The Euler equations for incompressible inviscid fluids may be written as
where is the velocity field, and is the pressure field. To avoid technicalities we will assume that both fields are smooth, and that is bounded. We will take the dimension to be at least two, with the three-dimensional case being of course especially interesting.
The Euler equations are the inviscid limit of the Navier-Stokes equations; as discussed in my previous post, one potential route to establishing finite time blowup for the latter equations when is to be able to construct “computers” solving the Euler equations, which generate smaller replicas of themselves in a noise-tolerant manner (as the viscosity term in the Navier-Stokes equation is to be viewed as perturbative noise).
Perhaps the most prominent obstacles to this route are the conservation laws for the Euler equations, which limit the types of final states that a putative computer could reach from a given initial state. Most famously, we have the conservation of energy
(assuming sufficient decay of the velocity field at infinity); thus for instance it would not be possible for a computer to generate a replica of itself which had greater total energy than the initial computer. This by itself is not a fatal obstruction (in this paper of mine, I constructed such a “computer” for an averaged Euler equation that still obeyed energy conservation). However, there are other conservation laws also, for instance in three dimensions one also has conservation of helicity
and angular momentum
(although, as we shall discuss below, due to the slow decay of at infinity, these integrals have to either be interpreted in a principal value sense, or else replaced with their vorticity-based formulations, namely impulse and moment of impulse). Total vorticity
is also conserved, although it turns out in three dimensions that this quantity vanishes when one assumes sufficient decay at infinity. Then there are the pointwise conservation laws: the vorticity and the volume form are both transported by the fluid flow, while the velocity field (when viewed as a covector) is transported up to a gradient; among other things, this gives the transport of vortex lines as well as Kelvin’s circulation theorem, and can also be used to deduce the helicity conservation law mentioned above. In my opinion, none of these laws actually prohibits a self-replicating computer from existing within the laws of ideal fluid flow, but they do significantly complicate the task of actually designing such a computer, or of the basic “gates” that such a computer would consist of.
Below the fold I would like to record and derive all the conservation laws mentioned above, which to my knowledge essentially form the complete set of known conserved quantities for the Euler equations. The material here (although not the notation) is drawn from this text of Majda and Bertozzi.
For reasons which may become clearer later, I will rewrite the Euler equations in the language of Riemannian geometry, in particular, using the abstract index notation of Penrose), and using the Euclidean metric on to raise and lower indices, and to define the covariant derivative through the Levi-Civita connection (which, in Cartesian coordinates, is just the usual partial derivative evaluated componentwise). The velocity field now is written as ; contracting against the metric gives a -form , which I will call the covelocity, and also write as . The Euler equations then become
In particular we have
which leads to the conservation of energy (1) upon integrating in space.
In the usual treatment of the Euler equations, it is common to introduce the material derivative
Here, we shall adopt the subtly different (but closely related) approach of using the material Lie derivative
where is the Lie derivative along the vector field . For scalar fields , the material Lie derivative is the same as the material derivative:
However, the two notions differ when applied to vector fields or forms, with the material Lie derivative having better covariance properties than the material derivative. When applied to vector fields , we have
Similarly, for -forms , we have
Since , the material Lie derivative of the velocity field is just the time derivative:
The material Lie derivative of the covelocity field is however more interesting:
Theorem 1 (Kelvin’s circulation theorem) Let be a time-dependent loop in which is transported by the flow (thus for any scalar function ). Then
Now we take an exterior derivative of the covelocity to obtain the vorticity
In abstract index notation, is the -form
As exterior derivatives commute with diffeomorphisms, they also commute with Lie derivatives, so in particular
(This fact was also interpreted as conservation of exterior momentum in this previous blog post.) This fact also follows from Kelvin’s circulation theorem, after first applying Stokes’ theorem to rewrite as for a spanning surface that is transported by the flow.
In two dimensions , the polar vorticity is just a scalar, which by abuse of notation is also denoted (in coordinates, ), and (7) becomes the well-known transport of scalar vorticity:
In three dimensions , is a vector field which by abuse of notation is also denoted (in coordinates, ), and (7) becomes the well-known vorticity equation:
at the initial time , then it continues to do so at all later times .
Now the exterior derivative of vanishes, so that is divergence-free, and so annihilates . We therefore conclude conservation of helicity (2). In fact we conclude the stronger statement that if is any time-dependent region in which is preserved by (i.e. it is the union of vortex lines) and is transported by the flow, then is conserved in time. This is consistent with Kelvin’s circulation theorem, since one can use Fubini’s theorem to compute the integral by first computing the integral of on each of the vortex lines in , and then integrating against on the space of vortex lines in (which is a two-dimensional space on which naturally descends to become an area form. All of these quantities are transported by the flow.
Finally, we consider the conservation of various moments of the velocity and vorticity. Here it is best to return to material derivatives instead of material Lie derivatives , basically because the flow along does not preserve the Euclidean metric or the flat connection , making the interchange of Lie derivatives with the integration of vector-valued quantities a little tricky.
Because we will be considering linear integrals of or rather than quadratic integrals, there can be some difficulty in ensuring absolute integrability of the integrals used; for instance, in three dimensions the Biot-Savart law suggests that could decay as slowly as , even if the vorticity is compactly supported. However, the vorticity transport equation (7) tells us (in any dimension) that if the vorticity is compactly supported at time zero, then it remains compactly supported at later times (with the support being transported by the flow). In practice, this means that we will be able to justify operations such as integration by parts if there is at least one factor of the vorticity present.
We begin with the total vorticity
which is well-defined as a -form thanks to the flat connection. Formally, if we write and integrate by parts, this vorticity should vanish; however if has slow decay then this is not necessarily the case. For instance, if is a smooth mollification of the 2D Biot-Savart kernel then the total vorticity is one (times the standard -form). In three dimensions, though, there is a trick that allows one to establish vanishing of the total polar vorticity
and hence also the total vorticity. Namely, if is the scaling vector field , then
and integration in parts (now involving the compactly supported vorticity ) gives the required vanishing. An application of Fubini’s theorem then shows that the total vorticity also vanishes in four and higher dimensions.
Now we look at total velocity
which (up to a scaling factor representing the density of the incompressible fluid) has the physical interpretation as the total momentum of the fluid. We have
which formally suggests that total velocity is conserved. However, in practice usually decays too slowly to justify this calculation, unless one works in a suitable principal value sense. We shall take a different tack, noting that
Thus, when has enough decay, one has
however, the right-hand side remains well defined even when decays slowly, assuming that the vorticity is compactly supported. It is thus natural to then define the impulse
in three dimensions, this would be . The above considerations suggest that the impulse should be another conserved quantity, and indeed it is. To see this, we first compute using (8):
and so it will suffice to show that is also a total derivative. But it is:
Finally, we look at the total angular momentum
Again, we have
which as before formally suggests that total angular momentum should be conserved. As with total momentum, in practice the velocity field decays too slowly to justify this calculation, unless one works carefully with principal value integrals (and uses quite precise asymptotics on the decay of at infinity). Once again, one can avoid these technicalities by recasting this quantity in terms of vorticity. Using to denote antisymmetrisation in the indices, we observe that
and so we have
when there is sufficient decay of the velocity field. Again, the right-hand side makes sense whenever the vorticity is compactly supported. If we then define the moment of impulse
then we expect this quantity to also be conserved by the flow. This is indeed the case, and can be verified by a rather lengthy calculation similar to that used to establish conservation of impulse; we omit the details here as they are rather tedious and unenlightening, with a key step being the establishment of the fact that is a total derivative, by manipulating the identity (9).