I’ve just uploaded to the arXiv my paper “On the universality of the incompressible Euler equation on compact manifolds“, submitted to Discrete and Continuous Dynamical Systems. This is a variant of my recent paper on the universality of potential well dynamics, but instead of trying to embed dynamical systems into a potential well ${\partial_{tt} u = -\nabla V(u)}$, here we try to embed dynamical systems into the incompressible Euler equations

$\displaystyle \partial_t u + \nabla_u u = - \mathrm{grad}_g p \ \ \ \ \ (1)$

$\displaystyle \mathrm{div}_g u = 0$

on a Riemannian manifold ${(M,g)}$. (One is particularly interested in the case of flat manifolds ${M}$, particularly ${{\bf R}^3}$ or ${({\bf R}/{\bf Z})^3}$, but for the main result of this paper it is essential that one is permitted to consider curved manifolds.) This system, first studied by Ebin and Marsden, is the natural generalisation of the usual incompressible Euler equations to curved space; it can be viewed as the formal geodesic flow equation on the infinite-dimensional manifold of volume-preserving diffeomorphisms on ${M}$ (see this previous post for a discussion of this in the flat space case).

The Euler equations can be viewed as a nonlinear equation in which the nonlinearity is a quadratic function of the velocity field ${u}$. It is thus natural to compare the Euler equations with quadratic ODE of the form

$\displaystyle \partial_t y = B(y,y) \ \ \ \ \ (2)$

where ${y: {\bf R} \rightarrow {\bf R}^n}$ is the unknown solution, and ${B: {\bf R}^n \times {\bf R}^n \rightarrow {\bf R}^n}$ is a bilinear map, which we may assume without loss of generality to be symmetric. One can ask whether such an ODE may be linearly embedded into the Euler equations on some Riemannian manifold ${(M,g)}$, which means that there is an injective linear map ${U: {\bf R}^n \rightarrow \Gamma(TM)}$ from ${{\bf R}^n}$ to smooth vector fields on ${M}$, as well as a bilinear map ${P: {\bf R}^n \times {\bf R}^n \rightarrow C^\infty(M)}$ to smooth scalar fields on ${M}$, such that the map ${y \mapsto (U(y), P(y,y))}$ takes solutions to (2) to solutions to (1), or equivalently that

$\displaystyle U(B(y,y)) + \nabla_{U(y)} U(y) = - \mathrm{grad}_g P(y,y)$

$\displaystyle \mathrm{div}_g U(y) = 0$

for all ${y \in {\bf R}^n}$.

For simplicity let us restrict ${M}$ to be compact. There is an obvious necessary condition for this embeddability to occur, which comes from energy conservation law for the Euler equations; unpacking everything, this implies that the bilinear form ${B}$ in (2) has to obey a cancellation condition

$\displaystyle \langle B(y,y), y \rangle = 0 \ \ \ \ \ (3)$

for some positive definite inner product ${\langle, \rangle: {\bf R}^n \times {\bf R}^n \rightarrow {\bf R}}$ on ${{\bf R}^n}$. The main result of the paper is the converse to this statement: if ${B}$ is a symmetric bilinear form obeying a cancellation condition (3), then it is possible to embed the equations (2) into the Euler equations (1) on some Riemannian manifold ${(M,g)}$; the catch is that this manifold will depend on the form ${B}$ and on the dimension ${n}$ (in fact in the construction I have, ${M}$ is given explicitly as ${SO(n) \times ({\bf R}/{\bf Z})^{n+1}}$, with a funny metric on it that depends on ${B}$).

As a consequence, any finite dimensional portion of the usual “dyadic shell models” used as simplified toy models of the Euler equation, can actually be embedded into a genuine Euler equation, albeit on a high-dimensional and curved manifold. This includes portions of the self-similar “machine” I used in a previous paper to establish finite time blowup for an averaged version of the Navier-Stokes (or Euler) equations. Unfortunately, the result in this paper does not apply to infinite-dimensional ODE, so I cannot yet establish finite time blowup for the Euler equations on a (well-chosen) manifold. It does not seem so far beyond the realm of possibility, though, that this could be done in the relatively near future. In particular, the result here suggests that one could construct something resembling a universal Turing machine within an Euler flow on a manifold, which was one ingredient I would need to engineer such a finite time blowup.

The proof of the main theorem proceeds by an “elimination of variables” strategy that was used in some of my previous papers in this area, though in this particular case the Nash embedding theorem (or variants thereof) are not required. The first step is to lessen the dependence on the metric ${g}$ by partially reformulating the Euler equations (1) in terms of the covelocity ${g \cdot u}$ (which is a ${1}$-form) instead of the velocity ${u}$. Using the freedom to modify the dimension of the underlying manifold ${M}$, one can also decouple the metric ${g}$ from the volume form that is used to obtain the divergence-free condition. At this point the metric can be eliminated, with a certain positive definiteness condition between the velocity and covelocity taking its place. After a substantial amount of trial and error (motivated by some “two-and-a-half-dimensional” reductions of the three-dimensional Euler equations, and also by playing around with a number of variants of the classic “separation of variables” strategy), I eventually found an ansatz for the velocity and covelocity that automatically solved most of the components of the Euler equations (as well as most of the positive definiteness requirements), as long as one could find a number of scalar fields that obeyed a certain nonlinear system of transport equations, and also obeyed a positive definiteness condition. Here I was stuck for a bit because the system I ended up with was overdetermined – more equations than unknowns. After trying a number of special cases I eventually found a solution to the transport system on the sphere, except that the scalar functions sometimes degenerated and so the positive definiteness property I wanted was only obeyed with positive semi-definiteness. I tried for some time to perturb this example into a strictly positive definite solution before eventually working out that this was not possible. Finally I had the brainwave to lift the solution from the sphere to an even more symmetric space, and this quickly led to the final solution of the problem, using the special orthogonal group rather than the sphere as the underlying domain. The solution ended up being rather simple in form, but it is still somewhat miraculous to me that it exists at all; in retrospect, given the overdetermined nature of the problem, relying on a large amount of symmetry to cut down the number of equations was basically the only hope.