On the universality of the incompressible Euler equation on compact manifolds



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.

Filed under: math.AP, math.DS, math.MG, paper Tagged: Euler equations, universality

Maryam Mirzakhani



I am totally stunned to learn that Maryam Mirzakhani died today yesterday, aged 40, after a severe recurrence of the cancer she had been fighting for several years.  I had planned to email her some wishes for a speedy recovery after learning about the relapse yesterday; I still can’t fully believe that she didn’t make it.

My first encounter with Maryam was in 2010, when I was giving some lectures at Stanford – one on Perelman’s proof of the Poincare conjecture, and another on random matrix theory.  I remember a young woman sitting in the front who asked perceptive questions at the end of both talks; it was only afterwards that I learned that it was Mirzakhani.  (I really wish I could remember exactly what the questions were, but I vaguely recall that she managed to put a nice dynamical systems interpretation on both of the topics of my talks.)

After she won the Fields medal in 2014 (as I posted about previously on this blog), we corresponded for a while.  The Fields medal is of course one of the highest honours one can receive in mathematics, and it clearly advances one’s career enormously; but it also comes with a huge initial burst of publicity, a marked increase in the number of responsibilities to the field one is requested to take on, and a strong expectation to serve as a public role model for mathematicians.  As the first female recipient of the medal, and also the first to come from Iran, Maryam was experiencing these pressures to a far greater extent than previous medallists, while also raising a small daughter and fighting off cancer.  I gave her what advice I could on these matters (mostly that it was acceptable – and in fact necessary – to say “no” to the vast majority of requests one receives).

Given all this, it is remarkable how productive she still was mathematically in the last few years.  Perhaps her greatest recent achievement has been her “magic wandtheorem with Alex Eskin, which is basically the analogue of the famous measure classification and orbit closure theorems of Marina Ratner, in the context of moduli spaces instead of unipotent flows on homogeneous spaces.  (I discussed Ratner’s theorems in this previous post.  By an unhappy coincidence, Ratner also passed away this month, aged 78.)  Ratner’s theorems are fundamentally important to any problem to which a homogeneous dynamical system can be associated (for instance, a special case of that theorem shows up in my work with Ben Green and Tamar Ziegler on the inverse conjecture for the Gowers norms, and on linear equations in primes), as it gives a good description of the equidistribution of any orbit of that system (if it is unipotently generated); and it seems the Eskin-Mirzakhani result will play a similar role in problems associated instead to moduli spaces.  The remarkable proof of this result – which now stands at over 200 pages, after three years of revision and updating – uses almost all of the latest techniques that had been developed for homogeneous dynamics, and ingeniously adapts them to the more difficult setting of moduli spaces, in a manner that had not been dreamed of being possible only a few years earlier.

Maryam was an amazing mathematician and also a wonderful and humble human being, who was at the peak of her powers.  Today was a huge loss for Maryam’s family and friends, as well as for mathematics.

[EDIT, Jul 16: New York times obituary here.]

[EDIT, Jul 18: New Yorker memorial here.]


Filed under: obituary Tagged: Maryam Mirzakhani