Thursday, March 20, 2014

Terence Tao Proposal

1.3. A program for establishing blowup for the true Navier-Stokes equations?

To summarise the strategy of proof of Theorem 1.4, a solution to a carefully chosen averaged version

                                                        

of the Euler equations is constructed which behaves like a “von Neumann machine” (that is, a self-replicating machine) in the following sense: at a given time  , it evolves as a sort of “quadratic computer”, made out of “quadratic logic gates”, which is “programmed” so that after a reasonable period of time , it abruptly “replicates” into a rescaled version of itself (being times smaller, and about  times faster), while also erasing almost completely the previous iteration of this machine. This replication process is stable with respect to perturbations, and in particular can survive the presence of a supercritical dissipation if the initial scale of the machine is sufficiently small.

This suggests an ambitious (but not obviously impossible) program (in both senses of the word) to achieve the same effect for the true Navier-Stokes equations, thus obtaining a negative answer to Conjecture 1.1. Define an ideal (incompressible, inviscid) fluid to be a divergence-free vector field u that evolves according to the true Euler equations 

                                                          
Somewhat analogously to how a quantum computer can be constructed from the laws of quantum mechanics (see e.g. [5]), or a Turing machine can be constructed from cellular automata such as Conway’s “Game of Life” (see e.g. [2]), one could hope to design logic gates entirely out of ideal fluid (perhaps by using suitably shaped vortex sheets to simulate the various types of physical materials one would use in a mechanical computer). If these gates were sufficiently “Turing complete”, and also “noise-tolerant”, one could then hope to combine enough of these gates together to “program” a von Neumann machine consisting of ideal fluid that, when it runs, behaves qualitatively like the blowup solution used to establish Theorem 1.4. Note that such replicators, as well as the concept of a universal constructor, have been built within cellular automata such as the “Game of Life”; see e.g. [1].

Once enough logic gates of ideal fluid are constructed, it seems that the main  difficulty executing the above program are of a “software engineering” nature, and would be in principle achievable, even if the details could be extremely complicated in practice. The main mathematical difficulty in executing this “fluid computing” program would thus be to arrive at (and rigorously certify) a design for logical gates of inviscid fluid that has some good noise tolerance properties. In this regard, ideas from quantum computing (which faces a unitarity constraint somewhat analogous to the energy conservation constraint for ideal fluids, albeit with the key difference of having a linear evolution rather than a nonlinear one) may prove to be useful.

It is worth pointing out, however, that even if this program is successful, it would only demonstrate blowup for a very specific type of initial data (and tiny perturbations thereof), and is not necessarily in contradiction with the belief that one has global regularity for most choices of initial data (for some carefully chosen definition of “most”, e.g. with overwhelming (but not almost sure) probability with respect to various probability distributions of initial data). However, we do not have any new ideas to contribute on how to address this latter question, other than to state the obvious fact that deterministic methods alone are unlikely to be sufficient to resolve the problem, and that stochastic methods (e.g. those based on invariant measures) are probably needed.

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