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

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