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?
,

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.)

 

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