lördag 9 februari 2019

Is Digital Computation a Form of Mathematics?

In the last two posts a resolution of the Clay Navier-Stokes Prize Problem is presented, a resolution based on digital computation. I have tried to get some comment on our proposed resolution from the group of pure mathematicians in charge of the problem including in particular its official formulation: Charles Fefferman, Terence Tao and Peter Constantin, to whom I refer as the Problem Committee.

Sorry to say, I can only report silence from the Problem Committee: no comment whatsoever!

How can we understand this state of affairs? Is it so that our resolution lacks scientific substance? No, it represents a true break-through unlocking the main difficulties of mathematical modeling and simulation of fluid flow and it is world-leading. No doubt about that!

The reason behind the silence is thus not lack of scientific interest, but probably rather the opposite: Our resolution being based on digital computation brings in a new kind of mathematics, which is different from that envisioned in the official formulation expressed in the frame of classical analytical theory of partial differential equations. It appears that the Problem Committee does not know how to react to this new kind of mathematics in the form of digital computation, and so silence is the only possible reaction, so far at least.

This connects to a wider question of the role of mathematics in physics including fluid mechanics with particular focus on the new role of digital computation.

Now, mathematics can be seen as different forms of computation with classical pde-theory expressed as symbolic computation by pen and paper, and the new kind expressed by a computer executing the symbolic computation represented in the computer code.

So I again ask about the view of the Problem Committee on the possibility of resolving the Clay Problem by digital computation. Is it thinkable?  Or can only a resolution in the form of symbolic computation with pen and paper be accepted?  Is digital computation a form of mathematics?

Tao does not give any hope that solution by symbolic computation with pen and paper is possible!

Apparently Fefferman would be willing to give the Prize to Tao for a proof of blow-up towards infinite velocities, but so far Tao has not succeeded. But even if one day he would succeed, that would only mean that the mathematical model is no good as a model of real fluid flow, since no observation of infinite velocities has been made, and why give a Prize for a discovery that a model is no good? More  meaningful maybe to give the Prize for a result about a mathematical model of physical significance, like the one we give?

PS1 A pure mathematician might say that digital computation cannot deliver an answer for all (smooth) data and so would lack the generality of an answer by symbolic computation valid for any (smooth) data. To meet this criticism we can add that our resolution exhibits a different form of universality: We show that lift and drag of a body only depends on the shape of the body for high Reynolds number flow beyond the drag crisis at Reynolds number around $5\times 10^5$, that is for a very wide range of flows. Lift and drag depending only on shape is a form of universality. And we can compute lift and drag of any given body, case by case, but of course we cannot get a result for all bodies in one computation.

PS2 The official problem formulation by Fefferman takes as a fact that a smooth unique solution can cease to exist only if velocities become unbounded (referred to as blow-up at some specific finite time). But this is probably a misconception, since smooth solutions may turn into non-smooth solutions because velocity gradients become unbounded, which is what happens as a shock forms in compressible flow and turbulence develops in incompressible flow, while velocities stay bounded.

The official problem formulation is thus filled with misconceptions, and requires reformulation to become meaningful as a Mathematics Prize Problem.