Last week, I had a chance to visit Edinburgh in part to serve as the external examiner on the PhD Thesis (papers) of Tim Candy. Tim is now Dr. Timothy Candy and has an exciting research program to develop as a postdoc at Imperial.

It turned out I had lucky timing since my visit overlapped with a visit by Oana Pocovnicu. I had a chance to hear her speak about her recent work on the Gross-Pitaevskii equation. I took some notes during Oana’s talk and they appear below.

Oana Pocovnicu

(joint work with Rowan Killip, Tadahiro Oh, and Monica Visan)

Edinburgh talk. 2012-05-21

  • Dynamics becomes more interesting with a nonvanishing condition at infinity.
  • This is the so-called energy critical case.

GP

$$
i \partial_t u + \Delta u = (|u|^2 – 1)u, u(0) = u_0
$$

The modulus will tend to 1 as $ |x| \rightarrow 1$.

Literature

  • $R$
    • Zhidkov 1987: introduced Zhidkov spaces.
    • Gall 2004. gGWP in $X^1 (R)$
  • $R^2, R^3$
    • Bethuel-Saut 1999 in $1+ H^1$.
    • Gourbet 2007
    • Gallo 2008
    • Gerard 2006 in the energy space.
  • $R^4$
    • Gerard 2006, small energy data such that $\nabla u \in L^2_t L^4_x.$

Remark: energy critical in $R^4$.

  • Gerard 2006 considered the energy space:

$$ E_{GP} = [ u = \alpha + v: |\alpha | =1, v \in \dot{H}^1, |v|^2 + 2 \Re (\overline{\alpha}v) \in L^2 (R^d)].
$$

Finite energy data do not have winding at spatial infinity. Therefore, to treat the finite energy case, it suffices to reduce the study to the setting where $u = 1 + v$ and $v$ satisfies…. She reduces the study to finite energy data so the set up excludes vortices right away.

Theorem (K-O-P-V):
GP is GWP in the energy space $E_{GP} (R^4)$.

Two ingredients:

  • GWP of energy-critical defocusing NLS on $R^4$.
  • Perturbation theory: We will treat the equation as a perturbation off the cubic NLS.

Scaling Invariance

  • Dilation invariance of solutions for cubic NLS is described.
  • Dependence of $\dot{H}^s$ in terms of the scaling parameter $\lambda$.
  • critical, subcritical, supercritical.
  • Cubic NLS on $R^4$ is critical in $\dot{H}^1$. Quintic NLS on $R^3$ is also critical in $\dot{H}^1$.

Strichartz Estimates

  • Dispersive decay estimate
  • Strichartz Norm; supremum over the admissible pairs.
  • $N(I \times R^d)$ is the dual space of the Strichartz space $S(I\times R^d)$.
  • Homogeneous Strichartz estimate
  • Inhomogeneous Strichartz estimate
  • Admissible pairs on $R^4: (\infty, 2), (2,4), (6, \frac{12}{5})$.
  • By Sobolev embedding, we have some nice Strichartz containments.

Energy Critical NLS

  • LWP. Cazenave-Weissler 1989
  • GWP for small data. She then describes this by passing through Strichartz and identifies:
    • If $\| \nabla e^{it \Delta } w_0 \|{L^6_t L^{12/5}_x}$ is small, we can close the argument.
    • The smallness of this expression can be insured by shrinking $T$, but this depends upon the profile properties not just upon the norm of the data.
    • GWP for small data follows.
  • Explains the blowup critereon showing that the spacetime $L^6$ norm controls the GWP+Scattering theory.

Main Results on defocusing energy-critical NLS

  • Bourgain 1999: GWP + Scattering, quintic NLS on $R^3$ with radial data.
    • induction on energy
    • localized Morawetz estimate
  • Grillakis 2000: global regularity for quintic NLS on $R^3$ with radial data.
  • CKSTT 2003: removed the radial assumption on $R^3$.
  • Ryckman-Visan 2007: GWP and scattering for cubic NLS on $R^4$.
  • Visan 2010: Simpler method for GWP+Scattering for cubic NLS on $R^4$, building on work of Dodson.
  • Kenig-Merle 2006: focusing energy-critical NLS on $R^3, R^4, R^4, R^5$. GWP+ Scattering for radial data with energy and kinetic energy smaller than those of the stationary solution.

Goal: prove existence of a global solution with control on the spacetime $L^6$.

  • Contradiction strategy.
  • Minimal blowup solution must exist.
  • Minimal blowup solutions mut be almost periodic. They are localized in physical and Fourier space.
  • Frequency localized Morawetz inequality. (only true for the minimal blowup solution). This is obtained by localizing in frequency the interaction Morawetz estimate.
  • This show that we have a smallness property on the spacetime $L^3$ norm on the high frequencies.
  • With some interpolation, we can then prove that the spacetime $L^6$ is bounded, contradicting the hypothesis.

Cubic NLS on $R^4$ (Visan)

(Original proof due to Ryckman-Visan but Visan recently simplified that following some ideas of Dodson.)

  • By contradiction and using concentration-compactness we have a minimal blowup solution.
  • There are only two scenarios. Rapid frequency cascade scenario; quasi-soliton scenario.

These are excluded using the long-time Strichartz estimates in the spirit of Dodson. The quasisoliton case is excluded using Morawetz.

Perturbation theory

Recalls the perturbation lemma from CKSTT, adapted to this problem.

She nicely describes the reduction to proving a local result on a time interval controlled by the energy. Once we have this type of local theory, we essentially convert the critical problem into one that behaves like the subcritical problem so GWP will follow.

Remarks on Proof

Subcritical quadratic terms in the Duhamel-Strichartz analysis on local intervals have a time factor. If this time factor is small enough, these subcritical terms can be absorbed. Oh, now I understand! The point here is that GP can be viewed as the energy-critical NLS plus some quadratic terms which don’t destroy energy conservation. This perspective guides the KOPV analysis. They show that the GP equation can be treated as a perturbation off the dilation invariant energy critical case.

Cubic-Quintic NLS with non-vanishing BC on $R^3$

They write $u=1+v$ and observe that $v$ satisfies energy critical NLS with subcritical lower order terms. The Hamiltonian is not sign definite so does not provide coercive control over the kinetic energy term. This is compensated for by using a lower order term $M(v)$, the $L^2$ norm of the real part of $v$. This quantity is not conserved. They show that it satisfies a Gronwall type estimate and that turns out to suffice.

Scattering for the GP equation in the case of large data

  • GP equation has traveling wave solutions that do NOT scatter.
  • Formation of traveling waves require a minimal energy in $R^d, d \geq 3$. Bethuel-Gravejat-Saut 2009, de Laire 2009.
  • Solutions with sufficiently small energy scatter. (Gustafson-Nakanish-Tsai 2006)
  • Can one prove scattering up to the minimal energy of a traveling wave?

Our goal is to fill in the gap. But, this problem does not seem too easy to attack, so we tried to apply these ideas on a simpler problem.

Killip-Oh-Pocovnicu-Visan

For a Cubic-Quintic NLS with zero boundary conditions (which has conserved mass and energy and has soliton solutions) the are working to show that if $v_0 \in H^1 (R^3)$ then scattering holds true if the mass is smaller than the mass of any soliton OR if it has positive energy, smaller than the enrgy of any solution.

(Final statement is a work in progress.)

 

,

Vladimir Arnold

I am at an interesting workshop in Edinburgh entitled Dynamical systems and classical mechanics: a conference in celebration of Vladimir Arnold 1937 – 2010.
Boris Khesin and Serge Tabachnikov have coordinated a Tribute to Vladimir Arnold which will soon appear in consecutive issues of the Notices of the AMS. These tributes were shared at the workshop and are also available here:

I will post below my notes from the talks. I apologize, especially to the speakers and readers (if any), for errors and typos.

Welcome

Remarks at introduction of workshop by S. Kuksin:

Kuksin highlights the openness of Arnold to discussions with students. A main message from Arnold “Mathematics must be interesting.”

Small program changes: Lai-Sang Yang speaks on Thursday at 1630. Laurent Stolovich speaks on Friday at 930.

Wednesday will have some short afternoon talks. Young people can either speak to me or Hakan Eliasson. We will make a page with a list of the talks.

Program

Monday 03 October


Alexander Shnirelman, Concordia University,
Some problems of the fluid mechanics

Alexander Shnirlmen

I am grateful and touched because of the invitation to speak at this conference. I had the influence of Arnold for several decades. I remember that the pace of sdiscoveries in his seminar was so fast. It reminds me a bit of the situation like 500 years ago. There was not enough time to examine in detail the discoveries as they developed. It is a bit similar to the discovery of continents.

1. Arnold’s presntation of basics of the fluid dynamics.

A lagrangian system whose configuariton space is a Lie group $G$ with unit element $e$. Take the kinetic energy defined by a right-invariant Riemanniain metric via
$$
E(u) = \frac{1}{2} \langle u , u \rangle = \frac{1}{2} (Au, u)$$
where $A: H \rightarrow H^*$ is the inertia operator. ….oh my this is too fast to follow while typing.

This is a survey of the Arnold approach to fluids. The critical points of the energy on each orbit are steady solutions.

Example: $SO(3)$. Surfaces $S$ are ellipsoids. levels lines. stable points.

We collapse to $n=2$. Take the quadratic form to be the $L^2$ norm. Discussing 2d Euler equation. Arnold called the surfaces isorotated velocity fields. There exists a unique stream function.

The second variation of $E$ on $S$ at the point $u$ is given by the quadratic form
$$
\delta^2 E (\phi) = \int_M (\nabla \phi)^2 + \frac{\nabla \psi}{(\nabla \Delta \psi)} (\Delta \phi)^2 dx.
$$
The solution $u$ is called Arnold Stable if this form is either poisitive or negative definite. What is $\phi$ here? It is a perturbation of $\psi$ and he writes on the board $\delta \psi = [ \psi, \phi ]$ (but with curly brackets).

2. Difficulties of the Arnold’s Approach

  1. The group $D$ is infinite-dimensional; its topology is usually stronger than topology defined by the Reimannian metric. Hence the existence and uniqueness of geodesics are not certain.
  2. The surfaces $S$ may be nonsmooth, and the partition of $H$ into these surfaces may be locally nontrivial.
  3. It is unclear whether the energy functional $E$ attains a maximum or a minimum on a given orbit.

3. Group as a Banach Manifold

Theorem (Lichtenstein, Giunter, …):

  1. For any initial velocity $u_0 \in X$, where $X$ is one of the above spaces (e.g. Holder, Sobolev, …), there exists $T>0$ and a unqique solution $u(x,t) \in X$ of the Euler equations with initial velocity $u_0$ defined for $|t| < T$.
  2. If $n=2, T = \infty$. (Volibner, Yudovic, Kato, ….)

Mentions Ebin-Marsden.

4. Mixing operators.

If a 2d domain $M$, the vorticity $\omega$ is transported by the flow; it is distorted and effectively missed. He considers a class of operators on $L^2$ given as integral kernel operators with positive kernel (so positive measures) and with marginals which are equal to 1.

He defines a partial order in $L^2$: $f \ll g$ if $f = Kg$ for some $K \in {\bf{K}}$. Now we write $u \ll v$ for two vector fields $ u, v \in V$ if $ curl u \ll curl v$.

By Zorn’s lemman, there exist a minimal flow wrt this ordering.

Theorem Minimal flows are Arnold stable.

Minimal flow is called energy excessive if $F’ \leq 0$; energy deficient if $F’ \geq 0$.

5. Long-time behavior of the flow; mixing of vorticity

Natural conjecture: minimal flows form an attractor. However, this is wrong.

Movie.

Shows a 2d torus. Vorticity on square patches. After some transition period, the solution becomes more or less periodic plus a constant velocity drift. In this experiment, the flow converges to time periodic flow whih is not a stable configuration. More detailed experiments show that the final flow is quasiperiodic with more details. Even possible to find almost periodic. These are not stable flows. There exists a wider class of flows which are attracting.

6. The evidence of irreversibility: Liapunov function.

Defines a Liapunov function. Existence of Liapunov function always shows that there is some kind of irreversibililty. Shows an elemtary example. For a free particle, we can consider $L(x, \dot{x}) = x \cdot \dot{x}$. (This is reminiscent of the Morawetz estimate.)

The first Liapunov functional for the fluid was found by V. Yudovic (1973). The LF is given by $L = \omega \omega_x \omega_y$. It turns out this combination satisfies that its time derivative is given by a square $(\omega^2 \omega_x^2)|_\pi \geq 0$.

7. Generalized minimal flow

Suppose $u_0 \longmapsto u$. Let’s close the orbit of the evolution in $L^2$.

Definition: A flow $u(t)$ with initial veloicty field $u_0$ is called a generalized minimal flow (GMF) if ….curl condition.

slide changes fast.

8. Construction of GMF by pseudoevolution

A process is decribed which produces GMF’s from a given seed data using the curl ordering.

Conjectures:

  1. The set $N$ is an attractor for the Euler equations in the ordinary sense.
  2. GMG are either statiornay or qaiperiodic with at most countable set of periouds.
  3. Stircktly speaking, we have not proven that the set $N$ is a a proper subset of $V$. e.g. there might exist flows which are not GMF. (This is analogous to the Landau damoping recently proved for the Vlasov-Poisson equation by Villani.)

9. Local regularity of partition into isovortical surfaces

The equation was addressed recently by V. Sverak and A. Choffrut (2010). Consider the distribution function for the vorticity
$$
\lambda (s) = {\mbox{mes}} [ x \in M : \omega (x) \leq s].
$$

Theorem (Sverak-Choffrut): Steady solutions close to a fixed Arnold stable one are in a smooth 1-1 correspondence with distribution functions $\lambda (s)$.

The proof is difficult and based on Nash-Schwarz implicit function theorem.

10. The structure of the exponential map.

  • Ebin-Marsden 1970
  • Ebin-Misiolek-Preston 2008
  • Misiolek 1993

John Mather, Princeton University,
Near a double resonance

John Mather

(blackboard talk)

I feel honored to speak here, especially since something I have been working on for a long time is called Arnold Diffusion. Some people speak about the problem. What is interesting to me is the program. Arnold posed many problems. There are two aspects that should be highlighted.

  1. They are very interesting.
  2. There is a chance that we could do them.

Generic quasiergodicity of Hamiltonian systems….a famous problem attributed to Boltzmann. This problem seems incredibly hard and seems to be inaccessible. In contrast, there appears to be hope for Arnold diffusion.

I apologize to those of you have heard this talk before. I want to speak about something that I announced in 2003 in Russian and it appeared in English in 2004. I announced some results at that time but, in the meantime I found some mistakes in the proof. I want to speak about corrections to those proofs.

This is a question about small perturbations of integrable systems. Usually, people discuss this in the setting of the Hamiltonian form. I prefer to approach it using the Lagrangian (equivalent form) since my method of approach is variational.

Lagrangian:
$$L(\theta, \dot{\theta}, t) = l_0 (\dot{\theta}) + \epsilon P(\theta, \dot{\theta}, t)$$

We consider here the case where $P(\theta, \dot{\theta}, t+1 ) = P(\theta, \dot{\theta}, t)$.

  • $\theta \in T^n$
  • $\dot{\theta} \in B^n \subset R^n $

We are looking for solutions of the Euler-Lagrange equation
$$
\frac{d}{dt} ( L_{\dot{\theta}} ) = L_\theta.
$$
Let’s assume that $d^2 l_0 >0$. (This is a strong restriction; we’d rather like to do it under the assumption that the determinant of the Hessian is nonzero. A great deal of the theory is developed under that assumption. For the methods I use, I need this stronger condition.)

The goal is to somehow show that the solutions go everywhere. That is too strong, but I can prove something along those lines in a special case.

The Lagrangian I am looking at is a small perturbation of an integrable system. For the integrable system, the E-L equation is
$$
\frac{d}{dt} ( L_{\dot{\theta}} ) = 0.
$$

Arnold Question: Are the orbits confined or do some go everywhere? (This is a vague question; certainly Arnold was more precise.)

In the case when $n=1$, the orbits are confined. This is a consequence of KAM theory. The KAM tori persist. There are Birkhoff regions of instability and the orbits are confined by the KAM tori.

In the case $n>1$, the expectation is that the orbits are not confined. There are results along these lines which show, in some cases, that this is the case. For the program we have in mind, we want to build methods which show this phenomena is somehow generic.

In the case $n=2$ (this is what I had announced in 2003): Generically (assuming the positive Hessian part), the orbits are not confined.

The methods that I use are variational. I want to say a little bit about those tools.

Consider $U_1, U_2, \dots, U_k$ open non-void sets in $B^2$. The construction guides the orbit to only be allowed to move in certain ways. The trick is to choose the conditions in such a ways so that when you minimize within those conditions, the solution stays inside the open set and doesn’t get pushed off to the boundary. This is a method that I introduced in the past in studying twist maps. The conditions are really complicated so you need a guide to tell you what the conditions are. The basic guide involves something called Aubry sets. These are sets contained inside $T^2_\theta \times T_t$. The phase space consists of $T^2 \times B^2 \times T$ and the state space is $T^2 \times T$. The Aubry sets are defined by a global minimizing condition. What shall I say about them? First of all, it is useful to consider invariant probability measures for the Lagrangian system. Let $\mu$ be a probability measure on $T^2 \times B^2 \times T$. You can then define a cohomoology class $c \in H^1 (T^2)$. This allows you to define an average action:
$$
A_c (\mu) = \int_{T^2} L d\mu – c
$$
…..I don’t ususally write it this way….scratches it out and writes instead
$$
A_c (\mu) = \int_{T^2 \times B^2 \times T} (L (\theta, \dot{\theta}, t) – c \dot{\theta}) d\mu (\theta , \dot{\theta}, t).
$$
We can then define
$$M_c = [~{\mbox{invariant probability measures that minimize}}~ A_c].$$
The support of $M_C \subset T^2 \times B^2 \times T$ and the map turns out to be injective.

Aubry set….defined by a minimizing condition.

Theorem: $M_c \subset Au_c$.

The conditions I build for the minimization process. First you have to know something about the Aubry sets, you can then state what the conditions are. The process was carried out successfully in the case of twist maps. I was able to prove that there could be wandering in the Birkhoff zones of instability. This procedure was also used by Cheng-Yan in the case of a priori unstable systems. They were able to prove a version of Arnold diffusion in these systems. One considers a rotator and a pendulum and take the product of the two. The diffusion is related to the unstable fixed point.

…trouble typing…..discussion of double resonance, with strong so that the resonant linear combination arises with control on the integer prefactors by a constant.

Massimiliano Berti, University of Naples Federico II,
Quasi periodic solutions of Hamiltonian PDEs

(Similar to what I saw in France at Ile de Berder, so I will watch rather than type…)

There was some discussion afterwards between me, Massimiliano and Walter Craig. I suggested that Massimiliano’s improvement of Walter’s pseudodifferential result might be reconsidered in the setting of the $MMT_{\alpha, \beta}$ models introduced by Majda-McLaughlin-Tabak. These models are slightly more general and consider parametrized $\alpha$-power dispersion relation with a $\beta$-smoothing operator inside the cubic nonlinearity. This might provide a generalized framework for investigating the relationship between dispersive smoothing and the derivative properties appearing in the nonlinearity. Admittedly, these are not directly physical models but the mathematical motivations for their study seem to keep appearing….

Andrei Agrachev, SISSA Trieste, The long-time behaviour of dissipative systems

Andrei Agrachev

A natural mechanical system on a Riemannian manifold. Traectories are curves on this manifold. The kinetic energy is the usual Riemannian length. The potential energy is a function on the manifold.

Hamiltonian $ = \frac{1}{2}|p|^2 + V(q)$
where $p \in T_q^* M$ and $|p| = \max [ \langle p, \xi \rangle: \xi \in T_q M, |\xi | =1 ]$

WE consider this system but with an isotropic dissipation:
$$
\dot{p} = – H_q – \alpha p, ~\alpha > 0$$
$$
\dot{q} = H_p.
$$

Toy example: $M=R, V(q) = b q$. All solutions converge to one particular solution. Eventually, the particle moves with a fixed velocity. If we perturb the V a little bit, the phase portrait will be very similar. There will be a limiting profile and the structure will be very similar.

Toy example: Pendulum. $V(q) = b \cos q, ~ \frac{\alpha^2}{4} < |b|.$ Then, we don’t have limiting behavior like that beffore. We have instead a vortex. The dissipation brings the trajectory down to the minimum of the potential energy. However, when we have $
\frac{\alpha^2}{4} > |b|,$ we have a different limiting configuration. We obtain a limiting potential “gradient” flow on the circle.

There is strong dissipation in life. The limitig dynamics of systems we observe, like a ship on the ocean, has a transitional period but eventually there is a balance.

We try to view things using the Eulerian viewpoint.

Definition: Potential stationary flow is a gradient vector field $\nabla u$ where $u \in C^2 (M)$ and $[d_q u: q \in M] \subset T^* M$ is an invariant submanifold of our system.

In particular, $\dot{\gamma}(t) = \nabla_{\gamma(t)} u$ implies that $t \longmapsto (d_{\gamma(t)}u, \gamma(t))$ is a solution.

Definition:
The curvature of the Hamiltonian $H$ at $p \in T_q^* M$ is a self-adjoint linear operator from the cotangent bundle to the cotangent bundle defined by the formula
$$
R^H_{(p,q)} \xi \cal{R} (\xi, p)p + (\nabla_q^2 V) \xi, ~ \xi \in T_q^* M,
$$
where $\nabla$ is the covariant derivative and $\cal{R}$ is the Riemannian curvature.

(This is a natural extension fromt he standard symplectic setting to the dissipative systems. Some interesting discussion…what is the connection….natural…only this kind of isotropic dissipative systems.)

Assume that $M$ is complete, $\cal{R}$ and $\nabla^2 V$ are uniformly bounded. (This follows if $M$ is compact.) Let $\Phi_t$ be the flow on the cotangent bundle. We consider a strip defined by $\Omega_c = [ (p,q) \in T^*M: |p| \leq c]$.

Theorem:

  • If $R^H_{(p,q)} < \frac{\alpha^2}{4} I, ~ \forall (p,q)$ such that $H(p,q) \leq \max V$, then $\exists$ a potential stationary flow $\nabla u$ such that
    $$
    \Phi_t (\Omega_c) \rightarrow [ d_q u : q \in M] ~{\mbox{as}}~ t \rightarrow + \infty$
    $$
    with an expoenetntial rate, $\forall c>0$.
  • $[d_q u: q \in M]$ is a normally stable submanifold of $\Phi_t$.
  • If $M$ is compact and $R^H_{(p,q)} < \frac{\kappa -1)\alpha^2}{\kappa^2} I$ then $ u \in C^k (M)$.
  • The map $(H,\alpha) \longmapsto u$ is continuous in the $C^2$-topology.

Smaller dissipation:
When dissipation is smaller, we have some hopeful hints. Discussion is moving a bit fast for me to type….Markov process…not an invariant measure but the measures can be propagated…

Interesting discussion about the use of measures in the presence of small dissipation limits as a device to probe the structure of the original Hamiltonian systems.

Slides stop….he still has about 10 minutes. He tries to explain the proof. This discussion is reminiscent of a principal theme I will try to convey in my talk. Infinite dimensional systems might be viewed as the envelope system of limits of finite-d systems. When we study the infinite-d system, one strategy of attack is to identify convenient finite-d systems which limit on the infinite-d system. One possible source of these convenient systems might be through appropriate choices of isotropic dissipative systems, which truncate high frequencies.

Some discussion striving to describe the “curvature” of a Hamiltonian system….family of vertical and horizontal Lagrangian distributions. Curvature of dissipative systems is easily accessed…..cheating a bit, but look in the paper for the details. Levi-Civita connection is tangent to the zero section.

Walter Craig, McMaster University,
The water wave problem as a Hamiltonian system

Walter Craig

I thought I would speak a bit about Arnold’s influence on my mathematical life. We were given his Mathematical Methods book in graduate school. His perspective has pervaded our approach to problems.

(joint work with Catherine Sulem; along with Alessandro Selvitella and Yun Wang)

Outline:

  • Two ODEs
  • Euler’s equations
  • Zakharov’s Hamiltonian
  • Partial Differential equations as Hamiltonian systems
  • Birkhoff Normal Forms
  • Implications of the normal form
  • The KdV scaling limit

Two ODEs

$$
\dot{z} = z^2, z(0) = \epsilon
$$
versus
$$
\dot{w} = w^3, w(0) = \epsilon
$$
…..ack slide changed and I missed the point.

Euler’s equations

Newton’s laws, Eulerian coordinates, incompressible fluid. We work on a pre-Columbian model of the earth.

Free surface water waves. We imagine that the velocity field is irrotational (oceanographers do this) so we can recast this as a potential flow. The bottom is not a sponge, so no penetration and we assume that the fluid velocity has no normal component at the bottom.

Free surface conditons: Kinetic BC at the top; Bernoulli condition.

Hamiltonian systems: Zakharov 1968. This was a poorly understood paper which has emerged as being very important.

Goals: explain this fact; use it to understand the PDEs.

Partial Differential equations as Hamiltonian systems

Zakharov’s Hamiltonian

The energy functional $H= K + P$ so it should be
$$
H = \int \int_{-h}^{\eta(x)} \frac{1}{2} |\nabla \phi |^2 dy dx + \int_x \frac{g}{2} \eta^2 dx.
$$

This is pretty clear but the difficulty is what are the choices of variables?

Zakharov’s choice:
$$ z = (\eta(x), \xi(x) = \phi(x, \eta(x))).$$
That is $\phi = \phi[\eta, \xi] (x,y).$

In these coordinates, we can realize the PDE for Euler flow for the free surface as a Hamiltonian system in Darboux coordinates. The subtlety is how to differentiate the Hamiltonian wrt the canonical coordinates.

Other Hamiltonian PDEs:

Boussinesq system; KdV equation; NLS; …

Dirichlet-Neumann oeprator:

  • Laplace’s equation on the fluid domain $-h < y < \eta(x)$.
    $$
    \xi(x) \longmapsto \phi(x, y) \longmapsto N \cdot \nabla \phi (1 + |\nabla_x \eta |^2)^{1/2} = G(\eta) \xi (x).
    $$
  • In Zakharov’s coordinates, we can express the Hamiltonian in terms of the D-N operator $G$ as
    $$ H(\eta, \xi) = \int \frac{1}{2} \xi G(\eta) \xi \frac{g}{2} \eta^2 dx.$$
  • The water wave system rewritten:
    $$ \partial_t \eta = G(\eta) \xi ,$$
    $$ \partial_t \xi = -g \eta – {\mbox{grad}}_\eta K. $$
    (This discussion is closely related to a variational formula of Hadamard from 1911, 1916)

Lemma (Properties of the Dirichlet-Neumann operator):
A singular integral operator $G(\eta)$ related to the Green’s function.

  1. Hermitian symmetric.
  2. $G(\eta) \geq 0$ and $G(\eta) 1 = 0$.
  3. $G(\eta): H^1_\xi \rightarrow L^2_\xi$ is analytic in $\eta$ for $\eta \in C^1$ [using a theorem of Christ-Journé (1987)]:
    $$ G(\eta) \xi = G^{(0)} \xi + G^{(1)}…$$

ack….slide changed.

Conservation Laws:

  • Mass is conserved. Mass is $\int \eta dx.$
  • Momentum is conserved. Momentum is $\int \eta \partial_x \xi dx.$
  • Energy is conserved.

(Poisson bracket calculations are quite direct.)

Taylor Expansion of the Hamiltonian:

From analyticity, we can expand around the stationary zero solution using the Taylor expansion of $G$.

Flow of the Harmonic oscillator. He is considering here the linearized problem and showing that we can explictly solve this problem using Fourier/superposition methods. We encounter a Fourier series with rotating phases. The typical solution is almost periodic.

Basic facts:

  • The flow preserves the (linearized) energy.
  • Actions are preserved. (This is the moment map.) Therefore, all Sobolev norms are preserved.
  • Phases involve linearly in time.

Basic Questions:

Add in the perturbations. Do any of those orbits persist? This turns out to be quite hard. In fact, it is challenging to show that any of them persist. The progress on these questions have been made using KAM theory. We know that there exist periodic solutions.

  • Do there exist quasiperiodic or almost periodic solutions?
  • Given a point $z^0$ in some phase space. Does the flow exist in M? This is hard.
  • Does it exist globally in time? This is basically open, although there are some recent advances. Do we have stability? Do we have a Nekhoroshev stability property?
  • Can you make the actions grow? Weak turbulence. Growth of Sobolev norms?

Birkhoff Normal Forms

Fix the dimension to $d=2$, so we have $x \in R$. We restrict to the periodic-in-$x$ case. We want to perform canonical transformations to move the Hamiltonian into a normal form at least in some neighborhood of the origin.

Conditions:

  • Make the transformation canonical.
  • Make the new Hamiltonian have the same linearization plus only resonant terms up to some order with a new truncation/residual error.
  • If $Z^{(3)} = 0$, we will have a chance to get longer existence intervals based on the analogy of the first ODEs at the beginning of the talk.

This transformation process is called the reduction to Birkhoff normal form.

Theorem (Craig-Sulem 2009):

Let $d=2$ (and $h = + \infty$) and fix $r>3/2$. Then, there exists a neighborhood of the ball at the origin in $H^r$ on which we have a Birkhoff normal form which kills off the quadratic nonlinear terms resulting a cubic equation.

Note: This transformation mixes the variables $\eta$ and $\xi$.

Outline of the proof: flying slides…..cohomological equation turns out to be a linear equation. THere are no nonzero $m=3$ resonances. It turns out to be rather challenging to show that the flow of the vector field exists.

Implications of the normal form

Long time existence theorem. Work in progress.

We should be able to build solutions that last for time intervals on the order of $\epsilon^{-2}$. I want this time so that I can study the NLS limit of the water wave equation. On this time scale, we would like to have a nice justification. This justification requires the desired long time existence result.

Wu 2009: Small Sobolev data lasts for exponentially long times.

Germain-Masmoudi-Shatah 2009:

For $d=3$, small Sobolev data exist globally in time.

The difference is that I am on a compact domain. Wu is in a dispersive situation.

Nathan Totz & Sijue Wu have done the NLS limit in the non-periodic case. Schneider-Wayne also have results in this direction.

The KdV scaling limit

pretty fast slide switching….but nice moves. He shows how the KdV Hamiltonian can emerge from the water wave Hamiltonian. Now perform the above sequence of transformations on the Birkhoff normal form for water waves. In the limit as the small parameter goes to zero, the water wave Hamiltonian collapses to the KdV Hamiltonian.

Tuesday 04 October


Yann Brenier, University of Nice,
From incompressible fluids to dust

Yann Brenier

It is a great honor for me to be here. I would like to discuss two issues that were familiar to V.I. Arnold. We saw fluids discussed yesterday in Shnirelman’s talk. Dust is a singularity of the Hamilton-Jacobi equation. (Nepecmponcka perestroika)

First part: euler equations and minimizing geodesics for volume preserving maps

  1. Euler equations of incompressible fluid mechanics
  2. Leas action principles
  3. Geometric analysis issues
  4. Minimizing geodesics: existence and uniqueness results for the pressure gradient

Euler’s Equation: Geometric Definition.

The fluid is moving inside a box denoted by $D$. We consider incompressible motion. This is viewed as a time dependent map $M_t: D \rightarrow D$. Points in $D$ are called $a$. $M_t$ is viewed as a map in the Hilbert space $H = L^2 (D, R^d)$, valued in the subset $VPM(D)$ of all Lebesgue measure-preserving maps.

Solutions of theEuler equatiosn, introduced in 1755, correspond to those curves $t \rightarrow M_t \in VPM (D)$ for which there exists a time dependent scalar function $p_t$ called the “pressure field” defined on D such that
$$
\frac{d^2}{dt^2}M_t + (\nabla p_t) \circ M_t =0
$$
where $\nabla$ is the gradient operator on $R^d$ (wrt Euclidean norm).

The Principle of Least Action.

Theorem: Assume $D$ convex. The $(M_t, p_t)$ be asolution of the E equations, with constant $\lambda$ such that
$$
\sum \frac{\partial^2 p_t}{\partial_i \partial_j} \xi_i \xi_j \leq \lambda |\xi|^2
$$
Then $M_t$ is the unique minimizer, among all curves along $VPM(D)$ that conincide with $M_t$ at the endpoints $t = t_0, t=t_1$ of the following action
$$
\frac{1}{2} \int_{t_0}^{t_1} \int_D | \frac{dM_t (x)}{dt}|^2 dx dt.
$$

In other words, such a curve is nothing but a (constant speed) geodesic along $VPM(D)$ wrt metric induced by $H = L^2 (D, R^d)$.

Arnold 1966, Ebin-Marsden 1970, Arnold-Khesin book 1998.

The Dual Action

Minimizing the actrion can be written as a saddle point problem, just by using a time-dependent Lagrange multiplier to relzs
$$
\inf_M \sup_p \int_{t_0}^{t_1} \int_D [\frac{1}{2} | \frac{dM_t (x)}{dt}|^2 - p_t (M_t (x)) + p_t (x) ] dx dt.
$$
This is trivially bounded fro below by
$$
\sup_p \inf_M (same)
$$
which naturally leads to a dual least action priciple.

Theorem: Using exactly the same conditions ($D$ convex and $(t_1 – t_0)^2 \lambda < \pi^2$, the pressure $p$ is the unique maximizer of the concave dual action
$$
I[p] = \int_D J_p (M_{t_0} (x), M_{t_1} (x)) dx + \int_{t_0}^{t_1} \int_D p_t (x) dx dt,
$$
where
$$
J_p (y,z) = \inf \int_{t} ( \frac{1}{2} |\frac{d\xi_t}{dt}|^2 – p_t (\xi_t)) dt
$$
where the infimum is taken over all curves $\xi_t \in D$ such that $\xi_{t_0} = y \in D, \xi_{t_1} = y \in D$.

The proof is elementary and follows from 1d Poincaré inequality.

Geometric Analysis Issues 1

  1. Density of diffeomorphisms in $VPM(D)$. $SDiff(D)$ is the set of volume preserving orientation preserving diffeomorphisms. This is more refined than the $VPM(D)$ condition. For $d \geq 2$, it turns out that $VPM$ is the $L^2$ closure of $SDiff$. The identification of the closure of $SDiff (D)$ for the a prior finer geoesic distance induced by $L^2$ is a much more difficult issue. For simple (say contractile) domains $D$, this closure is still $VPM(D)$ for $d \geq 3$ (but defintely not for $d=2$) as show by Shnirelman in his land mark paper (Math USSR Sb 1985). These results have striking consequences: in particular maps of form
    $$ M(x) = (h(x_1), x_2, x_3)
    $$ where $h$ is any Lebesgue-measure preserving map of the unit interval, are in the closure of $SDiff([0,1]^3)$. (Thus, even though we are interested in volume preserving maps, we have to open our eyes to all Lebesgue measure preserving map of the unit interval. This is a much much richer class than $SDiff$!)
  2. Density of permutations in $VPM(D)$. Another interesting subset of $VPM([0,1]^3)$ is made of all “permutations” of all dyadic divisions of the unit cube in sub-cubes of equal volumes. You divide the buce into dyadic sub-cubes, like a Rubick’s cube. To every permutation, you define a permutation which shuffles the cubes. It turns out that the union of these permutations taken over all scales defines a dense set of maps in $VPM$! He shows some remarkable gifs where dust appears related to the orientation reversal.
  3. Geodesic completeness. Big issue….global well-posedness of E.
  4. Minimizing Geodesics. (Shnirelman Math USSR Sb 1986) The 3d case turns out to be “easy” with a crucial use of the convex structure of the dual problem. The case $d=2$ is clearly linked to symplectic geometry and seems extremely difficult: a fascinating strategy has been developed by Shnirelman, by adding braid constraints to the minimization problem, which certainly deserves further investigations.

Approximate Minimizing Geodesics:

fast slide…The existence of such approximations is in no way trivial and is a consequence of a key density result due to Shnirelman (GAFA 1994).

Main Theorem: Let us assume $D$ to be convex, with $ d \geq 3$, fix $t_0 = 1, ~ t_1 = 1$ and consider maps $M_0, M_1 \in VPM (D)$. Then there is aunique pressure gradient $\nabla p_t$ such that for all $\epsilon$-minimizing geodesics, we have in the sense of distributions
$$ \frac{d^2 M_t^\epsilon}{dt^2} \circ (M^\epsilon_t)^{-1} + \nabla p_t \rightarrow 0, \epsilon \rightarrow 0.$$

Yann insists that this has “nothing to do with geodesic completeness”. I am confused…..

Minimizing Geodesics: Final Comments

  1. Uniqueness of the pressure gradient. This is a remarkable feature of the theory. There is no equivalent result for finite-d configuration spaces such as $SO(3)$,on which geodesic curves (for appropriate metrics) correspond to the motion of solid bodies in classical mechanics. WE believe this strange phenomenon to be the consequence of the “hidden convexity” of the problem in dimension 3 and more.
  2. Limited regularity of the pressure gradient.

Some references.

Second Part: From incompressible fluids to dust

  1. Gravitating particles as a natural approximation of Euler incompressible fluids
    ….fast slides…..jet lag….fascinating….Yann is a fast thinker…

Penalization of the Euler Action

Use a penalty method to try to approximate “geodesics” on discrete sets using permutations.

Monge-Ampere (instead of Poisson) nonlinear correction to the classical Newton gravitation.

Rafael de la Llave, Georgia Institute of Technology,
Arnold diffusion in a-priori unstable Hamiltonian systems of high dimension

Rafael de la Llave

(joint work with Delshams, de la Llave, T.M. Seara)

(Related collaborators: Elisaget Canalias, Marian Gidea, Gemma Huguet, Vadim Kaloshin, …)

Instability for a priori unstable Hamiltonian systems

We consider a periodic in tim perturbation of $n$ pendula and a $d$-dimensional rotor described by non-autonomous Hamitonian,
$$
H(p,q, I, \phi, t , \epsilon) = P(p,q) + h(I) + \epsilon Q (p,q, I, phi, t, \epsilon)
$$
with $$P(p,q) = \sum P_j (p_j, q_j), ~ P_j = \pm (\frac{1}{2} p_j^2 + V_j (q_j)).
$$

Elemntary and regularity assumptions.

  • H1: Assume that the functions $h, V_j, Q$ are $C^r$ in their corresponing domains with $ r \geq r_0$ sufficiently large.
  • H2: Assume that the potentials $V_j$ have non-degenerate local maxima, say at $q_j = 0$, each of which gives rise to a homoclinc orbit of the pendulum $P_j$: They are penduli, they have critical points and have homoclinc connections.

    “The enemies to this problem are the KAM and the Nekoroshev. Of course, they are my friends in other talks…”

  • H3: The mapping $I \rightarrow \omega(I) = …$ ack slide change.
  • H4: The function $Q$ is assumed to be a trigonometric polynomial. (This can be removed)

Remark: [Delshams-Llave-S06], [Delshams-Huguet09], [Gidea-Llave06].

Melnikov Potential:

Poncare-Arnold-Melnikov. This basically measures the effect of the perturbation on a homoclinic orbit at first order. (Big integral….too long to type this fast.)

  • H5: Assume that the system of equations
    $$
    \frac{\partial}{\partial \tau} L(\tau, I, \phi, s) = 0
    $$ admits a nondegenerate solution. This allows us to eliminate the $\tau$ in terms of the other variables.

Poincaré reduced function

  • H6:
  • H7: $Q$ satisfies some nondegeneracy assumptions.
  • H8: You don’t want the resonances to be flat.

….I can’t really keep up….so I will just watch.

Tenyson 83 Probed many mechanisms of diffusion by observing numerically.

Chirikov 82

Vadim Kaloshin, University of Maryland,
Hausdorff dimension of oscillatory motions for three body problems

Vadim Kaloshin

(blackboard talk)

I have a deep admiration of V.I. Arnold. He was a “god of mathematics and still is.” Here is a list of topics that have occupied my interest for research. They are all explicitly linked with ideas of Arnold.

  1. My undergraduate thesis was on a topic called prevalence, a notion of probability one in infinite dimensional spaces. My project emerged from Arnold’s note that one could understand genericity through this notion.
  2. Hilbert-Arnold Problem. This was my first problem I studied with Ilyashenko. This was motivated by the 2nd part of Hilbert 16th problem. You look at a family of vector fields on $x \in S^2, ~ \epsilon \in B^k$. You consider $\dot{x} = v(x, \epsilon)$. Generic fanily of $C^\infty$ v. fields has $LC(\epsilon) < \infty.$
  3. Growth of the number of periodic points. $M$ is a compact manifold. You look at $f: M \rightarrow M; f \in Diff (M)$. You look at $P_n (f) = {\mbox{Number}} [x: f^n x = x]$. How quickly $P_n(f)$ generically?
  4. Arnold Diffusion.

As you can see, most of my research is either inspired by or directly leads from questions suggested by Arnold as interesting directions.

Qualitative analysis of 3-body problem. Let $q_i \in R^d, ~ d =2,3$ are point masses. Each point has mass $m_i$. The Newton law then gives the dynamical law
$$
m_i \frac{d^2}{dt^2} q = – \sum m_i m_j \frac{q_i – q_j}{|q_i – q_j |^3}.
$$

Kepler motions:

2 Body problem (2BP).
$m_0 q_0 + m_1 q_1 = 1$.
$$
H(q, \dot{q}) = \frac{\dot{q}^2}{2} – \frac{1}{|q|}
$$
Three cases for the 2BP:

  • $H<0$ either circular or elliptic.
  • $H=0$ parabolic; escapes to infinity with zero velocity.
  • $H<0$ hyperbolic; escapes to infinity with nonzero velocity.

Three Body Problem: (Sun-Jupiter-Comet)

Four types of motion:

  • B: $\sup_{\pm t>0} |q_i (t)| < K < + \infty$
  • Parabolic: escapes to infinity with zero velocity at infinity.
  • Hyperbolic: escapes to infinity with nonzero velocity at infinity.
  • Oscillatory: $\limsup_{t \rightarrow \infty} |q_i (t)| = \infty; ~ \liminf_{t \rightarrow \infty} |q_i (t)| < + \infty $.

What kind of behavior is possible in the future? What kind of behavior is possible in the past?

A famous result of (missed the names…) showed that there are solutions with any of these four behaviors in either direction of time infinity.

Shows a table. He reports that every one of the boxes (except one) in the table have been shown to have positive measure. He will focus on the case whether the situation with oscillatory motions in the past and in the future. We want to know whether this event has positive measure or not. There was a conjecture of Kolmogorov: He conjectured that this was expected to have measure zero.

Shows two papers by Alexeev. The French version has no attribution to Kolmogorov. The English version has attribution to Kolmogorov. Katok says you should attribute this to Kolmogorov. The English version was published after his death.

(joint work with A. Gonodetski; preprint)

Kolmogorov conjecture: $Mes OS = 0$.

Main Result 1. Often 2 degree of freedom 3 body problem have Hausdorff Dimension maximal possible.

Leading Idea: Build a “fat” Cantor set $\Lambda \ni \infty$ with ergodic dynamics.

If one could produce an ergodic component with positive measure, one could perhaps prove a counterxample to Kolmogorov’s conjecture.

Second version of the main result:

Remark: 2 degrees of freedom Hamiltonian dynamics locally reduces to a 2 dimensional area preserving map. My analysis will concern those maps since they have some advantages, for example I can draw pictures.

Newhouse domains in dissipative setting. Suppose $f:M^2 \rightarrow M^2$. Suppose $f$ has a homoclinic tangency (HT). If $f$ has a saddle point then $f^k p = p$ such that unstalbe and stable manifolds satisfy….. RETURN HERE….Newhouse domains…..

Main Result 2: 2 dof 3BPs have Newhouse domains.

Remark: Duarte proved a conservative Newhouse phenomenon. (20 year interval between Newhouse and Duarte.)

Newhouse domains give rise to striking dynamical examples.

1st Model (Sitnikov): There is a beautiful book by Moser which gives an example of oscillatory motions. You have two bodies $q_0, q_1$ in elliptic orbits with eccentricity $e$. The masses $m_0 = m_1 = 1.$ The third body $q_2 = (0,0,z)$ lies on the $z$ axis.
$$
H(t, z, \dot{z}) = \frac{\dot{z}^2}{2} – \frac{1}{\sqrt{z^2 + r_e^2 (t)}}.
$$

Theorem 1 (GK): $\exists$ open nonvoid $\cal{N} \subset (0,1)$ such that for a generic $e \in \cal{N}$ we have $HD(OS) = 3$. $\cal{N}$ is a subset of a Newhouse domain.

2nd Model (Restricted planar circular 3BP): $m_2 = 0$ (restricted, comet). $q_0, q_1$ move in circular orbits. Motions are planar. In a rotating frame, these masses are fixed. Set $m_0 + m_1 =1$ and choose $m_0 = \mu, ~ m_1 = 1 – \mu$. The parameter $\mu$ is called the mass ratio. Form the so called Jacobi constant $J(x,y, \dot{x}, \dot{y}) = \frac{\dot{x}^2 + \dot{y}^2}{2} – [\frac{{x}^2 + {y}^2}{2} + \frac{1-\mu}{d_0} + \frac{\mu}{d_1}]$.

(Here $d_0$ represents the distance from $q_0$ to $q_2$ measured in the rotating frame. $d_1$ relative to $q_1$. )

Theorem 2 (GK): $\exists ~ J_0$ such that $\forall ~ J > J_0$ then $\exists ~ \cal{N}J \subset (0,1)$ with property generic $\mu \in \cal{N}J$ and $HD(OS \cap J(…) = J^*) = 3.$

Meta Theorem. Let $[H_\delta]$ be a 1-parameter family of Hamiltonian systems of 2 dof (or 1.5 dof) satisfy Hypothesis:

  • H1: As $\delta \rightarrow 0$, the limiting Hamiltonian $H_0$ is integrable with a separatrix loop.
  • H2: For $\delta \neq 0$, we want the separatrix to split tranversally.
  • H3: Melnikov function satisfies a certain open condition.

Then for a generic $\delta$ in an open nonempty set, $H_\delta$ has a hyperbolic (nonzero Lyapunov exponent) set of HD = 3.

Ideas from proof (Sitnikov):

$e = 0, ~ H = \frac{\dot{z}^2}{2} – \frac{1}{\sqrt{z^2 + 0.25}}$. He draws a picture in the $(z, \dot{z})$-plane and identifies the region $H<0$ and the region $H>0$ and highlights the “separatrix loop”. He glues the points at infinity at $ z = \pm \infty$ to highlight this as a separatrix loop. Constructing oscillatory motions corresponds to building orbits that come arbitrarily close to this point.

$C^2 – \lambda$ – Lemma…. why do we need this? We need quadratic tangency. This is an explicit system so we can’t use genericity.

Marc Chaperon, Université Paris 7,
Generalised Hopf bifurcations

Marc Chaperon

It’s a great honor to be here. I admired Vladimir Arnold very much. We liked each other. What I will speak about appears in the MMF v11(3) in memory of Arnold. The reason I became a mathematician was because of Thom but the reaosn why I persisted was probably because of ARnold. It was amazing how much energy he had. When he came to Paris, he knew more about it than I did, more than most natives. He was some kid of wunderkind and remained so his whole life.

Motivation: interest in the coupling of oscillators.

Chenciner-Iooss 1979

….I’m a bit tired so stopped typing.

Anton Zorich, University of Rennes,
Lyapunov exponents of the Hodge bundle

Anton Zorich

(joint work with Alex Eskin and Maxim Kontsevich)

I am jealous towards my colleagues. I can’t claim this work was motivated by work of Arnold. But I can report that he was constantly interested in this topic. I enormously regret that, now that the story is complete, I can not tell it to Arnold.

Motivations. Consider a billiard in the plane with $Z^2$-periodic rectangular obstacles.

Theorem (Delcroiz, Hubert, Lelivre 2011): For almost all parameters of the problem, the billiard trajectory ecapes to infinity with a rate of $t^{2/3}$.

How can we capture this $2/3$? The obstacles that can appear in this story must involve rectangles.

Exponents like this have appeared in work by Giovanni Forni.

Geometric interpretation of multiplicative ergodic theorem:

Consider a vector bundle endowed with a flat connecton over a manifold $X^n$. Having a flow on the base, we can take a fiber of the vector bundle and transport it along a trajectory of the flow. When the trajectory comes close to the starting poitn, we identify the fibers using the connection and we get a linear transformation of the fiber. The multiplicative ergodic theorem says that when the flow is ergodic a “matrix of mean monodromy” along the flow.

Moduli spaces of Abelian differentials.

Abstract version and a concrete version…..slides are pretty dense and mving a bit fast.

Teichmuller discs.

Teichmuller geodesic flow:

Teichmuller geodesic flow acts in the modulie space of pairs (complex structure, holomorphic quadratic differential.) Away from the zeros of a quadratic differential $q$ one can find a local coordinate $z$ on the underlying Riemann surface in which $q = (dz)^2$. This distinguished local coordinates defines a flat metric $|dz|^2$, which has a canonical singularities at the points where the quadratic differential has zeroes. Teichmuller geodesic flow acts as a uniform contraction in teh vertical direction and unform expansion in the horizongal direction.

(This is like Asteroids on a much richer surface than the 2-torus!)

The unraveled quotient space can be viewed as a polygon with parallel sides identified. There are some rich combinatorics available by cutting and regluing.

Hodge bundle and Gauss-Manin connection:

This reduces things down to $g-1$ Lyapunov exponents. To compute these exponents appear to be out of reach in most every dynamical system.

Siegel-Veech constant:

Closed regular geodesics on flat surfaces appear in families of parallel closed geodesics sharing the same lenght. Every such family fills a mximal cylinder having conical points on each of the boundary components. Denote by $N_{area} (S, L)$ the sum of areas of all cylinders spanned by geodesics of length at most $L$.

Theorem (Veech-Vorobets): For every $SL(2, R)$-invariant finite ergodic measure the following ratio is constant (ie.e does not depend on the value of a positive parameter L):
$$
\frac{1}{\pi L^2} \int N_{area} (S, L) d\nu_1 = c_{area} (d\nu_1 ).
$$
(The integration here is over the entire family of flat surfaces. Here $\nu_1$ is the invariant measure on the space of these surfaces.)

The constant $c_{area}$ is called the Siegel-Veech constant.

Eskin-Masur have a similar theorem for fixed $S$ where you take the limit $L \rightarrow \infty$.

What happens for the torus? For most of the tori, you can’t find closed small geodesics. However, inside the family of flat tori, there are some with very narrow cylinders with closed geodesics.

Eskin-Masur-AZ

Eskin-Okounkov computed the volumes explicitly.

Main Theorem: The sum of the Lyapunov exponents can be expressed as a sum of two (explicitly computable) constants.

V. Knizhnik

This stuff is amazing, truly beautiful. But, I can’t keep up with the typing….

Big advance by Eskin-Mirzakhani is in redaction.

Wednesday 05 October


Jacques Féjoz, Université Paris- Dauphine & Observatoire de Paris,
Diffusion along mean motion resonance in the restricted three-body problem

Jacques Féjoz

(blackboard talk)

(joint work w M. Guardia, V. Kaloshin and P.Roldan)

This work would not exist w/o the marvelous 1964 paper of Arnold. This talk is closely related to the talk that Rafael gave yesterday and also to Vadim’s talk. I will try to concentrate on other aspects.

In the solar system, there is one priviledged place where we should look for instabilities. It is called the Asteroid Belt. It is located between Mars and Jupiter. Dust particles in this part of the solar system never condensed to form additional planets. Instead, there remain nearly 2 million particles ranging from microscopic to larger asteroids, some having a size of a few hundred kilometers in diameter. If you look to the current distribution of the asteroids in this belt, it gives a quite precise idea about the stability and instability zones between Jupiter and Mars.

In 1857, an American mathematician and astronomer named Daniel Kirkwood observed that there are gaps inside the belt where there are basically no asteroids and other zones where there are lots of asteroids. These gaps correspond to orbital resonances with Jupiter. Since the orbital frequency can be read off from Kepler’s third law….he draws a graph with vertical axis as the number of asteroids and the horizontal axis is the semi-major axis between the Mars and Jupiter radii. He highlights a gap appearing at the zone located in 3:1 resonance with the Jupiter orbit. Why are there these gaps? The conjectural explanation is that an asteroid is in resonance with Jupiter (which has a mass of about 1/1000 of the mass of the sun) then the eccentricity will be unstable. If the eccentricity goes through large variations, then its perihelon will be at size $a(1 – e)$ from the sun. Therefore, the asteroid will get closer and closer to Mars. Due to this very close encounter with Mars, the dynamics will transform so that the principal force acting on it will be due to gravity from Mars rather than with the sun. In this talk, I would like to focus on the first step in this scenario. Namely, why should the eccentricity vary a lot when the asteroid is in resonance with the Jupiter orbit?

Planar restricted 3-body problem: Sun, Jupiter, Asteroid. Restricted means we take the limit when $m_{asteroid} = 0$. Practically speaking, this means that we imagine the Sun and Jupiter take place along the 2 body motion and they are not influenced by the motion of the asteroid. Let’s normalize two things. Set $\mu = mass_{jupiter}$ and the mass of the Sun is $1 – \mu$.

There are 3 “small” parameters.

  • The mass of Jupiter $\mu \rightarrow 0$. The asteroid motion collapses then to an integrable 2 body problem.
  • The eccentricity $e_0$ of Jupiter. This parameter is slightly more subtle. The limiting dynamics as $e_0 \rightarrow 0$ is not integrable. We then obtain the circular restricted problem in which the two primary objects orbit on a circle centered at the center of mass. The problem restricts from 2.5 dof down to 2 dof.
  • Semimajor axis $a$ of the Asteroid. When we let $a$ go to zero, or to infinity, we encounter 2 body problems. In one limit, the Sun dominates the asteroid motion and Jupiter is irrelevant. In the other limit, the asteroid essentially sees the gravity of a combined mass of the Jupiter and Sun.

In the problem inside our solar system:

  • $ \mu = \frac{1}{1000}$
  • $a = (\frac{p}{q})^{2/3}$. This implies that the periods of the asteroid and Jupiter satisfy $ \frac{T}{T_j} = \frac{p}{q}$.
  • We will restrict attention to $0 < e_0 \ll 1$. This allows us to view the problem as a singular perturbation of the restricted circular problem. There is also a computational reason for doing this. Part of the proof will require some numerical computations. Our strategy was to make these calculations as simple and convincing as we possibly could. All the numerical computations are done on the circular problem and boil down to checking for zeros of a one variable function. A final reason is that the real eccentricity of Jupiter is $\frac{1}{20}$. There is hope that we could in fact claim our theorem for the real value. This will require some quantifications which in principle we could extract.

Theorem: Set $\mu = \frac{1}{1000}$. Fix $\frac{p}{q} = 7$ (chosen for incidental reasons; we expect this can be relaxed; so this asteroid is outside of Jupiter corresponding more closely with Uranus….nice discussion). Assume $0 < e_0 \ll 1$. There exists a solution and a time T with $e(0) < e_{min} = 0.48$ and $e(T)> e_{max} = 0.67$ and all the while the asteroid radius $a(t) \thicksim (\frac{p}{q})^{2/3}$. Thus we have a $(p:q)$ orbital resonance with Jupiter.

What is the time scale of $T$? Conjecturally, we have $T \thicksim – \frac{\ln \mu e_0}{\mu^{3/2} {e_0}}$.

What are the units of time? Year of Jupiter.

Circular Problem:

$$
H = \frac{|p|^2}{2} – \frac{1}{|q|} + \frac{1}{|q|} – [ \frac{1-\mu}{|q + \mu|} - \frac{\mu}{|q - (1 -\mu)|}].
$$
This formulation views the principal force as provided by a fictitious mass at the origin perturbed by the separation of the Jupiter and Sun masses.

Delaunay Coordinates (written in notation of Poincaré): $(L, l, G, g) $

  • $L = \sqrt{a}$
  • $G = \sqrt{a}\sqrt{1 – e^2}$ (angular momentum)
  • $g$ is an angle to Jupiter.
  • $l$ is the angle of the asteroid advanced past Jupiter.

If we set $\mu =0$, the perturbing term vanishes and we are left with a degenerate 2BP. Understanding this limit does not bring much light to the problem.

Assume $\mu >0$. The first natural idea is to average out the fast orbital angle remaining inside this Hamiltonian. This involves first making a change of variable. The averaging process leads to an integral. This is a transcendent process. However, when $e_0, e \ll 1$, you can use the Laplace coefficients to make some computations by hand. However, this computation is not much help because we are interested in proving diffusion in the eccentricities. This calculation does give some intuition which suggests that the perturbation looks generic and the degeneracy observed in teh $\mu =0$ limit is broken.

Fact (Numerical): There exists a normally hyperbolic cylinder foliated by periodic orbits $\gamma_e$, $0.48 < e < 0.67$. This is related to an idea of R. Moeckel from the late 90s. He draws some surfaces intersecting and explains that there are some small splitting issues requiring high precision arithmetics. We limited our attention to higher eccentricities to avoid these issues. Morally, the larger value of eccentricity the faster the diffusion. As we go above $0.67$ we get closer to the Euler relative equilibrium $L_{1,2}$.

In order to lower the number of dimensions, it is perhaps a good idea to consider a Poincaré return map to $[g = 0]$. This means we are looking at what happens every time the ellipse of the asteroid is aligned along the line connecting the Sun and Jupiter. He draws a helix of 7 levels high above an ellipse. Then he slices the helix with a vertical plane and identifies the intersection points as 7 normally hyperbolic invariant cylinders. It is now time to introduce the analogs of the “inner” and “outer” maps that Rafael introduced in the more general case yesterday.

OK, mostly pictures now….

Discussion: There is no steepness in this Hamiltonian so Nekoroshev’s theorem does not apply.

Boris Khesin, University of Toronto,
Optimal transport and geodesics on diffeomorphism groups

Boris Khesin

(blackboard talk)

(joint work with J. Lennels, G. Misiolek, S. Preston)

Plan:

  • Euler equation on $SDiff$. Otto’s calculus.
  • $SDiff \subset Diff$ (numerous applications in optimal transportation)
  • $L^2$ and $H^1$ metrics.

I. Arnold’s approach to the Euler equation

This was discussed in Shnirelman and Brenier’s talks so I can perhaps be brief.

Definition:
Consider $v$ to be a velocity field on a manifold $(M, (,))$ which is divergence free. The Euler equation is then given as
$$
\partial_t v + (v \cdot \nabla) v = – \nabla p, ~ \nabla \cdot v = 0.
$$
Here $\nabla$ is the covariant derivative. If there is a boundary, we assume that $v$ is parallel to $\partial M$.

Draws a picture and identifies the Lie algebra $g$ at the identity. He explains how to use Lie algebra transportation along a geodesic in $G$.

Theorem (Arnold, Bombshell in the 60s): The Euler equation may be viewed as a the geodesic equation on $G = SDiff (M)$ (the group of volume presernving differomorphisms) w.r.t right invariant energy $L^2$-metric given on $g = Lie(G)$ by
$$
E(v) = \frac{1}{2} \int_M (v,v) \mu.

Remark: Other groups and energies give Euler top, Kirchoff equations for motion of rigid body in a fluid, KdV, Camassa-Holm, MHD, Landau-Lifschitz equation, …

II. Geometry of full diffeo group

This is the point of view rather common in optimal transport. This discussion unifies these two perspectives.

Consider $Diff$, the group of diffeomorphisms on $M$. Inside this group, we have the subgroup of volume preserving diffeomorphisms $SDiff$. The notion of $SDiff$ requires the volume form. Ebin-Marsden. He views $Diff$ as a space of fibers over the space of densities. Once you specify the volume form, this induces the fibration over the densities. Fibers $=F_\nu = [g \in Diff: g_* \mu = \nu]$.

$\exists$ “natural” $L^2$-type metric on $Diff$ for flat $M$:
$$
l^2 [g(t, \cdot)] = \int_0^1 ( \int_M (\partial_t g, \partial_t g) \mu) dt.
$$

Remark: $\forall ~M$,

$$
(v \circ g, v \circ g){L^2} = \intM (v \circ g, v \circ g)_{g(x)} \mu(x).
$$

Properties:

  • Not right invariant on $Diff$
  • Is right invariant on $SDiff$, because the Jacobian term arising from the change of variables disappears.
  • “flat” for a flat $M$: $Diff \thicksim L^2 [ g(x)]$…pre-Hlibert.
  • geodesics in $Diff(M) \iff $ solutions of the Burgers equation:
    $$
    \partial_t g (t,x) = v(t, g(t,x)); ~ \partial_t v + (v \cdot \nabla) v = 0. (*)
    $$
    This differs from the Euler equation since the right side is zero. It also does not require the zero divergence condition.This formulation appears in the paper of Ebin-Marsden.
  • Geodesics which are orthogonal to $SDiff \iff $ potential solutions $v_0 = \nabla \phi$.

The proof of (*) follows from the chain rule and the fact we are considering geodesics.
$$ 0 = \partial_t^2 g = \partial_t (v(t, g(t,x)))$$
$$ = (\partial_t v + (v \cdot \nabla v))(t, g(t,x))$$.

Remark: Geodesics on $SDiff$ are constrained within the larger family $Diff$ to remain on the subgroup. This requires imposing a force to keep the evolution within $SDiff$. This force is the pressure.

What I am describing right now is called Otto’s Calculus which arose in the study of optimal transportation.

Remark: It turns out there is a natural metric on the space of densities. We can introduce a measurement of the cost to move one density $\mu$ to another $\nu$. The natural metric is called the Wasserstein-Kantorovich $L^2$-metric on densities:
$$
dist(\mu, \nu) = \inf_{g_* \mu = \nu } \int_M |x – g(x)|^2 \mu(x).
$$
This is the cost of transporting $\mu$ to $\nu$.

Theorem (F. Otto):
$$(Diff, L^2) \longmapsto (Densities, dist)$$
is a Riemannian submersion.

Corollary: Geodesics in the space of densities starting at $\mu$ are in 1:1 correspondence with horizontal geodesics in $Diff$ starting at the identity.

This is the picture behind the scenes driving the proofs of many theorems.

Applications:

  • Conjugate points along the base of densities correspond to focal points inside the space $Diff$. P. Lee, A. Agrachev and a former student (I missed the name…)
  • Asymptotic Directions $\iff$ geodesics with higher than ususal $(\epsilon^3)$-tangency.
    Theorem: Asymptotic directions to $SDiff$ must satisfy
    $$\nabla \cdot v = 0,$$
    $$ \nabla \cdot (v \cdot \nabla) v = 0.$$
    (These are called the Bao-Ratiu equations and arise naturally from this perspective.)

Theorem (K-Misiolek): For $M$ of dimension 2 (surface), ${\overline{K}} \neq 0, ~ \forall x \in M$ there do not exist asymptotic directions. (For $K > 0$ this result was called Palmer’s theorem.)

So asymptotic directions are “rare.”

What happens if we consider slightly more general metrics instead of $L^2$. Recently, there was interest in $H^1$ metrics so let me say a few words about that.

dimension 1 2 3
$SDiff$ Rot $H(x,y)$ $[Jac = 1]$
Density $[\hat{f}(x)]$ $[f(x,y)]$ $[f(x,y,z)]$

III. $H^1$-(right-invariant) metrics on $Diff$

article

$$
(v,v){H^1} = a \| v \|{L^2}^2 + b \| \delta v^\flat \|{L^2}^2 + c \| d v^\flat \|{L^2}^2
$$

where $ v \in Vect \rightarrow v^\flat \in \Omega^1 (M)$. So, we have terms involving $\nabla \cdot v$ and another involving $curl v$, etc.

Theorem: The Euler-Arnold equations are

$(n = 1)$

  • $b = 0 \implies$ Burgers (KdV for Virasaro)
  • $a = b = 1 \implies $ Camassa-Holm equation: $v_t – v_{txx} = -3 v v_x + 2 v_x v_{xx} + v v_{xxx} $\nabla \cdot
  • $a=0 \implies$ Hunter-Saxton equation: $u_{txx} = – 2 u_x u_{xx} – u u_{xxx}$

For other $n$, one can write the corresponding equation. Various other equations arise such as Euler-$\alpha$ and many others….

There exists one metric which has nicer properties than others.

IV $H^1$-metric on Densities

$(a = c = 0, b = \frac{1}{4} \neq 0)$. So, we are considering the metric $\| v \|_{\dot{H}^1}^2 = \frac{1}{4} \int |\nabla \cdot u |^2 \mu.$

Theorem: For any compact $M$ there exists an isometry $\Phi: Densities \rightarrow U \subset S_\rho^\infty$ (an infinite dimensional sphere) where $\rho = \sqrt{vol(M)}$.

Remark: The dimension 1 case was observed by Lennels. At first, we thought this was a special case but turns out to be general and produces some nice insights.

  • Geodesics on densities are great circles on the sphere.
  • They are solutions of a high dimensional Hunter-Saxton equation (completely integrable system)

Proof: $\Phi: \eta \in Diff \rightarrow f = \sqrt{Jac (\eta)}$. Then
$$
\int_M f^2 \mu = \int_M (Jac (\eta)) \mu = \int_{\eta(M)} \mu = vol(M)$.
$$
This very metric on the sphere arised earlier in probability theory and is known as the Hellinger distance, aka Fisher-Rao metric. All these objects come together from this point of view.

article

Bassam Fayad, IMJ CNRS,
Smooth linearization of commuting circle diffeomorphisms

Bassam Fayad

(blackboard talk)

My work was completely influenced by Arnold’s papers and ideas. I had very close friends who were his students. They were completely venerating hi. I like to think that I am representing them here. They are two. These students report that what they miss most is long discussions with Arnold who had patience and knowledge and he fills you with interest.

I work on small denominators.

Mixing tori. Impossible on $T^2$. He drew a cube representing a 3-torus. The statement was not so explicit.

He connects this to the original motivation of Kolmogorov which revolved around an interest in finding mixing.

Circle Diffeomorphisms:

$$
f(\theta) = \theta + \tilde{\alpha} + \phi(\theta)
$$

$\rho (f) = \alpha$ is the (irrational) rotation number.

Denjoy theory. If $f \in C^2$ then $ f = h R_\alpha h^{-1}$ where $h$ is a homeomorphism of the circle.

What is the regularity of the homeomorphism?

Siegel. Arnold.

Arnold showed that if $\alpha$ is Diophantine and $f$ is close to $R_\alpha$ then $ f $ is analytic and $h$ is analytic.

What is diophantine?

  • $\alpha \in DC (\gamma, \tau)$ if $ |\alpha – \frac{p}{q}| > \frac{\gamma}{q^{2 +\tau}}$
  • Best approximations. $\| k \alpha \| = \inf_l | k\alpha – l|$. The sequence of best approximations is defined by
    $$ \| q_n \alpha \| < \| q \alpha \|$$
    for all $ q < q_{n+1}, ~q \neq q_n$.

Linearized equation:

$\phi (x) = \psi (x + \alpha ) – \psi (x)$.

The global problem remained open and was conjected by Arnold. Even without the closeness condition, Arnold conjectured that the rotation number being diophantine was all that was required to ensure the analyticity of the homeomorphism.

Herman 1976 ($H$-class of numbers), Yoccoz 1981 (all Diophantine numbers)

If $f = h R_\alpha h^{-1}$ with $h$ a homeomorphism and $\alpha$ is irrational and you have $f \circ g = g \circ f$ then $g = h R_{\rho{g}} h^{-1}$. (My quotation of the Qualifiers might be wrong here….be CAREFUL….)

Why does commutation imply higher regularity, more rigidity? The idea emerges from a paper by Moser 1981 who proved KAM smooth linearization of $f,g$ commuting if $(\alpha, \beta) \in SDC$

$(\alpha, \beta)$ are SDC (Simultaneous Diophantine Condition) if $\max(\|k \alpha\|, \| k \beta \|) \geq \frac{\gamma}{k^\nu}$.

Applying these techniques, you can show: If $(\alpha, \beta) \notin SDC$ then $\exists ~ (f,g)$ commuting then $h$ is not absolutely continuous.

Theorem (K. Khanin, F): $(\alpha, \beta) \in SDC$ and $ f \circ g = g \circ f$ in $Diff^\infty \implies h \in Diff^\infty$.
KF Paper: Annals 2009

Very clever pivots in a case-by-case analysis. Some pigeon holes. Make friends with your enemy.

Thursday 06 October


Chong-Qing Cheng, Nanjing University,
One way to Cross Complete Resonance

Nice introductory discussion of Arnold Diffusion, placing the principal settings studied so far in context. Mentions that there is a “definition” of Arnold Diffusion in v3 of Arnold’s book.

Some nice pictures suggesting the mechanism.

A big issue to overcome is that there are uncountably many barrier functions. One way is to study the regularity. This will impliy the finiteness of the Hausdorff dimension.

Resonance path. KAM iteration at complete resonant point. Very nice pictures of the Aubry set!

Interesting discussion following the talk between Cheng and Mather, comparing their respective strategies.

James Ellis Colliander, University of Toronto,
Hamiltonian PDEs

I spoke so I didn’t type.

Artur Avila, IMJ, CNRS, 
Global theory of one frequency Schrödinger operators

Artur Avila

(blackboard talk)

This topic can be introduced in several ways. I try to present this work in a way that is connected to the work of Arnold.

KAM-persistence of quasiperiodic motion.

One theorem of Arnold: $f$ analytic diffeo of $T = R/Z$ orientation preserving has a rotation number $\rho$.

Theorem: If $\rho$ is Diophantine and $f$ is close to translation then $f$ is linearizable (analytically conjugated to tranlation).

He also makes a conjecture. This should be global. This means that the hypothesis “f close to translation” should not be necessary. This was proved by Herman in a breakthrough work introducing new techniques, and eventually completely resolved by Yoccoz. These results are now understood in a new framework called renormalization.

What is the situation in higher dimensions? Not every $T^2$ has a notion of translation number. Suppose we have a diffeo that has a rotation number and is close to translation. What can be said? In higher dimensions, the local theorem survives. Herman asked: which aspect of the global theorem will survive? It’s a paper of Herman with a very long title…

“Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnolʹd et de Moser sur le tore de dimension 2.”

Example:

$$(x,y) \longmapsto (x + \alpha, A(x) \cdot y)$$

$A(x)$ will be some projective action. In particular, I will imagine that $A(x) \in SL(2;R)$. So, I am viewing the second coordinate as an element of $PR^2$.

$A(x)$ is a matrix ($E – \lambda v(x), -1$) (top row) and (1,0) in bottom row. Here the parameters are $E \in R$, $v$ is a trig polynomial, $\lambda >0$.

Rotation number?

In the first slot, it is clear that the number is $\alpha$. In the second slot, it depends on all the parameters. It turns out that in this case, it is well-defined and is montonic wrt $E$. Diffeos of the circle have those regions called Arnold Tongues and there are similar structures here…rationality condition…draws a Cantor-like set. He draws a “vertical curve” in the $(E, \lambda)$ plane and along that curve we have $\rho = constant$.

Theorem (Herman):

  • $\lambda \ll 1$, KAM works
  • $\lambda \gg 1$, Lyapunov exponent $LE >0$. Independent of E

$(\alpha, A)^n = (n \alpha, A_n)$. $A_n (x) = A(x + (n-1) \alpha) … A(x)$.

$$L = \lim \frac{1}{n} \int \ln \| A_n (x ) \| dx \geq 0. $$

If $(\alpha, A)$ is conjugate to translation then $LE = 0$.

OK, so what is the obstruction to globalization? IS $LE$ the only obstruction to conjugacy?

$LE = 0$, continuity argument. Goldstein-Schlag, Bourgain-Jitormskaya

At the endpoint of the supremum of the good parameter, we can not have analytic conjugacy. What broke down? You might think it is just that we lose analyticity. But this turns out to not be the case because in this context topological conjugacy $\implies$ analytic conjugacy. So, it is possible to have $LE =0$ while losing even the topological conjugacy.

$A$ is analytic, extends to a neighborhood $[|\Im x | < \epsilon]$. $A_n (x)$ is defined on the same neighborhood. Maybe the LE changes a bit as we move off the real axis? You see that if $(\alpha, A)$ is analytic conjugate to translation then this kind of complexified LE
$$
\lim \frac{1}{n} \int_{Im x = a} \ln \| A_n (x) \| dx = 0.
$$
is still zero. Now, you have some kind of necessary condition that is implied by analytic conjugacy.

Theorem: $(\alpha, \rho)$ diophantine. $(\alpha, A)$ is analytically conjugate to translation $\iff ~ LE =0$ for $|\Im x|$ sufficiently small.

The method of proof of this theorem is quite interesting, but I won’t do that right now. You can’t just apply KAM theorem. It is closer to a different theorem. You need an a priori bound on the renormalization. In fact, you have that the complexified LE is behaving subexponentially…lots of work to be done to get conjugacy. What I want to do instead is to connect with the title of the talk.

$l^2 (Z)$

$
(Hu)n = u{n+1} + u_{n-1} + v(n \alpha ) u_n.
$

$$ v: T \rightarrow R, ~ analytic.
$$

$$ Hu = Eu. $$

This generates a one-parameter family of cocycles. $H \iff $ one-parameter family of cocycles. Those cocycles have been studied a lot for similar reasons why dynamicists should interested in cocycles. They are the simplest class with $LE>0$, are consistent with KAM theory, and have rich dynamical behavior (e.g transition from absolutely continuous and discrete spectra, etc.)

(spectrum should usually be thought to be a Cantor set, similar to the situation of those Arnold tongues.)

Local Theories:

When $v$ is small. This is often quantified by writing $\lambda v$ with $\lambda$ small. This was developed by Dinaburg-Sinai, Eliasson, Bourgain, Jitormiriskaya, Avila, Fayad, Krikorian, …. From the beginning, it started with $\lambda$ small so that you can apply KAM theory. An end conclusion here is that the spectral measures are absolutely continuous. sigh… What does this imply? You start with some state that lives in this lattice and you let it evolve. What happens here is that it spreads at the fastest possible transport. This is called ballistic motion, so it moves like the free problem.

Eliasson: not the same speed?….Avila:….in average over time, you can see that it is the same speed whenever there is continuous spectrum. Eliasson:….oh you time average, I see…. Craig:….seems you need some smoothnes. Avila:..(eagerly)….which kind of smoothness do you need? Craig:….need to integrate by parts. Avila:….I have some weak smoothness. Craig:….maybe can help. Avila:…I expect it will help but don’t know how to use it yet.

When $\lambda \gg 1$, there is a different theory. Sinai, Frohlic-Spencer, Eliasson, Bourgain, Goldstein, Schlag. This theory corresponds to the situation where $LE > 0$. (Typically…some almost every conditions), the spectral measure is pure point. This means that the infinite matrix is diagonalizable and the quantum dynamics is quasiperiodic. It is known that the eigenfunctions decay exponentially, so they are better localized than required by $L^2$. Two approaches to this: KAM and more recent interactive techniques like renormalization. With Bourgain, Goldstein, Schlag, new nonperturbative techniques developed based upon the assumption that $LE >0$. It might be possible to understand the dynamics across the entire parameter space.

He draws an egg. He draws a line along the egg representing strength of nonlinearity. When the nonlinearity is small, we KAM-like behavior. Both regions can be shown to be open. When the nonlinearity is big, we $LE >0$. The egg has an “region” in between. Does that intermediate region have a non-empty interior?

Natural global questions:

  • What is the behavior of typical one frequency Schrodinger operator?
  • In particular, is there an influence of other behaviors of cocycles?
  • You might be optimistic and hope to prove there is some kind of phase transition between the KAM-like and $LE >0$ regimes?
  • Describe the phase transition as “interface-like”? This would go in the direction of showing that there are not these other types of dynamics of cocycles.

This was basically blocked for some time. But, recently, well maybe not recently, it was in 2008, there emerged some new ideas to approach these questions. Large parts of the emerging program have been carried out.

Center your attention on $LE$ and its dependence upon parameters. It’s good to get some kind of target to focus your attention upon. There will be three regimes:

  • Supercritical: $LE >0$ (leads to localization; point spectrum)
  • Critical: otherwise
  • Subcritical: $LE = 0$ in a complex neighborhood $[|Im x| < \epsilon]$. (show that it is KAM-like; AC spectrum)

The main parts of the program. Study the critical “interface”. Study how it intersects one-parameter families and so on. This would be a kind of geometric approach to begin to understand the dynamics across the parameter space.

What parts are completed? Several parts…..here is a main theorem.

Theorem: $H$ is a typical one-frequency Schrödinger operator. $H = H_+ \oplus H_-$. $\sigma(H_+) \cap \sigma(H_+) = \emptyset$. $H_+ $ is supercritical, localization. $H_-$ is subcritical and KAM-like, AC spectrum.

What about the growth of critical energies? There are no critical energies. This is a bit surprising. As I said before, I had a picture moving up from the KAM region and we encounter a transition point. It would be natural to expect that you will face a critical energy. This turns out to not be the case. The critical interface has zero measure, a kind of Cantor set, inside a codimension 1 subspace. This means it won’t intersect a typical one-parameter family. All this takes place inside the geometric analysis of the parameter space.

$\alpha$ diophantine, subcritical is $KAM$-like. Even thought I don’t have the conjugacy, I still have good control on the dynamics. This is the concept of almost-reducibiilty.

What is the meaning of typical? $\alpha$ is almost every. We also have $v$. What’s a zero measure set in infinite dimensions? It involves some kind of concept called prevalence and Gaussian measures. My good set of $v$’s has the following property:
$\forall ~ v_0, $ for almost every $\lambda_n \in [-1, 1]^Z$ then $v_0 + \sum \lambda_n^{\epsilon_n} e^{2 \pi i n \cdot x}$ where $\epsilon_n = \frac{1}{(n!)^2}$.

Mikhail Sevryuk, Russia Academy of Sciences, 
The reversible context 2 in KAM theory for lower dimensional tori

A review of KAM theory. “Meta-reason” for the ubiquity of invariant tori carrying conditionally periodic motions: any finite dimensional connected compact Abelian Lie group is a torus.

Various structures (contexts):

  • Hamiltonian
  • reversible
  • volume preserving
  • dissipative (no structure at all)

Key advances highlighted here.

  • Moser 1966
  • Herman 1988

Lai-Sang Young, New York University,
Toward a smooth ergodic theory for infinite dimensional systems

Lai-Sang Young

(Scanned hand-written marker slides! Cool….we party like its 1999!)

Aim of larger project:

  • Extend finite-d nonuniform hyperbolic theory (=ergodic theory of chaotic systems) to $\infty$-d
  • New phenomena
  • include settings related to some PDEs. (principles should include nonempty set of PDEs.)

The body of finite-d stuff that I have in mind does not include Hamiltonian systems. Instead, we are looking at problems which include some dissipation. So the invariant sets we are looking at are like attractors, etc.

Today’s talk:

  1. Reduction to finite-d via $W^c$-inf and $W^\epsilon$-foliations. Upshot: notion of “a.e.” initial conditions in $\infty $ dimensions.
  2. Example of strange attractors from Hopf bifiurcation + forcing. Illustration of how to leverage finite-d techniques
  3. Lyapunov exponents, periodic orbits and horseshoes for semiflows on Hilbert spaces (extend Katok’s results for finite-d diffeos.)

Some background info:

Givne an evolutionary PDE, view this as an “ode” on a function space. I want to see it as a dynamical system.

$$
\frac{du}{dt} + Au = F(u)
$$
where $u \in X, ~ A $ is a linear operator, $F$ is the nonlinear part.

To define (smooth) dynamical system, need $(X, \| \cdot \|)$ such that

  1. $u(0) \in X \implies u(t) \in X ~ \forall t \geq 0$,
  2. $ t \longmapsto u(t), ~ t \geq 0$, continuous,
  3. Smoothness of the time-t map $f^t : (X, \| \cdot \|) \longmapsto (X, \| \cdot \|)$ which maps $ u(0) \longmapsto u(t)$.

for nonunif hyperbolic theory, generally require $C^{1 + \alpha}$.

Model setting:

$X$ is a Banach space. $A$ is an operator on $X$. Assume $A$ is “sectorial” or self-adjoint w eigenvalues on $[a, \infty)$.

Q: Can I just think of $A = \Delta$? A: Yes. (OK, I’ll think that way….knowing that there are generalizations.)

A sample result:

Theorem (Henry ~80):

Discussion….skip it….just know that we are not talking about an empty set.

“Solution” means mild solution.

I. Reduction via center manifolds

$W^c$ can be local, global, or “medium size”

Constantin-Foias-Nicolanenko 86, Chow, Sell, Mallet-Paret, Lu, …

Think of $f$ as the time one flow-map associated to this dynamical system on $X$.

(A1) Reference Splitting: $X = E^c \oplus E^s$, closed subspace, not invariant.
(A2) Absorbing “slab”: $\forall ~ R ~ \exists ~ R’$ such that $f( E^c \times B^s (0,R)) \subset E^c \times B^s (o, R’)$.
(A3) INvariant cones ….lots written on slide here, can’t keep up with that. Some nice pictures to explian what is going on.

$E^s$ is vertical, $E^c$ is horizontal.

….questions….is the center manifold infinite dimensional?…..this is just the setting. I’ll be precise about the theorems soon.

some spectral assumptions.

Theorem 1 (Existence of $W^c$): $\exists ~ ! ~ W^c = graph(h^c), ~ h^c : E^c \rightarrow E^s, ~C^{1+\alpha}$, invariant.

Theorem 2 (Existence of $W^s$ foliations): slide slid up…..

Theorem 3 (Absolute continuity of $W^s$-foliations): (Zeng Lian, Chongchun Zeng, LSY): Assume $dim(W^c) < \infty. Then $W^s$-foliation is absolutely continuous.

Strange Attractors arsising from periodically forced Hopf bifurcations.

(joint work w. Kening Lu and Qiudong Wang)

Result for ODE in 2D $\rightarrow $ Corresponding equation for evolution equation in Hilbert space $\rightarrow $ Application to a specific PDE.

We have an unforced system with a parameter which is undergoing a “generic” supercritical Hopf bifurcation at $\mu = 0$.

  • $\mu < 0$
  • $ \mu = 0$
  • $ \mu > 0$.

Normal form. Introduce the twist number, expressed in terms of coefficients appearing in the normal form.

Forced system. Periodically, we kick it and then let it relax. It doesn’t have to be a kick. It just needs to relax in between the applications of the forcing.

Theorem (LWY): …I read it rather than type it…. there is some number you can calculate that as to be pretty big. We have a big kick period. Then you have a strange attractor with complicated dynamics. The attractor has an SRB measure.

An SRB measure is an important concept in finite-d and is the first challenge to bring it to infinite dimensions. If you look at a Hamiltonian system with flowmap $\phi_t$. Let $m$ be the Liouville measure. Assume ergodic. Then $\forall$ cts $g$ we find
$$
\frac{1}{T} \int_0^T g( \phi_t) x dt \rightarrow \int g dm
$$
for $m-a.e.$ x. (Birkhoff Ergodic Theorem)

Now suppose you have an attractor. (Sinai-Ruelle-Bowen). An invariant measure $m$ is called SRB or physical measure if the same convergence takes place for Lebesgue-a.e. In this setting $m$ is completely singular compared to the ambient Lebesgue measure. This is considered to be the analog of the Liouville theorem for dissipative systems.

I am claiming that these attractors support these measures.

Example, nice pictures.

Kick can be quite general.

Lyapunov exponents and $W^u, ~ W^s$-manifolds

Ruelle, Mané, Thieullen 80s, Lian-Lu (later)

Cocycle set up. Biggest differences w. finite-d:

  1. $\Phi (x)$ generally not onto (possibly 1:1)
  2. “Essential spectrum” – Lyapunov exponent is not defined.

Kuratowski measure of noncompactness.

Extension of Katok’s results….moving a bit fast here.

Friday 06 October


(Alas, my flight departure time will force me to miss out on hearing these talks.)

Laurent Stolovitch, CNRS, Université de Nice, 
Smooth Gevrey normal forms of vector fields near a fixed point

Laurent Stolovitch

Claude Viterbo, Ecole Normale Supérieure, Symplectic Homogenization

Claude Viterbo

Anatoly Neishtadt, Loughborough University
Averaging, passages through resonances, and captures into resonance in dynamics of charged particles

Anatoly Neishtadt

Galina Perelman: 2 soliton collision in NLS

$$
i \psi_t = – \psi_{xx} + F(|\psi|^2) \psi, ~ x \in R
$$
where $F(\xi) = -2 \xi + O (\xi^2), ~ \xi \rightarrow 0.$

This family of equations has solitary wave solutions
$$
e^{i \theta(x,t) \phi (x – b(t), E)}
$$
where $\theta(x,t) = \omega t + \gamma + v \frac{x}{2}, ~b(t) =vt + c$ (all reall parameters). The profile $\phi$ is the associated ground state, which is $C^2$, decays exponentially, is even, …

If I set $\epsilon^2 = E$ and write $\phi(y, \epsilon^2) = \epsilon \hat{\phi}(\epsilon, \epsilon).$ We have then that $\hat{\phi}(z, \epsilon) = \phi_0 (z) + O(\epsilon^2)$ where $\phi_0$ is the standard soliton for cubic NLS. A calculation shows that
$$\| \phi(\cdot, \epsilon^2) \|_{H^1} = O(\epsilon^{1/2}).$$ Let’s collect the parameters $\sigma = (\beta, E, b, v) \in R^4.$

The question I’d like to address:

Question: As $t \rightarrow -\infty$, suppose that $\psi(t) = w(\cdot, \sigma_0 (t)) + w(\cdot, \sigma_1 (t)) + o_{H^1} (1)$. Because of the galilean invariance we can arrange so that $\sigma_0$ does not move and we assume that $v_1 > 0$. So, we can arrange this data to have completely decoupled solitons as $t \rightarrow – \infty$. The question is then to understand the soliton collision and also what happens afterwards.

Perturbative regime:
$$\epsilon^2 = E_1 \ll 1, E_0 \thicksim 1, v_1 \thicksim 1.$$

Collision Scenario:

  1. $w(\cdot, \sigma_0 (t))$ is ‘preserved’.
  2. $w(\cdot, \sigma_1 (t))$ splits into two outgoing waves of the cubic NLS. The splitting is controlled by the linearized operator associated to the large soliton $w_{\sigma_0}$.

Collision: $|t| \lesssim \epsilon^{-1-\delta}, ~ \delta > 0$.
pre-interaction: $t leq – \epsilon^{-1-\delta}$
post-interaction: $t leq – \epsilon^{-1-\delta}$

She draws a picutre:

Long wide soliton to the left of a big soliton at the origin before the collision. After the collision the small soliton splits into two waves, one moving left and one moving right. The big soliton at the origin is drawn not centered at the origin.

$s = s(\frac{v_1}{2}), r = r (\frac{v_1}{2})$ where $s(k), r(k)$ are the translation and reflection coefficients of the linearized operator corresponding to $w(\cdot, \sigma_0 (t))$. Here we have $|s|^2(k) + |r|^2 (k) =1$. The only trace of nonlinearity appears in the phase.

This phenomena has been observed before by Holmer-Mazuola-Zworski and earlier by physicists.
H-M-Z conisdered the cubic NLS with an external delta potential. For small incoming solitons, they have observed the small soliton splitting caused by the Dirac function potential.

Hypotheses:

(H0): $F \in C^\infty, F(\xi) = – 2 \xi + O(\xi^2), \xi \rightarrow 0.$
$F(\xi ) \geq – C\xi^q, C>0, q<2, \xi \geq 1$. (GWP in $H^1$)
$\exists !$ ground state.

Linearization around $w(x, \sigma(t)) = e^{i\theta} \phi(x – b(t), E)$. We substitute $\psi = w + f$ and expand to obtain the following equation for $f$:

$$
i {\bf{f}}_t = L(E) {\bf{f}}.
$$

Here ${\bf{f}}$ is a (column) vector $(f, \overline{f})$.
$$
L(E)= (-\partial_y^2 + E) \sigma_3 + V(E).
$$
Here $\sigma_3$ is the Pauli matrix and $V$ is a certain matrix involving $V_1 = F(\phi^2) + F’ (\phi^2) \phi^2$ and $V_2 = F’ (\phi^2) \phi^2$.

She draws a spectral plane. Essential spectrum along real line in region $|x| > E$ and some eigenvalues drawn as x’s inside the gap and one above and below the real line on the imaginary axis. 0 is an eigenvalue. We have two explicit eigenfunctions $\xi_0$ and $\xi_1$.

$M(E)$ is the generalzied null space of $L(E)$. We have the following equivalence:

$$\sigma(L(E)) \subset R, {\mbox{dim}} M(E) = 4 \iff \frac{d}{dE} \| \phi(E) \|_2^2 > 0.$$

These conditions imply the orbital stability of $\Phi$.

$Lf = \lambda f, ~\lambda \geq E, \lambda = E + k^2, ~ k \in R$. If $k^2 + I \notin \sigma_p (L(E))$ then $\exists ~! ~ f(x,k) = s(k) e^{i k x} (1, 0)^t + O(e^{-\gamma x})$ as $ x\rightarrow + \infty, ~ \gamma > 0$ and $f(x,k) = e^{ikx} (1,0)^t + r(k) e^{-ikx}(1,0)^t + O(e^{\gamma x}), x \rightarrow – \infty$.

$w(x,\sigma, t), ~ j=0,1$ normalized as before.

(H1):
$$\frac{d}{dE} \| \phi(E) \|2^2 |{E=E_0} > 0$$

(H2): $\epsilon^2 = E_1$ sufficiently small

(H3): $M(E + \frac{v_1^2}{4}) \notin \sigma_p (L(E_0))$ (Nobody knows how to prove no embedded eignevalues.)

Proposition: $\exists ~! ~ \psi \in C(R, H^1)$ such that ….

Theorem: For $\epsilon^{-1-\delta} \leq t \leq \delta \epsilon^{-2} | \ln \epsilon |$

$$ \psi (t) = w (\cdot, \sigma(t)) + \psi_+ (t) + \psi_{-} (t) + h(t)$$

  1. $\sigma(t) = (\beta(t), E_0, b(t), v_0), ~V_0 = \epsilon \kappa$ where $\kappa$ is an explicit constant and
    $$
    |\beta(t) – \beta_0 (t)|, |b(t) – v_0 t| \leq C \epsilon^2 t.
    $$
  2. $\Psi_{\pm} (x,t) = ….ack too fast to type…
    $$\Psi_{\pm}$$

is expressed as an explicit phase times a function $S^{\pm}$ which solves cubic NLS emerging from data built using thre reflection, transmission coefficients and $\phi_0 (y)$.
3. error estimates in terms of $\epsilon.$


Edriss Titi: Loss of smoothness in 3d Euler Equations

(joint work with Claude Bardos)

Overview:

  1. Background
    • Euler
    • Classical
    • Nonuniqueness: De Lellis – Sh…
  2. Shear flow
    • DiPerna Majda example: weak limit of Euler solutions whose limit is not a solution
    • Illposedness of Euler in C^{0,\alpha}
  3. Vortex sheets induced by 3d shear flows
    • Examples
    • Differences between 2d and 3d Kelvin-Helmholtz problems
    • Comments on numerics

Euler equations

Euler equations on the 3-torus. $\omega$ is the vorticity. Recast using Biot-Savart.

Vorticity stretching term distinguishes 2d and 3d.

Classical Wellposedness:

  • global existence and uniquenes for initial data $\omega_0 \in L^\infty$.
    This result is due to Yudovich (1963). Some extension….
  • For data in $C^{1,\alpha}$, Euler equations are short time well-posed and the solution conserves energy. [Lictenstein (1925)]
  • The same result holds the context of Sobolev spaces $H^s, ~ s > \frac{5}{2}$. (Basically same result in more modern spaces)

Question: Does there exist a regular solution (say in $C^{1,\alpha}$) of the 3d Euler equation that becomes singular in a finite time (blows up problem)? This is in osome sense as difficult as the millenium problem. There are different opinions….

“I spoke with Necas about this…near end of his life…on Wendesday’s he thinks it blows up and on Thursdays he thinks no…so he has bad dreams about it…”

DeLellis-Szekelyhidi: There exists a set of initial data $u_0 \in L^2 (\Omega)$ (not explicitly constructed, Baire argument) for which the Cauchy problem has, for the same inital data, an infinite family of weak solutiosn of the 3d Euler equations: a residual set in the space $C(R; L^2_{weak} (\Omega))$.

These are also in $L^\infty$ so they have finite energy. (Built on Shnirelman and others….). This is a breakthrough…but it is not so physical. Maybe a selection mechanism….for NS we don’t have such a result. Leray solutions are not known to be unique. Any result like this for NS would be extremely important….connect it with turbulence. The lack of uniqueness, according to Leray, relates to turbulence.

Shear flows:

$$u(x,t) = (u_1 (x_2), 0, u_3 (x_1 – t u_1 (x_2))).$$

For $u_1, u_3 \in C^1$, the above shear flow is a classical solution of the Euler equations with pressure $p=0$. Yudovich used these to show the existence of solutions with exponentially growing high regularity norms.

This example due to DiPerna-Majda (1987).

Theorem (DiPerna-Lions): Norm explosion in $W^{1,p}$ for Euler, for any $p \geq 1$.

Idea of the proof:
$ \partial_{x_2} u_3 (x_1 – t u_1 (x_2))=…$

Theorem: The shear flow is a weak solution of the Euler equations in the sense of distribtuions in $R^3$, provided $u_1, u_3 \in L^2_{loc} (R^3)$. On the periodic box, we can do same thing and in this case we have finite energy.

Why do I stress the finite energy? This relates to the Onsager conjecture.

Theorem [Ill-posedness of the Euler equations in $C^{0,\alpha}$]:

The shear flow with $C^{1,\alpha}$ components $u_1, u_3$. However, for $u_1, u_3 \in C^{0,\alpha}$ then the above shear flow is always in $C^{0, \alpha^2}$ which is a much larger space. We instantly lose the $C^{0, \alpha}$. There exists a shear flow which starts in $C^{0, \alpha}$ which, at any positive time, is not in $C^{0, \beta}$ for any $\beta > \alpha^2$.

This family of solutions is compactly supported in space and time.

Other spaces and optimal spaces:

There are many layers of spaces between these H”older spaes. He writes a tower of inclusions between $C^{1,\alpha } \subset C^{0, \alpha}$. In fact, there is well-posedness
[Pak and Park] vs. failure of wp in $B^1_{\infty, infty}$ (Zygmund class) and failure in certain Triebel-Lizorkin spaces.

Weak limit of oscillating initial data:

DiPerna-Majda example…

Shear flow with vorticity interface. Vortex sheet flows are irrotational off an interface. To build such solutions he takes $u_1, u_3$ as (parametrized) Heaviside functions.

…wow…this talk is coming pretty fast, slides are changing…I stop typing and start to just try to keep up.

Numerical investigation of blowup for the 3d Euler

John Gibbon gave a talk a few years ago on the history of these investigations. Tom Hou and Bob Kerr are competing and disagreeing in this direction….is there a singularity…maybe not?

Question: Does the soluton of the following PDE blow up?
$$
\partial_t u – \nu \Delta u = |\nabla u |^4?
$$

What would you try numerically to determine if it blows up or not? You can even collapse it to the corresponding 1d problem?

Postlude Discussion:
Yudovich explored the DiPerna-Lions shear flow examples to see that norms measuring high regularity can grow exponentially in time. Chemin has studied the vortex patch and shown some measures of regularity of the boundary of the patch grow doubly exponentially fast. It was not explicitly clear to me yet how to relate Chemin’s rough patch boundary example to the growth of norms measuring regularity of the solution. Also, Chemin’s examples emerge from non-smooth initial data. I remain interested in the question: Does there exist nice data for 2D Euler which evolves with high regularity norms growing doubly exponentially?


Benoit Grébert: Hamiltonian Interpolation for Approximation of PDEs.

(joint work [Grébert-Faou] with Erwan Faou)

Aim:

Take a PDE with solution u. Consider a numerical approximation $u^n$ built with a symplectic integrator which approximates $u(nh)$. We build a hamiltonian $H_h$ such that
$$u^n = \Phi_{Hh}^{nh}(u_0) + very ~small.$$

I am concerned with the long time behavior of the numerical trajectory.

My concern right now is not in estimating the quality of the approximation. Instead, I want to understand the numerical flow.

Outline:

  1. Finite dimensional Context (ODE)
  2. PDE Context
  3. Ideas of the proof (time permitting)

Finite Dimensional Context

We go back to Moser’s theorem. A discrete symplectic map close to the identity can be approximated by a Hamiltonian flow. Consider an analytic symplectic map
$$
R^{2n} \ni (p,q) \longmapsto \Psi(p,q) \in R^{2n}
$$
with $\Psi = Id + O(\epsilon)$. Then $\exists~ H_\epsilon$ such that
$$\Psi = \Phi_{H\epsilon}^\epsilon + O(e^{-\frac{1}{c\epsilon}}).$$
([Moser 1968], [Benettin-Giorgilli 1994])

Numerical Context: Suppose I have a Hamiltonian ODE system
$$ (\dot{p}, \dot{q}) = X_H (p,q)
$$
and an associated numerical discrete-time-step symplectic integrator
$$
(p_n, q_n)= \Psi_h^n (p_0, q_0).
$$
We then have that $\Psi_{h} = \Phi_{Hh} + O(e^{-1/ch}).$ We obtain that $H_h (p_n, q_n) = H_h (p_0, q_0) + n e^{-1/ch}$. So, we are observing that the modified energy is essentially conserved for exponentially long times.

Backward Error Analysis

PDE Context

$$ H = H_0 + P$$
Here we imagine $H_0$ is the linear part and P is the nonlinear part. As an example, consider the cubic NLS on $T^d$. We can treat other equations as well. Let’s recall the Hamiltonian formalism in the Fourier variables:

Expand $u$ to get
$$
u = \sum \xi_j e^{ijx}, ~ {\overline{u}}= \sum \eta_j e^{-ijx}.
$$
We can then write, for each $j \in Z^d$,
$$
{\dot{\xi}} = -i \frac{\partial H}{\partial \eta}
$$
$$
{\dot{\eta}} = i \frac{\partial H}{\partial \xi}.
$$
For the cubic NLS case, we obtain
$$
H = \sum |j|^2 \xi_j \eta_j + \sum^* \xi_{k_1} \xi_{k_2} \eta_{l_1} \eta_{l_2}
$$
where $\sum^*$ is the sum over all the parameters subject to the constraint $k_1 + k_2 = l_1 + l_2$.

The problem we face here is that the linear part is unbounded, and we have infinitely many dimensions as first obstructions in passing from the ODE to the PDE context.

Splitting Method:
$$\Phi_{P+H0 } \thicksim \Phi_p^h \circ \Phi_{H0}^h =^? \Phi_{Hh}^h.$$

First naive idea: Use the Baker-Campbell-Haussdorf formula. We can then expand as a Lie series…
to write
$$\Phi_p^h \circ \Phi_{H0}^h = e^{h\mathcal{L}p}e^{h\mathcal{L}H0} = e^{h\mathcal{Hh}}
$$
with $H_h = H_0 + P + \frac{h}{2}{ P, H_0 } + \dots.
$$

To proceed, we will need conditions $small = h^N C(N,, \| num sol \|_H^N)$ NOT FAIR! So we need to work harder.

First Idea:
Replace $hH_0$ by $A_0$ by cutting off to low frequencies. We can splt and impose the CFL condition.
Midpoint + split. He considers different cutoffs.

We then consider $\Phi_p^h \circ \Phi_{A_0}^1$.

Second Idea: Use the Wiener Algebra. Space of functions with Fourier coefficients in $l^1$.


Theorem (Grébert-Faou): For the approximation scheme $\Phi_p^h \circ \Phi_{A_0}^1$ there exists a (polynomial) modified energy $H_h$ such that

$$
\| \Phi_p^h \circ \Phi_{A_0}^1 (\xi, \eta) – \Phi_{Hh}^h(\xi, \eta) \|_{l^1} \leq h^{N+1} (cN)^N
$$

uniformly for $\|(\xi, \eta)\|_{l^1} \leq M.$

So, assuming that the numerical trajectory is bounded in $l^1$ (as opposed to the stronger claim that it is bounded in $H^k$ for $k$ large) then
$$
H_h (u^n) = H_h (u_0) + Cn h^{N+1}.
$$


Of course, I have to explain: what is $N$? This is related to a regularization condition. We know that $N = \frac{r-2}{r_0 – 2}$ where $r_0$ is the degree of $P$ (so 4 for cubic NLS). The parameter $r$ is determined by the condition:
$ \forall ~ j = 1, \dots, r$ and for any $j$-tuple of integers $(k_1, \dots, k_j) \in Z^d$, we have
$$|\lambda_{k1} \pm \lambda_{k2} \pm \dots \pm \lambda_{kj}| \leq 2 \pi.$$

CFL: $|\lambda_k | \leq C$.

He describes some examples where $N = 3, 4$ and $N=7$.

For cubic NLS, we end up obtaining
$$Hh = \frac{1}{h} A_0 + Z_1 + h Z_2 + \dots
$$
where
$$
Z_1 = \sum^* \frac{e^{i(\lambda_{k1} \pm \lambda_{k2} \pm \dots \pm \lambda_{kj})}}{e^{i(\lambda_{k1} \pm \lambda_{k2} \pm \dots \pm \lambda_{kj})} – 1}.
$$
You can now see how the zero divisor issue emerges and is resolved.


,


Sergei Kuksin (École Polytechnique): Nonlinear Schrödinger Equation

We consider Hamiltonian PDE. This is of course very interesting. In physics, there is a class of pdes which is also of interest:

Hamiltonian PDE = small damping + small forcing

Why is it so important?

  1. This class contains a very important equation: Navier-Stokes.
    $$
    \dot{u} + (u\cdot \nabla) u + \nabla p = \epsilon \Delta u + force; ~ \nabla \cdot u = 0.
    $$
    We are interested in cases $d = 2,3.$ For $d=3$, this problem seems impossible. So, let’s collapse to the 2d case.
  2. Nonlinear Schrödinger equation with some damping and forcing
    $$
    \dot{u} + i \Delta u – i |u|^2 u = \epsilon \Delta u + force.
    $$
    Similarly, we might want to study the PKdV equation
    $$
    \dot{u} + u_{xxx} + u u_x = \epsilon u_{xx} + force.
    $$

We are interested in the small viscosity $\epsilon \ll 1$ and $t \rightarrow \infty$ extremes. At least we want to study $t \gtrsim \epsilon^{-1}$.

Two papers on my web page:

We introduce the slow time $\tau = \epsilon t$.

Perturbations of linear Hamiltonian PDEs

$$
\frac{\partial u}{\partial \tau} + i \epsilon^{-1} (- \Delta u + V(x)u ) = \Delta u – \gamma_R |u|^{2p}u – i \gamma_I |u|^{2q} u + (random force).
$$
Both of the parameters $\gamma > 0$ and satisfy $\gamma_R^2 + \gamma_I^2 =1$. The parameters $p,q$ are natural numbers, possibly 0. WE weill look at the case $d=1$ on $x \in [0,\pi]$ with Dirichlet boundary conditions.

Some more information about the random force,
$$
(random force) = \frac{d}{d\tau} \sum_{j=1}^\infty b_j \beta_j (\tau) e_j (x)
$$
Here the $\beta_j$ are complex valued standard, independent random variables.

We will work in the Sobolev space $H^2$.

Theorem 1: If $u_0 \in H^1$ then $\exists ~! ~ u^\epsilon (\tau, x)$ such that
$$
E ( \|u\|1^2 + \int0^\tau \| u(s)\|_2^2 ds) < \infty.
$$

Let $u_0^\omega \in H^1$ be a random.

  • Let $\mathcal{P}(u_0^\omega) = \mu$ denote the measure in H^1
  • Calculate $u^\omega (\tau)$.

Definition: A measure $\mu$ is called a stationary measure if $\forall ~ \tau$ we have $\mathcal{P} (\mu_\tau ) = \mu.$

Bogolyubov-Krylov: A stationary measure almost always exists.

Theorem 2 (Hairer, Odasso, AS): If $b_j \neq 0 ~ \forall ~j$ then $\exists ~ !$ stationary measure $\mu^\epsilon.$ For any solution $u(\tau)$, we have
$$
\mbox{dist} (\mathcal{P}(u(\tau)), \mu^\epsilon) \rightarrow 0 ~\mbox{as}~ \tau \rightarrow 0.
$$

The measure $\mu_\epsilon$ depends upon the force but not on the data.

Fourier Tranform

For the operator $A = – \Delta + V(x)$ consider the eignefunctions $\phi_1, \phi_2, \dots$ with associated eigenvalues $\lambda_1, \lambda_2, \dots$. Assume that

  1. $\lambda_1 >0$
  2. $\lambda \cdot s \neq 0 ~ \forall s \in {\mathbb{Z}}^\infty, ~ 0 <|s| < \infty.$

For any $u \in H^1$, we can expand $u$ w.r.t. the basis and denote the associated coefficients by $v_1, v_2, \dots$. The Fourier transform is the map $u \longmapsto v$ and the inverse goes the other way.

We can pass from $v_j$ to polar coordinates $I_j, \phi_j$. He recasts the dynamics w.r.t the polar coordinate variables and started speaking about averaging lemmas.

Effective Equations

These objects are somehow analogs of the kinetic equations in the theory of weak turbulence….some notation….I want to understand this better….an average of the nonlinear potential energy term. This is a semilinear heat equation with a nonlocal heat equation. The term proportional to $\gamma_I$ does not influence the effective equation. This equation really takes complete control when $\epsilon$ is very small.

The advance obtained here uses randomness in the forcing. “I expect that the effective equation is relevant even without the randomness but I don’t know how to prove it.”


J. Colliander: Numerical Simulations of Radial Supercritical Defocusing Waves

My slides


F. Bouchet (ENS-Lyon): Invariant measures

Collaborators:

  • A. Venaille
  • E. Simonnet
  • H. Morita
  • M. Corvellec

Physical phenomena. I am interested in self-organization in turbulent flows. Examples: stripes and spots on Jupiter. Ocean currents. Height differences in ocean surface. Stable jets.

I will mainly speak about the 2d Navier-Stokes equation with random forcing. This is not such a good model for these phenomena. There are others that are quite similar that might be better to describe the phenomena listed above like the quasigeostrophic and shallow water layer models.

Equilibrium will be related to 2D Euler. For 2D, we have the vorticity-stream formulation. Steady solutions to the Euler equation satisfying $\omega = f(\psi)$ or, equivalently, ${\bf{u}} \cdot \nabla \omega = 0,$ play a crucial role in describing the dynamics. Degeneracy: what is the selection mechanism leading to $f$? The main advance is that $f$ can be predicted using equilibrium statistical mechanics ideas.

Outline:

  1. Invariant measures of the 2D Euler equation
    • Equilibrium stat mech
    • applications of equilibrium stat mec
    • invariant measures of the 2d euler equation
  2. Irreversible relaxation of the 2D Euler equations
    • irreversibility in fluid mechanics
    • …..slide switched….ack
  3. 2D stochastic Navier-Stokes equation: non-equilibrium phase transitions

Statisitical mechanics for 2d and geopphysical flows.

Statistical equilibrium. very old idea. famous contributions

  • Onsager 1949
  • Joyce-Montgomery 1970
  • Caglioti Marhioro plvirenti lions 1990
  • Robert-Sommeria 1991
  • Miller 1991
  • Eyink-Spohn 1994

Robert-Sommeria-Miller (RSM) theory:

The most probable vorticity field. We want to measure the number of microscopic fields $\omega$ which correspond to a probabiility $\rho$. The number of such configuarations is quantified by the Boltzmann-Gibbs Entropy. This is the mixing entropy. Microcanonical RSM variational problem. Critical points are startionary flows of the QG model.

Microcanonical measures for Hamiltonian systems:

  • Hamilton’s equations
  • Liouville Theorem
  • Define the microcanonical measures which are the natural invariant measures taking into account the constraints in the dynamics.

Detailed Liouvilles thEorem for 2D Euler:

Lee 1952, Kraichnan JFM 1975, Robert 2000

We want to take into account the casimirs and the constraints. He describes a limiting process based on galerkin approximations. Mean field behavior? Large deviations? Sanov theorem?

……lots of discussion…..ideas vs. proofs…..nontribvial…what’s going on? Audience is confusing me…speaker seems clear.

Young measures….entropy…

The claim is that the theory he and his collaborators hav developed explains the emergence and stabiltiy of coherent structures like the great spot on Jupiter. Similar statements about ocean structures.

Are microcanonical measures invariant measures for the 2D Euler dynamics? Is the setof invariant Young measures for the 2D Euler dynamics larger than the set of microcanonical measures?

Two conjectures:

  • Weak perturbations of the 2D Euler equations close to steady states converge to invariant Young measures.
  • The 2D Euler equations converge to invariant Young measures.

Wave breaking is an irreversible mechanism in fluids that does not require viscosity.


Sebastian Reich: Data Assimilation

Data Assimilation

Nature Physical Laws

Measurements Model

        Data

        Optimal prediction

He drew arrows between these frameworks of understanding and highlights the assembly of processing at the data assimilation level.

Sequential Data Assimilation in a nutshell.

Model + Observations $\longmapsto$ Prediction

Ingredients of Data Assimilation:

  1. Mathematical and numerical model. solutions and their undertainties caused by approximation errors as well as state and parameter undertainties.
  2. Data/observations with measurements as well as approximation (forward operators) errors –> Inverse problems
  3. Numerical approximations to the data assimilation problem within a statistical (Bayesian) framework, assessment of the induced predictions and their uncertainties.

Mathematical problem statement

Consider an evolution problem for which the initial state is treated as a random variable with some given probability density function. For simplicity assume finite-d phase space. The uncertainty in the initial conditions will generally lead to unpredictability over long time intervals. Weather prediction is a nice example.

To counterbalance this increase in uncertainty, we collect observations at discrete times subject to some random measurement errors. We wish to find a trajectory that makes optimal use of the available information in terms of initial data, observations and model dynamics. The task of data assimilation is to combine the model, the measurements and then we want to make the optimal prediction.

Theoretical solution

i) Model dynamics

Lift the dynamics to the level of the Liouville equation on the probability distribution function.

ii) Data assimilation

Assimilate data using Bayes’ theorem
$$\pi (x|y) \thicksim \pi(y|x) X \rho_{pr} (x).
$$

Here $\pi(x|y)$ is the know conditional PDF (likelihood) for observing $y$ given a state $x$. Given an actual measurement, we can correct and proceed.

Under Bayes’ theorem, we always reuce uncertainty.

Ensemble Prediction…

ack….slides are changing fast.

Particle filter. We give better weight to points that are closer to the observed data. If we repeat this a few times, there will be very few particles contributing to the final answer.

Assimilation as a continuous deformation of probability: McKean-Vlasov

We can think of Bayes theorem as an optimal transportation problem.

Crisan-Xiong 2010 did something similar in the context of continuous time filter problem.

Otto 2001 for an application in gradient flow dynamics.

We started with an ODE, spoon fed the measurement data to update the dynamics, and encounter a more complicated dynamical description of the system. We encounter a McKean-Vlasov system, a modified Liouville equation, which is closed by an elliptic PDE.

Numerical filter implementations will now rely on appropriate approximations to the lliptic PDE. We use the ensemble of solutions to define an appropriate statistical model and then solve via numerics or by quadrature.

Obvious choices for the numerical version of $\rho$ include a Gaussian PDF parameterized by the ensemble mean and covariance matri (ensemble Kalmna filter) or Gaussian mxture modes.


N. Faou: 2d Submarines

2D Euler equation on 2-torus….I was a bit tired and did not type notes during this talk.


 

,

These are notes from a meeting entitled Advanced Numerical Studies in Nonlinear PDEs in Edinburgh, Scotland.

Walter Craig (McMaster): Water Wave Interactions

I’m an analyst but I’m going to talk about numerics and experiments as well as analysis. We will discuss the problem of water waves and then I’ll talk about two specific settings in which the theory has led to good and quite elegant numerics and the numerics have started to answer some questions.

(joint work with P. Guyenne and C. Sulem)

Outline

  • Free surface water waves
  • Hamiltonian PDEs
  • Periodic Traveling wave patterns
  • Solitary wave Interactions
  • The KdV scaling limit

Free surface water waves

Euler’s equations of hydrodynamics, incompressible and irrotational flow. This is therefore given as a potential flow. The irrotational assumption is really an oceanographers assumption. Of course, there is vorticity but we follow the models of oceanographers.

The fluid domain is $-h < y < \eta (x,t)$. So, the domain is changing. Free surface boundary conditions hold on $y = \eta (x,t)$.

Zakharov’s Hamiltonian

  • The energy functional

$$ H = K+P $$
$$
K = \int_x \int_{-h}^{\eta(x)} \frac{1}{2} |\nabla \phi|^2 dy dx.
$$
$$
P = \int_x \frac{g}{2} \eta^2 dx.
$$

This could also include surface tension effects.

  • Zakharov’s choice of variables,
    $$
    z = ( \eta(x), \xi(x) = \phi(x, \eta(x))),
    $$
    for which we consider $\phi = \phi[\eta, \xi] (x,y)$.
  • Express the energy in terms of $\xi$ and $\eta$. This involves the Dirichlet-Neumann operator $G(\eta)$.

Dirichlet-Neumann operator

  • Laplace’s equation on the fluid domain: $\Delta \phi = 0$ subject to bottom Neumann boundary condition. Free surface boundary data $\phi (x, \eta(x)) = \xi(x)$, for which the D-N operator is given by
    $$
    \xi(x) \longmapsto \phi(x,y) \longmapsto N \cdot \nabla \phi (1+ |\nabla_x \eta|^2)^{1/2} := G(\eta) \xi(x).
    $$
  • In these coordinates, we can rewrite the boundary conditions in a new (and nicer) form. This reexpresses the water wave problem as a Hamiltonian system in Darboux coordinates.

Hamiltonian PDEs

  • KdV is a Hamiltonian PDE with a different symplectic structure.
  • Other Hamiltonian PDEs
    • shallow water equations
    • Boussinesq
    • KP
    • NLS
    • Dysthe equation

Many of these problems arise in scaling limits of the water wave problem.

Lemma (Properties of D-N operator):

  1. $G(\eta) \geq 0$ and $G(\eta) 1 = 0$.
  2. $G(\eta)^* = G(\eta)$ Hermitian Symmetric
  3. $G(\eta): H^1_\xi \rightarrow L^2_\xi$ is analytic in $\eta$ for $\eta \in C^1$. There is an operator valued power series expansion of $G(\eta)$ (using a theorem of Christ-Journé 1987).
  4. Some explicit calculations of the Taylor expansion (I couldn’t keep up….)
  5. Conservation Laws
    • Mass: $M = \int \eta dx$ (He shows the calculation using properties of $G$.)
    • Momentum: Similar calculation
    • Energy: Easy since the commutator of $H$ with itself vanishes.
  6. Taylor expansion of the Hamiltonian
  7. Linearized equations; comparison with the harmonic oscillator.

Periodic Traveling wave patterns

  • Can I find traveling wave solutions?
    $$ \eta(x,t) = \eta ( x-tc); \xi(x,t) = \xi(x – tc) $$
  • Spatially periodic, $\Gamma \subset {\mathbb{R}^{d-1}}$.
    $$
    \eta(x + \gamma, \cdot) = \eta(x, \cdot), \xi(x+\gamma, \cdot) = \xi(x, \cdot), ~ \forall \gamma \in \Gamma.
    $$

On such domains, we can use the Fourier tranform.

Rk: Notice this is a mathematician imposing a period rather than the physics making that selection. More can be said in this direction, but let’s proceed this way.

Rk: These (time independent) traveling wave patterns can be imagined to emerge in transient interactions in seas. The nonlinear actions create large amplitudes and this might be related to the phenomena of freak waves.

Equations for traveling waves.

Periodic traveling wave patterns are critical points of the Hamltonian on the variety $I = const$, with Lagrange multiplier $c \in {\mathbb{R}^{d-1}}.$

This leads to a bifurcation problem.

brief history (dimension $d=2$):

  • Levi-Civita 1925; existence of traveling waves
  • Struik 1926; traveling waves case
  • Zeidler 1971
  • Beale 1979
  • Jones-Toland 1985

brief history (dimension $d=3$):

  • Reeder Shinbrot 1981
  • Sun 1986
  • Craig-Nicholls 2000
  • Iooss-Plotnikov-Toland 2000 (small divisor problem)

He shows a picture from the wave tank at Penn State. He then shows some numerics which are trying to model those observations and they look beautiful.

Kuksin Question: Stability of these patterns?

Craig Answer: This is a very good question. I don’t know results like that. This is related to Benjamin-Feir. McLean showed instability for $d = 3$. Some further discussion….We need the Bloch theory of stability for these wave patterns. This appears to be difficult analytically so might need some numerical studies at first. There are instability zones….

Solitary wave Interactions

Solitary waves in 2-dimensions (Friedreichs-Hyers 1954, Amick-Fraenkel-toland 1980s)

  • Head-on collisions of solitons.

The numerics reveal some inelasticity in the collision. We’d like to understand those. If we make the amplitude of the solitons bigger, the dispersive ripples are more visible.

The KdV scaling limit

Titi’s Question: Can we reduce to the surface equations including rotation?

Craig’s Answer: Yes and No. You can make a rotation depending purely on y and impose that. Then it is reducible. But this is rather artificial. There is stuff that happens in the middle which is not a surface effect. Therefore, this problem requires a more complete analysis of the Euler equation and will not collapse to a system on the surface.


Sergey Nazarenko (Warwick): Assumptions, Techniques, Cahllenges in Wave Turbulence

This is not so much about new result. Instead, this is an attempt by a physicist trying to explain wave turbulence ideas being explored by physicists to mathematicians. My view is that there is a lot of interesting work to be done. Lots of open problems….

What is wave turublence?

He shows a picture of a relatively calm seashore from Nice. He emphasizes there is a wide range fo scales in these problems. WT is a statistical system of nonlinear waves.

Examples:

  • Water waves
  • Waves in rotating and stratified fluids (internal and inertial waves, Rossby waves)
  • Plasma waves
  • Waves in Bose-Einstein condensates
  • Kelvin waves on quantized vortex filnments
  • MHD turbulence in interstellar turbulence and solar wind
  • Nonlinear optics
  • Solids: phonons, spin waves. Kinetics of phonons in weakly anharmonic crystals is a first example of study in tis direction (1920s). I didn’t catch the name….

He shows a picutre of a wave take of Lukaschuk.

Waves in fusion plasmas. Shows a picture of a Tokamak. Drift wave turbulence causes anomalous heat and particle loss – major problem for fusion. The devices have grown larger and larger basically to carry out the confinement for a longer period of time.

MHD turbulence in astrophysics. He shows some data from the Ulysses/Swoops (los alamos) solar wind studies.

Bose Einstein Condensates Nazarenko-Onorato 2006:

  • Inverse cascade – condensation
  • Condensate strongly affects WT

Quantum Turbulence (see Lvov et. al. 2007) (Superfluid turbulence)

  • Kelvin waves on quantized vortex filaments
  • Interaction with hydro eddies (vortex bundles) is important
  • Kelvin Wave Turbulence

Optical Turbulence

  • Bortolozzo et. al 2008
  • This project studies nonlinear corrections (coming from the optical physics) which are included beyond the 1d NLS model.

Kuksin Question: Which corrections? Can you write them down?

Nazareknko: Something like a DNLS correction…not so clear.

Ingredients in the approach

He writes $NLS_3^\pm (T^d)$ and comments that this is a physically reasonable model but we are really interested in the study in infinite space with finite energy density.

He reexpresses the NLS equation in Fourier language.

Set of wave modes: amplitudes and phases.

N-mode joint probability density function. Some notation….probability…sectors in the wave modes setting.

Random Phase (RP) and Random Phase Amplitude (RPA) systems

RP:

All phases are independent random variables such that uniformly distributed on $S^1$.

RPA:

  1. All amplitudes and all phases are independent random variables.
  2. All phases are uniformly distributed on $S^1$.

Note: RPA does not mean Gaussian. Nevertheless, we have obtained successful closures without assuming the Gaussian statistics.

Frog Jumps!

  • expanding in small nonlinearity
  • Assuming RP at $t=0$.
  • Taking limit of a large box followed by the limit of small nonlinearity.

(The order of these steps is important.)

Evolution of joint PDF? We can derive the evolution equation under these assumptions. The derivation is rather systematic, in fact it is perhaps rigorous.

Mathematical Challenges:

  • WT is formally derived for $t=0$.
  • Does it work at the long time of nonlinear evolution?
  • Does RPA survive over this time?
  • Adding forcing and dissipation: will WT describe the steady state?

Hmmmm….This RPA condition at $t=0$ reminds me a bit of the assumption of product wave function in the QMB theory. The dynamics in the Hartree derivation might drive the multiparticle wave function away from the product case. Here we have a dynamic that might drive us away from the RPA condition.

Evolution of 1-mode PDF.

Kinetic equation (Hasselmann 1962).

Kolmogorov-Zakharov state.

  • Explained a steady state spectrum corresponding to energy cascade.
  • Exact solution of the asymptotic closure.

Numerics and Analysis of KE.

  • What is the role of KZ solutions with respect to the thermodynamic Rayleigh-Jeans state?
  • Similar issues for the classical Boltzmann equation.

Zakharov was awarded the 2003 Dirac Medal for “putting the theory of wave turbulence on a firm mathematical ground”! What is it that we want to do?


Gregor Tanner (Nottingham): A wave chaos approach towards describing the vibro-acoustic response of engineering structures

(Joint work with D. Chappel, Stefano Gianai, Hanya Ben Hamdin, Dmitrii Maksimov)

This talk is more directed toward engineering applications. inuTech is an industrial collaborator.

Overview:

  • Introduction – the need for numerical short wavelenght methods in vibroacoustics
  • From wave equations to the Liouville equation
  • Solving the Liouville equation – a boundary integral approach (Dynamical Energy Analysis – DEA)
  • Tackling the Midfrequency problem – hybrid methods
  • Numerical results

Aim: predicting wave intensity distributions for the vibro-acoustical response of mechanical structures. Think of a car. Companies like Bombardier and Airbus use these methods. It is a difficult problem. You want these structures to be quite and with no noise in the interior.

Where is the problem?

Techniques:

  • Low frequencies – wavelength around the size of the object
  • Finite Element method
  • Boundary elemtn method
  • plane wave methods

High Frequencies:

  • Ray tracing
  • Statistical energy analysis

Midfrequency problems:

  • Structures with large variations in the local wavelength. (Large variations in the stiffness of components, ie body frame and side panels.)
  • Hybrid methods. Try to connect exact numerical methods with the statistical methods.

Short wavelength approximations – from wave chaos to statistical methods

  • Wave chaos -short wavelength asymptotics
    • Keller
    • Gutzwiller
    • Berry
    • Bogolmolny
    • Smilansky
  • Nonlinear dynamics – thermodynamic formalism
    • Ruelle
    • Arnold
    • Sinai
    • Eckmann
    • Cvitanovic – chaosbook.org
  • Wave transport – statistical methods in vibro-acoustics
    • Lyon – SEA (1967 paper)
    • Langley – WIA
    • Heron
    • Weaver – diffusion equation
    • Le Bot – radiative transformation

Linear wave equation. WKB ansatz. Hamiltoninan equations for the amplitude and phase. Characteristics of JH; nonlinear ODE; Liouville equation (linear).

Linear wave –> WKB –> HJ equation –> Liouville Equation

Think of polygonal billiards, not necessarily convex.

We want to understand the influence of a source (transfmitting at frequency $\omega$) at one location on the wave amplitude at another point. He writes this as a green’s function $G(r, r_0, \omega)$.

Small wavelength limit, so low frequency waves.

Write things as sums over all paths.

Perron-Frobenius operator…

Tandem Satellite images of coastline of Madagascar

I started wondering about connections between these ideas and quantum ergodicity…

Typical Wave Function in a stadium Billiard

Bouncing Ball Modes

Postlude:

I had a nice conversation with Gregor after the break. I learned from him about microlasers. The idea is to build a circular region out of a lasing material. We energize the material somehow with hopes to excite the whispering gallery mode. The laser light propagates near the boundary but can be arranged to exit the medium by raising the curvature at a specific location. These appear to be rather hard to control to create a unidirectional beam. Since the losses take place all along the boundary, there is very little power in the output beam. Some web searching revealed an advance made by the Capasso group at Harvard.
Elliptic Notched Microlaser Cavity Drawing
Elliptic notched microlaser cavity SEM photograph.
Microlaser Cavity (Artistic Rendition)
Schematic Image
Artistic Rendering


David Dritschel (St. Andrews): CLAM, The Combined Lagrangian Advection Method

(Many many collaborators)

Courbet's "The Wave"

I’ll be speaking a bit about a numerical method. I’ll focus mostly on the results we’ve obtained to understand the large scale atmospheres, like Jupiter and perhaps also the ocean.

The numerical method (CLAM) emerges from a Lagrangian method from the 50s for studying fluid dynamics. Zabusky then built from these developments to develop new methods in plasmas. We’ve been extending these ideas to treat certain geophysical fluid flows.

The atmosphere and the oceans are extremely complex, turbulent flows. Accurate computer simulation is immensely difficult to achieve. However, much of this difficulty is inherent in the computational methods employed:

  • None take direct advantage of the natural inherent Lagrangian advection of dynamical, chemical and biological tracers. (Exploit Lagrangian Descriptions.)
  • None seek to separate slow vortical (eddying) and fast wave-like motions and use appropriate, optimal numerical methods for each. (Slow Rossby waves interacting with fast waves inertial-gravity waves.)

We can build the mathematical theory of the separation into the numerical methods and this will lead to better predictions.

Contour Advection (CASL) Dritschel & Ambaum 1997

geostropic and hydrostatic balances are basic features for describing atmospheric wave dynamics.

This talk reminds me somehow of Bourgain’s high/low method for proving low regularity GWP.

The idea is to use the advection of the vorticity to resolve some (especially relevant) sub-grid scales.


Dugald Duncan: IDE equation

Overview:

  • full IDE equation and how it looks like, where it arises
  • Linear part of the IDEbehaviour and approximation
  • the full problem – behanviour and approximation
  • examples

$$
u_t = \sigma \int_\Omega J(x-y) [u(y,t) - u(x,t)]dy + f(u) dx ~ \forall x \in \Omega, t>0.
$$
Typically, $f(u) = u – u^3$. This should be contrasted with the Allen-Cahn equation

$$
u_t = \sigma \Delta u + f(u) dx ~ \forall x \in \Omega, t>0.
$$

There are no spatial derivatives. Therefore, there are now boundary conditions. Instead, this is some kind of integral dynamical equation. It is similar to the Cahn-Allen equation.

This equation is also related to sandpiles, neurons, phase transitions.

Other variations recently: Rossi, Perez-Llanos, Andreu, Mazon, Toledo et. al. They study a nonlocal version of the $p$-laplacian.

Linear IDE:

  • Ignore the nonlinear reaction term for now and take $\sigma \geq 0$ and $\Omega \subset {\mathbb{R}}$:
    $$ u_t = Lu.$$
  • L is a linear operator – partly a convolution:
    $$
    Lu = \int_\Omega J(x-y) [u(y,t) - u(x,t)]dy = J * u
    $$
    …ack slide changed….

Discontinuities don’t move. The solution collapses to the average value. Ther eis acomparison principle.

Snapshots of linear behavior.

He does a Fourier analysis of the behavior of plane waves. Instead of having an $\omega^2$, we have $$\hat{J} (\omega) – \hat{J} (0) \thicksim \frac{\omega^2}{2} \frac{d^2}{d \omega^2} {\widehat{J}} (\omega).$$

Peter Bates and Paul Fife did some of the earliest analysis on this equation.

,