onsdag 27 april 2016

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.  

tisdag 26 april 2016

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.  


måndag 25 april 2016

Reformulation of Clay Navier-Stokes Problem Needed 3

The official formulation of the Clay Navier-Stokes Problem does not include any reference to the Reynolds number $Re =\frac{UL}{\nu}$ with $U$ a typical flow speed, $L$ length scale, and $\nu$ viscosity scale, and thereby makes no distinction between laminar/smooth flow at small Reynolds numbers and turbulent/non-smooth flow at large Reynolds numbers.

Since no bound on the Reynolds number is given, it can only mean that the Reynolds number can have any size, in particular be arbitrarily large. By normalizing $U$ and $L$ to unity, the viscosity thus can be arbitrarily small (or normalizing  viscosity and length scale to unity and letting $U$ become large), which means that the Clay problem includes the incompressible Euler equations as the incompressible Navier-Stokes equations with vanishingly small viscosity.

This is precisely what the book Computational Turbulent Incompressible Flow is about! As a basic example from the book, let us consider flow around a sphere under vanishing viscosity depicted in the following pictures:

We see a distinct large-scale separation pattern developing consisting of 4 tubes of counter-rotating flow attaching to the rear of of the sphere, which are dissolved into turbulent flow further down-stream. We see that the length of the tubes increases with decreasing viscosity, which is consistent with Kolmogorov's conjecture that the total amount of turbulent dissipation stays roughly constant under vanishing viscosity (along with total drag), requiring the surfaces of intense dissipation of the 4 tube pattern to extend further downstream.

We thus discover a vanishing viscosity solution to the incompressible Euler equations, which is fundamentally different from the formal exact solution in the form of potential flow, which is symmetric in the flow direction with symmetric attachment and separation without the 4tube gross pattern, a formal exact solution which is unstable at separation and thus not a limit of Navier-Stokes solutions.

The official problem formulation does not fit well with this situation, since stability aspects are left out and the presence of vanishing viscosity solutions is hidden, and the Clay Navier-Stokes problem thus asks for a reformulation away from the deadlock of the present (meaningless) formulation.



Reformulation of Clay Navier-Stokes Problem Needed 2

The official formulation of the Clay Navier-Stokes Problem is unfortunate by not mentioning the Reynolds number $Re =\frac{UL}{\nu}$ with $U$ a typical flow speed, $L$ length scale, and $\nu$ viscosity scale, and thereby making no distinction between laminar/smooth flow at small Reynolds numbers and turbulent/non-smooth flow at large Reynolds numbers.

Instead the problem is formulated so as lead people to seek a solution in the form of blow-up (or not blow-up) to infinity of flow speeds in finite time, following a simple methodology borrowed from ordinary differential equations saying that as long as a solution trajectory is finite it can be continued for some time. But this is not the real issue for a partial differential equation like Navier-Stokes, where the essential distinction is instead between laminar/smooth and turbulent/non-turbulent flow.

The result is a problem formulation which is meaningless because it is both unphysical and unmathematical, and as such cannot be given a meaningful answer.

The problem is owned by a small group of pure mathematicians including Fefferman and Tao, who refuse to participate in any form of discussion about the problem and its formulation. This is not in the interest of anybody outside this group and thus not in the general interest of mathematics as science, which must be the interest of mr Clay in particular...

 

torsdag 21 april 2016

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



    onsdag 20 april 2016

    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".   

    tisdag 19 april 2016

    P=NP? vs Escher's Descending-Ascending Monks vs 2nd Law


    There is a connection to turbulent Euler solutions as counterexample to P=NP in Eschers's descending-ascending Monks, who walk around either descending or ascending all the time, yet coming back to where they started again and again. If you believe this is an illusion, then you have a good chance in understanding the counterexample.

    Recall that turbulent dissipation forward/backward always means loosing energy and thus descending all the time. And then you cannot get back to where you started, unless you believe in the illusion of Escher's monks...Thus you cannot recover the initial state from a later state.

    P=NP? vs 2nd Law vs Turbulent Dissipation

    This is a continuation of the argument from recent posts that the 2nd Law expressed by turbulent solutions to the incompressible Euler equations represents a counterexample to P=NP. The basic ingredients are the following:
    1. Exact smooth solutions of the Euler equations such as potential flow are unstable and thus are not computable globally in time under finite precision.
    2. Turbulent weak solutions can be computed globally in time with polynomial work. Such solutions have substantial turbulent dissipation independent of resolution and thus are substantially irreversible in time. More precisely, the irreversibility with substantial turbulent dissipation independent of resolution makes recovery of an initial state u(0) at time 0 from a computed solution u(T) at later time T, impossible with polynomial work.
    3. The problem Q of recovering u(0) from u(T) is thus appears to be notP, while it is possible to check if a recovery candidate v(0) produces u(T) upon solution forward in time with polynomial work.
    4. It is the unavoidable substantial turbulent dissipation which creates substantial irreversibility under polynomial work and thus shows that Q is notP, while forward solution with polynomial work makes Q NP and thus a counterexample to P=NP.
    5. It is the unavoidable substantial turbulent dissipation which is the reality of the irreversibility of the 2nd Law and makes into a dictate, which cannot be escaped with polynomial computational work and thus neither in physical reality, with physics as a form of analog computation with finite precision, which by the necessary limitations of physical reality can only involve polynomial work.  
    6. The connection to physics as analog computation and the 2nd Law as a law of physics as computation, gives the P=NP? problem a more clear meaning by offering a sharp distinction between P=possible=physical and notP=impossible=unphysical, than in the standard setting of polynomial vs exponential growth, which may be impossible to clearly separate in any form of real computation. 
    7. The formulation of P=NP? as the question if it is easy to go forward from a position u(0) to end up at a position u(T), is it then possible to back-track to an initial position v(0) possibly different from u(0) such that going from v(0) will end up X? For example going down a tree branching one way or the other to X, it is possible to back-track to a possibly different initial position forward connected to X. In any case the P=NP? problem has a definite flavour of irreversibility as forward-easy and backward-not easy.