 proved the existence of so-called weak solutions to the Navier–Stokes equations, satisfying the equations in mean value, not pointwise.[3]
- Terence Tao in February 2014 announced a finite time blowup result for an averaged version of the 3-dimensional Navier-Stokes equation. He writes that the result formalizes a "supercriticality barrier" for the global regularity problem for the true Navier-Stokes equations, and claims that the method of proof in fact hints at a possible route to establishing blowup for the true equations.[4]
Attempt at solution
Classical solutions
In 2013, Mukhtarbay Otelbaev of the Eurasian National University in Astana, Kazakhstan, proposed a solution. As an attempt to solve an important open problem, the proof was immediately inspected by others in the field, who found at least one serious flaw.[5] Otelbaev is attempting to fix the proof, but other mathematicians are skeptical.
Alternative solutions
Terence Tao in 18 March, 2007announced three possible strategies of an alternative solutions if one wants to solve the full Millennium Prize problem for the 3-dimensional Navier-Stokes equation. Strategy 1 “Solve the Navier-Stokes equation exactly and explicitly (or at least transform this equation exactly and explicitly to a simpler equation)” is used in these works:
- Navier –Stokes First Exact Transformation (3 pages)
- Navier –Stokes Second Exact Transformation (5 pages)
The author of these brief notes Alexandr Kozachok has offered (in February 2008 – Internet , in 2008, 2010, 2012 – INTERNATIONAL CONFERENCE reports, in November 2013 and February 2014 - INTERNATIONAL journal) two exact transformations to the simpler equations. These transformations are executed by well-known classical methods of mathematical physics. Therefore not only some professionals, but also educational, social and many other sites have republished or paid attention to these works .
Notes
References
- ^ a b c Official statement of the problem, Clay Mathematics Institute.
- ^ Ladyzhenskaya, O. (1969), The Mathematical Theory of Viscous Incompressible Flows (2nd ed.), New York: Gordon and Breach.
- ^ Leray, J. (1934), "Sur le mouvement d'un liquide visqueux emplissant l'espace", Acta Mathematica, 63: 193–248, doi:10.1007/BF02547354
- ^ Finite time blowup for an averaged three-dimensional Navier-Stokes equation Blog post by Terence Tao, with link to Arxiv preprint of paper.
- ^ Moskvitch, Katia (5 August 2014). "Fiendish million-dollar proof eludes mathematicians". Nature. doi:10.1038/nature.2014.15659.
External links
- The Clay Mathematics Institute's Navier–Stokes equation prize
- Why global regularity for Navier–Stokes is hard — Possible routes to resolution are scrutinized by Terence Tao.
- Navier–Stokes existence and smoothness (Millennium Prize Problem) A lecture on the problem by Luis Caffarelli.