20

Summary of Mathematical Aspects

My feeling is that many mathematicians and graduate students are intrigued
by what they hear about problems of the mechanics of an incompressible
fluid, but don’t study them because they don’t know enough physics and
fear to make fools of themselves. What they don’t know, I think, is how
abysmally little we actually know about fluids, and how it would be hard to
act more the fool than many have already done. I believe the difficulty arises
first, over an inflated nomenclature that burdens the subject and, second,
over a lack of understanding about how ignorant we can be in our technolog-
ical society and how close to the surface many problems lie.... In spite of the
profound mathematical methods we use to attack the problems, we know
very little about fluids, we can tell the physicist almost nothing of what he
wants to know, and interesting problems abound. (Marwin Shinbrot, 1973)

Our consciousness does not reflect the molecular chaos of the phenomena
but exerts an integrating function with respect both space and time, from
results the apparent homogeneity and continuity of the phenomena. (Weyl)

Blind fate could never make all the planets move one and the same way in
orbs concentric. (Newton)

20.1 Outputs of
-weak Solutions

We have introduced the concept of
-weak solutions to the NS equations. To es-
timate the difference in output of two
-weak solutions ˆ

u and ˆ

w, M (ˆ

u)

−M( ˆ

w),

with M (ˆ

u)

≡ ((ˆu, ˆ

ψ)) defined by a function ˆ

ψ, we introduced a linearized dual

problem with coefficients depending on u and w and estimated derivatives
of the solution of the dual problem in a corresponding stability factor S



( ˆ

ψ),

to get

|M(ˆu) − M( ˆ

w)

| ≤ 2
S



( ˆ

ψ).

We next noted that a G2-solution ˆ

U is an C

U

hR( ˆ

U )

-weak solution, and

this way we obtained an a posteriori output error estimate of the form

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20 Summary of Mathematical Aspects

|M(ˆu) − M( ˆ

U )

| ≤ (
+ C

U

hR( ˆ

U )

)S



G2

( ˆ

ψ),

with S



G2

( ˆ

ψ) a corresponding stability factor, and

G2

= C

U

hR( ˆ

U )

. Sim-

plifying, assuming
small (and C

U

≤ 1), the a posteriori error estimate took

the form

|M(ˆu) − M( ˆ

w)

| ≤ hR( ˆ

U )

S

0

( ˆ

ψ),

(20.1)

and the corresponding stopping criterion was

hR( ˆ

U )

S

0

( ˆ

ψ)

≤ T OL.

If the stability factor S

0

( ˆ

ψ) is not too large and the tolerance T OL not too

small, then we may be able to reach the stopping criterion with available
computer power.

We have pointed out a basic feature of the a posteriori error estimate result-

ing from the properties of G2, namely the presence of the factor h multiplying
the residual R( ˆ

U ). If S

0

( ˆ

ψ) is not too large, this means that we may reach the

stopping criterion without the residual R( ˆ

U ) being pointwise small. We may

thus compute an accurate mean value output from a discrete solution with
a pointwise large residual. In a turbulent flow we may expect (and actually
see in computations) that pointwise R( ˆ

U )

∼ h

−1/2

. This evidence strongly

indicates that the mere idea of a pointwise solution to a turbulent flow will
have to be refuted. As already pointed out above, this is in direct opposition
to the Clay Institute formulation of its Prize Problem concerning existence,
regularity and uniqueness of pointwise solutions to the NS equations.

20.2 Chaos and Turbulence

We have been led to the following essential aspects of a dynamical system with
chaotic solutions such as the NS equations: (i) strong sensitivity of pointwise
outputs, (ii) weak sensitivity of mean value outputs, and (iii) weak sensitivity
of stability factors.

To identify these features for a given dynamical system, we would first

compute one trajectory u(t) pointwise. We would then solve the corresponding
dual problem linearized at u(t) with data corresponding to pointwise output
to find a large stability factor, and with data corresponding to a mean value
output to find a stability factor which is not large. This would give evidence
of (i) and (ii). In particular we would get the information that the mean value
output would be insensitive to solution perturbations, and thus that we could
expect to be able to compute the mean value output from only one solution
trajectory.

There would be one piece of information missing, namely (iii) which rep-

resents insensitivity of the mean value stability factor to the choice of solution
trajectory underlying the linearization in the dual problem. To get evidence
of this insensitivity, we would have to compute a couple of different solutions

20.4 Irreversibility

171

u(t) by introducing some perturbations and then solve the corresponding dual
problems. The evidence would then be that the corresponding stability fac-
tors would be insensitive to the perturbations. In particular, we would get the
signal that the more precise nature of the perturbations would be insignificant.

In this book we present evidence that turbulent flow has the features

(i)-(iii) and thus carries the basic features of the type of chaos we suggest
above. The result is that a mean-value output may be observable/computable
to a tolerance of interest under statistical perturbations of input of unknown
nature, while a point value is not.

In a turbulent flow a lot of detailed information is destroyed in dissipation,

which thermodynamically connects to a substantial increase of entropy. In
order for a mean value in turbulent flow to be well defined, it cannot have
other than a weak dependence on the destroyed information, and indeed we
observe this to be a real phenomenon since we find mean value aspects of
turbulent flow to be computable without resolving all details of the flow. Thus
certain aspects of turbulent flows may be computable, in fact, sometimes more
easily computable than laminar flows, which may show a stronger dependence
on details.

This is in contrast to a conventional standpoint, where turbulent flow may

seem to be uncomputable without turbulence models, which are difficult if
not impossible to design. In this book thus we give concrete evidence that
turbulent flow is computable, in fact often computable on a PC within hours.

20.3 Computational Turbulence

We have pointed out that the secret of computational turbulence is to under-
stand how it may be possible to compute mean value outputs, while point-
value outputs are not computable. We have noted that this can be explained
by the stability properties of the dual solution, which by cancellation effects is
smaller for mean-value outputs than for point-values. Thus we may say that
the secret lies in the cancellation in the dual problem, which may be observed
to take place by simply computing the dual solution. We may also analyze
the cancellation effect in simple model problems, but it seems impossible to
mathematically analyze this cancellation effect in any realistic situation. Thus
we may get a glimpse of the secret, but we seem to be unable to capture the
whole truth by mathematical analysis. Our lives may carry a similar secret:
we may observe what we experience/compute as we go along and we may
understand some aspects, but the full truth will remain hidden.

20.4 Irreversibility

We have unfolded the secret of irreversibility in reversible systems in the
special case of incompressible inviscid flow governed by the Euler equations

172

20 Summary of Mathematical Aspects

solved by G2. We have seen that the irreversibility is a necessary consequence
of the non-existence of stable pointwise solutions of the Euler equations and
the dissipative nature of G2 when computing approximate solutions. We may
phrase our result as a proof of the 2nd Law of thermodynamics from the 1st
Law combined with finite precision in the form of G2. We have remarked that
EG2 is a parameter-free mathematical model of (a part of) the World in the
spirit of Einstein.