energy dynamics across scales and wave turbulence in ...stolo/conf/hiver15/hani15.pdf · energy...

Energy dynamics across scales and wave turbulence inHamiltonian PDE.

Zaher Hani

School of MathematicsGeorgia Institute of Technology

Ecole d’hiver ”Dynamics and PDE”Saint Etienne de Tinee

February 13, 2015

Zaher Hani (GeorgiaTech) Energy Cascades and wave turbulence 1 / 33

Introduction: The general problematic

Energy dynamics in Hamiltonian systems

General Problematic: Understand energy transfer and redistributionin Hamiltonian systems:

pk = − ∂H∂qk

, qk =∂H

∂pk; 1 6 k 6 N.

Here, (pk(t), qk(t))k are functions of time and H(p,q) is theHamiltonian function. N = # of degrees of freedom (d.o.f.) 6∞.

Finite dimensional example: problem of heat transfer.

Particularly, we will focus on Hamiltonian PDE of dispersive type,where the problem translates into understanding energy transferacross scales.

This is a major scientific problem appearing in oceanography, plasmacontrol, Bose-Einstein condensation, etc.



Energy dynamics in Hamiltonian systems

Suppose that the energy is initially injected in a small fraction of allthe available degrees of freedom (d.o.f.).

Guiding Questions

Q1) Will it stay as concentrated, or will the energy migrate towards other degreesof freedom?

Q2) What are the dynamical mechanisms or obstructions to this energy transfer?

Q3) Can one put laws of statistical physics that allow identifying steady states ofenergy distributions both in equilibrium and non-equilibrium configurations?

Q4) Can one reconcile the dynamics of the system with theassumptions/conclusions of the statistical physics just postulated? Likegiving a dynamical path to thermalization?



Answer to Q1): Completely integrable systems

In finite dimensions, these are systems for which energy is notexchanged between degrees of freedom (after a canonical change ofvariable). Motion there is quasi-periodic (linear motion on a finitedimensional torus). In infinite dimensions, a similar statement has tobe made with a bit of caution because things start depending on thespace one is working in (e.g. Szego equation).

For dispersive equations on Rd , dispersion translates into decay, whichtypically leads to scattering (at least in defocusing nonlinearities).Moser: Scattering can be regarded as a form of complete integrability(in particular, no strong energy transfer between scales).

We won’t be interested in systems that are completely integrable. Ofparticular importance are nonlinear dispersive equations on compactdomains (possibly very large).



The intuition from finite dimensions (ODE world)

Concerning Q2): Dynamical systems theory identifies obstructionsand mechanisms for energy transfer in finite dimensions.

I KAM tori (1954) are the most well-known obstructions: Persistence ofquasi-periodic behavior.

I Arnold diffusion (1964) is a mechanism of energy transfer: solutionsmigrating from a quasi-periodic torus to another.

Concerning Q3): Statistical mechanics provides a systematic theoryto deal with macroscopic (collective) properties of systems withmany d.o.f., based on empirically valid principles (like entropyincrease, ergodicity, etc.)

Concerning Q4): Few results justifying the conclusions of statisticalmechanics from a dynamical principles (a.k.a. FPU Paradox).Though, see Sinai, Ruelle, Lanford, [Gallagher-S. Raymond- et al.]...The problem becomes more tractable in presence of a stochasticforcing (e.g. fluctuation-dissipation models, cf. Eckmann, Rey-Bellet,Hairer, Mattingly, etc).



Our model: Cubic nonlinear Schrodinger (NLS)equation

i∂tv(t, x) + ∆v(t, x) = |v(t, x)|2v(t, x), x ∈ T2

L := [0, L]× [0, L]

v(0) = v0,

E [u(t)] := 12

∫T2L|∇u(t)|2dx + 1

4

∫T2L|u(t)|4dx= E [u(0)].

Finite energy solutions exist globally (Bourgain ’93). Sign ofnonlinearity is not important for us; Also, dimension could be higher.

Aim: Understand energy dynamics of small solutions.

Ansatz v(t, x) = εu(t, x) with ‖u0‖L2(T2L) ∼ 1 Weak nonlinearity.

i∂tu(t, x) + ∆u(t, x) = ε2|u(t, x)|2u(t, x)

u(0) = u0,(NLSε)



Fourier pictureu(t, x) = 1

L2

∑K∈Z2/L cK (t)e2πiK .x ; cK (t) :=

∫T2Lu(t, x)e−2πiK .x dx .

i∂tcK (t)− 4π2|K |2cK (t) =ε2

L4

∑(K1,K2,K3)∈SK

cK1(t)cK2(t)cK3(t)

where SK = (K1,K2,K3) ∈ (Z2L)3 : K1 − K2 + K3 = K.

Define aK (t) := e4π2i |K |2tcK (t), (Interaction representation picture):

i∂taK (t) =ε2

L4

∑(K1,K2,K3)∈SK

aK1(t)aK2(t)aK3(t)e−4π2iΩt (NLSε)

Ω =|K1|2 − |K2|2 + |K3|2 − |K |2.Resonant interactions: R(K ) = S(K ) ∩ Ω = 0 are most important.

i∂trK =ε2

L4

∑(K1,K2,K3)∈RK

rK1(t)rK2(t)rK3(t) (RNLS)



Outline

2 Energy Cascades: Mechanisms of energy transfer in dispersive systems

3 Wave turbulence theory: Statistical physics of dispersive waves

4 Mathematical Attempts to justify wave turbulence


Infinite Energy cascades

Energy cascade

The energy of the system moves its concentration zones betweencharacteristically different length-scales.

Forward (or direct) Cascade of energy: Migration of energy from lowto high frequency concentration zones (small scales).

Q2) ↔ Construct solutions exhibiting this forward cascade of energy.

Growth of Sobolev norms: Movement of energy to high-frequencyregions leads to the increase in the Hs Sobolev norms for s > 1

‖u(t)‖Hs(Td ) =∑|α|6s

‖∇αu‖L2(Td ) ∼

∑n∈Zd

(1 + |n|2)s |u(n)|21/2

.



Problem (Bourgain; GAFA 2000)

Does there exist global solutions to cubic NLS whose Hs norm (s > 1)exhibits infinite growth in time, i.e.

lim supt→+∞

‖u(t)‖Hs = +∞?

Polynomial upper bounds on the growth obtained by Bourgain, Staffilani,Sohinger.

Theorem (CKSTT; Inventiones 2008)

Let s > 1 and d > 2. For any δ 1 and K 1, there exists a solutionu(t) of cubic NLS on Td and a time T such that

‖u(0)‖Hs 6 δ but ‖u(T )‖Hs > K .

See also [Guardia-Kaloshin], [Haus-Procesi], [Guardia-Haus-Procesi].



Infinite growth on Td?

Going from such a finite growth result to infinite growth requiresperforming intricate constructions near very large initial data, whereone does not have much control on all the possible interactions.

It turns out that if one removes some unmanageable interactions fromNLS, such constructions can be done.

Theorem (H. ARMA 2012)

There exists solutions to the resonant NLS (RNLS) on Td (d > 2) thatexhibit infinite growth of high Sobolev norms. The same is true for afamily of systems approximating (NLS) arbitrarily closely.

Bourgain constructed earlier some examples of artificial nonlinearitiesthat exhibit infinite growth. Also, see Gerard-Grellier for genericinfinite growth of Szego.

For the cubic NLS equation on Td , the infinite cascade questionremains open.



Unbounded orbits for cubic NLS on R× Td

(i∂t + ∆)u = λ|u|2u, u|t=0 = u0, t ∈ R, x ∈ R× Td

Theorem (H., Pausader, Tzvetkov, Visciglia)

Let us fix d > 2 and s > 30. For any ε > 0, there exists u0 ∈ Hs(R× Td)with ‖u0‖Hs(R×Td ) < ε such that the corresponding solution of (NLS)satisfies

lim supt→∞

‖u(t)‖Hs = +∞.

Moreover, there exists a sequence tk such that tk →∞ and

‖u(tk)‖Hs(R×Td ) & exp(c(log log tk)1/2).

Such growth cannot happen for d = 0, 1.



One quick word about the proof

Main new component. Precise description of the asymptotic behavior:

Theorem (Modified scattering to resonant dynamics; HPTV)

Solutions u(t) of NLS on R× Td (initially small in some norm S+)exhibits modified scattering to the dynamics of its resonant system in thefollowing sense: there exists a solution G (t) of that system such that

e−it∆U(t)→ G (π log t) as t → +∞.

Remark: Smallness of initial data and convergence occur in Banachspaces containing HN . Modification to the scattering operator is nota simple phase correction.

Surprise! Solution of the resonant NLS on Td can be embedded(lifted) into solutions of the resonant system on R× Td . Now, use[H., 2012] to conclude (to get explicit rate, revisit using[Guardia-Kaloshin] tools).


Wave turbulence theory

Outline

1 Introduction: The general problematic







Peierls (1929), Hasselman (1962), Zakharov (1966), Benney (1969)...

Basic idea: Derive effective equations for the dynamics to understandthe collective behavior of the frequency modes aK (t) (K ∈ Z2

L).

Fundamental Equation is the Wave Kinetic Equation (WKE) effective dynamics for |aK (t)|2. There exists n(K , t) : R2 → R+

(wave spectrum) satisfying

∂tn(K ) =

∫∫∫n(K1)n(K2)n(K3)n(K )

(1

n(K1)− 1

n(K2)+

1

n(K3)− 1

n(K )

)δ(K1 − K2 + K3 − K )δ(|K1|2 − |K2|2 + |K3|2 − |K |2)dK1dK2dK3.

1 Equilibrium Stationary solutions: a.k.a. Raleigh-Jeans spectran(K ) = |K |−2 and n(K ) = 1.

2 Non-equilibrium stationary solutions: KZ spectra (Zakharov ’66) fordirect and inverse cascade.



BUT...

Problem: The theory is not rigorously justified yet, despite empiricalsuccesses (e.g. oceanography)! This prevents from making anybackward logical conclusions from (WKE) to the original dispersivesystem (NLS here).

Formal derivation: Start with random initial data, so aK (0) areindependent random variable (K ∈ Z2

L).

To find the equation for n(K , t) := E|aK (t)|2 take ∂t and use NLS E(aK1aK2aK3aK4)!! a very complicated infinite hierarchy.

To get a closed equation for n(K , t), the following limits/assumptionsare made in a rather cavalier way:

1 Phase and amplitude (quasi-)randomness (a.k.a propagation of chaos).2 Large-box limit (L→∞), (a.k.a. thermodynamic limit).3 weak-nonlinearity limit (ε→ 0), (a form of time averaging).


Mathematical Attempts to justify wave turbulence

Outline

1 Introduction: The general problematic






First attempt: Thermodynamics limits (Faou-Germain-H.)

In a first attempt to understand some of the limits performed in waveturbulence closures, we can start by rigorously taking the weaklynonlinear ε→ 0 and thermodynamic limit (L→∞) of NLS.

Another motivation is trying to understand the different effectivedynamics sustained by NLS.

We derive this equation rigorously and obtain quantitative estimateson its relevance for the NLS dynamics in the thermodynamic limit[F-G-H].



Sketch of [F-G-H]: First argue formally

ε→ 0 limit: Approximate the NLS dynamic with that of (RNLS).This is done by a normal forms transformation. Upshot is

i∂taK“ = ”ε2

L4

∑(K1,K2,K3)∈RK

aK1(t)aK2(t)aK3(t)

R(K ) =(K1,K2,K3) ∈ Z2L : K1 − K2 + K3 = K ,

Ω := |K1|2 − |K2|2 + |K3|2 − |K |2 = 0.

L→∞ limit: Understand equidistribution of lattice points on thevariety Ω = 0. We start by reparametrizing: Let (K1,K2,K3) ∈ R(K ). Set Ni = Ki − K (i = 1, 2, 3) N2 = N1 + N3 & |N2|2 = |N1|2 + |N3|2 ⇒ N1 ⊥ N3.



Parametrization of rectangles in Z2/L

i∂taK (t) =ε2

L4

∑N1,N3∈Z2

LN1⊥N3

aK+N1(t)aK+N1+N3(t)aK+N3(t)

N1 = α(p, q)/L with α ∈ N and (p, q) ∈ Z2 satisfyingg. c. d(|p|, |q|) = 1. Then N3 = β(−q, p)/L for some β ∈ Z.

A lattice point J ∈ Z2L is called visible if J = (p, q)/L with

g. c. d(|p|, |q|) = 1.

Writing N1 = αJ and N3 = βJ⊥, with J visible, one obtains



Resonant NLS in new coordinates

i∂ta(K ) =ε2

L4

∑α∈N,β∈Z

∑J∈Z2

Lvisible

a(K +

N1︷︸︸︷αJ )a(K +

N3︷︸︸︷βJ⊥)a(K +

N2︷︸︸︷αJ + βJ⊥)

Passing to the large box limit (L→∞) amounts to replacing theabove sums by integrals.

To do this we need information about the equidistribution of visiblelattice points+quantitative error estimates.



Co-prime equidistribution

Equidistribution: L−2∑

K∈Z2Lu(K )→

∫R2 u(z)dz as L→∞ provided

say that u is sufficiently well-behaved (like u ∈ L1, ∇u ∈ L1).

Key point: Density of visible lattice points in Z2L is 1

ζ(2) = 6π2 . I.e.

L−2#J ∈ Z2L ∩ Ω : J visible → Vol Ω

ζ(2) as L→∞ (classical).

Proposition (Co-prime equidistribution)

Suppose that u is sufficiently nice (say |u|+ |∇u| ∈ 〈K 〉−2−δL∞(R2)),then for L 1∣∣∣∣∣∣∣∣L

−2∑

J∈Z2/LJ visible

u(J)− 1

ζ(2)

∫R2

u(z)dz

∣∣∣∣∣∣∣∣ = O(log L

L), ζ(2) =

π2

6.



Continuum limit

Using this info., we get (pretending that aK is smooth in K !)

i∂ta(K , t)“ = ”1

T ∗

∫ 1

−1

∫R2

a(K + λz)a(K + λz + z⊥)a(K + z⊥)dz dλ

where T ∗def= ζ(2)L2

2ε2 log L∼ L2

ε2 log L( ε−2!).

Reparametrizing time t = T ∗τ , we get formally that

i∂τa(K , τ)“ = ”

∫ 1

−1

∫R2

a(K + λz)a(K + λz + z⊥)a(K + z⊥) dz dλ.



Continuum limit

In the upshot, we get (first at a formal level)

i∂ta(K , t) =ε2

L4

∑(K1,K2,K3)∈RK

aK1(t)aK2(t)aK3(t)

L→∞−→ 1

T ∗

∫ 1

−1

∫R2

a(K + λz)a(K + λz + z⊥)a(K + z⊥)dz dλ

where T ∗def= ζ(2)L2

2ε2 log L∼ L2

ε2 log L( ε−2!).

Reparametrizing time t = T ∗τ , we get formally that

i∂τa(K , τ) =

∫ 1

−1

∫R2

a(K + λz)a(K + λz + z⊥)a(K + z⊥) dz dλ.



The Continuous Resonant equation (CR)

i∂tg(ξ, t) =T (g , g , g)(ξ, t); ξ ∈ R2

T (g , g , g)(ξ, t) =

∫ 1

−1

∫R2

g(ξ + λz , t)g(ξ + λz + z⊥)g(ξ + z⊥) dz dλ.

(CR)

Here g : Rt × R2ξ → C. The equation seems to be new.

It is Hamiltonian (like NLS):

H(g) =1

2

∫ 1

−1

∫R2ξ×R2

z

g(ξ)g(ξ + λz)g(ξ + λz + z⊥)g(ξ + z⊥) dz dλ dξ

=1

2

∫Rs

∫R2x

|e is∆R2g(x)|4dsdx → L4t,x Strichartz norm!



Fourier transform

Theorem (Invariance under Fourier transform)

If g(t) is a solution of (CR), the so is g(t) := Fg(t). Moreover,

H(f ) = H(f ) for any function f ∈ L2.



Invariance of Harmonic oscillator eigenspaces

The quantum harmonic oscillator H = −∆ + |x |2 admits anorthonormal basis of eigenvectors for L2(R2).

The eigenspaces Ek correspond to the eigenvalue 2k (k = 1, 2, ...).They are k-dimensional and are spanned by k − th order Hermite

functions (e.g. E0 = Spane−|x|2

2 ).

Theorem

The spaces Ek are invariant by the nonlinear flow of (CR), i.e. if g0 ∈ Ek ,then g(t) ∈ Ek for all t ∈ R.



Solutions

(CR) is globally regular and has many explicit stationary solutions.

Gaussian family: For any α > 0,

g(t, z) =e iπ2

2te−

π2|z|2 for any α > 0, z ∈ R2

g(t, z) =ei π

2

2(2n)!

4n(n!)2 tzne−π2|z|2 , z ∈ R2 = C, n > 0.

Many other interesting dynamics were later discovered and studiedboth deterministically and probabilistically. See recent works[Germain, H., Thomann, 2015].

“Raleigh-Jeans” solution

g(t, ξ) =e ict

|ξ|solves (CR) corresponds to n(ξ) = |ξ|−2of (WKE).



Rigorous approximation resultTheorem (Faou-Germain-H.)

Let g(t, ξ) be a sufficiently “nice”a solution of the (CR) equation on aninterval [0,M](M arbitrary). Suppose we start with an NLS solution suchthat aK (0) = g0(K ). If L is large enough, and if ε is sufficiently small,

‖aK (t)− g(t

T ∗,K )‖X 6 C (log L)−1 for all 0 6 t 6 MT ∗ .

aenough to have |g0|+ |∇g0| . |ξ|−2−κ.

On the unit torus T2, (CR) becomes the equation for high-frequencyenvelopes for NLS.

Proof combines tools from 1) Dynamical systems (normal forms), 2)analytic number theory (Mobius inversion formula), and 3) harmonicanalysis (sharp periodic Strichartz estimates).



Estimates on resonant sums

∥∥∥∥∥∥∑R(K)

aK1 bK2 cK3

∥∥∥∥∥∥X (Z2

L)

6 C (L)‖aK‖X (Z2L)‖bK‖X (Z2

L)‖cK‖X (Z2L) (*)

Formal argument gives that C (L) ∼ L2 log L if aK, bK, cK are“smooth”.

If X = 〈K 〉−σ`2L (Sobolev space), (*) is equivalent to the (still open!)

∥∥∥e it∆T2PNφ∥∥∥L4t,x ([0,1]×T2)

???︷︸︸︷6 C (logN)1/4‖φ ‖L2(T2).

We prove (*) in the space X σ = 〈K 〉−σ`∞ for σ > 2 with the sharpconstant L2 log L. Corollary: This corresponds to a periodicStrichartz-type estimates at critical L2−scaling.



An independent derivation from Hermite-Schrodinger(H.-Thomann)

(CR) also appears naturally as an asymptotic system: Consideri∂tu +Hptu = ±|u|2uHpt = −∆R3 + |x ′|2, (x1, x

′) ∈ R× R2.(HS)

Cigar-shaped trap: This model is at the basis of experimentalobservations of dark solitons in Bose-Einstein condensates (cf.Kevrekidis et al.).

Theorem (H.-Thomann 2014)

Solutions of (HS) with initial data u0 (6 ε0 in some Banach space)exhibit modified scattering to the dynamics of (CR) in the following sense:there exists a solution g(t) of “ (CR)” such that

e−itHptu(t)→ g(π log t) as t → +∞.



Remarks

This theorem allows to translate any interesting dynamics of (CR)into that for (HS), like stationary solutions. Interestingly, physicistsarrived at some of those solutions using different heuristic arguments(Bose-Einstein vortices zne−

π2|z|2). (CR) gives a rigorous derivation

which allows to study the dynamics many vortices. Also leads toother stationary solutions.

With J. Shatah, we study the derivation of analogues of (CR) forother dispersive equations (higher-degree nonlinearities and higherdimensions). There, one needs the Hardy-Littlewood circle method tohandle the number theoretic aspect.



Thanks!

Thank you for your attention!


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