The existence of solutions to the (incompressible) Navier-Stokes is one of the Clay Institutes Millennium Prizes. Each of the problems is a wickedly difficult mathematical problem, and the Navier-Stokes existence proof is no exception. The interest in the problem has been enlivened by a claim that the problem has been solved. Terry Tao has made a series of stunning posts to his blog outlining majestically the mathematical beauty and difficulty associated with the problem. In my mind, the issue is whether it really matters to the real world, or whether it is formulated in a manner that leads to utility.

I fear the answer no. We might have enormous mathematical skill applied to a problem that has no meaning to reality.

The key word I left out so far is “incompressible”, and incompressible is not entirely physical. An incompressible fluid is impossible. Really. It is impossible, but before you stop reading out of disgust with me let me explain why.

One characteristic of incompressibility is the endowment of infinitely fast sound waves into the Navier-Stokes equations. By infinite, I mean infinite, sound is propagated everywhere instantaneously. This is clearly unphysical (a colleague of mine from Los Alamos deemed the sound waves to be “superluminal”). It violates a key principle known as causality.

More properly, incompressibility is an approximation to reality, and there is a distinct possibility that this approximation causes the resulting equations to depart from reality in essential ways for a given application. For very many applications incompressibility is enormously useful as an approximation, but like all approximations it has limits on its utility. Some of these limitations are well known. Mathematically this makes the equations set elliptic. This ellipticity is at the heart of why the problem remains unsolved to this day. Real fluids are not elliptic, real fluids have finite speed of sound. In fact, compressibility may hold the key to solving the most important real physical problem in fluid mechanics, turbulence.

Turbulence is the chaotic motion of fluids that arise when a fluid is moving fast enough and the inertial force of the flow exceeds the viscous force to a certain degree. Turbulence is characterized by the loss of perceptible dependence of the solution on the initial data, carries with it a massive loss of predictability. Turbulence is enormously important to engineer, and science in general. The universe is full of turbulence flows, as is the Earth’s atmosphere and ocean. Turbulence also causes the loss of efficiency of almost any machine built by engineers. It also drives mixing from stars to car engines to the cream in the coffee cup sitting next to my right hand.

There does exist a set of equations that does have the physical properties the incompressible equations lack, the compressible Navier-Stokes equations. The problem is that the Millennium prize doesn’t focus on this equation set, but rather it focuses upon the incompressible version. The question is whether or not going from compressible to incompressible has changed something essential about the equations, and if that is something that is essential to understanding turbulence. There is a broadly stated assumption about turbulence; it is contained in the solution of the incompressible Navier-Stokes equations (see for example the very first page of Frisch’s book “Turbulence: The Legacy of A. N. Kolmogorov”). In other words, it is contained inside an equation set that has undeniably unphysical aspects. Implicitly, the belief is that these details do not matter to turbulence. I counter that we really don’t understand turbulence well enough to make that leap.

Incompressible flow is a meaningful oft used approximation for very many engineering applications where the large-scale speed of the fluid flow is small. This parameter is the Mach number, and if the Mach number is low (less than 0.1 to 0.3) it is assumed that the flow can be taken to be incompressible. Incompressibility is a useful approximation that allows the practical solution of many problems. Turbulence is ubiquitous and particularly relevant in this low speed limit. For this reason scientists have believed that ignoring sound waves is reasonable and turbulence can be tackled with the incompressible approximation.

It is worth point out that turbulence is perhaps one of the most elusive topics known. It has defied progress for a century with only a trickle of advances. No single person has brought more understanding to bear on turbulence than the Russian scientist Kolmogorov. His work has established some very fundamental scaling laws that get right to the heart of the problem with incompressibility.

He established an analytical estimate that flows achieve at very high Reynolds numbers (Reynolds number is the ratio of inertial forces to dissipative forces), the 4/5 law. Basically this law implies that turbulence flows are dissipative and the rate of dissipation is not determined by the value of the viscosity. In other words, as the Reynolds number becomes large its precise value does not matter to the rate of dissipation. The implications of this are massive and get to the heart of the issue with incompressibility. This law implies that the flow has discontinuous dissipative solutions (gradients that become infinitely steep). These structures would be strongly analogous to shock waves where they would appear to step functions at large scales, and any transition would take place at infinitesimally small distances. These structures have eluded scientists both experimentally and mathematically. I believe part of the reason for evasion has been the insistence on incompressibility.

Compressible flows have no such problems. The flows physically and mathematically readily admit discontinuous solutions. It is not really a challenge to see this, and shock waves form naturally. Shock waves form at all Mach numbers, including the limit of zero Mach number. These structures actually dissipate energy at the same rate as Kolmogorov’s law would indicate (this was first observed by Hans Bethe in 1942, Kolmogorov’s proof occurred in 1941, there was no indication that they knew of each other’s work). It is worth looking at Bethe’s derivation closely. In the limit of zero Mach number, the compressible flow equations are dissipation free if the expansion is taken to second-order. It is only when the third-order term in the asymptotic expansion is considered that dissipation arises and shock looks different than an adiabatic smooth solution. The question is whether in taking the limit of incompressibility has removed the desired behavior from the Navier-Stokes equations.

I believe the answer is yes. More explicitly shock phenomena and turbulence are assumed to be largely independent except for more compressible flows where classical shocks are found. The question is whether the fundamental nature of shocks changes continuously as the Mach number goes to zero. We know that shocks continue to form all the way to a Mach number of zero. In that limit, the dissipation of energy is proportional to the third power of the jump in variables (velocity, density, pressure). This dependence matches the scaling associated with turbulence in the above-mentioned 4/5 law. For shocks, we know that the passage of any shock wave creates entropy in the flow. A question worth asking is “How does this bath of nonlinear sound waves leading to pressure act, and do these nonlinear features work together to produce the effects of turbulence?” This is a very difficult problem, and we may have made it more difficult by focusing on the wrong set of equations.

There have been several derivations of the incompressible Navier-Stokes equations from the compressible equations. These are called the zero-Mach number equations and are an asymptotic limit. Embedded in their derivation is the assumption that the equation of state for the fluid is adiabatic, no entropy is created. This is the key to the problem. Bethe’s result is that shocks are non-adiabatic. In the process of deriving the incompressible equations we have removed the most important behavior visa-vis turbulence, the dissipation of energy by purely inertial forces.

The problem with all the existing work I’ve looked at is that entropy creating compressible flows is not considered in the passage to the zero Mach limit. In the process one of the most interesting and significant aspects of a compressible fluid is removed because the approximation doesn’t go far enough. It is possible, or maybe even likely that these two phenomena are firmly connected. A rate of dissipation independent of viscosity is a profound aspect of fluid flows. We understand intrinsically how it arises from shock waves, and its presence with turbulence remains mysterious. It implies a singularity, a discontinuous derivative, which is exactly what a shock wave is. We have chosen to systematically remove this aspect of the equations from incompressible flows. Is it any wonder that the problem of turbulence is so hard to solve? It is worth thinking about.

Mukhtarbay Otelbayev of Kazakhstan claims that he has proved the Navier-Stokes existence and smoothness problem. I don’t know whether he has or not, I don’t have the mathematical chops to deliver that conclusion. What I’m asking is if he has, does it really matter to our understanding of real fluid dynamics? I think it might not matter either way

Choro Tukembaev

said:See

http://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/

ibid the comment Victor Ivrii, F.R.S.C., June 20, 2014

Bill Rider

said:Choro,

I am a regular reader of Terry’s blog, and find it to be exceptional with that article no exception. This point is subtle. Terry’s work on the incompressible Navier-Stokes could be perfect and utterly correct, but still say nothing about turbulence. Equations are an abstraction and their connection to reality in question. Really that is where I’m going. I’m questioning the validity of incompressibility for studying turbulence. One can still study the equations and even solve the Clay prize and do very little for turbulence if the equations are not the right ones. This is the core of the discussion not taking place, the one that is needed.

Choro Tukembaev

said:Omurov created a new method for solving sixth problems, including the N – dimensional case, which has been suggested by Professor Tao

http://terrytao.wordpress.com/2014/02/04/finite-time-blowup-for-an-averaged-three-dimensional-navier-stokes-equation/

Compressible fluid beyond the sixth problem. Turbulence begins to appear in an incompressible fluid, which was previously known in hydrodynamics (see unsolved classical problems of hydrodynamics). Omurov gave the solution of unsolved classical problems on the basis of a new method that can be found in his monograph

http://literatura.kg/articles/?aid=2030

http://literatura.kg/articles/?aid=2042

Bill Rider

said:Thanks. I don’t believe any of this addresses the relevance of this problem to actual fluid dynamics.

Choro Tukembaev

said:If we substitute the solution into the Navier-Stokes equations, we get an identity (The solution of Omurov – Navier-Stokes equations – identity). The real fluid is studied in the form of the equations of motion and the equations of divergence. Many of the parameters of a real fluid is discarded, so it is necessary to solve the Navier-Stokes equations. Turbulence – this is another urgent problem, which is associated with a compressible fluid. Let announced prize of one million dollars for the turbulence, then it will be necessary to solve a new problem. Poem (in Russian) decorate your life

Математика – царица наук,

Вдохновенье – мать творчества,

Без него математик, как паук,

Неведающий зодчества,

Эх, страдалец одиночества!

A mathematician who’s not also a little bit a poet, will never be a perfect mathematician (Karl Weierstrass).

Bill Rider

said:Care to translate?

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Jed Brown

said:To further illustrate how the Millennium Problem is a game divorced from reality, recall Giovanni Prouse’s 1979 well-posedness result for a variational inequality respecting the speed of light. Nobody cared. Franco Brezzi put it well in this lively letter rejecting Claes Johnson’s (outrageous) paper submitted to M3AS:

http://www.nada.kth.se/~jhoffman/pmwiki/pmwiki.php?n=M3as.Brezzitojohnson

Choro Tukembaev

said:Existence and uniqueness theorem is proved (MR0621026: Prouse, Giovanni) for the speed of light. Hence, the Navier-Stokes equations are valid in the relativistic region, and not only for the speed of sound. However, it is a known fact! This occurs because the Navier-Stokes equations are derived from the conservation laws.

Choro Tukembaev

said:In classical hydrodynamics movement is determined by the speed of sound. Your counterexample applies to electromagnetic waves, and therefore, it’s the other equations. In order to accelerate the body to the speed of light, it is necessary to overcome the strength of intermolecular bonding (van der Waals). Then the volume will start to disintegrate into particles (molecules and ions, ie splashes of champagne, figuratively speaking), which will radiate electromagnetic waves at a characteristic wavelength, depending on the speed and chemical properties of the particles. That’s what happens when the speed of sound is replaced by the speed of light. The solution of the Navier-Stokes want to substitute on splash of champagne. But where is the fluid? Liquid long lost, when the speed of sound is replaced by the speed of light. Best master of such illusions is David Copperfield.

Choro Tukembaev

said:See

http://www.businessinsider.com.au/navier-stokes-millennium-prize-problems-2014-10

http://terrytao.wordpress.com/2014/02/04/finite-time-blowup-for-an-averaged-three-dimensional-navier-stokes-equation/

http://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/

Choro Tukembaev

said:In connection with the sixth problem of the Millennium, as is known, statement of the problem was given by Charles Fefferman, please pay attention to the article

T. D. Omurov, Existence and uniqueness of a solution of the nD Navier-Stokes equation, Advances and Applications in Fluid Mechanics 19(3), 589-604 (July 2016)

A fundamental problem in mathematics is to decide whether such smooth, physically reasonable solutions exist for the Navier-Stokes equations. We restrict attention here to incompressible fluids filling all of Rn. The Cauchy problem for the nD Navier-Stokes equations is reduced to the integral equations of Volterra and Volterra-Abel, investigation of which, allows us to solve positively question on uniqueness and smoothness of the solution.

See

http://www.pphmj.com/abstract/10041.htm

With kind regards,

Choro Tukembaev.

Bill Rider

said:The key assumption no one ever addresses is the viability of incompressibility as an appropriate physical approximation. For turbulence I suspect it is not.

Choro Tukembaev

said:We prove the existence of quasi-smooth solution on time and on the basis of the uniqueness of the proposed method. This transformation not only linearizes inertial members, and allows you to find the connection between speed and pressure. On the other hand, the proposed method transforms study the problem in the system of operator equations in the integral form. The purpose of Fefferman – indicate solvability in the general formulation for an incompressible fluid. Problem for turbulent solutions as viscosity when small-parameter (solutions prove behavior when setting the viscosity tends to zero), this is not required. As is known, if the studied problem is reduced to the integral form, classifying the viscosity parameter, you can build a critical and not critical decision on the basis of integral equations in different spaces. Therefore, in paragraph 3 of Article considered viscosity as a small parameter, which is used in turbulence research. This is done in [8] for the 3D-equations: see list of references in the article.

Choro Tukembaev

said:The article

T. D. Omurov, Existence and uniqueness of a solution of the nD Navier-Stokes equation, Advances and Applications in Fluid Mechanics 19(3), 589-604 (July 2016)

is available (is free) on the website

http://www.literatura.kg/uploads/existence_and_uniqueness_of_a_solution_of_the_nd_navier-stokes_equation.pdf

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Choro Tukembaev

said:For 3D Euler equations the answer see

T.D. Omurov, (2013), Navier-Stokes problem for Incompressible fluid with viscosity // Varia Informatica 2013. Ed. M. Milosz, PIPS Polish Lublin, pp.137-158.

Read pages 147-151. There assessed the proximity of solutions of the Euler and NS when the viscosity tends to zero.

Choro Tukembaev

said:To understand the singularity, see also for 3D Euler equations see

T.D. Omurov, (2013), Navier-Stokes problem for Incompressible fluid with viscosity // Varia Informatica 2013. Ed. M. Milosz, PIPS Polish Lublin, pp.137-158.

Read pages 147-151. There assessed the proximity of solutions of the Euler and NS when the viscosity tends to zero.