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I have just uploaded to the arXiv my paper “On the universality of the incompressible Euler equation on compact manifolds, II. Non-rigidity of Euler flows“, submitted to Pure and Applied Functional Analysis. This paper continues my attempts to establish “universality” properties of the Euler equations on Riemannian manifolds , as I conjecture that the freedom to set the metric ought to allow one to “program” such Euler flows to exhibit a wide range of behaviour, and in particular to achieve finite time blowup (if the dimension is sufficiently large, at least).

In coordinates, the Euler equations read

where is the pressure field and is the velocity field, and denotes the Levi-Civita connection with the usual Penrose abstract index notation conventions; we restrict attention here to the case where are smooth and is compact, smooth, orientable, connected, and without boundary. Let’s call an *Euler flow* on (for the time interval ) if it solves the above system of equations for some pressure , and an *incompressible flow* if it just obeys the divergence-free relation . Thus every Euler flow is an incompressible flow, but the converse is certainly not true; for instance the various conservation laws of the Euler equation, such as conservation of energy, will already block most incompressible flows from being an Euler flow, or even being approximated in a reasonably strong topology by such Euler flows.

However, one can ask if an incompressible flow can be *extended* to an Euler flow by adding some additional dimensions to . In my paper, I formalise this by considering warped products of which (as a smooth manifold) are products of with a torus, with a metric given by

for , where are the coordinates of the torus , and are smooth positive coefficients for ; in order to preserve the incompressibility condition, we also require the volume preservation property

though in practice we can quickly dispose of this condition by adding one further “dummy” dimension to the torus . We say that an incompressible flow is *extendible to an Euler flow* if there exists a warped product extending , and an Euler flow on of the form

for some “swirl” fields . The situation here is motivated by the familiar situation of studying axisymmetric Euler flows on , which in cylindrical coordinates take the form

The base component

of this flow is then a flow on the two-dimensional plane which is not quite incompressible (due to the failure of the volume preservation condition (2) in this case) but still satisfies a system of equations (coupled with a passive scalar field that is basically the square of the swirl ) that is reminiscent of the Boussinesq equations.

On a fixed -dimensional manifold , let denote the space of incompressible flows , equipped with the smooth topology (in spacetime), and let denote the space of such flows that are extendible to Euler flows. Our main theorem is

Theorem 1

- (i) (Generic inextendibility) Assume . Then is of the first category in (the countable union of nowhere dense sets in ).
- (ii) (Non-rigidity) Assume (with an arbitrary metric ). Then is somewhere dense in (that is, the closure of has non-empty interior).

More informally, starting with an incompressible flow , one usually cannot extend it to an Euler flow just by extending the manifold, warping the metric, and adding swirl coefficients, even if one is allowed to select the dimension of the extension, as well as the metric and coefficients, arbitrarily. However, many such flows can be *perturbed* to be extendible in such a manner (though different perturbations will require different extensions, in particular the dimension of the extension will not be fixed). Among other things, this means that conservation laws such as energy (or momentum, helicity, or circulation) no longer present an obstruction when one is allowed to perform an extension (basically this is because the swirl components of the extension can exchange energy (or momentum, etc.) with the base components in a basically arbitrary fashion.

These results fall short of my hopes to use the ability to extend the manifold to create universal behaviour in Euler flows, because of the fact that each flow requires a different extension in order to achieve the desired dynamics. Still it does seem to provide a little bit of support to the idea that high-dimensional Euler flows are quite “flexible” in their behaviour, though not completely so due to the generic inextendibility phenomenon. This flexibility reminds me a little bit of the flexibility of weak solutions to equations such as the Euler equations provided by the “-principle” of Gromov and its variants (as discussed in these recent notes), although in this case the flexibility comes from adding additional dimensions, rather than by repeatedly adding high-frequency corrections to the solution.

The proof of part (i) of the theorem basically proceeds by a dimension counting argument (similar to that in the proof of Proposition 9 of these recent lecture notes of mine). Heuristically, the point is that an arbitrary incompressible flow is essentially determined by independent functions of space and time, whereas the warping factors are functions of space only, the pressure field is one function of space and time, and the swirl fields are technically functions of both space and time, but have the same number of degrees of freedom as a function just of space, because they solve an evolution equation. When , this means that there are fewer unknown functions of space and time than prescribed functions of space and time, which is the source of the generic inextendibility. This simple argument breaks down when , but we do not know whether the claim is actually false in this case.

The proof of part (ii) proceeds by direct calculation of the effect of the warping factors and swirl velocities, which effectively create a forcing term (of Boussinesq type) in the first equation of (1) that is a combination of functions of the Eulerian spatial coordinates (coming from the warping factors) and the Lagrangian spatial coordinates (which arise from the swirl velocities, which are passively transported by the flow). In a non-empty open subset of , the combination of these coordinates becomes a non-degenerate set of coordinates for spacetime, and one can then use the Stone-Weierstrass theorem to conclude. The requirement that be topologically a torus is a technical hypothesis in order to avoid topological obstructions such as the hairy ball theorem, but it may be that the hypothesis can be dropped (and it may in fact be true, in the case at least, that is dense in all of , not just in a non-empty open subset).

The Boussinesq equations for inviscid, incompressible two-dimensional fluid flow in the presence of gravity are given by

where is the velocity field, is the pressure field, and is the density field (or, in some physical interpretations, the temperature field). In this post we shall restrict ourselves to formal manipulations, assuming implicitly that all fields are regular enough (or sufficiently decaying at spatial infinity) that the manipulations are justified. Using the material derivative , one can abbreviate these equations as

One can eliminate the role of the pressure by working with the *vorticity* . A standard calculation then leads us to the equivalent “vorticity-stream” formulation

of the Boussinesq equations. The latter two equations can be used to recover the velocity field from the vorticity by the Biot-Savart law

It has long been observed (see e.g. Section 5.4.1 of Bertozzi-Majda) that the Boussinesq equations are very similar, though not quite identical, to the three-dimensional inviscid incompressible Euler equations under the hypothesis of axial symmetry (with swirl). The Euler equations are

where now the velocity field and pressure field are over the three-dimensional domain . If one expresses in polar coordinates then one can write the velocity vector field in these coordinates as

If we make the axial symmetry assumption that these components, as well as , do not depend on the variable, thus

then after some calculation (which we give below the fold) one can eventually reduce the Euler equations to the system

where is the modified material derivative, and is the field . This is almost identical with the Boussinesq equations except for some additional powers of ; thus, the intuition is that the Boussinesq equations are a simplified model for axially symmetric Euler flows when one stays away from the axis and also does not wander off to .

However, this heuristic is not rigorous; the above calculations do not actually give an embedding of the Boussinesq equations into Euler. (The equations do match on the cylinder , but this is a measure zero subset of the domain, and so is not enough to give an embedding on any non-trivial region of space.) Recently, while playing around with trying to embed other equations into the Euler equations, I discovered that it is possible to make such an embedding into a *four*-dimensional Euler equation, albeit on a slightly curved manifold rather than in Euclidean space. More precisely, we use the Ebin-Marsden generalisation

of the Euler equations to an arbitrary Riemannian manifold (ignoring any issues of boundary conditions for this discussion), where is a time-dependent vector field, is a time-dependent scalar field, and is the covariant derivative along using the Levi-Civita connection . In Penrose abstract index notation (using the Levi-Civita connection , and raising and lowering indices using the metric ), the equations of motion become

in coordinates, this becomes

where the Christoffel symbols are given by the formula

where is the inverse to the metric tensor . If the coordinates are chosen so that the volume form is the Euclidean volume form , thus , then on differentiating we have , and hence , and so the divergence-free equation (10) simplifies in this case to . The Ebin-Marsden Euler equations are the natural generalisation of the Euler equations to arbitrary manifolds; for instance, they (formally) conserve the kinetic energy

and can be viewed as the formal geodesic flow equation on the infinite-dimensional manifold of volume-preserving diffeomorphisms on (see this previous post for a discussion of this in the flat space case).

The specific four-dimensional manifold in question is the space with metric

and solutions to the Boussinesq equation on can be transformed into solutions to the Euler equations on this manifold. This is part of a more general family of embeddings into the Euler equations in which passive scalar fields (such as the field appearing in the Boussinesq equations) can be incorporated into the dynamics via fluctuations in the Riemannian metric ). I am writing the details below the fold (partly for my own benefit).

Next week (starting on Wednesday, to be more precise), I will begin my class on Perelman’s proof of the Poincaré conjecture. As I only have ten weeks in which to give this proof, I will have to move rapidly through some of the more basic aspects of Riemannian geometry which will be needed throughout the course. In particular, in this preliminary lecture, I will quickly review the basic notions of infinitesimal (or microlocal) Riemannian geometry, and in particular defining the Riemann, Ricci, and scalar curvatures of a Riemannian manifold. (The more “global” aspects of Riemannian geometry, for instance concerning the relationship between distance, curvature, injectivity radius, and volume, will be discussed later in this course.) This is a review only, in particular omitting any leisurely discussion of examples or motivation for Riemannian geometry; it is impossible to compress this subject into a single lecture, and I will have to refer you to a textbook on the subject for a more complete treatment (I myself am using the text “Riemannian geometry” by my colleague here at UCLA, Peter Petersen).

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