IAS School of Mathematics

 

Workshop web page

Friday: 2012-03-16

  • 9:00 – 10:00 James Colliander, University of Toronto, “Big frequency cascades in the cubic nonlinear Schroedinger flow on the 2-torus” abstract
  • 10:15 – 11:15 Marcel Guardia, IAS, “Growth of Sobolev norms for the cubic defocusing nonlinear Schroedinger equation in polynomial time” abstract
  • 11:30 – 12:30 Yann Brenier, University of Nice, “Approximate geodesics on groups of volume preserving diffeomorphisms and adhesion dynamics” abstract

James Colliander: Big frequency cascades in the cubic nonlinear Schrödinger flow on the 2-torus

(chalk talk)

(joint work with M. Keel, G. Staffilani, H. Takaoka, T. Tao)

I prepared slides but decided to give a chalk talk. The slides are located here: http://uoft.me/nls-cascade. The paper discussed in this talk is located here.

(See also: The thesis of Zaher Hani has advanced along these lines and is surveyed on his slides from the Ilde de Berder Workshop.)

The construction of the frequency civilization is partly conveyed by the following cartoon. Notice that the underachieving child frequency in the cartoon is always sent to the zero frequency. This violates the injectivity requirements used in our construction of the set $\Lambda$.

cartoon_construction

The next cartoon is meant to convey a traveling wave through the generations in the civilization. This wave is constructed by concatenating heteroclinic orbits in the toy model evolution.

wave_generations

The idea that the orbits could be concatenated reminded my coauthors of this famous commercial:

http://www.youtube.com/watch?v=KXA8g90g7so

 

 

Marcel Guardia: Growth of Sobolev norms for the cubic defocusing nonlinear Schrödinger equation in polynomial time

(joint work with Vadim Kaloshin; we have a preprint; slides from the talk; 32 pages)

This talk is strongly related with the previous talk.

$NLS_3^+ (T^2)$. Energy and Mass are conserved. The problem is globally well-psed in time Bourgain 1993.

Transfer of Energy

  • Fourier series of $u$.
  • Can we have a transfer of energy to higher and higher modes ass $ t \rightarrow + \infty$?
  • This is quantified with the growth of Sobolev norms.

We need to move mass toward high frequencies in a careful way to satisfy the mass and energy constraints.

Theoreom (Bourgain 1993): As $t \rightarrow + \infty$, the $H^s$ norm is upper bounded by $\leq t^{2(s-1)+} \| u(0) \|_{{H^s}}.$

This result has been improved or applied to other Hamiltonian PDEs by various authors.

Question (Bourgain 2000): Are there solutions $u$ such that for $ s>1$ such that
$$\| u(t)\|s \rightarrow \infty $$
as $ t \rightarrow + \infty? $ Moreover, he conjectured that the growth should be subpolynomial in time: $ \| u(t)\|
{H^s} \ll t^\epsilon$.

The second part was partly conjectured because of insights related to Nekoroshev type theorems for NLS.

Kuksin studied the growth of Sobolev norms for NLS for large initial condition. For such data, a change of coordinates recasts the dynamics into

$$
- i \dot{w} = – \delta \Delta w + |w|^2 w, ~ \delta \ll 1.
$$

Theorem (CKSTT 2010): $\exists $ big frequency cascades in the $NLS_3^+ (T^2)$ flow.

The solutions have small intial mass and energy. They remain small as time involves whereas the s-Sobolev norm grows considerably.

Theorem: (long statement, I’m reading instead of typing.)

The mass is small but the $H^s$ norm is initially large. They can then grow it up to any big threshold over a polynomially related time interval.

Remark: One might view this equation as a perturbation (when the data is small) of the (integrable) linear Schr”odinger. It is well know that the Nekoroshov type results for PDEs often loses the exponential estimates and becomes polynomial. Our result is consistent with this.

Remark: Our result deals with a different regime than the Bourgain subpolynomial conjecture. Our result is rather fast, but it could perhaps slow down over infinite time. Our construction involves a finite number of modes. If we try to build something on an infinite number of modes, the transfer mechanism might slow down.

Comments:

  • One can tensor this up to obtain similar results on $T^d, d \geq 2$.
  • We can obtain more detailed information about the distribution of the Sobolev norm of the solution $u$, among its Fourier modes when $t = T$. In particular, the high Sobolev norm is carried by two high achievers at the last stage. The high Sobolev norm is essentially localized in two modes.

Main Ideas in the Proof:

  • $I$-team introduced a finite-d toy model.
  • This toy model approximates well certain solutions of NLS
  • Our contribution is the analysis of the toy model. Using dynamical system tools, and a careful choice of the initial conditions, we find a faster motion.
  • The solutions of NLS can be proven to approximate well the solutions for the toy model for long time.

Reduction to the toy model.

  • $FNLS$
  • $RFNLS$
  • Construct $\Lambda$.
  • Toy Model ODE

For $N$ big enough, the set $\Lambda$ can be chosen to have the “wide diaspora property.” This is partly why we don’t have an infinite cascade. The construction only involves a finite number of modes. We want to quantify everything in terms of the number $N$ of generations. At the end we have $N \thicksim \log K$. We have to quantify everything.

Toy Model Theorem: There exists an orbit in the toy model which moves from the first generation to the last. Their statement includes quantifications! They compute the time of this transfer process.

To make things happen quickly, they want to make the transfers as fast as possible. This development uses a different orbit construction than the one performed by CKSTT.

Dynamics of the Toy Model:

  • ODE explicitly written out.
  • Each 4-d plane is invaraint.
  • Dynamics in each 4-d plane is given by a simple Hamiltonian involving nearest neighbor interactions.

Nice picture of invariant planes intersecting to form something like a polyhedra with a curve following along nearby invariant lines. “Of course, we are not in the plance but we are nearby it.” Of course to do this, we need to understand the dynamics in each of these planes. To obtain these orbits, we use hyperbolicity. But these planes have certain normal positive Lyapunov exponents so one has to be very careful. If we just move away from these planes, we lose control.

Dynamics in $L_j$:

  • To construct such orbits, we need to understand dynamics in each $L_j$.
  • Hamiltonian $h_j$ and $M_j(b_j, b_{j+1}) = |b_j|^2 + |b_{j+1}|^2$…..ack slide changed.
  • Contains two periodic orbits.
  • Periodic orbits in $L_j$ are hyperbolic.
  • Stable and unstable invariant manifolds of the periodic orbits coincide.
  • Call $\gamma_j$ the heteroclinic connection between the two dimensional manifold asymptotic to $T_j$ as $ t \rightarrow – \infty$ and asymptotic to $T_{j+1}$ as $ t \rightarrow \infty$.

Key Problem: The Shadowing

(nice picture)

  • We put sections transveral to the flow.
  • We study local maps: dynamics close to the periodic orbits $T_j$. Global maps: study dynamics close to the heteroclinic connections $\gamma_j$.

Local and Global Maps:

  • Shadowing for global map is basically applying (refined) Gronwall estimates.
  • Local map is more delicate: periodic orbits are of mixed type. Hyperbolic eigenvalues are resonant.
  • This resonance complicates the analysis of the local maps.

We need to choose very carefully which orbits we study.

The Model Problem:

  • After some reductions, we have a Hamiltonian of the form:

$$
H(p,q) = p_1 q_1 + p_2 q_2 + H_4 (p,q)
$$
where $H_4$ is a degree 4 homogenous polynomial, the variables “1” correspond to the variable $b_{j-1}$ ….slide changed.

Analysis of map from a section $\Sigma_+$ to $\Sigma_-$.

Dynamics of the linear saddle (Kill the $H_4$ and see what happens.).

Dynamics of the resonant saddle:

  • System is not well approximated by its linear part due to the resonance.
  • For typical initial conditions, we have a resonat affect creating logarithmic (in $\delta$ ) corrections to the transfer across hetereoclinic connections.
  • We need $~N$ transitions.
  • The number of logarithms becomes exponential in $N$.
  • We need to stay close to the periodic orbits to control the shadowing
  • This implies we need to start….slide change

We use the beautiful Shilnikov trick. The worst term that was developing with logarithms is now computed more accurately in terms of some function $g(p_0, q_0)$. This transfers the resonant saddle dynamics into essentially the dynamics of the linear saddle, provided that we carefully choose the domain of the map. This is kind of delicate and needs to be iterated through compositions.

Composing the local and the global maps:

  • We need to compose the local and global maps.
  • We define sets $U_j$ in the transversal secions and we show that the dynamics moves one into the other. (This is the “perfect shot”.)
  • To avoid deviations at each local map, we need to impose a restriction at every step.
  • “Product-like” step.

Product-like structure sets.

  • We start with a polydisk.
  • At each step, we impose a condition on the mode $b_{j-1}$.
  • Inductively, we rstrict the domain on previous domains involving conditions on previous mode involving the Shilnikov function $g$.
  • Since the restricitons involve a different mode at each step, the conditions are compatible.

Composing the local and global maps produces the toy model result. The detailed discussion partly explains the time quantification.

Approximating solutions of NLS:

  • Last step obtain a solution of NLS close to the solution of the toy model.
  • We modify the set $\Lambda$ from the $I$-tema so that the modes out of $\Lambda$ only gets influenced by few modes in $\Lambda$.
  • Each $b_j$ is excited only for a short period of time.
  • A mode out of $\Lambda$ only receives mass from $\Lambda$ during a short time.
  • This implies that the spreading of mass to modes out of $\Lambda$ is very slow.
  • We obtain an orbit for NLS that undergoes the growth of Sobolev nroms in polynomial time.

Yann Brenier: Approximate geodesics on groups of volume preserving diffeomorphisms and adhesion dynamics

(chalk talk; but here are the slides.)

It’s a good time for all of us to thank the organizers for this meeting. (Applause!)

Related to a question posed by Shnirelman from 1985.

System of interacting particles along the real line with sticky collisions. When the particles hit, they merge and continue with the same momentum. This is an inelastic, sticky collision. This is clearly

  1. dissipative
  2. nonreversible in time

Shnirelman’s Question (1985): Can we modify the action principle to handle these dissipative collisions?

Unfortunately, the paper is hard to find. You can think of the collision in a higher dimensional space and keep track of the energy in the extra variables.

G. Wolansky (2008 ?)

In this talk, I want to provide some ideas that come from ideal fluids. This seems strange because this problem is highly compressible, etc.

This talk is about a proposal for a modified action suggested by ideal fluid mechanics.

Arnold’s geometric interpretation (1966) of Euler equation for incompressible fluids (1755).

Let $D = [0,1]^3$. Let $VPM (D) = [ volume ~ preserving ~ maps ~ of ~ D]$. This may be viewed as a subset of $H = L^2 (D, R^3)$. Geodesics along VPM are (formally) the solutions of the Euler equations.

There is a discrete subset of $VPM (D)$ are the permutation maps $S =P_N (D)$. Partition the unit cube into a collection of $N$ subcubes $Q_i$ each with center of mass $A_i$. You would like to do some kind of discrete fluid mechanics by exchanging these cubes. There is a folklore of approximating geodesics with these kinds of maps. This is used in some works in computational geometry. How to define approximate geodesics along $P_N (D)$?

More generally, let $H$ be a Euclidean (or Hilbert) space. You have a closed bounded subset $S$. Introduce a potential
$$\Phi [x] = \frac{d^2}{2} (x,s) = \inf_{s \in S} \frac{|x-s|^2}{2} = \frac{|x|^2}{2} – R(x).
$$
Here $R$ is the Legendre transform:
$$
R(x) = \sup_{s \in S} (x|s) – \frac{1}{2} |s|^2.
$$
Convex, Lipschitz, usually not smooth.

Approximate minimizing geodesics are found by minimizing between two given points $A, B \in H$ by
$$
\int_0^1 (\frac{1}{2} |\frac{dx}{dt} (t)|^2 + \frac{1}{2\epsilon} \Phi [x(t)] ) dt
$$
satisfying $X(0) = A, X(1) = B$. If $S$ is a smooth manifold this converges to geodesics Rubin-Ungar 1957 (Yann’s birth year!).

These ideas were applied by David Ebin to fluids.

A simpler example than the one appearing in Shnirelman’s question…

Take $H = R^2$. Let $S$ be the St. George cross. He writes the coordinate axes in $R^2$ in red and forecasts that a joke will soon come up…

Whenever $\Phi$ is smooth about $X$, we have $\nabla \Phi (x) = x – \pi_S (x)$ (the closes point to $x$ inside $S$, not necessarily unique). The bad set $N$ where differentiability fails is both meager and has lebesgue measure zero in finite-d case. This has to do with the regularity of Lipschitz functions.

What is the bad set related to the St. George cross? Of course, it is the St. Andrew cross, the flag of Scotland! (He draws that in blue.) You can also reverse the picture so that the bad set becomes the St. George cross if you prefer to view it that way…..

If $x \in H \backslash N$, we have $\phi (x) = \frac{1}{2} |x – \pi_S (x)|^2 = \frac{1}{2} |\nabla \phi (x)|^2$.

Look at the action (for simplicity $\epsilon = 1$) for a “good curve” $ t \rightarrow x(t)$. Namely a curve for which $x(t) \in H \backslash N$ for a.e. time, the action reads
$$
\int_0^t (\frac{1}{2} |\frac{dx}{dt}|^2 + \frac{1}{2} |\nabla \Phi [x(t)]|^2 ) dt.
$$

So, obvious minimizers are those good curves that satisfy the first order equation
$$
(FO) ~ \frac{dx}{dt} = \nabla \Phi [x] = x – \nabla R [x].
$$

This is a so-called gradient flow of a Lipschitz convex function (up to the first term which can be absorbed). These objects have been studied.

The theory of maximal monotone operators does the job (cf H. Brezis book) in the sense that this is completely well-posed in $H$. We know from that theory that $ x \in C(R_+; H)$, Lipschitz in $t$, and
$$
\frac{dx}{dt} (t+0) = x(t) – {d^0 R[x]}
$$
which is sometimes called the minimal selection gradient or “mean” gradient (studied in the Italian school).

Example. Differentiate $|x|$. The subgradient fills in the vertical line. The minimal gradient has value zero at $x=0$. This is a nice theory but it gives us very bad curves.

If you start on this St. George cross example, he describes the dynamics and interprets this as a dissipative mechanism. This has little to do with the action principle but it does have dissipation. So, we might take some inspiration from this example….this is a proposal for a modified action.

Modified action:

$$
\int_0^t \frac{1}{2} |\frac{dx}{dt} – d^0 \Phi [x(t)]|^2 dt.
$$
Minimizers of the modified action are very likely to be bad curves.

Some rats were confined in a box by electric shocks and another which is very hot. But, if you dig a small channel between the other two boxes. It turns out the rats can survive longer by moving back and forth between the two boxes. I hope it is not a true story….

The dissipation is not incompatible with the arrow of time if you order the data.

Now, I’d like to go back to permutations and fluids. What kid of equation do I get?

Remember the box, broken up into the subcubes. Consider the set $S$ to be the permutations of all the centers. Let $H$ denote $R^{dN}$. In the $d=1$ case, you get a friendly approximate geodesic equation through the classical (nonmodified) action. We are then describing $N$ particles on the line.
$$ \epsilon \frac{d^2 x_i}{d t^2} = x_i – \frac{1}{2N} \sum_{j=1}^N ~{\mbox{sgn}} (x_i – x_j).
$$
This is like a gravitating parallel pancackes according to Newton gravity plus a repulsive background. This type of model was studied by people like Zeldovich. The repulsive effect is natural in that context. By approximating the incompressible Euler this way, it is nice that you get a model that is reasonable from the point of gravity.

In higher dimensions, the model is NOT consistent with Newtonian gravitation but is instead consistent with a Monge-Ampere correction to Newton’s gravitation. You get something like $\Delta \phi = \rho -1$ and then eventually find something like $ {\mbox{det}} (I + D^2 \phi) = \rho.$ I am not yet certain if this is geometrically reasonable. It is related to Born-Infeld correction to Maxwell’s equations.

So, what is the point? If you modify the action, you can recover interaction with sticky collision.

This is the so-called “Dust” in the Russian literature. These are elementary ideas that explain why matter has clumped in cosmology. Sluggish motions in the early universe moves like honey. Tiny fluctuations of qunatum origin and these create a Jeans instability which tends to concentrate matter. This is at a very large scale and concentrated on a llower dimensional fractal set.

IAS School of Mathematics

 

Workshop web page

Thursday: 2012-03-15

  • 9:00 – 10:00 Peter Topalov, Northeastern University, “Qualitative features of periodic solutions of KdV” abstract
  • 10:15 – 11:15 Jiansheng Geng, Nanjing University, “Invariant tori for the nonlinear lattice one-dimensional Schroedinger equations with real analytic potential” abstract
  • 11:30 – 12:30 Massimiliano Berti, UNINA, “Quasi periodic solutions of Hamiltonian PDEs” abstract
  • 2:30 – 3:30 Ralph Saxton, University of New Orleans, “The generalized inviscid Proudman Johnson equation” abstract
  • 4:30 – 5:30 Dongho Chae, Sungkyunkwan University, “On the blow-up problem for the Euler equations and the Liousville type results in the fluid equations” abstract

Peter Topalov: Qualitative features of periodic solutions of KdV

Peter Topalov

I need to do some detailed setup to expose the ideas I want to describe. We will discuss the KdV equation.

$$ q_t – 6 q q_x + q_{xxx} = 0.$$

Let’s impose periodic boundary conditions. We impose the initial condition $q|{t=0} = q0 \in H^N (T) $. This parameter $N$ will change at different times in the context of the talk, depending upon the theorem we are considering.

$$
H_{KdV} (q) = \int_0^1 (q^3 + \frac{(q_x)^2}{2} ) dx.
$$

What is the symplectic (more precisely the Poisson) structure? The phase space where the evolution will happen in $H^N$. For two functions $F,G: H^N \rightarrow R,$ we have the Gardner bracket
$$
{ F, G } = \int_0^1 \partial_q F \partial_x (\partial_q G) dx.
$$
(Ack….I am having trouble making curly brackets show up in the Gardner bracket even when I try to escape using a slash.)

Linearizing around $q=0$, we find $q_t =q_{xxx}$ which we can solve explicitly to find the evolution for the Fourier coefficient:
$$
\dot{\hat{q_k}} = – (2 k \pi i)^3 \hat{q_k} = (2 k \pi)^3 i \hat{q_k}.
$$
We can solve this directly to find
$$
\hat{q_k}(t) = \hat{q_k} e^{i (2k\pi)^3 t}.
$$
He draws a collection of complex Fourier planes and draws circles representing the motions of the Fourier coefficients.

Let’s see what the Poisson structure looks like when the dynamics are viewed in terms of the Fourier coefficients.

We compute the Gardner bracket of two Fourier coefficients:
$$
{ \hat{q_k}, \hat{q_l} } = \int_0^1 e^{-2 k \pi i x} (e^{-2 k \pi i x})’ dx = – 2 l \pi i \delta_{k, -l}.
$$
(missing curly brackets on left side.)

We fix attention to zero mean initial data. We will look at $H^N_0$ where the subscript reminds us that we are looking at the zero mean setting.

We define $z_k = \frac{\hat{q_k}}{\sqrt{|k| \pi}}$ and then observe that $z_k = x_k + i y_k$ gives us Darboux coordinates $x_k, y_k$.

We have a mapping $\Phi_L : H^N_0 \rightarrow h^{N+\frac{1}{2}}$. Let’s see why this $\frac{1}{2}$. We take an element of phase space $q$ and apply $\Phi_L$ and this takes us to the associated Darboux coordinates $z_k = \frac{\hat{q_k}}{\sqrt{|k| \pi}}$ and the division by $|k|$ explains the $\frac{1}{2}.$

Remarks about this map $\Phi_L$:

  1. diffeomorphism
  2. canonical
  3. linearizes the flow

Return to KdV.

Theorem 1: $\exists ~ \Phi: H^N_0 \rightarrow h^{N + \frac{1}{2}}$ such that

  1. $\Phi$ is a diffeomorphism;
  2. $\Phi$ is canonical;
  3. $z_k (t) = z_k e^{i \omega_k (q) t}.$
  4. (New) $\Phi = \Phi_L + A; ~\Phi^{-1} = \Phi_L^{-1} + B$ where $A$ is 1-smoothing. What this means is that $A, B$ are bounded maps such that
    $$A: H^N_0 \rightarrow h^{N + \frac{3}{2}};$$
    $$ B: h^{N + \frac{1}{2}} \rightarrow H^{N+1}.$$

In 3. the phases depend only upon the initial data but for some reason I don’t want to write $q_0$ right now.

1., 2., 3. were proven by Kappeler-Poschel-Makarov for $N \geq 0$. For the interval $-1 \leq N \leq 0$, 1.,2.,3. was established by Kappeler-Topalov.

Item 4. is new and recently proven by Kappeler-Schad-Topalov (I didn’t catch the name…). This advance may be viewed as a globalization of a local statement obtained by Kuksin-Perelman.

Consider the KdV evolution moving through phase space. We can also consider the linearized evolution. We are interested in the difference. Denote by $S_t (q)$ the KdV evolution. We can do something a little bit different:
$$
S_t (q) – \sum_{k \neq 0} (\hat{q_k} e^{i \omega_k (q) t}) e^{2k \pi i x} = R_t (q).
$$

Theorem 2:

  1. $R_t: H^N_0 \rightarrow H^{N+1}_0$ is continuous (even analytic on the Casimir $[q] = 0$).
  2. $\forall ~ q \in H^N_0$, we can consider the orbit $[R_t(q): t \in R] \subset H^{N+1}_0$ is relatively compact.
  3. $\forall ~ M > 0, [R_t (q): t \in R, \| q \|{H^N} \leq M] \subset H^{N+1}0$ is bounded.

In particular, from 2., the norms are relatively bounded.

I want to say something about the proof. The overview involves an expansion of the flow maps using the structure in Theorem 1, item 4. The core of the analysis is in the spectral theory of the Shcrodinger operator.

Jiansheng Geng: Invariant tori for the nonlinear lattice one-dimensional Schroedinger equations with real analytic potential

(joint work with J. You an Z. Zhao)

We study a nonlinear Schrodinger equation on the lattice and show there exist quasiperiodic solutions.
$$
i \dot{q_n} + \delta( q_{n+1} – q_n) + V_n q_n + |q_n|^2 q_n = 0, n \in Z.$$

Here $\delta $ is small. $V_n (x) = V(n \tilde{\alpha} + x)$ with $V$ a nonconstant real analytic function on R/Z and $\alpha$ satisfying a Diophantine equation.

Eliasson 1997, Acta

Slides moving fast…

Theorem: For small enough $\delta$, this equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions for a.e. $x \in R/Z$.

Also works in the nonlinear case.

Choffrut: What is Whitney smooth? A: Some discussion… Kaloshin: The function is defined on a Cantor set and you need to define what it means to be smooth. You can’t differentiate so you have to do something to understand smoothness….this is the idea of Whitney smooth.

Related works (Linear case):

  • Belissard-Lima-Scoppola 1983 CMP
  • Fröhlich-Spencer-Wittwer 1990 CMP
  • Chulaevsky-Dinaburg 1993 CMP
  • Eliasson 1997 Acta

Related works (Nonlinear case):

Töplitz-Lipschitz property

  • Eliasson-Kuksin 2010
  • Geng-Xu-You 2011

Slides are quite dense, too technical for me to convey here. Abstract KAM theorem.

Massimiliano Berti: Quasi periodic solutions of Hamiltonian PDEs

Nonlinear Wave Equation

$$ (NLW):~ u_{tt} – \Delta u + V(x) u = \epsilon f( \omega t, x, u).$$

$\omega$ diophantine.

Question: Do $\exists$ quasiperiodic solutions of NLW ro $\epsilon \neq 0$?

Linear wave equation: ($\epsilon = 0$.)

Solutions are built by superposition.

  • Eigenfunctions are orthonormal in $L^2$: “Normal Modes”
  • Eignevalues $\lambda_j \rightarrow + \infty$: the $\sqrt{\lambda_j}$ are the “Normal frequencies”.

All these linear soutions are periodic. Their superpositions are quasiperiodic. Do these persist when we turn on the nonlinearity.

We look for quasiperiodoc solutions. This leads to an equation for qp solutions:

$$
(\omega \cdot \partial_\phi)^2 u – \Delta u _ V(x)u = f.
$$

We can approach this existence question as a bifurcation problem.

We make a NON-RESONANT assumption:

$$ | (\omega \cdot l)^2 – \lambda_j | \geq \frac{\gamma}{1 + |l|^\gamma}, ~ \forall (l,j). $$
The inverse operator is unbounded so the classical implicit function theorem fails. We need a replacement, some kind of Quadratic scheme.

We use a Nash-Moser IFT: Newton method + “smoothing”

The advantage is the rapid convergence. The disadvantage is that we have to invert in a whole neighborhood of the expected solution.

Literature

$d=1$

  • Kuksin 89, Wayne 90; 2nd order Melnikov non-resonance conditions OK. Dirichlet conditions to ensure simplicity of eignevalues.
  • Craig-Wayne 93 periodic solutions
  • Bourgain 94 quasiperiodic solutions

Lyapunov-Schmidt, f analytic, Netwon Method. 1st order Melnikov conditions.

$d \geq 2$

  • Eigenvalues of $\Delta + V(x)$ appear in clusters of increasing size.
  • If $d \geq 2$, the eigenfunctions of $-Delta + V(x)$ are NOT localized wrt exponentials! (**Feldman-Knönner-Trubowitz**)

Often, these issues motivate the study of “pseudo-PDEs.”

Newton Method

  • Bourgain 98 Annals 05 Annals
  • Wang 10, 11

KAM theory

  • …Processi, Berti…Craig-Wayne…ack slide changged.

Nash-Moser

Eliasson 89

Berti-Bolle 2011 (to appear in JEMS)

Existence: (Summary of statements; slides are more precise)
Under some conditions on $f$, there exists a Cantor like set $C_\epsilon$ of asymptotically full Lebesgue measure. “This is a classical KAM-like statement.”

Regularity:

The Cantor-like set is not technical, e.g. CKSTT 2010 Inventiones.

Pre-assigned direction of tangential frequencies

  • Geng-Ren 2010
  • Berti-Biasco CMP 2011
  • Bambusi-Berti-Magistrelli, JDE 2011

Weaker non-resonance condition

simpler technique

Many of these results should carry over to spheres, Zoll manifolds, Lie groups, homogenous spaces: symmertries and properties of eigenfunctions and eigenvalues are key properties! Related to Birkhoff normal form results by Bambusi, Delort, Grebert, Szeftel for spheres and Zoll manifolds.

For periodic solutions, see Berti-Procesi Duke 2011

Idea of Proof

Small divisors.

Töplitz matrices.

Difficulties:

  • T has only a polynomial decay off the diagonal.

Smoothing operators; finite-d projectors. TAME estimates are needed. We need estimates on the inverse operator on high regularity Sobolev spaces. Counterexaple of Lojaciewitz-Zehnder! This example shows identifies a parameter boundary in the Newton iteration scheme.

Step 1. $L^2$-estimates: Lower bounds for the eigenvalues.

Step 2. “Separation Properties” of small divisors

Locations where the divisors are small become more and more rare. There emerge “irrational” conditions on the slope $\omega$. These conditions are not needed for the Schrödinger equation. The dispersive relationship is different and helps you here.

KAM

Nash-Moser via the 1st Melnikov conditions. This is in some sense the minimal assumption. This approach works well in case of multiple eigenvalues. However, it has the disadvantage that it requires studying the linearized equation with non-constant coefficients.

Other strategy: impose stronger nonresonant conditions of 2nd Melnikov type (as usual in KAM). This has the advantage that we have a linearized equation with constant coefficients. There exists a torus and a reducible normal form.

Question: Do quasiperiodic solutions persist for nonlinearities which involve derivatives? Important physical applications.

  • Kuksin 1998
  • Kappeler-Pöschel 2003
  • Liu-Yuan 2010 for Hamiltonian DNLS (Benjamin-Ono)

Theorem (Berti-Biasco-Procesi 2011): DNLW has a Cantor-like family of quasiperiodic solutions. These qp solutions have zero Lyapunov exponents and the linearized equations can be reduced to constant coefficients.

Ideas of proof. View this as an infinite dimensional Hamiltonian system. Use conservation of momentum (Geng-You).

Birkhoff Normal form step, reduction to action-angle variables. Then apply an abstract infinite-d KAM theorem.

The Hamiltonian vector field is BOUNDED and “Quasi-Töplitz”.

  • Procesi-Xu 2011 (introduced Quasi-Töplitz)
  • Eliasson-Kuksin (similar notion Töplitz-Lipschitz)

Quasi-Töplitz functions

see slides….there is an algebraic closure property of this class under the normal form manipulations.

DNLW

Not Hamiltonian but “reversible” PDE. This is a relaxed setting but which rules out certain nonlinearities like $y_t^3$.

Real coefficients condition which excludes $y_x^3$.

Moser, Arnold, Sevriuk. Algebra of classical reversible KAM theory works out on this PDE as well. The asymptotic expansion of the normal frequencies controlled similarly as in the Hamiltonian case, in analogy with the quasi-Töplitz framework.

Ralph Saxton: The generalized inviscid Proudman Johnson equation

Ralph Saxton

(joint work with Aleajandro Sarria)

This is the Proudman-Johnson (PJ) equation:

$$ (\partial_t + u \partial_x) \partial_x u = \lambda u_x^2 – (\lambda+1) \int_0^1 u_x^2 .$$

This equation comes from the n-dimensional Euler equations. The solutions we consider coming from Euler are unbounded as we go toward spatial infinity so these are infinite energy. He describes some further modeling assumptions culminating into a collapse of Euler into the Proudman-Johnson equation.

History:

  • Childress, Lerley, Spiegel, Young 1989
  • Saxton-Tiglay 2008, Okamoto 2009
  • Okamoto-Zhu 2000
  • wunsch 2009
  • A. Constantin 2000

Diverse phenomena as $\lambda$ varies.

Dongho Chae: On the blow-up problem for the Euler equations and the Liouville type results in the fluid equations

Dongho Chae

Contents

  1. On the blowup problem for Euler
  2. Liouville type equations for fluids

Euler Blowup Problem

Euler 1757

Euler equation on $R^N$.

Kato, Temam, Bouguignon-Brezis. Local existence in $H^m (R^3)$ with $m>5/2$. Do singularities form?

  • Beale-Kato-Majda 84 Critereon: If there is blowup at time $ {T^*}$ then$$
    \int_0^{T^*} \| \omega (s) \|_{L^\infty} ds = \infty.
    $$
  • Constantin-Fefferman-Majda 1996 critereon.
  • Refinements using Triebel-Lizorkhin type spaces. Interpolations.

On the self-similar blowup scenarios:

  • Self-similar blowup is a popular scenario in search for finte time singularity in nonlinear PDE.
  • E has a scaling property:
    $$ v^{\lambda, \alpha} = \lambda^\alpha v(\lambda x, \lambda^{\alpha + 1}t), ~ p = \lambda^{2\alpha} (same).$$

We consider the possibility of self-similar blowups for E.

Energy conservation suggests choosing $\alpha = \frac{N}{2}

Substitute a self-similar ansatz into E to obtain a system called SSE, the self-similar Euler equation. In the Navier-Stokes case, this system is called the Leray system. Leray asked if there exist self-similar blowup solutons for the Navier-Stokes equations in 1930.

Negative answers to Leray’s questions.

  • $V \in L^3 (R^3)$. Necas-Ruzicka-Sverak 1996
  • $V \in L^p (R^3), p>3$. Tsai 1998.
  • Theproofs rely upon maximum principle based arguments, which are not available in the context of the Euler equation.

Theorem (Chae 2007):

Let $V$ be a solution of SSE satisfy

  1. $V \in [C^1 (R^3)]^3$ vanishing near infinity.
  2. There exists $p_1 >0$ such that $\Omega – \nabla \times V \in \bigcap_{0<p<p_1} L^p (R^3).$ Then $V=0.$

The proof of this theorem used the “back to label map” due to Constantin. Recently, I found a much simpler elementary proof.

Chae-Shvydkoy 2012

This is the Euler version of the Navier-Stokes $L^3$ result of Necas-Ruzicka-Sverak.

Nonexistence of asymptotically self-similar blowup Giga-Kohn 1985. See Chae 2007.

Liouville Type Results for Navier-Stokes

Compare with Galdi. Slides are very detailed, provides a survey of the field.

IAS School of Mathematics

Simonyi Hall

Workshop web page

Happy Einstein Birthday!

Albert Einstein

Wednesday: 2012-03-13

  • 9:00 – 10:00 Wilfrid Gangbo, Georgia Institute of Technology, “Lifting absolutely continuous curves from P(Td) to P2(Rd)” abstract
  • 10:15 – 11:15 Jonatan Lenells, Baylor University, “Geometry of diffeomorphism groupos, complete integrability and optimal transport” abstract
  • 11:30 – 12:30 David Ebin, SUNY, “Groups of diffeomorphisms and geodesics on them” abstract
  • 2:30 – 3:30 Susan Friedlander, University of Southern California, “Well / Ill-posedness results for the magneto-geostrophic equations: the importance of being even”. abstract

Wilfrid Gangbo: Lifting absolutely continuous curves from $P(T^d)$ to $P_2(R^d)$

Wilfrid Gangbo

(chalk talk; joint work with A. Tudorascu)

This work extends earlier work on the space of probability measures on the torus $P(T)$ to analogous results on $P(T^d)$. The earlier work used the embedding $P(T) \rightarrow L^2 (0,1)$ but we don’t have this embedding in the higher dimensional case.

Let $P_2 (R^d)$ be the set of Borel measures on $R^d$ with finite second moment $\int |x|^2 \mu (dx) < \infty$. We say $\mu_0 \thicksim \mu_1$ if and only if $\int F d\mu_0 = \int F d \mu_1 ~ \forall F \in C(T^d), ~ \forall F \in C(R^d), ~ \forall F(x+z) = F(x), z \in Z^d$.

I define $P(T) = P(T^d)/\thicksim$.

Let $\gamma$ be a measure on $R^d \times R^d$ which satisfy $\pi_1$ # $ \gamma = \mu_0$ and $\pi_2$ # $ \gamma = \mu_1$.
More generally, we write
$$W^2_2 (\mu_0, \mu_1) = \inf_\gamma \int_{R^d \times R^d} |x-y|^2 \gamma (dx, dy).$$

Problem: Data: $v:(0,T) \times T^d \rightarrow R^d$ and $t \rightarrow \sigma_t \in P(T^d)$. Assume that $\partial_t \sigma_t + \nabla \cdot (\sigma v) = 0$ (in the sense of distributions). Can we find $t \rightarrow \hat{\sigma_t} \in P_2 (R^d)$ and $\hat{v}: (0,T) \times T^d \rightarrow R^d$ such that $\partial_t \hat{\sigma} + \nabla \cdot (\hat{\sigma} \hat{v}) = 0$ in the sense of distributions. Here $\hat{\sigma_t} \thicksim \sigma_t.$

Such a lift becomes important if I want to associate the rotation number. I want to write
$$
\frac{d}{dt} \int_{T^d} x d \sigma_t = \int_{T^d} v_t d \sigma_t$.
$$

Weak KAM: (A small fraction of what is known) $M= T^d$. Let $h: T^* M \rightarrow R$. Let $w(z_0, z_1) = z_0 (J z_1)$ where $J$ is the usual matrix satisfying $J^2 = – Id$. Let $X_h$ denote the associated Hamiltonian vector field:
$$
\dot{\phi} = X_h (\phi), \phi_0 = (x_0, p_0).
$$
The associated flow is denoted $\phi_t = (x_t, p_t).$

Existence of weak Lagrangian Tori: $\overline{h}: R^d \rightarrow R$ is the effective Hamiltonian, $ c \in R^d$.

  1. $\exists ~ u \in C(t^d)$ with $h(x, c+ \nabla u) = \overline{h} (c)$ (viscosity)
  2. $u_c (x) = c \cdot x + u(t).$ ($\partial u_c$ is invariant under $\phi$.)
  3. $\forall ~ x_0 \in T^d ~ \exists v_0 $ such that if $(x_t, p_t) = \phi_t$ then $\forall ~ T$
    $$
    u(x_T) – u(x_0) = \int_0^T [l(x,\dot{x}) + c \cdot \dot{x} + \overline{h} (c)] dt$$
    $$
    \lim_{t \rightarrow \infty} \frac{\hat{x_t}}{t} = – \nabla \overline{h} (c).
    $$

General Fact: $M \rightarrow $ compact.

Specific to finite $d$:

Given $x \in W^{1,2} (0,T, T^d)$ and take two lifts $\hat{x}, \hat{y} \in W^{1,2, R^d}$. We then find that
$$
\hat{x_t} – \hat{y_t} = n \in Z^d
$$
because we have
$$
\lim_{t \rightarrow \infty} \frac{\hat{x_t}}{t} = \lim_{t \rightarrow \infty} \frac{\hat{y_t}}{t}
$$

Obstacle in infinite dimensions:

Let $M_0 = P(T^d)$ and in the sense of distributions we have
$$\partial_t \sigma + \nabla \cdot (\sigma v) = 0.$$

If $\nabla \cdot (\sigma w) = 0$ then $v+w$ is another velocity.

(wash board…)

Let $\mu \in P_2 (R^d)$ and define the tangent space $T_\mu P_2 (R^d)$ and also the space $T_\mu P(T^d)$. These are defined with $L^2$ closures.

Pseudo symplectic form: ….going faster and I’m not keeping up with the typing.

Theorem (Gangbo-Kun-Pacuni 2011):

  1. $\Omega$ is a closed skew symmetric nondegenerate 2-form.
  2. $\exists ~ X_H$ such that $ -dH = \Omega (X_H, \cdot)$.
  3. $\dot{f} = X_H (f) \iff \partial_t f + \nabla_x (vf) = \nabla_v (f (\nabla V + \nabla W * \rho)).$

This is a nonlinear Vlasov equation.

I want to state the analog of the weak KAM theorem in our context.

Theorem Let $\overline{H}$ be the effective Hamiltonian of $H$ restricted to $R^d$ and let $c \in R^d$.

  1. $\exists ~ U: P(T^d) \rightarrow R$ such that (in viscosity sense)
    $$ H(\mu, c + \nabla_w H) = \overline{H} (c). $$
  2. Given $\sigma_0 \in P(T^d) ~ \exists ~ v_0: T^d \rightarrow R^d$ such that if $f_0 = \sigma_0 \delta_{{v0}}$ then
    $$
    f_t = \sigma_t \delta_{{vt}},
    $$
    $$
    U(\sigma_t) – U(\sigma_0) = \int_0^T [ L(\sigma_t, v_t) + \int_{R^d} v_t \cdot c d\sigma_t + \overline{H} (c)] dt.
    $$
  3. We also have
    $$
    | \frac{1}{T} \int_0^T dt \int_{R^d} v_t d\sigma_t +\nabla \overline{H} (c) | \leq \frac{const}{\sqrt{T}}.
    $$

Corollary: If $(\hat{\sigma}, \hat{v_t})$ is an appropriate lift then
$$\lim_{T \rightarrow \infty} \frac{1}{T} \int_0^T ( \int_{R^d} x d\sigma_t) dt = – \nabla \overline{H} (c).$$

Questions:

Mather: This has connections with fluids?

Answer: Kinetic theory. Consider the system $ \partial_t^2 x = \frac{1}{N} \sum_{j=1}^N W(x_i – x_j) – \nabla V(x_i)$. When we consider the $N \rightarrow \infty$ limit, we can move the weak KAM theory from this $N$ particle system to the infinite particle case by moving to the setting of measures. This framework lets us prove convergence of discrete models to the PDE case.

Jonatan Lenells: Geometry of diffeomorphism groups, complete integrability and optimal transport

Jonatan Lennells

(pdf slides; Happy $\Pi$ day!; Einstein’s birthday)

(joint work with B. Khesin, G. Misiolek, S. Preston)

Outline

  • A new equation
  • Geometry of $Diff(M)$
  • A sphere
  • Optimal Transport
  • Geometric Statistics

A new equation

$$ \rho_t + u \cdot \nabla \rho + \frac{1}{2} \rho^2 = \frac{- \int_M \rho^2 d \mu}{2 \mu(M)}.
$$

  • $M$ is a compact Riemannian manifold.
  • $\mu(M)$ is the volume of $M$.
  • This is an exciting equation because it is completely integrable for any $M$.
  • This is a geodesic equation on $Diff(M)/Diff_\mu (M)$.
  • Describes $\dot{H}^1$-optimal transport
  • reduces to the Hunter-Saxton equation for $M=S^1$. (derived in the context of liquid crystals in the early 90s.)

Euler-Arnold Equations. Summary of those ideas.

abc-metric. You can add some other terms to the original $L^2$ inner product involving $L^2$ inner products involvling codifferentials and the musical isomorphism. A lot of different equations arise as you take different values of the parameters. Writing down the associated abc Euler-Arnold equation generates a big equation which can be specialized into various equations. To obtain the $\dot{H}^1$ metric, we cancel away the terms associated with factors a and c. We simplify by setting $a=0, b= \frac{1}{4}, c = 0$.

The equation induced by these choices has some degeneracy issues. These can be resolved by quotienting out part of the phase space. The function $u$ is not uniquely determined but its coset is uniquely determined. (Similar issues arise in Hunter-Saxton.) The equation we are considering here is a geodesic equation on a (quotiented) Diffeomorphism group.

Jacobian determinant.

A sphere

Theorem (Khesin-Misiolek-Lennels-Preston): The map which takes the coset $[\eta]$ to its associated Jacoobian $\sqrt{Jac_\mu \eta}$ is an isometry onto a subset of the sphere.

This isometry lets them transport all the questions about the geodesic equation on this complicated Diff phase space into corresponding questions about geodesics on the sphere. Since we understand the sphere well, we can conjugate results there using the mapping to obtain explicit solution formulae for the geodesic equation. Magical integrability!

Preceding works.

Khesin-Misiolek 2003: Showed Hunter-Saxton may be viewed within the Euler-Arnold framework.

Lennels 2006: Recognized the image of the map as a portion of the sphere.

Optimal Transport

Optimal Transport. Wasserstein distance between two probability measures.

Moser 1965, Ebin-Marsden 1970, Otto 2001, also Benamou-Brenier.

The $\dot{H}^1$ optimal distance induces what they call the spherical Hellinger distance since it resembles the Hellinger distance used in probability theory

Geometric Statistics

Statistical model.

Fisher-Rao information metric.

Theorem (KMLP): The $\dot{H}^1$ metric coincides with the Fisher-Rao metric when restricted to any k-dimensional submanifold of the (quotiented) $Diff$.

This is another reason why we think this metric is important. It arises from many different points of view.

Summary

  • We found a new integrable PDE.
  • The PDE is a geodesic equation on a quotieted diff with $\dot{H}^1$ metric.
  • A sphere
  • One can understand what is going on using the optimal transportation point of view using the $\dot{H}^1$ metric.
  • This metric coincides with a basic metric arising in geometric statistics.

Open problems

  • Global weak solutions of the PDE? All solutions break in finite time because you hi the boundary of the diffeomorphism coset. However, there is no problem when you view the dynamics on the sphere. The motion along the great circle may be continued. This type of development has taken place already in the context of the Hunter-Saxton equation. The process appears to be more complicated in this more general context since the Jacobian can vanish.
  • Transfer results from geometric statistics into this diffeomorphism quotient. Then reinterpret these objects in the setting of PDE.
    Amari-Nagaoka 2000 alpha-connections, dual connections.
  • Develop an optimal transport theory based on the $\dot{H}^1$ theory.
  • Find a Lax pair.
  • Find a bi-Hamiltonian structure.
  • Analyze the associated two-component equation (c.f. Lennels-Zhao 2011). There is a 2-component Hunter-Saxton so that object suggests we might find a corresponding generalization. This has been observed by LZ.

David Ebin: Groups of diffeomorphisms and geodesics on them

David Ebin

(joint work with Stephen Preston)

Maps from a manifold to itself. Discussion of various topologies of such maps.

  • Volume preserving maps. Diffeomorphisms (Volumorphisms)
  • Even dimensional manifolds with a symplectic form. We can consider the maps which preserve the symplectic form. (Symplectomorphisms)
  • For odd dimensional manifolds, we can consider maps which preserve the contact form. (Contactomorphisms)

In all these cases, we can discuss the geodesics…..ack low battery.

Boothby-Wang fibration.

Susan Friedlander: Well / Ill-posedness results for the magneto-geostrophic equations: the importance of being even

Susan Friedlander

(joint work with Vlad Vicol, Walter Rusin, Francisco Gancedo, Weiran Sun)

Homage to Oscar Wilde…

Amain theme is that there is a difference in behavior of solutions in Active Scalar Equations when the associated Fourier multiplier is even versus odd.

Active Scalar Equations; Incompressible Fluids

$$\theta_t + u \cdot \nabla \theta =0. $$
$$ \nabla \cdot u = 0. $$

$$ u = O [\theta], ~ PDO $$

$R^d$ or $T^d$. Even or odd Fourier multiplier symbol. The results I’ll describe are not influenced by the presence of a physical boundary. The emphasis will be on examining the influence of the operator O on the properties of the PDE.

Consider $u_j = \partial_i T_{ij} \theta, ~ \nabla \cdot u = $. Here $T_{ij}$ is a $d \times d$ Calderon-Zygmund operators.

  • ODD Symbol: Locally well-posed in Sobolev spaces. Commutator in energy estimates. **Chae et. al, Friedlander-Vicol.
  • EVEN Symbol: Lipschitz ill-posed in Sobolev spaces. Friedlander-Vicol; Nonuniqueness for $L^\infty$-weak solutions. Techniques from convext integration. Shuydkoy.

Recent reviews of results for certain active scalar equations. “Regularity and blowup for active scalars” Kiselev 2010.

SQG Equation: $u = R^\perp \theta$, symbol $\frac{i (k_2, -k_1)}{|k|}$.

Constantin-Majda-Tabak 1994, Resnick 1995 (Chicago thesis; unpublished), Wu, Cordoba, Chae, Iyer, Ju, Fefferman

The SQG equation had been known in the geophysics community before its introduction to the mathematical community by Constantin et.al.

Local existence for smooth initial data BUT global existence of smooth solutions is OPEN (just as it is open for 3D Euler). Cordoba and Fefferman have ruled out the existence of certain solution scenarios.

“Modified” SQG Equation (Okhitani)

Insert a power of $(-\Delta)^{1/2} = \Lambda$ in the map $\theta \rightarrow u$ so that
$$
u = \nabla^\perp \Lambda^{\beta-2} \theta
$$
where $1 < \beta \leq 2. Chae, Constantin, Cordoba, Ganceda, Wu 2011. Local existence of smooth solutions in $H^s$, global existence of weak solutions.

Note: result holds more generally when the symbol is ODD and order $\leq 1$.

IPM equation: singular integral operator with EVEN symbol

Darcy’s law.

$$ u = R^\perp R_1 \theta. $$

Cordoba-Gancedo-Orive 2007

Regular initial data, local existence, weak solutions, SQG and IPM present different behaviours. Global existence of smooth solutions is OPEN.

Even symbol:
$$( \frac{k_1 k_2}{|k|^2}, \frac{-k_1^2}{|k|^2}).$$

There is very different behaviors among these equations for rough data. “Patch type initial data”

SIPM equations (even, unbounded)

$$ u = R^\perp R_1 \Lambda^\beta \theta$$

Friedlander-Gancedo-Sun-Vicol (2012)

  • Locally Lipschitz ill-posed in $H^s, ~ s>2$. Proved for $0 < \beta \leq 2$ in $T^d \times [0,\infty]$
  • Locally well-posed for some “patch-type” weak solutions. Proved for $0<\beta <1 $ in $R^2$.

(Discussion: the notions of “wellposedness” changes between the previous two bullet points.)

Symbol: $k_1 k^\perp |k|^{\beta -2}$

Magnetogeostrophic (MG) equations

Friedlander-Vicol 2011

Long symbol $M$, even, unbounded, 3D. $ u = M\theta$. Here $M$ is a vector operator that defines a 3-vector $u$.

Cauchy problem is ill-posed in Hadamard sense in Sobolev spaces. There is no Lipschitz solution map.

Ill-posedness: singular, even, symbol

Active scalar equation. Special direction with index $d$, often associated with gravity. A list of many conditions on the $d$th component $S_d$ of the Fourier multiplier operator….allowing them to build eigenfunctions to show Lipschitz failure.

Definition: Locally Lipschitz $(X,Y)$ well-posed.

$$\| \theta_1 (\cdot, t) – \theta_1 (\cdot, t)\|X \leq K \| \theta1 (\cdot, 0) -\theta_2 (\cdot, 0) \|_Y.
$$

The spaces $X,Y$ are often chosen to be $(H^r, H^S)$.

Theorem: Under the many assumptions on $S_d$, the active scalar equation is Lipschitz $(H^r, H^s)$-illposed for any $r > R, s \geq r+1$.

Linear problem

Linearize around $\theta_0 = \sin m x_d $. Write out a Fourier series. Crank out a recurrence relation.

Continued fractions, characteristic equation. These ideas where used by Michalkin and Sinai to show unstable eigenvalues for the shear flow for Navier-Stokes equations.

Ill-posedness of the nonlinear problem

Follows a proof by contradiction.

Effects of dissipation: MG

Dissipation: $ \nu (-Delta)^{1/2}$

Using De Giorgi techniques, Caffarelli-Vasseur proved critical SQG. What can we say about the MG equation?

  • Case $1/2 M \gamma < 1$: LWP in $H^s$, for $ s> \frac{5}{2} + (1 – 2\gamma)$. Well-prepared initial data.
  • Case $ 0 < \gamma < 1/2$: Diffusion is too weak to overcome the continued fraction construction.
  • Case $\gamma = 1/2$: Unique global solution when the initial data and source are small in a suitable sense, then there exists a unique golbal solution. However, if the data are large in this respect then we can run the ill-posedness construction. This reveals a very precise dichotomy.

 

IAS School of Mathematics

This image taken from IAS web site.

 

Workshop web page

Tuesday: 2012-03-13

  • 9:00 – 10:00 Laszlo Szekelyhidi, University of Leipzig, “The h-principle for the Euler equations” abstract
  • 10:15 – 11:15 Camilo de Lellis, University of Zurich, “The h-principle for the Euler equations Part 2” abstract
  • 11:30 – 12:30 Vladimir Sverak, University of Minnesota, “On the long-time dynamics of some infinite-dimensional Hamiltonian systems” abstract
  • 2:30 – 3:30 Antoine Choffrut, University of Leipzig, “On the local structure of the set of stationary flows to the 2D incompressible Euler equations” abstract
  • 4:30 – 5:30 Thomas Kappeler, University of Zurich, “Symplectic techniques for integrable PDEs” abstract

Laszlo Szekelyhidi: The h-principle for the Euler equations

Lazlo Szekelyhidi

(chalk talk; joint work w. Camillo De Lellis)

“What I will speak about has nothing to do with symplectic and nothing to do with dynamics.” Hofer: very good.

Euler

$$\partial_t v + \nabla \cdot (v \otimes v) + \nabla p = 0; \nabla \cdot v = 0.$$

The spatial dimension $n=2,3$, certainly $>1$. We will speak about weak solutions $v \in L^2_{loc} (T^n \times [0,T])$ and we throw all derivatives onto test functions.

Why look at weak solutions? The equations tell you conservation of mass and momentum. The derivation is done from a continuum analysis so this formulation is natural. For $n=3$, another reason is the relationship with turbulence (K41, O49). The story starts with anomalous dissiipation. The observation is that if you consider $NS_\nu$ with small $\nu$ then formally, the dissipation rate
$$
\nu \int |\nabla v|^2 dx > \epsilon.
$$
This is observed in experiments as $\nu \rightarrow 0$. If you plot $\log k$ vs. $\log E(k)$ there are three different regimes: a low frequency regime related to the geometry of the domain, an inertial range with slope $-5/3$ and then a rapid dissipation at high frequencies. Bob Kohn once said that a log-log plot always looks linear. ….discussion with Peter and Camillo….”Bob Kohn is not here so let’s leave him alone.” K41 looked at ensemble averages and he derived the $-5/3$ scaling law. O49 was thinking of a single solution. He said that if we beleive in this kind of log-log picture then in the intermediate regime we are far from dissipation so the nonlinear term is responsible for the $-5/3$ decay. Is it possible to see a single solution that displays this type of decay. If you translate this point of view into a regularity statement and look for $ v \in L^\infty_t C^\alpha_x$ then when $\alpha > 1/3$ we have energy conservation and for $\alpha < 1/3$ then you have anomalous dissipation possible. Klainerman: Is this $1/3$ easy to see? Discussion: Yes, it is just scaling, look at Fourier coefficients….

Eyink, Constantin-E-Titi solved the $\alpha > 1/3$ part of Onsager’s conjecture. There is basically nothing known in the $\alpha < 1/3$.

Spencer: Uniqueness in that range? Answer: No you need Lipschitz to see uniqueness so there remains a big gap.

Studying weak solutions puts us ina different framework than the study of smooth solutions, long time behavior for 2D,etc. This is a different world.

Theorem (Scheffer-Shnirelman): There exists a nontrivial weak solution with compact support in time.

This solution can be thought of as having initial data zero, then it is not zero and after a while, it is zero again. (This solution is far from regularity $1/3$.)

h-principle (Gromov)

This theorem can be viewed as a statement of the form of the h-principle. This principle should be viewed as a different tpe of statement related to Hadamard ill-posedness.

Theorem (Nash-Kuiper): Any strictly short smooth embedding of (compact) $M^n \rightarrow R^{n+1}$ can be uniformly approximated by $c^1$ isometric embeddings.

For a geometer, this is viewed as a completely wrong theorem. It seemingly contradicts the classical rigidity of the 2-sphere. Any isometric embedding of the 2-sphere into $R^3$ is the standard embedding. However, that theorem requires curvature so needs $C^2$.

Berti: What is short? Answer: Distances in the image are shorter than distances in the domain. So, Lipschitz with constant less than 1.

Two conditions: A topological global condition, an embedding. A local condition, isometric.

  • global: embedding
  • local: isometric

General statement. If you can satisfy the global constraint, you can twist it satisfy the local statement. Another example is Gromov’s theorem
saying 2 forms can be converted into symplectic forms.

An idea of the proof of this statement (Nash): $M^n \rightarrow R^{n+2}$. This is basically about a single chart so let’s look instead at $\Omega \subset R^n$ and we consider the embedding $\Omega \rightarrow R^{n+2}$. We have
$$
\nabla u^T \nabla u = g
$$
and strictly short means $g – \nabla u^T \nabla u >0$. We can make wrinkles. Wrinkling is written as a spiral
$$\tilde{u} (x) = u(x) + \frac{a(x)}{\lambda} (\sin (\lambda x \cdot \xi) \zeta (x) + \cos (\lambda x \cdot \xi) \eta (x)),$$
where $\zeta, \eta$ are unit normal to $u(\omega)$. This looks like a “telephone cord”. What happens in orthogonal directions? How does this affect the metric? This is something you can calculate:
$$
\nabla \tilde{u}^T \nabla \tilde{u} = \nabla u^T \nabla u + a^2 (x) \zeta \otimes \zeta + O(\frac{1}{\lambda}).
$$
This means I have a lot of freedom to change the metric in a fixed given direction. This allows me to write $g – \nabla u^T \nabla u $ as a sum of terms $\sum a_j^2 (x) \xi^j \otimes \xi^j.$ It is important here that the $\xi$ does not depend upon $x$. This allows me to achieve a reduction in the difference

$$\| g – \nabla u^T \nabla u \|_0 = O(\frac{1}{\lambda}),$$

$$\| u – \tilde{u} \|_0 = O(\frac{1}{\lambda}), $$

$$\| u – \tilde{u} \|1 \thicksim \| a \|0 \thicksim \| g – \nabla u^T \nabla u \|_0^{1/2}.$$

Lipschitz isometries

$\Omega \rightarrow R^n$

Kirchlein Baire Category Method.

Consider the space $X = [ u \in Lip (\Omega): \nabla u^T \nabla u \leq Id]$ endowed with the supremum norm. The observation of Kirchlein is that $\nabla \cdot X \rightarrow L^1$ is Baire-1. Consider 1-Libschitz maps converging to the zero function. With this same argument, I can take any function and add corrugations.

Mather: What is Baire-1? Answer: It is a pointwise limit of continuous maps.

A corollary of Baire Category theorem. The points of continuity is dense. Despite the troubles with the corrugation possibility, most maps in this space are points of continuity. The only places where you can’t improve is where the gradient is already maximizing. As a consequence, most maps in this space are isometric.

This is an argument which can be generalized quite a bit and can be applied to the Euler equations.

Example. The original system (O). (Tartar-DiPerna ideas)

$$\sum_{i=1}^n A_i \partial_i z = 0, ~in D’$$
$$ z(x) \in K, ~a.e. ~x$$

and we want to move to same relaxed condition (R) but with $z(x) \in K^{\Lambda}$, which he refers to as the convex hull of $K$. The principle is that most (in the sense of Baire Category) solutions of the relaxed setting R are solutions of the original problem O.

Question: What is the set $K^\Lambda$? This is the “wave cone”. Let $z: R^n \rightarrow R^d$ here. $\Lambda = [ \hat{z}: \exists \xi \in S^{n-1} ~s.t.~ \sum_i A_i \xi_i \hat{z} = 0]$. So, these are the directions in which we can oscillate while maintaining the conservation law and keeping the constitutive relations intact. So, that is $\Lambda$ and then the $\Lambda$-convex hull is like this. $z \notin K^\Lambda$ if $\exists ~ f ~ \Lambda$-convex so that $f(z) > 0, f|_K \leq 0$. You can separate. This general point of view was developed by Tartar, DiPerna.

Klainerman: what is the “h”? Answer: In this setting “h” stands for homotopy but that is not so present in this discussion. My view of the weak version of the h-principle is that there are a lot of solutions which have less regularity. De Lellis: Gromov woud say that you can take your short map and homotopize it while maintaining that it is an isometry, except at the endpoint.

Application to Euler

Now, you can write the Euler equations in this form by renaming the nonlinearity as a new variable. He shows how to do this by renaming some variables, interprets the associated $K$ and $K^\Lambda$.

Theorem (DL-Sz): Let $\overline{e}$ be a given function on $T^n \times [0,T]$ and let $(\overline{v}, \overline{u}, \overline{q})$ be smooth strict subsolution. Then $\exists ~ v_k \in L^\infty$, a sequence of weak solutions of Euler such that $v_k \rightharpoonup \overline{v}$ in $L^\infty$ and $\frac{|v_k|^2}{2} = \overline{e}$ a.e. $(x,t)$.

This is the “local part” of the h-principle. Given one subsolution, I can construct a solution by adding these waves. More or less, what Scheffer-Shnirelman have done is to take 0 as the subsolution.

….I am very much running out of time….so just to state one more theorem which touches the initial data.

Admissibility. Those weak solutions for which the $L^2$ norm is nonincreasing. Under this type of assumption, you have the weak-strong uniqueness. Any such strong solution is unique within the larger class of weak solutions emerging from the same initial data.

Theorem: Let $n=2$. Let $v_0(x)$ be the shear flow (he indicates this graphically wiht an interface and an arrow to the right above and an arrow to left below). $\exists~$ infinitely many admissible weak solutions.

He draws the interface with a thickened interface of some growing-with-time size outside of which we have the share flow.

Among all the selection criteria you might be considering for restoring uniqueness among the weak solutions, you could ask for maximally dissipating, you could choose the shear flow itself. Or you could ask for the one which has the fastest interface thickening. It is not yet clear which is the physically relevant selection critereon.

Questions:

Sverak: If you take a sequence of smooth solutions onverging to your data, is there any relation to your solution? Answer: It depends how it converges. Sverak: The best you can with continuous. Answer: I’m not sure.

Camilo de Lellis: The h-principle for the Euler equations Part 2

(continuation of previous talk; chalk talk)

I’ll start by mentioning some related results in the literature.

Survey article: D-Sz posted on web in 2011 contains all this literature.

  • Wiedeman: Global existence of weak solutions for any $L^2$ initial data in $R^3$. This would have been a fantastic theorem if it had not been too many solutions. This is the global analog of what was outlined before. Kappeler: ARe they adminssible? Answer: No. We are not anywhere near the blowup problem.
  • Sz-Wiedemann: You can approximate any measure-valued solutions (a la DiPerna-Majda) with exact solutions.
  • Sz-Wiedemann: The set of “bad” initial data is $L^2$-dense.

These techniques can also be applied to other equations.

  • Incompressible porous medium equations. Some class of active scalar equations. Caddoba-Faoco-Grancedo, Shydkoy, Sz.
  • Compressible Euler. D-Sz, Chiararoli, Chiadaroli-D. (Higher dimensional conservation laws have a striking nonuniqueness, contrast with the entropy conditions in 1d)

The fact that there exist $C^1$ isometric embeddings uniformly approximating any short smooth embedding was surprising.

Theorem (D-Sz 2011): For any given $e: [0,1] \rightarrow R^+$ smooth. Then there exists a $C^0$ solution of incompressible Euler in $T^3 \times [0,1]$ such that
$$
e(t) = \frac{1}{2} \int \frac{|v|^2}{2} (x,t) dx.
$$

We are moving towards the lower part of the conjecture of Onsager. I can’t claim this is saying anything about turbulence, but it does speak to the issues of dynamics of solutions viewed on Fourier coefficients.

Remark: It seems we can reach some Holder regularity, something explicit like $\frac{1}{500}$.

Shnirelman: Can you say something about modulus of regularity about this solution? Answer: You can work out something. It would be painful to work out. Finding a Holder exponent is achieved through an iteration process.

The process uses smoothness properties of $e(\cdot)$.

….change gears.

Borisov ‘50: If $v \in C^{1,\frac{2}{3} + \epsilon}$ is an isometric embedding of a positively curved connected 2d surface in $R^3$ then the image is convex.

This gives you rigidity. This is a local theorem. The global theorem would say that the sphere has an isometric rigidity. Borisov also had an announcement…..never published his proof.

Borisov (1965—>2004): h-principle for 2d analytic surfaces in $R^3$ if $u \in C^{0, \frac{1}{13} – \epsilon}$.

Conti; D-Sz: Rigidity and h-principle in general dimension (with better exponenets). For 2d the exponent is $\frac{1}{7}$.

He describes a double iteration scheme. He emphasizes that the parameter $\xi_0$ appearing inside the $\sin$ and $\cos$ is independent of $x$.

There are always successive one dimensional layers in any convex integration, in any h-principle appliation. To improve these constructions and obtain the $C^0$ statement we need to replace these constructions by higher dimensional generalizations. This is a direction suggested also by Gromov in his book.

…slowing down on typing….I’m just going to watch this.

Vladimir Sverak: On the long-time dynamics of some infinite-dimensional Hamiltonian systems

Vladimir Sverak

2 main examples

  1. $w_t + u \nabla w =0, ~ w = \curl u, \nabla \cdot u = 0, x \in T^2
  2. $NLS_3 (T^2)$

Thanks to Shnirelman, Kuksin, Choffrut, Keel, Polacik (slide changed fast might be incomplete).

Speculative picutre in Fourier space.

There is an interesting possibilty that on the macroscopc scale the dynamics might be simpler than in finite dimensional cases. Why? If you look at the dynamics on the Fourier space, at time $t=0$, we impose some nice initial data and as tiem evolves, some part of the solution moves toward infinite frequency. Everything in this direction is very hard to establish. There are some obstructions to this behavior. For example, KAM and complete integrability block this phenomena. We consider here “generic solutions”. Even though the initial data might be very complicated, we might end up with complexity moving into the high Fourier modes and what remains will be dictated by the conservation laws. This should be contrasted with finite-d systems. If one truncates the PDE in a naive way, we will initially see the same type of picture (motion toward infinite frequencies) and then we will hit the frequency cutoff and there will eventually be a thermalization. The “complexity” has “nowhere to go”.

Let’s look at defocusing $NLS_3$. We have, for this equation, three basic conserved quantities:

  • Energy
  • Momentum
  • Mass

Variational principal. We can think of the equation as a variatonal principal: minimize E subject to the mass=m and momentum=p constraints. We can compare this situation to 2D Euler. (In 3D there is the possibility that everything goes to infinity.) The $L^2$ norm and momentum provide constraints and the minimization principle gives you some nontrivial solutions even in the linear case.

The most optimisitic scenario for the transfer to high frequencies is that for a generic solution over a long period of time, the solution will spend most of its time (in some weak topology) near this manifold of minimizers of $E$ subject to the constraints given by parameters $p,m$. The natural “phase space” for solutions is $H^1$ and we view the manifold $M(p,m) subset H^1$. Ergodicity.

The variationa principle may be viewed from the statistical mechanics point of view. Consider a finite-d truncation via Dirichlet prjection. If we believe in stat mech in this scenario, we can use the microcanonical ensemble and look at the set of all points in our phase space where our energy lies between $E$ and $E + \delta$. We have a natural volume measure on this space. We can similarly $\delta$-thicken around the momentum and mass level sets. We can then hope that the solution will concentrate onto a measure living on these subspaces. This is provably rigorous in the linear case. In the limit as the truncation parameter goes to infinity, the energy $E$ is “forgotten” while the mass and momentum constraints are “remembered”. Another way to look at it is to follow the rule-of-thumb that the energy is equidistributed across possible states. Our initial energy is finite and we are equidistributing it across more and more states. Therefore, the temperature (energy per state) and the whole solution will weakly converge to zero. So, as $N \rightarrow \infty$, we concentrate on low frequencies consistent with the constraints and the rest of the solution goes to zero temperature.

Comparison with Gibbs measure. (Lebowitz, Bourgain,…) In this construction, one exponentiates the Hamiltonian and interprets this as a density with respect to the Wiener measure. For the Gibbs measure, $\langle E \rangle = + \infty$ and $\beta > 0$. These functions live on function spaces with infinite energy. The measure is concentrated on functions with low regularity and infinite energy. Remarkably, the dynamics is still well-defined by the PDE (Bourgain).

A rigorous result. NLS defines a dynamical system on the space $(X, w)$, where $X = [ \psi \in H^1, \| \psi \|_{H^1}\leq C]$. Here “w” denotes the weak topology. This is OK because Bourgain has shown well-posedness on $H^s, s<1$ (Bourgain).

Theorem: The $\Omega$-limit set (wrt the weak topology) contains solutions for which no movement to high frequencies goes to infinity. These are precompact in $H^1$.

These are the “end states” which solutions approach in the weak topology.

Question: Does every $\psi \in \Omega_+ (\psi_0)$ have this property?

Heuristics: If $\psi \in \Omega_+ (\psi_0)$ then the high frequency part has already separated.

Interpretation:

The solutions do not quite approach $M(p,m)$. there is probably some escape….ack slide changed.

2D incompressible Euler

the situation is quite similar. The difference between 2D Euler and NLS is that, in some sense, 2D Euler is a Poisson system instead of a Hamiltonian system. The system is a Hamiltonian system on symplectic leaves. Formally, 2D Euler should be viewed as a family of Hamiltonian systems and the orbit takes place on the leaf.

Fourier representation. Stream function. Energy. Conserved quantities associated with vorticity transport.

Natural “phase space”…. $L^\infty (T^2)$ with a weak * topology. Yudovich has shown the evolution is well-posed in $L^\infty (T^2)$. We have the necessary well-posedness pieces in place. Importantly for us, in this context of developing fine structures in teh flow, we have the stability result: If the data converges weak start then the same is true at later times. this follows from the proof of Yudovich.

Analogy with NLS. He draws a table.

  • Phase space: $H^1$ …. $L^\infty$
  • Weakly continuous: Momentum, mass …. $ E = \int -\frac{1}{2} \omega.$
  • Lower semicontinues: Energy….. ack slide changed.

Fourier picture. Variational principle, related to the notion of “mixing” introduced by A. Shnirelman). We minimize the energy subject to the constraints associated to the weakly continuous quantities. This produces a steady state solution (whih depends on $f_0$).

More geometric picture. We know that a good topology is the weak * topology on $L^\infty$. We can therefore study the weak * closure of the orbit.

Example: Data that looks like $\omega_0 = \chi_A – \chi_B.$ Here $B = T^2 \backslash A, ~ |A| = |B|$.

Onsager 1947, Montgomery-Johce, 1970s, Miller, Robert 1990s, Turkington 1990s, closely related to Shnirelman’s notion of “mixing”.

Looks similar to Ising model, except that the interaction is long-range. “Most-probable” configuration for a given energy $E$? But we specify here the number of configuration cells that have sign +1, and how many have -1. We can then do the usual statistical mechanics calculations.

He draws another analogy diagram.

  • NLS …. 2D Euler
  • $C^N$…..Ising config
  • Classical Maxwell Boltzmann microcannical ensemble picture……fermions (generalized) with long range interaction.

Full ergodicity seems….ack.

A more geometric picture for Euler (a sketch).

Geometric finite-d approximations inside the $SDiff (T^2)$ viewpoint. Remarkable fact learned from Khesin book showing that we have finite-d geometric approximations using $SU(N)$.

Determining the “end states” for Euler.

Calcuating the “entropy” etc., one gets different answers depending upon how one counts.

In Euler on the torus, the temperature is not zero but is instead negative as was observed by Onsager. All these predictions suggest that hte “end states” consist of shear flows.

An example where transfer to high frequencies was proved: Landau damping.

The only situation where this was rigorously proved wiath the Vlasov-Poisson system.

  • Landau 1946
  • Caglioti-Maffei
  • Hwang-Velazquez 2008
  • Mouhot-Villani 2009

Is there an analog of Landau dampoing possible for 2D Euler?

  1. Linear Landau damping: Yes (Sverak)
  2. Nonlinear case: probably yes, but seems more difficult than the Vlasov-Poisson case.

Is there an analog for this in the 2d NLS case? This seems more difficult in the NLS case. Transfer to high frequencies for dispersive equations:

  • 2d NLS: “I-team”
  • Various model situations: Bourgain,…

Questions:
1. Can you formulate a stability statement related to your question about the “end states” in the Schrodinger setting? Answer: I expect there should be stability statements around the energy minimizers subject to mass and momentum constraints.
2. Do you expect corresponding statements for the focusing problem (when you don’t expect blowup)? Yes, but there will be a weak convergence to soliton instead of the “breather” energy minimizers.

In discussion after the talk, Sverak suggested that these minimizers are not localized onto single Fourier modes and instead appear to be some kind of “breather” solution. These objects should have variational stability properties resembling corresponding statements about solitons and built along the motif of Arnold Stability results as appearing in the book by Khesin.

T. Oh reported to me that some studies like those suggested in this talk appear in work of Chatterjee-Kirkpatrick.

Antoine Choffrut: On the local structure of the set of stationary flows to the 2D incompressible Euler equations

Antoine Choffrut

arXiv

(chalk talk)

The Euler flow evolves on symplectic leaves. If you start on one of these leaves, then the Euler flow stays on the leaf. The result I want to prevent is the following. Suppose you have a stationary solution on one leaf, then the other stationary solutions are located on a curve that passes through the leaves. There is a 1-1 correspondence between the stationary solutions and the leaves.

This talk will be a bit more elementary than the other talks. Some aspects were forecasted by Preston and Sverak in earlier talks in this workshop.

Consider a (potato shaped) domain $\Omega$. The Euler equation $ \partial_t u + (u \cdot \nabla) u _ \nabla p =0, \nabla \cdot u =0, u \cdot N =0 (\partial \Omega).$

In 2D, we introduce $u = (u^1, u^2)$ and the vorticity $w =\partial_x u^2 – \partial_y u^1$ and we can derive the vorticity equation
$$ \partial_t w + u \cdot \nabla w = 0.$$

How do you recover $u$ from $w$? and vice versa? Introduce, in 2D, the stream function $u = \nabla^\perp \psi$.

$\psi|_{\partial \Omega} =$ locally constant. We have $\Delta \psi = w$.

Kelvin: A curve $c_0$ at time $t =0$ evolves along the flow to a curve $c_t$ at a later time. We obtain that $\int_{\Gamma_i} \frac{\partial \psi}{\partial N} = \gamma_i$. $\Delta \psi = w$ and some boundary conditions….

I can rewrite the vorticity equation as $\partial_t w + {\psi, w } = 0$ where the bracket is defined as the area of the parelellogram determined by $\nabla w$ and $\nabla \psi$.

Transport interpretation.

Coadjoint orbit.

Theorem (Choffrut-Sverak; GAFA 2012): Let $\overline{w}$ be a “non-degenerate” steady state. Then there exists a $C^\infty$-nhbhd $W$ of $\overline{w}$ such that any coadjoint orbit intersecting $W$ contains exactly one steady state in $W$.

The proof is by an inverse function theorem. I need to say how I am going to implement this function theorem. How do I describe my orbits? How do I describe my steady states? I want to show these are in one-to-one correspondence.

….lots of discussion…..smooth dependence….lots of chatter…..speaker needs to be able to describe more.

Characterization of steady states. The transport equation tells me that there is no dependence of $w$ on $t$ so that $w (\eta_t (x)) = w(x)$, $w = F(\psi)$ provided $\nabla \psi \neq 0$. I impose that $\nabla \psi \neq 0$ in $\Omega$ and I assume that $\Omega$ is an annular domain (with exactly one hole). Steady states are exactly parametrized by $F$.

Characterization of coadjoint orbits.

Transport $\implies A(0) = | [x \cdot w (x) < c ]| = |[x: w \circ \zeta (x) < c]|$

Correspondence.

  • Steady states ……… F
  • Orbits …………. A

Steady states can be characterized as conditional critical points of
$$
E(w) = \frac{1}{2} \int_\Omega | \nabla \psi |^2
$$ (restricted to an orbit.)

Recall $\Delta \phi = w$.
Calculating the first variation leads us to $\delta E = \int \nabla \psi \cdot \nabla \phi = – \int \psi { \alpha, w} = 0 $ (using a permutation property.) Since this is true for all $\alpha$, we find that ${\psi, w} = 0$. Similar ideas come up in the Arnold stability theorem.

Euler as a geodesic configuration space. Lagrangian least action principle. Marchioro-Pulvarenti, Chemin.

Clairaut, Noether, Lie group with (left) invariant metric.

That was the Lagrangian formulation on the tangent space. The real action takes place in the cotangent bundle where we have a Hamiltonian formalism.

Thomas Kappeler: Symplectic techniques for integrable PDEs

Thomas Kappeler

(pdf slides)

Aim: Survey of recent results on integraple PDEs obtained by symplectic techniques. The model equation is the $NLS_3^{\pm}(T)$.

Topics:

  • Phase portrait/ space of orbits; construction of normal coordinates
  • Asymptotic properties of solutions
  • KAM theorem

NLS as a Hamiltonian system.

Phase space: $L^2_C$. Pairs of functions. More generally, a function space.

There is a canonical Poisson bracket

$$ {F,G}(\phi) = -i \int_0^1 \partial_1 F \partial_2 G – \partial_2 F \partial_1 G) \phi dx. $$

Defocusing NLS. Focusing NLS.

Defocusing NLS as an integrable PDE on T. Grebert-Kappeler-Poschel “The defocusing NLS equation and its normal form” to appear in EMS.

Focusing NLS as integrable PDE: only few results.

Part 1. Review of NF for defocusing NLS.

Theorem (GKP): There exists a canonical map (closely related to the Fourier transform) which reveals that defocusing NLS may be viewed as a system of infintely many coupled oscillators.

Steps of proof:

Local part. We have to construct these coordinates $x_n (\phi)$ and $y_n (\phi)$. How to build these coordinates?

Global part. We have a global chart.

Zakharov-Shabat operator (ZS)

  • Lax pair for NLS
  • Periodic spectrum of $L(\phi)$ on [0,2].

Counting Lemma: He describes the spectrum with a picture on the board. Floquet theory.

  • characteristic function
  • two-sheeted spectral curve

von Neumann-Wigner 1929

The eigenvalues come in pairs. Asymptotically, they are like $n\pi + l_n^2$.

Construction of actions/angles

  • Choose cycles $a_n$
  • Cycles induce (i) actions and (ii) 1-forms.
  • 1-forms induce angles.

Birkhoff coordinates

Euclidean versions of these action angle coordinates.

Important features of construction

  • same cycles $a_n$ were used to define $I_n$ and the 1-forms $\beta_n$.
  • Cycles/1-forms are defined on the spectral curve and not on phase space.

Part 2. Normal Form for fNLS

  • There do not exist global Birkhoff coordinates.
  • The associated Zakharov-Shabat operator $L$ is not necessarily self-adjoint.
  • Symmetries of $spec_p (L(\phi))$.
  • $spec_p (L(\phi))$ can be described.

We believe that one can characterize the (local) existence of Birkhoff coordinates near $\phi \iff$ spectral properties of the Zakharov-Shabat operator.

Standard Potentials

Describes how to carry out the construction. The focusing case requires local constructions.

The discussion here is very precise and I don’t think I can convey more than is available on the slides.

Part 4. KAM for defocusing NLS

  • Defocusing NLS is integrable on all of $L^2$.
  • Question: KAM on $L^2$, not only near equilibrium point 0?
  • Defocusing NLS frequencies can be expressed using the Birkhoff coordinates.

Proposition:
Can check the Kolmogorov and Melnikov conditions and then conclude by analyticity.

Near resonances

  • $\omega_j – \omega_{-j} = O(1)$ for $ j \rightarrow \pm \infty$ jeopardizes measure estimate of standard KAM theorem for integrable PDE.
  • Ways to overcome difficulties:
  • Craig-Wayne-Bourgain method. Bourgain, IMRN 95
  • Kuksin-Poschel, Berti
  • Restrict the perturbations Geng-You CMP 06, JDE 05

Kappeler-Liang JDE 12

 

I’m participating in a workshop at the Institute for Advanced Study on Symplectic Dynamics. This is part of the special concentration this year organized by Helmut Hofer. I’ll try to write notes on the talks I hear. Apologies to the speakers for typos and misquotations…. Comments (pending approval) are open for suggestions and edits.

IAS School of Mathematics

Simonyi Hall

Workshop web page

  • 10:15 – 11:15 Peter Constantin, Princeton University, “Long time, vanishing viscosity limits” abstract
  • 11:30 – 12:30 Steve Preston, University of Colorado, “The inextensible string as a toy model of fluids” abstract
  • 2:30 – 3:30 Emanuele Caglioti, University of Rome, “Long time behavior of solutions of Vlasov-like equations” abstract
  • 4:30 – 5:30 Roberto Camassa, University of North Carolina, “Large amplitude internal waves and their stability” abstract

Peter Constantin: Long time, vanishing viscosity limits

Peter Constantin

(pdf slides)

Navier-Stokes equations. $\nu$ multiplies the damping term. Studying it in $R^d$ or $T^d$, $d=2,3$. We are interested in $T \rightarrow \infty$ and $\nu \rightarrow 0$. (Reynolds number $\frac{UL}{\nu}$.)

Limits: selected stationary statistical solutions.

$$\lim_{Re \rightarrow \infty} \lim_{T \rightarrow \infty} \frac{1}{T} \int_0^T \phi( S^{NS} (t) ) dt.$$

Anomalous dissipation of energy:

$$
\lim_{\nu \rightarrow 0} \lim_{T \rightarrow \infty} \frac{\nu}{T} \int_0^T \int_{R^3} |\nabla u (x,t)|^2 dx dt = \epsilon > 0.$$

(K41)

2d Navier Stokes

No anomalous dissipation of energy. Enstrophy balance. Anomalous dissipation of enstrophy? Kraichnan 68: Yes. Bernard 00: add (extra…to be described…discussion….friction from the bottom) damping and then no.

Constantin-Ramos 07: Bernard was right. Heuristics.

We are talking about weak solutions. 2 reasons to talk about these. The singularities really describe the phenomena, e.g. shocks in hyperbolic conservation laws. Deduce from the few conservation laws that solutions exist and they are nice and smooth but we can’t so we content ourselves with what we can build.

Active Scalars

$$\partial_t \theta + u \cdot \nabla \theta = 0$$

Examples: 2d Euler, SQG.

$$ u = \Lambda^\gamma R^\perp \theta. $$

(Here $R$ is the Riesz transform, $\Lambda = \sqrt{-\Delta}$.)

Weak solutions for SQG. Resnick 95. Almost Lipschitz on $L^2$. Euler and NS does not have this weak continuity property. We don’t have uniqueness. If you have unique weak solutions, then you are golden. I’m going to talk about statistical solutions eventually.

Theorem (Chae-C-Cordoba-Gancedo-Wu): Let $\theta_0 \in L^2 (T^2)$. There exists a global $L^2$-weak solutions of the generalized SQG.

….too fast to type. Generalizes result of Resnick. Remarks…fast. Friedlander-Vicol. Moffat.

A commutator estimate

Summarizes the ideas of the proof. Stream function representation. Cancellations….end up with a commutator….end up with a weak vs. strong.

Damped Driven Navier-Stokes Equation

Standard Navier-Stokes with forcing and an added term $+ \gamma u$, which represents the damping.

You get some a priori bounds which are independent of the viscosity by following the usual arguments.

Stationary Solutions (a warm up)

$$ \gamma \omega + u \cdot \nabla \omega – \nu \Delta \omega – g =0.$$

Absence of anomalous dissipation

Theorem:
$$ \lim_{\nu \rightarrow 0} \nu \int_{R^2} |\nabla \omega^\nu |^2 dx = 0.$$

The key idea is to extract extra structure by taking limits.

Statistical Stationary Solutions

Definition: A stational statisitcal solution (SSS) of the damped, driven NS equation on the phase space of vorticity is a probability meausre $\mu^\nu$ on $L^2 (R^d$ such that )…..some conditions indicating that the equation holds in a weak sense against “cylindrical test functions”.

Ideas of proof….pretty fast….hard to keep up while typing. He emphasizes the role of a nonlinear function $\beta$ which he uses to map from “$L^2$ to $L^\infty$”

One Key idea. Use $J(u \omega) – J(u) J(\omega)$ when you are considering a mollifier $J$. This is a useul trick. “Quadratic flux formula”

The hard part of the proof is to prove the enstrophy balance.

The delicate dance of the $\beta$ parameter reminds me of the role of the smoothing operator $I_N$ in the $I$-method. I’d like to understand this better….

The punch line is that you only need this “old language” for the proof. At the end, we obtain a proof of the absence of anomalous dissipation. I’d like to prove corresponding results for quasigeostrophic.

Remarks:

  • The existence of weak solutions for damped, driven Euler equations: Barcilon-Constantin-Titi
  • For absence of anomalous dissipation: Renormalized weak solutions. Enstrophy balance.
  • Results for gSQG. Mentions Caffarelli-Vasseur, Nazorov-Volberg-? on this equation.

Questions/Discussion


A suggested direction: Stability implies regularity. If you assume that NS solutions are $L^2$ stable then you should be able to prove regularity. Conjecture: Fix viscosity. Imagine you have a time horizon and you have a constant. Lipschitz regularity of the flow map implies regularity.

Steve Preston: The inextensible string as a toy model of fluids

Steve Preston

(pdf slides talk; joint work with Ralph Saxton)

The original idea that the inextensiblestring might have soemthing to do with fluids was due to V. Yudovich. This problem was introduced to the speaker by A. Shnirelman. The part of fluids I want to have a toy model for is the geometric viewpoint.

Geometric aspects of fluid mechanics

At a point in a fluid, you have a velocity felid and we minagine this vector pushes a fluid element along that direction. You put an $L^2$-Riemannian metric on the space of maps $C^\infty (M, M)$ and you define geodesics wrt this metric.

Volumorphisms. He views this as a “submanifold” in the space of all maps. A lot of things break down here but formally we think of this curve of volume preserving maps inside the larger flat space of all maps. Hodge decomposition allows us to decompose an arbitrary map into one along the volumorphism submanifold and another involving a divergence.

Misiolek-Himonas

Ebin-Marsden

Riemannian exponential map.

Eulerian viewpoint is easy to formulate, but the Lagrangian form is more convenient for geometry. He emphasized the loss of dierivatives.

“Smoothness of the exponential map is the most basic requirement for doing infinite-dimensional Riemannian geometry rigorously.”

There is a gap between the topology and geometry of volumorphisms.

  • Topology. We want the volumorphisms to be a smooth submanifold of the space of smooth maps from $M$ to $M$. We can make this rigorous if we enlarge to Sobolev $H^s$ diffeomorphisms with $s > \frac{1}{2} {\mbox{dim}}(M) + 1$ to ensure that $\eta \in C^1$. For smaller $s$, we don’t get a smooth submanifold.
  • Geometry. The metric is defined only in terms of $L^2$ distance. So geometrycially, we should consider measureable maps $\eta: M \rightarrow M$ which preserve the measure. These may not even be bijections so that manifold structure fails (not all tangent spaces are isomorphic).

If $M$ is three-dimensional, the $L^2$ closure of smooth volumorhpisms is the space of all measure-preserving measureable maps (Shnirelman). Fluids should not behave like that! Or maybe they could come close?

In 2d, we don’t understand this so well.

Lagrangian Averaged Euler equations

We introduce a parameter $\alpha$ and define an alternative Riemannian metric which has an $H^1$ inner product (time $\alpha$) and we consider this only on the Volumorphism “submanifold”. The associated geodesic equation leads to the LAE-$\alpha$. “The idea is to average over small scales of a fluid; as $\alpha \rightarrow 0$ we expect the solutions to approach the usual Euler equation solutions.” When these were introduced, it was expected that 3d global existence would be easier than for Euler but this has not turned out to be the case.

To better understand what is going on, we want simpler one dimensional examples.

  • Camassa-Holm: $ u_t – u_{txx} + 3 u u_x – 2 u_x u_{xx} – u u_{xxx} = 0$
  • Constantin-Lax-Majda: $\omega_t = \omega H \omega, ~ u_x = H \omega$$

In the CLM equation $H$ is the Hilbert transform. Modified: $\omega_t – \frac{1}{2} u \omega_x = \omega u_x, ~ u_x = H \omega$.
This is a geodesic equation on the space of diffeomorphisms on $S^1$ with a right-invariant $H^{1/2}$ metric.

….quick transition to the inextensible curves…..moving faster….

Unit speed parametrization. This constraint forces the curve the generate its own tension to satisfy the constraint. This is similar to the pressure which results from the zero divergence condition.

Geodesic equation for $L^2$ string

Orthogonal acceleration $\implies$

$$ \eta_{tt} = \partial_x (\sigma \eta_x)$$

where
$$
\sigma_{xx} – |\eta_{xx}|^2 \sigma = – |\eta_{xt}|^2.
$$
“Here the tension $\sigma$ is analogous to the pressure in the Euler equation, determined nonlocally by a purely spatial differential equation.

WE can think of this as an approximation of the nonlinear wave quation, in the same way as the incompressible Euler equation is an approximation of the compressible Euler equation.”

(This is an interesting nonlocal equation so it should be mentioned on the developing nonlocal equations wiki.)

Finite-d approximate model through a system of rigid rods oscillating like coupled pendula.

In $R^2$, we can write $\eta_x = (\cos \theta, \sin \theta)$ (which enforces the constraint) and rewrite the equations. Some discussion of boundary conditions. The boundary conditions then determine nonlinear constraints on the parameter $\theta$.

Movie:

  • $L^2$ whip. The whip cracks. The curvature becomes singular. Thess-Zikanov-Nepomnyashchy The loops make the crack!

So, I’d like to revisit the Lagrangian averaging in the setting of the inextensible string. He again adds the $H^1$ inner product to the Riemannain metric on the space of curves. This produces a new geodesic equation, which does not resemble the earlier wave equation. It does still have the nonlocal $\sigma$,…technical slide….

Preston-Saxton, DCDS-A (to appear) Some GWP results.

Movie:

  • Same initial condition and the loops don’t get pinched.

Emanuele Caglioti: Long time behavior of solutions of Vlasov-like equations

Emanuele Caglioti

(pdf slide talk)

Outline

one slide, overview of the talk.

Vlasov-Poisson and 3d Euler

The Vlasov equation
$$\partial_t f + v \partial_x f + F_f \partial_v f = 0$$

Here $f(x,v): S^1 \times R$ is the phase space density:
$$
\rho (x) = \int dv f(x,v)
$$
is the space density
$$
F_f (x) = \int dy \rho(y) \mathcal{F} (x-y)
$$
is the force.

Vlasov-Poisson Equation (VPE)

$\mathcal{F} = \partial_x \mathcal{V}, ~ \mathcal{V} = \partial_{xx}^{-1} \delta$. For $x \in [0, 2 \pi)$….ack slide change….

2d Euler equation is another example. The vorticity $\omega$ is transported along the flow.

The density $f(x,v,t)$ is transported along the trajectories of an Hamiltonian system:
$$
\dot{x} = v, \dot{v} = F_f (x).
$$

The hamiltonian is a functional of the density $f$ itself: self-consistent force field. Therefore the area of the level sets of $f$ is conserved. Same for 2D Euler the vorticity is tranported along the trajectories of an Hamiltonian system.

Mather: What does that mean “self-consistent force field”? Answer: Some discussion about the mean field limit and statistical mechanics. Here is a link to related discussion.

Possible behaviors

Stable stationary solutions of VPE.

Marchioro-Pulvirenti (1986)

$f = g(v) $ is an example.

BGK waves

Bernstein, Greene, Kruskal (1957)

$ f = f_0 (x - u_0 t, v - u_0).$

These solutions satisfy some conditions related to the Hamiltonian.

I try to make a parallel with 2D Euler.

Any radial vorticity: $ \omega = g(\rho), ~ \rho = \sqrt{x^2 + y^2}$
is a stationa soution of 2D Euler.

Kirchoff (1876) showed that elliptiacl patches are rotating solutions of 2D Euler. The patch is stable if $a < 3b$ (parameters refer to the geometry of the ellipse.)

Landau Damping

Landau, on the basis of the analysis of the VPE linearized around an equilibrium conjectured that for initial data close to equilibrium
$$
f_0 = f(v) + \epsilon g(x,v)
$$
asymptotically the electric field will vanish and the phase space density will become homogeneous. The linear case has ben fully characterized (Maslov- and Fedoryuk.)

Existence of a class of damped analytic solutions has been proved with a scattering approach by Caglioti-Maffei (1998). Hwuang-Velasquez (2009) extended this result to situations close to equilibrium solutions. Initial data cannot be characterized.

Mouhot-Villani (2009) proved that close to equilibrium initil data are exponentially damped (Landau Damping). The result is proved in an analytic framework (also Gevray type regularity).

Lin-Zeng (Recently) have shown that BGK exists for small regularity: $W^{s,p}, ~ s< 1 + \frac{1}{p}$. Therefore, there is NO LANDAU DAMPING for small regularity. This is interesting to me….a long time behavior which is dependent upon regularity.

Matthaeus (1991) simulation.

….Constantin: this is probably hyperviscosity. ok, but the point he wants to convey is not dependent on this issue.
Shnirelman: this is a very robust picture. When you simulate NS on 2D torus, it always emerges that there are two vortices like this.

Possible limiting behaviors.

  • Stationary (stable) solution for VPE and for 2D Euler
  • time periodic solutions for BGK and Kirkhoff Ellipses for 2D Euler
  • Landau damping
  • Is it possible to prove damping to BGK solutions? Many people are working on this.

(This theory seems to be quite analogous to the state of the art in nonlinear Schrodinger and wave equations.)

What can we say in general?

Shnirelman ICM 2010 Mixing operators in $L^2$. Shnirelman’s construction. Bistochastic operators.

Partial ordering, minimal flows.

It is possible to prove that minimal flows are stationary stable solutions of 2D Eler.

A first conjecture: The set of minimal flows is an attractor for the 2D Euler flow (essentially Landau damping conjecture)

Motivation: If the fluid does not go to stationary solutions level lines of vorticity are stretched and stretched and therefore the solution reaches a minimal element.

The conjecture is probably wrong because more complicated behaviors are expected from simulations.

Brenier: True with probability 1. Caglioti: probably not.

Generalized minimal flows (Shnirelman):

The Navier-Stokes equation with random forcing in the null viscosity limit has an attractor which is concentrated on generalized minimal flows. We reformulate this conjecture in the language of Landau Damping from Vlasov.

A strictly related conjecture.

Given $f_0$, let us define
$$
\Omega (f_0) = [ \mbox{weak limit points of} f(x,v,t): t \rightarrow + \infty ]
$$

Then it is reasonable that generically:

  • $S(g) \geq S(f_0)$
  • If $g_1$ and $g_2$….ack slide changed.

Construction of periodic solutions for the HMF model

Morita-Kaneco PRL (2006)

See also Antoniazzi-Fanelli-Barre-Chavanis-Dauxois-Ruffo, PRE (2007)

work in progress with D. Benedetto and P. Butta.

Conclusions:

We might reasonably expect that asymptitocally the dynamics will become a simple motion. It might even be chaotic but with a few degrees of freedom involved.

Roberto Camassa: Large amplitude internal waves and their stability


Article about Carolina Wave tank

(Collaboration with A. Almgren, S. Chen, R. Tiron, C. Viotti.)

Somewhat soft…..filled with movies…..suitable for the end of the day.

Outline

Motivation: practical (quantitative) vs. “paradigm” (qualitative) significance of simple models of wave motion in fluids?

Three examples:

  • Two layer Euler vs. strongly nonlinear models
  • Wave induced instabilities
  • Integral equations:

Introduction. Fluids lab experiments. Cool movie.

Stratified incompressible Euler equations. 2 layer model. As you saw in the movie, diffusion can be ignored on the time scale of the movie.

Grue et. al JFM 1999 (Norway experimental group)

Stanton-Ostrovsky 1998 (Oregon Coast), 150m

ASIAEX 2004 (340m)

Helfrisch-Melville 2006

These papers reveal that large amplitude internal waves exist.

Miyata 1998, Choi-Camassa JFM 1999

Shallow water strongly nonlinear models.

Pictures….pictures….movies.

Richardson Number.

Taylor-Goldstein Eigenvalue problem controls stability.