Turing completeness of Euler equations

To prove (or disprove) that the Navier-Stokes equations blow up at finite time, lot of efforts from geometers and topologists are recently garnering attention, after Tao introduced the fluid computer program in 2014.

From a physicist perspective, you see a jet of liquid pinching off at a certain time and position, and Eggers especially has used beautiful asymptotics (approximation of NS at tiny scale) to build an approximate PDE model for the dynamics near pinch-off. However, mathematically speaking, this asymptotic construction is not self-consistent mathematically, as Escuriaza, Gregory Seregrin, Sverak proved it in 2003.

Tao’s fluid program is, on a better end, discretely self-similar in that the parcels of fluid (in a physicist language) are programmed as abstract, disconnected blocks which work on certain energy conservation and mass conservation rules. Non-linear term in NS accounts for the coupled frequency components of the equation and any energy input for one frequency component will be diffused to other frequency components in the system. In a typical NS analysis, this diffusion will lead to decrease in velocity to a level that finite-time blowup is not possible. Tao’s program aim to prove the blowup by building an artificial discrete self-similar fluid program which waits for a certain frequency component to remain in equilibium and then transfer the energy to other frequency component. In such a way, finite-time blowup is shown. Now with such a program, nothing has been proved or disproved, except that the course of this research is shifted towards trying to prove these PDEs as Turing complete which then can be linked to showing blowup at finite time on programming it.

Turing completeness means that an algorithm can be encoded in Turing machine with initial conditions to yield an output which can halt the program at a certain state. Essentially, the machine has a finite set of states Q, initial state q0, a transition function. The transition function converts the present state and present symbol to a new state and moves the tape to a new symbol for next instruction.

Now, recently a paper by Cardona et al. (2021) proved that Euler equations (varying with time) are Turing complete, which means that given an initial velocity field, it is certain and proved that encoding such initial state on a manifold and running the program (that Eva built), velocity field will abruptly come to a halt at a certain (velocity) state. Now, where this field actually lies, is it blowed up or not, or is it going to blow up or not, is indeed the next task of the work and surely can be done; hence the question of universality is potentially answerable. Even if it is answerable, it would be in a manifold other than R³. But manifold other than R³ is not scalar invariant, hence the blow up can’t be proven because as length scales go smaller, the metric structure becomes flat and flat.

The real question is:

Can we construct a hybrid-manifold which is scalar invariant in one part and has some interesting geometry in other part? The goal is to construct a geometry (ref. 53:00) which leads the equations to blow up in R3, in finite-time.


Few points:
1. Any geometry shouldn’t have any contact endpoints that might cause effects on singularity, so consider geometry which is free, infinite (no corners)
2. Idea to put velocity field on a metric constructed in a manifold!