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We now approach conformal maps from yet another perspective. Given an open subset {U} of the complex numbers {{\bf C}}, define a univalent function on {U} to be a holomorphic function {f: U \rightarrow {\bf C}} that is also injective. We will primarily be studying this concept in the case when {U} is the unit disk {D(0,1) := \{ z \in {\bf C}: |z| < 1 \}}.

Clearly, a univalent function {f: D(0,1) \rightarrow {\bf C}} on the unit disk is a conformal map from {D(0,1)} to the image {f(D(0,1))}; in particular, {f(D(0,1))} is simply connected, and not all of {{\bf C}} (since otherwise the inverse map {f^{-1}: {\bf C} \rightarrow D(0,1)} would violate Liouville’s theorem). In the converse direction, the Riemann mapping theorem tells us that every open simply connected proper subset {V \subsetneq {\bf C}} of the complex numbers is the image of a univalent function on {D(0,1)}. Furthermore, if {V} contains the origin, then the univalent function {f: D(0,1) \rightarrow {\bf C}} with this image becomes unique once we normalise {f(0) = 0} and {f'(0) > 0}. Thus the Riemann mapping theorem provides a one-to-one correspondence between open simply connected proper subsets of the complex plane containing the origin, and univalent functions {f: D(0,1) \rightarrow {\bf C}} with {f(0)=0} and {f'(0)>0}. We will focus particular attention on the univalent functions {f: D(0,1) \rightarrow {\bf C}} with the normalisation {f(0)=0} and {f'(0)=1}; such functions will be called schlicht functions.

One basic example of a univalent function on {D(0,1)} is the Cayley transform {z \mapsto \frac{1+z}{1-z}}, which is a Möbius transformation from {D(0,1)} to the right half-plane {\{ \mathrm{Re}(z) > 0 \}}. (The slight variant {z \mapsto \frac{1-z}{1+z}} is also referred to as the Cayley transform, as is the closely related map {z \mapsto \frac{z-i}{z+i}}, which maps {D(0,1)} to the upper half-plane.) One can square this map to obtain a further univalent function {z \mapsto \left( \frac{1+z}{1-z} \right)^2}, which now maps {D(0,1)} to the complex numbers with the negative real axis {(-\infty,0]} removed. One can normalise this function to be schlicht to obtain the Koebe function

\displaystyle  f(z) := \frac{1}{4}\left( \left( \frac{1+z}{1-z} \right)^2 - 1\right) = \frac{z}{(1-z)^2}, \ \ \ \ \ (1)

which now maps {D(0,1)} to the complex numbers with the half-line {(-\infty,-1/4]} removed. A little more generally, for any {\theta \in {\bf R}} we have the rotated Koebe function

\displaystyle  f(z) := \frac{z}{(1 - e^{i\theta} z)^2} \ \ \ \ \ (2)

that is a schlicht function that maps {D(0,1)} to the complex numbers with the half-line {\{ -re^{-i\theta}: r \geq 1/4\}} removed.

Every schlicht function {f: D(0,1) \rightarrow {\bf C}} has a convergent Taylor expansion

\displaystyle  f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots

for some complex coefficients {a_1,a_2,\dots} with {a_1=1}. For instance, the Koebe function has the expansion

\displaystyle  f(z) = z + 2 z^2 + 3 z^3 + \dots = \sum_{n=1}^\infty n z^n

and similarly the rotated Koebe function has the expansion

\displaystyle  f(z) = z + 2 e^{i\theta} z^2 + 3 e^{2i\theta} z^3 + \dots = \sum_{n=1}^\infty n e^{(n-1)\theta} z^n.

Intuitively, the Koebe function and its rotations should be the “largest” schlicht functions available. This is formalised by the famous Bieberbach conjecture, which asserts that for any schlicht function, the coefficients {a_n} should obey the bound {|a_n| \leq n} for all {n}. After a large number of partial results, this conjecture was eventually solved by de Branges; see for instance this survey of Korevaar or this survey of Koepf for a history.

It turns out that to resolve these sorts of questions, it is convenient to restrict attention to schlicht functions {g: D(0,1) \rightarrow {\bf C}} that are odd, thus {g(-z)=-g(z)} for all {z}, and the Taylor expansion now reads

\displaystyle  g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots

for some complex coefficients {b_1,b_3,\dots} with {b_1=1}. One can transform a general schlicht function {f: D(0,1) \rightarrow {\bf C}} to an odd schlicht function {g: D(0,1) \rightarrow {\bf C}} by observing that the function {f(z^2)/z^2: D(0,1) \rightarrow {\bf C}}, after removing the singularity at zero, is a non-zero function that equals {1} at the origin, and thus (as {D(0,1)} is simply connected) has a unique holomorphic square root {(f(z^2)/z^2)^{1/2}} that also equals {1} at the origin. If one then sets

\displaystyle  g(z) := z (f(z^2)/z^2)^{1/2} \ \ \ \ \ (3)

it is not difficult to verify that {g} is an odd schlicht function which additionally obeys the equation

\displaystyle  f(z^2) = g(z)^2. \ \ \ \ \ (4)

Conversely, given an odd schlicht function {g}, the formula (4) uniquely determines a schlicht function {f}.

For instance, if {f} is the Koebe function (1), {g} becomes

\displaystyle  g(z) = \frac{z}{1-z^2} = z + z^3 + z^5 + \dots, \ \ \ \ \ (5)

which maps {D(0,1)} to the complex numbers with two slits {\{ \pm iy: y > 1/2 \}} removed, and if {f} is the rotated Koebe function (2), {g} becomes

\displaystyle  g(z) = \frac{z}{1- e^{i\theta} z^2} = z + e^{i\theta} z^3 + e^{2i\theta} z^5 + \dots. \ \ \ \ \ (6)

De Branges established the Bieberbach conjecture by first proving an analogous conjecture for odd schlicht functions known as Robertson’s conjecture. More precisely, we have

Theorem 1 (de Branges’ theorem) Let {n \geq 1} be a natural number.

  • (i) (Robertson conjecture) If {g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots} is an odd schlicht function, then

    \displaystyle  \sum_{k=1}^n |b_{2k-1}|^2 \leq n.

  • (ii) (Bieberbach conjecture) If {f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots} is a schlicht function, then

    \displaystyle  |a_n| \leq n.

It is easy to see that the Robertson conjecture for a given value of {n} implies the Bieberbach conjecture for the same value of {n}. Indeed, if {f(z) = a_1 z + a_2 z^2 + a_3 z^3 + \dots} is schlicht, and {g(z) = b_1 z + b_3 z^3 + b_5 z^5 + \dots} is the odd schlicht function given by (3), then from extracting the {z^{2n}} coefficient of (4) we obtain a formula

\displaystyle  a_n = \sum_{j=1}^n b_{2j-1} b_{2(n+1-j)-1}

for the coefficients of {f} in terms of the coefficients of {g}. Applying the Cauchy-Schwarz inequality, we derive the Bieberbach conjecture for this value of {n} from the Robertson conjecture for the same value of {n}. We remark that Littlewood and Paley had conjectured a stronger form {|b_{2k-1}| \leq 1} of Robertson’s conjecture, but this was disproved for {k=3} by Fekete and Szegö.

To prove the Robertson and Bieberbach conjectures, one first takes a logarithm and deduces both conjectures from a similar conjecture about the Taylor coefficients of {\log \frac{f(z)}{z}}, known as the Milin conjecture. Next, one continuously enlarges the image {f(D(0,1))} of the schlicht function to cover all of {{\bf C}}; done properly, this places the schlicht function {f} as the initial function {f = f_0} in a sequence {(f_t)_{t \geq 0}} of univalent maps {f_t: D(0,1) \rightarrow {\bf C}} known as a Loewner chain. The functions {f_t} obey a useful differential equation known as the Loewner equation, that involves an unspecified forcing term {\mu_t} (or {\theta(t)}, in the case that the image is a slit domain) coming from the boundary; this in turn gives useful differential equations for the Taylor coefficients of {f(z)}, {g(z)}, or {\log \frac{f(z)}{z}}. After some elementary calculus manipulations to “integrate” this equations, the Bieberbach, Robertson, and Milin conjectures are then reduced to establishing the non-negativity of a certain explicit hypergeometric function, which is non-trivial to prove (and will not be done here, except for small values of {n}) but for which several proofs exist in the literature.

The theory of Loewner chains subsequently became fundamental to a more recent topic in complex analysis, that of the Schramm-Loewner equation (SLE), which is the focus of the next and final set of notes.

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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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Let {P(z) = z^n + a_{n-1} z^{n-1} + \dots + a_0} be a monic polynomial of degree {n} with complex coefficients. Then by the fundamental theorem of algebra, we can factor {P} as

\displaystyle  P(z) = (z-z_1) \dots (z-z_n) \ \ \ \ \ (1)

for some complex zeroes {z_1,\dots,z_n} (possibly with repetition).

Now suppose we evolve {P} with respect to time by heat flow, creating a function {P(t,z)} of two variables with given initial data {P(0,z) = P(z)} for which

\displaystyle  \partial_t P(t,z) = \partial_{zz} P(t,z). \ \ \ \ \ (2)

On the space of polynomials of degree at most {n}, the operator {\partial_{zz}} is nilpotent, and one can solve this equation explicitly both forwards and backwards in time by the Taylor series

\displaystyle  P(t,z) = \sum_{j=0}^\infty \frac{t^j}{j!} \partial_z^{2j} P(0,z).

For instance, if one starts with a quadratic {P(0,z) = z^2 + bz + c}, then the polynomial evolves by the formula

\displaystyle  P(t,z) = z^2 + bz + (c+2t).

As the polynomial {P(t)} evolves in time, the zeroes {z_1(t),\dots,z_n(t)} evolve also. Assuming for sake of discussion that the zeroes are simple, the inverse function theorem tells us that the zeroes will (locally, at least) evolve smoothly in time. What are the dynamics of this evolution?

For instance, in the quadratic case, the quadratic formula tells us that the zeroes are

\displaystyle  z_1(t) = \frac{-b + \sqrt{b^2 - 4(c+2t)}}{2}

and

\displaystyle  z_2(t) = \frac{-b - \sqrt{b^2 - 4(c+2t)}}{2}

after arbitrarily choosing a branch of the square root. If {b,c} are real and the discriminant {b^2 - 4c} is initially positive, we see that we start with two real zeroes centred around {-b/2}, which then approach each other until time {t = \frac{b^2-4c}{8}}, at which point the roots collide and then move off from each other in an imaginary direction.

In the general case, we can obtain the equations of motion by implicitly differentiating the defining equation

\displaystyle  P( t, z_i(t) ) = 0

in time using (2) to obtain

\displaystyle  \partial_{zz} P( t, z_i(t) ) + \partial_t z_i(t) \partial_z P(t,z_i(t)) = 0.

To simplify notation we drop the explicit dependence on time, thus

\displaystyle  \partial_{zz} P(z_i) + (\partial_t z_i) \partial_z P(z_i)= 0.

From (1) and the product rule, we see that

\displaystyle  \partial_z P( z_i ) = \prod_{j:j \neq i} (z_i - z_j)

and

\displaystyle  \partial_{zz} P( z_i ) = 2 \sum_{k:k \neq i} \prod_{j:j \neq i,k} (z_i - z_j)

(where all indices are understood to range over {1,\dots,n}) leading to the equations of motion

\displaystyle  \partial_t z_i = \sum_{k:k \neq i} \frac{2}{z_k - z_i}, \ \ \ \ \ (3)

at least when one avoids those times in which there is a repeated zero. In the case when the zeroes {z_i} are real, each term {\frac{2}{z_k-z_i}} represents a (first-order) attraction in the dynamics between {z_i} and {z_k}, but the dynamics are more complicated for complex zeroes (e.g. purely imaginary zeroes will experience repulsion rather than attraction, as one already sees in the quadratic example). Curiously, this system resembles that of Dyson brownian motion (except with the brownian motion part removed, and time reversed). I learned of the connection between the ODE (3) and the heat equation from this paper of Csordas, Smith, and Varga, but perhaps it has been mentioned in earlier literature as well.

One interesting consequence of these equations is that if the zeroes are real at some time, then they will stay real as long as the zeroes do not collide. Let us now restrict attention to the case of real simple zeroes, in which case we will rename the zeroes as {x_i} instead of {z_i}, and order them as {x_1 < \dots < x_n}. The evolution

\displaystyle  \partial_t x_i = \sum_{k:k \neq i} \frac{2}{x_k - x_i}

can now be thought of as reverse gradient flow for the “entropy”

\displaystyle  H := -\sum_{i,j: i \neq j} \log |x_i - x_j|,

(which is also essentially the logarithm of the discriminant of the polynomial) since we have

\displaystyle  \partial_t x_i = \frac{\partial H}{\partial x_i}.

In particular, we have the monotonicity formula

\displaystyle  \partial_t H = 4E

where {E} is the “energy”

\displaystyle  E := \frac{1}{4} \sum_i (\frac{\partial H}{\partial x_i})^2

\displaystyle  = \sum_i (\sum_{k:k \neq i} \frac{1}{x_k-x_i})^2

\displaystyle  = \sum_{i,k: i \neq k} \frac{1}{(x_k-x_i)^2} + 2 \sum_{i,j,k: i,j,k \hbox{ distinct}} \frac{1}{(x_k-x_i)(x_j-x_i)}

\displaystyle  = \sum_{i,k: i \neq k} \frac{1}{(x_k-x_i)^2}

where in the last line we use the antisymmetrisation identity

\displaystyle  \frac{1}{(x_k-x_i)(x_j-x_i)} + \frac{1}{(x_i-x_j)(x_k-x_j)} + \frac{1}{(x_j-x_k)(x_i-x_k)} = 0.

Among other things, this shows that as one goes backwards in time, the entropy decreases, and so no collisions can occur to the past, only in the future, which is of course consistent with the attractive nature of the dynamics. As {H} is a convex function of the positions {x_1,\dots,x_n}, one expects {H} to also evolve in a convex manner in time, that is to say the energy {E} should be increasing. This is indeed the case:

Exercise 1 Show that

\displaystyle  \partial_t E = 2 \sum_{i,j: i \neq j} (\frac{2}{(x_i-x_j)^2} - \sum_{k: i,j,k \hbox{ distinct}} \frac{1}{(x_k-x_i)(x_k-x_j)})^2.

Symmetric polynomials of the zeroes are polynomial functions of the coefficients and should thus evolve in a polynomial fashion. One can compute this explicitly in simple cases. For instance, the center of mass is an invariant:

\displaystyle  \partial_t \frac{1}{n} \sum_i x_i = 0.

The variance decreases linearly:

Exercise 2 Establish the virial identity

\displaystyle  \partial_t \sum_{i,j} (x_i-x_j)^2 = - 4n^2(n-1).

As the variance (which is proportional to {\sum_{i,j} (x_i-x_j)^2}) cannot become negative, this identity shows that “finite time blowup” must occur – that the zeroes must collide at or before the time {\frac{1}{4n^2(n-1)} \sum_{i,j} (x_i-x_j)^2}.

Exercise 3 Show that the Stieltjes transform

\displaystyle  s(t,z) = \sum_i \frac{1}{x_i - z}

solves the viscous Burgers equation

\displaystyle  \partial_t s = \partial_{zz} s - 2 s \partial_z s,

either by using the original heat equation (2) and the identity {s = - \partial_z P / P}, or else by using the equations of motion (3). This relation between the Burgers equation and the heat equation is known as the Cole-Hopf transformation.

The paper of Csordas, Smith, and Varga mentioned previously gives some other bounds on the lifespan of the dynamics; roughly speaking, they show that if there is one pair of zeroes that are much closer to each other than to the other zeroes then they must collide in a short amount of time (unless there is a collision occuring even earlier at some other location). Their argument extends also to situations where there are an infinite number of zeroes, which they apply to get new results on Newman’s conjecture in analytic number theory. I would be curious to know of further places in the literature where this dynamics has been studied.

I’ve just uploaded to the arXiv my paper “On the universality of the incompressible Euler equation on compact manifolds“, submitted to Discrete and Continuous Dynamical Systems. This is a variant of my recent paper on the universality of potential well dynamics, but instead of trying to embed dynamical systems into a potential well {\partial_{tt} u = -\nabla V(u)}, here we try to embed dynamical systems into the incompressible Euler equations

\displaystyle  \partial_t u + \nabla_u u = - \mathrm{grad}_g p \ \ \ \ \ (1)

\displaystyle  \mathrm{div}_g u = 0

on a Riemannian manifold {(M,g)}. (One is particularly interested in the case of flat manifolds {M}, particularly {{\bf R}^3} or {({\bf R}/{\bf Z})^3}, but for the main result of this paper it is essential that one is permitted to consider curved manifolds.) This system, first studied by Ebin and Marsden, is the natural generalisation of the usual incompressible Euler equations to curved space; it 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 Euler equations can be viewed as a nonlinear equation in which the nonlinearity is a quadratic function of the velocity field {u}. It is thus natural to compare the Euler equations with quadratic ODE of the form

\displaystyle  \partial_t y = B(y,y) \ \ \ \ \ (2)

where {y: {\bf R} \rightarrow {\bf R}^n} is the unknown solution, and {B: {\bf R}^n \times {\bf R}^n \rightarrow {\bf R}^n} is a bilinear map, which we may assume without loss of generality to be symmetric. One can ask whether such an ODE may be linearly embedded into the Euler equations on some Riemannian manifold {(M,g)}, which means that there is an injective linear map {U: {\bf R}^n \rightarrow \Gamma(TM)} from {{\bf R}^n} to smooth vector fields on {M}, as well as a bilinear map {P: {\bf R}^n \times {\bf R}^n \rightarrow C^\infty(M)} to smooth scalar fields on {M}, such that the map {y \mapsto (U(y), P(y,y))} takes solutions to (2) to solutions to (1), or equivalently that

\displaystyle  U(B(y,y)) + \nabla_{U(y)} U(y) = - \mathrm{grad}_g P(y,y)

\displaystyle  \mathrm{div}_g U(y) = 0

for all {y \in {\bf R}^n}.

For simplicity let us restrict {M} to be compact. There is an obvious necessary condition for this embeddability to occur, which comes from energy conservation law for the Euler equations; unpacking everything, this implies that the bilinear form {B} in (2) has to obey a cancellation condition

\displaystyle  \langle B(y,y), y \rangle = 0 \ \ \ \ \ (3)

for some positive definite inner product {\langle, \rangle: {\bf R}^n \times {\bf R}^n \rightarrow {\bf R}} on {{\bf R}^n}. The main result of the paper is the converse to this statement: if {B} is a symmetric bilinear form obeying a cancellation condition (3), then it is possible to embed the equations (2) into the Euler equations (1) on some Riemannian manifold {(M,g)}; the catch is that this manifold will depend on the form {B} and on the dimension {n} (in fact in the construction I have, {M} is given explicitly as {SO(n) \times ({\bf R}/{\bf Z})^{n+1}}, with a funny metric on it that depends on {B}).

As a consequence, any finite dimensional portion of the usual “dyadic shell models” used as simplified toy models of the Euler equation, can actually be embedded into a genuine Euler equation, albeit on a high-dimensional and curved manifold. This includes portions of the self-similar “machine” I used in a previous paper to establish finite time blowup for an averaged version of the Navier-Stokes (or Euler) equations. Unfortunately, the result in this paper does not apply to infinite-dimensional ODE, so I cannot yet establish finite time blowup for the Euler equations on a (well-chosen) manifold. It does not seem so far beyond the realm of possibility, though, that this could be done in the relatively near future. In particular, the result here suggests that one could construct something resembling a universal Turing machine within an Euler flow on a manifold, which was one ingredient I would need to engineer such a finite time blowup.

The proof of the main theorem proceeds by an “elimination of variables” strategy that was used in some of my previous papers in this area, though in this particular case the Nash embedding theorem (or variants thereof) are not required. The first step is to lessen the dependence on the metric {g} by partially reformulating the Euler equations (1) in terms of the covelocity {g \cdot u} (which is a {1}-form) instead of the velocity {u}. Using the freedom to modify the dimension of the underlying manifold {M}, one can also decouple the metric {g} from the volume form that is used to obtain the divergence-free condition. At this point the metric can be eliminated, with a certain positive definiteness condition between the velocity and covelocity taking its place. After a substantial amount of trial and error (motivated by some “two-and-a-half-dimensional” reductions of the three-dimensional Euler equations, and also by playing around with a number of variants of the classic “separation of variables” strategy), I eventually found an ansatz for the velocity and covelocity that automatically solved most of the components of the Euler equations (as well as most of the positive definiteness requirements), as long as one could find a number of scalar fields that obeyed a certain nonlinear system of transport equations, and also obeyed a positive definiteness condition. Here I was stuck for a bit because the system I ended up with was overdetermined – more equations than unknowns. After trying a number of special cases I eventually found a solution to the transport system on the sphere, except that the scalar functions sometimes degenerated and so the positive definiteness property I wanted was only obeyed with positive semi-definiteness. I tried for some time to perturb this example into a strictly positive definite solution before eventually working out that this was not possible. Finally I had the brainwave to lift the solution from the sphere to an even more symmetric space, and this quickly led to the final solution of the problem, using the special orthogonal group rather than the sphere as the underlying domain. The solution ended up being rather simple in form, but it is still somewhat miraculous to me that it exists at all; in retrospect, given the overdetermined nature of the problem, relying on a large amount of symmetry to cut down the number of equations was basically the only hope.

I’ve just uploaded to the arXiv my paper “On the universality of potential well dynamics“, submitted to Dynamics of PDE. This is a spinoff from my previous paper on blowup of nonlinear wave equations, inspired by some conversations with Sungjin Oh. Here we focus mainly on the zero-dimensional case of such equations, namely the potential well equation

\displaystyle  \partial_{tt} u = - (\nabla F)(u) \ \ \ \ \ (1)

for a particle {u: {\bf R} \rightarrow {\bf R}^m} trapped in a potential well with potential {F: {\bf R}^m \rightarrow {\bf R}}, with {F(z) \rightarrow +\infty} as {z \rightarrow \infty}. This ODE always admits global solutions from arbitrary initial positions {u(0)} and initial velocities {\partial_t u(0)}, thanks to conservation of the Hamiltonian {\frac{1}{2} |\partial_t u|^2 + F(u)}. As this Hamiltonian is coercive (in that its level sets are compact), solutions to this equation are always almost periodic. On the other hand, as can already be seen using the harmonic oscillator {\partial_{tt} u = - k^2 u} (and direct sums of this system), this equation can generate periodic solutions, as well as quasiperiodic solutions.

All quasiperiodic motions are almost periodic. However, there are many examples of dynamical systems that admit solutions that are almost periodic but not quasiperiodic. So one can pose the question: are the dynamics of potential wells universal in the sense that they can capture all almost periodic solutions?

A precise question can be phrased as follows. Let {M} be a compact manifold, and let {X} be a smooth vector field on {M}; to avoid degeneracies, let us take {X} to be non-singular in the sense that it is everywhere non-vanishing. Then the trajectories of the first-order ODE

\displaystyle  \partial_t u = X(u) \ \ \ \ \ (2)

for {u: {\bf R} \rightarrow M} are always global and almost periodic. Can we then find a (coercive) potential {F: {\bf R}^m \rightarrow {\bf R}} for some {m}, as well as a smooth embedding {\phi: M \rightarrow {\bf R}^m}, such that every solution {u} to (2) pushes forward under {\phi} to a solution to (1)? (Actually, for technical reasons it is preferable to map into the phase space {{\bf R}^m \times {\bf R}^m}, rather than position space {{\bf R}^m}, but let us ignore this detail for this discussion.)

It turns out that the answer is no; there is a very specific obstruction. Given a pair {(M,X)} as above, define a strongly adapted {1}-form to be a {1}-form {\phi} on {M} such that {\phi(X)} is pointwise positive, and the Lie derivative {{\mathcal L}_X \phi} is an exact {1}-form. We then have

Theorem 1 A smooth compact non-singular dynamics {(M,X)} can be embedded smoothly in a potential well system if and only if it admits a strongly adapted {1}-form.

For the “only if” direction, the key point is that potential wells (viewed as a Hamiltonian flow on the phase space {{\bf R}^m \times {\bf R}^m}) admit a strongly adapted {1}-form, namely the canonical {1}-form {p dq}, whose Lie derivative is the derivative {dL} of the Lagrangian {L := \frac{1}{2} |\partial_t u|^2 - F(u)} and is thus exact. The converse “if” direction is mainly a consequence of the Nash embedding theorem, and follows the arguments used in my previous paper.

Interestingly, the same obstruction also works for potential wells in a more general Riemannian manifold than {{\bf R}^m}, or for nonlinear wave equations with a potential; combining the two, the obstruction is also present for wave maps with a potential.

It is then natural to ask whether this obstruction is non-trivial, in the sense that there are at least some examples of dynamics {(M,X)} that do not support strongly adapted {1}-forms (and hence cannot be modeled smoothly by the dynamics of a potential well, nonlinear wave equation, or wave maps). I posed this question on MathOverflow, and Robert Bryant provided a very nice construction, showing that the vector field {(\sin(2\pi x), \cos(2\pi x))} on the {2}-torus {({\bf R}/{\bf Z})^2} had no strongly adapted {1}-forms, and hence the dynamics of this vector field cannot be smoothly reproduced by a potential well, nonlinear wave equation, or wave map:

On the other hand, the suspension of any diffeomorphism does support a strongly adapted {1}-form (the derivative {dt} of the time coordinate), and using this and the previous theorem I was able to embed a universal Turing machine into a potential well. In particular, there are flows for an explicitly describable potential well whose trajectories have behavior that is undecidable using the usual ZFC axioms of set theory! So potential well dynamics are “effectively” universal, despite the presence of the aforementioned obstruction.

In my previous work on blowup for Navier-Stokes like equations, I speculated that if one could somehow replicate a universal Turing machine within the Euler equations, one could use this machine to create a “von Neumann machine” that replicated smaller versions of itself, which on iteration would lead to a finite time blowup. Now that such a mechanism is present in nonlinear wave equations, it is tempting to try to make this scheme work in that setting. Of course, in my previous paper I had already demonstrated finite time blowup, at least in a three-dimensional setting, but that was a relatively simple discretely self-similar blowup in which no computation occurred. This more complicated blowup scheme would be significantly more effort to set up, but would be proof-of-concept that the same scheme would in principle be possible for the Navier-Stokes equations, assuming somehow that one can embed a universal Turing machine into the Euler equations. (But I’m still hopelessly stuck on how to accomplish this latter task…)

Fifteen years ago, I wrote a paper entitled Global regularity of wave maps. II. Small energy in two dimensions, in which I established global regularity of wave maps from two spatial dimensions to the unit sphere, assuming that the initial data had small energy. Recently, Hao Jia (personal communication) discovered a small gap in the argument that requires a slightly non-trivial fix. The issue does not really affect the subsequent literature, because the main result has since been reproven and extended by methods that avoid the gap (see in particular this subsequent paper of Tataru), but I have decided to describe the gap and its fix on this blog.

I will assume familiarity with the notation of my paper. In Section 10, some complicated spaces {S[k] = S[k]({\bf R}^{1+n})} are constructed for each frequency scale {k}, and then a further space {S(c) = S(c)({\bf R}^{1+n})} is constructed for a given frequency envelope {c} by the formula

\displaystyle  \| \phi \|_{S(c)({\bf R}^{1+n})} := \|\phi \|_{L^\infty_t L^\infty_x({\bf R}^{1+n})} + \sup_k c_k^{-1} \| \phi_k \|_{S[k]({\bf R}^{1+n})} \ \ \ \ \ (1)

where {\phi_k := P_k \phi} is the Littlewood-Paley projection of {\phi} to frequency magnitudes {\sim 2^k}. Then, given a spacetime slab {[-T,T] \times {\bf R}^n}, we define the restrictions

\displaystyle  \| \phi \|_{S(c)([-T,T] \times {\bf R}^n)} := \inf \{ \| \tilde \phi \|_{S(c)({\bf R}^{1+n})}: \tilde \phi \downharpoonright_{[-T,T] \times {\bf R}^n} = \phi \}

where the infimum is taken over all extensions {\tilde \phi} of {\phi} to the Minkowski spacetime {{\bf R}^{1+n}}; similarly one defines

\displaystyle  \| \phi_k \|_{S_k([-T,T] \times {\bf R}^n)} := \inf \{ \| \tilde \phi_k \|_{S_k({\bf R}^{1+n})}: \tilde \phi_k \downharpoonright_{[-T,T] \times {\bf R}^n} = \phi_k \}.

The gap in the paper is as follows: it was implicitly assumed that one could restrict (1) to the slab {[-T,T] \times {\bf R}^n} to obtain the equality

\displaystyle  \| \phi \|_{S(c)([-T,T] \times {\bf R}^n)} = \|\phi \|_{L^\infty_t L^\infty_x([-T,T] \times {\bf R}^n)} + \sup_k c_k^{-1} \| \phi_k \|_{S[k]([-T,T] \times {\bf R}^n)}.

(This equality is implicitly used to establish the bound (36) in the paper.) Unfortunately, (1) only gives the lower bound, not the upper bound, and it is the upper bound which is needed here. The problem is that the extensions {\tilde \phi_k} of {\phi_k} that are optimal for computing {\| \phi_k \|_{S[k]([-T,T] \times {\bf R}^n)}} are not necessarily the Littlewood-Paley projections of the extensions {\tilde \phi} of {\phi} that are optimal for computing {\| \phi \|_{S(c)([-T,T] \times {\bf R}^n)}}.

To remedy the problem, one has to prove an upper bound of the form

\displaystyle  \| \phi \|_{S(c)([-T,T] \times {\bf R}^n)} \lesssim \|\phi \|_{L^\infty_t L^\infty_x([-T,T] \times {\bf R}^n)} + \sup_k c_k^{-1} \| \phi_k \|_{S[k]([-T,T] \times {\bf R}^n)}

for all Schwartz {\phi} (actually we need affinely Schwartz {\phi}, but one can easily normalise to the Schwartz case). Without loss of generality we may normalise the RHS to be {1}. Thus

\displaystyle  \|\phi \|_{L^\infty_t L^\infty_x([-T,T] \times {\bf R}^n)} \leq 1 \ \ \ \ \ (2)

and

\displaystyle  \|P_k \phi \|_{S[k]([-T,T] \times {\bf R}^n)} \leq c_k \ \ \ \ \ (3)

for each {k}, and one has to find a single extension {\tilde \phi} of {\phi} such that

\displaystyle  \|\tilde \phi \|_{L^\infty_t L^\infty_x({\bf R}^{1+n})} \lesssim 1 \ \ \ \ \ (4)

and

\displaystyle  \|P_k \tilde \phi \|_{S[k]({\bf R}^{1+n})} \lesssim c_k \ \ \ \ \ (5)

for each {k}. Achieving a {\tilde \phi} that obeys (4) is trivial (just extend {\phi} by zero), but such extensions do not necessarily obey (5). On the other hand, from (3) we can find extensions {\tilde \phi_k} of {P_k \phi} such that

\displaystyle  \|\tilde \phi_k \|_{S[k]({\bf R}^{1+n})} \lesssim c_k; \ \ \ \ \ (6)

the extension {\tilde \phi := \sum_k \tilde \phi_k} will then obey (5) (here we use Lemma 9 from my paper), but unfortunately is not guaranteed to obey (4) (the {S[k]} norm does control the {L^\infty_t L^\infty_x} norm, but a key point about frequency envelopes for the small energy regularity problem is that the coefficients {c_k}, while bounded, are not necessarily summable).

This can be fixed as follows. For each {k} we introduce a time cutoff {\eta_k} supported on {[-T-2^{-k}, T+2^{-k}]} that equals {1} on {[-T-2^{-k-1},T+2^{-k+1}]} and obeys the usual derivative estimates in between (the {j^{th}} time derivative of size {O_j(2^{jk})} for each {j}). Later we will prove the truncation estimate

\displaystyle  \| \eta_k \tilde \phi_k \|_{S[k]({\bf R}^{1+n})} \lesssim \| \tilde \phi_k \|_{S[k]({\bf R}^{1+n})}. \ \ \ \ \ (7)

Assuming this estimate, then if we set {\tilde \phi := \sum_k \eta_k \tilde \phi_k}, then using Lemma 9 in my paper and (6), (7) (and the local stability of frequency envelopes) we have the required property (5). (There is a technical issue arising from the fact that {\tilde \phi} is not necessarily Schwartz due to slow decay at temporal infinity, but by considering partial sums in the {k} summation and taking limits we can check that {\tilde \phi} is the strong limit of Schwartz functions, which suffices here; we omit the details for sake of exposition.) So the only issue is to establish (4), that is to say that

\displaystyle  \| \sum_k \eta_k(t) \tilde \phi_k(t) \|_{L^\infty_x({\bf R}^n)} \lesssim 1

for all {t \in {\bf R}}.

For {t \in [-T,T]} this is immediate from (2). Now suppose that {t \in [T+2^{k_0-1}, T+2^{k_0}]} for some integer {k_0} (the case when {t \in [-T-2^{k_0}, -T-2^{k_0-1}]} is treated similarly). Then we can split

\displaystyle  \sum_k \eta_k(t) \tilde \phi_k(t) = \Phi_1 + \Phi_2 + \Phi_3

where

\displaystyle  \Phi_1 := \sum_{k < k_0} \tilde \phi_k(T)

\displaystyle  \Phi_2 := \sum_{k < k_0} \tilde \phi_k(t) - \tilde \phi_k(T)

\displaystyle  \Phi_3 := \eta_{k_0}(t) \tilde \phi_{k_0}(t).

The contribution of the {\Phi_3} term is acceptable by (6) and estimate (82) from my paper. The term {\Phi_1} sums to {P_{<k_0} \phi(T)} which is acceptable by (2). So it remains to control the {L^\infty_x} norm of {\Phi_2}. By the triangle inequality and the fundamental theorem of calculus, we can bound

\displaystyle  \| \Phi_2 \|_{L^\infty_x} \leq (t-T) \sum_{k < k_0} \| \partial_t \tilde \phi_k \|_{L^\infty_t L^\infty_x({\bf R}^{1+n})}.

By hypothesis, {t-T \leq 2^{-k_0}}. Using the first term in (79) of my paper and Bernstein’s inequality followed by (6) we have

\displaystyle  \| \partial_t \tilde \phi_k \|_{L^\infty_t L^\infty_x({\bf R}^{1+n})} \lesssim 2^k \| \tilde \phi_k \|_{S[k]({\bf R}^{1+n})} \lesssim 2^k;

and then we are done by summing the geometric series in {k}.

It remains to prove the truncation estimate (7). This estimate is similar in spirit to the algebra estimates already in my paper, but unfortunately does not seem to follow immediately from these estimates as written, and so one has to repeat the somewhat lengthy decompositions and case checkings used to prove these estimates. We do this below the fold.

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I’ve just posted to the arXiv my paper “Finite time blowup for Lagrangian modifications of the three-dimensional Euler equation“. This paper is loosely in the spirit of other recent papers of mine in which I explore how close one can get to supercritical PDE of physical interest (such as the Euler and Navier-Stokes equations), while still being able to rigorously demonstrate finite time blowup for at least some choices of initial data. Here, the PDE we are trying to get close to is the incompressible inviscid Euler equations

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

\displaystyle \nabla \cdot u = 0

in three spatial dimensions, where {u} is the velocity vector field and {p} is the pressure field. In vorticity form, and viewing the vorticity {\omega} as a {2}-form (rather than a vector), we can rewrite this system using the language of differential geometry as

\displaystyle \partial_t \omega + {\mathcal L}_u \omega = 0

\displaystyle u = \delta \tilde \eta^{-1} \Delta^{-1} \omega

where {{\mathcal L}_u} is the Lie derivative along {u}, {\delta} is the codifferential (the adjoint of the differential {d}, or equivalently the negative of the divergence operator) that sends {k+1}-vector fields to {k}-vector fields, {\Delta} is the Hodge Laplacian, and {\tilde \eta} is the identification of {k}-vector fields with {k}-forms induced by the Euclidean metric {\tilde \eta}. The equation{u = \delta \tilde \eta^{-1} \Delta^{-1} \omega} can be viewed as the Biot-Savart law recovering velocity from vorticity, expressed in the language of differential geometry.

One can then generalise this system by replacing the operator {\tilde \eta^{-1} \Delta^{-1}} by a more general operator {A} from {2}-forms to {2}-vector fields, giving rise to what I call the generalised Euler equations

\displaystyle \partial_t \omega + {\mathcal L}_u \omega = 0

\displaystyle u = \delta A \omega.

For example, the surface quasi-geostrophic (SQG) equations can be written in this form, as discussed in this previous post. One can view {A \omega} (up to Hodge duality) as a vector potential for the velocity {u}, so it is natural to refer to {A} as a vector potential operator.

The generalised Euler equations carry much of the same geometric structure as the true Euler equations. For instance, the transport equation {\partial_t \omega + {\mathcal L}_u \omega = 0} is equivalent to the Kelvin circulation theorem, which in three dimensions also implies the transport of vortex streamlines and the conservation of helicity. If {A} is self-adjoint and positive definite, then the famous Euler-Poincaré interpretation of the true Euler equations as geodesic flow on an infinite dimensional Riemannian manifold of volume preserving diffeomorphisms (as discussed in this previous post) extends to the generalised Euler equations (with the operator {A} determining the new Riemannian metric to place on this manifold). In particular, the generalised Euler equations have a Lagrangian formulation, and so by Noether’s theorem we expect any continuous symmetry of the Lagrangian to lead to conserved quantities. Indeed, we have a conserved Hamiltonian {\frac{1}{2} \int \langle \omega, A \omega \rangle}, and any spatial symmetry of {A} leads to a conserved impulse (e.g. translation invariance leads to a conserved momentum, and rotation invariance leads to a conserved angular momentum). If {A} behaves like a pseudodifferential operator of order {-2} (as is the case with the true vector potential operator {\tilde \eta^{-1} \Delta^{-1}}), then it turns out that one can use energy methods to recover the same sort of classical local existence theory as for the true Euler equations (up to and including the famous Beale-Kato-Majda criterion for blowup).

The true Euler equations are suspected of admitting smooth localised solutions which blow up in finite time; there is now substantial numerical evidence for this blowup, but it has not been proven rigorously. The main purpose of this paper is to show that such finite time blowup can at least be established for certain generalised Euler equations that are somewhat close to the true Euler equations. This is similar in spirit to my previous paper on finite time blowup on averaged Navier-Stokes equations, with the main new feature here being that the modified equation continues to have a Lagrangian structure and a vorticity formulation, which was not the case with the averaged Navier-Stokes equation. On the other hand, the arguments here are not able to handle the presence of viscosity (basically because they rely crucially on the Kelvin circulation theorem, which is not available in the viscous case).

In fact, three different blowup constructions are presented (for three different choices of vector potential operator {A}). The first is a variant of one discussed previously on this blog, in which a “neck pinch” singularity for a vortex tube is created by using a non-self-adjoint vector potential operator, in which the velocity at the neck of the vortex tube is determined by the circulation of the vorticity somewhat further away from that neck, which when combined with conservation of circulation is enough to guarantee finite time blowup. This is a relatively easy construction of finite time blowup, and has the advantage of being rather stable (any initial data flowing through a narrow tube with a large positive circulation will blow up in finite time). On the other hand, it is not so surprising in the non-self-adjoint case that finite blowup can occur, as there is no conserved energy.

The second blowup construction is based on a connection between the two-dimensional SQG equation and the three-dimensional generalised Euler equations, discussed in this previous post. Namely, any solution to the former can be lifted to a “two and a half-dimensional” solution to the latter, in which the velocity and vorticity are translation-invariant in the vertical direction (but the velocity is still allowed to contain vertical components, so the flow is not completely horizontal). The same embedding also works to lift solutions to generalised SQG equations in two dimensions to solutions to generalised Euler equations in three dimensions. Conveniently, even if the vector potential operator for the generalised SQG equation fails to be self-adjoint, one can ensure that the three-dimensional vector potential operator is self-adjoint. Using this trick, together with a two-dimensional version of the first blowup construction, one can then construct a generalised Euler equation in three dimensions with a vector potential that is both self-adjoint and positive definite, and still admits solutions that blow up in finite time, though now the blowup is now a vortex sheet creasing at on a line, rather than a vortex tube pinching at a point.

This eliminates the main defect of the first blowup construction, but introduces two others. Firstly, the blowup is less stable, as it relies crucially on the initial data being translation-invariant in the vertical direction. Secondly, the solution is not spatially localised in the vertical direction (though it can be viewed as a compactly supported solution on the manifold {{\bf R}^2 \times {\bf R}/{\bf Z}}, rather than {{\bf R}^3}). The third and final blowup construction of the paper addresses the final defect, by replacing vertical translation symmetry with axial rotation symmetry around the vertical axis (basically, replacing Cartesian coordinates with cylindrical coordinates). It turns out that there is a more complicated way to embed two-dimensional generalised SQG equations into three-dimensional generalised Euler equations in which the solutions to the latter are now axially symmetric (but are allowed to “swirl” in the sense that the velocity field can have a non-zero angular component), while still keeping the vector potential operator self-adjoint and positive definite; the blowup is now that of a vortex ring creasing on a circle.

As with the previous papers in this series, these blowup constructions do not directly imply finite time blowup for the true Euler equations, but they do at least provide a barrier to establishing global regularity for these latter equations, in that one is forced to use some property of the true Euler equations that are not shared by these generalisations. They also suggest some possible blowup mechanisms for the true Euler equations (although unfortunately these mechanisms do not seem compatible with the addition of viscosity, so they do not seem to suggest a viable Navier-Stokes blowup mechanism).

Throughout this post we shall always work in the smooth category, thus all manifolds, maps, coordinate charts, and functions are assumed to be smooth unless explicitly stated otherwise.

A (real) manifold {M} can be defined in at least two ways. On one hand, one can define the manifold extrinsically, as a subset of some standard space such as a Euclidean space {{\bf R}^d}. On the other hand, one can define the manifold intrinsically, as a topological space equipped with an atlas of coordinate charts. The fundamental embedding theorems show that, under reasonable assumptions, the intrinsic and extrinsic approaches give the same classes of manifolds (up to isomorphism in various categories). For instance, we have the following (special case of) the Whitney embedding theorem:

Theorem 1 (Whitney embedding theorem) Let {M} be a compact manifold. Then there exists an embedding {u: M \rightarrow {\bf R}^d} from {M} to a Euclidean space {{\bf R}^d}.

In fact, if {M} is {n}-dimensional, one can take {d} to equal {2n}, which is often best possible (easy examples include the circle {{\bf R}/{\bf Z}} which embeds into {{\bf R}^2} but not {{\bf R}^1}, or the Klein bottle that embeds into {{\bf R}^4} but not {{\bf R}^3}). One can also relax the compactness hypothesis on {M} to second countability, but we will not pursue this extension here. We give a “cheap” proof of this theorem below the fold which allows one to take {d} equal to {2n+1}.

A significant strengthening of the Whitney embedding theorem is (a special case of) the Nash embedding theorem:

Theorem 2 (Nash embedding theorem) Let {(M,g)} be a compact Riemannian manifold. Then there exists a isometric embedding {u: M \rightarrow {\bf R}^d} from {M} to a Euclidean space {{\bf R}^d}.

In order to obtain the isometric embedding, the dimension {d} has to be a bit larger than what is needed for the Whitney embedding theorem; in this article of Gunther the bound

\displaystyle d = \max( n(n+5)/2, n(n+3)/2 + 5) \ \ \ \ \ (1)

 

is attained, which I believe is still the record for large {n}. (In the converse direction, one cannot do better than {d = \frac{n(n+1)}{2}}, basically because this is the number of degrees of freedom in the Riemannian metric {g}.) Nash’s original proof of theorem used what is now known as Nash-Moser inverse function theorem, but a subsequent simplification of Gunther allowed one to proceed using just the ordinary inverse function theorem (in Banach spaces).

I recently had the need to invoke the Nash embedding theorem to establish a blowup result for a nonlinear wave equation, which motivated me to go through the proof of the theorem more carefully. Below the fold I give a proof of the theorem that does not attempt to give an optimal value of {d}, but which hopefully isolates the main ideas of the argument (as simplified by Gunther). One advantage of not optimising in {d} is that it allows one to freely exploit the very useful tool of pairing together two maps {u_1: M \rightarrow {\bf R}^{d_1}}, {u_2: M \rightarrow {\bf R}^{d_2}} to form a combined map {(u_1,u_2): M \rightarrow {\bf R}^{d_1+d_2}} that can be closer to an embedding or an isometric embedding than the original maps {u_1,u_2}. This lets one perform a “divide and conquer” strategy in which one first starts with the simpler problem of constructing some “partial” embeddings of {M} and then pairs them together to form a “better” embedding.

In preparing these notes, I found the articles of Deane Yang and of Siyuan Lu to be helpful.

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I’ve just uploaded to the arXiv my paper Finite time blowup for high dimensional nonlinear wave systems with bounded smooth nonlinearity, submitted to Comm. PDE. This paper is in the same spirit as (though not directly related to) my previous paper on finite time blowup of supercritical NLW systems, and was inspired by a question posed to me some time ago by Jeffrey Rauch. Here, instead of looking at supercritical equations, we look at an extremely subcritical equation, namely a system of the form

\displaystyle \Box u = f(u) \ \ \ \ \ (1)

 

where {u: {\bf R}^{1+d} \rightarrow {\bf R}^m} is the unknown field, and {f: {\bf R}^m \rightarrow {\bf R}^m} is the nonlinearity, which we assume to have all derivatives bounded. A typical example of such an equation is the higher-dimensional sine-Gordon equation

\displaystyle \Box u = \sin u

for a scalar field {u: {\bf R}^{1+d} \rightarrow {\bf R}}. Here {\Box = -\partial_t^2 + \Delta} is the d’Alembertian operator. We restrict attention here to classical (i.e. smooth) solutions to (1).

We do not assume any Hamiltonian structure, so we do not require {f} to be a gradient {f = \nabla F} of a potential {F: {\bf R}^m \rightarrow {\bf R}}. But even without such Hamiltonian structure, the equation (1) is very well behaved, with many a priori bounds available. For instance, if the initial position {u_0(x) = u(0,x)} and initial velocity {u_1(x) = \partial_t u(0,x)} are smooth and compactly supported, then from finite speed of propagation {u(t)} has uniformly bounded compact support for all {t} in a bounded interval. As the nonlinearity {f} is bounded, this immediately places {f(u)} in {L^\infty_t L^2_x} in any bounded time interval, which by the energy inequality gives an a priori {L^\infty_t H^1_x} bound on {u} in this time interval. Next, from the chain rule we have

\displaystyle \nabla f(u) = (\nabla_{{\bf R}^m} f)(u) \nabla u

which (from the assumption that {\nabla_{{\bf R}^m} f} is bounded) shows that {f(u)} is in {L^\infty_t H^1_x}, which by the energy inequality again now gives an a priori {L^\infty_t H^2_x} bound on {u}.

One might expect that one could keep iterating this and obtain a priori bounds on {u} in arbitrarily smooth norms. In low dimensions such as {d \leq 3}, this is a fairly easy task, since the above estimates and Sobolev embedding already place one in {L^\infty_t L^\infty_x}, and the nonlinear map {f} is easily verified to preserve the space {L^\infty_t H^k_x \cap L^\infty_t L^\infty_x} for any natural number {k}, from which one obtains a priori bounds in any Sobolev space; from this and standard energy methods, one can then establish global regularity for this equation (that is to say, any smooth choice of initial data generates a global smooth solution). However, one starts running into trouble in higher dimensions, in which no {L^\infty_x} bound is available. The main problem is that even a really nice nonlinearity such as {u \mapsto \sin u} is unbounded in higher Sobolev norms. The estimates

\displaystyle |\sin u| \leq |u|

and

\displaystyle |\nabla(\sin u)| \leq |\nabla u|

ensure that the map {u \mapsto \sin u} is bounded in low regularity spaces like {L^2_x} or {H^1_x}, but one already runs into trouble with the second derivative

\displaystyle \nabla^2(\sin u) = (\cos u) \nabla^2 u - (\sin u) \nabla u \nabla u

where there is a troublesome lower order term of size {O( |\nabla u|^2 )} which becomes difficult to control in higher dimensions, preventing the map {u \mapsto \sin u} to be bounded in {H^2_x}. Ultimately, the issue here is that when {u} is not controlled in {L^\infty}, the function {\sin u} can oscillate at a much higher frequency than {u}; for instance, if {u} is the one-dimensional wave {u = A \sin(kx)}for some {k > 0} and {A>1}, then {u} oscillates at frequency {k}, but the function {\sin(u)= \sin(A \sin(kx))} more or less oscillates at the larger frequency {Ak}.

In medium dimensions, it is possible to use dispersive estimates for the wave equation (such as the famous Strichartz estimates) to overcome these problems. This line of inquiry was pursued (albeit for slightly different classes of nonlinearity {f} than those considered here) by Heinz-von Wahl, Pecher (in a series of papers), Brenner, and Brenner-von Wahl; to cut a long story short, one of the conclusions of these papers was that one had global regularity for equations such as (1) in dimensions {d \leq 9}. (I reprove this result using modern Strichartz estimate and Littlewood-Paley techniques in an appendix to my paper. The references given also allow for some growth in the nonlinearity {f}, but we will not detail the precise hypotheses used in these papers here.)

In my paper, I complement these positive results with an almost matching negative result:

Theorem 1 If {d \geq 11} and {m \geq 2}, then there exists a nonlinearity {f: {\bf R}^m \rightarrow {\bf R}^m} with all derivatives bounded, and a solution {u} to (1) that is smooth at time zero, but develops a singularity in finite time.

The construction crucially relies on the ability to choose the nonlinearity {f}, and also needs some injectivity properties on the solution {u: {\bf R}^{1+d} \rightarrow {\bf R}^m} (after making a symmetry reduction using an assumption of spherical symmetry to view {u} as a function of {1+1} variables rather than {1+d}) which restricts our counterexample to the {m \geq 2} case. Thus the model case of the higher-dimensional sine-Gordon equation {\Box u =\sin u} is not covered by our arguments. Nevertheless (as with previous finite-time blowup results discussed on this blog), one can view this result as a barrier to trying to prove regularity for equations such as {\Box u = \sin u} in eleven and higher dimensions, as any such argument must somehow use a property of that equation that is not applicable to the more general system (1).

Let us first give some back-of-the-envelope calculations suggesting why there could be finite time blowup in eleven and higher dimensions. For sake of this discussion let us restrict attention to the sine-Gordon equation {\Box u = \sin u}. The blowup ansatz we will use is as follows: for each frequency {N_j} in a sequence {1 < N_1 < N_2 < N_3 < \dots} of large quantities going to infinity, there will be a spacetime “cube” {Q_j = \{ (t,x): t \sim \frac{1}{N_j}; x = O(\frac{1}{N_j})\}} on which the solution {u} oscillates with “amplitude” {N_j^\alpha} and “frequency” {N_j}, where {\alpha>0} is an exponent to be chosen later; this ansatz is of course compatible with the uncertainty principle. Since {N_j^\alpha \rightarrow \infty} as {j \rightarrow \infty}, this will create a singularity at the spacetime origin {(0,0)}. To make this ansatz plausible, we wish to make the oscillation of {u} on {Q_j} driven primarily by the forcing term {\sin u} at {Q_{j-1}}. Thus, by Duhamel’s formula, we expect a relation roughly of the form

\displaystyle u(t,x) \approx \int \frac{\sin((s-t)\sqrt{-\Delta})}{\sqrt{-\Delta}} \sin(1_{Q_{j-1}} u(s)) (x)\ ds

on {Q_j}, where {\frac{\sin((s-t)\sqrt{-\Delta})}{\sqrt{-\Delta}}} is the usual free wave propagator, and {1_{Q_{j-1}}} is the indicator function of {Q_{j-1}}.

On {Q_{j-1}}, {u} oscillates with amplitude {N_{j-1}^\alpha} and frequency {N_{j-1}}, we expect the derivative {\nabla_{t,x} u} to be of size about {N_{j-1}^{\alpha+1}}, and so from the principle of stationary phase we expect {\sin(u)} to oscillate at frequency about {N_{j-1}^{\alpha+1}}. Since the wave propagator {\frac{\sin((s-t)\sqrt{-\Delta})}{\sqrt{-\Delta}}} preserves frequencies, and {u} is supposed to be of frequency {N_j} on {Q_j} we are thus led to the requirement

\displaystyle N_j \approx N_{j-1}^{\alpha+1}. \ \ \ \ \ (2)

 

Next, when restricted to frequencies of order {N_{j}}, the propagator {\frac{\sin((s-t)\sqrt{-\Delta})}{\sqrt{-\Delta}}} “behaves like” {N_{j}^{\frac{d-3}{2}} (s-t)^{\frac{d-1}{2}} A_{s-t}}, where {A_{s-t}} is the spherical averaging operator

\displaystyle A_{s-t} f(x) := \frac{1}{\omega_{d-1}} \int_{S^{d-1}} f(x + (s-t)\theta)\ d\theta

where {d\theta} is surface measure on the unit sphere {S^{d-1}}, and {\omega_{d-1}} is the volume of that sphere. In our setting, {s-t} is comparable to {1/N_{j-1}}, and so we have the informal approximation

\displaystyle u(t,x) \approx N_j^{\frac{d-3}{2}} N_{j-1}^{-\frac{d-1}{2}} \int_{s \sim 1/N_{j-1}} A_{s-t} \sin(u(s))(x)\ ds

on {Q_j}.

Since {\sin(u(s))} is bounded, {A_{s-t} \sin(u(s))} is bounded as well. This gives a (non-rigorous) upper bound

\displaystyle u(t,x) \lessapprox N_j^{\frac{d-3}{2}} N_{j-1}^{-\frac{d-1}{2}} \frac{1}{N_{j-1}}

which when combined with our ansatz that {u} has ampitude about {N_j^\alpha} on {Q_j}, gives the constraint

\displaystyle N_j^\alpha \lessapprox N_j^{\frac{d-3}{2}} N_{j-1}^{-\frac{d-1}{2}} \frac{1}{N_{j-1}}

which on applying (2) gives the further constraint

\displaystyle \alpha(\alpha+1) \leq \frac{d-3}{2} (\alpha+1) - \frac{d-1}{2} - 1

which can be rearranged as

\displaystyle \left(\alpha - \frac{d-5}{4}\right)^2 \leq \frac{d^2-10d-7}{16}.

It is now clear that the optimal choice of {\alpha} is

\displaystyle \alpha = \frac{d-5}{4},

and this blowup ansatz is only self-consistent when

\displaystyle \frac{d^2-10d-7}{16} \geq 0

or equivalently if {d \geq 11}.

To turn this ansatz into an actual blowup example, we will construct {u} as the sum of various functions {u_j} that solve the wave equation with forcing term in {Q_{j+1}}, and which concentrate in {Q_j} with the amplitude and frequency indicated by the above heuristic analysis. The remaining task is to show that {\Box u} can be written in the form {f(u)} for some {f} with all derivatives bounded. For this one needs some injectivity properties of {u} (after imposing spherical symmetry to impose a dimensional reduction on the domain of {u} from {d+1} dimensions to {1+1}). This requires one to construct some solutions to the free wave equation that have some unusual restrictions on the range (for instance, we will need a solution taking values in the plane {{\bf R}^2} that avoid one quadrant of that plane). In order to do this we take advantage of the very explicit nature of the fundamental solution to the wave equation in odd dimensions (such as {d=11}), particularly under the assumption of spherical symmetry. Specifically, one can show that in odd dimension {d}, any spherically symmetric function {u(t,x) = u(t,r)} of the form

\displaystyle u(t,r) = \left(\frac{1}{r} \partial_r\right)^{\frac{d-1}{2}} (g(t+r) + g(t-r))

for an arbitrary smooth function {g: {\bf R} \rightarrow {\bf R}^m}, will solve the free wave equation; this is ultimately due to iterating the “ladder operator” identity

\displaystyle \left( \partial_{tt} + \partial_{rr} + \frac{d-1}{r} \partial_r \right) \frac{1}{r} \partial_r = \frac{1}{r} \partial_r \left( \partial_{tt} + \partial_{rr} + \frac{d-3}{r} \partial_r \right).

This precise and relatively simple formula for {u} allows one to create “bespoke” solutions {u} that obey various unusual properties, without too much difficulty.

It is not clear to me what to conjecture for {d=10}. The blowup ansatz given above is a little inefficient, in that the frequency {N_{j+1}} component of the solution is only generated from a portion of the {N_j} component, namely the portion close to a certain light cone. In particular, the solution does not saturate the Strichartz estimates that are used to establish the positive results for {d \leq 9}, which helps explain the slight gap between the positive and negative results. It may be that a more complicated ansatz could work to give a negative result in ten dimensions; conversely, it is also possible that one could use more advanced estimates than the Strichartz estimate (that somehow capture the “thinness” of the fundamental solution, and not just its dispersive properties) to stretch the positive results to ten dimensions. Which side the {d=10} case falls in all come down to some rather delicate numerology.

I’ve just uploaded to the arXiv my paper Finite time blowup for a supercritical defocusing nonlinear wave system, submitted to Analysis and PDE. This paper was inspired by a question asked of me by Sergiu Klainerman recently, regarding whether there were any analogues of my blowup example for Navier-Stokes type equations in the setting of nonlinear wave equations.

Recall that the defocusing nonlinear wave (NLW) equation reads

\displaystyle \Box u = |u|^{p-1} u \ \ \ \ \ (1)

 

where {u: {\bf R}^{1+d} \rightarrow {\bf R}} is the unknown scalar field, {\Box = -\partial_t^2 + \Delta} is the d’Alambertian operator, and {p>1} is an exponent. We can generalise this equation to the defocusing nonlinear wave system

\displaystyle \Box u = (\nabla F)(u) \ \ \ \ \ (2)

 

where {u: {\bf R}^{1+d} \rightarrow {\bf R}^m} is now a system of scalar fields, and {F: {\bf R}^m \rightarrow {\bf R}} is a potential which is homogeneous of degree {p+1} and strictly positive away from the origin; the scalar equation corresponds to the case where {m=1} and {F(u) = \frac{1}{p+1} |u|^{p+1}}. We will be interested in smooth solutions {u} to (2). It is only natural to restrict to the smooth category when the potential {F} is also smooth; unfortunately, if one requires {F} to be homogeneous of order {p+1} all the way down to the origin, then {F} cannot be smooth unless it is identically zero or {p+1} is an odd integer. This is too restrictive for us, so we will only require that {F} be homogeneous away from the origin (e.g. outside the unit ball). In any event it is the behaviour of {F(u)} for large {u} which will be decisive in understanding regularity or blowup for the equation (2).

Formally, solutions to the equation (2) enjoy a conserved energy

\displaystyle E[u] = \int_{{\bf R}^d} \frac{1}{2} \|\partial_t u \|^2 + \frac{1}{2} \| \nabla_x u \|^2 + F(u)\ dx.

Using this conserved energy, it is possible to establish global regularity for the Cauchy problem (2) in the energy-subcritical case when {d \leq 2}, or when {d \geq 3} and {p < 1+\frac{4}{d-2}}. This means that for any smooth initial position {u_0: {\bf R}^d \rightarrow {\bf R}^m} and initial velocity {u_1: {\bf R}^d \rightarrow {\bf R}^m}, there exists a (unique) smooth global solution {u: {\bf R}^{1+d} \rightarrow {\bf R}^m} to the equation (2) with {u(0,x) = u_0(x)} and {\partial_t u(0,x) = u_1(x)}. These classical global regularity results (essentially due to Jörgens) were famously extended to the energy-critical case when {d \geq 3} and {p = 1 + \frac{4}{d-2}} by Grillakis, Struwe, and Shatah-Struwe (though for various technical reasons, the global regularity component of these results was limited to the range {3 \leq d \leq 7}). A key tool used in the energy-critical theory is the Morawetz estimate

\displaystyle \int_0^T \int_{{\bf R}^d} \frac{|u(t,x)|^{p+1}}{|x|}\ dx dt \lesssim E[u]

which can be proven by manipulating the properties of the stress-energy tensor

\displaystyle T_{\alpha \beta} = \langle \partial_\alpha u, \partial_\beta u \rangle - \frac{1}{2} \eta_{\alpha \beta} (\langle \partial^\gamma u, \partial_\gamma u \rangle + F(u))

(with the usual summation conventions involving the Minkowski metric {\eta_{\alpha \beta} dx^\alpha dx^\beta = -dt^2 + |dx|^2}) and in particular exploiting the divergence-free nature of this tensor: {\partial^\beta T_{\alpha \beta}} See for instance the text of Shatah-Struwe, or my own PDE book, for more details. The energy-critical regularity results have also been extended to slightly supercritical settings in which the potential grows by a logarithmic factor or so faster than the critical rate; see the results of myself and of Roy.

This leaves the question of global regularity for the energy supercritical case when {d \geq 3} and {p > 1+\frac{4}{d-2}}. On the one hand, global smooth solutions are known for small data (if {F} vanishes to sufficiently high order at the origin, see e.g. the work of Lindblad and Sogge), and global weak solutions for large data were constructed long ago by Segal. On the other hand, the solution map, if it exists, is known to be extremely unstable, particularly at high frequencies; see for instance this paper of Lebeau, this paper of Christ, Colliander, and myself, this paper of Brenner and Kumlin, or this paper of Ibrahim, Majdoub, and Masmoudi for various formulations of this instability. In the case of the focusing NLW {-\partial_{tt} u + \Delta u = - |u|^{p-1} u}, one can easily create solutions that blow up in finite time by ODE constructions, for instance one can take {u(t,x) = c (1-t)^{-\frac{2}{p-1}}} with {c = (\frac{2(p+1)}{(p-1)^2})^{\frac{1}{p-1}}}, which blows up as {t} approaches {1}. However the situation in the defocusing supercritical case is less clear. The strongest positive results are of Kenig-Merle and Killip-Visan, which show (under some additional technical hypotheses) that global regularity for such equations holds under the additional assumption that the critical Sobolev norm of the solution stays bounded. Roughly speaking, this shows that “Type II blowup” cannot occur for (2).

Our main result is that finite time blowup can in fact occur, at least for three-dimensional systems where the number {m} of degrees of freedom is sufficiently large:

Theorem 1 Let {d=3}, {p > 5}, and {m \geq 76}. Then there exists a smooth potential {F: {\bf R}^m \rightarrow {\bf R}}, positive and homogeneous of degree {p+1} away from the origin, and a solution to (2) with smooth initial data that develops a singularity in finite time.

The rather large lower bound of {76} on {m} here is primarily due to our use of the Nash embedding theorem (which is the first time I have actually had to use this theorem in an application!). It can certainly be lowered, but unfortunately our methods do not seem to be able to bring {m} all the way down to {1}, so we do not directly exhibit finite time blowup for the scalar supercritical defocusing NLW. Nevertheless, this result presents a barrier to any attempt to prove global regularity for that equation, in that it must somehow use a property of the scalar equation which is not available for systems. It is likely that the methods can be adapted to higher dimensions than three, but we take advantage of some special structure to the equations in three dimensions (related to the strong Huygens principle) which does not seem to be available in higher dimensions.

The blowup will in fact be of discrete self-similar type in a backwards light cone, thus {u} will obey a relation of the form

\displaystyle u(e^S t, e^S x) = e^{-\frac{2}{p-1} S} u(t,x)

for some fixed {S>0} (the exponent {-\frac{2}{p-1}} is mandated by dimensional analysis considerations). It would be natural to consider continuously self-similar solutions (in which the above relation holds for all {S}, not just one {S}). And rough self-similar solutions have been constructed in the literature by perturbative methods (see this paper of Planchon, or this paper of Ribaud and Youssfi). However, it turns out that continuously self-similar solutions to a defocusing equation have to obey an additional monotonicity formula which causes them to not exist in three spatial dimensions; this argument is given in my paper. So we have to work just with discretely self-similar solutions.

Because of the discrete self-similarity, the finite time blowup solution will be “locally Type II” in the sense that scale-invariant norms inside the backwards light cone stay bounded as one approaches the singularity. But it will not be “globally Type II” in that scale-invariant norms stay bounded outside the light cone as well; indeed energy will leak from the light cone at every scale. This is consistent with the results of Kenig-Merle and Killip-Visan which preclude “globally Type II” blowup solutions to these equations in many cases.

We now sketch the arguments used to prove this theorem. Usually when studying the NLW, we think of the potential {F} (and the initial data {u_0,u_1}) as being given in advance, and then try to solve for {u} as an unknown field. However, in this problem we have the freedom to select {F}. So we can look at this problem from a “backwards” direction: we first choose the field {u}, and then fit the potential {F} (and the initial data) to match that field.

Now, one cannot write down a completely arbitrary field {u} and hope to find a potential {F} obeying (2), as there are some constraints coming from the homogeneity of {F}. Namely, from the Euler identity

\displaystyle \langle u, (\nabla F)(u) \rangle = (p+1) F(u)

we see that {F(u)} can be recovered from (2) by the formula

\displaystyle F(u) = \frac{1}{p+1} \langle u, \Box u \rangle \ \ \ \ \ (3)

 

so the defocusing nature of {F} imposes a constraint

\displaystyle \langle u, \Box u \rangle > 0.

Furthermore, taking a derivative of (3) we obtain another constraining equation

\displaystyle \langle \partial_\alpha u, \Box u \rangle = \frac{1}{p+1} \partial_\alpha \langle u, \Box u \rangle

that does not explicitly involve the potential {F}. Actually, one can write this equation in the more familiar form

\displaystyle \partial^\beta T_{\alpha \beta} = 0

where {T_{\alpha \beta}} is the stress-energy tensor

\displaystyle T_{\alpha \beta} = \langle \partial_\alpha u, \partial_\beta u \rangle - \frac{1}{2} \eta_{\alpha \beta} (\langle \partial^\gamma u, \partial_\gamma u \rangle + \frac{1}{p+1} \langle u, \Box u \rangle),

now written in a manner that does not explicitly involve {F}.

With this reformulation, this suggests a strategy for locating {u}: first one selects a stress-energy tensor {T_{\alpha \beta}} that is divergence-free and obeys suitable positive definiteness and self-similarity properties, and then locates a self-similar map {u} from the backwards light cone to {{\bf R}^m} that has that stress-energy tensor (one also needs the map {u} (or more precisely the direction component {u/\|u\|} of that map) injective up to the discrete self-similarity, in order to define {F(u)} consistently). If the stress-energy tensor was replaced by the simpler “energy tensor”

\displaystyle E_{\alpha \beta} = \langle \partial_\alpha u, \partial_\beta u \rangle

then the question of constructing an (injective) map {u} with the specified energy tensor is precisely the embedding problem that was famously solved by Nash (viewing {E_{\alpha \beta}} as a Riemannian metric on the domain of {u}, which in this case is a backwards light cone quotiented by a discrete self-similarity to make it compact). It turns out that one can adapt the Nash embedding theorem to also work with the stress-energy tensor as well (as long as one also specifies the mass density {M = \|u\|^2}, and as long as a certain positive definiteness property, related to the positive semi-definiteness of Gram matrices, is obeyed). Here is where the dimension {76} shows up:

Proposition 2 Let {M} be a smooth compact Riemannian {4}-manifold, and let {m \geq 76}. Then {M} smoothly isometrically embeds into the sphere {S^{m-1}}.

Proof: The Nash embedding theorem (in the form given in this ICM lecture of Gunther) shows that {M} can be smoothly isometrically embedded into {{\bf R}^{19}}, and thus in {[-R,R]^{19}} for some large {R}. Using an irrational slope, the interval {[-R,R]} can be smoothly isometrically embedded into the {2}-torus {\frac{1}{\sqrt{38}} (S^1 \times S^1)}, and so {[-R,R]^{19}} and hence {M} can be smoothly embedded in {\frac{1}{\sqrt{38}} (S^1)^{38}}. But from Pythagoras’ theorem, {\frac{1}{\sqrt{38}} (S^1)^{38}} can be identified with a subset of {S^{m-1}} for any {m \geq 76}, and the claim follows. \Box

One can presumably improve upon the bound {76} by being more efficient with the embeddings (e.g. by modifying the proof of Nash embedding to embed directly into a round sphere), but I did not try to optimise the bound here.

The remaining task is to construct the stress-energy tensor {T_{\alpha \beta}}. One can reduce to tensors that are invariant with respect to rotations around the spatial origin, but this still leaves a fair amount of degrees of freedom (it turns out that there are four fields that need to be specified, which are denoted {M, E_{tt}, E_{tr}, E_{rr}} in my paper). However a small miracle occurs in three spatial dimensions, in that the divergence-free condition involves only two of the four degrees of freedom (or three out of four, depending on whether one considers a function that is even or odd in {r} to only be half a degree of freedom). This is easiest to illustrate with the scalar NLW (1). Assuming spherical symmetry, this equation becomes

\displaystyle - \partial_{tt} u + \partial_{rr} u + \frac{2}{r} \partial_r u = |u|^{p-1} u.

Making the substitution {\phi := ru}, we can eliminate the lower order term {\frac{2}{r} \partial_r} completely to obtain

\displaystyle - \partial_{tt} \phi + \partial_{rr} \phi= \frac{1}{r^{p-1}} |\phi|^{p-1} \phi.

(This can be compared with the situation in higher dimensions, in which an undesirable zeroth order term {\frac{(d-1)(d-3)}{r^2} \phi} shows up.) In particular, if one introduces the null energy density

\displaystyle e_+ := \frac{1}{2} |\partial_t \phi + \partial_r \phi|^2

and the potential energy density

\displaystyle V := \frac{|\phi|^{p+1}}{(p+1) r^{p-1}}

then one can verify the equation

\displaystyle (\partial_t - \partial_r) e_+ + (\partial_t + \partial_r) V = - \frac{p-1}{r} V

which can be viewed as a transport equation for {e_+} with forcing term depending on {V} (or vice versa), and is thus quite easy to solve explicitly by choosing one of these fields and then solving for the other. As it turns out, once one is in the supercritical regime {p>5}, one can solve this equation while giving {e_+} and {V} the right homogeneity (they have to be homogeneous of order {-\frac{4}{p-1}}, which is greater than {-1} in the supercritical case) and positivity properties, and from this it is possible to prescribe all the other fields one needs to satisfy the conclusions of the main theorem. (It turns out that {e_+} and {V} will be concentrated near the boundary of the light cone, so this is how the solution {u} will concentrate also.)

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