two dimensional water waves in holomorphic coordinateslmichel/confgilles/slides/tataru.pdfform...

Two dimensional water waves in holomorphiccoordinates

Daniel Tataru

University of California, Berkeley

Nice, June 2014

This is joint work with Mihaela Ifrim and, in part, with John Hunter

D. Tataru (UC Berkeley) 2-d water waves Nice, June 2014 1 / 30

Two dimensional water-waves

The setting:

inviscid incompressible fluid flow (Euler equations)

infinite bottom

free boundary

irrotational

with gravity or surface tension

periodic or nonperiodic setting

Question:

Study long time solutions (i.e. the stability of the trivial steady state)in two settings:

lifespan bounds for small data

global solutions for small localized data


The standard formulation

Fluid domain: Ω(t), free boundary Γ(t).Velocity field u, pressure p, gravity g, surface tension σ.Euler equations in Ω(t):

ut + u · ∇u = ∇p− gjdiv u = 0

curl u = 0

u(0, x) = u0(x)

Boundary conditions on Γt:∂t + u · ∇ is tangent to

⋃Γt (kinematic)

p = −2σH on Γt (dynamic)

H = mean curvature of the boundary.


Irrotational flows

Velocity potential

u = ∇φ, ∆φ = 0 in Ωt

Dynamic boundary condition:

φt +1

2|∇φ|2 + y = 0 on Γt

Equations reduced to the boundary in Eulerian formulation:∂tη −G(η)φ = 0

∂tφ+ gη − σH(η) +1

2|∇φ|2 − 1

2

(∇η∇ψ +G(η)ψ)2

1 + |∇η|2= 0.

H(η) = div

(∇η

1 + |∇η|2

), G(η) = Dirichlet to Neuman operator


Overview:

Features:

Hamiltonian structure

dispersive character

few resonant interactions (g/t)

symmetries: translation, scaling (g/t)

gauge freedom = choice of coordinates (e.g. Euler or Lagrange)

Difficulties:

quasilinear evolution

degenerate higher order hyperbolic evolution

nonlocal equation

quadratic interactions

many nonresonant interactions (g+t)

solitons (g+t)


Overview:Earlier work:

Local well-posedness in Sobolev spaces: Nalimov ’74, Yoshihara,Wu, Lannes, Coutand-Shkoller, Lindblad, Shatah-Zeng ...Alazard-Burq-Zuily.

Long time solutions for small localized data (g): Wu ’09 (almostglobal) Ionescu-Pusateri ’13, (global), Alazard-Delort ’13 (global)

Global solutions for small localized data in 3-d:Germain-Masmoudi-Shatah ’10(g) , ’12 (t), Wu ’10 (g).

Our goals:

To use the formulation of the equations in holomorphiccoordinates (Nalimov ’74) in order to provide a simpler approachto the local problem

To introduce a modified energy method, improving on the normalform method, which yields an easier route to long time solutions

To develop a more robust and simpler way, based on wave-packets,in order to obtain global solutions.


Holomorphic coordinates:Holomorphic coordinates:

Z : =z ≤ 0 → Ωt, α− iβ → Z(α+ iβ)

Boundary condition at infinity:

Z(α)− α→ 0 (nonperiodic) Z(α)− α periodic (periodic)

Free boundary parametrization:

Z : R→ Ωt, α→ Z(α)

Perturbation of steady state:

W = Z − α

Holomorphic velocity potential (v = ∇φ):

Q = φ+ iψ


Water wave equations in holomorphic coords.

P - Projection onto negative wavenumbers

Equations for (W = Z − α,Q):Wt + F (1 +Wα) = 0,

Qt + FQα + P

[|Qα|2

J

]− igW+ iσP

[Wαα

J1/2(1 +Wα)− Wαα

J1/2(1 + Wα)

]= 0.

where

F = P

[Qα − Qα

J

], J = |1 +Wα|2.

Conserved energy (Hamiltonian):

E(W,Q) =

∫=(QQα) +

1

2g(|W |2 −<(W 2Wα)

)+

1

4σ(J

12 − 1−<Wα) dα

Symmetries:

Translations in α and t.

Scaling (W (t, α), Q(t, α))→ (λ−2W (λt, λ2α), λ−3Q(λt, λ2α))


Physical parameters:

R =Qα

1 +Wα- velocity field on the free boundary

b = 2<P[

R

1 + Wα

]- advection coeff. (high freq. velocity limit)

a = 2=P [RRα] > 0 - normal derivative of the pressure = g + a

Alternate quasilinear system for diagonal variables (W = Wα, R):Wt + P [bWα] + P

[(1 + W)Rα

1 + W

]= G

Rt + P [bRα] + iP

[g + a

1 + W

]+ iσ

1

1 + WP

[Wα

J1/2(1 + W)

]α

= K,

where

G = (1 + W)P

[Rα

1 + W+

RWα

(1 + W)2

]+ [P,W]

(Rα

1 + W+

RWα

(1 + W)2

)K = − P

[RRα

]+ iσP

[Wα

J1/2(1 + W)

]α

represent perturbative terms in the equation.D. Tataru (UC Berkeley) 2-d water waves Nice, June 2014 9 / 30

Cubic lifespan bounds

Theorem

Consider the two dimensional differentiated water wave equation withinitial data in (W0, R0) ∈ H1 of size ε,

‖(W0, R0)‖H0∩H1 . ε (gravity waves)

respectively

‖(W0, R0)‖H0∩H2 . ε (capillary waves)

Then the solutions have a lifespan of at least

Tε ≈ ε−2

Proof idea: quasilinear modified energy method

Bounds for these and higher norms propagate on same timescale.

The result applies both in the periodic and nonperiodic setting.


Normal forms and long time existenceQuestion: Find improved lifespan estimates for small data solutions.

(i) Equations with quadratic nonlinearities:

d

dtE(u) . ‖u‖E(u)

For data ‖u(0)‖ = ε 1 this leads by Gronwall to a lifespan Tε ≈ ε−1

(ii) Equations with cubic nonlinearities:

d

dtE(u) . ‖u‖2E(u)

For data ‖u(0)‖ = ε 1 this leads by Gronwall to a lifespan Tε ≈ ε−2

(iii) Normal form method (Shatah ’85): transform an equation with aquadratic nonlinearity into one with a cubic one via a normal formtransformation,

u→ v = u+B(u, u) + higher


Dispersive character (gravity waves)Linearization around zero:

∂tw + rα = 0

∂tr − igw = 0

Energy:‖(w, r)‖2H0 = g‖w‖2L2 + ‖r‖2

H12

Dispersion relation:τ = ±

√g|ξ|, ξ ≤ 0

Group velocity of waves:

v = ±√g

2√|ξ|

Quasilinear model: (∂t + b∂α)w + rα = 0

(∂t + b∂α)r − i(g + a)w = 0


Dispersive character (gravity waves)Linearization around zero:

∂tw + rα = 0

∂tr − igw = 0

Energy:‖(w, r)‖2H0 = g‖w‖2L2 + ‖r‖2

H12

Dispersion relation:τ = ±

√g|ξ|, ξ ≤ 0


v = ±√g

2√|ξ|

Bilinear resonant interactions:

|ξ|12 ± |η|

12 ± |ξ + η|

12 = 0 =⇒ ξη(ξ + η) = 0

No resonances away from frequency zero !D. Tataru (UC Berkeley) 2-d water waves Nice, June 2014 13 / 30

Normal forms for gravity water waves

Existence of a normal form transformation is related to the absence ofresonant bilinear interactions. For 2-d gravity waves in holomorphiccoordinates, such a normal form transformation exists and is given by

W = W − 2M<WWα,

Q = Q− 2M<WQα,

The normal variables solve an equation of the form∂tW + Qα = cubic and higher

∂tQ− iW = cubic and higher

However, the cubic and higher nonlinearities also contain higherderivatives, so one cannot close the energy estimates. This is related tothe fact that the normal form transformation is not invertible, andfurther to the fact that the water wave equation is quasilinear, ratherthan semilinear.


Dispersive character (capillary waves)Linearization around zero:

∂tw + rα = 0

∂tr + iσwαα = 0

Energy:‖(w, r)‖2H1 = σ‖wα‖2L2 + ‖r‖2

H12

Dispersion relation:

τ = ±√σ|ξ|

32 , ξ ≤ 0


v = ±3

2

√σ|ξ|

Quasilinear model: (∂t + b∂α)w + rα = 0

(∂t + b∂α)r − iσJ−12wαα = 0


Dispersive character (capillary waves)Linearization around zero:

∂tw + rα = 0

∂tr + iσwαα = 0

Energy:‖(w, r)‖2H1 = σ‖wα‖2L2 + ‖r‖2

H12

Dispersion relation:

τ = ±√σ|ξ|

32 , ξ ≤ 0


v = ±3

2

√σ|ξ|

Bilinear resonant interactions:

|ξ|32 ± |η|

32 ± |ξ + η|

32 = 0 =⇒ ξη(ξ + η) = 0

No resonances away from frequency zero !D. Tataru (UC Berkeley) 2-d water waves Nice, June 2014 16 / 30

Normal forms for capillary water waves

Existence of a normal form transformation is related to the absence ofresonant bilinear interactions. For 2-d capillary waves in holomorphiccoordinates, such a normal form transformation exists and is given by

W = W + L1(W,W ) + L0(Q,Q)

Q = Q+ L1(W,Q)

where L0, L1 stand for bilinear paraproduct type operators of order0, 1. The normal variables solve an equation of the form

∂tW + Qα = cubic and higher

∂tQ− iW = cubic and higher

as before. Again, the cubic and higher nonlinearities also containhigher derivatives, so one cannot close the energy estimates.


The modified energy methodIdea: Modify the energy rather than the equation in order to get cubicenergy estimates.

Step 1: Construct a cubic normal form energy

EnNF (W,Q) = (quadratic+ cubic)(‖W (n)‖2L2 + ‖Q(n)‖2H

12)

Thend

dtEnNF (W,Q) = quartic+ higher

Here higher derivatives arise on the right, making it impossible to close.

Step 2: Switch EnNF (W,Q) to diagonal variables EnNF (W, R).

Step 3: To account for the fact that the equation is quasilinear,replace the leading order terms in EnNF (W, R) with their naturalquasilinear counterparts to obtain a good cubic quasilinear energyEn(W, R).D. Tataru (UC Berkeley) 2-d water waves Nice, June 2014 18 / 30

Energy estimates for gravity waves

We use the control norms:

A = ‖W‖L∞ + ‖W(1 + W)−1‖L∞ + ‖D12R‖L∞ , (scale invariant)

B = ‖D12W‖BMO + ‖Rα‖BMO (controlled by ‖(W, R)‖H1)

and construct energy functionals En(W, R) with these properties:

(i) Energy equivalence:

En(W, R) = (1 +O(A))‖(W, R)‖2Hn

(ii) Cubic scale invariant energy estimate:

d

dtEn(W, R) .A ABE

n(W, R)

Similar H0 bound for linearized eqn (w, r) = (w, q −Rw).


Energy estimates for capillary waves

We use the control norms:

A = ‖W‖L∞ + ‖W(1 + W)−1‖L∞ + ‖D−12R‖L∞ , (scale invariant)

B = ‖D32W‖BMO + ‖Rα‖BMO (controlled by ‖(W, R)‖H2)

and construct energy functionals En(W, R) with these properties:

(i) Energy equivalence:

En(W, R) = (1 +O(A))‖(W, R)‖2Hn

(ii) Cubic scale invariant energy estimate:

d

dtEn(W, R) .A ABE

n(W, R)


Water waves on the real line: heuristics #1Dispersive decay for the linear equation

Wt +Qα = 0

Qt − iW = 0

with smooth localized data of size ε:

|W |+ |Q| . εt−12

This shows that one would expect our control norms to decay like

A,B ≈ εt−12

Henced

dtE . ε2t−1E

which provides uniform bounds up to an exponential time

Tε = ecε−2


Water waves on the real line: heuristics #2Wt + Qα = cubic(W , Q) + higher

Qt − iW = cubic(W , Q) + higher

Based on the dispersion relation and the group velocity of waves

τ = ±√|ξ|, v = ± 1

2√|ξ|

for small data localized near 0, one expects asymptotics

(W , Q) ≈ γ(t, α/t)t−12 e

it2

4α

(t

2α, 1

)with a better function γ. Substituting in the above equations, we getthe asymptotic evolution

γ(t, v) = ct−1γ(t, v)|γ(t, v)|2 +O(εt−1−), γ(1, α) = O(ε)

The global boundedness of gamma follows.D. Tataru (UC Berkeley) 2-d water waves Nice, June 2014 22 / 30

New tool: scaling symmetryScaling symmetry for gravity waves:

(W (t, α), Q(t, α))→ (λ−2W (λt, λ2α), λ−3Q(λt, λ2α))

Generator:

S(W,Q) = ((S − 2)W, (S − 3)Q), S = t∂t + 2α∂α

S(W,Q) solves the linearized equation, good energy estimates.

Scaling symmetry for capillary waves:

(W (t, α), Q(t, α))→ (λ−1W (λ32 t, λα), λ−

12Q(λ

32 t, λα)).

Generator:

S(W,Q) = ((S − 1)W, (S − 1

2)Q), S =

3

2t∂t + α∂α.

S(W,Q) solves the linearized equation, good energy estimates.D. Tataru (UC Berkeley) 2-d water waves Nice, June 2014 23 / 30

Global gravity waves for small localized dataWeighted function space WH for localized data:

‖(W,Q)(t)‖2WH = ‖(W,Q)‖2H0 + ‖(W, R)‖2H6 + ‖AS(W,Q)‖2H0

Theorem

Assume that the initial data for the water wave equation satisfies:

‖(W,Q)(0)‖WH . ε

Then the solution exists globally in time, with energy bounds

‖(W,Q)(t)‖WH . εtCε2

and pointwise decay

A(t) +B(t) .ε√t

Original result by Ionescu-Pusateri and Alazard-Zuily (with longerproofs and with more regularity)

.D. Tataru (UC Berkeley) 2-d water waves Nice, June 2014 24 / 30

Global capillary waves for small localized data

Weighted function space WH for localized data:

‖(W,Q)(t)‖2WH = ‖(W,Q)‖2Hδ∩H10 + ‖S(W,Q)‖2Hδ∩H1

Theorem

Assume that the initial data for the water wave equation satisfies:

‖(W,Q)(0)‖WH . ε

Then the solution exists globally in time, with energy bounds

‖(W,Q)(t)‖WH . εtCε2

and pointwise decay

C(t) := ‖(D12W,Q)‖W 4,∞ .

ε√t


Bootstrap argument for gravity waves:1. Energy estimates:

‖(W,Q)(t)‖WH . e∫ T0 AB(s)ds‖(W,Q)(0)‖WH . εtCε

2

based on the cubic energy functionals

2. Initial pointwise bounds:

(A+B)(t) . t−12ω(t, α/t)‖(W,Q)(t)‖WH, ω(t, v) = t−

118 +(v+v−1)−

12

akin to the Klainerman-Sobolev inequalities, only harderuses the normal form variablesbetter decay for v away from 1, reduces problem to t−δ < v < tδ.

3. Final pointwise bounds:

A(t) +B(t) . εt−12

Uses the method of testing with wave-packets to derive anasymptotic equation.


Bootstrap argument for capillary waves

All is similar except for the energy estimates, which are done in twoways, both for the full equation and for the linearization:

1. High frequency energy estimates (e.g. in H10)

Uses the modified energy metod to obtain a cubic energyfunctional.

2. Low frequency energy estimates (i.e. in Hδ)Uses the normal form metod to obtain a cubic energy functional.

Not all cubic terms in the normal form equation are good, but thebad ones are nonresonant and can be eliminated with a furthercubic normal form correction.


Asymptotic equations in NLS context

A. Hayashi-Naumkin, refined by Kato-Pusateri; derive an asymptoticequation for the Fourier transform of the solutions,

d

dtu(t, ξ) = λit−1 u(t, ξ)|u(t, ξ)|+OL∞(t−1−ε).

B. Lindblad-Soffer; derive an asymptotic equation in the physicalspace along rays,

(t∂t + x∂x)u(t, x) = λitu(t, x)|u(t, x)|2 +OL∞(t−ε).

C. Deift-Zhou used the inverse scattering method to obtain longrange asymptotics

D. Ifrim-T. wave packet testing: test the NLS solution with anapproximate wave packet type linear wave.


Testing by wave packets

Wave packets:

q = vχ

(v(α− vt)√

t

)eit2

4α , w = −iqt

The γ function:γ(t, v) = 〈(W , Q), (w,q)〉H0

Good approximation (also for derivatives):

(W , Q) = γ(t, α/t)t−12 e

it2

4α

(t

2α, 1

)+O(t−

58 )

Asymptotic equation for γ:

γ(t, v) = ct−1γ(t, v)|γ(t, v)|2 +O(εt−1−), γ(1, α) = O(ε)


Happy birthday Gilles !


two dimensional water waves in holomorphic coordinateslmichel/confgilles/slides/tataru.pdfform...

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