Mr Clay and a Meaningless Navier-Stokes Prize Problem

Turbulent flow around a landing gear as non-smooth solution of the 3d incompressible Navier-Stokes equations, by CTLab KTH. Watch also turbulent flow around an airplane in landing configuration. To argue that these flows are smooth would be a meaningless abuse of mathematical language.

The Clay Institute of Mathematics (CMI) founded by Landon T. Clay celebrated the new Millennium by setting up 7 Prize Problems each worth $1 million,  presented in beautiful words:

• The Clay Mathematics Institute (CMI) grew out of the longstanding belief of its founder, Mr. Landon T. Clay, in the value of mathematical knowledge and its centrality to human progress, culture, and intellectual life....
• ...to further the beauty, power and universality of mathematical thinking...deepest, most difficult problems... achievement in mathematics of historical dimension; 
• ...to elevate in the consciousness of the general public the fact that, in mathematics, the frontier is still open and abounds in important unsolved problems...
• Problems have long been regarded as the life of mathematics.  A good problem is one that defies existing methods...whose solution promises a real advance in our knowledge. 

I have long argued that since the Navier-Stokes Prize Problem is formulated without including the fundamental aspects of wellposedness and turbulence, it misses these values and thus is not a good Prize Problem. Here is my argument again:

Consider the incompressible Navier-Stokes equations with viscosity $\nu >0$ in the case of (very) large Reynolds number $Re =\frac{UL}{\nu}$ with $U$ global flow speed and $L$ global length scale. Assume $U=L=1$ and thus $\nu$ (very) small. Such flows are observed physically and computationally to be turbulent with substantial velocity fluctuations $u\sim \nu^\frac{1}{4}$ on a smallest spatial scale $\epsilon\sim\nu^\frac{3}{4}$ with corresponding substantial viscous dissipation $\sim 1$.  For the jumbojet in the above simulation $Re\approx 10^8$ and the smallest scale a fraction of a millimeter. The heuristic argument to this effect goes as follows:

A: Breakdown to smaller scales only takes place for sufficiently large local Reynolds number (of size 100 or more), which gives the following relation for the fluctuations $u$ on the smallest scale $\epsilon$:

• $\frac{u\epsilon}{\nu}\sim 1$.

B: Substantial dissipation on smallest scale $\epsilon$ means 

• $\nu (\frac{u}{\epsilon})^2\sim 1$.

Combination of A and B gives $u\sim \nu^\frac{1}{4}$ and $\epsilon\sim\nu^\frac{3}{4}$ as stated. This can be viewed to express Lipschitz-Hölder continuity with exponent $\frac{1}{3}$ and thus that turbulent solutions for (very) small $\nu$ are non-smooth, because they are $Lip^{\frac{1}{3}}$ on (very) small scales. 

The existence of such turbulent solutions can mathematically be proved by standard methods by regularization on scales much smaller than $\epsilon$, which does not change the solution but the NS equation. 

For smooth data such solutions to regularized NS could formally be proved to be smooth in the sense of the formulation of the NS Prize Problem by Fefferman, but this would be in conflict with the observation that solutions are non-smooth  ($Lip^{\frac{1}{3}}$) on (very) small scales $\sim\nu^\frac{3}{4}$. 

The only mathematically and physically reasonable way to resolve this conflict of definitions, would be to view turbulent solutions to be non-smooth ($Lip^{\frac{1}{3}}$ on very small scales), and thus as weak solutions, with weakly small but strongly large Euler residuals, and the aspect of wellposedness would then be of focal interest.

Computational sensitivity (stability) analysis shows that turbulent weak solutions, are weakly wellposed in the sense that solution mean-values are not highly sensitive to perturbations of data (while point-values are).

Stability analysis further shows that globally smooth solutions with derivatives of unit size for smooth data of unit size, are unstable and thus are not physical solutions. 

The net result is that the present formulation of the NS Prize Problem is meaningless from both mathematical and physical point of view. A meaningful formulation must include wellposedness and turbulence as key issues, with existence settled by standard techniques, and a meaningful resolution would have to offer mathematical evidence of weak wellposedness and features of turbulence.

I have asked Terence Tao, as a world leading mathematician working on the Prize Problem, about his views on the aspects I have brought up, and will report his response. I have earlier many times asked Fefferman the same thing but the only response I get is "To me my formulation is meaningful". 

What would then Mr Clay say if he understood that the NS Prize Problem is not meaningful     

outside a small group of mathematicians (which may contain just one person), when comparing to the mission to which he donated his Prize:

•  the value of mathematical knowledge and its centrality to human progress, culture, and intellectual life....
• ...to further the beauty, power and universality of mathematical thinking...deepest, most difficult problems... achievement in mathematics of historical dimension. 

PS It is remarkable (or deplorable) that my repeated request to start a discussion about the formulation of the Prize problem is met with complete silence from those in charge of the problem. If my view-points are silly, that could be said by those who know better. If they are not silly, maybe even relevant, then it would be silly (or deplorable) to not say anything.  In either case, silence is not reasonable and it is tiresome to keep silent under increasing pressure from the outside world to say something... 

Tao on Clay Navier-Stokes and Turbulence?

Terence Tao is working on the Clay Navier-Stokes Prize Problem and in a recent post considers  Kolmogorov's power law for turbulence. A heuristic derivation goes as follows: The smallest spatial scale $\epsilon$ of a fluctuation $u$ of turbulent incompressible flow of small viscosity $\nu >0$ is determined by a local Reynolds number condition

• $\frac{u\epsilon}{\nu}\sim 1$.

Assuming the smallest scale carries a substantial part of the total dissipation gives 

• $\nu(\frac{u}{\epsilon})^2\sim 1$.

Combination gives 

• $u\sim \nu^{\frac{1}{4}}$
• $\epsilon\sim\nu^{\frac{3}{4}}$ 

suggesting that the turbulent solution is Lipschitz continuous with exponent $\frac{1}{3}$. 

My question to Tao posed as a post comment is if according to the Clay problem formulation, such a $Lip^\frac{1}{3}$ turbulent solution with smallest scale $\nu^\frac{3}{4}$ is to be viewed as a smooth solution for any small $\nu >0$?

Answer to My Question about Formulation of Clay Navier-Stokes Prize Problem

Here is the response from the Clay Mathematics Institute on my message that the formulation of the Navier-Stokes Prize Problem does not include the fundamental aspect of wellposedness required for a mathematical model of a physics phenomenon to be meaningful:

Dear Dr Johnson,

Thank you for your interest in the Millennium Prize Problems. Complete details can be found athttp://www.claymath.org/millennium-problems.

As a matter of policy, the Clay Mathematics Institute does not join in discussion of the formulation of the Millennium Prize Problems, nor does it comment on potential solutions.  I am afraid that we have nothing to add to what is said on the CMI's website.

Best wishes,

Anne Pearsall (Mrs)
Administrative Assistant
Office of the President, Clay Mathematics Institute

Andrew Wiles Building

Radcliffe Observatory Quarter

Woodstock Road

Oxford OX2 6GG, UK

OK, so we learn that the Administrative Assistant of the President of the Clay Mathematics Institute, not the President himself,  "is afraid that we have nothing to add" and that the Institute "does not join in discussion of the formulation of the Millennium Prize Problems". 

Yes, this is indeed something to be afraid of, in particular if mr Clay himself understands that the formulation of the NS problem is unfortunate in the sense of lacking meaning to physics, and as a meaningless problem cannot have a meaningful solution. 

The fact that my question about the meaningfulnness of the NS Problem in its present formulation, is met by compact silence, may be interpreted as a silent acknowledgement that the formulation indeed is meaningless, and that it is purposely so in order to reserve the problem to meaningless mathematics and guarantee that, in Newton's words, "little smatterers" are kept out.

Wellposedness vs the Clay Navier-Stokes Problem?

In a sequence of posts I have argued that the omission of wellposedness in the Official Description of the Clay Navier-Stokes Prize Problem by Charles Fefferman, makes the problem meaningless. To support this I quote from Wellposedness and Physical Possibility by B. Gyenis:

Well posedness is widely held to be an essential feature of physical theories. Consider the following remarks of Mikhail M. Lavrentiev, Alan Rendall, and Robert M. Wald – leading experts in their respective fields of physics – intended as motivations for the continuous dependence condition:

• One should remember that the main goal of solving mathematical problems is to describe certain physical processes in mathematical terms. In this case the initial data are obtained experimentally; and since measurements cannot be absolutely precise, the data contain mea- surement errors. For a mathematical model to describe a real physical process, the problem should be supplemented with some additional requirements reflecting, in a physical sense, the fact that the solution should have only small variations under slight changes of initial data or, to put it conventionally, the stability of the solution under small perturbations in the data. (Lavrentiev et al.; 2003, p. 6) 
• The condition of continuity is sometimes called Cauchy stability. The reason for including it is as follows. If PDE are to be applied to model phenomena in the natural world it must be remembered that measurements are never exact but always associated with some error. As a consequence it is impossible to know initial data for a problem exactly and so if solutions depend on the initial data in an uncontrollable way the model cannot make useful predictions. Cauchy stability guarantees that this does not happen and thus represents a necessary condition for the application of PDE to the real world. (Rendall; 2008, p. 134) 
• If a theory can be formulated so that “appropriate initial data” may be specified (possibly subject to constraints) such that the subsequent dynamical evolution of the system is uniquely determined, we say that the theory possesses an initial value formulation. How- ever, even if such a formulation exists, there remain further properties that a physically viable theory should satisfy. First, in an appropriate sense, “small changes” in initial data should produce only correspondingly “small changes” in the solution over any fixed compact region of spacetime. If this property were not satisfied, the theory would lose essentially all predictive power, since initial conditions can be measured only to a finite accuracy. It is generally assumed that the pathological behavior which would result from the failure of this property does not occur in physics. [...]2 (Wald; 1984, p. 224) 

These remarks express a sentiment widely shared among physicists: wellposedness is a necessary condition for models to describe real physical processes. Lack of wellposedness would be pathological and it “does not occur in physics,” at least not in describing forward time propagation of physical processes.

OK, so leading experts of physics consider wellposedness to a necessary requirement for a mathematical model of some physical phenomena to be meaningful. The Navier-Stokes equations is the basic model of fluid mechanics, and as such requires some form of wellposedness to be meaningful.

The leading mathematical expert Charles Fefferman formulates the Clay Navier-Stokes problem without reference to wellposedness and thus apparently considers wellposedness to not be a central aspect. But doing so Fefferman separates the mathematics of Navier-Stokes equations from physics, which goes against the reason of formulating a Prize Problems about a mathematical model of fundamental importance in physics.  

When I ask The Clay Institute and Fefferman to give a comment concerning these facts, I get zero response. I think my viewpoints are reasonable and essential and thus worthy of some form of answer.

Wellposedness and Turbulence Not Part of Clay Navier-Stokes Problem!

A central aspect of the mathematical theory of partial differential equations, such as the incompressible Navier-Stokes equations, concerns wellposedness, which is the sensitivity of solutions with respect to perturbations of data in suitable quantitative form. Without wellposedness in some form solutions have no permanence and meaning, since they can change arbitrarily subject to virtually nothing.

But the Official Description of Clay Navier-Stokes Prize Problem does not include the aspect of wellposedness.

A central aspect of incompressible flow described by the Navier-Stokes equations, is turbulence. 

But the Official Description of the Clay Navier-Stokes Prize Problem does not include any aspect of turbulence.

The Official Description is thus questionable, to say the least, from both mathematical and physical point of view, by leaving out what is fundamental.

When I point this out to Charles Fefferman who has formulated the Official Description of the problem, and Luis Cafarelli who gives a video presentation thereof, and Peter Constantin who acts as referee to evaluate proposed solutions and Terence Tao who works to solve the problem and to the President of the Clay Institute, I get no reaction but silence.

This is not reasonable, since the Navier-Stokes equations and the mathematics thereof belongs to us all and thus must be open to public discussion, in particular so when it has been elevated to a Millennium Prize Problem of importance to humanity.

I sent to following renewed request to the people involved to reveal their cards:

Dear Colleagues:

I try to get a response from you concerning my questioning of the Official Description of Clay Navier-Stokes Prize Problem expressed here. 

I get no response but compact silence. I don't think this is in the interest of a Clay Prize Problem as of concern to a wide mathematical and scientific community and not secluded to a very small closed circle.

The omission of both wellposedness and turbulence in the Official Description lacks rationality from both mathematical and physical point of view, and irrationality is against the principles of mathematics and physics.

I hope you can see that my questioning requires a response from you in your respective roles.

Sincerely, Claes Johnson

PS I raised the same question a couple of  years ago, and the only response then on my question how the Prize problem could be meaningful without including the aspect of wellposednes , was Fefferman's short reply: "It is meaningful to me". I think this answer misses that fact that science is not only a private thing.

Crisis in Mathematics Education in France like in Sweden

Mathematics education is falling freely also in France as reported on images des Maths (in my translation):

• The many problems present in mathematics education today is of concern to everybody. Results are falling since 1990.
• Many people speak thereof but few do anything about it.
• The debate is troublesome in the community of mathematicians, and even more so for the general public.
• The phenomenon has several causes: 
• One is the training of mathematics teachers. It was better before.

This is the same analysis as in Sweden based on the following postulates:

1. The training of math teachers was good before and math eduction was then working.
2. Today math education does not work anymore and the reason can only be that the training of math teachers is not as good as before.
3. Hence what is needed is re-training of math teachers to the old standard.

Billions of tax-payer money is now spent in Sweden spent on re-training in collegial form, where teachers without "good" education "lift" each other into the old level of training. 

Of course, the result is small and much money and effort is lost. 

What is forgotten, both in France and Sweden, is that in our computer time, math education has a new role to play and the old role is outdated and cannot be resurrected. Very few people in the math community are willing to face this reality, and the result is that the fall of math education continues to new low levels each year, in France and Sweden alike. 

BodySoul Mathematical Simulation Technology Translated to Chinese

Today I recieved the following letter from Zhimin Zhang (with copy to Qun Lin as a leading Chinsese applied mathematician):

Dear Professor Johnson,

First, I would like to apologize for taking almost 4 years to get back to you about your book. The reason was that Professor Lin wanted to understand and "digest" your book more before talking with you.

To make a long story short, he likes your book very much and has organized a group of Ph.D. students to translate your book into Chinese. It is a book with more than 1600 pages and that is why it takes almost 4 years to complete. Now Professor Lin wants me to ask for your permission to publish the Chinese translation of your book. In addition, if you have updated your book, we would like to have the new version and update our translation.

We look forward to your favorable response, Zhimin.

I replied that I was glad to hear this and suggested to set up a formal agreement about the use of BodySoul Mathematical Simulation Technology in China. I will report what comes out of this.

I recall that the book is censored at KTH, and so apparently Sweden has stricter censorship than China.

The number of fresh engineering students each year is 1000 at KTH, while it is 10.000.000 in China.

Almost Dictatorial Consensus in Germany

An internal memo On the situation in the field of meteorology-climatology of the German Meteorological Society reveals a growing and widespread worry over the suppression of scientific views under almost dictatorial consensus:

• ….how certain developments are becoming cemented into their scientific fields (foremost climatology) which from a scientific point of view simply cannot be accepted and do not comply to their professional ethics.
• In meteorology-climatology every one includes a highly visible army of organized, little known persons; in Germany this is almost the entire public! 
• The changes that have taken place in science as a result have in our opinion (and that of others) led to very negative impacts on the quality standards of science. 
• For example expressed and disseminated meteorological flaws can hardly be contained and cannot be corrected publicly at all. Yet our meteorological scientists do not speak up.
• And it is hardly perceived that behind these developments – admittedly – there is also a political objective for the transformation of society, whether one wants it or not. Currently global sustainable change is the same thing.
• Meteorology-climatology is playing a decisive role this political action. The – alleged – CO2 consensus here is serving as a lever within the group that consists of known colleagues who deal with climate, but also consists of a large number of climate bureaucrats coming from every imaginable social field. Together both groups consensually have introduced a binding dogma into this science (which is something that is totally alien to the notion of science).
• This is not the first time such a thing has happened in the history of science. Here although this dogma came about through democratic paths (through consensus vote?), in the end it is almost dictatorial. 
• Doubting the dogma is de facto forbidden and is punished? In climatology the doubt is about datasets or results taken over from hardly verifiable model simulations from other parties. Until recently this kind of science was considered conquered – thanks to our much celebrated liberty/democratic foundation!
• The constant claim of consensus among so-called climatologists, who relentlessly claim man-made climate change has been established, attempts to impose by authority an end to the debate on fundamental questions. 
• Thus a large number of scientist colleagues end up being ostracized, and thus could lead to the prompting of actions that would have considerable burdens on the well-intended society. Such a regulation and the resulting incalculable consequence it would have for all people would in our view – and that of many meteorological specialists we know - be irresponsible with respect to our real level of knowledge in this field.
• We must desire in general, and also in our scientific field, a return to an international scientific practice that is free of pre-conceptions and cemented biased opinions. 
• This must include the freedom of presenting (naturally well-founded) scientific results, even when these do not correspond to the mainstream (e.g. the IPCC requirements).

The bullying of Lennart Bengtsson is a recent example of violation of scientific/democratic principles  in the name of "almost dictatorial consensus". Another is KTH-gate. Where is Western society heading?

Towards Computational Solution of Clay Navier-Stokes Problem 3

The formulation of the Clay Navier-Stokes Prize problem is unfortunate, or more precisely both mathematically and physically meaningless, because the following two completely fundamental aspects are not included:

1. wellposedness
2. turbulence.

To see the effect consider exterior flow with a slip boundary condition, which allows a unique stationary smooth near-solution as potential flow with a Navier-Stokes residual, which scales with the viscosity $\epsilon$. Smooth potential flow thus offers a solution to the NS equations with a

vanishingly small residual under vanishingly small viscosity. But potential flow is not stable since it  under small perturbation develops into a completely different turbulent solution. In other words, potential flow is not wellposed in any sense and thus not a physical solution.

The present problem formulation without 1 and 2 does not allow unphysical smooth potential flow to be distinguished from physical turbulent flow. The result is that the Clay NS problem has no meaningful solution and does not serve the purpose of a Prize problem.

Note that the Clay NS problem is introduced with the following description of the essence of the problem and its importance to humanity:

• Waves follow our boat as we meander across the lake, and turbulent air currents follow our flight in a modern jet. 
• 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.

But turbulence is not an issue in the official formulation. The secret to unlock is turbulence, but that is not part of the problem formulation. Something is weird here. I have pointed that out to the President of Clay Mathematics Institute and will report the reaction. Here is the letter:

President

Clay Mathematics Institute

I want to convey the information that the formulation of the Clay Navier-Stokes problem is incorrect both mathematically and physically, because the fundamental aspects of (i) wellposedness and (ii) turbulence, are not included, as exposed in detail in the following sequence of blog posts:

http://claesjohnson.blogspot.se/search/label/clay%20problem

The result is that the problem cannot be given a meaningful solution and thus does not serve well as a Prize problem. Evidence is given by the fact that no progress towards a solution has been made.

I have tried to engage Charles Fefferman, who has formulated the problem, Peter Constantin, who acts as a referee, and Terence Tao, who is working on the problem, into a discussion, but I get no response.

I hope this way to stimulate discussion, which I think would be more constructive than no discussion.

Sincerely, Claes Johnson 

Towards Computational Solution of Clay Navier-Stokes Problem 2

This is a continuation of a previous post: The basic energy estimate which is easily proved analytically by multiplying the momentum equation by the velocity $u_\epsilon$ and integrating, reads for $T>0$ with $Q =\Omega\times (0,T)$:

• $\int_\Omega\vert u_\epsilon (x,T)\vert^2\, dx +\int_{Q}\epsilon\vert\nabla u_\epsilon (x,t)\vert^2\, dxdt =\int_\Omega\vert u^0(x)\vert^2\, dx$

or in short notation with obvious meaning:

• $U(T) + D_\epsilon (U) = U(0)$,

which expresses a balance of kinetic energy $U(T)$ at time $T$ and dissipation $D_\epsilon (U)$ over the time interval $(0,T)$ summing up to initial kinetic energy $U(0)$. 

Computations with small $\epsilon$ (compared to data as $\Omega$ and $U(0)$) produce turbulent solutions characterized by 

•  $D_\epsilon (U) =\alpha U(0)$ where $\alpha$ is not small,

that is solutions with substantial (turbulent) dissipation. For turbulent solutions $\vert \nabla u\vert$ is large, typically scaling with $\epsilon^{-\frac{1}{2}}$, even if initial data is smooth, which can be viewed as an expression of non-smoothness.

The basic energy estimate can thus be used to signify non-smoothness by substantial turbulent dissipation. The Clay problem can thus be reduced to the question of proving that the dissipation term is  substantial in the basic energy estimate. 

Evidence to this effect is given by computation. Analytical evidence can be given by the following argument: Smooth laminar solutions have small dissipation but smooth laminar solutions are all unstable. If the dissipation remained small it would mean that an unstable solution would remain smooth and unstable, which is not possible under perturbation. 

The dissipation therefore must be substantial in the basic energy estimate and only a non-smooth solution can exist (and does exist by computation). An answer to the Clay problem may thus be possible along the following lines, assuming the viscosity is small and data are smooth:

1. Solutions exist for all time and do not cease to exist by blow-up.
2. Solutions become non-smooth (turbulent) in finite time. 
3. Solutions cannot stay smooth for all time, because any smooth solution is unstable. 
4. Solutions are weakly well-posed in the sense that solution mean-values are stable to perturbations, because of a cancellation effect in turbulent solutions which is not present for smooth solutions.  

The group of mathematicians in charge of the problem (Fefferman, Constantin and Tao) do not answer my repeated requests to open a discussion about the formulation of the problem and possible approaches to solution. This is not helpful to progress. Mathematicians apparently want to have a heaven of their own, where they can explain phenomena which have no scientific relevance, but this is a dangerous strategy in the long run, because without connection to science funding may cease.