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Let ${u: {\bf R}^3 \rightarrow {\bf R}^3}$ be a divergence-free vector field, thus ${\nabla \cdot u = 0}$, which we interpret as a velocity field. In this post we will proceed formally, largely ignoring the analytic issues of whether the fields in question have sufficient regularity and decay to justify the calculations. The vorticity field ${\omega: {\bf R}^3 \rightarrow {\bf R}^3}$ is then defined as the curl of the velocity:

$\displaystyle \omega = \nabla \times u.$

(From a differential geometry viewpoint, it would be more accurate (especially in other dimensions than three) to define the vorticity as the exterior derivative ${\omega = d(g \cdot u)}$ of the musical isomorphism ${g \cdot u}$ of the Euclidean metric ${g}$ applied to the velocity field ${u}$; see these previous lecture notes. However, we will not need this geometric formalism in this post.)

Assuming suitable regularity and decay hypotheses of the velocity field ${u}$, it is possible to recover the velocity from the vorticity as follows. From the general vector identity ${\nabla \times \nabla \times X = \nabla(\nabla \cdot X) - \Delta X}$ applied to the velocity field ${u}$, we see that

$\displaystyle \nabla \times \omega = -\Delta u$

and thus (by the commutativity of all the differential operators involved)

$\displaystyle u = - \nabla \times \Delta^{-1} \omega.$

Using the Newton potential formula

$\displaystyle -\Delta^{-1} \omega(x) := \frac{1}{4\pi} \int_{{\bf R}^3} \frac{\omega(y)}{|x-y|}\ dy$

and formally differentiating under the integral sign, we obtain the Biot-Savart law

$\displaystyle u(x) = \frac{1}{4\pi} \int_{{\bf R}^3} \frac{\omega(y) \times (x-y)}{|x-y|^3}\ dy. \ \ \ \ \ (1)$

This law is of fundamental importance in the study of incompressible fluid equations, such as the Euler equations

$\displaystyle \partial_t u + (u \cdot \nabla) u = -\nabla p; \quad \nabla \cdot u = 0$

since on applying the curl operator one obtains the vorticity equation

$\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u \ \ \ \ \ (2)$

and then by substituting (1) one gets an autonomous equation for the vorticity field ${\omega}$. Unfortunately, this equation is non-local, due to the integration present in (1).

In a recent work, it was observed by Elgindi that in a certain regime, the Biot-Savart law can be approximated by a more “low rank” law, which makes the non-local effects significantly simpler in nature. This simplification was carried out in spherical coordinates, and hinged on a study of the invertibility properties of a certain second order linear differential operator in the latitude variable ${\theta}$; however in this post I would like to observe that the approximation can also be seen directly in Cartesian coordinates from the classical Biot-Savart law (1). As a consequence one can also initiate the beginning of Elgindi’s analysis in constructing somewhat regular solutions to the Euler equations that exhibit self-similar blowup in finite time, though I have not attempted to execute the entirety of the analysis in this setting.

Elgindi’s approximation applies under the following hypotheses:

A model example of a divergence-free vector field obeying these properties (but without good decay at infinity) is the linear vector field

$\displaystyle X(x) = (x_1, x_2, -2x_3) \ \ \ \ \ (5)$

which is of the form (3) with ${u_r(r,x_3) = r}$ and ${u_3(r,x_3) = -2x_3}$. The associated vorticity ${\omega}$ vanishes.

We can now give an illustration of Elgindi’s approximation:

Proposition 1 (Elgindi’s approximation) Under the above hypotheses (and assuing suitable regularity and decay), we have the pointwise bounds

$\displaystyle u(x) = \frac{1}{2} {\mathcal L}_{12}(\omega)(|x|) X(x) + O( |x| \|\omega\|_{L^\infty({\bf R}^3)} )$

for any ${x \in {\bf R}^3}$, where ${X}$ is the vector field (5), and ${{\mathcal L}_{12}(\omega): {\bf R}^+ \rightarrow {\bf R}}$ is the scalar function

$\displaystyle {\mathcal L}_{12}(\omega)(\rho) := \frac{3}{4\pi} \int_{|y| \geq \rho} \frac{r y_3}{|y|^5} \omega_{r3}(r,y_3)\ dy.$

Thus under the hypotheses (i), (ii), and assuming that ${\omega}$ is slowly varying, we expect ${u}$ to behave like the linear vector field ${X}$ modulated by a radial scalar function. In applications one needs to control the error in various function spaces instead of pointwise, and with ${\omega}$ similarly controlled in other function space norms than the ${L^\infty}$ norm, but this proposition already gives a flavour of the approximation. If one uses spherical coordinates

$\displaystyle \omega_{r3}( \rho \cos \theta, \rho \sin \theta ) = \Omega( \rho, \theta )$

then we have (using the spherical change of variables formula ${dy = \rho^2 \cos \theta d\rho d\theta d\phi}$ and the odd nature of ${\Omega}$)

$\displaystyle {\mathcal L}_{12}(\omega) = L_{12}(\Omega),$

where

$\displaystyle L_{12}(\Omega)(\rho) = 3 \int_\rho^\infty \int_0^{\pi/2} \frac{\Omega(r, \theta) \sin(\theta) \cos^2(\theta)}{r}\ d\theta dr$

is the operator introduced in Elgindi’s paper.

Proof: By a limiting argument we may assume that ${x}$ is non-zero, and we may normalise ${\|\omega\|_{L^\infty({\bf R}^3)}=1}$. From the triangle inequality we have

$\displaystyle \int_{|y| \leq 10|x|} \frac{\omega(y) \times (x-y)}{|x-y|^3}\ dy \leq \int_{|y| \leq 10|x|} \frac{1}{|x-y|^2}\ dy$

$\displaystyle \leq \int_{|z| \leq 11 |x|} \frac{1}{|z|^2}\ dz$

$\displaystyle = O( |x| )$

and hence by (1)

$\displaystyle u(x) = \frac{1}{4\pi} \int_{|y| > 10|x|} \frac{\omega(y) \times (x-y)}{|x-y|^3}\ dy + O(|x|).$

In the regime ${|y| > 2|x|}$ we may perform the Taylor expansion

$\displaystyle \frac{x-y}{|x-y|^3} = \frac{x-y}{|y|^3} (1 - \frac{2 x \cdot y}{|y|^2} + \frac{|x|^2}{|y|^2})^{-3/2}$

$\displaystyle = \frac{x-y}{|y|^3} (1 + \frac{3 x \cdot y}{|y|^2} + O( \frac{|x|^2}{|y|^2} ) )$

$\displaystyle = -\frac{y}{|y|^3} + \frac{x}{|y|^3} - \frac{3 (x \cdot y) y}{|y|^5} + O( \frac{|x|^2}{|y|^4} ).$

Since

$\displaystyle \int_{|y| > 10|x|} \frac{|x|^2}{|y|^4}\ dy = O(|x|)$

we see from the triangle inequality that the error term contributes ${O(|x|)}$ to ${u(x)}$. We thus have

$\displaystyle u(x) = -A_0(x) + A_1(x) - 3A'_1(x) + O(|x|)$

where ${A_0}$ is the constant term

$\displaystyle A_0 := \int_{|y| > 10|x|} \frac{\omega(y) \times y}{|y|^3}\ dy,$

and ${A_1, A'_1}$ are the linear term

$\displaystyle A_1 := \int_{|y| > 10|x|} \frac{\omega(y) \times x}{|y|^3}\ dy,$

$\displaystyle A'_1 := \int_{|y| > 10|x|} (x \cdot y) \frac{\omega(y) \times y}{|y|^5}\ dy.$

By the hypotheses (i), (ii), we have the symmetries

$\displaystyle \omega(y_1,y_2,-y_3) = - \omega(y_1,y_2,y_3) \ \ \ \ \ (6)$

and

$\displaystyle \omega(-y_1,-y_2,y_3) = - \omega(y_1,y_2,y_3) \ \ \ \ \ (7)$

and hence also

$\displaystyle \omega(-y_1,-y_2,-y_3) = \omega(y_1,y_2,y_3). \ \ \ \ \ (8)$

The even symmetry (8) ensures that the integrand in ${A_0}$ is odd, so ${A_0}$ vanishes. The symmetry (6) or (7) similarly ensures that ${\int_{|y| > 10|x|} \frac{\omega(y)}{|y|^3}\ dy = 0}$, so ${A_1}$ vanishes. Since ${\int_{|x| < y \leq 10|x|} \frac{|x \cdot y| |y|}{|y|^5}\ dy = O( |x| )}$, we conclude that

$\displaystyle \omega(x) = -3\int_{|y| \geq |x|} (x \cdot y) \frac{\omega(y) \times y}{|y|^5}\ dy + O(|x|).$

Using (4), the right-hand side is

$\displaystyle -3\int_{|y| \geq |x|} (x_1 y_1 + x_2 y_2 + x_3 y_3) \frac{\omega_{r3}(r,y_3) (-y_1 y_3, -y_2 y_3, y_1^2+y_2^2)}{r|y|^5}\ dy$

$\displaystyle + O(|x|)$

where ${r := \sqrt{y_1^2+y_2^2}}$. Because of the odd nature of ${\omega_{r3}}$, only those terms with one factor of ${y_3}$ give a non-vanishing contribution to the integral. Using the rotation symmetry ${(y_1,y_2,y_3) \mapsto (-y_2,y_1,y_3)}$ we also see that any term with a factor of ${y_1 y_2}$ also vanishes. We can thus simplify the above expression as

$\displaystyle -3\int_{|y| \geq |x|} \frac{\omega_{r3}(r,y_3) (-x_1 y_1^2 y_3, -x_2 y_2^2 y_3, x_3 (y_1^2+y_2^2) y_3)}{r|y|^5}\ dy + O(|x|).$

Using the rotation symmetry ${(y_1,y_2,y_3) \mapsto (-y_2,y_1,y_3)}$ again, we see that the term ${y_1^2}$ in the first component can be replaced by ${y_2^2}$ or by ${\frac{1}{2} (y_1^2+y_2^2) = \frac{r^2}{2}}$, and similarly for the ${y_2^2}$ term in the second component. Thus the above expression is

$\displaystyle \frac{3}{2} \int_{|y| \geq |x|} \frac{\omega_{r3}(r,y_3) (x_1 , x_2, -2x_3) r y_3}{|y|^5}\ dy + O(|x|)$

giving the claim. $\Box$

Example 2 Consider the divergence-free vector field ${u := \nabla \times \psi}$, where the vector potential ${\psi}$ takes the form

$\displaystyle \psi(x_1,x_2,x_3) := (x_2 x_3, -x_1 x_3, 0) \eta(|x|)$

for some bump function ${\eta: {\bf R} \rightarrow {\bf R}}$ supported in ${(0,+\infty)}$. We can then calculate

$\displaystyle u(x_1,x_2,x_3) = X(x) \eta(|x|) + (x_1 x_3, x_2 x_3, -x_1^2-x_2^2) \frac{\eta'(|x|) x_3}{|x|}.$

and

$\displaystyle \omega(x_1,x_2,x_3) = (-6x_2 x_3, 6x_1 x_3, 0) \frac{\eta'(|x|)}{|x|} + (-x_2 x_3, x_1 x_3, 0) \eta''(|x|).$

In particular the hypotheses (i), (ii) are satisfied with

$\displaystyle \omega_{r3}(r,x_3) = - 6 \eta'(|x|) \frac{x_3 r}{|x|} - \eta''(|x|) x_3 r.$

One can then calculate

$\displaystyle L_{12}(\omega)(\rho) = -\frac{3}{4\pi} \int_{|y| \geq \rho} (6\frac{\eta'(|y|)}{|y|^6} + \frac{\eta''(|y|)}{|y|^5}) r^2 y_3^2\ dy$

$\displaystyle = -\frac{2}{5} \int_\rho^\infty 6\eta'(s) + s\eta''(s)\ ds$

$\displaystyle = 2\eta(\rho) + \frac{2}{5} \rho \eta'(\rho).$

If we take the specific choice

$\displaystyle \eta(\rho) = \varphi( \rho^\alpha )$

where ${\varphi}$ is a fixed bump function supported some interval ${[c,C] \subset (0,+\infty)}$ and ${\alpha>0}$ is a small parameter (so that ${\eta}$ is spread out over the range ${\rho \in [c^{1/\alpha},C^{1/\alpha}]}$), then we see that

$\displaystyle \| \omega \|_{L^\infty} = O( \alpha )$

(with implied constants allowed to depend on ${\varphi}$),

$\displaystyle L_{12}(\omega)(\rho) = 2\eta(\rho) + O(\alpha),$

and

$\displaystyle u = X(x) \eta(|x|) + O( \alpha |x| ),$

which is completely consistent with Proposition 1.

One can use this approximation to extract a plausible ansatz for a self-similar blowup to the Euler equations. We let ${\alpha>0}$ be a small parameter and let ${\omega_{rx_3}}$ be a time-dependent vorticity field obeying (i), (ii) of the form

$\displaystyle \omega_{rx_3}(t,r,x_3) \approx \alpha \Omega( t, R ) \mathrm{sgn}(x_3)$

where ${R := |x|^\alpha = (r^2+x_3^2)^{\alpha/2}}$ and ${\Omega: {\bf R} \times [0,+\infty) \rightarrow {\bf R}}$ is a smooth field to be chosen later. Admittedly the signum function ${\mathrm{sgn}}$ is not smooth at ${x_3}$, but let us ignore this issue for now (to rigorously make an ansatz one will have to smooth out this function a little bit; Elgindi uses the choice ${(|\sin \theta| \cos^2 \theta)^{\alpha/3} \mathrm{sgn}(x_3)}$, where ${\theta := \mathrm{arctan}(x_3/r)}$). With this ansatz one may compute

$\displaystyle {\mathcal L}_{12}(\omega(t))(\rho) \approx \frac{3\alpha}{2\pi} \int_{|y| \geq \rho; y_3 \geq 0} \Omega(t,R) \frac{r y_3}{|y|^5}\ dy$

$\displaystyle = \alpha \int_\rho^\infty \Omega(t, s^\alpha) \frac{ds}{s}$

$\displaystyle = \int_{\rho^\alpha}^\infty \Omega(t,s) \frac{ds}{s}.$

By Proposition 1, we thus expect to have the approximation

$\displaystyle u(t,x) \approx \frac{1}{2} \int_{|x|^\alpha}^\infty \Omega(t,s) \frac{ds}{s} X(x).$

We insert this into the vorticity equation (2). The transport term ${(u \cdot \nabla) \omega}$ will be expected to be negligible because ${R}$, and hence ${\omega_{rx_3}}$, is slowly varying (the discontinuity of ${\mathrm{sgn}(x_3)}$ will not be encountered because the vector field ${X}$ is parallel to this singularity). The modulating function ${\frac{1}{2} \int_{|x|^\alpha}^\infty \Omega(t,s) \frac{ds}{s}}$ is similarly slowly varying, so derivatives falling on this function should be lower order. Neglecting such terms, we arrive at the approximation

$\displaystyle (\omega \cdot \nabla) u \approx \frac{1}{2} \int_{|x|^\alpha}^\infty \Omega(t,s) \frac{ds}{s} \omega$

and so in the limit ${\alpha \rightarrow 0}$ we expect obtain a simple model equation for the evolution of the vorticity envelope ${\Omega}$:

$\displaystyle \partial_t \Omega(t,R) = \frac{1}{2} \int_R^\infty \Omega(t,S) \frac{dS}{S} \Omega(t,R).$

If we write ${L(t,R) := \int_R^\infty \Omega(t,S)\frac{dS}{S}}$ for the logarithmic primitive of ${\Omega}$, then we have ${\Omega = - R \partial_R L}$ and hence

$\displaystyle \partial_t (R \partial_R L) = \frac{1}{2} L (R \partial_R L)$

which integrates to the Ricatti equation

$\displaystyle \partial_t L = \frac{1}{4} L^2$

which can be explicitly solved as

$\displaystyle L(t,R) = \frac{2}{f(R) - t/2}$

where ${f(R)}$ is any function of ${R}$ that one pleases. (In Elgindi’s work a time dilation is used to remove the unsightly factor of ${1/2}$ appearing here in the denominator.) If for instance we set ${f(R) = 1+R}$, we obtain the self-similar solution

$\displaystyle L(t,R) = \frac{2}{1+R-t/2}$

and then on applying ${-R \partial_R}$

$\displaystyle \Omega(t,R) = \frac{2R}{(1+R-t/2)^2}.$

Thus, we expect to be able to construct a self-similar blowup to the Euler equations with a vorticity field approximately behaving like

$\displaystyle \omega(t,x) \approx \alpha \frac{2R}{(1+R-t/2)^2} \mathrm{sgn}(x_3) (\frac{x_2}{r}, -\frac{x_1}{r}, 0)$

and velocity field behaving like

$\displaystyle u(t,x) \approx \frac{1}{1+R-t/2} X(x).$

In particular, ${u}$ would be expected to be of regularity ${C^{1,\alpha}}$ (and smooth away from the origin), and blows up in (say) ${L^\infty}$ norm at time ${t/2 = 1}$, and one has the self-similarity

$\displaystyle u(t,x) = (1-t/2)^{\frac{1}{\alpha}-1} u( 0, \frac{x}{(1-t/2)^{1/\alpha}} )$

and

$\displaystyle \omega(t,x) = (1-t/2)^{-1} \omega( 0, \frac{x}{(1-t/2)^{1/\alpha}} ).$

A self-similar solution of this approximate shape is in fact constructed rigorously in Elgindi’s paper (using spherical coordinates instead of the Cartesian approach adopted here), using a nonlinear stability analysis of the above ansatz. It seems plausible that one could also carry out this stability analysis using this Cartesian coordinate approach, although I have not tried to do this in detail.

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 ${(M,g)}$, as I conjecture that the freedom to set the metric ${g}$ 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

$\displaystyle \partial_t u^k + u^j \nabla_j u^k = - \nabla^k p \ \ \ \ \ (1)$

$\displaystyle \nabla_k u^k = 0$

where ${p: [0,T] \rightarrow C^\infty(M)}$ is the pressure field and ${u: [0,T] \rightarrow \Gamma(TM)}$ is the velocity field, and ${\nabla}$ denotes the Levi-Civita connection with the usual Penrose abstract index notation conventions; we restrict attention here to the case where ${u,p}$ are smooth and ${M}$ is compact, smooth, orientable, connected, and without boundary. Let’s call ${u}$ an Euler flow on ${M}$ (for the time interval ${[0,T]}$) if it solves the above system of equations for some pressure ${p}$, and an incompressible flow if it just obeys the divergence-free relation ${\nabla_k u^k=0}$. 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 ${M}$. In my paper, I formalise this by considering warped products ${\tilde M}$ of ${M}$ which (as a smooth manifold) are products ${\tilde M = M \times ({\bf R}/{\bf Z})^m}$ of ${M}$ with a torus, with a metric ${\tilde g}$ given by

$\displaystyle d \tilde g^2 = g_{ij}(x) dx^i dx^j + \sum_{s=1}^m \tilde g_{ss}(x) (d\theta^s)^2$

for ${(x,\theta) \in \tilde M}$, where ${\theta^1,\dots,\theta^m}$ are the coordinates of the torus ${({\bf R}/{\bf Z})^m}$, and ${\tilde g_{ss}: M \rightarrow {\bf R}^+}$ are smooth positive coefficients for ${s=1,\dots,m}$; in order to preserve the incompressibility condition, we also require the volume preservation property

$\displaystyle \prod_{s=1}^m \tilde g_{ss}(x) = 1 \ \ \ \ \ (2)$

though in practice we can quickly dispose of this condition by adding one further “dummy” dimension to the torus ${({\bf R}/{\bf Z})^m}$. We say that an incompressible flow ${u}$ is extendible to an Euler flow if there exists a warped product ${\tilde M}$ extending ${M}$, and an Euler flow ${\tilde u}$ on ${\tilde M}$ of the form

$\displaystyle \tilde u(t,(x,\theta)) = u^i(t,x) \frac{d}{dx^i} + \sum_{s=1}^m \tilde u^s(t,x) \frac{d}{d\theta^s}$

for some “swirl” fields ${\tilde u^s: [0,T] \times M \rightarrow {\bf R}}$. The situation here is motivated by the familiar situation of studying axisymmetric Euler flows ${\tilde u}$ on ${{\bf R}^3}$, which in cylindrical coordinates take the form

$\displaystyle \tilde u(t,(r,z,\theta)) = u^r(t,r,z) \frac{d}{dr} + u^z(t,r,z) \frac{d}{dz} + \tilde u^\theta(t,r,z) \frac{d}{d\theta}.$

The base component

$\displaystyle u^r(t,r,z) \frac{d}{dr} + u^z(t,r,z) \frac{d}{dz}$

of this flow is then a flow on the two-dimensional ${r,z}$ 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 ${\rho}$ that is basically the square of the swirl ${\tilde u^\rho}$) that is reminiscent of the Boussinesq equations.

On a fixed ${d}$-dimensional manifold ${(M,g)}$, let ${{\mathcal F}}$ denote the space of incompressible flows ${u: [0,T] \rightarrow \Gamma(TM)}$, equipped with the smooth topology (in spacetime), and let ${{\mathcal E} \subset {\mathcal F}}$ denote the space of such flows that are extendible to Euler flows. Our main theorem is

Theorem 1

• (i) (Generic inextendibility) Assume ${d \geq 3}$. Then ${{\mathcal E}}$ is of the first category in ${{\mathcal F}}$ (the countable union of nowhere dense sets in ${{\mathcal F}}$).
• (ii) (Non-rigidity) Assume ${M = ({\bf R}/{\bf Z})^d}$ (with an arbitrary metric ${g}$). Then ${{\mathcal E}}$ is somewhere dense in ${{\mathcal F}}$ (that is, the closure of ${{\mathcal E}}$ has non-empty interior).

More informally, starting with an incompressible flow ${u}$, 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 “${h}$-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 ${u}$ is essentially determined by ${d-1}$ independent functions of space and time, whereas the warping factors ${\tilde g_{ss}}$ are functions of space only, the pressure field is one function of space and time, and the swirl fields ${u^s}$ 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 ${d>2}$, 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 ${d=2}$, 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 ${x^i}$ (coming from the warping factors) and the Lagrangian spatial coordinates ${a^\beta}$ (which arise from the swirl velocities, which are passively transported by the flow). In a non-empty open subset of ${{\mathcal F}}$, 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 ${M}$ 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 ${M = ({\bf R}/{\bf Z})^d}$ case at least, that ${{\mathcal E}}$ is dense in all of ${{\mathcal F}}$, not just in a non-empty open subset).

These lecture notes are a continuation of the 254A lecture notes from the previous quarter.
We consider the Euler equations for incompressible fluid flow on a Euclidean space ${{\bf R}^d}$; we will label ${{\bf R}^d}$ as the “Eulerian space” ${{\bf R}^d_E}$ (or “Euclidean space”, or “physical space”) to distinguish it from the “Lagrangian space” ${{\bf R}^d_L}$ (or “labels space”) that we will introduce shortly (but the reader is free to also ignore the ${E}$ or ${L}$ subscripts if he or she wishes). Elements of Eulerian space ${{\bf R}^d_E}$ will be referred to by symbols such as ${x}$, we use ${dx}$ to denote Lebesgue measure on ${{\bf R}^d_E}$ and we will use ${x^1,\dots,x^d}$ for the ${d}$ coordinates of ${x}$, and use indices such as ${i,j,k}$ to index these coordinates (with the usual summation conventions), for instance ${\partial_i}$ denotes partial differentiation along the ${x^i}$ coordinate. (We use superscripts for coordinates ${x^i}$ instead of subscripts ${x_i}$ to be compatible with some differential geometry notation that we will use shortly; in particular, when using the summation notation, we will now be matching subscripts with superscripts for the pair of indices being summed.)
In Eulerian coordinates, the Euler equations read

$\displaystyle \partial_t u + u \cdot \nabla u = - \nabla p \ \ \ \ \ (1)$

$\displaystyle \nabla \cdot u = 0$

where ${u: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}^d_E}$ is the velocity field and ${p: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}}$ is the pressure field. These are functions of time ${t \in [0,T)}$ and on the spatial location variable ${x \in {\bf R}^d_E}$. We will refer to the coordinates ${(t,x) = (t,x^1,\dots,x^d)}$ as Eulerian coordinates. However, if one reviews the physical derivation of the Euler equations from 254A Notes 0, before one takes the continuum limit, the fundamental unknowns were not the velocity field ${u}$ or the pressure field ${p}$, but rather the trajectories ${(x^{(a)}(t))_{a \in A}}$, which can be thought of as a single function ${x: [0,T) \times A \rightarrow {\bf R}^d_E}$ from the coordinates ${(t,a)}$ (where ${t}$ is a time and ${a}$ is an element of the label set ${A}$) to ${{\bf R}^d}$. The relationship between the trajectories ${x^{(a)}(t) = x(t,a)}$ and the velocity field was given by the informal relationship

$\displaystyle \partial_t x(t,a) \approx u( t, x(t,a) ). \ \ \ \ \ (2)$

We will refer to the coordinates ${(t,a)}$ as (discrete) Lagrangian coordinates for describing the fluid.
In view of this, it is natural to ask whether there is an alternate way to formulate the continuum limit of incompressible inviscid fluids, by using a continuous version ${(t,a)}$ of the Lagrangian coordinates, rather than Eulerian coordinates. This is indeed the case. Suppose for instance one has a smooth solution ${u, p}$ to the Euler equations on a spacetime slab ${[0,T) \times {\bf R}^d_E}$ in Eulerian coordinates; assume furthermore that the velocity field ${u}$ is uniformly bounded. We introduce another copy ${{\bf R}^d_L}$ of ${{\bf R}^d}$, which we call Lagrangian space or labels space; we use symbols such as ${a}$ to refer to elements of this space, ${da}$ to denote Lebesgue measure on ${{\bf R}^d_L}$, and ${a^1,\dots,a^d}$ to refer to the ${d}$ coordinates of ${a}$. We use indices such as ${\alpha,\beta,\gamma}$ to index these coordinates, thus for instance ${\partial_\alpha}$ denotes partial differentiation along the ${a^\alpha}$ coordinate. We will use summation conventions for both the Eulerian coordinates ${i,j,k}$ and the Lagrangian coordinates ${\alpha,\beta,\gamma}$, with an index being summed if it appears as both a subscript and a superscript in the same term. While ${{\bf R}^d_L}$ and ${{\bf R}^d_E}$ are of course isomorphic, we will try to refrain from identifying them, except perhaps at the initial time ${t=0}$ in order to fix the initialisation of Lagrangian coordinates.
Given a smooth and bounded velocity field ${u: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}^d_E}$, define a trajectory map for this velocity to be any smooth map ${X: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}^d_E}$ that obeys the ODE

$\displaystyle \partial_t X(t,a) = u( t, X(t,a) ); \ \ \ \ \ (3)$

in view of (2), this describes the trajectory (in ${{\bf R}^d_E}$) of a particle labeled by an element ${a}$ of ${{\bf R}^d_L}$. From the Picard existence theorem and the hypothesis that ${u}$ is smooth and bounded, such a map exists and is unique as long as one specifies the initial location ${X(0,a)}$ assigned to each label ${a}$. Traditionally, one chooses the initial condition

$\displaystyle X(0,a) = a \ \ \ \ \ (4)$

for ${a \in {\bf R}^d_L}$, so that we label each particle by its initial location at time ${t=0}$; we are also free to specify other initial conditions for the trajectory map if we please. Indeed, we have the freedom to “permute” the labels ${a \in {\bf R}^d_L}$ by an arbitrary diffeomorphism: if ${X: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}^d_E}$ is a trajectory map, and ${\pi: {\bf R}^d_L \rightarrow{\bf R}^d_L}$ is any diffeomorphism (a smooth map whose inverse exists and is also smooth), then the map ${X \circ \pi: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}^d_E}$ is also a trajectory map, albeit one with different initial conditions ${X(0,a)}$.
Despite the popularity of the initial condition (4), we will try to keep conceptually separate the Eulerian space ${{\bf R}^d_E}$ from the Lagrangian space ${{\bf R}^d_L}$, as they play different physical roles in the interpretation of the fluid; for instance, while the Euclidean metric ${d\eta^2 = dx^1 dx^1 + \dots + dx^d dx^d}$ is an important feature of Eulerian space ${{\bf R}^d_E}$, it is not a geometrically natural structure to use in Lagrangian space ${{\bf R}^d_L}$. We have the following more general version of Exercise 8 from 254A Notes 2:

Exercise 1 Let ${u: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}^d_E}$ be smooth and bounded.

• If ${X_0: {\bf R}^d_L \rightarrow {\bf R}^d_E}$ is a smooth map, show that there exists a unique smooth trajectory map ${X: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}^d_E}$ with initial condition ${X(0,a) = X_0(a)}$ for all ${a \in {\bf R}^d_L}$.
• Show that if ${X_0}$ is a diffeomorphism and ${t \in [0,T)}$, then the map ${X(t): a \mapsto X(t,a)}$ is also a diffeomorphism.

Remark 2 The first of the Euler equations (1) can now be written in the form

$\displaystyle \frac{d^2}{dt^2} X(t,a) = - (\nabla p)( t, X(t,a) ) \ \ \ \ \ (5)$

which can be viewed as a continuous limit of Newton’s first law ${m^{(a)} \frac{d^2}{dt^2} x^{(a)}(t) = F^{(a)}(t)}$.

Call a diffeomorphism ${Y: {\bf R}^d_L \rightarrow {\bf R}^d_E}$ (oriented) volume preserving if one has the equation

$\displaystyle \mathrm{det}( \nabla Y )(a) = 1 \ \ \ \ \ (6)$

for all ${a \in {\bf R}^d_L}$, where the total differential ${\nabla Y}$ is the ${d \times d}$ matrix with entries ${\partial_\alpha Y^i}$ for ${\alpha = 1,\dots,d}$ and ${i=1,\dots,d}$, where ${Y^1,\dots,Y^d:{\bf R}^d_L \rightarrow {\bf R}}$ are the components of ${Y}$. (If one wishes, one can also view ${\nabla Y}$ as a linear transformation from the tangent space ${T_a {\bf R}^d_L}$ of Lagrangian space at ${a}$ to the tangent space ${T_{Y(a)} {\bf R}^d_E}$ of Eulerian space at ${Y(a)}$.) Equivalently, ${Y}$ is orientation preserving and one has a Jacobian-free change of variables formula

$\displaystyle \int_{{\bf R}^d_F} f( Y(a) )\ da = \int_{{\bf R}^d_E} f(x)\ dx$

for all ${f \in C_c({\bf R}^d_E \rightarrow {\bf R})}$, which is in turn equivalent to ${Y(E) \subset {\bf R}^d_E}$ having the same Lebesgue measure as ${E}$ for any measurable set ${E \subset {\bf R}^d_L}$.
The divergence-free condition ${\nabla \cdot u = 0}$ then can be nicely expressed in terms of volume-preserving properties of the trajectory maps ${X}$, in a manner which confirms the interpretation of this condition as an incompressibility condition on the fluid:

Lemma 3 Let ${u: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}^d_E}$ be smooth and bounded, let ${X_0: {\bf R}^d_L \rightarrow {\bf R}^d_E}$ be a volume-preserving diffeomorphism, and let ${X: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}^d_E}$ be the trajectory map. Then the following are equivalent:

• ${\nabla \cdot u = 0}$ on ${[0,T) \times {\bf R}^d_E}$.
• ${X(t): {\bf R}^d_L \rightarrow {\bf R}^d_E}$ is volume-preserving for all ${t \in [0,T)}$.

Proof: Since ${X_0}$ is orientation-preserving, we see from continuity that ${X(t)}$ is also orientation-preserving. Suppose that ${X(t)}$ is also volume-preserving, then for any ${f \in C^\infty_c({\bf R}^d_E \rightarrow {\bf R})}$ we have the conservation law

$\displaystyle \int_{{\bf R}^d_L} f( X(t,a) )\ da = \int_{{\bf R}^d_E} f(x)\ dx$

for all ${t \in [0,T)}$. Differentiating in time using the chain rule and (3) we conclude that

$\displaystyle \int_{{\bf R}^d_L} (u(t) \cdot \nabla f)( X(t,a)) \ da = 0$

for all ${t \in [0,T)}$, and hence by change of variables

$\displaystyle \int_{{\bf R}^d_E} (u(t) \cdot \nabla f)(x) \ dx = 0$

which by integration by parts gives

$\displaystyle \int_{{\bf R}^d_E} (\nabla \cdot u(t,x)) f(x)\ dx = 0$

for all ${f \in C^\infty_c({\bf R}^d_E \rightarrow {\bf R})}$ and ${t \in [0,T)}$, so ${u}$ is divergence-free.
To prove the converse implication, it is convenient to introduce the labels map ${A:[0,T) \times {\bf R}^d_E \rightarrow {\bf R}^d_L}$, defined by setting ${A(t): {\bf R}^d_E \rightarrow {\bf R}^d_L}$ to be the inverse of the diffeomorphism ${X(t): {\bf R}^d_L \rightarrow {\bf R}^d_E}$, thus

$\displaystyle A(t, X(t,a)) = a$

for all ${(t,a) \in [0,T) \times {\bf R}^d_L}$. By the implicit function theorem, ${A}$ is smooth, and by differentiating the above equation in time using (3) we see that

$\displaystyle D_t A(t,x) = 0$

where ${D_t}$ is the usual material derivative

$\displaystyle D_t := \partial_t + u \cdot \nabla \ \ \ \ \ (7)$

acting on functions on ${[0,T) \times {\bf R}^d_E}$. If ${u}$ is divergence-free, we have from integration by parts that

$\displaystyle \partial_t \int_{{\bf R}^d_E} \phi(t,x)\ dx = \int_{{\bf R}^d_E} D_t \phi(t,x)\ dx$

for any test function ${\phi: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}}$. In particular, for any ${g \in C^\infty_c({\bf R}^d_L \rightarrow {\bf R})}$, we can calculate

$\displaystyle \partial_t \int_{{\bf R}^d_E} g( A(t,x) )\ dx = \int_{{\bf R}^d_E} D_t (g(A(t,x)))\ dx$

$\displaystyle = \int_{{\bf R}^d_E} 0\ dx$

and hence

$\displaystyle \int_{{\bf R}^d_E} g(A(t,x))\ dx = \int_{{\bf R}^d_E} g(A(0,x))\ dx$

for any ${t \in [0,T)}$. Since ${X_0}$ is volume-preserving, so is ${A(0)}$, thus

$\displaystyle \int_{{\bf R}^d_E} g \circ A(t)\ dx = \int_{{\bf R}^d_L} g\ da.$

Thus ${A(t)}$ is volume-preserving, and hence ${X(t)}$ is also. $\Box$

Exercise 4 Let ${M: [0,T) \rightarrow \mathrm{GL}_d({\bf R})}$ be a continuously differentiable map from the time interval ${[0,T)}$ to the general linear group ${\mathrm{GL}_d({\bf R})}$ of invertible ${d \times d}$ matrices. Establish Jacobi’s formula

$\displaystyle \partial_t \det(M(t)) = \det(M(t)) \mathrm{tr}( M(t)^{-1} \partial_t M(t) )$

and use this and (6) to give an alternate proof of Lemma 3 that does not involve any integration in space.

Remark 5 One can view the use of Lagrangian coordinates as an extension of the method of characteristics. Indeed, from the chain rule we see that for any smooth function ${f: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}}$ of Eulerian spacetime, one has

$\displaystyle \frac{d}{dt} f(t,X(t,a)) = (D_t f)(t,X(t,a))$

and hence any transport equation that in Eulerian coordinates takes the form

$\displaystyle D_t f = g$

for smooth functions ${f,g: [0,T) \times {\bf R}^d_E \rightarrow {\bf R}}$ of Eulerian spacetime is equivalent to the ODE

$\displaystyle \frac{d}{dt} F = G$

where ${F,G: [0,T) \times {\bf R}^d_L \rightarrow {\bf R}}$ are the smooth functions of Lagrangian spacetime defined by

$\displaystyle F(t,a) := f(t,X(t,a)); \quad G(t,a) := g(t,X(t,a)).$

In this set of notes we recall some basic differential geometry notation, particularly with regards to pullbacks and Lie derivatives of differential forms and other tensor fields on manifolds such as ${{\bf R}^d_E}$ and ${{\bf R}^d_L}$, and explore how the Euler equations look in this notation. Our discussion will be entirely formal in nature; we will assume that all functions have enough smoothness and decay at infinity to justify the relevant calculations. (It is possible to work rigorously in Lagrangian coordinates – see for instance the work of Ebin and Marsden – but we will not do so here.) As a general rule, Lagrangian coordinates tend to be somewhat less convenient to use than Eulerian coordinates for establishing the basic analytic properties of the Euler equations, such as local existence, uniqueness, and continuous dependence on the data; however, they are quite good at clarifying the more algebraic properties of these equations, such as conservation laws and the variational nature of the equations. It may well be that in the future we will be able to use the Lagrangian formalism more effectively on the analytic side of the subject also.

Remark 6 One can also write the Navier-Stokes equations in Lagrangian coordinates, but the equations are not expressed in a favourable form in these coordinates, as the Laplacian ${\Delta}$ appearing in the viscosity term becomes replaced with a time-varying Laplace-Beltrami operator. As such, we will not discuss the Lagrangian coordinate formulation of Navier-Stokes here.

This coming fall quarter, I am teaching a class on topics in the mathematical theory of incompressible fluid equations, focusing particularly on the incompressible Euler and Navier-Stokes equations. These two equations are by no means the only equations used to model fluids, but I will focus on these two equations in this course to narrow the focus down to something manageable. I have not fully decided on the choice of topics to cover in this course, but I would probably begin with some core topics such as local well-posedness theory and blowup criteria, conservation laws, and construction of weak solutions, then move on to some topics such as boundary layers and the Prandtl equations, the Euler-Poincare-Arnold interpretation of the Euler equations as an infinite dimensional geodesic flow, and some discussion of the Onsager conjecture. I will probably also continue to more advanced and recent topics in the winter quarter.

In this initial set of notes, we begin by reviewing the physical derivation of the Euler and Navier-Stokes equations from the first principles of Newtonian mechanics, and specifically from Newton’s famous three laws of motion. Strictly speaking, this derivation is not needed for the mathematical analysis of these equations, which can be viewed if one wishes as an arbitrarily chosen system of partial differential equations without any physical motivation; however, I feel that the derivation sheds some insight and intuition on these equations, and is also worth knowing on purely intellectual grounds regardless of its mathematical consequences. I also find it instructive to actually see the journey from Newton’s law

$\displaystyle F = ma$

to the seemingly rather different-looking law

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

$\displaystyle \nabla \cdot u = 0$

for incompressible Navier-Stokes (or, if one drops the viscosity term ${\nu \Delta u}$, the Euler equations).

Our discussion in this set of notes is physical rather than mathematical, and so we will not be working at mathematical levels of rigour and precision. In particular we will be fairly casual about interchanging summations, limits, and integrals, we will manipulate approximate identities ${X \approx Y}$ as if they were exact identities (e.g., by differentiating both sides of the approximate identity), and we will not attempt to verify any regularity or convergence hypotheses in the expressions being manipulated. (The same holds for the exercises in this text, which also do not need to be justified at mathematical levels of rigour.) Of course, once we resume the mathematical portion of this course in subsequent notes, such issues will be an important focus of careful attention. This is a basic division of labour in mathematical modeling: non-rigorous heuristic reasoning is used to derive a mathematical model from physical (or other “real-life”) principles, but once a precise model is obtained, the analysis of that model should be completely rigorous if at all possible (even if this requires applying the model to regimes which do not correspond to the original physical motivation of that model). See the discussion by John Ball quoted at the end of these slides of Gero Friesecke for an expansion of these points.

Note: our treatment here will differ slightly from that presented in many fluid mechanics texts, in that it will emphasise first-principles derivations from many-particle systems, rather than relying on bulk laws of physics, such as the laws of thermodynamics, which we will not cover here. (However, the derivations from bulk laws tend to be more robust, in that they are not as reliant on assumptions about the particular interactions between particles. In particular, the physical hypotheses we assume in this post are probably quite a bit stronger than the minimal assumptions needed to justify the Euler or Navier-Stokes equations, which can hold even in situations in which one or more of the hypotheses assumed here break down.)

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

$\displaystyle (\partial_t + u_x \partial_x+ u_y \partial_y) u_x = -\partial_x p \ \ \ \ \ (1)$

$\displaystyle (\partial_t + u_x \partial_x+ u_y \partial_y) u_y = \rho - \partial_y p \ \ \ \ \ (2)$

$\displaystyle (\partial_t + u_x \partial_x+ u_y \partial_y) \rho = 0 \ \ \ \ \ (3)$

$\displaystyle \partial_x u_x + \partial_y u_y = 0 \ \ \ \ \ (4)$

where ${u: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}^2}$ is the velocity field, ${p: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}}$ is the pressure field, and ${\rho: {\bf R} \times {\bf R}^2 \rightarrow {\bf R}}$ 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 ${D_t := \partial_t + u_x \partial_x + u_y \partial_y}$, one can abbreviate these equations as

$\displaystyle D_t u_x = -\partial_x p$

$\displaystyle D_t u_y = \rho - \partial_y p$

$\displaystyle D_t \rho = 0$

$\displaystyle \partial_x u_x + \partial_y u_y = 0.$

One can eliminate the role of the pressure ${p}$ by working with the vorticity ${\omega := \partial_x u_y - \partial_y u_x}$. A standard calculation then leads us to the equivalent “vorticity-stream” formulation

$\displaystyle D_t \omega = \partial_x \rho$

$\displaystyle D_t \rho = 0$

$\displaystyle \omega = \partial_x u_y - \partial_y u_x$

$\displaystyle \partial_x u_y + \partial_y u_y = 0$

of the Boussinesq equations. The latter two equations can be used to recover the velocity field ${u}$ from the vorticity ${\omega}$ by the Biot-Savart law

$\displaystyle u_x := -\partial_y \Delta^{-1} \omega; \quad u_y = \partial_x \Delta^{-1} \omega.$

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

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

$\displaystyle \nabla \cdot u = 0$

where now the velocity field ${u: {\bf R} \times {\bf R}^3 \rightarrow {\bf R}^3}$ and pressure field ${p: {\bf R} \times {\bf R}^3 \rightarrow {\bf R}}$ are over the three-dimensional domain ${{\bf R}^3}$. If one expresses ${{\bf R}^3}$ in polar coordinates ${(z,r,\theta)}$ then one can write the velocity vector field ${u}$ in these coordinates as

$\displaystyle u = u^z \frac{d}{dz} + u^r \frac{d}{dr} + u^\theta \frac{d}{d\theta}.$

If we make the axial symmetry assumption that these components, as well as ${p}$, do not depend on the ${\theta}$ variable, thus

$\displaystyle \partial_\theta u^z, \partial_\theta u^r, \partial_\theta u^\theta, \partial_\theta p = 0,$

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

$\displaystyle \tilde D_t \omega = \frac{1}{r^4} \partial_z \rho \ \ \ \ \ (5)$

$\displaystyle \tilde D_t \rho = 0 \ \ \ \ \ (6)$

$\displaystyle \omega = \frac{1}{r} (\partial_z u^r - \partial_r u^z) \ \ \ \ \ (7)$

$\displaystyle \partial_z(ru^z) + \partial_r(ru^r) = 0 \ \ \ \ \ (8)$

where ${\tilde D_t := \partial_t + u^z \partial_z + u^r \partial_r}$ is the modified material derivative, and ${\rho}$ is the field ${\rho := (r u^\theta)^2}$. This is almost identical with the Boussinesq equations except for some additional powers of ${r}$; thus, the intuition is that the Boussinesq equations are a simplified model for axially symmetric Euler flows when one stays away from the axis ${r=0}$ and also does not wander off to ${r=\infty}$.

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 ${r=1}$, 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

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

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

of the Euler equations to an arbitrary Riemannian manifold ${(M,g)}$ (ignoring any issues of boundary conditions for this discussion), where ${u: {\bf R} \rightarrow \Gamma(TM)}$ is a time-dependent vector field, ${p: {\bf R} \rightarrow C^\infty(M)}$ is a time-dependent scalar field, and ${\nabla_u}$ is the covariant derivative along ${u}$ using the Levi-Civita connection ${\nabla}$. In Penrose abstract index notation (using the Levi-Civita connection ${\nabla}$, and raising and lowering indices using the metric ${g = g_{ij}}$), the equations of motion become

$\displaystyle \partial_t u^i + u^j \nabla_j u^i = - \nabla^i p \ \ \ \ \ (9)$

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

in coordinates, this becomes

$\displaystyle \partial_t u^i + u^j (\partial_j u^i + \Gamma^i_{jk} u^k) = - g^{ij} \partial_j p$

$\displaystyle \partial_i u^i + \Gamma^i_{ik} u^k = 0 \ \ \ \ \ (10)$

where the Christoffel symbols ${\Gamma^i_{jk}}$ are given by the formula

$\displaystyle \Gamma^i_{jk} := \frac{1}{2} g^{il} (\partial_j g_{lk} + \partial_k g_{lj} - \partial_l g_{jk}),$

where ${g^{il}}$ is the inverse to the metric tensor ${g_{il}}$. If the coordinates are chosen so that the volume form ${dg}$ is the Euclidean volume form ${dx}$, thus ${\mathrm{det}(g)=1}$, then on differentiating we have ${g^{ij} \partial_k g_{ij} = 0}$, and hence ${\Gamma^i_{ik} = 0}$, and so the divergence-free equation (10) simplifies in this case to ${\partial_i u^i = 0}$. The Ebin-Marsden Euler equations are the natural generalisation of the Euler equations to arbitrary manifolds; for instance, they (formally) conserve the kinetic energy

$\displaystyle \frac{1}{2} \int_M |u|_g^2\ dg = \frac{1}{2} \int_M g_{ij} u^i u^j\ dg$

and 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 specific four-dimensional manifold in question is the space ${{\bf R} \times {\bf R}^+ \times {\bf R}/{\bf Z} \times {\bf R}/{\bf Z}}$ with metric

$\displaystyle dx^2 + dy^2 + y^{-1} dz^2 + y dw^2$

and solutions to the Boussinesq equation on ${{\bf R} \times {\bf R}^+}$ 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 ${\rho}$ appearing in the Boussinesq equations) can be incorporated into the dynamics via fluctuations in the Riemannian metric ${g}$). I am writing the details below the fold (partly for my own benefit).

I’ve been meaning to return to fluids for some time now, in order to build upon my construction two years ago of a solution to an averaged Navier-Stokes equation that exhibited finite time blowup. (I recently spoke on this work in the recent conference in Princeton in honour of Sergiu Klainerman; my slides for that talk are here.)

One of the biggest deficiencies with my previous result is the fact that the averaged Navier-Stokes equation does not enjoy any good equation for the vorticity ${\omega = \nabla \times u}$, in contrast to the true Navier-Stokes equations which, when written in vorticity-stream formulation, become

$\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u + \nu \Delta \omega$

$\displaystyle u = (-\Delta)^{-1} (\nabla \times \omega).$

(Throughout this post we will be working in three spatial dimensions ${{\bf R}^3}$.) So one of my main near-term goals in this area is to exhibit an equation resembling Navier-Stokes as much as possible which enjoys a vorticity equation, and for which there is finite time blowup.

Heuristically, this task should be easier for the Euler equations (i.e. the zero viscosity case ${\nu=0}$ of Navier-Stokes) than the viscous Navier-Stokes equation, as one expects the viscosity to only make it easier for the solution to stay regular. Indeed, morally speaking, the assertion that finite time blowup solutions of Navier-Stokes exist should be roughly equivalent to the assertion that finite time blowup solutions of Euler exist which are “Type I” in the sense that all Navier-Stokes-critical and Navier-Stokes-subcritical norms of this solution go to infinity (which, as explained in the above slides, heuristically means that the effects of viscosity are negligible when compared against the nonlinear components of the equation). In vorticity-stream formulation, the Euler equations can be written as

$\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u$

$\displaystyle u = (-\Delta)^{-1} (\nabla \times \omega).$

As discussed in this previous blog post, a natural generalisation of this system of equations is the system

$\displaystyle \partial_t \omega + (u \cdot \nabla) \omega = (\omega \cdot \nabla) u \ \ \ \ \ (1)$

$\displaystyle u = T (-\Delta)^{-1} (\nabla \times \omega).$

where ${T}$ is a linear operator on divergence-free vector fields that is “zeroth order” in some sense; ideally it should also be invertible, self-adjoint, and positive definite (in order to have a Hamiltonian that is comparable to the kinetic energy ${\frac{1}{2} \int_{{\bf R}^3} |u|^2}$). (In the previous blog post, it was observed that the surface quasi-geostrophic (SQG) equation could be embedded in a system of the form (1).) The system (1) has many features in common with the Euler equations; for instance vortex lines are transported by the velocity field ${u}$, and Kelvin’s circulation theorem is still valid.

So far, I have not been able to fully achieve this goal. However, I have the following partial result, stated somewhat informally:

Theorem 1 There is a “zeroth order” linear operator ${T}$ (which, unfortunately, is not invertible, self-adjoint, or positive definite) for which the system (1) exhibits smooth solutions that blowup in finite time.

The operator ${T}$ constructed is not quite a zeroth-order pseudodifferential operator; it is instead merely in the “forbidden” symbol class ${S^0_{1,1}}$, and more precisely it takes the form

$\displaystyle T v = \sum_{j \in {\bf Z}} 2^{3j} \langle v, \phi_j \rangle \psi_j \ \ \ \ \ (2)$

for some compactly supported divergence-free ${\phi,\psi}$ of mean zero with

$\displaystyle \phi_j(x) := \phi(2^j x); \quad \psi_j(x) := \psi(2^j x)$

being ${L^2}$ rescalings of ${\phi,\psi}$. This operator is still bounded on all ${L^p({\bf R}^3)}$ spaces ${1 < p < \infty}$, and so is arguably still a zeroth order operator, though not as convincingly as I would like. Another, less significant, issue with the result is that the solution constructed does not have good spatial decay properties, but this is mostly for convenience and it is likely that the construction can be localised to give solutions that have reasonable decay in space. But the biggest drawback of this theorem is the fact that ${T}$ is not invertible, self-adjoint, or positive definite, so in particular there is no non-negative Hamiltonian for this equation. It may be that some modification of the arguments below can fix these issues, but I have so far been unable to do so. Still, the construction does show that the circulation theorem is insufficient by itself to prevent blowup.

We sketch the proof of the above theorem as follows. We use the barrier method, introducing the time-varying hyperboloid domains

$\displaystyle \Omega(t) := \{ (r,\theta,z): r^2 \leq 1-t + z^2 \}$

for ${t>0}$ (expressed in cylindrical coordinates ${(r,\theta,z)}$). We will select initial data ${\omega(0)}$ to be ${\omega(0,r,\theta,z) = (0,0,\eta(r))}$ for some non-negative even bump function ${\eta}$ supported on ${[-1,1]}$, normalised so that

$\displaystyle \int\int \eta(r)\ r dr d\theta = 1;$

in particular ${\omega(0)}$ is divergence-free supported in ${\Omega(0)}$, with vortex lines connecting ${z=-\infty}$ to ${z=+\infty}$. Suppose for contradiction that we have a smooth solution ${\omega}$ to (1) with this initial data; to simplify the discussion we assume that the solution behaves well at spatial infinity (this can be justified with the choice (2) of vorticity-stream operator, but we will not do so here). Since the domains ${\Omega(t)}$ disconnect ${z=-\infty}$ from ${z=+\infty}$ at time ${t=1}$, there must exist a time ${0 < T_* < 1}$ which is the first time where the support of ${\omega(T_*)}$ touches the boundary of ${\Omega(T_*)}$, with ${\omega(t)}$ supported in ${\Omega(t)}$.

From (1) we see that the support of ${\omega(t)}$ is transported by the velocity field ${u(t)}$. Thus, at the point of contact of the support of ${\omega(T_*)}$ with the boundary of ${\Omega(T_*)}$, the inward component of the velocity field ${u(T_*)}$ cannot exceed the inward velocity of ${\Omega(T_*)}$. We will construct the functions ${\phi,\psi}$ so that this is not the case, leading to the desired contradiction. (Geometrically, what is going on here is that the operator ${T}$ is pinching the flow to pass through the narrow cylinder ${\{ z, r = O( \sqrt{1-t} )\}}$, leading to a singularity by time ${t=1}$ at the latest.)

First we observe from conservation of circulation, and from the fact that ${\omega(t)}$ is supported in ${\Omega(t)}$, that the integrals

$\displaystyle \int\int \omega_z(t,r,\theta,z) \ r dr d\theta$

are constant in both space and time for ${0 \leq t \leq T_*}$. From the choice of initial data we thus have

$\displaystyle \int\int \omega_z(t,r,\theta,z) \ r dr d\theta = 1$

for all ${t \leq T_*}$ and all ${z}$. On the other hand, if ${T}$ is of the form (2) with ${\phi = \nabla \times \eta}$ for some bump function ${\eta = (0,0,\eta_z)}$ that only has ${z}$-components, then ${\phi}$ is divergence-free with mean zero, and

$\displaystyle \langle (-\Delta) (\nabla \times \omega), \phi_j \rangle = 2^{-j} \langle (-\Delta) (\nabla \times \omega), \nabla \times \eta_j \rangle$

$\displaystyle = 2^{-j} \langle \omega, \eta_j \rangle$

$\displaystyle = 2^{-j} \int\int\int \omega_z(t,r,\theta,z) \eta_z(2^j r, \theta, 2^j z)\ r dr d\theta dz,$

where ${\eta_j(x) := \eta(2^j x)}$. We choose ${\eta_z}$ to be supported in the slab ${\{ C \leq z \leq 2C\}}$ for some large constant ${C}$, and to equal a function ${f(z)}$ depending only on ${z}$ on the cylinder ${\{ C \leq z \leq 2C; r \leq 10C \}}$, normalised so that ${\int f(z)\ dz = 1}$. If ${C/2^j \geq (1-t)^{1/2}}$, then ${\Omega(t)}$ passes through this cylinder, and we conclude that

$\displaystyle \langle (-\Delta) (\nabla \times \omega), \phi_j \rangle = -2^{-j} \int f(2^j z)\ dz$

$\displaystyle = 2^{-2j}.$

Inserting ths into (2), (1) we conclude that

$\displaystyle u = \sum_{j: C/2^j \geq (1-t)^{1/2}} 2^j \psi_j + \sum_{j: C/2^j < (1-t)^{1/2}} c_j(t) \psi_j$

for some coefficients ${c_j(t)}$. We will not be able to control these coefficients ${c_j(t)}$, but fortunately we only need to understand ${u}$ on the boundary ${\partial \Omega(t)}$, for which ${r+|z| \gg (1-t)^{1/2}}$. So, if ${\psi}$ happens to be supported on an annulus ${1 \ll r+|z| \ll 1}$, then ${\psi_j}$ vanishes on ${\partial \Omega(t)}$ if ${C}$ is large enough. We then have

$\displaystyle u = \sum_j 2^j \psi_j$

on the boundary of ${\partial \Omega(t)}$.

Let ${\Phi(r,\theta,z)}$ be a function of the form

$\displaystyle \Phi(r,\theta,z) = C z \varphi(z/r)$

where ${\varphi}$ is a bump function supported on ${[-2,2]}$ that equals ${1}$ on ${[-1,1]}$. We can perform a dyadic decomposition ${\Phi = \sum_j \Psi_j}$ where

$\displaystyle \Psi_j(r,\theta,z) = \Phi(r,\theta,z) a(2^j r)$

where ${a}$ is a bump function supported on ${[1/2,2]}$ with ${\sum_j a(2^j r) = 1}$. If we then set

$\displaystyle \psi_j = \frac{2^{-j}}{r} (-\partial_z \Psi_j, 0, \partial_r \Psi_j)$

then one can check that ${\psi_j(x) = \psi(2^j x)}$ for a function ${\psi}$ that is divergence-free and mean zero, and supported on the annulus ${1 \ll r+|z| \ll 1}$, and

$\displaystyle \sum_j 2^j \psi_j = \frac{1}{r} (-\partial_z \Phi, 0, \partial_r \Phi)$

so on ${\partial \Omega(t)}$ (where ${|z| \leq r}$) we have

$\displaystyle u = (-\frac{C}{r}, 0, 0 ).$

One can manually check that the inward velocity of this vector on ${\partial \Omega(t)}$ exceeds the inward velocity of ${\Omega(t)}$ if ${C}$ is large enough, and the claim follows.

Remark 2 The type of blowup suggested by this construction, where a unit amount of circulation is squeezed into a narrow cylinder, is of “Type II” with respect to the Navier-Stokes scaling, because Navier-Stokes-critical norms such ${L^3({\bf R}^3)}$ (or at least ${L^{3,\infty}({\bf R}^3)}$) look like they stay bounded during this squeezing procedure (the velocity field is of size about ${2^j}$ in cylinders of radius and length about ${2^j}$). So even if the various issues with ${T}$ are repaired, it does not seem likely that this construction can be directly adapted to obtain a corresponding blowup for a Navier-Stokes type equation. To get a “Type I” blowup that is consistent with Kelvin’s circulation theorem, it seems that one needs to coil the vortex lines around a loop multiple times in order to get increased circulation in a small space. This seems possible to pull off to me – there don’t appear to be any unavoidable obstructions coming from topology, scaling, or conservation laws – but would require a more complicated construction than the one given above.

Throughout this post, we will work only at the formal level of analysis, ignoring issues of convergence of integrals, justifying differentiation under the integral sign, and so forth. (Rigorous justification of the conservation laws and other identities arising from the formal manipulations below can usually be established in an a posteriori fashion once the identities are in hand, without the need to rigorously justify the manipulations used to come up with these identities).

It is a remarkable fact in the theory of differential equations that many of the ordinary and partial differential equations that are of interest (particularly in geometric PDE, or PDE arising from mathematical physics) admit a variational formulation; thus, a collection ${\Phi: \Omega \rightarrow M}$ of one or more fields on a domain ${\Omega}$ taking values in a space ${M}$ will solve the differential equation of interest if and only if ${\Phi}$ is a critical point to the functional

$\displaystyle J[\Phi] := \int_\Omega L( x, \Phi(x), D\Phi(x) )\ dx \ \ \ \ \ (1)$

involving the fields ${\Phi}$ and their first derivatives ${D\Phi}$, where the Lagrangian ${L: \Sigma \rightarrow {\bf R}}$ is a function on the vector bundle ${\Sigma}$ over ${\Omega \times M}$ consisting of triples ${(x, q, \dot q)}$ with ${x \in \Omega}$, ${q \in M}$, and ${\dot q: T_x \Omega \rightarrow T_q M}$ a linear transformation; we also usually keep the boundary data of ${\Phi}$ fixed in case ${\Omega}$ has a non-trivial boundary, although we will ignore these issues here. (We also ignore the possibility of having additional constraints imposed on ${\Phi}$ and ${D\Phi}$, which require the machinery of Lagrange multipliers to deal with, but which will only serve as a distraction for the current discussion.) It is common to use local coordinates to parameterise ${\Omega}$ as ${{\bf R}^d}$ and ${M}$ as ${{\bf R}^n}$, in which case ${\Sigma}$ can be viewed locally as a function on ${{\bf R}^d \times {\bf R}^n \times {\bf R}^{dn}}$.

Example 1 (Geodesic flow) Take ${\Omega = [0,1]}$ and ${M = (M,g)}$ to be a Riemannian manifold, which we will write locally in coordinates as ${{\bf R}^n}$ with metric ${g_{ij}(q)}$ for ${i,j=1,\dots,n}$. A geodesic ${\gamma: [0,1] \rightarrow M}$ is then a critical point (keeping ${\gamma(0),\gamma(1)}$ fixed) of the energy functional

$\displaystyle J[\gamma] := \frac{1}{2} \int_0^1 g_{\gamma(t)}( D\gamma(t), D\gamma(t) )\ dt$

or in coordinates (ignoring coordinate patch issues, and using the usual summation conventions)

$\displaystyle J[\gamma] = \frac{1}{2} \int_0^1 g_{ij}(\gamma(t)) \dot \gamma^i(t) \dot \gamma^j(t)\ dt.$

As discussed in this previous post, both the Euler equations for rigid body motion, and the Euler equations for incompressible inviscid flow, can be interpreted as geodesic flow (though in the latter case, one has to work really formally, as the manifold ${M}$ is now infinite dimensional).

More generally, if ${\Omega = (\Omega,h)}$ is itself a Riemannian manifold, which we write locally in coordinates as ${{\bf R}^d}$ with metric ${h_{ab}(x)}$ for ${a,b=1,\dots,d}$, then a harmonic map ${\Phi: \Omega \rightarrow M}$ is a critical point of the energy functional

$\displaystyle J[\Phi] := \frac{1}{2} \int_\Omega h(x) \otimes g_{\gamma(x)}( D\gamma(x), D\gamma(x) )\ dh(x)$

or in coordinates (again ignoring coordinate patch issues)

$\displaystyle J[\Phi] = \frac{1}{2} \int_{{\bf R}^d} h_{ab}(x) g_{ij}(\Phi(x)) (\partial_a \Phi^i(x)) (\partial_b \Phi^j(x))\ \sqrt{\det(h(x))}\ dx.$

If we replace the Riemannian manifold ${\Omega}$ by a Lorentzian manifold, such as Minkowski space ${{\bf R}^{1+3}}$, then the notion of a harmonic map is replaced by that of a wave map, which generalises the scalar wave equation (which corresponds to the case ${M={\bf R}}$).

Example 2 (${N}$-particle interactions) Take ${\Omega = {\bf R}}$ and ${M = {\bf R}^3 \otimes {\bf R}^N}$; then a function ${\Phi: \Omega \rightarrow M}$ can be interpreted as a collection of ${N}$ trajectories ${q_1,\dots,q_N: {\bf R} \rightarrow {\bf R}^3}$ in space, which we give a physical interpretation as the trajectories of ${N}$ particles. If we assign each particle a positive mass ${m_1,\dots,m_N > 0}$, and also introduce a potential energy function ${V: M \rightarrow {\bf R}}$, then it turns out that Newton’s laws of motion ${F=ma}$ in this context (with the force ${F_i}$ on the ${i^{th}}$ particle being given by the conservative force ${-\nabla_{q_i} V}$) are equivalent to the trajectories ${q_1,\dots,q_N}$ being a critical point of the action functional

$\displaystyle J[\Phi] := \int_{\bf R} \sum_{i=1}^N \frac{1}{2} m_i |\dot q_i(t)|^2 - V( q_1(t),\dots,q_N(t) )\ dt.$

Formally, if ${\Phi = \Phi_0}$ is a critical point of a functional ${J[\Phi]}$, this means that

$\displaystyle \frac{d}{ds} J[ \Phi[s] ]|_{s=0} = 0$

whenever ${s \mapsto \Phi[s]}$ is a (smooth) deformation with ${\Phi[0]=\Phi_0}$ (and with ${\Phi[s]}$ respecting whatever boundary conditions are appropriate). Interchanging the derivative and integral, we (formally, at least) arrive at

$\displaystyle \int_\Omega \frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0}\ dx = 0. \ \ \ \ \ (2)$

Write ${\delta \Phi := \frac{d}{ds} \Phi[s]|_{s=0}}$ for the infinitesimal deformation of ${\Phi_0}$. By the chain rule, ${\frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0}}$ can be expressed in terms of ${x, \Phi_0(x), \delta \Phi(x), D\Phi_0(x), D \delta \Phi(x)}$. In coordinates, we have

$\displaystyle \frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0} = \delta \Phi^i(x) L_{q^i}(x,\Phi_0(x), D\Phi_0(x)) \ \ \ \ \ (3)$

$\displaystyle + \partial_{x^a} \delta \Phi^i(x) L_{\partial_{x^a} q^i} (x,\Phi_0(x), D\Phi_0(x)),$

where we parameterise ${\Sigma}$ by ${x, (q^i)_{i=1,\dots,n}, (\partial_{x^a} q^i)_{a=1,\dots,d; i=1,\dots,n}}$, and we use subscripts on ${L}$ to denote partial derivatives in the various coefficients. (One can of course work in a coordinate-free manner here if one really wants to, but the notation becomes a little cumbersome due to the need to carefully split up the tangent space of ${\Sigma}$, and we will not do so here.) Thus we can view (2) as an integral identity that asserts the vanishing of a certain integral, whose integrand involves ${x, \Phi_0(x), \delta \Phi(x), D\Phi_0(x), D \delta \Phi(x)}$, where ${\delta \Phi}$ vanishes at the boundary but is otherwise unconstrained.

A general rule of thumb in PDE and calculus of variations is that whenever one has an integral identity of the form ${\int_\Omega F(x)\ dx = 0}$ for some class of functions ${F}$ that vanishes on the boundary, then there must be an associated differential identity ${F = \hbox{div} X}$ that justifies this integral identity through Stokes’ theorem. This rule of thumb helps explain why integration by parts is used so frequently in PDE to justify integral identities. The rule of thumb can fail when one is dealing with “global” or “cohomologically non-trivial” integral identities of a topological nature, such as the Gauss-Bonnet or Kazhdan-Warner identities, but is quite reliable for “local” or “cohomologically trivial” identities, such as those arising from calculus of variations.

In any case, if we apply this rule to (2), we expect that the integrand ${\frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0}}$ should be expressible as a spatial divergence. This is indeed the case:

Proposition 1 (Formal) Let ${\Phi = \Phi_0}$ be a critical point of the functional ${J[\Phi]}$ defined in (1). Then for any deformation ${s \mapsto \Phi[s]}$ with ${\Phi[0] = \Phi_0}$, we have

$\displaystyle \frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0} = \hbox{div} X \ \ \ \ \ (4)$

where ${X}$ is the vector field that is expressible in coordinates as

$\displaystyle X^a := \delta \Phi^i(x) L_{\partial_{x^a} q^i}(x,\Phi_0(x), D\Phi_0(x)). \ \ \ \ \ (5)$

Proof: Comparing (4) with (3), we see that the claim is equivalent to the Euler-Lagrange equation

$\displaystyle L_{q^i}(x,\Phi_0(x), D\Phi_0(x)) - \partial_{x^a} L_{\partial_{x^a} q^i}(x,\Phi_0(x), D\Phi_0(x)) = 0. \ \ \ \ \ (6)$

The same computation, together with an integration by parts, shows that (2) may be rewritten as

$\displaystyle \int_\Omega ( L_{q^i}(x,\Phi_0(x), D\Phi_0(x)) - \partial_{x^a} L_{\partial_{x^a} q^i}(x,\Phi_0(x), D\Phi_0(x)) ) \delta \Phi^i(x)\ dx = 0.$

Since ${\delta \Phi^i(x)}$ is unconstrained on the interior of ${\Omega}$, the claim (6) follows (at a formal level, at least). $\Box$

Many variational problems also enjoy one-parameter continuous symmetries: given any field ${\Phi_0}$ (not necessarily a critical point), one can place that field in a one-parameter family ${s \mapsto \Phi[s]}$ with ${\Phi[0] = \Phi_0}$, such that

$\displaystyle J[ \Phi[s] ] = J[ \Phi[0] ]$

for all ${s}$; in particular,

$\displaystyle \frac{d}{ds} J[ \Phi[s] ]|_{s=0} = 0,$

which can be written as (2) as before. Applying the previous rule of thumb, we thus expect another divergence identity

$\displaystyle \frac{d}{ds} L( x, \Phi[s](x), D\Phi[s](x) )|_{s=0} = \hbox{div} Y \ \ \ \ \ (7)$

whenever ${s \mapsto \Phi[s]}$ arises from a continuous one-parameter symmetry. This expectation is indeed the case in many examples. For instance, if the spatial domain ${\Omega}$ is the Euclidean space ${{\bf R}^d}$, and the Lagrangian (when expressed in coordinates) has no direct dependence on the spatial variable ${x}$, thus

$\displaystyle L( x, \Phi(x), D\Phi(x) ) = L( \Phi(x), D\Phi(x) ), \ \ \ \ \ (8)$

then we obtain ${d}$ translation symmetries

$\displaystyle \Phi[s](x) := \Phi(x - s e^a )$

for ${a=1,\dots,d}$, where ${e^1,\dots,e^d}$ is the standard basis for ${{\bf R}^d}$. For a fixed ${a}$, the left-hand side of (7) then becomes

$\displaystyle \frac{d}{ds} L( \Phi(x-se^a), D\Phi(x-se^a) )|_{s=0} = -\partial_{x^a} [ L( \Phi(x), D\Phi(x) ) ]$

$\displaystyle = \hbox{div} Y$

where ${Y(x) = - L(\Phi(x), D\Phi(x)) e^a}$. Another common type of symmetry is a pointwise symmetry, in which

$\displaystyle L( x, \Phi[s](x), D\Phi[s](x) ) = L( x, \Phi[0](x), D\Phi[0](x) ) \ \ \ \ \ (9)$

for all ${x}$, in which case (7) clearly holds with ${Y=0}$.

If we subtract (4) from (7), we obtain the celebrated theorem of Noether linking symmetries with conservation laws:

Theorem 2 (Noether’s theorem) Suppose that ${\Phi_0}$ is a critical point of the functional (1), and let ${\Phi[s]}$ be a one-parameter continuous symmetry with ${\Phi[0] = \Phi_0}$. Let ${X}$ be the vector field in (5), and let ${Y}$ be the vector field in (7). Then we have the pointwise conservation law

$\displaystyle \hbox{div}(X-Y) = 0.$

In particular, for one-dimensional variational problems, in which ${\Omega \subset {\bf R}}$, we have the conservation law ${(X-Y)(t) = (X-Y)(0)}$ for all ${t \in \Omega}$ (assuming of course that ${\Omega}$ is connected and contains ${0}$).

Noether’s theorem gives a systematic way to locate conservation laws for solutions to variational problems. For instance, if ${\Omega \subset {\bf R}}$ and the Lagrangian has no explicit time dependence, thus

$\displaystyle L(t, \Phi(t), \dot \Phi(t)) = L(\Phi(t), \dot \Phi(t)),$

then by using the time translation symmetry ${\Phi[s](t) := \Phi(t-s)}$, we have

$\displaystyle Y(t) = - L( \Phi(t), \dot\Phi(t) )$

as discussed previously, whereas we have ${\delta \Phi(t) = - \dot \Phi(t)}$, and hence by (5)

$\displaystyle X(t) := - \dot \Phi^i(x) L_{\dot q^i}(\Phi(t), \dot \Phi(t)),$

and so Noether’s theorem gives conservation of the Hamiltonian

$\displaystyle H(t) := \dot \Phi^i(x) L_{\dot q^i}(\Phi(t), \dot \Phi(t))- L(\Phi(t), \dot \Phi(t)). \ \ \ \ \ (10)$

For instance, for geodesic flow, the Hamiltonian works out to be

$\displaystyle H(t) = \frac{1}{2} g_{ij}(\gamma(t)) \dot \gamma^i(t) \dot \gamma^j(t),$

so we see that the speed of the geodesic is conserved over time.

For pointwise symmetries (9), ${Y}$ vanishes, and so Noether’s theorem simplifies to ${\hbox{div} X = 0}$; in the one-dimensional case ${\Omega \subset {\bf R}}$, we thus see from (5) that the quantity

$\displaystyle \delta \Phi^i(t) L_{\dot q^i}(t,\Phi_0(t), \dot \Phi_0(t)) \ \ \ \ \ (11)$

is conserved in time. For instance, for the ${N}$-particle system in Example 2, if we have the translation invariance

$\displaystyle V( q_1 + h, \dots, q_N + h ) = V( q_1, \dots, q_N )$

for all ${q_1,\dots,q_N,h \in {\bf R}^3}$, then we have the pointwise translation symmetry

$\displaystyle q_i[s](t) := q_i(t) + s e^j$

for all ${i=1,\dots,N}$, ${s \in{\bf R}}$ and some ${j=1,\dots,3}$, in which case ${\dot q_i(t) = e^j}$, and the conserved quantity (11) becomes

$\displaystyle \sum_{i=1}^n m_i \dot q_i^j(t);$

as ${j=1,\dots,3}$ was arbitrary, this establishes conservation of the total momentum

$\displaystyle \sum_{i=1}^n m_i \dot q_i(t).$

Similarly, if we have the rotation invariance

$\displaystyle V( R q_1, \dots, Rq_N ) = V( q_1, \dots, q_N )$

for any ${q_1,\dots,q_N \in {\bf R}^3}$ and ${R \in SO(3)}$, then we have the pointwise rotation symmetry

$\displaystyle q_i[s](t) := \exp( s A ) q_i(t)$

for any skew-symmetric real ${3 \times 3}$ matrix ${A}$, in which case ${\dot q_i(t) = A q_i(t)}$, and the conserved quantity (11) becomes

$\displaystyle \sum_{i=1}^n m_i \langle A q_i(t), \dot q_i(t) \rangle;$

since ${A}$ is an arbitrary skew-symmetric matrix, this establishes conservation of the total angular momentum

$\displaystyle \sum_{i=1}^n m_i q_i(t) \wedge \dot q_i(t).$

Below the fold, I will describe how Noether’s theorem can be used to locate all of the conserved quantities for the Euler equations of inviscid fluid flow, discussed in this previous post, by interpreting that flow as geodesic flow in an infinite dimensional manifold.

As we are all now very much aware, tsunamis are water waves that start in the deep ocean, usually because of an underwater earthquake (though tsunamis can also be caused by underwater landslides or volcanoes), and then propagate towards shore. Initially, tsunamis have relatively small amplitude (a metre or so is typical), which would seem to render them as harmless as wind waves. And indeed, tsunamis often pass by ships in deep ocean without anyone on board even noticing.

However, being generated by an event as large as an earthquake, the wavelength of the tsunami is huge – 200 kilometres is typical (in contrast with wind waves, whose wavelengths are typically closer to 100 metres). In particular, the wavelength of the tsunami is far greater than the depth of the ocean (which is typically 2-3 kilometres). As such, even in the deep ocean, the dynamics of tsunamis are essentially governed by the shallow water equations. One consequence of these equations is that the speed of propagation ${v}$ of a tsunami can be approximated by the formula

$\displaystyle v \approx \sqrt{g b} \ \ \ \ \ (1)$

where ${b}$ is the depth of the ocean, and ${g \approx 9.8 ms^{-2}}$ is the force of gravity. As such, tsunamis in deep water move very fast – speeds such as 500 kilometres per hour (300 miles per hour) are quite typical; enough to travel from Japan to the US, for instance, in less than a day. Ultimately, this is due to the incompressibility of water (and conservation of mass); the massive net pressure (or more precisely, spatial variations in this pressure) of a very broad and deep wave of water forces the profile of the wave to move horizontally at vast speeds. (Note though that this is the phase velocity of the tsunami wave, and not the velocity of the water molecues themselves, which are far slower.)

As the tsunami approaches shore, the depth ${b}$ of course decreases, causing the tsunami to slow down, at a rate proportional to the square root of the depth, as per (1). Unfortunately, wave shoaling then forces the amplitude ${A}$ to increase at an inverse rate governed by Green’s law,

$\displaystyle A \propto \frac{1}{b^{1/4}} \ \ \ \ \ (2)$

at least until the amplitude becomes comparable to the water depth (at which point the assumptions that underlie the above approximate results break down; also, in two (horizontal) spatial dimensions there will be some decay of amplitude as the tsunami spreads outwards). If one starts with a tsunami whose initial amplitude was ${A_0}$ at depth ${b_0}$ and computes the point at which the amplitude ${A}$ and depth ${b}$ become comparable using the proportionality relationship (2), some high school algebra then reveals that at this point, amplitude of a tsunami (and the depth of the water) is about ${A_0^{4/5} b_0^{1/5}}$. Thus, for instance, a tsunami with initial amplitude of one metre at a depth of 2 kilometres can end up with a final amplitude of about 5 metres near shore, while still traveling at about ten metres per second (35 kilometres per hour, or 22 miles per hour), and we have all now seen the impact that can have when it hits shore.

While tsunamis are far too massive of an event to be able to control (at least in the deep ocean), we can at least model them mathematically, allowing one to predict their impact at various places along the coast with high accuracy. (For instance, here is a video of the NOAA’s model of the March 11 tsunami, which has matched up very well with subsequent measurements.) The full equations and numerical methods used to perform such models are somewhat sophisticated, but by making a large number of simplifying assumptions, it is relatively easy to come up with a rough model that already predicts the basic features of tsunami propagation, such as the velocity formula (1) and the amplitude proportionality law (2). I give this (standard) derivation below the fold. The argument will largely be heuristic in nature; there are very interesting analytic issues in actually justifying many of the steps below rigorously, but I will not discuss these matters here.