Publicistklubben och klimatfrågan

Tege Tornvall föreslog Publicistklubbens ordförande Anna Hedenmo att ordna en debatt om klimathotets realitet och ev. bristande information om observerade klimatfakta. På detta svarade Hedenmo så här:

Hej Tege,

Jag har förstått att du vill ha ett mer utförligt svar på ditt mail om klimatfrågan. Det ska du få.

Eftersom en absolut majoritet av ledande klimatforskare i dag är eniga om att klimatförändringarna är en följd av mänsklig påverkan, anser jag att en PK-debatt på det temat varken är seriös eller meningsfull. Samma inställning finner du i de flesta större medier, sen ett antal år.

I andra frågor, som flyktingfrågan eller metoo, har jag som du kanske känner till tidigare efterlyst en mer tillåtande debatt, utan åsiktskorridor. Men när det gäller klimatfrågan ser jag ingen anledning att debattera det som i dag måste betraktas som ett faktum.

Vi i Publicistklubben får ständigt förslag på debattämnen. En del tar vi fasta på, andra väljer vi bort. Du är med andra ord inte den enda medlemmen som inte har fått gehör för sina förslag.

Jag beklagar om du inte upplever att du får något för medlemsavgiften. Men kanske kan du söka något av våra stipendier?

Hälsningar
Anna Hedenmo

Med anledning av detta beklämmande och nonchalanta bemötande av ett seriöst förslag har jag skickat följande brev till Hedenmo med begäran om svar.

Till Anna Hedenmo

ordf i Publicistklubben

Tege Tornvall meddelar att Du avfärdat hans förslag till debatt i PK om klimatfrågan, med motivering att: 

• "jag ser ingen anledning att debattera det som i dag måste betraktas som ett faktum".

Jag ställer nu följande frågor till Dig, som jag önskar får besvarade för publicering på min välbesökta blogg http://claesjohnson.blogspot.com :

1. Du välkomnar på Publicistklubbens hemsida med "vår uppgift är att slå vakt om det fria ordet, som en livsviktig del av demokratin". Om Du nu värnar om det fria ordet, hur kan Du censurera debatt om mänsklighetens ödesfråga nummer 1: klimatfrågan? 
2. Vad är det som "i dag måste betraktas som ett faktum"?
3. Vad är innebörden av “måste” i Ditt uttalande? Vem uttalar detta “måste"?

Hälsningar 

Claes Johnson

prof em i tillämpad matematik KTH

Hedenmos svar kommer att publiceras så snart det inkommit. 

Nedan är Hedenmos icke-svar på mina frågor i form av osorterade vaga påståenden, som Hedenmo samlat ihop utifrån sina kunskaper om klimatet och som hon tydligen "måste" komma dragande med, följt av mitt avslutande brev. 

Hej Claes och Tege,

Nedan ser ni fakta i frågan. 

Nu sätter jag punkt för denna mailkonversation.

Beträffande Publicistklubben, så är det helt frivilligt att vara medlem. 

Hälsningar

Anna Hedenmo 

Jordens klimat håller på att värmas upp och det är en konsekvens av människans utsläpp av koldioxid och andra växthusgaser. Koldioxiden i atmosfären absorberar den utgående värmestrålningen. Ökande halter innebär därför en ökad absorption och planeten värms upp. Det påverkar jordens vädersystem, leder till smältande glaciärer, havsnivåhöjningar (både till följd av termisk expansion och smältande glaciärer), samt en ökande avsmältning av havsisen i Arktis och shelf-isarna i Antarktis. Det ökande energiinnehållet i atmosfären leder till fler och mer extrema väderhändelser. Regionalt kan klimatet bli såväl blötare som torrare och perioder med extremvärme blir allt vanligare. På längre sikt kan avsmältningen av de stora isarna på Grönland och Antarktis kraftigt komma att påverka havsnivåerna. 

Den globala CO2-utsläppskurvan planade ut mellan 2014—2016 men har sedan 2017 börjat öka igen. En annan viktig indikator är koncentrationen av CO2 i atmosfären. Den mäts direkt i atmosfären sedan 1957 och med hjälp av borrkärnor i is (för att få längre mätserier). Före industrialiseringen låg den på 280 ppm. I dag ligger den på 405 ppm. Det kommer att ta lång tid för minskande utsläpp att ge ett utslag i sjunkande koncentrationer av CO2. Utsläpp av koldioxid har funnits i miljontals år. Vid till exempel en skogsbrand släpps stora mängder CO2 ut i atmosfären. När sedan växter och träd tar upp kolet igen skapas ett naturligt kretslopp. Men när man eldar kol/olja/gas frigörs kol som under miljontals år funnits i jordskorpan och som inte är en del av det naturliga kretsloppet. 

Sedan 1900 har jordens genomsnittliga temperatur ökat med 1,0 grad men med variation mellan enskilda år. I vissa regioner, som Arktis går uppvärmningen mycket snabbare. Sverige är delvis ett arktiskt land och i norra Sverige går klimatförändringarna just nu snabbare än det globala genomsnittet. Uppvärmningen går snabbare över land än över hav. Den globala uppvärmningen sker snabbare över norra halvklotet än södra halvklotet.

Som jämförelse kan nämnas att skillnaden mellan dagens globala medeltemperatur och den globala medeltemperaturen under de kallaste perioderna under den senaste istiden är 5-7 grader. Detta ger ett perspektiv på det faktum att en uppvärmning över 1,5 eller 2 grader kan få mycket stora konsekvenser. 

Den globala havsnivåhöjningen, havet har höjts med 20 cm senaste 100 åren. Haven absorberar värme vilket också gör att havet expanderar. Dessutom bidrar inlandsisars och glaciärers avsmältning. Haven blir också allt mer försurade i takt med ökande koncentrationer av koldioxid. Detta tillsammans med stigande havsvattentemperaturer bidrar till blekning av koraller och kan få andra mycket stora konsekvenser för ekosystemen i haven. 

Forskarna har inte uppmätt fler extrema väderhändelser, men de är intensivare och kraftigare än förr. 2018 var ett extremt väderår. Det går till historien som det fjärde varmaste sedan mätningarna inleddes. 20 av de 22 senaste åren har varit de varmaste sedan mätningarna inleddes. Varje fraktion av en grad leder till extremare väder händelser som mer intensiva orkaner, torka och konsekvenser som skogsbränder, översvämningar, döende korallrev, ökning av smittsamma sjukdomar. https://public.wmo.int/en/media/press-release/wmo-climate-statement-past-4-years-warmest-record

I oktober i år publicerade FN:s klimatpanel IPCC sin 1,5 c-rapport beställd av världens regeringar 2015. I den beskrivs konsekvenserna av 1,5 c graders uppvärmning av jordens medeltemperatur, som enligt IPCC med stor sannolikhet kommer inträffa 2030—2052. 

Forskarna uppmanar nu världens länder att tredubbla sina ansträngningar att fasa ut förbränningen av olja, naturgas och kol. Tidsfönstret krymper snabbt, och en minskning på 70 procent bör ske till 2030, enligt forskarnas rekommendationer.


Här är mitt svar till Anna:

Tack Anna för Ditt svar på mitt brev, som jag lagt upp min blogg.

Dock svarar Du inte på mina frågor utan kommer istället med en samling mer eller mindre lösa påståenden (s k fakta) som Du på något sätt samlat ihop efter de kunskaper i ämnet Du besitter. 

Vad man kan säga om detta är att varje påstående skulle kunna tas till utgångspunkt för en diskussion med syfte att klargöra dess signifikans och avgöra dess sanningshalt. 

Men Du censurerar varje sådan diskussion hos PK, och Du avslutar vår mailkonversation.

För att reda ut det hela föreslår jag att Du bjuder in mig eller Tege till Min Sanning (om klimatet)!

Vänliga hälsningar
Claes

PS Publicistklubben inbjuder till debatt den 11 mars över ämnet:

• Vad kan politiken göra för att värna journalistiken i tider av filterbubblor och misstro mot etablerade medier?

Frågan kan också formuleras: Hur kan censuren göras mer effektiv? 

Planck's Desperate Mad Ugly Ad Hoc Trick: The Quantum

Planck's reasoning was mad, but his madness has that divine quality that only the greatest transitional figures can bring to science. (Abraham Pais in The Science and Life of Albert Einstein)

...the whole procedure was an act of despair because a theoretical interpretation had to be found at any price, no matter how high that might be... (Planck on the statistical mechanics basis of his radiation law)

Sabine Hossenfelder on Backreaction gives praise to the new book Breakfast with Einstein by Chad Orzel:

• Physics is everywhere, that is the message of Chad Orzel’s new book “Breakfast with Einstein,” and he delivers his message masterfully.
• In contrast to many recent books about physics, Orzel stays away from speculation, and focuses instead on the many remarkable achievements that last century’s led to.

Planck was not happy with his desperate mad ugly ad hoc trick of the quantum

Chapter 2 of the book has the title The Heating Element: Planck's Desperate Trick with the objective of describing the birth of quantum mechanics attributed to Planck's (ugly ad hoc) trick of avoiding the apparent ultraviolet catastrophe of classical wave mechanics by introducing the concept of a smallest package of energy named quantum: 

• This “quantum hypothesis” does the necessary trick of cutting off the amount of light at high frequencies—exactly where the ultraviolet catastrophe happens. 
• Planck initially introduced the quantum hypothesis thinking it was a “desperate mathematical trick.” 
• Despite the many successes of his formula and the personal fame it brought him, Max Planck himself was never particularly satisfied with his quantum theory.
• He regarded the quantum hypothesis as an ugly ad hoc trick, and he hoped that someone would find a way to get from basic physical principles to his formula for the spectrum without resort- ing to that quantum business. 
• Once the idea was out there, though, other physicists picked it up and ran with it, most notably a certain patent clerk in Switzerland—leading to a complete and radical transformation of all of physics.

I have presented an alternative theory based on finite precision computation, which meets Planck's wish of explaining the black body spectrum from basic classical wave mechanics physics, which is presented on Computational BlackBody Radiation. Why not take a look, and see if you get enlightened? By a physical theory of blackbody radiation.

The idea of finite precision computation is the same as that used in a new explanation of the the 2nd law of thermodynamics discussed in the previous post on Boltzmann and his explanation based on (ugly ad hoc) statistics.

The master of ugly ad hoc tricks is Roger Stone as documented in his new book Stone's Rules. Such tricks can take you to the top of both science and politics! They can give you fame, but evidently not happiness. Another master of this game was the patent clerk in Switzerland, who also was unhappy with his theories, in particular the theory of the quantum he picked up from Planck, which gave him such immense fame:

• If I would be a young man again and had to decide how to make my living, I would not try to become a scientist or scholar or teacher. I would rather choose to be a plumber or a peddler in the hope to find that modest degree of independence still available under present circumstances.
• All these fifty years of conscious brooding have brought me no nearer to the answer to the question, “What are light quanta?”. Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken.
• For the most part I do the thing which my own nature drives me to do. It is embarrasing to earn so much respect and love for it. 
• Why is it that nobody understands me, and everybody likes me? (Einstein in New York Times, March 12, 1944) 

PS1 Often a truth about science, or rather a truth about a shortcoming of some scientific theory, is more honestly expressed in popular science, as the truth of the ugly ad hoc science of the quantum in Orzel's book (because the audience is supposed to be ignorant), than in some professional scientific context hiding the shortcoming in some cover-up (because the audience is supposed to be knowledgable and critical). Therefore it is interesting to read popular science also for a scientist.

PS2 Planck's desperate ugly ad hoc trick (which originates from Boltzmann) has caused a lot of confusion among physicists. For example, quantum mechanics, which is not understood by any serious honest physicist,  is supposed to have some mysterious connection to the quantum of energy of Planck, but the fact is that quantum mechanics is based on Schrödinger's equation, which is a continuum mechanical model and not a discrete model build from small packets of energy. The confusion is exhibited in Real Quantum Mechanics offering a new form of and new view on Schrödinger's equation with the common confusion eliminated.  But it is not easy to get a discussion going on the fundamentals of quantum mechanics, since the confusion is so monumental resulting from a  desperate mad ugly ad hoc trick, supposed to be the foundation of modern physics. No wonder that physics is in crisis. See also Dr Faustus of Modern Physics.

PS3 Recall that it was Einstein who introduced the idea that light is made of discrete chunks of energy $h\nu$ as photons with $h$ Planck's constant in Joulesecond and $\nu$ frequency, in his heuristic Law of the photoelectric effect $h\nu + W = eU$ with $W$ work to release an electron
and $eU$ in electron volt eV with $U$ the stopping potential in volt and e the charge of an electron. I argue in Mathematical Physics of BlackBody Radiation and related blogg posts that the Law is to viewed as a frequency threshold condition, which has no relation to any idea of light as consisting of discrete photons or light quanta, which according to the above quote was also the view of the late Einstein.

The Law shows that Planck's constant $h$ appears as a conversion between energy related to light frequency $\nu$ (in Joule) and electron energy (in eV), for which Einstein received the Nobel Prize in Physics in 1921 with explicit mention that he did not get the Prize for his theories of relativity). 

PS4 Schrödinger's equation connects energy related to light frequency and electron energy and it is thus no wonder that the Planck constant appearing in Schrödinger's equation is the same as that in the Law of the photoelectric effect. Mathematical Physics of BlackBody Radiation also gives evidence that the Law of the photoelectric effect is a consequence of Schrödinger's equation, within a continuum model without photon particles and reference to Einstein's heuristic argument that a photon of sufficient energy can kick out an electron. 

 

Boltzmann 175 vs 2nd Law by Finite Precision Computation

Ludwig Boltzmann 1844-1906

Lubos on the Reference Frame recalls the 175th birthday of Ludwig Boltzmann:

• Yesterday, Ludwig Eduard Boltzmann would have had a chance to celebrate his 175th birthday if he hadn't killed that chance by hanging himself at age of 62...
• Boltzmann's reason powering the suicide were intellectually driven frustrations.
• If he were resurrected and if he were around, he would probably ask me whether there's a reasonable chance that the people will get more reasonable when it comes to the ideas required for his new statistical picture of thermodynamics and physics in general. I would probably answer "No" and he would hang himself again. 

Lubos then enters into a defence of Boltzmann's 2nd law based on statistics and the related
Copenhagen interpretation of quantum mechanics with electrons randomly jumping around atom kernels, something which Einstein and Schrödinger never accepted. 

The problem Boltzmann tried to solve, with its tragic ending, is how formally reversible systems can show to have irreversible solutions. Boltzmann showed that you can hang yourself, but he could not un-hang himself, and he sought the explanation in statistics. Unsuccessfully according to Lubos, because still today people cannot understand what he was saying, about entropy and a 2nd law based on nonsensical statistics saying that something with a higher probability is more likely to happen.  I think this is not because people/scientists are stupid, which Lubos claims, but because what Boltzmann says makes sense only to Lubos.

I have presented a different explanation based on finite precision computation. This says that the reason that you cannot un-do things is lack of precision, and that all physics as well as digital computation is realised in finite precision. This means that you can enter a labyrint (the woods/world) with finite precision, like taking a step forward in time, but you cannot find your way out of the labyrint or retrace your path through the woods, go back in in time, because your are limited by finite precision. The arrow of time is an expression of finite precision computational physics. This is meaningful physics and different from Boltzmann's empty idea that the world moves form less probable to more probable states or from more ordered to less ordered states (with a start/bigbang as most ordered state inexplicable). 

I thus offer an explanation of the 2nd law of thermodynamics presented in many earlier blog posts 

and the book Computational Thermodynamics explaining that finite precision solutions of formally reversible system like the Euler equations of fluid mechanics can show to be irreversible, e.g. by the emergence of turbulence. This directly connects to a resolution of the Clay Navier-Stokes Problem reported in previous posts. 

The catch is that formally reversible systems can have irreversible solutions if precision is finite, and of course precision cannot be infinite, not in digital computation and neither in the physical world.

Hamming and Tartar on Clay Navier Stokes Problem

Richard Hamming (1915-98)

The mathematician Richard Hamming said:

• Mathematics is an interesting intellectual sport but it should not be allowed to stand in the way of obtaining sensible information about physical processes.

An example is given in the official formal formulation of the Clay Navier-Stokes Problem by Fefferman, which does not mention the world turbulence, which in the informal presentation is central:

• 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.

Everybody, except Fefferman, understands that turbulence is the secret hidden in Navier-Stokes and that the Clay problem, to be more than an intellectual sport standing in the way for sensible information, should ask about a mathematical theory unlocking the secret of turbulence. 

The informal presentation is sensible, while the formal presentation is nothing but an intellectual sport, which neither has practitioners since no progress towards a resolution has been made since 2000, or rather since 1932 when Leray proved existence of weak solutions with uniqueness or wellposedness left completely open.  

We have presented a resolution to a reformulated Clay Problem offering sensible information about the physical process of turbulence, by computation. We hope there are some sensible people that can show a reaction to our resolution. We show by computation that weak solutions exist, are non-smooth/turbulent and have wellposed mean-values such as drag and lift. 

Hamming also said:

• The purpose of computing is insight, not numbers. … 
• [But] sometimes … the purpose of computing numbers is not yet in sight.

Yes, we find that being able to compute (turbulent) solutions to Navier-Stokes equations opens to gain insight into the nature and manifestation of turbulence.  DFS is in sight and gives insight! 

So, what insight has DFS brought? Here is one major revelation:

• bluff body flow = potential flow + turbulent 3d rotational slip separation.

Bluff body flow is thus computable by DFS, which offers a revolutionary new capacity to CFD with a vast field of applications for all sorts of vehicles or life moving through air and water, and the fluid mechanics is understandable!

Also note what the mathematician Luc Tartar says in the presentation of his book on Navier-Stokes:

• To an uninformed observer, it may seem that there is more interest in the Navier-Stokes equation nowadays, but many who claim to be interested show such a lack of knowledge about continuum mechanics that one may wonder about such a superficial attraction. 
• Could one of the Clay Millennium Prizes be the reason behind this renewed  interest?
• Reading the text of the conjectures to be solved for winning that particular prize leaves the impression that the subject was not chosen by people interested in continuum mechanics, as the selected questions have almost no physical content.
• The problems seem to have been chosen in the hope that they will be solved by specialists of harmonic analysis...
• I  hope that this particular set of lecture notes...may help the readers understand a little more about the physical content of the equation, and also its limitations, which many do not seem to be aware of.

And as before: the pure mathematicians Fefferman, Constantin and Tao in charge of the problem formulation refuse to participate in any form of discussion.  Why? Lack of knowledge about continuum mechanics, with focus instead on harmonic analysis?

And remember:

• What is computable is understandable. (Pythagoras)

Luc Tartar

   

From Equation to Solution

This is a continuation of the previous post on the role of functional analysis, more precisely the role of the finite element method as a form of computational functional analysis.

We start with the basic partial differential equation of physics and mechanics, Poisson's equation:

• $-\Delta u(x) = f(x)$ for $x\in\Omega$,
• $u(x)=0$ for  $x\in\Gamma$, 

where $\Omega$ is a domain in space with boundary $\Gamma$, $f(x)$ is a given function defined on $\Omega$ and $u(x)$ is the solution to the equation defined on $\Omega$ and $\Gamma$. The game is: Given $f(x)$ find $u(x)$ satisfying Poisson's equation.

We can think of the differential equation $-\Delta u(x)=f(x)$ as expressing force balance at the point $x$ with $u(x)$ the deflection of an elastic membrane under a transversal force or load $f(x)$, in case $\Omega$ is two-dimensional.  There are endless other interpretations.

So far so good, the partial differential equation $-\Delta u=f$ captures complex physics in very compact beautiful mathematical form, and so is marvellous, but there is one caveat: The formulation of the equation gives no clue to how to determine the solution $u(x)$. The equation is like a rebus without any hint of resolution.

It is here that functional analysis enters by offering a reformulation of the differential equation $-\Delta u =f$ into variational form: Find $u\in V$ such that

• $\int_\Omega \nabla u\cdot\nabla v\, dx = \int_\Omega fv\, dx$ for all $v\in V$,       (1) 

where $V$ is a collection (function space) of possible solutions, from which a best possible solution $u(x)$ is determined by the relation (1). Formally (1) is obtained by multiplying the differential equation $-\Delta u=f$ on both sides with an arbitrary function $v\in V$ and integrating over $\Omega$ using integration by parts to see that (using that $v=0$ on $\Omega$)  

• $-\int\Delta uv\, dx =\int_\Omega\nabla u\cdot\nabla v\, dx$. 

In the finite element method the space $V$ consists of piecewise polynomial functions over a triangulation of $\Omega$ and (1) is a linear system of algebraic equations, which can be solved by Jacobi iteration or Gaussian elimination. 

The differential equation as unsolvable rebus has thus been reformulated into variational form which allows a best possible solution to be computed by standard linear algebra software.  Here functional analysis enters in the variational formulation and the construction of the finite element space $V$.

The great thing is now that the same method works for virtually any (partial) differential equation, in particular the differential equations of science and technology: Reformulating the differential equation into variational form allows computation of best possible (approximate) solution. 

This is realised in the FEniCS Project which is software automating the whole process consisting of 

• reformulation into variational form, 
• construction of finite element space $V$,
• computation of solution by linear algebra. 

The crown jewel is automated computation of best possible solution of Navier-Stokes equations which we claim resolves the Clay Navier-Stokes Problem and makes turbulent flow computable and thus understandable, for the first time. And this is only the beginning of a FEniCS revolution.

We understand that the differential equation $-\Delta u(x)=f(x)$ expresses local force balance (at the point x), while the solution $u(x)$ comes out as a global effect depending on $f(y)$ for all $y$ and not just $f(x)$. This means that to determine $u(x)$ requires computation collecting many local inputs to one global output.

The mathematics of Jacobi iteration then corresponds to the physics of relaxation where the system reacts to reduce force imbalance. Gaussian elimination (or even better multi-grid) is more efficient than Jacobi iteration, which allows mathematics to take a short-cut to solution compared to physical relaxation.

PS The Navier-Stokes-Euler equations for incompressible flow contains the equation

• $\nabla\cdot u=0$ 

expressing the incompressibility, together with an equation expressing force balance according to Newton's 2nd law. The equation $\nabla\cdot u=0$ does not express force balance and appears more like a regulation stipulating a certain property of the solution (incompressibility) than a true law of physics like Newton's 2nd law. In DFS (near) incompressibility is instead expressed as a pressure law of basic form

• $\Delta p=\frac{\nabla\cdot u}{\delta}$  

where $\delta > 0$ is a small parameter, with the effect of forcing $\nabla\cdot u$ to be small by pressure as an expression of some physics. The lesson is that a differential equation without solution procedure is only half of the story.  Stating laws without means of enforcing the laws may be empty.

Banach and DFS and Clay Navier-Stokes Problem

This is an exercise in preparation for participation in a film about the Polish mathematician Stefan Banach who advanced functional analysis as mathematics describing relations between functions or analogies between analogies. My punch line is that the finite element method, as the subject of my work, is (nothing but) computational functional analysis following the spirit of Banach.

The crown of my work, together with Johan Hoffman and Johan Jansson, is Direct Finite Element Simulation DFS as solution of the Navier-Stokes-Euler equations without turbulence model or complicated wall model from a principle of best possible solution, in a situation where there is no exact solution. DFS brings revolutionary new capacity to Computational Fluid Dynamics CFD, which we (as a show case) claim resolves the Clay Navier-Stokes Problem by computation.

Functional analysis was formed by the mathematician Hilbert at the switch to modernity around 1900, with contributions from the Swedish mathematician Fredholm, and was further developed by Banach starting in 1920.  A prime objective was to justify mathematical models in the form of partial differential equations of solid and fluid mechanics and electromagnetics formulated during the 19th century by Laplace, Fourier, Navier, Stokes and Maxwell, by answering basic questions concerning existence and uniqueness of solutions, as well a construction of solutions by computation.

The basic element of functional analysis is a collection of functions named Hilbert space or Banach space equipped with a structure or geometry generalising that of ordinary three dimensional space. The solution of a given partial differential equation is then an element of a suitably chosen Hilbert or Banach space in basic cases determined by a principle of energy minimisation. The differential equation, which is impossible to solve directly by symbolic computation with pen and paper,  is thus reformulated into a minimisation problem over a function space, which allows construction of solutions as a limits of functions with decreasing energy computed according to the Banach Contraction Mapping Theorem.

Starting in the 1950s this form of computational functional analysis has been developed under the name of the finite element method into a universal method for computing solutions of the differential equations of science and engineering bringing revolutionary new capacities.  This success story was darkened only by Navier-Stokes-Euler equations of fluid mechanics, which were believed to demand computational power beyond anything which could be envisioned, the reason being the phenomena of turbulence and thin boundary layers involving small scales too costly to resolve computationally, the impossibilities presented in NASA CFD Vision 2030.

We show that with DFS the NASA CFD Vision 2030 is realised already today. By computational functional analysis in the spirit of Banach.

DFS and functional analysis gives a new perspective on differential equations representing ideal physics, however with uncomputable or non-existing exact solutions as in the case of Navier-Stokes-Euler,  and reformulations in terms of functional analysis with computable approximate solutions representing real physics.

Kolmogorov/Onsager: Turbulent Velocity 1/3 Hölder Continuous

Let me here recall the derivation by a scaling argument of the law of Kolmogorov/Onsager stating that fully developed turbulent velocities are Hölder continuous with exponent 1/3.

If $dx$ is smallest scale in space and $du$ the corresponding variation of velocity u, then we have with $\nu >0$ the (small) viscosity:

• $\nu (du/dx)^2 \sim 1$ (finite rate of turbulent dissipation)
• $\frac{du\times dx}{\nu}\sim 1 $ (Reynolds number on smallest scale $\sim 1$).

We solve to get $dx\sim \nu^{\frac{3}{4}}$ and $du\sim \nu^{\frac{1}{4}}$ and so $du\sim dx^{\frac{1}{3}}$ showing Hölder continuity 1/3.

The idea is that the flow will by instability develop smaller and smaller structures until the local Reynolds number becomes so small ($\approx 1000$) that this cascade stops on a smallest scale generating the bulk of the turbulent dissipation.

We see that velocity gradients $\frac{du}{dx}\sim \nu^{-\frac{1}{2}}$ are large, since $\nu$ is small, and so velocities are non-smooth.

The official formulation of the Clay Navier-Stokes Prize Problem by Fefferman asks about existence of smooth solutions. By the above argument this question cannot have a positive answer and so the question does not serve well as a Prize Problem.

A pure mathematician may counter this argument by claiming that a velocity with very large gradients still can be smooth, just with very large derivatives. And so even a turbulent solution of the Navier-Stokes equations can be viewed to be smooth, just with very large derivatives, and so asking for existence of smooth solutions in fact can be meaningful and so the Prize Problem in fact is meaningful. I think this means twisting the logic and terminology, which is not in the spirit of meaningful mathematics, pure and applied.

Is Digital Computation a Form of Mathematics?

In the last two posts a resolution of the Clay Navier-Stokes Prize Problem is presented, a resolution based on digital computation. I have tried to get some comment on our proposed resolution from the group of pure mathematicians in charge of the problem including in particular its official formulation: Charles Fefferman, Terence Tao and Peter Constantin, to whom I refer as the Problem Committee.

Sorry to say, I can only report silence from the Problem Committee: no comment whatsoever!

How can we understand this state of affairs? Is it so that our resolution lacks scientific substance? No, it represents a true break-through unlocking the main difficulties of mathematical modeling and simulation of fluid flow and it is world-leading. No doubt about that!

The reason behind the silence is thus not lack of scientific interest, but probably rather the opposite: Our resolution being based on digital computation brings in a new kind of mathematics, which is different from that envisioned in the official formulation expressed in the frame of classical analytical theory of partial differential equations. It appears that the Problem Committee does not know how to react to this new kind of mathematics in the form of digital computation, and so silence is the only possible reaction, so far at least.

This connects to a wider question of the role of mathematics in physics including fluid mechanics with particular focus on the new role of digital computation.

Now, mathematics can be seen as different forms of computation with classical pde-theory expressed as symbolic computation by pen and paper, and the new kind expressed by a computer executing the symbolic computation represented in the computer code.

So I again ask about the view of the Problem Committee on the possibility of resolving the Clay Problem by digital computation. Is it thinkable?  Or can only a resolution in the form of symbolic computation with pen and paper be accepted?  Is digital computation a form of mathematics?

Tao does not give any hope that solution by symbolic computation with pen and paper is possible!

Apparently Fefferman would be willing to give the Prize to Tao for a proof of blow-up towards infinite velocities, but so far Tao has not succeeded. But even if one day he would succeed, that would only mean that the mathematical model is no good as a model of real fluid flow, since no observation of infinite velocities has been made, and why give a Prize for a discovery that a model is no good? More  meaningful maybe to give the Prize for a result about a mathematical model of physical significance, like the one we give?

PS1 A pure mathematician might say that digital computation cannot deliver an answer for all (smooth) data and so would lack the generality of an answer by symbolic computation valid for any (smooth) data. To meet this criticism we can add that our resolution exhibits a different form of universality: We show that lift and drag of a body only depends on the shape of the body for high Reynolds number flow beyond the drag crisis at Reynolds number around $5\times 10^5$, that is for a very wide range of flows. Lift and drag depending only on shape is a form of universality. And we can compute lift and drag of any given body, case by case, but of course we cannot get a result for all bodies in one computation.

PS2 The official problem formulation by Fefferman takes as a fact that a smooth unique solution can cease to exist only if velocities become unbounded (referred to as blow-up at some specific finite time). But this is probably a misconception, since smooth solutions may turn into non-smooth solutions because velocity gradients become unbounded, which is what happens as a shock forms in compressible flow and turbulence develops in incompressible flow, while velocities stay bounded.

The official problem formulation is thus filled with misconceptions, and requires reformulation to become meaningful as a Mathematics Prize Problem.

Wellposedness of Navier-Stokes/Euler: Clay Problem

This is a continuation of the previous post proposing a resolution of the Clay Navier-Stokes Millennium Problem with further remarks on the aspect of wellposedness identified by
Hadamard in 1902 as being necessary in order for a mathematical model to have physical meaning and relevance. The Navier-Stokes equations serve as the basic mathematical model of fluid mechanics and the Clay Problem can be viewed to reduce to the question of wellposedness, since the existence of (weak) solutions was established by Leray in 1932.

And this is the question we give an answer: We show that weak solutions are computable (exist) and are non-smooth/turbulent with wellposed mean value outputs. We do this by solving a (dual) linearized problem with certain data and show a bound of the dual solution (here for lift of a jumbojet) in terms of the data, which we refer to as assessment of stability, and which translates to an error bound on output of a computed solution in terms of its Navier-Stokes residual, showing that the output is well determined under the presence of small disturbance.

The dual linearized problem has a reaction term with coefficient $\nabla u$ with $u$ a computed velocity. The reaction term drives both exponential growth and decay with its trace being zero by incompressibility. The wellposedness of  computed turbulent solutions is reflected by cancellation effects from the reaction term with exponential growth balanced by exponential deacy from  oscillations of turbulent solutions.

We thus argue that we have resolved the Clay Problem by showing that weak solutions are computable/exist and show to be non-smooth/turbulent with wellposed mean-value outputs. In particular we show that lift and drag are wellposed and thus reveal the secret of flight.

It remains to be seen if our resolution will be accepted by the group of pure mathematicians owning the problem including Charles Fefferman responsible for the official problem formulation, Peter Constantin and Terence Tao. One thing is notable: Fefferman’s formulation does not involve the aspect of wellposedness and so missses the heart of the problem, if Navier-Stokes is viewed as a mathematical model of fluid mechanics, which is clearly emphasized in the official problem presentation:

• 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.

All of this is presented in detail in this book supplied as evidence to the Clay problem committee with complementing material listed in the previous post. In particular the book contains a study of the (dual) linearized Navier-Stokes/Euler equations, a topic which for some reason has not attracted the attention of mathematicians despite its fundamental importance from mathematical point of view. In short, we feel that we have made substantial progress toward a mathematical theory which unlocks the secrets vidden in the Navier-Stokes equations, including the Secret of Flight.

Concerning the view of the problem committee recall the opening statement in the opening article Euler Equations, Navier-Stokes Equations and Turbulence by Peter Constantin
(in this book):

• In 2004 the mathematical world will mark 120 years since the advent of turbulence theory. In his 1884 paper Reynolds introduced the decomposition of turbulent flow into mean and fluctuation and derived the equations that describe the interaction between them. The Reynolds equations are still a riddle. They are based on the Navier-Stokes equations, which are a still a mystery. The Navier-Stokes equations are a viscous regularization of the Euler equations, which are still an enigma. Turbulence is a riddle wrapped in a mystery inside an enigma.

In other words, total confusion in the committee in charge of problem formulation and evaluation of proposed resolutions. In particular, Fefferman formulates the problem as the questions of existence and smoothness, forgetting wellposedness, and claims that his problem was solved by standard pde-theory long ago in the case of two space dimensions and evidently has in mind a similar resolution in three dimensions by som ingenious new estimate derived by a clever pure mathematician. But wellposedness is essential also in two space dimensions and so Fefferman exposes the gulf between pure mathematics and mathematics of fluid mechanics, which is not helpful to science.

Fefferman would probably say that wellposedness is a consequence of smoothness, but this is not necessarily so since assessment of smoothness may involve stability factors of arbitrary size and so may say nothing about wellposedness.  But of course questions like this have to remain in the mist since the problem committee is not open to any form of discussion.

Solution of the Clay Navier-Stokes Problem by Computer-Assisted Proof

This is a reminder of the resolution of the Clay Navier-Stokes Millennium Problem which I have presented together with Johan Hoffman and Johan Jansson in different forms over the years:

• Is the Navier-Stokes Problem Well Posed?
• Clay Navier-Stokes Problem Solved
• The Truth about the Clay Navier-Stokes Problem
• Computational Turbulent Incompressible Flow
• Adaptive Modeling of Turbulent Flow with Residual Based Turbulent Kinetic Energy Dissipation
• Towards a Parameter-Free Method for High Reynolds Number Flow...
• Time-resolved Adaptive Direct FEM Simulation of High-lift Aircraft Configurations
• Icarus Simulation
• New Theory of Flight (article in JMFM)
• Old posts on the Clay Problem.

Hopefully our suggested resolution will now be considered by the Clay Institute.

The Clay problem concerns existence of smooth solutions to Navier-Stokes equations as formulated by Charles Fefferman. No progress towards a solution using techniques of analytical mathematics has been reported in the literature since the problem formulation in 2000.  

Terence Tao has attempted to construct by analytical methods a solution which shows blow-up towards infinite velocities in finite time and thus would give a negative answer to the question of 

existence of smooth solutions for all smooth data. But Tao has not yet (fully) succeeded. 

We suggest to seek an answer instead by a computational method in the form of Direct Finite Element Simulation DFS on a sequence of finite element meshes with mesh size tending to zero. 

DFS is a Galerkin method stabilised by weighted least squares control of the Navier-Stokes residual R(U, P) with U velocity and P pressure. DFS introduces turbulent dissipation as an effect of residual least squares stabilisation and can be seen as a solver of the Euler equations (Navier-Stokes with vanishing viscosity) with an automatic turbulence model.

DFS produces on a given mesh a piecewise linear (U, P) with residual R(U,P) which is small in a weak sense (H-1) by construction (Galerkin orthogonality). The key point is then that the residual R(U, P) shows to be large in a strong sense (L2) as an expression of non-smoothness 

of turbulent solutions.

DFS produces/constructs/computes solutions to Euler/Navier-Stokes which show to be non-smooth/turbulent even if initiated as smooth potential solutions.  

DFS thus puts on the table for inspection a sequence of approximate solutions of Navier-Stokes equations with residuals tending to zero weakly in H-1, while showing blow up in L2 as an expression of non-smoothness of turbulent flow (finite rate of turbulent dissipation). DFS offers simulation/prediction of a very large range of important engineering applications in aero and hydro-mechanics of slightly viscous flow.

DFS shows Navier-Stokes equations to have non-smooth turbulent solutions and thus gives an answer to the Clay Problem. 

Stability analysis in computational form through an associated dual solution gives the further information that mean-value quantities such as lift and drag are computable by DFS with error tending to zero as the square root of the mesh-size.  But point-wise quantities are not computable to arbitrary precision. 

Finally, stability analysis shows that any smooth solution is unstable (as conjectured by Birkhoff) and thus cannot persist over time. Potential solutions are examples of smooth solutions, which thus do not persist over time but turn into non-smooth turbulent solutions. 

Our suggested resolution of the Clay Problem is based on computing approximate solutions to Navier-Stokes/Euler equations, which show to be non-smooth turbulent. 

We thus compute solutions of Navier-Stokes equations and put them on the table for anyone to check that they are non-smooth turbulent. 

Will this convince a jury of mathematicians used to analytical methods? Is it thinkable that for example Tao would give our argument a moment of scrutiny? We argue that we contribute the following basic elements to the scientific discussion of Navier-Stokes/Euler equations in the case of slightly viscous flow:

1. DFS computes solutions of Navier-Stokes/Euler without user-specied turbulence model. DFS thus solves the basic open problem of designing a mathematical model of turbulence.
2. Inspection of computed solutions shows them to be non-smooth/turbulent. It is concluded that solutions of Navier-Stokes/Euler for slightly viscous flow are non-smooth turbulent, which gives the Clay problem an answer.
3. Slightly viscous flow is identified as flow with Reynolds number larger than say $10^6$ associated with a reduction of drag resulting from delayed separation due to an effective slip boundary condition.  

Remark 1 Tao discusses Onsager's conjecture that Navier-Stokes solutions are (less smooth than) Hölder 1/3  resulting in finite rate of turbulent dissipation. DFS solutions typically show to be Hölder 1/3 with gradients $\nabla U$ blowing up like $h^{-0.5}$ with variations $h^{0.25}$ on scales of size $h^{0.75}$ consistent with weigthed least squares stabilisation $\int h\vert\nabla U\vert^2dx\sim 1$.

Here is one DFS Navier-Stokes solution put on the table for inspection showing turbulent flow with finite rate of turbulent dissipation around a jumbojet (with finite drag and lift):

Remark 2 Sabine Hossenfelder reminds us in Quanta Magazine about

• The End of Theoretical Physics As We Know It:
• Computer simulations and custom-built quantum analogues are changing what it means to search for the laws of nature.

Yes, computational techniques are changing the way physics is done and so also the mathematical physics of fluid mechanics and the related mathematics. The formulation, meaning and practical utility of a mathematical model for som physical phenomena, typically in the form a (differential) equation like Navier-Stokes equations,  closely connects to techniques for computing solutions and thus it is natural to expect that questions concerning the nature of solutions can be answered by computation with thus the computer offering a powerful new tool for mathematical modeling and analysis. The Clay Navier-Stokes problem can be seen as the outstanding open problem of classical continuum physics. This is the problem of predicting turbulent flow, which can now be viewed to be solved by computation.

Remark 3 The Clay problem is officially presented in the following words:

• 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. 

We see the connection with turbulence and we see that we can indeed solve the Navier-Stokes equations and thus predict turbulent flow representing world-unique breakthrough of making NASA Vision 2030 Grand Challenge into a reality already today. We are also proud to see that our New Theory of Flight indeed unlocks secrets hidden in the equations. We thus believe that we have something substantial to contribute which is worthy of consideration. But it is a new kind of science with new standards and so reviewers must be open-minded.

Remark 4 Recall that we argue that the problem formulation by Fefferman is unfortunate by not including the aspect of well-posedness, which is very well understood by mathematicians since Hadamard to be a necessary quality for physical relevance. One can thus argue that the Navier-Stokes problem essentially concerns the question of well-posedness and that our resolution is to give this question a positive answer: Computed solutions show to be non-smooth/turbulent and as such show to be well-posed physical solutions with stable mean-value outputs (such as lift and drag persisting over time making flight possible). We also argue that a solution initiated as smooth does not have stable (mean-value) outputs persisting over time and thus is not well-posed.

We thus give a positive answer to the Clay problem formulated as a question of well-posedness.
In short: Computed solutions show to be non-smooth/turbulent and well-posed with stable outputs. A solution initialised as smooth (for example as potential flow) is unstable and develops over time into a non-smooth/turbulent solution.

It is the oscillating nature of turbulent solutions which make them well-posed as expressed by the dual solution which over an oscillating velocity field shows little growth. On the other hand, a smooth solution is not oscillating and thus can give the dual solution consistent growth into non-wellposedness.

Remark 5 With Clay problems in mind, we are led to a counterexample to the  P = NP problem in the form of Turbulent Irreversible Solutions of the Euler Equations. Take a look and see if you buy the argument!

Remark 6 DFS can be seen as an incompressible Euler solver with automatic turbulence model which produces a non-smooth turbulent solution with finite rate of turbulent dissipation. DFS applied to the compressible Euler equations typically produce solutions with dissipative (energy-consuming) shocks.  DFS for Euler thus introduces dissipation from residual stabilisation as automatic turbulence/shock modeling.

Remark 7 Recall that the discussion involves the following elements:

1. Physics (of fluid particles): (i) Newton's 2nd Law + (ii) Incompressibility. 
2. Mathematics: Formulation of (i) + (ii) in terms of Calculus = Navier-Stokes/Euler.
3. Computation: DFS as solver of Navier-Stokes/Euler.

Physical fluid particles move so as to satisfy (i) and (ii), while DFS computes motion of virtual fluid particles from a mathematical principle best possible satisfaction of (i) and (ii) in the form of Navier-Stokes/Euler. 

DFS can be given an interpretation in physical terms through Navier-Stokes/Euler residuals and DFS can thus be viewed to be the physical model to be analysed, rather than the Navier-Stokes/Euler equations in conventional Calculus form which occurs in the problem formulation by Fefferman. 

Concerning the mathematical formulation of an equation/model describing som physics the technique  for solving the equation usually connects to the formulation of the equation, and so solution and formulation are intertwined and often cannot be separated. 

This is the case for Navier-Stokes/Euler where the classical approach of first seeking to formulate a turbulence model and then solve the corresponding equations, has failed. With DFS we instead directly aim at solving Navier-Stokes/Euler in original formulation without turbulence model and where the computational technique of residual stabilisation automatically produces the turbulence model.

DFS expresses best possible satisfaction of (i)+(ii) on a given mesh with piecewise linear velocity-pressure, which is not exact satisfaction. Similarly, a physical fluid can be expected to seek best possible satisfaction of (i)+(ii), which may not mean exact satisfaction of e.g. incompressibility.

We argue that DFS can be a more meaningful object of study than the Navier-Stokes/Euler equations in a conventional strong or weak formulation asking for interpretations. 

Remark 8 Our proposed solution of the Clay problem has the form of an open-source computer program (Unicorn FEniCS/HPC), which upon execution delivers a solution of Navier-Stokes/Euler showing to be non-smooth/turbulent. When executed on a sequence of automatically generated adaptive meshes with decreasing mesh size, mean-value quantities such as lift and drag are seen to converge to specific values within tolerances which can be estimated by duality, and can be made as small as desired. Residuals of computed solutions are seen to tend to zero weakly while becoming large strongly as expression of non-smoothness.

Our solution can be seen as  a computer-assisted proof similar to the celebrated computer-assisted proof of the four-color theorem and the Feit-Thompson theorem on group classification.

The FEniCS/HPC code is open-source and thus available for inspection, evaluation and testing by anyone. It uses the automated modeling of FEniCS and as such can be expected to be correct, or at least possible to be made correct.

It is thus possible to check if our claim of having solved the Clay problem is correct or not. The question is if anyone with connection to Clay is willing to make the check.

Remark 9 The standard following Prandtl is to specify the boundary condition for Navier-Stokes to be a no-slip condition $U=0$ asking both tangential and normal velocity to vanish, while for Euler only asking the normal velocity to vanish (non-penetration) leaving the tangential velocity free as a slip condition.

However, it is more natural from physical point of view to specify slip also for Navier-Stokes as an expression of small friction for slight viscosity, as mixed Dirichlet-Neumann condition. This is what we do, which opens a whole new perspective for computation without requirement of resolving thin boundary layers beyond the capacity of any forseeable  computer.  DFS thus solves Navier-Stokes /Euler with a slip boundary condition and so allows prediction of virtually any slightly viscous flow at affordable cost.

With slip we do not make a distinction between Navier-Stokes with slight viscosity and Euler with formally zero viscosity, since in the numerics it is the residual stabilisation which introduces the main viscosity/dissipation and not a vanishingly small standard viscosity term.

Altogether, we argue that we have given a resolution of the Clay problem by computation offering

1. Accurate prediction for arbitrary data (geometry and forcing) at affordable computational cost.
2. Understanding of the nature of solutions of Navier-Stokes/Euler from observations of computed solutions. In particular we observe that computed solutions can be described as non-smooth turbulent dissipative Euler solutions with Hölder continuity 1/3 in accordance with Onsager's conjecture and Kolmogorov's 4/5 law. 

Remark 10 Recall that we consider the official formulation by Fefferman to be incorrect from mathematical point by not including the crucial aspect of well-posednedness (here and here). It may well be that the question posed by Fefferman (existence of smooth solution for all time for all smooth data) will be impossible to answer, since mathematical techniques for proving global smoothness will remain hidden to humans together with techniques for construction of blow-up.

We suggest to reformulate the problem into a question of wellposedness of weak solutions for which a positive answer is offered by DFS.  This is the relevant question from physical point of view and then also from mathematical point of view, since Navier-Stokes is a mathematical equation with physical meaning.

Remark 11 We argue that only a notion of approximate solution of Navier-Stokes/Euler is meaningful, and this is what DFS delivers and which upon inspection shows to be non-smooth/turbulent with undetermined point-values of velocity and pressure, but with mean-value outputs such as lift and drag computable with quantitative error control.  In particular,
we argue that it does not make much sense to ask about exact solutions in a situation where solutions are non-smooth without well determined point-values and thus have the form of distributions which to be defined require the specification of a wealth/infinity of integrals weighted with smooth test functions. In short, does it make any sense to ask for exact specification of mean-values requiring a wealth of information. Isn't it more reasonable to be satisfied with specification of a piecewise linear DFS velocity-pressure, which is an approximate solution with error controled output? What more you could you ask for?

Remark 12 Our resolution includes the following ingredients:

1. Open-source computer code FEniCS/Dolphin (about 100.000 lines) for automated discretization of the Navier-Stokes/Euler equations in standard analytical form into a system of algebraic equations in piecewise linear DFS velocity-pressure on a given finite element mesh (millions of mesh points) expressing asking residuals to vanish weakly combined with weighted least squares stabilisation.
2. Open-source computer code PETc (about 100.000 lines) for automated computation of DFS solution.
3. Open source computer code FEniCS/Unicorn for quality assessment of computed DFS solution as quantitative measure of accuracy of chosen (mean-value) output by computation of a dual solution expressing sensitivity of output with respect to DFS residuals. 
4. Open-source code FEniCS/Unicorn for automated mesh adaptivity to reach specify output accuracy.  

The codes express a massive volume of analytical mathematics and execution of the codes massive volume of computational work.  Our resolution is the result of a combination of analytical mathematics and brute computational force with the goal/scope of delivering answers to "all that can be asked for".  It is not to be expected that non-linear pde-theory within the frame of Fefferman's problem formulation, can deliver anything near this volume of information.

Remark 13 There is only one notable mathematical result for Navier-Stokes/Euler in the literature and that is the existence proof by Leray from 1934 of weak solutions, however without any information on uniqueness/wellposedness. Leray gives a short mathematically simple argument based on basic energy estimate everyone knows. And after Leray basically no progress! No existence of unique strong solutions and nothing about wellposedness of weak solutions.

What we do is to continue Leray's work by (i) computing weak solutions and (ii) assessing wellposed of weak solutions. From the pictures above of computed solutions it is clear that Fefferman's question about existence of unique smooth solutions has a negative answer, and so the remaining question concerns wellposedness of weak solutions, a question we answer.  

PS For perspective browse this talk on the Clay problem by Titi, where at the end the question of computer-assisted proof is raised, and we learn that Titi believes it will take 5000 years to compute solutions to Navier-Stokes/Euler. We know that the reality today is that it takes hours.

Here is a discussion of the relevance of the problem formulation by Fefferman.