A common concern is the concept of real number and the set of real numbers $\mathcal{R}$ as the playground for most of modern mathematics. Wildberger takes a critical look on how these concepts are introduced in standard texts noting that basic difficulties are swept under the rug. View in particular this episode: Real numbers as Cauchy sequences does not work.
BodySoul takes a constructive approach viewing the natural numbers 1, 2, 3,..., to be constructed by repetition of the operation +1, the integers as solutions to equations $x+n=m$ with $n$ and $m$ natural numbers, the rational numbers as solutions to equations $q*x=p$ denoted $x=\frac{p}{q}$ with $p$ and $q\neq 0$ integers, while the real number $\sqrt{2}$ is defined as the positive solution to the equation $x^2=2$ or $x*x=2$.
Recall that the Pythagorean society based on the concepts of natural and rational number, collapsed when it became public that $\sqrt{2}$ is not a rational number. Modern mathematics is based on the concept of $\mathcal{R}$ as the set of all real numbers. Wildberger concludes that all attempts to bring rigour into the foundations of mathematics as the virtue of modern mathematics including Dedekind cuts, equivalence classes of Cauchy sequences and infinite sequences of decimal expansions, have failed. The trouble with all these attempts is the resort to infinities in different form. What will be the fate of the society of modern mathematics when this fact becomes public?
In the constructive approach of BodySoul there is no need to introduce infinities: In particular it is sufficient to work with rational numbers as finitely periodic decimal expansions or even more restrictive as finite decimal expansions, which makes perfect sense to anybody. But it requires making the notion of solution of an equation like $x*x=2$ precise, that is making precise the meaning of the equality sign $=$.
We then have to make the distinction between exact equality or more precisely logical identity denoted $\equiv$ and numerical equality denoted by the usual equality $=$ as something different to be defined. We thus have $A\equiv A$ while writing $A=B$ would mean that $B$ is not identical to $A$ but equal in some restricted meaning to be defined.
We then understand that $x\equiv\frac{1}{3}$ as exact solution to the equation $3*x=1$, while $x=0.333333333$ is a solution in a restricted meaning. We meet the same situation as concerns the solution to the equation $x*x=2$ with $x=1.414$ and $x=1.41421356$ as solutions in a restricted sense, or approximate solutions of different quality or accuracy.
To measure the quality of a given approximate solution $x$ to the equation $x*x=2$, it is natural to evaluate the residual $res(x)=x*x-2$ and then from the value of $res(x)$ seek to evaluate the quality of $x$. This can be measured by the derivative $f^\prime (x)=2*x$ of the function $f(x)=x*x-2$, noting that a different approximate solution $\bar x$ is connected to $x$ by the mean-value theorem
from which the quality of approximate solutions can be measured in terms of the residuals with $\frac{1}{2}$ as sensitivity factor.
This analysis generalises to to approximate solution to equations $f(x)=0$ for general functions $f(x)$ with the derivate $\frac{1}{f^\prime (x)}$ expressing residual sensitivity. In particular we see that if $f^\prime (x)$ is small the sensitivity is large asking the residual to be very small to reach precision in $x$.
But this argument is not central in modern mathematics where the notion of exact solution to an equation is viewed as the ideal. The exact/ideal solution to the equation $x*x=2$ would thus be viewed as a non-periodic infinite decimal expansion, which would require an infinite amount work to be determined, thus involving the infinities which Wildberger questions. The equality sign in this setting comes without quality measure in finite terms as an unattainable (Platonic) ideal.
In the setting of the algebraic equation $x*x=2$ the notion of an ideal solution may not cause much confusion, but for more general equations such as partial differential equations it has generated a lot of confusion because the quality aspect of approximate solutions is missing. The quality of an ideal solution is infinite beyond measurement but also beyond construction.
There is a notion in modern mathematical analysis of partial differential equations named well-posedness with connects to the sensitivity aspect of approximate solutions, but it has received little attention in quantitative terms.
We may compare the Pythagoreans facing the equation $x*x=2$ with a notion of ideal solution, and modern mathematics hitting a wall confronted with the Clay Math Institute Millennium Problem on ideal solutions of Navier Stokes equations.
PS Recall the famous Kronecker quote: "God made the integers, all the rest is the work of man". So the power of an almighty God was not enough to proceed and also make the real numbers. What are the prospects that man can succeed?
ChatGPT informs me that there is no mathematical proof of 2nd Law of thermodynamics and so it is simply an empirical law albeit:
supported by a wealth of empirical evidence
deeply ingrained in our understanding of how physical systems behave
while mathematical frameworks like statistical mechanics provide a basis for explaining the law, the principle itself is derived from the consistency of these explanations with real-world observations.
ChatGPT is useful in the sense that it reports what it has learned from physics literature, while it is not intelligent enough to cover up like a real theoretical physicist, who would never admit anything like that.
In any case we learn that that there is no mathematical proof/explanation of the 2nd Law.
It means that this could be added to the list of Clay Millennium Problems, or even better replace the closely related Navier-Stokes problem, which is still open without any progress to a solution, see previous post.
What do you think? Is there a mathematical proof? Or not?
PS Recall that Boltzmann's H-theorem stating a steady progress to a Maxwellian equilibrium in a dilute gas attempted as a mathematical proof of the 2nd Law, is based on Boltzmann's Stosszahlansatz asking two particles about to collide to be uncorrelated as an assumption of statistical nature, which however cannot be verified nor assumed to be true in any generality.
This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding.
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.
The problem is still open. No solution is even in sight after 24 years. No progress at all.
Another main open problem of mathematical physics is the 2nd Law of Thermodynamics, which in particular applies to the flow of fluids such as water and air as governed by Navier-Stokes equations.
It is thus possible to view the Clay Navier-Stokes Problem as an instance of the 2nd Law of Thermodynamics, and so be reformulated into:
Mathematical proof of the 2nd Law of Thermodynamics for fluids. (P)
This version has a more obvious significance and it is possible that a solution can be found and so increase understanding in the spirit of Clay.
If a certain norm of the solution stays bounded (or grows at a controlled rate), then the solution stays regular.
Taken in the contrapositive, they assert that if a solution blows up at a certain finite time T, then certain norms of the solution must also go to infinity.
The certain norm can be the $L^3$ norm over the spatial domain of the velocity $u(t)$ for $t<T$, which thus would serve as an indicator for solution of the Navier-Stokes Millennium Problem.
I have in sequence of posts expressed criticism of the formulation of the Millennium Problem which can be condensed into:
The normalisation to unit viscosity means that the physics of turbulence, which appears for small viscosity and bounded flow velocities, is missing. This makes the Millennium Problem into a purely academic mathematical problem without significance to the real world of fluid flow (and so probably violating the intention of Mr Clay).
More precisely, a turbulent solution would not be a classical regular solution nor a solution with unbounded velocities, thus a solution outside the present formulation of the Millennium Problem.
It is thus natural to ask for a reformulation of the Navier-Stokes Problem, in particular since no advance towards a solution in the original formulation has been made since 2000. Compare with this article on Knowino.
PS This post was triggered by a new comment to the original post by Terence from 2019. See also my next post here.
$\frac{\partial u}{\partial t}+u\cdot\nabla u+\nabla p -\nu\Delta u =0$,
$\nabla\cdot u=0$,
where $u(x,t)$ is velocity and $p(x,t)$ pressure depending on a space coordinate $x\in R^3$ and time coordinate $t\ge 0$, $\nu$ is a positive (constant) viscosity, and an initial velocity is given at $t=0$.
The question posed in the official formulation of the problem is:
Do smooth solutions exist for all time (global in time)?
Or do solutions cease to exist at some finite time (finite time break down)?
Mathematician have been struggling with this problem since the equations were formulated in the 1830s, however with little progress, in particular after 2000.
The present main assault to solve the problem is led by Terence Tao as the most able mathematician on Earth. Tao approaches the problem along a well traced path based on a theorem stating that if velocities are (suitably) bounded, then smooth solutions existing for small time if initial data are smooth, will not cease to exist and thus exist for all time.
In short: Bounded solutions will stay smooth. And the other way around: The only way smooth solutions may cease to exist is through velocities blowing up to infinity.
In a recent article Tao seeks to give this purely heavily qualitative result (with very little information) a quantitive form (with hopefully more information). The basic result is stated as Theorem 1.2 taking the basic form: If velocities by are bounded by some positive constant A, then first derivatives of velocity and vorticity are bounded by constants of size:
exp exp exp A $= e^{e^{e^A}}$.
In short, if velocities are bounded, then so are gradients (and similarly higher derivates) and so a solution initialised as smooth will stay smooth.
But the bound on the derivatives with the triple exponent makes no sense. From any reasonable point of view the bound is infinite and thus says nothing about smoothness.
In this approach to the Clay problem made by mathematicians it appears that reason is gone: If a smooth solution can have basically infinitely large derivatives, then the concept of smoothness is twisted away from any reasonable meaning. Is the idea to solve the Clay problem with meaningless mathematics, to report that it has been solved, once and for all?
In several previous posts I have indicated a different approach to resolve the Clay problem in a meaningful way. Take a look. The basic insight is that smooth solutions of Navier-Stokes equations in general develop into turbulent solutions which are not smooth. But this does not appear to be something a (pure) mathematician can accept, and then not the Clay Prize committee, even if this is the truth. Is this as an expression of crisis of modern mathematics? Or not at all?
So when is fluid flow turbulent non-smooth? The answer is: When viscous dissipation is of the same size as kinetic energy. More precisely, the basic energy estimate for Navier-Stokes equations reads:
for $T\gt 0$ with on the left side kinetic energy at time $T$ plus total dissipated viscous energy balancing kinetic energy at initial time $t=0$. Here $u$ is normalised to be of size 1 and the viscosity $\nu $ is smaller than $10^{-6}$, as a typical case when solutions turn turbulent. With a smooth initial solution the viscous dissipation starts out as very small and then grows as turbulence develops with kinetic energy transformed into viscous dissipation with large velocity gradients (of size $\nu^{-1/2}\ge 10^3)$. This is reality very far from the triple exponential world of Tao, but mathematicians do not seem to be willing to listen to reason...I have asked Tao for comment...
On top of the triple exponentials Tao scales the equations so that viscosity is 1 which means that fluid
velocity is boosted with another big factor making the argument even more unphysical and then also unmathematical if meaning is intended.
A Navier-Stokes solution initialised as smooth does not turn non-smooth from velocities blowing up to infinity, but from gradients of velocities becoming large as expression of turbulence which is non-smooth flow. It is very difficult to understand why this not something that Tao understands very well.
PS The Navier-Stokes problem was formulated by pure mathematicians to be solved by pure mathematicians by methods of pure mathematics. Since no progress has been made and none is in sight, my expressed view is that the problem should be reformulated to make sense for a wider scientific community including applied and computational mathematics.
Computational software used by developers of airplanes do not seem to allow simulation of the dynamics of stall and so the impact of stall on aircraft design and safety assessment must be done solely by expensive and time-consuming experiments in flight, and then also the design of the apparently needed control system. This may show to have been insufficient to make the plane safe, something which FAA did not have the capacity to check.
PS1 The Jas 39 Gripen is a fighter airplane designed and manufactured by the Swedish aerospace company Sabb, which is inherently unstable (relaxed stability) to allow quick turns and thus needs to be stabilized by automatic active control of small canards (wings) ahead of the main wings. The plane crashed twice from pilot-induced oscillations with a the pilot seeking to compensate an over-reacting control system, in 1989 during landing and in 1993 over central Stockholm. A commercial aircraft is not designed to be unstable, but to be positively stable.
PS3Boeing is working on an update of MCAS to get 737 Max approved by FAA to fly again, but the problem may be bigger than just a software update asking for a redesign of the airplane, see here.
Mathematics is an interesting intellectual sport but it should not be allowed to stand in the way of obtaining sensible information about physical processes.
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)
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 reformulationof 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 $\Omega$)
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.
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 thefinite 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 equationsof 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.
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.
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?
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.
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.
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:
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.
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:
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.
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.
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 1Tao 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):
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 isverywell 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 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:
Physics (of fluid particles): (i) Newton's 2nd Law + (ii) Incompressibility.
Mathematics: Formulation of (i) + (ii) in terms of Calculus = Navier-Stokes/Euler.
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.
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
Accurate prediction for arbitrary data (geometry and forcing) at affordable computational cost.
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:
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.
Open-source computer code PETc (about 100.000 lines) for automated computation of DFS solution.
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.
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.
Let me end with a few words about the significance of the problems posed here.
Fluids are important and hard to understand....our understanding is at a very primitive level.
Standard methods from PDE appear inadequate to settle the problem.
Instead, we probably need some deep, new ideas.
Yes, fluids are hard to understand for a pure mathematician and the understanding appears to be on a very primitive level, and this has led to an unfortunate formulation of the problem leading into a fruitless search for either (i) blowup into infinite fluid velocities in finite time as non-smoothness, or (ii) not blowup as smoothness of solutions.
But for a fluid the distinction between smoothness and non-smoothness concerns the size of velocity gradients. It is a well-known fact since long that compressible flow may exhibit non-smoothness in the form of shocks with large velocity gradients but without large velocities.
The Clay Problem concerns incompressible flow (at unbounded Reynolds numbers), which does not form shocks but instead becomes turbulent for large Reynolds number with again large velocity gradients as expression of non-smoothness, and (most likely) without large velocities.
The clue to solve the Clay Problem offered in the official formulation by searching for infinite velocities in fluid flow, which Tao has picked up in recent attempts to solve the problem, thus appears to be misleading and as such is not helpful to mathematics as science.
Fefferman is asking for some new ideas, but closes the door to any form of communication with computational turbulence as a new idea towards understanding and resolution of the problem.
For initial data $u^0(x)$ not assumed to be small, it is known that (A) and (B) hold (also for $\nu = 0$) if the time interval $[0,∞)$ is replaced by a small time interval $[0,T)$, with $T$ depending on the initial data.
For a given initial $u^0(x)$, the maximum allowable $T$ is called the “blowup time.” Either (A) and (B) hold, or else there is a smooth, divergence-free $u^0(x)$ for which (1), (2), (3) have a solution with a finite blowup time.
For the Navier–Stokes equations ($ν > 0$), if there is a solution with a finite blowup time $T$, then the velocity $u_i(x,t)),1≤i≤3$ becomes unbounded near the blowup time.
We read that Fefferman claims that the distinction between (i) YES or (ii) NO to the question of existence+regularity for the Navier-Stokes equations, is between (i) bounded flow velocity for all time and (ii) unbounded velocity for some "blowup time" $T$.
Fefferman here uses the same distinction as in the classical theory of ordinary differential equations (odes) based on a (correct) mathematical analysis showing that the only way a solution trajectory can cease to exist, is to tend to infinity in finite time.
But this argument cannot be generalised to partial differential equations (pdes), because a smooth solution to a pde can cease to exist as a smooth solution because of unbounded derivatives of the solution, without the solution itself becoming infinite (as required in the ode case).
The basic distinction for Navier-Stokes is instead between (i) laminar/smooth flow and (ii) turbulent/non-smooth for all time without blowup to infinity of the velocity, where non-smooth means large velocity gradients.
The official formulation of the problem is unfortunate by (incorrectly) claiming that the question can be reduced to a question of infinite velocities at finite blowup time. The Clay problem thus needs to be reformulated, since an incorrectly formulated problem can only lead in a wrong direction.
In an effort to solve the Clay Navier-Stokes problem as formulated by Fefferman, Terence Tao in recent work seeks to construct a solution to the incompressible Euer equations with velocities becoming infinite in finite time, but does "not quite achieve" the goal.
Let me present some evidence indicating that the goal cannot be achieved. To this end we compare the incompressible Euler equations:
$\frac{\partial u}{\partial t}+u\cdot\nabla u+\nabla p =0$
$\nabla\cdot u=0$
with (i) vector-Burgers as a model of very compressible flow:
$\frac{\partial u}{\partial t}+u\cdot\nabla u=0$
and (ii):
$\frac{\partial u}{\partial t}+u\cdot\nabla u+\nabla p =0$,
$\delta\Delta p=\nabla\cdot u$
with $\delta >0$ a small constant, as a model of slightly compressible flow.
For Burgers equation and so for (i), velocities may become discontinuous corresponding to the development of shocks over time, but velocities do not tend to infinity.
In case (ii) solving for the pressure p gives the following equation along a streamline $x(t)$:
which formally gives a bound on the possible growth of velocity in terms of $\frac{1}{\delta}$ preventing blow-up to infinity.
We conclude that neither very compressible nor slightly compressible flow appears to accommodate blow-up to infinite velocity. Is it then the incompressibility which will squeeze the flow to infinite pressure driving flow velocity to infinity? Far-fetched in my view.
On the other hand, we have strong evidence that Euler solutions become turbulent with substantial turbulent dissipation from large velocity gradients, while velocity does not spike to infinity. Again, the formulation of the Clay Navier-Stokes problem without reference to turbulence, appearently leads mathematicians into meaningless dead ends.
The formulation of the Clay Millennium Problem about global smoothness of solutions to the incompressible Navier-Stokes equations by Charles Fefferman, circumvents the phenomenon of turbulence as the most important aspect of fluid flow from both mathematical and physical point of view. The result is a problem which is both meaningless and without solution, and thus cannot serve well as a Clay Millennium Problem.
The unfortunate formulation by Fefferman comes out in the recent attempts by Terence Tao to construct a solution with local blow-up of fluid speed to infinity in finite time. Tao thus seeks a negative answer to global smoothness by constructing solutions with flow speed going to infinity locally. But he does not succeed and there is no reason to expect that he ever will, because the viscous term in Navier-Stokes dominates the convective term on small scales. Tao working in conjunction with Fefferman, thus is led into a fruitless direction.
The question of global smoothness in Fefferman's formulation, should better be replaced by a question of turbulence with turbulent flow defined as flow with velocity $u(x,t)$ with given initial data $u(x,0)$ such that for a positive constant $C$ (which is not small)
for all small viscosities $\nu > 0$. Note that by renormalizing initial data, the effective value of the viscosity can always be made as small as desired, see PS below. Non-turbulent = laminar flow then has a constant $C$ which tends to zero with $\nu$.
A turbulent solution would then correspond to a non-smooth solution in Fefferman's formulation, and then a laminar = non-turbulent solution to a smooth solution, and a Clay problem about global existence of laminar solutions would have a negative answer: For any given positive viscosity, there is data such that the corresponding Navier-Stokes solution becomes turbulent in finite time. Or turned the other way: For many initial data there is a viscosity such that the corresponding solution becomes turbulent in finite time.
Another aspect of Fefferman's unfortunate formulation is that the flow is supposed to fill all of space, or be periodic in space, which means that the completely crucial presence of flow boundary and choice of boundary condition, is neglected. There can be no rational reason to formulate a mathematical problem presented as being connected to physical reality of importance to humanity, in a way that makes any such connection meaningless.
PS1 renormalisation goes as follows: If $u(x,t)$ satisfies
$\frac{\partial u}{\partial t}+u\cdot\nabla u-\nu\Delta u = 0$,
then $\bar u=\frac{u}{\alpha}$ with $\alpha >0$ satisfies
$\alpha\frac{\partial\bar u}{\partial t}+\bar u\cdot\nabla\bar u-\alpha\nu\Delta\bar u = 0$
and thus $\bar u$ with renormalisation of time $t=\alpha\bar t$ satisfies:
$\frac{\partial\bar u}{\partial\bar t}+\bar u\cdot\nabla\bar u-\bar\nu\Delta \bar u = 0$
with $\bar\nu =\bar\alpha\nu$ arbitrarily small with $\alpha >0$.
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 in the official formulation of the problem by Fefferman there is nothing about turbulence! Instead, it appears that the problem is deliberately cleverly formulated so as to completely exclude this fundamental aspect from the discussion, by focussing on blow-up instead of turbulence as expression of non-smoothness. I have many times tried to get this across to people spending time on seeking a solution and to mr Clay ready to spend money on a solution, so far with little success. But for every year without solution the lack of meaning of the present problem formulation may become more understood. Maybe time is now ripe for a revision of the formulation of the problem, mr Clay?
In any case, waving with turbulence and then excluding turbulence is not correct science.
PS3 Here is copy of a letter to Tao:
Hi Terence
I see that you seek to construct solutions with blow-up to Euler/Navier-Stokes equations in an effort to solve the Clay Navier-Stokes problem as formulated by Fefferman. I have already tried to get across to you and Fefferman that the present formulation is not correct from scientific point of view (as I see it), since turbulence is named as the main unresolved mystery of the Navier-Stokes equations in the presentation of the problem by the Clay Institute, yet the formulation by Fefferman is made so as to exclude turbulence from the discussion.
I would appreciate if you could give your view on this apparent contradiction. I would also be happy if you could comment on the reformulation of the problem including turbulence suggested here:
I understand that you may want to discard my proposal because your time is limited, but working on a problem without meaningful answer may represent even more loss of time.
We compute an approximate turbulent solution $U_h$ to the Euler equations using G2 on a given mesh with mesh size $h$ characterised by substantial turbulent dissipation. We ask if with $U_h$ given, it is possible to construct a function $\hat U_h$ which solves the Navier-Stokes equations for some viscosity $\nu_h$?
This can be answered by pointwise computing the Euler residual $E(\hat U_h)$ of a regularisation $\hat U_h$ of $U_h$ together with the Laplacian $\Delta\hat U_h$ and then defining
$\hat h =\frac{E(\hat U_h)}{\Delta U_h}$.
If it turns out that $\hat h >0$, then we have a function $\hat U_h$ which exactly solves the Navier-Stokes equations with a viscosity $\hat h$, and if $U_h$ is turbulent, so will $\hat U_h$ be.
Depending on the variation of $\hat h$, we could argue that we have constructed an exact solution to a modified Navier-Stokes equation (with constant viscosity), with the modification depending on the variation of the computed $\hat h$, a solution which is turbulent and thus non-smooth.
This argument has a connection to that presented by Terence Tao in a setting of modified Euler/Navier-Stokes equations. The difference is that we use a computed solution of great complexity instead the analytical solution of less complexity constructed by hand by Tao.
There is strong evidence from experimental observation and computing that solutions to the Navier-Stokes equations with small viscosity, are always turbulent and thus that the Clay problem about global existence of smooth solutions has a negative answer. Thus it seems pretty clear that computational evidence can settle the Clay problem, but this may not be accepted by a jury of mathematicians trained in analytical mathematics developed before the computer. It may be that without computational evidence the problem may stay unsolved for ever, or that an answer by analytical mathematics becomes so particular that the very meaning of the problem is lost.
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