The Euler equations for incompressible inviscid fluids may be written as

$\displaystyle \partial_t u + (u \cdot \nabla) u = -\nabla p$

$\displaystyle \nabla \cdot u = 0$

where ${u: [0,T] \times {\bf R}^n \rightarrow {\bf R}^n}$ is the velocity field, and ${p: [0,T] \times {\bf R}^n \rightarrow {\bf R}}$ is the pressure field. To avoid technicalities we will assume that both fields are smooth, and that ${u}$ is bounded. We will take the dimension ${n}$ to be at least two, with the three-dimensional case ${n=3}$ 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 ${n=3}$ 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

$\displaystyle \int_{{\bf R}^n} |u|^2\ dx \ \ \ \ \ (1)$

(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

$\displaystyle \int_{{\bf R}^3} u \cdot (\nabla \times u)\ dx \ \ \ \ \ (2)$

and (formally, at least) one has conservation of momentum

$\displaystyle \int_{{\bf R}^3} u\ dx$

and angular momentum

$\displaystyle \int_{{\bf R}^3} x \times u\ dx$

(although, as we shall discuss below, due to the slow decay of ${u}$ 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

$\displaystyle \int_{{\bf R}^3} \nabla \times u\ dx$

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 ${\eta_{ij}}$ on ${{\bf R}^n}$ to raise and lower indices, and to define the covariant derivative ${\nabla_i}$ through the Levi-Civita connection (which, in Cartesian coordinates, is just the usual partial derivative ${\partial_i}$ evaluated componentwise). The velocity field ${u}$ now is written as ${u^i}$; contracting against the metric ${\eta}$ gives a ${1}$-form ${u_i := \eta_{ij} u^j}$, which I will call the covelocity, and also write as ${u^*}$. The Euler equations then become

$\displaystyle \partial_t u^i + u^j \nabla_j u^i = - \nabla^i p$

$\displaystyle \nabla_i u^i = 0.$

In particular we have

$\displaystyle \partial_t |u|^2 = 2 u_i \partial_t u^i$

$\displaystyle = - 2 u_i u^j \nabla_j u^i - 2 u_i \nabla^i p$

$\displaystyle = - \nabla_j (u^j u_i u^i) - 2 \nabla^i (u_i p)$

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

$\displaystyle D_t = \partial_t + u \cdot \nabla.$

Here, we shall adopt the subtly different (but closely related) approach of using the material Lie derivative

$\displaystyle {\cal D}_t := \partial_t + {\cal L}_u$

where ${{\cal L}_u}$ is the Lie derivative along the vector field ${u}$. For scalar fields ${f}$, the material Lie derivative is the same as the material derivative:

$\displaystyle {\cal D}_t f = D_t f = \partial_t f + u^i \nabla_i f.$

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 ${X^i}$, we have

$\displaystyle {\cal L}_u X^i = u^j \nabla_j X^i - X^j \nabla_j u^i$

and so

$\displaystyle {\cal D}_t X^i = D_t X_i - X^j \nabla_j u^i.$

Similarly, for ${1}$-forms ${\lambda_i}$, we have

$\displaystyle {\cal L}_u \lambda_i = u^j \nabla_j \lambda_i + \lambda_j \nabla_i u^j,$

and similarly for ${2}$-forms ${\omega_{ij}}$ we have

$\displaystyle {\cal L}_u \omega_{ij} = u^k \nabla_k \omega_{ij} + \omega_{kj} \nabla_i u^k + \omega_{ik} \nabla_j u^k \ \ \ \ \ (3)$

leading to similar formulae comparing ${{\cal D}_t}$ and ${D_t}$ for forms.

Since ${{\cal L}_u u = 0}$, the material Lie derivative of the velocity field ${u^i}$ is just the time derivative:

$\displaystyle {\cal D}_t u^i = \partial_t u^i = - u^j \nabla_j u^i - \nabla^i p.$

The material Lie derivative of the covelocity field ${u_i}$ is however more interesting:

$\displaystyle {\cal D}_t u_i = \partial_t u_i + u^j \nabla_j u_i + u^j \nabla_i u_j$

$\displaystyle = - \nabla_i p + \nabla_i( \frac{1}{2} |u|^2 )$

$\displaystyle = \nabla_i( - p + \frac{1}{2} |u|^2 ).$

In particular, we see that the material Lie derivative of the covelocity is a gradient:

$\displaystyle {\cal D}_t u^* = d( -p + \frac{1}{2} |u|^2 ). \ \ \ \ \ (4)$

Since the integral of a gradient along any closed loop is zero, we obtain

Theorem 1 (Kelvin’s circulation theorem) Let ${\gamma: [0,T] \times S^1 \rightarrow {\bf R}^n}$ be a time-dependent loop in ${{\bf R}^n}$ which is transported by the flow (thus ${\partial_t F(t,\gamma(t,s)) = {\cal D}_t F( t, \gamma(t,s))}$ for any scalar function ${F: [0,T] \times {\bf R}^n \rightarrow {\bf R}}$). Then

$\displaystyle \partial_t \int_\gamma u^* = 0.$

Now we take an exterior derivative of the covelocity ${u^*}$ to obtain the vorticity

$\displaystyle \omega := d u^*.$

In abstract index notation, ${\omega}$ is the ${2}$-form

$\displaystyle \omega_{ij} = \nabla_i u_j - \nabla_j u_i.$

As exterior derivatives commute with diffeomorphisms, they also commute with Lie derivatives, so in particular

$\displaystyle {\cal D}_t \omega = d( {\cal D}_t u^* ).$

Since ${{\cal D}_t u^*}$ was a gradient, its exterior derivative vanishes, and we thus have transport of vorticity:

$\displaystyle {\cal D}_t \omega = 0. \ \ \ \ \ (5)$

(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 ${\int_\gamma u^*}$ as ${\int_S \omega}$ for a spanning surface ${S}$ that is transported by the flow.

If we let ${\hbox{vol}}$ be the usual volume ${n}$-form on ${{\bf R}^n}$, then the divergence-free nature of ${u}$ (and the time-independent nature of ${\hbox{vol}}$) implies that ${\hbox{vol}}$ is also transported by the flow:

$\displaystyle {\cal D}_t \hbox{vol} = 0. \ \ \ \ \ (6)$

If we thus define the polar vorticity ${*\omega}$ to be the ${n-2}$-vector that is the Hodge star of ${\omega}$ with respect to this volume form, thus

$\displaystyle *\omega( v ) \hbox{vol} = \omega \wedge v$

for all ${n-2}$-forms ${v}$, then we see from (5), (6) that the polar vorticity is also transported by the flow:

$\displaystyle {\cal D}_t *\omega = 0. \ \ \ \ \ (7)$

In two dimensions ${n=2}$, the polar vorticity ${*\omega}$ is just a scalar, which by abuse of notation is also denoted ${\omega}$ (in coordinates, ${\omega=\partial_1 u_2 - \partial_2 u_1}$), and (7) becomes the well-known transport of scalar vorticity:

$\displaystyle D_t \omega = 0.$

In three dimensions ${n=3}$, ${*\omega = \omega^i}$ is a vector field which by abuse of notation is also denoted ${\omega}$ (in coordinates, ${\omega = \nabla \times u}$), and (7) becomes the well-known vorticity equation:

$\displaystyle D_t \omega^i = \omega^j \nabla_j u^i.$

From (7) we also see that the vortex lines are transported by the flow; in fact we have the stronger statement that if ${\gamma: [0,T] \times {\bf R} \rightarrow {\bf R}^3}$ is transported by the flow and obeys

$\displaystyle \partial_s \gamma^i(t,s) = \omega^i( \gamma(t,s) )$

at the initial time ${t=0}$, then it continues to do so at all later times ${t}$.

In three dimensions, we may contract the polar vorticity ${\omega^i}$ against the covelocity ${u_i}$ to obtain a scalar ${u \cdot \omega}$. We may then combine (7) and (4) to obtain

$\displaystyle {\cal D}_t (u \cdot \omega) = \omega^i \partial_i( -p + \frac{1}{2} |u|^2 ) = {\cal L}_{*\omega}( -p + \frac{1}{2} |u|^2 ).$

Now the exterior derivative of ${\omega = du^*}$ vanishes, so that ${*\omega}$ is divergence-free, and so ${{\cal L}_{*\omega}}$ annihilates ${\hbox{vol}}$. We therefore conclude conservation of helicity (2). In fact we conclude the stronger statement that if ${\Omega}$ is any time-dependent region in ${{\bf R}^3}$ which is preserved by ${\omega}$ (i.e. it is the union of vortex lines) and is transported by the flow, then ${\int_\Omega (u \cdot \omega)\ \hbox{vol}}$ is conserved in time. This is consistent with Kelvin’s circulation theorem, since one can use Fubini’s theorem to compute the integral ${\int_\Omega (u \cdot \omega)\ \hbox{vol}}$ by first computing the integral of ${u^*}$ on each of the vortex lines in ${\Omega}$, and then integrating against ${\omega}$ on the space of vortex lines in ${\Omega}$ (which is a two-dimensional space on which ${\omega}$ 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 ${D_t}$ instead of material Lie derivatives ${{\cal D}_t}$, basically because the flow along ${u}$ does not preserve the Euclidean metric ${\eta_{ij}}$ or the flat connection ${\nabla}$, making the interchange of Lie derivatives with the integration of vector-valued quantities a little tricky.

Because we will be considering linear integrals of ${u}$ or ${\omega}$ 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 ${u = -\nabla \times \Delta^{-1} \omega}$ suggests that ${u}$ could decay as slowly as ${1/|x|^2}$, 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

$\displaystyle \int_{{\bf R}^n} \omega_{ij}\ d\hbox{vol},$

which is well-defined as a ${2}$-form thanks to the flat connection. Formally, if we write ${*\omega = du^*}$ and integrate by parts, this vorticity should vanish; however if ${u}$ has slow decay then this is not necessarily the case. For instance, if ${u}$ is a smooth mollification of the 2D Biot-Savart kernel ${\frac{1}{2\pi} (-\frac{x_2}{|x|^2},\frac{x_1}{|x|^2})}$ then the total vorticity is one (times the standard ${2}$-form). In three dimensions, though, there is a trick that allows one to establish vanishing of the total polar vorticity

$\displaystyle \int_{{\bf R}^3} \omega^i\ d\hbox{vol}$

and hence also the total vorticity. Namely, if ${x^i}$ is the scaling vector field ${x \cdot \nabla}$, then

$\displaystyle \omega^i = \omega^j \nabla_j x^i$

$\displaystyle = \nabla_j (\omega^j x^i )$

and integration in parts (now involving the compactly supported vorticity ${\omega^j}$) gives the required vanishing. An application of Fubini’s theorem then shows that the total vorticity also vanishes in four and higher dimensions.

In any dimension, though, the total vorticity (and hence also total polar vorticity) is conserved. Indeed, from (5) and (3) we have

$\displaystyle D_t \omega_{ij} = - \omega_{kj} \nabla_i u^k - \omega_{ik} \nabla_j u^k$

$\displaystyle = -\nabla_i(\omega_{kj} u^k) - \nabla_j( \omega_{ik} u^k ) + u^k (\nabla_i \omega_{kj} + \nabla_j \omega_{ik})$

$\displaystyle = -\nabla_i(\omega_{kj} u^k) - \nabla_j( \omega_{ik} u^k ) - u^k \nabla_k \omega_{ji}$

$\displaystyle = -\nabla_i(\omega_{kj} u^k) - \nabla_j( \omega_{ik} u^k ) - \nabla_k (u^k \omega_{ji})$

where we have used the vanishing ${d\omega=0}$ of the exterior derivative of vorticity, as well as the divergence-free nature of ${u}$. This expresses ${D_t \omega}$ as a total derivative

$\displaystyle D_t \omega_{ij} = -\nabla_i(\omega_{kj} u^k) - \nabla_j( \omega_{ik} u^k ) - \nabla_k (u^k \omega_{ji}), \ \ \ \ \ (8)$

giving conservation of total vorticity.

Now we look at total velocity

$\displaystyle \int_{{\bf R}^n} u^i\ d\hbox{vol},$

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

$\displaystyle D_t u^i = - \nabla^i p$

which formally suggests that total velocity is conserved. However, in practice ${u}$ 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

$\displaystyle x^i \omega_{ij} = x^i \partial_i u_j - x^i \partial_j u_i$

$\displaystyle = -(n-1) u_j + \partial_i( x^i u_j ) - \partial_j( x^i u_i ).$

Thus, when ${u}$ has enough decay, one has

$\displaystyle \int_{{\bf R}^n} u^i\ d\hbox{vol} = \frac{-1}{n-1} \int_{{\bf R}^n} x^i \omega_{ij}\ d\hbox{vol};$

however, the right-hand side remains well defined even when ${u}$ decays slowly, assuming that the vorticity is compactly supported. It is thus natural to then define the impulse

$\displaystyle \frac{-1}{n-1} \int_{{\bf R}^n} x^i \omega_{ij}\ d\hbox{vol};$

in three dimensions, this would be ${\frac{1}{2} \int_{{\bf R}^3} x \times \omega}$. The above considerations suggest that the impulse should be another conserved quantity, and indeed it is. To see this, we first compute using (8):

$\displaystyle D_t( x^i \omega_{ij} ) = (D_t x^i) \omega_{ij} + x^i D_t \omega_{ij}$

$\displaystyle = u^i \omega_{ij} - x^i \nabla_i( \omega_{kj} u^k ) - x^i \nabla_j( \omega_{ik} u^k ) - \nabla_k (u^k \omega_{ji}))$

$\displaystyle = u^i \omega_{ij} -\nabla_i( x^i \omega_{kj} u^k ) - \nabla_j ( x^i \omega_{ik} u^j ) - \nabla_k ( x^i u^k \omega_{ji})$

$\displaystyle + n \omega_{kj} u^k + \omega_{jk} u^j + u^i \omega_{ji},$

and so it will suffice to show that ${u^i \omega_{ij}}$ is also a total derivative. But it is:

$\displaystyle u^i \omega_{ij} = u^i (\nabla_i u_j - \nabla_j u_i)$

$\displaystyle = \nabla_i (u^i u_j) - \nabla_j(\frac{1}{2} u^i u_i ). \ \ \ \ \ (9)$

Finally, we look at the total angular momentum

$\displaystyle \int_{{\bf R}^n} u_j x_k - u_k x_j\ d\hbox{vol}.$

Again, we have

$\displaystyle D_t( u_j x_k - u_k x_j ) = (D_t u_j) x_k - (D_t u_k) x_j + u_j D_t x_k - u_k D_t x_j$

$\displaystyle = - (\nabla_j p) x_k + (\nabla_k p) x_j + u_j u_k - u_k u_j$

$\displaystyle = - \nabla_j(px_k) + \nabla_k(px_j)$

which as before formally suggests that total angular momentum should be conserved. As with total momentum, in practice the velocity field ${u}$ decays too slowly to justify this calculation, unless one works carefully with principal value integrals (and uses quite precise asymptotics on the decay of ${u}$ at infinity). Once again, one can avoid these technicalities by recasting this quantity in terms of vorticity. Using ${(a_{jk})_{[j,k]} := a_{jk}-a_{kj}}$ to denote antisymmetrisation in the ${jk}$ indices, we observe that

$\displaystyle x^i \omega_{ij} x_k - x^i \omega_{ik} x_j = (x^i \omega_{ij} x_k)_{[j,k]}$

$\displaystyle = ( x^i x_k \partial_i u_j - x^i x_k \partial_j u_i )_{[j,k]}$

$\displaystyle = ( \partial_i(x^i x_k u_j) - \partial_j (x^i x_k u_i) - n x_k u_j - x^i u_i \eta_{jk} )_{[j,k]}$

$\displaystyle = ( \partial_i(x^i x_k u_j) - \partial_j (x^i x_k u_i) )_{[j,k]} - n(u_j x_k - u_k x_j)$

and so we have

$\displaystyle \int_{{\bf R}^n} u_j x_k - u_k x_j\ d\hbox{vol} = \frac{-1}{n} \int_{{\bf R}^n} x^i \omega_{ij} x_k - x^i \omega_{ik} x_j\ d\hbox{vol}$

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

$\displaystyle \frac{-1}{n} \int_{{\bf R}^n} x^i \omega_{ij} x_k - x^i \omega_{ik} x_j$

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 ${(u^i \omega_{ij} x_k)_{[j,k]}}$ is a total derivative, by manipulating the identity (9).