Meaningless Clay Navier-Stokes Problem Solved by AI

OpenAI announces a proof of existence of a solution to the Navier-Stokes equations (but not its numerical values), which starting from zero under smooth forcing ceases to exist in finite time: 

• We’re sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. We’re sharing both a writeup of the proof and a formalization in Lean.

Charles Fefferman, who formulated the problem in precise mathematical terms, is along with other leading mathematicians such as Terence Tao, happy that the understanding of fluid motion has now taken a big leap forward by mathematical analysis, even if the development of the singularity cannot be followed in any precise terms. Something goes wrong but what and how is hidden.

There is a further problem in this happy moment, which I have complained about over the years: Fefferman's formulation misses the essence of the physics of fluid motion, namely turbulence. The Clay problem is sold as concerned with basic aspects of fluid motion,  but does not address the most fundamental problem of all of turbulence. Fefferman's formulation directs the interest away from physics, and the unhappy result is that solution now presented by AI covering 167 pages cannot be read to learn anything, simply a mess of formulas and theorems. 

This is certainly a memento for mathematics: AI can now produce proofs of an endless number of mathematical problems without real meaning, proofs which cannot be understood by mathematicians in detail only verified formally by Lean. What will be the result?

Numerical mathematics offers a solution to the fundamental problem of turbulence, thus a different solution to a different problem formulation. See tags to this post starting with this post from 2013.

Recall that slightly viscous flow is unstable from shear and stretch and so develops into non-smooth turbulent flow which however does not break down like the Clay solution. So the solution of physical interest is non-smooth and non-singular, which is not captured in Fefferman's dichotomy of smooth or singular.  

Turbulence is an extreme form of the design of a complex world with a variety of phenomena on different scales: Develop growth from instability + curb growth to allow continued existence, not captured by Fefferman's formulation.  

PS1 When I 20 years ago complained to Fefferman that his formulation lacked true interest from physics point of view, he returned that it was enough that the problem was interesting to him.

PS2 Note that the AI solution is a proof of the existence of a very special function (unknown to details) which is a solution with a very specific particular forcing. This is not the real setting which is to study solutions under general forcing. 

PS3 Here is an interesting catch of the AI proof of existence of a singular solution. Computational solutions can be constructed for general data including turbulence and any such solution can be viewed as an AI proof of existence performed by a computer according to strict mathematical principles, including evaluation of quality. The whole process can be seen as an AI proof of existence of a solution for each given set of data. It would be strange to not consider that as a solution to the essence of the Clay problem albeit not captured in Fefferman's formulation. 

PS4 Allowing AI as computational process into the Clay problem game, we may compare the Open AI proposal as an analytical AI proof of non-existence in a very special case, with an computational AI proof of existence for any data, except one. Which proposal would you give the money to? Or 50-50? Note that the estimated cost of the OpenAI solution is several million dollars, so the Prize money will not suffice to cover, what remains is fame at price of a couple million dollars, fine for OpenAI but not for a poor pure mathematician. 

PS5 The verification by Lean in principle requires each step to be verified from logic and previous axioms/therorems/steps, which is overwhelming and cannot be done. Compare with a numerical solution produced in a number of computational steps, where a verification of solution quality can be made without verifying each step (involving round-off which propagates) because the solution produced is known. Not so with the singular solution proved to exist by AI and so only stepwise check is available (which is more impossible than possible). 

PS6 The size of the forcing appears to scale with the square root of the viscosity which means that the constructed solution is not turbulent. Another sign that the problem formulation misses the essence of Navier-Stokes. How could it go so wrong for so many mathematicians?

2nd Law vs Clay Millennium Problem on Navier-Stokes Equations

The Clay Institute Millennium Problem on Navier-Stokes equations is introduced as follows:

• 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.

A resolution to (P) is presented in recent posts on the 2nd Law.   

I have sent the following letter to the President of the Clay Institute and will report reaction:

Dear President 

No progress towards a solution to the Clay Millennium Problem on Navier-Stokes equations has been 

made over a period of 24 years. A reformulation into a problem which possibly can be solved may better 

meet the stated Clay objective of increasing understanding. 

Thus I suggest a reformulation into a mathematical proof of the 2nd Law of Thermodynamics for fluids as 

expressed here:

https://claesjohnson.blogspot.com/2024/03/2nd-law-vs-clay-millennium-problem-on.html

Sincerely

Claes Johnson

prof em applied mathematics Royal Institute of Technology Stockholm

Solving the Clay Navier-Stokes Problem with Meaningless Mathematics?

The Clay 2000 Millennium Navier-Stokes problem concerns solutions to the incompressible Navier-Stokes equations:

• $\frac{\partial u}{\partial t}+u\cdot\nabla u+\nabla p -\nu\Delta u =0$,
• $\nabla\cdot u=0$,

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:

• $\int \vert u(x,T)\vert^2 dx +2\int_0^T\int\nu\vert\nabla u(x,t)\vert^2dxdt =\int\vert u(x,0)\vert^2 dx$

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. 

Reformulation of Clay Navier-Stokes Problem Needed 5

Fefferman concludes the official formulation of the Clay Navier-Stokes Problem with:

• 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.  

Reformulation of the Clay Navier-Stokes Problem 4

The official formulation of the Clay Navier-Stokes Problem by Fefferman includes the following statements (with (A) and (B) global existence+regularity and (1)-(3) Navier-Stokes equations):

• 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 a lecture about the problem, Cafarelli falls in the trap of Fefferman.  

Velocity Blow-up to Infinity for Incompressible Euler?

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 $x(t)$:

• $\frac{du(x(t))}{dt} + \frac{1}{\delta}\nabla\Delta^{-1}\nabla\cdot u(x(t),t)=0$ 

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.

Reformulation of Clay Navier-Stokes Problem Needed 1

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)

• $\frac{\nu \int\vert\nabla u(x,t)\vert^2 dx}{\int \vert u(x,t)\vert^2 dx} > C$ for $t>0$    

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 $\alpha >0$.

PS2 The Clay Navier Stokes problem is presented by the Clay Mathematical Institute as follows:

• 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:

http://claesjohnson.blogspot.se/2016/04/reformulation-of-clay-navier-stokes.html

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.

Best regards

Claes

Turbulent Euler Solutions and the Clay Navier-Stokes Problem 2

This is a continuation from the previous post:

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.

Turbulent Euler Solutions and the Clay Navier-Stokes Problem 1

Turbulent solutions of the incompressible Navier-Stokes equations with viscosity $\nu >0$ can be characterised as having substantial turbulent dissipation, that is, satisfying for all sufficiently small positive $\nu$ with normalisation of the velocity $u$ (for $t>0$ say):

• $\int\nu\vert \nabla u(x,t)\vert^2\, dx > C$ 

where $C$ is a positive constant.

Dimension analysis suggests that turbulent solutions are non-smooth Hölder continuous with exponent 1/3 on a smallest scale in space of size $\nu^{\frac{3}{4}}$ with $\vert\nabla u\vert\sim \nu^{-1/2}$. 

We view such solutions as approximate weak solutions of the Euler equations (formally corresponding to $\nu =0$), or turbulent Euler solutions, thus characterised by substantial turbulent dissipation. Stability analysis and computation strongly suggest that all smooth solutions to the Navier-Stokes with small $\nu$ and the Euler equations, become turbulent over time, see Computational Turbulent Incompressible Flow.

Terence Tao struggles to analytically construct solutions to the incompressible Euler equations with blow up in finite time, which could possibly show blow-up also for Navier-Sokes,  but does "not fully achieve" the goal, which is to answer the Clay Navier-Stokes problem. 

Let us compare our approach based on stability analysis/computation with that of Tao based on analytical construction of solution with blow-up. We thus give evidence that (i) turbulent solutions can be computed over global time, (ii) all smooth solutions become turbulent because of inherent instability, while Tao seeks to (iii) construct a very specific solution with blow-up for Euler and Navier-Stokes.

We see that our approach is complementary to that of Tao, or the other way around: (i)-(ii) concerns the general problem and gives life to solutions after blow-up as turbulent solutions, while (iii) concerns a very specific problem without life after blow-up.  

The evidence of (i)-(ii) consists of stability analysis + high performance computation, while (iii) is 

based on analytical computation by hand.  It may be that (i)-(iii) together capture the core aspects of the Clay Navier-Stokes problem using different forms of mathematics and "proofs".   

Counterexample to P = NP with Relation to 2nd Law

Let me here present the counterexample to P = NP from the previous post in more concise form.

We consider the Algorithm of solving the incompressible Euler equations over a time interval (0,T) by the stabilized finite element method G2.  We observe that solutions are turbulent with turbulent dissipation which remains substantial (positive) as the discretisation is refined.

We know that exact laminar solutions (such as potential solutions) of the Euler equations are unstable and as a result all solutions turn turbulent over time. We know that given an initial state u(0), the Algorithm delivers a final state u(T) at later time T > 0 with local mean-values of u(T) of certain size computable to a certain precision with polynomial work.

We pose the problem Q of computing the initial state u(0) from a computed final state u(T).  We ask of this can be done with polynomial work, that is, we ask if Q is P?

For any reconstruction candidate v(0), the corresponding v(T) can be computed with polynomial work, which allows check if v(T) is sufficiently close to u(T) to say that v(0) is a acceptable reconstruction of u(0). In other words, Q is NP.

We know that solving the Euler equations backward in time, cannot lead to an acceptable  reconstruction of u(0) from u(T) if the tolerance is small enough, because the substantial turbulent dissipation necessarily being introduced computing u(T) from u(0) in forward time, cannot be reversed. Instead additional substantial turbulent dissipation is introduced in backward time, independent of the work invested.

This means that a substantial gap will remain between original image u(0) and any reconstruction v(0) computed backward in time from u(T), independent of the work invested in the backward process.

Q is thus notP if Q is backward time-stepping. Is it possible to reconstruct u(0) in some other way?
Simply testing all u(0) is certainly exponential and the question is then if it is possible to restrict the testing or shooting? The complexity of turbulent flow with each preimage u(0) giving a computed image u(T) of great local mean-value variability, means that substantial restriction seems impossible and so Q appears to be notP by simply testing all possibilities or shooting in all directions.

The remaining possibility would be some iterative method with further restriction of testing by successive improvement of shooting. We thus ask if there is some thinkable iterative method to solve an identification problem of solving an equation of the form Eu(0) = u(T) with Eu(0) the solution of the Euler equations with initial data u(0) evaluated at time T.

Here E represents an operator which is locally exponentially both unstable and stable, and any attempt to solve such an equation iteratively would seem to fail because the spectrum (of a linearization) of E will be spread over the whole complex plane.

We conclude that solving Q by (a) backward time stepping, (b) restricted guessing, (c) iterative shooting, appears to be clearly notP.  Are there any other possibilities? If not, then Q would be notP.

We summarise: Turbulent Euler solutions have the very special property of:

1. Unavoidable substantial irreversibility from unavoidable substantial turbulent dissipation, independent of resolution.
2. Great output variability resulting from local exponential instability-stability. 
3. Forward problem P up to local mean values.
4. Backward time stepping notP because of substantial irreversibility.
5. Iterative shooting notP because of spectrum with no restriction. 

Altogether, turbulent Euler appears to give a counterexample to P = NP.

At the same time it gives meaning to the 2nd Law as an easy-forward and difficult-backward problem, which is not based on ad hoc introduction of diffusion or probability (which is the common way of giving "meaning" to the 2nd Law).

At the same time turbulent Euler gives a counterexample to the Clay Problem of global smoothness of Navier-Stokes. 

The Euler equations thus represent a most remarkable mathematical problem, which cannot be solved exactly, but in computational form gives both meaning to the 2nd law and shows limits of computation and mathematics.

It is perhaps not so surprising that several seemingly unrelated and seemingly very difficult problems,  all have a common answer relating to turbulence as an observable phenomenon of reality, which cannot be simply an unresolvable mystery for ever hidden to human understanding.

I have tried over a long to get some understanding for the issues presenting themselves in the Euler equations, without too much success:

• I appears that mathematicians are not keen to bring in stability or well-posedness, although the mathematician Hadamard told them to do that, and the result is dead-lock. 
• It appears that computer scientists prefer integers before real numbers and physics, and the result is a dead-lock with man-made algorithms. 
• It appears that physicists prefer to speculate about multiversa and quantum foam beyond rationale.
• It appears that fluid mechanicians are stuck in infinitely thin Prandtl boundary layers beyond  computability.   

But there is hope: reality is there to discover and understand.