Analysis & Applied Math Seminar 2013-03-21

Speaker: Zaher Hani

Institution: New York University

Abstract: Inspired by the general paradigm of weak turbulence theory, we consider the 2D cubic nonlinear Schrödinger equation on a box of size L with periodic boundary conditions. In an appropriate “large box regime” (L very large), we derive a continuum equation on ℝ2 that governs the dynamics of the discrete frequency modes over nonlinear time scales. This equation turns out to satisfy many surprising symmetries and conservation laws, as well as several families of explicit solutions. (This is joint work with Erwan Faou (INRIA, France) and Pierre Germain (Courant Institute, NYU)).

Zaher Hani

Introduction

 

Setup

2d cubic defocsusing or focusing NLS on a box of size $L$. Energy and mass conservation. NLS is GWP for small data in $H^s$ for $ s \geq 1$. We are not concerned with the existence issue. We are working in the setting of global-in-time solutions.

Physical and mathematical setup: weak nonlinearity.

  • Aim: understand out-of-equilibrium dnamics of small solutions. e.g. CKSTT
  • Take small data….nonlinear time scale is $\epsilon^{-2}$ where we imagine $ u \thicksim \epsilon v$ and we study $v$ with an $\epsilon^2$ coefficient and the data is of the size 1.
  • Fourier ansatz, transfer dynamics onto the coefficients. Comments on the $L^2$ norm dependence upon the box size parameter $L$.
  • Express the dynamics in terms of the $a_k (t)$.
  • Define the 4-frequency convolution hypersurface. He calls that $S_K$.
  • Interaction Representation. Conjugate by the fast linear dynamic….remove the linear dynamics. The new Fourier variable is called $\tilde{a}_k (t).$

All that has happened was a change of variables enabling us to view the dynamics on the Fourier side.

Weak turbulence “paradigm”

  • Aim: statistical description of out-of-equilibrium dynamics of small solutions (Zakharov 60s, Kolmogorov 50s)
  • RPA rand phase and amplitude.
  • $n(K,t) = {\mathbb{E}} |a_k (t)|^2$ is the wave spectrum or mass density.
  • propagation of chaos assumption. True at $t=0$, but not propagated.
  • Roughly, we have three main steps:
    1. Statistical and time averaging.
    2. large-box limit $L \rightarrow \infty$.
    3. weak nonlinearity limit $\epsilon \rightarrow 0$ to arrive at a continuum equation for $n(K), ~ K \in {\mathbb{R}}^2.$

The Kolmogorov-Zakharov kinetic equation. Long convolution equation localized on the convolution hypersurvface and further localized on the resonant set.

  • Admits explicit stationary solutions called KZ spectra. These solutions are thought to offer some explanation to some cascade phenomena.
  • Non-rigorous, KZ spectra are not integrable, negliects some finite-size effects, some numerical discrepancies, the appearance of some coherent structures called “quasi-solitons” even in defocusing problems.

A New Limiting equation

Statistical averaging was causing problems. Let’s dispense with that but still take the large box and weak nonlinearity limits.

Resonant cutoff/normal forms transformation. He goes to the board and describes the separation. On the non-resonant portion, he makes a stationary phase type integration by parts, and then makes a crude estimate using the equation. This shows the non-resonant portion contributes at size $\epsilon^4 L^2$. Therefore, we concentrate our attention on the resonant terms.

(slide 12/37 and we are 20 minutes into the talk….)

He analyzes the convolution + resonance condition and identifies orthogonality properties based on the pythagorean relationship among frequencies.

Parametrization of rectangles in $\mathbb{Z}^2 / L$.

A lattice point $ J \in \mathbb{Z}^2 / L$ is called visible if $z = (p,q)/L$ with $gcd (|p|, |q|) =1. $ These points can be connected by a straight to the origin without hitting another point in the lattice.

Some new coordinates involving an $\alpha$ and $\beta$.

Co-prime equidistribution:

You can, in certain circumstances, replace sums by corresponding integrals with bounds. A classical number theory result establishes the density of visible lattice points in $\mathbb{Z}^2 / M$ is $\frac{6}{\pi^2}$. This lets you translate equidistribution into a co-prime equidistribution statement enabling us to replace sums by corresponding integrals with bounds.

Q:…interesting, I wonder to what extent similar ideas can be used on the sums we have omitted earlier in the argument. Perhaps those sums can also be represented as integrals with appropriate bounds?

Continuum limit.

Following these formal arguments leads to an integral equation resembling the KZ equation. Q: What are the differences/similarities with the KZ equation? One difference is that it preserves the Hamiltonian structure and has a positive definite Hamiltonian. He calls this equation $*$.

Symmetries lead to conserved quantities.

He writes the trilinear term in the equation as $\mathcal{T}(f,g,h)$.

  • Hamiltonian
  • Mass
  • Momentum
  • Position
  • Second momentum
  • Kinetic energy
  • Angular momentum

A scaling property.

Invariance under Fourier transform.

If $g$ solves $*$ then $\hat{g}$ also solves $*$.

Properties of the continuum equation

Boudedness properties and well-posedness. He reports on LWP and GWP properties of the equation $*$.

Gaussian family is a family of explicit stationary solutions. Gaussians are the unique maxima of the Hamiltonian functional.

Heavy tailed solutions.

Are there more?

Invariance of Harmonic oscillator eigenspaces. Hermite polynomials. The associated linear spans are invariant under the nonlinear flow $*$. The Hamiltonian of the harmonic oscillator is an integral of motion so the two flows commute and you get this easily.

Question: Is this equation $*$ completely integrable?

Rigorous Approximate Results

In analogy to the CKSTT cascade result, there is a reduction to an equation related to NLS. Can we transfer information from $*$ back to learn something about NLS?

Three difficulties:

  1. Pass to the resonant sum.
  2. Obtain good discrete to continuum error estimates.
  3. Trilinear estimates on resonant sums.

….discussion of these issues…. identifies the small nonlinearity regime characterized by the condition
$$ \epsilon^4 L^2 \ll \frac{\epsilon^2 \log L}{L^2}.$$

Möbius inversion formula.

Convergence Theorem: ….long statement. He gets a convergence statement on an interval that is longer than the nonlinear time scale by a factor $ M \leq \log \log L$.

Further Questions

  • Numerical study comparing NLS and $*$.
  • Other explicit solutions of $*$? Cascading solutions? Videos.
  • Is $*$ completely integrable?
  • Similar continuum limit for other equations?
,

 

My former colleague Larry Guth (now at MIT) visited us recently and gave a beautiful colloquium talk. The Department has recently deployed a video streaming service so we are able to share Larry’s talk with the world. We look forward to sharing other videos in the future.

 

Here is the video:

 

 

Unexpected applications of polynomials in combinatorics

by Larry Guth | MIT
Time: 16:10  (Wednesday, Jan. 23, 2013)
Location: BA6183, Bahen Center, 40 St George St
Abstract:
In the last five years, several hard problems in combinatorics have been solved by using polynomials in an unexpected way. In some cases, the proofs are very short, and I will present a complete proof in the lecture. One of the problems is the joints problem. Given a set of lines in $R^3$, a joint is a point that lies in three non-coplanar lines. Given $L$ lines in $R^3$, how many joints can there be? Another problem is the distinct distance problem in the plane. If P is a set of points in the plane, the distance set of $P$ is the set of all distances from one point of $P$ to another. If $P$ is a set of $N$ points in the plane, how small can the distance set of $P$ be? The proofs involve studying a set of points in a vector space by finding a polynomial of controlled degree that vanishes at the points, and then using the geometry of the zero-set to understand the combinatorial properties of the points. The goal for the talk is to give an overview of this new method.

 

 

 

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

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

Oana Pocovnicu

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

Edinburgh talk. 2012-05-21

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

GP

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

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

Literature

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

Remark: energy critical in $R^4$.

  • Gerard 2006 considered the energy space:

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

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

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

Two ingredients:

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

Scaling Invariance

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

Strichartz Estimates

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

Energy Critical NLS

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

Main Results on defocusing energy-critical NLS

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

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

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

Cubic NLS on $R^4$ (Visan)

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

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

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

Perturbation theory

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

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

Remarks on Proof

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

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

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

Scattering for the GP equation in the case of large data

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

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

Killip-Oh-Pocovnicu-Visan

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

(Final statement is a work in progress.)

 

,

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.