A Variant of the Ill-Posed Clay Navier-Stokes Millennium Problem

I have long argued that the Clay Navier-Stokes Millennium Problem is ill-posed (here and here and previous post) and as such does not have a good answer. The consequence is that nobody will ever get the prize for solving the Navier-Stokes problem, which was not the intention of Clay.

To illustrate the ill-posed nature of the Navier-Stokes problem, let me consider the following related problem which could have been a Clay problem similar to the Navier-Stokes problem: Prove (A) or (B) with

(A) Existence and Uniqueness of Smooth Solutions to the Backward Heat Equation. The backward heat equation with smooth initial data admits a unique smooth solution over finite time.

(B = not A) Breakdown of Backward Heat Equation Solution:  The backward heat equation with smooth initial data does not admit a unique smooth solution over finite time.

To attempt a solution consider the backward heat equation in one space dimension on the interval $(0,\pi )$: Find a smooth function $u(x,t)$ satisfying

• $\frac{\partial u}{\partial t} + \frac{\partial^2 u}{\partial x^2}=$ for $0< x < \pi$ and $0 < t < 1$,
• $u(x,0) = u^0(x)$ for $0 < x < \pi$,

where the initial data $u^0(x)$ is given by

• $u^0(x)=\sum_{n=1}^{\infty}c_n\sin(nx)$,

with certain coefficients $c_n$ rapidly decaying to zero as $n$ tends to infinity. Formally the solution $u(x,t)$ is given by the solution formula

• $u(x,t)=\sum_{n=1}^{\infty}c_n\exp(tn^2)\sin(nx)$. 

where the series converges along with a certain number of derivatives if $c_n$ tends to zero with $n$ sufficiently fast, for example, $c_n\le \exp(n^2)n^{- N}$ for some natural number $N$. 

We could view the formula as a proof of (A), that is, existence of a unique smooth solution for sufficiently smooth initial data. On the other hand, we know (with Hadamard) that the backward heat equation is ill-posed in the sense that an infinitesimal perturbation of the initial data $u^0$ will make the solution blow up (or break down) and thus (B = not A) must hold. We thus have evidence of both (A) and (not A), which shows that the problem formulation is such that no answer can be given.

Unfortunately for mr Clay, the formulation of the Navier-Stokes problem suffers from the same defect.  In other words, the problem formulation is ill-posed and should be reformulated to make sense, to mr Clay and the mathematical world.  It is difficult to understand why the insight of Hadamard was completely neglected by Charles Fefferman in his problem formulation, very difficult.

PS Of course my article was rejected and Fefferman did not want to discuss the issue.

Computational Solution of the Clay Navier-Stokes Problem

The computations underlying the New Theory of Flight to soon be presented to the world in Journal of Mathematical Fluid Mechanics, suggest the following resolution of the Clay Millenium Problem on existence or non-existence of smooth solutions to the incompressible Navier-Stokes equations with smooth data:

• Consider exterior flow, governed by the incompressible Navier-Stokes equations with viscosity $\nu >0$, around a given solid body with smooth boundary with given smooth flow at infinity as given smooth data. Let $N$ be any given finite amount of computation as number of flops. Then there is a choice of small viscosity $\nu$ such that it is impossible to compute an approximate solution of the Navier-Stokes equations by time-stepping with a Navier-Stokes residual which is small (e.g smaller than 1) in the entire fluid domain within the limit of computation $N$. 

This suggests a resolution in the negative: For given smooth data there is no smooth solution to the Navier-Stokes equations if $\nu$ is chosen small enough. 

The argument is that a smooth solution should be computable by time-stepping an approximate solution with small residual within a given amount of computation $N$.  An observed impossiblity of time-stepping an approximate solution with small pointwise residual within the given limit $N$, then gives evidence of non-existence of a smooth solution.

Note that interpreting a smooth solution as possible to compute by time-stepping with small residual, introduces an aspect of stability into the notion of smoothness. This is necessary since a potential solution has a small Navier-Stokes residual for $\nu$ small, yet is not computable by time-stepping because of instability. In the formulation of the Clay problem this point is missed, which makes the problem ill-posed and without good answer. The result is that the Clay $1million Prize will never be given out, against the intention of Clay.

Another resolution based the theory of flight goes as follows: Existence of smooth solutions of the Navier-Stokes equations with small viscosity would make flight impossible. Since flight is observed to be possible, smooth solutions do not exist. 

Poster for The Secret of Flight Revealed in Heidelberg

Here is the poster for my talk in Heidelberg Dec 4:

Pisa: Sverige Kan Ta Täten med IT-Matematik som Nytt Skolämne

Sverige ligger slaget till marken av Pisa-undersökningen av 15 åriga skolelevers prestationer i matematik, Folkpartiet med Jan Björklund i katedern är knäckt och Alliansen går mot en säker valförlust:

• Skolverket: Läget är allvarligt.
• SvD: Sverige rasar fortare än något annat land i den stora Pisa-rapporten om kunskaper i matematik.
• DN: Resultatet av årets Pisa-undersökning var nedslående. Att svenska 15-åringars resultat rasar i jämförelse med andra länder är ett reslutat av en skola som misskötts länge, säger Lärarnas Riksförbund.
• Expressen: De svenska elevernas resultat har kraftigt försämrats....I årets huvudämne matematik har resultatet rasat...Utvecklingen i Sverige är den sämsta bland OECD-länderna.
• Lotta Gröning: Vi bevittnar en skola i fritt fall.
• Frida Boisen: En havererad ruin till skolväsende.
• K G Bergström: Detta kan avgöra valet.
• osv osv

Men det finns ett annat sätt att tolka svenska elevers dåliga prestationer i det traditionella matematikämne som testas i Pisa-undersökningen: Svenska elever kan i mindre grad än många andra länders elever motiveras att ägna sin energi åt något som upplevs som både krångligt och meningslöst.

Många svenska elever frågar sig varför man skall ägna år av möda att hantera bråkräkning eller använda formeln för lösning av en andragradsekvation, eller att ställa upp räta linjens ekvation i alla de tusen fall som skall betas av. 

När eleven frågar varför, så har läraren inget svar på detta, annat än att om man inte lyckas få godkänt i matematik så är det kört, och det är inte roligt. Läraren har nämligen i sin egen lärarutbildning inte fått någon annan information än att matematik är både roligt och nyttigt, så tycker alla elever egentligen, om det nu bara inte går snett på vägen nånstans, för då fortsätter det så och då är det kört. Tanken är att om man (läraren, föräldern, stödpersonalen, specialpedagogen, mormor och morfar, osv) bara kan fånga upp precis första gången det går snett och sedan fortsätta med denna styrning, så kommer alla elever att tycka att matematik är både roligt och nyttigt och då klara åtminstone godkänt. 

Men det finns ett annat sätt att närma sig denna problematik och då se Pisa-undersökningen i nytt ljus. Tänk om elevens upplevelse av matematik som krångligt och meningslöst, är en adekvat känsla eftersom den traditionella matematikundervisning som meddelas har tappat sin mening i dagens IT-samhälle, där matematiken har en annan och i själva verket mycket viktigare roll än på den gamla goda tiden, då huvuduppgiften var bråkräkning och lösning av andragradsekvationer, något som gav tillträde till goda samhällspositioner om det klarades väl. 

Tänk om det är så? I så fall ligger svenska elever före i sin verklighetsuppfattning, och skulle mycket väl kunna stimuleras av rolig, intressant och användbar IT-Matematik (se 61 bloggposter) som ersättning för det traditionella matematikämnet. Sverige skulle då kunna vända förlust till seger, på samma sätt som Sverige skulle kunna ha gjort i play-off mot Portugal om bara inte tron på den egna förmågan hade sviktat. 

Jag har försökt få både Alliansen och S att tänka i dessa banor, dock utan någon framgång hittills. Men kanske kan Pisa-undersökningen medverka till att skapa den krismedvetenhet som krävs för att våga öppna ögonen och tänka konstruktivt framåt, istället för uppgivet bakåt med resultat att ännu mer resurser satsas på ett i grunden meningslöst projekt.

Sverige kan aldrig konkurrera med Kina vad gäller kateder-matematik av igår, men Sverige kan ta täten vad gäller IT-Matematik idag och imorgon.

Jag skall göra ett nytt försök att nå Björklund eller Löven. Valet kan avgöras av skolfrågan och där finns IT-Matematik som en joker. Tänk om någon vågade tro på sin egen förmåga att tänka konstruktivt och använde den, lite grann åtminstone. Men om ämnet är matematik så drabbas nästan alla av black-out under traditionens tryck och tycker att dom korkade, fastän dom inte är det, och tankeförmågan upphör.Tänk...

New Theory of Flight Presented at HGS MathComp von Neumann Lecture in Heidelberg

Tomorrow I will present the new theory of flight revealing The Secret of Flight, developed together with Johan Hoffman and Johan Jansson at KTH, in the IWR-Colloquium / HGS MathComp von Neumann Lecture at the University of Heidelberg on the invitation by prof Rolf Rannacher, see poster.

At the same time I expect our article New Theory of Flight submitted to Journal of Mathematical Fluid Mechanics edited by Galdi and Rannacher, to be accepted for publication and thus be presented to the scientific world after having been rejected by Journal of AIAA.

I consider this to be the highlight of my career, having recently passed 70 and understanding that discovering a new theory resolving a main mystery of science, can happen only once in a life time.

How interesting it would have been if von Neumann could have been present, at least by Skype from somewhere out there... In any case it was with the help of the von Neumann computer that the Secret was revealed...

Here is a quote by von Neumann connecting to the presentation:

• The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work - that is correctly to describe phenomena from a reasonably wide area. Furthermore, it must satisfy certain esthetic criteria - that is, in relation to how much it describes, it must be rather simple.

Von Neumann could have added that the Euler/Navier-Stokes equations for slightly viscous flow is a prime  example of a mathematical model which is simple and describes virtually all of aerodynamics.   

Konrad Zuse on The World as Clock with Finite Precision

                                      Konrad Zuse pondering if physics is digital? If so, time has a direction.

I have recently discovered that the idea which I have pursued in different pieces of work, the idea to view physics as a Clock with Finite Precision, was expressed in 1969 by the German computer pioneer (and civil engineer) Konrad Zuse (who constructed the first working computer named Z3 in 1941) in the remarkable article Calculating Space on Digital Physics, starting out with the following experience which I share with Zuse:

• The work which follows stands somewhat outside the presently accepted method of approach, and it was for this reason rather difficult to find a publisher ready to undertake publication of such a work. 

Here are some highlights from Calculating Space, connecting in particular to my book The Clock and the Arrow: A Brief Theory of Time about the 2nd law of thermodynamics and the direction of time:

• It is obvious to us today that numerical calculations can be successfully employed in order to illuminate physical relationships. 
• To what extent are the realizations gained from study of calculable solutions useful when applied directly to the physical models? Is nature digital, analog or hybrid? And is there essentially any justification for asking such a question? 
• The examples of digitalization of fields and particles which have been preented are in their present unfinished form still far removed from being able to serve in the formulation of physical rules. Nevertheless, they give a rough impression of the possibilities for using the tools of the automaton theory to answer physical questions.
• The question to what extent it is possible to consider the entire universe as a finite automaton depends on the assumption which we make in relation to its dimensions. 
• An infinite information content is required for an unlimited spacetime element. It is practically impossible to simulate such a model with computers because of the necessity of infinite number of places required. 
• The sources of error are correspondingly great in the extremely large number of collisions between gas molecules, and these errors quickly lead to deviations from theoretical processes.
• This means that the better the causality rule is approximated in the reverse time direction, the more calculations we must be prepared to carry out in our model. This leads to the result that simulations of universal systems with causality functioning in both time directions belong to the category of “unsolvable” problems.
• Of course, it can be said that this is true only for calculating simulative models. But this result should encourage us to reconsider the matter. Are we justified in assuming a model of nature for which no calculable simulation is possible? 
• From this point of view, it appears that the frequently advanced argument of determination in both time directions should be fundamentally reexamined.
• But if Zuse didn’t hit upon the concept of universal computation (as Turing did), he was interested in another very deep question, the question of the nature of nature: “Is nature digital?” He tended toward an affirmative answer. (from Afterword)

Traditional TextBook Is Dead: Long Live New WebBook

• Textbook publisher predicts death of “dead tree” books within 3 years.
• Death of the Text Book - and the 50-pound Book Bag
• Facebook Co-founder Chris Hughes: The textbook is dead: 
• In the next five to seven years, the textbook is no longer going to be the basic building block of education.
• On a similar note, Nobel Laureate Harry Kroto said he now regards Wikipedia as more accurate than printed textbooks and far less costly for both students and educators. In preparing his GEOSET lectures, Kroto said, he never uses a single textbook.

The traditional university course, in mathematics and physics in particular, is based on a traditional printed textbook, which defines the course, its subject, questions and answers.

The traditional printed textbook is cut in stone and defines knowledge by exclusion: What is not in the textbook is not of concern. The world is defined by what is inside the book and that is what defines the exam questions. The student is supposed to go inside the closed world of the textbook and not look out and get distracted. The textbook answers specific questions posed in the book, and other questions should not be asked.

The previous post reported about an education leader at Chalmers University of Technology who complained that today students no longer read textbooks; instead they check out lecture notes based on the book if any,  ask a study mates, search the web, look at a web-lecture from MIT or Khan Academy. Since they do not use the text book, they do not need to buy it either and thus can save some money.

The traditional textbook will be replaced by something on the web, some form of WebBook and the question is what that may be?

One thing seems clear and that is that the WebBook will be open to the exterior, in contrast to the traditional text book forming its own closed world without window.

The WebBook will be formed by inclusion through links to the exterior. The traditional textbook is based on exclusion and does not have links to the exterior.

The new web student will take a new active role and independently search and collect material to bring into the course, instead of passively relying on the words of the book transmitted through the teacher.

The web teacher will have to give up the omnipotent role of understanding everything by understanding everything in the book defining the world, but not necessarily much more, and join the student in search and collection of stuff to bring into the course.

These new roles are now forming and it will be very interesting to follow this process.