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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).

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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.

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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.

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