Problem 6 in the Clay million dollar prize series concerns the Navier Stokes equations.

They are simultaneously the most simple to state (one line of math),

the most beautiful (they make the leaves tumble on a windy day),
and the most difficult (just expand this equation into its components!).

I would like to see Navier stokes expanded using interval arithmetic.
This creates the added value of being able to associate certainty with the flow at every point in the field.

Highly uncertain flow is subject to chaos.

NS rewritten in interval arithmetic looks like:

where (p,p’) = p +/- p’. read in the ordinary sense, a quantity p plus or minus its associated uncertainty.

Where the quantity is a vector, so is the uncertainty.

Interval Arithmetic

The rules for adding, subtracting, multiplying and dividing in interval arithmetic are:

(a, a’) + (b, b’) = (a + b, a’ + b’)

(a, a’) – (b, b’) = (a – b, a’ + b’)

(a, a’) * (b, b’) = (a * b, 2(a’ * |b| + a * |b’|))

(a, a’) / (b, b’) = (a / b, 2(a’ / |b| + a * |b’|))

This is very beautiful math, since it propagates what is “knowable” as part of the algebra.

A related equation, the second law of thermodynamics is involved.
The second law appears to give the arrow of time its point.

Any solution to Navier-Stokes must also satisfy the Clausius-Duhem entropy inequality.

Second Law Considerations

Currently the second law of thermodynamics is an inequality.

It is was originally my conjecture that to solve Clay problem 6, one must first
complete the second law by adding an information term that completes the
entropy integral and makes it an equality constraint.

An increase in entropy is a measure of the degree to which the system has gone from a more ordered,
to less ordered state. There is thermodynamic entropy as defined by J. Willard Gibbs et. Al. and
there is logical entropy as defined by Richard Feynman. Thermo entropy has units of
energy/unit of molecular kinetic energy (temp). Logical entropy is the log of the ratio of the number
of ways a system can be ordered and is dimensionless.

 
In any event the notion of entropy of a system is similar to the notion of chaos, and chaos is similar to noise
and noise can be characterized as uncertainty, and uncertainty can be swept along as a scalar term using interval arithmetic.
So when we talk entropy we are really talking about certainty, and we have to use numbers of the form

a + bi, where a is the value, b is how much it could vary and I is a unit that identifies the uncertainty.

Interval numbers are not complex numbers, the rules are different for their arithmetic as we saw above from the four basic ops.

Vectors of the form ai + bj are to forms like a + bi like namespaces of the form index0, index1 are to index, index1

Sidebars:

1)       if someone ever asks you for an answer that has a numerical value, you should report your certainty with it.
This is a good habit of intellectual honesty. Examples include age, weight and your speedometer’s values in routine traffic stops.

2)       If a policeman ever writes down your speed on a ticket without also reporting an uncertainty, you should ask the judge to throw out the ticket.

3)       You might want the upper uncertainty to be different than the lower uncertainty, rather than trying to center uncertainty.

One could represent a system’s degree of in-for-med-ness not as a scalar but rather as a vector field of forcing functions with potentially infinite complexity.

If this bothers you as it does me, you might want to use the fact that increasing a systems informed-ness decreases the value b of uncertainty at every point in its vector fields and just run with that.

History

The mathematics of finite differences came after those of continuous fields mainly because sand pounders started making computers chips lately. If finite differences had come first (as in Newton and Leibniz not bothering to take the limit as Dx goes to zero), then we might have gotten to figuring this out quicker and never have bothered to do continuum assumption mathematics. The reason I make this unclear remark is that if we are carrying finite differences around, the uncertainty math and interval arithmetic fits right in. If you feel lost, do not despair.

Consider interval addition. It makes sense, the new number is just the sum of the originals and the sum of the uncertainties. Uncertainties are always positive or zero, if you are perfectly certain. Negative uncertainty is something for smarter people than I to chase after. If your uncertainty is larger than the number itself then that is interesting too. The derivation of the multiplication version is so beautiful you must try it yourself lest I spoil the fun. Just draw the 2D plane. The absolute value signs are necessary to guarantee that uncertainties are not negative.

Toy Example

Solutions to the common quadratic equation ax² + bx + c = 0, are not closed with respect to the real numbers,
that is, it is possible to specific real numbers a, b and c such that no real value of x will satisfy the equation.

Let’s solve the quadratic equation using interval arithmetic with real numbers. Neglecting uncertainty in x itself which complicates the solution the answer is:

(-(b,b’) +/- sqrt( (b,b’)^2 – 4 (a,a’) (c,c’)))/2(a,a’)

The interesting thing is what happens when the discriminant:

(b,b’)^2 – 4 (a,a’) (c,c’) is negative.

It can be the case that the center values produce a positive or zero result, but including the uncertainty creates the possibility of an undefined result with respect to the real numbers. This is very interesting.

Conclusions and a Conjecture

Solutions to the Navier stokes equations depend on the boundary conditions that define the flow conditions coming into and out of some characteristic control volume.

An informal Statement of the Clay problem is:

Show that a box named Navier Stokes has smooth outputs for all time for smooth inputs.

My response is:

The shape of the outputs depend on the shape of the box in which Navier Stokes is enforced.

My conjecture is that there exists a box in which non-smooth outputs exist for smooth inputs and these outputs include shock waves, turbulence and other chaotic phenomena that we observe in much simpler equations such as non-linear pendulums. The scale of these phenomena is just the scale associated with the uncertainty. Reynold’s number is the dimensionless parameter associated with producting the expected behavior of the flow. It would be an interesting exercise to relate Reynold’s number to uncertainty in the rewritten Navier stokes.