2nd Law for Cosmology

A mathematical model of the Universe can take the form of Euler's equations for a gas supplemented with Newton's law of gravitation as stated in Chap 32 Cosmology of Computational Thermodynamics.  

Computational solutions of these equations satisfy the following evolution equations as laws of thermodynamics depending on time $t$ 

• $\dot K(t)=W(t)-D(t)-\dot\Phi (t)$     (1)
• $\dot E(t)=-W(t)+D(t)$,                  (2)

where $K(t)$ is total kinetic energy, $E(t)$ total internal energy (heat energy), $W(t)$ is total work, $D(t)\ge 0$ is total turbulent dissipation, $\Phi (t)$ is total gravitational energy and the dot signifies differentiation with respect to time. Adding (1) and (2) gives the following total energy balance:

• $K(t)+E(t)-\Phi(t)= constant.$          (3)

Further (1) and (2) express an irreversible transfer of energy from kinetic to internal energy with $D(t)>0$, and so serve as a 2nd Law for Cosmology giving time a direction. Recall that the theoretical challenge is to tell/show why turbulent dissipation is unavoidable. 

Computations may start from a hot dense state at $t=0$ which is seen to expand/cool (run code) (Big Bang) to maximal size and then contract/warm back to a hot dense state (Big Crunch) (run code) in an irreversible sequence of expansions/contractions until some final stationary equilibrium state with $E(\infty )=P(\infty )$. Compare with post from 2011.

The 2nd Law in a World of Finite Precision

Let there be a World of Finite Precision.

Here is a summary of aspects of the 2nd Law of Thermodynamics discussed in recent posts: 

• 2nd Law gives an arrow of time or direction of time. 
• A dissipative system satisfies a 2nd Law.
• A dissipative system contains a diffusion mechanism decreasing sharp gradients by averaging. 
• Averaging is irreversible since an average does not display how it was formed. 
• Averaging/diffusion destroys ordered structure/information irreversibly. 
• Key example: Destruction of large scale ordered kinetic energy into small scale unordered kinetic energy as heat energy in turbulent viscous dissipation.    

To describe the World, it is not sufficient to describe dissipative destruction, since also processes of construction are present. These are processes of emergence where structures like waves and vortices with velocity gradients are formed in fluids, solid ordered structures are formed by crystallisation and living organisms develop. 

The World then appears as combat between anabolism as building of ordered structure and metabolism as destruction of ordered structure into unordered heat energy. 

The 2nd Law states that destruction cannot be avoided. Perpetual motion is impossible. There will always be some friction/viscosity/averaging present which makes real physical processes irreversible with an arrow of time. 

The key question is now why some form of friction/viscosity/averaging cannot be avoided? There is no good answer in classical mathematical physics, because it assumes infinite precision and with infinite precision there is no need to form averages since all details can be kept. In other words, in a World of Infinite Precision there would be no 2nd Law stating unavoidable irreversibility, but its existence would not be guaranteed.  

But the World appears to exist and then satisfy a 2nd Law and so we are led to an idea of an Analog World of Finite Precision, which possible can be mimicked by a Digital World of Finite Precision (while a possibly non-existing World of infinite precision cannot). 

The Navier-Stokes equation for a fluid/gas with positive viscosity as well as Boltzmann's equations for a dilute gas are dissipative systems satisfying a 2nd Law with positive dissipation. But why positive viscosity? Why positive dissipation?

The Euler equations describe a fluid with zero viscosity, which formally in infinite precision is a system without dissipation violating the 2nd Law.  

We are led to consider the Euler equations in Finite Precision, which we approach by digital computation to find that computational solutions are turbulent with positive turbulent dissipation independent of mesh size/precision once sufficiently small. We understand that the presence of viscosity/dissipation is the result of a necessary averaging to avoid the flow to blow-up from increasing large velocity gradients emerging form convection mixing high and low speed flow. 

We thus explain the emergence of positive viscosity in a system with formally zero viscosity as a necessary mechanism to allow the system to continue to exist in time. 

The 2nd Law thus appears as being a mathematical necessity in an existing World of Finite Precision.   

The mathematical details of this scenario in the setting of Euler's equations id described in the books Computational Turbulent Incompressible Flow, Computational Thermodynamics and Euler Right.

PS A related question is if in a World of Very Limited Human Intelligence a WW3 destruction is necessary?

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

Physical 2nd Law Without Statistics and Entropy

Why is there a 2nd Law of Thermodynamics?

In the book Computational Thermodynamics a 2nd Law of Thermodynamics is formulated as follows in the setting of Euler's equations for a compressible fluid/gas with vanishing viscosity (with quantities integrated in space): 

• $\frac{dK}{dt} - W=Q\ge 0$,               (2nd Law)

where $K$ is kinetic energy, $W$ is mechanical work of variable sign (positive in expansion) and $Q>0$ is turbulent dissipation as a positive quantity adding to internal heat energy. In this formulation all quantities involved have physical meaning and no notion of entropy of unclear physical meaning is present (and so is not needed). Neither is any statistics involved. 

The 2nd Law states a limit to transformation between kinetic energy and work with turbulent dissipation appearing as a loss in the form of heat energy, which according to a stability analysis in the book, always is present locally in space and time. The loss is thus inevitable. The loss is also irreversible, since time reversal violates the sign of (2nd Law) and so gives an arrow of time. Kinetic energy once converted to heat energy cannot be retrieved as an expression of the term internal energy. 

The above formulation of the 2nd Law closely connects to classical formulations preceding that of Boltzmann, who introduced statistics and entropy. A new aspect is that it is based on Euler's equations as a precise mathematical model, in particular on computational solution in a certain precise weak-strong sense reflecting that strong solutions do not exist. This gives the 2nd Law a precise computational mathematical meaning in the presence of finite precision, which can be interpreted in physical terms so can be referred to as a Physical 2nd Law. 

The reason that heat energy as micro-scale unordered kinetic energy once created in turbulent dissipation, cannot be retrieved into macro -cale ordered kinetic energy, is that it requires a very high precision which cannot be met with finite precision physics/computation.  

The Physical 2nd Law thus appears as a resolution in the digital age of a basic problem of physics, which could not be resolved within classical analytical mathematics nor statistics. 

This post connects to earlier posts on Wolfram's recently presented resolution also based on computation but in fundamentally different form. 

Modern physicists have since long left the 2nd Law behind as a trivial no-problem not asking for any resolution although it has been an outstanding open problem of physics, and have so proceeded to new orchards of string theory and multi-versa, which however have not delivered any fruits and so a return to basics could possibly be of some interest to todays fundamental physicists, or not?

Why Euler CFD is a Zero-Cost Parameter Free ToE

The fact that for slightly viscous incompressible bluff body flow with Reynolds number $Re>500.000$, Euler CFD with slip boundary condition (without boundary layers) serves as a parameter-free essentially zero-cost computational model without dependence on Reynolds number as a Theory of Everything ToE in the spirit of Einstein, depends on two key circumstances:

1. Finite rate of turbulent dissipation effectively independent of $Re$ after transition to turbulence (e.g $Re >200$ in isotropic turbulence) (reference).
2. Slip serving as effective boundary condition for $Re > 500.000$ (NACA0012 previous post).

Here 1. reflects Kolmogorov's conjecture and means that a mesh size of around 1/200 of gross dimension is sufficient to capture turbulence. Further, 2. reflects that Euler CFD with slip can capture drag and lift with little dependence on $Re$ beyond drag crisis around $Re=500.000$.  

Altogether, Euler CFD can capture drag and lift beyond drag crisis with a mesh size of around 1/200 of gross dimension thus at essentially zero computational cost (because no thin boundary layers have to be resolved). 

The fact the drag and lift coefficients do not include dependence on $Re$ (with $Re =\frac{UL}{\nu}$,  $U$ typical flow speed, $L$ typical length scale and $\nu$ typical viscosity) and yet can serve as measures of drag and lift for $Re$ beyond drag crisis, gives observational evidence that indeed drag and lift have a very weak dependence on $Re$ beyond drag crisis, as shown in the previous post (see reference showing drag independence for $Re>10000$ for a collection of blunt bodies).  Also recall that drag and rate of turbulent dissipation balance, and so observed independence of drag beyond drag crisis supports 1.   

   

Euler CFD as Parameter Free CFD as ToE

The Euler equations in velocity-pressure $(u,p)$ and $(x,t)$-coordinates are invariant under a rescaling of velocity $u$ into $\bar u =\frac{u}{U}$ with $U$ a reference speed such as free stream speed in bluff body flow with corresponding rescaling of pressure $p$ into $\bar p=\frac{p}{U^2}$ and time $t$ into $\bar t =Ut$ without rescaling of space with thus $\bar x = x$. The scaling of pressure with $U^2$ conforms with Bernoulli's Law and the scaling of drag force $\sim C_DU^2$ from a drag coefficient $C_D$. The propulsion power to balance drag thus scales with $U^3$. The Euler equations are thus formally invariant under change of velocity scale as an expression of formally zero viscosity or infinite Reynolds number.

The basic energy estimate of Euler CFD expresses a balance between rate of loss of kinetic energy and computational residual-based turbulent dissipation of the basic simplified form $C\frac{h}{\vert u\vert}|\vert u\cdot\nabla u\vert^2$ both scaling with $u^3$. The propulsion power is balanced by the rate of loss of kinetic energy and so by turbulent dissipation. The drag coefficient can thus alternatively be computed from total turbulent dissipation.

The (remarkable) fact that the drag coefficient $C_D$ does not include dependence of the Reynolds number $Re$, expresses observations that drag depends little on $Re$ beyond drag crisis, which connects to Kolmogorov's conjecture of finite limit of turbulent dissipation as well as mesh and stabilisation independence in computation. The functionality of the drag coefficient supports Euler's Dream that Euler CFD offers a Theory of Everything ToE for slightly viscous incompressible flow with independence of $Re$ beyond drag crisis. Since total drag shows little dependence on $Re$ while in principle it has a contribution from skin friction with a skin friction coefficient (scaling with $U^2$) decreasing with $Re$, the skin friction contribution appears to be small, in contradiction to a common conception of major contribution: If major drag indeed would come from skin friction, then drag would decrease with increasing $Re$, but it does not beyond drag crisis.

Notice that the Navier-Stokes equations with constant viscosity $\nu$ with turbulent dissipation intensity $\nu\vert\nabla u\vert^2$ scaling with $u^2$, are not velocity scale invariant and thus carry a dependence on $Re$ possibly making computational solution impossible for large $Re$. 

Recall that the definition of $Re =\frac{UL}{\nu}$ with $U$ a reference speed and $L$ a reference length and $\nu$ a viscosity is not well determined and so independence of mean value quantities such as drag, lift and pitch moment is a necessary requirement to make CFD predictable.

Here is experimental evidence that $C_D$ for NACA0012 at zero angle of attack does not depend on $Re$ beyond drag crisis:

Notice the reduction of $C_D$ by a factor 2 from $Re =100.00$ to $Re > 500.00$ as an expression of drag crisis.

Euler Was Right, Prandtl Was Wrong II

I am working on a new article to be expanded to a book with the title Euler Was Right, Prandtl Was Wrong  which can be seen as a summary of my work on fluid dynamics for 30 years together with former students Johan Hoffman, Johan Jansson and Anders Szepessy. In short, our work shows that the following prophetic declaration by Euler from 1755, indeed is fully correct:

• My two equations contain all of the theory of fluid mechanics. It is not the principles of mechanics we lack to pursue this analysis but only Analysis (computation), which is not sufficiently developed for this purpose...We have to wait until the age of the computer to solve the equations.

And yes, we now live in the age of the computer and then Euler's two equations as a parameter-free model can be solved in the form of Euler CFD (Computational Fluid Dynamics) and so open a whole new world of turbulent flow to prediction, analysis and control, without any further need of mathematical modeling with parameter fitting. 

Euler CFD is to be compared with Prandtl CFD as the Standard CFD developed during the 20th century based on Prandtl's boundary layer theory including complicated wall and turbulence models with many parameters, which does not offer true predictive computation, as the legacy of the declared Father of Modern Fluid Mechanics. 

Take look and see what you think. This post directly connects to the discussion in recent posts with Doug McLean representing Standard CFD. See also previous post.

A ToE for Fluid Mechanics

Einsteins ideal as a Theory of Everything ToE is a mathematical model of physics without any parameters. 

The standard model of particle physics contains 18 parameters. It is a very complicated model. To determine the parameters experimentally is impossible.

The standard model of isotropic linear elasticity contains 2 parameters. This is a very simple model but for a non- isotropic body the number of parameters includes 18 parameters. 

To be a useful model the values of its parameters must be supplied as input determined from experiments or more basic model, which in general is very difficult. The 2 parameters of isotropic linear elasticity can be determined from simple tests, but the 18 parameters for non-isotropic linear elasticity are difficult to determine, not to speak of non-linear elasticity and all the parameters of the standard model. 

Are there any parameter-free models of physics? A basic example is a circle described as the set of points in a plane with a certain distance to a given mid-point from which the value of Pi can be computed as the quotient between circumference and diameter. That is a very simple model. Is there any model of more complex physics which is parameter-free? 

Yes, there is one, and maybe this is the only one: Euler's equations for incompressible fluid flow are expressed in terms of velocity and pressure without any parameter: Input is geometry, in/out-flow conditions and external forces, but no parameter, since viscosity is set to zero.   

The remarkable thing is now that the drag and lift of a body moving through a slightly viscous fluid like air and water can accurately be predicted by computing turbulent solutions to the Euler equations with only geometry of the body as input. This is like computing the ratio of circumference/diameter of a circle (that is computing Pi), but just more astounding. Drag and lift coefficients (scaling with $speed^2$) of a body only depend on the geometry of the body! No parameter input needed! See Computational Turbulent Incompressible Flow and Breakthrough of predictive simulation.

The Euler equations for incompressible flow is a ToE for slightly viscous incompressible flow like air (subsonic) and water.  This is remarkable. Is this is the only ToE in physics.

Well, Newton's law of gravitation contains the gravitational constant G connecting gravitational force to mass as parameter, but may be viewed as a ToE in the sense of correctly predicting that all bodies independent of composition move the same way subject to gravitation. 

PS Von Neuman famously claimed that he (in principle) could model an elephant with 4 parameters, and make it wiggle its trunk with a 5th, but in practice how would he determine the parameters?  Elephant experiments are costly and cumbersome.

Euler Was Right, Prandtl Was Wrong I

Euler vs Prandtl

In 1755 the great mathematician Euler formulated the Euler equations for slightly viscous nearly incompressible flow (of air and water) with the following prophetic declaration:

• My two equations contain all what is contained in the theory of fluid mechanics. It is not the principles of mechanics we lack to pursue this analysis but only Analysis (computation), which is not sufficiently developed for this purpose.  

Euler's equations are formulated in terms of fluid velocity and fluid pressure depending on space and time as an expression of force balance (Newton's 2nd Law) and incompressibility complemented by a slip boundary condition with only pressure forces from a solid wall meeting the fluid, that is, with zero skin friction allowing the tangential flow velocity to be non-zero restricting only the normal flow velocity to be zero on a wall.  Euler's equations are parameter-free (formally zero viscosity), thus meeting Einstein's ideal of a mathematical model. The only force acting on fluid particles is pressure and shear forces are assumed to be negligible.  Euler made the assumption about zero skin friction from experiments showing very small skin friction in slightly viscous flow with massive evidence in modern times. 

Eulers adversary d'Alembert quickly crushed Euler's grand plan by showing that Euler's equations admitted certain solutions (potential solutions) showing zero net forces (drag, lift) of a body moving through air or water, in direct contradiction to observation. This was coined d'Alembert's Paradox which from start, as expressed by Chemistry Nobel Laureate Hinshelwood:

• separated practical fluid mechanics (hydraulics) describing phenomena (drag, lift), which cannot be explained, from theoretical fluid mechanics explaining phenomena (zero drag, lift), which cannot be observed.       

Zero lift is incompatible with flight and so d'Alembert's Paradox had to be resolved, in particular after powered human flight was shown to be possible by the Wright brothers in 1903, and so the young fluid mechanician Prandtl presented a resolution in a sketchy 8-page conference contribution in 1904, where he discriminated potential flow with zero skin friction claiming that a real fluid always meets a solid wall with zero tangential velocity named no-slip.  Prandtl thus "resolved" d'Alembert's Paradox by declaring that Euler's equations with slip had to be replaced by the Navier-Stokes equations including small viscosity and no-slip. But no-slip was an ad hoc assumption which Prandtl could not justify since the exact nature of the microscopic contact between fluid and wall was unknown to him and so has remained into our days. 

Prandtl in 1904 with his self-built fluid test channel resolving d'Alembert's Paradox.

Anyway, the scientific community was by Prandtl relieved from a main headache making theory of fluid mechanics into a joke and accordingly Prandtl was named Father of Modern Fluid Mechanics based on the Navier-Stokes equations with no-slip and not Euler's equations with slip. 

But there was one main caveat: The Navier-Stokes equations with no-slip have solutions with boundary layers so thin that computational resolution is impossible with any forseeable computational power.  Prandtl's resolution thus came with the cost of making Computational Fluid Dynamics CFD into an impossibility asking for resolution of atomistic scales in a macroscopic setting.

In 2010, Hoffman and Johnson published in Journal of Mathematical Fluid Mechanics a different resolution of d'Alembert's paradox showing that the reason zero-drag/lift of potential flow cannot observed, is that potential flow (in fact any laminar flow) is unstable and thus turns into turbulent flow. This was shown by computing turbulent solutions to Eulers equations with slip with drag and lift in close correspondence to observations supported by stability analysis, as exposed in detail in the book Computational Turbulent Incompressible Flow. As a spin off a New Theory of Flight was developed revealing the true Secret of Flight in physical terms, very different from the unphysical lifting line theory advocated by Prandtl. 

Since then massive evidence has been accumulated by Johan Jansson showing that computing turbulent solutions of Euler's equations with slip opens basically all of slightly viscous nearly incompressible flow to predictive simulation without parameter input and need to resolve thin no-slip boundary layers, thus with readily available computing power, all along Euler's prophecy. More evidence: HighLift Workshop.

Euler was thus right, and he understood that he just had to wait for computing power to see his prophecy become true. It took 250 years, but now it is here.

It means that Prandtl was wrong claiming drag and lift to be effects of thin no-slip boundary layers thereby making CFD into an impossibility. 

Question

How will the fluid dynamics community react to replacing Prandtl by Euler as Father of Modern Fluid Mechanics thus changing CFD from impossible to possible? 

Further Important Facts

Turbulent solutions to Euler's equations are computed as best possible approximate solutions in the sense of having residuals which are small in a weak sense and not too large in a strong sense, in a situation when all solutions with small residual in a strong sense (laminar solutions) are unstable and do not persist over time. We thus face a new situation where only turbulent flow is computable and laminar not, as an expression of the fluctuating nature of turbulence, as seen in a waving flag showing the only motion which can persist. The control of the residual in strong sense introduces a viscous effect as a form of turbulent viscosity set by computation alone without need to model or measure turbulent viscosity beyond human comprehension.  

Euler was a mathematician while Prandtl as Father of Modern Fluid Mechanics was more of an engineer. Replacing Prandtl by Euler means freeing the full power of mathematics with computation in a rare example of parameter-free mathematical model with very rich applicability.

Standard CFD under a Planck dictate of no-slip has developed complicated wall models as well as turbulence models including many parameters, and an agreement has been made to adjust parameters to give  50% or more of total drag to skin friction. Turbulent Euler computations with zero skin friction show correct drag in a large variety of situations, which is incompatible with the 50% skin friction from standard CFD.

Total drag consists of pressure drag and skin friction drag. Turbulent Euler computations show that pressure drag dominates skin friction by a factor of at least 10, and so standard CFD claiming 50% skin friction must underestimate pressure drag by a factor 2. The CFD community is now wrestling under this contradiction. The investments in standard CFD are huge and will loose their value if Euler is allowed to take over from Prandtl...Compare with posts on Prandtl Medal.

Incompressible flow is well captured by the Euler equations  for Reynolds numbers (scaling with 1/viscosity) larger than about 500.000 associated with the so called drag crisis when drag of a bluff body drastically decreases with a factor 2-3 as the boundary condition effectively turns into slip from limited velocity strains, with late separation and small wake of low pressure, in particular with lift/drag around 15 for a wing allowing flight at affordable power.

Euler vs Navier-Stokes: What is viscosity?

The Navier-Stokes equations connect fluid velocity strains (derivatives in space) with shear forces through a positive coefficient of viscosity $\nu$ as a parameter to be supplied as input, assumed to be constant independent of fluid velocity in the basic case, but in general with a very complex unknown non-linear dependence on local flow velocities. Formally $\nu =0$ in the parameter-free Euler's equations.

In slightly viscous flow the coefficient of viscosity is small with a Reynolds number $Re = \frac{UL}{\nu}$ beyond drag crisis (bigger than 100.000- 500.000) with $U$ typical flow speed and $L$ typical spatial scale L. 

The Navier-Stokes equations can be complemented by a (skin) friction boundary condition with a friction parameter $\beta$ connecting (tangential) shear stress to tangential flow velocity, with slip corresponding to $\beta =0$ and effective no-slip for $\beta >1$, thus covering a range from slip to no-slip with important effects on flow separation and drag (as exposed in Computational Turbulent Incompressible Flow). 

To determine the viscosity as input to the Navier-Stokes equation experimentally or theoretically has shown to be virtually impossible in the case of slightly viscous flow, which is always partially turbulent with a very complex expression of viscosity. Using Navier-Stokes equations for true prediction of slightly viscous flow has not been shown to be possible. With parameter fitting in viscosity models standard CFD can match measured drag, but generally fail in blind tests without prior knowledge of the correct value to match.

Computing turbulent solution to the Euler equations includes automatic modeling of viscosity

through weighted strong residual control as a dissipative effect with a complex flow dependence beyond viscous shear stress.  It appears as a solution to the open problem of turbulence modeling. In particular, size of the strong residual measures the turbulent dissipation as a mesh independent quantity meeting Kolmogorov's conjecture. 

The Navier-Stokes equation model (1823) with constant positive viscosity is generally viewed to be a better/more complete model then the Euler equations (1755) with formally zero viscosity. This was picked up by Prandtl in 1904 using in particular no-slip from the presence of positive viscosity as a way to discriminate potential flow and get around d'Alembert's paradox. But the more complete model showed to be boundary layer uncomputable and asking for parameter input and so non-predictive, while the basic Euler model showed to be more useful by being both computable (no boundary layers) and predictive as parameter free. 

The ultimate quest for a physicist is to find a Theory of Everything ToE as a parameter free model explaining all of basic physics. Computing turbulent solutions to the Euler equations is a ToE for fluid mechanics. 

Spiral Galaxy Formation in Extended Newtonian Gravitation

1. Cosmological Model 

This is a continuation of previous posts on dark matter and The Universe as Weakly Compressible Gas subject to Pressure and Gravitational Forces, which post we recall:

We consider a cosmological model in the form of Euler's equations for a compressible gas subject to Newtonian gravitation: Find $(\rho ,m, e ,\phi ,p)$ depending on a Euclidean space coordinate $x$ and time $t$, such that for all $(x,t)$:

• $\dot\rho + \nabla\cdot (\rho u ) =0$       (or $\frac{D\rho}{Dt} = -\rho\nabla\cdot u$)
• $\dot m +\nabla\cdot (mu) +\nabla p + \rho\nabla\phi =0$
• $\dot e +\nabla\cdot (eu) +p\nabla\cdot u +\rho\nabla\cdot m=0$,

where $\rho$ is mass density, $u=\frac{m}{\rho}$ is matter velocity, $p$ is pressure, $\phi$ is gravitational potential, and $e$ is internal energy as the sum of heat energy $\rho T$ with $T$ temperature and gravitational energy $\rho\phi$, and the dot indicates time differentiation and

• $\frac{D\rho}{Dt}=\dot\rho +u\cdot\nabla\rho$

is the convective time derivative of $\rho$, see Many-Minds Relativity 20.3 and Computational Thermodynamics Chap 32.

These equations express conservation of mass $\rho$, conservation of momentum $m$ with $\nabla p$ pressure force and $-\nabla\phi$ gravitational force, and conservation of internal energy $e$. These laws of conservation are complemented with constitutive laws connection $p$ and $\phi$ to density, of the following form:

A1: Weakly compressible gas ($\delta$ small positive constant):

• $\Delta p =\frac{\nabla\cdot u}{\delta}= - \frac{1}{\delta\rho}\frac{D\rho}{Dt}$

or

A2: Compressible perfect gas ($0 < \gamma < 1 $):

• $p=\gamma \rho T$.

B: Newton's law of gravitation:

• $\Delta\phi =\rho$ with $\phi =0$ at infinity.            

We observe

1. Similarity of $\nabla p$ and $\nabla\phi$ in momentum equation. 
2. Similarity between A1 and B connecting $\Delta p$ to $-\frac{D\rho}{Dt}$ (or $-\rho$) and $\Delta\phi$ to $\rho$.
3. $p \ge 0$ and $\phi \le 0$.

Here 1. can be seen as the Equivalence Principle (equality of heavy and inertial mass) expressing that there is no difference between gravitational and other forces (pressure) in Newton's 2nd law expressing conservation of momentum.

Further, 2. expresses that the constitutive laws A1 and B both can be viewed as action at distance if $\rho$ is viewed as the cause, but represent local action of differentiation if $\rho$ is viewed as the effect. 

For a weakly compressible gas described by A1, there is no need per se to identify a cause-effect relation between $p$ and $\rho$; it is enough to say that $p$ and $\rho$ are connected in a certain way expressing a form of "perfect harmony". 

In the same way, there is no need per se to identify a cause-effect relation between $\phi$ and $\rho$; it is enough to say that $\phi$ and $\rho$ are connected in certain way expressing a form of  "perfect harmony" in the spirit of Leibniz.

The relation $\Delta\phi =\rho$ is explored in Newtonian Matter and Antimatter with $\Delta\phi > 0$ identifying matter and $\Delta\phi < 0$ antimatter, with dark matter where $\Delta\phi$ is smooth and visible matter where $\Delta\phi$ is singular, typically as a sum of multiples of delta functions representing matter in point form.  We refer to such a model as Extended Newtonian Gravitation. 

2. Galaxy Formation

We start from a spherical distribution of matter of low density of dark matter (a halo) with $\Delta\phi$ a smooth function, which we assume to be in static equilibrium with the the gravitational force balanced by a weak pressure force with $\nabla p = - \rho\nabla\phi$. 

Starting from this halo of low density dark matter, we assume that some visible matter (stars) is formed by concentration of dark matter by gravitational attraction into point masses with $\rho$ becoming large locally with the result that the gravitational force $\rho\nabla\phi$ can no longer be balanced by a weak pressure force $-\nabla p$. This is an effect of the different action of pressure and gravitational force, with pressure scaling with surface and gravitational force with volume.

The combined effect of the presence of a halo of dark matter and gravitational collapse of visible matter as a system of point masses, may then create a spiral galaxy of visible matter surrounded by a halo of dark matter, which is the standard view of the nature of a spiral galaxy, with in particular a characteristic distribution of velocity of visible matter as roughly independent of the distance to the galaxy center as an effect of the dark matter halo. 

It thus appears that an extended Newtonian model with $\Delta\phi$ of variable sign and concentration may be sufficient to explain essential aspects of galaxy formation, for which Einstein's equation equation is useless.   

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.

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.