the water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf ·...

Introduction Formulation The Model Problem The Full Problem

The water-waves problem with surface tension

H. Christianson (joint work with V. Hur (UIUC) and G. Staffilani(MIT))

Department of MathematicsMassachusetts Institute of Technology


Outline

1 IntroductionGoals of talkWhat is the water-waves problem?History

2 FormulationDispersive EquationsWater-waves as a dispersive equation

3 The Model ProblemEnergy estimatesDispersion Estimate

4 The Full ProblemEnergy estimates and LWPMicrolocal DispersionMain Results


Goals

Inspiration?

Figure: Author: Mila,http://home.comcast.net/∼milazinkova/Fogshadow.html


Goals

Inspiration??

... (water waves), which are easily seen by everyone andwhich are used as an example of waves in elementarycourses... are the worst possible example.... They have allthe complications that waves can have.

Richard Feynman1

1R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics,Addison-Wesley, 1963, Section 51-4.


Goals

An old topic


Goals

Goals of this talk

Describe water-waves problem


Goals

Goals of this talk


Brief historical account


Goals

Goals of this talk



Recast problem as a dispersive equation


Goals

Goals of this talk




Dispersive equation can be approached by energy methods andFourier transform methods


Goals

Goals of this talk




Dispersive equation can be approached by energy methods andFourier transform methods

Statement of results


Water-waves

The Problem

Incompressible Euler equations in 2d:{∂t u + (u · ∇)u = −∇p + (0,−g),

∇ · u = 0.


Water-waves

The Problem


∇ · u = 0.

u(x , y , t) is velocity field, p(x , y , t) is pressure, g ≥ 0 is gravity


Water-waves

The Problem


∇ · u = 0.


Assume irrotational:∇× u = 0


Water-waves

The Problem


∇ · u = 0.


Assume irrotational:∇× u = 0

All the “action” occurs on the surface


Water-waves

The Picture

air

fluid

x

y

y = η(t , x)

Figure: The “free boundary” setup


Water-waves

Boundary Conditions I

Assume interface moves with the particle velocity:

u · n = 0, n normal vector


Water-waves




Assume pressure “jump” at interface is proportional to meancurvature:

[p] = Sκ, κ curvature


Water-waves




Assume pressure “jump” at interface is proportional to meancurvature:

[p] = Sκ, κ curvature

S ≥ 0 is surface tension.


Water-waves

Boundary Conditions II

Assume not much happens at large depths:

u → 0 as |(x , y)| → ∞


Water-waves

Boundary Conditions II

Assume not much happens at large depths:

u → 0 as |(x , y)| → ∞

Assume flat surface at infinity

y → 0 as |x | → ∞


Water-waves

The guiding principle

Water waves are affected by two things, gravity and surface tension.


Water-waves


Water waves are affected by two things, gravity and surface tension.From a distance, you don’t see the surface tension


Water-waves


Water waves are affected by two things, gravity and surface tension.From a distance, you don’t see the surface tensionOn a small scale, surface tension is non-negligible (for us S > 0) andhelps “regularize” the solution.


Water-waves


Water waves are affected by two things, gravity and surface tension.From a distance, you don’t see the surface tensionOn a small scale, surface tension is non-negligible (for us S > 0) andhelps “regularize” the solution.Can this be made rigorous?


History

A little history I

Early history is patchy:

Euler (1757,1761) hydrodynamic equations


History

A little history I



Laplace (1776) static surface tension


History

A little history I




Lagrange (1781, 1786) velocity potential


History

A little history I





Cauchy (1815) and Poisson (1816) studied as an initial valueproblem


History

A little history I






Weber-Weber (1825) Careful experimental data


History

A little history I






Weber-Weber (1825) Careful experimental data

Airy (1841) linearized gravity waves, ...


History

A little history II

More recently, the water-waves problem has been studied as an initialvalue problem


History

A little history II

More recently, the water-waves problem has been studied as an initialvalue problemWhat might we ask about an initial value problem?


History

A little history II

More recently, the water-waves problem has been studied as an initialvalue problemWhat might we ask about an initial value problem? Well-posedness:


History

A little history II


Existence of solution on some time interval (local or global intime)?


History

A little history II



Uniqueness of solution on the interval of existence?


History

A little history II




Continuous dependence on initial conditions?


History

A little history II





In addition, we may ask “what does the solution look like”?


History

A little history II





In addition, we may ask “what does the solution look like”? Properties:


History

A little history II






How regular is the solution compared to initial data?


History

A little history II






How regular is the solution compared to initial data?

If the solution exists for long times, what are the long-timeasymptotics?


History

A little history III

This problem is very nonlinear (quasi-linear) → most results so far arein local-well-posedness theory using sophisticated nonlinear energyestimates.


History


This problem is very nonlinear (quasi-linear) → most results so far arein local-well-posedness theory using sophisticated nonlinear energyestimates.S = 0 (gravity waves) hard!


History



small data in Sobolev spaces: Nalimov (’74), Yosihara (’82,’83),Craig (’85); analytic category: Kano-Nishida (’79),Sulem-Sulem-Bardos-Frisch (’81), many more...


History




Beale-Hou-Lowengrub (’93): linear problem well-posed ⇐⇒∇p · n < 0 (Taylor-Young inequality)


History




Beale-Hou-Lowengrub (’93): linear problem well-posed ⇐⇒∇p · n < 0 (Taylor-Young inequality)

Wu (’97): interface nonself-intersecting =⇒ Taylor-Younginequality holds


History

A little history IV

What about global/almost global well-posedness?


History

A little history IV

What about global/almost global well-posedness? Obviously muchharder! Not many results so far.


History

A little history IV


Wu (’09) almost global-well-posedness for 2d gravity waves


History

A little history IV


Wu (’09) almost global-well-posedness for 2d gravity waves

Germain-Masmoudi-Shatah (’09) and Wu (’09)global-well-posedness for 3d gravity waves


History

A little history V

S > 0 (our case) Taylor-Young inequality always holds


History

A little history V

S > 0 (our case) Taylor-Young inequality always holdsSurface tension acts as a regularizing force. What kind of qualitativeproperties can we prove?


History

A little history V


Spirn-Wright (’09) Dispersion for linearized water-waves problemwith surface tension


History

A little history V



C-Hur-Staffilani (’08-’09) 2d local-well-posedness, parametrixconstruction, Strichartz estimates, local smoothing


History

A little history V




Alazard-Burq-Zuily (’09) nd irregular bottomlocal-well-posedness, 2d local smoothing


History

A little history V




Alazard-Burq-Zuily (’09) nd irregular bottomlocal-well-posedness, 2d local smoothing

Note: Dispersive estimates from invariant vector field identitiesused by Wu in 2d AGWP and 3d GWP (’09); andGermain-Masmoudi-Shatah for 3d GWP (’09) (S = 0)


Dispersive Equations

What is a dispersive equation?

A dispersive equation means that a plane-wave solution to thisequation travels at different speeds depending on the spatialmomentum.




A dispersive equation means that a plane-wave solution to thisequation travels at different speeds depending on the spatialmomentum.A plane-wave is of the form ei(λt+xξ) (Fourier decomposition).




A dispersive equation means that a plane-wave solution to thisequation travels at different speeds depending on the spatialmomentum.A plane-wave is of the form ei(λt+xξ) (Fourier decomposition).Stationary phase shows this wave is traveling at speed ∂ξλ withmomentum ξ, as we will see in some examples next.



Schrödinger equation

The Schrödinger equation{

(i∂t − ∂2x )u = 0,

u(0, x) = u0(x),

has solution

u(t , x) = (2π)−1∫

eitξ2eixξu0(ξ)dξ,





(i∂t − ∂2x )u = 0,

u(0, x) = u0(x),

has solution

u(t , x) = (2π)−1∫


so λ(ξ) = ξ2.





(i∂t − ∂2x )u = 0,

u(0, x) = u0(x),

has solution

u(t , x) = (2π)−1∫


so λ(ξ) = ξ2.The phase function is ϕ = tξ2 + xξ, so the exponential oscillates a lotunless ∂ξϕ = 0, or the phase is stationary.





(i∂t − ∂2x )u = 0,

u(0, x) = u0(x),

has solution

u(t , x) = (2π)−1∫


so λ(ξ) = ξ2.The phase function is ϕ = tξ2 + xξ, so the exponential oscillates a lotunless ∂ξϕ = 0, or the phase is stationary.Main contribution to integral is when 2tξ + x = 0, so the speeddepends on ξ.



Transport equation

The transport equation{

(∂t − ∂x)u = 0,

u(0, x) = u0(x),

is not dispersive.



Transport equation


(∂t − ∂x)u = 0,

u(0, x) = u0(x),

is not dispersive.Solution is

u(t , x) = (2π)−1∫

eitξeixξu0(ξ)dξ,

so λ(ξ) = ξ.



Transport equation


(∂t − ∂x)u = 0,

u(0, x) = u0(x),

is not dispersive.Solution is

u(t , x) = (2π)−1∫

eitξeixξu0(ξ)dξ,

so λ(ξ) = ξ.Applying stationary phase idea gives

∂ξϕ = t + x = 0,

the speed of propagation is just −1.



Other dispersive equations

The wave equation (dimension n ≥ 2) (∂2t − ∆)u = 0 is degenerately

dispersive





dispersiveSolution is

u(t , x) = (4π)−n∫

ei(t|ξ|+x·ξ)(u0 +u1

i|ξ|) + ei(−t|ξ|+x·ξ)(u0 −

u1

i|ξ|)dξ






u(t , x) = (4π)−n∫


i|ξ|) + ei(−t|ξ|+x·ξ)(u0 −

u1

i|ξ|)dξ

but stationary phase implies ±t ξ|ξ| + x = 0, so propagation depends

on direction but not magnitude of momentum.






u(t , x) = (4π)−n∫


i|ξ|) + ei(−t|ξ|+x·ξ)(u0 −

u1

i|ξ|)dξ



KdV equation is dispersive






u(t , x) = (4π)−n∫


i|ξ|) + ei(−t|ξ|+x·ξ)(u0 −

u1

i|ξ|)dξ




Benjamin-Ono equation is dispersive






u(t , x) = (4π)−n∫


i|ξ|) + ei(−t|ξ|+x·ξ)(u0 −

u1

i|ξ|)dξ




Benjamin-Ono equation is dispersive

Heat equation is dissipative but not dispersive



Dispersive properties I

What do we gain from dispersion?




What do we gain from dispersion? Two very robust tools are energyestimates and dispersion estimates.




What do we gain from dispersion? Two very robust tools are energyestimates and dispersion estimates.Energy estimates:





For Schrödinger equation,

ddt

‖u(t , ·)‖2L2 = 0.





For Schrödinger equation,

ddt

‖u(t , ·)‖2L2 = 0.

For wave equation,

ddt

(‖∂x u(t , ·)‖2L2 + ‖∂tu(t , ·)‖2

L2) = 0.



Dispersive properties II

A common dispersion estimate is a time dependent L1 → L∞

estimate.





estimate.For Schrödinger equation, phase ϕ = tξ2 + xξ.





estimate.For Schrödinger equation, phase ϕ = tξ2 + xξ. If u0 is a reasonablefunction, stationary phase lemma =⇒ major contribution to integralis at ∂ξϕ = 0 with a prefactor of |∂2

ξϕ|−1/2 = (2t)−1/2, so we have the

dispersion estimate

|u(t , x)| ≤ Ct−1/2‖u0‖L1 .


Water-waves formulation

New Formulation I

Based on Ambrose-Masmoudi (’05).

2The Birkhoff-Rott integral arises as the interfacial limit of the Biot-Savart law torecover the velocity from the vorticity distribution.



New Formulation I

Based on Ambrose-Masmoudi (’05). Parameterize by arclengthα ∈ R. Evolution of a point (x(t , α), y(t , α)) is

(x , y)t = Un + T t




New Formulation I


(x , y)t = Un + T t

Renormalize arclength so |(x , y)α| = 1




New Formulation I


(x , y)t = Un + T t

Renormalize arclength so |(x , y)α| = 1 T is determined




New Formulation I


(x , y)t = Un + T t


Evolution of U is governed by Birkhoff-Rott integral2. U = W · n,where

W =

∫γ(t , α′)

z(t , α) − z(t , α′)dα′,

with z(t , α) = x(t , α) + iy(t , α).




New Formulation I


(x , y)t = Un + T t


Evolution of U is governed by Birkhoff-Rott integral2. U = W · n,where

W =

∫γ(t , α′)

z(t , α) − z(t , α′)dα′,

with z(t , α) = x(t , α) + iy(t , α).

Here γ is the vortex sheet strength, the amplitude of adistribution supported on the surface interface




New Formulation II

Introduce modified tangential velocity u (in terms of γ) and tangentialangle θ,



New Formulation II

Introduce modified tangential velocity u (in terms of γ) and tangentialangle θ, use Hilbert transform to approximate Birkhoff-Rott integral



New Formulation II

Introduce modified tangential velocity u (in terms of γ) and tangentialangle θ, use Hilbert transform to approximate Birkhoff-Rott integral equivalent system

{∂tu = S

2 ∂2αθ − gθ − u∂αu + r1(t , α),

∂tθ = −u∂αθ + H∂αu + r2(t , α).



New Formulation II


{∂tu = S

2 ∂2αθ − gθ − u∂αu + r1(t , α),

∂tθ = −u∂αθ + H∂αu + r2(t , α).

The rj are “nicer” than the explicit terms



New Formulation II


{∂tu = S

2 ∂2αθ − gθ − u∂αu + r1(t , α),

∂tθ = −u∂αθ + H∂αu + r2(t , α).

The rj are “nicer” than the explicit terms

H is Hilbert transform: Hf (ξ) = −isgn(ξ)f (ξ).



New Formulation III

Differentiate utt and back substitute



New Formulation III

Differentiate utt and back substitute second equivalent system:



New Formulation III


∂2t u − S

2 H∂3αu + gH∂αu = −2u∂t∂αu − u2∂2

αu + R(u, ∂t u)

u(0, α) = u0(α),

ut(0, α) = u1(α)

(2.1)



New Formulation III


∂2t u − S

2 H∂3αu + gH∂αu = −2u∂t∂αu − u2∂2

αu + R(u, ∂t u)

u(0, α) = u0(α),

ut(0, α) = u1(α)

(2.1)

R is again “controlled” by the explicit terms.



New Formulation III


∂2t u − S

2 H∂3αu + gH∂αu = −2u∂t∂αu − u2∂2

αu + R(u, ∂t u)

u(0, α) = u0(α),

ut(0, α) = u1(α)

(2.1)

R is again “controlled” by the explicit terms.Equation for θ is a nonlinear transport equation weakly coupled to(2.1).


Energy estimates

A Dispersive Equation!

Model case: take g = 0, S/2 = 1, all nonlinear terms zero:


Energy estimates



∂2t u − H∂3

αu = 0

u(0, α) = u0(α),

ut(0, α) = u1(α)

(3.1)


Energy estimates



∂2t u − H∂3

αu = 0

u(0, α) = u0(α),

ut(0, α) = u1(α)

(3.1)

Fourier transform of equation is (−τ2 + |ξ|3)u = 0, so dispersionrelation is τ = ±|ξ|3/2


Energy estimates



∂2t u − H∂3

αu = 0

u(0, α) = u0(α),

ut(0, α) = u1(α)

(3.1)

Fourier transform of equation is (−τ2 + |ξ|3)u = 0, so dispersionrelation is τ = ±|ξ|3/2

We will use both energy conservation and a dispersion estimate.


Energy estimates

Energy conservation

Energy conservation comes from multiplying equation by u andintegrating by parts:

E(t) :=

∫(|ut |

2 − uH∂3αu)dα


Energy estimates

Energy conservation


E(t) :=

∫(|ut |

2 − uH∂3αu)dα

satisfiesE ′ = 0


Energy estimates

Energy conservation


E(t) :=

∫(|ut |

2 − uH∂3αu)dα

satisfiesE ′ = 0

E controls 1 time derivative and 3/2 space derivative of u inL∞

t L2α


Energy estimates

Energy conservation


E(t) :=

∫(|ut |

2 − uH∂3αu)dα

satisfiesE ′ = 0

E controls 1 time derivative and 3/2 space derivative of u inL∞

t L2α

Shows ‖u(t)‖2H3/2

α

+ ‖ut(t)‖2L2

αis controlled by ‖u0‖

2H3/2 + ‖u1‖

2L2


Dispersion Estimate

Dispersion estimate I

Solve via Fourier transform:

u(t , α) =1

4π

∫ei(α−β)ξ

(eit|ξ|3/2

(u0(β) +

u1(β)

i|ξ|3/2

)

+ e−it|ξ|3/2

(u0(β) −

u1(β)

i|ξ|3/2

))dξdβ


Dispersion Estimate



u(t , α) =1

4π

∫ei(α−β)ξ

(eit|ξ|3/2

(u0(β) +

u1(β)

i|ξ|3/2

)

+ e−it|ξ|3/2

(u0(β) −

u1(β)

i|ξ|3/2

))dξdβ

If u0 and u1 have compact support in ξ > 0, compute dispersionestimate via stationary phase: We write u = u+ + u− with


Dispersion Estimate



u(t , α) =1

4π

∫ei(α−β)ξ

(eit|ξ|3/2

(u0(β) +

u1(β)

i|ξ|3/2

)

+ e−it|ξ|3/2

(u0(β) −

u1(β)

i|ξ|3/2

))dξdβ


u±(t , α) =

∫K±

0 (t , α, β)u0(β) + K±1 (t , α, β)u1(β)dβ.


Dispersion Estimate



u(t , α) =1

4π

∫ei(α−β)ξ

(eit|ξ|3/2

(u0(β) +

u1(β)

i|ξ|3/2

)

+ e−it|ξ|3/2

(u0(β) −

u1(β)

i|ξ|3/2

))dξdβ


u±(t , α) =

∫K±

0 (t , α, β)u0(β) + K±1 (t , α, β)u1(β)dβ.

K±0 (t , α, β) =

∫eiϕ±(t,ξ,α,β)

1suppu0dξ,

and similarly for K1.


Dispersion Estimate



u(t , α) =1

4π

∫ei(α−β)ξ

(eit|ξ|3/2

(u0(β) +

u1(β)

i|ξ|3/2

)

+ e−it|ξ|3/2

(u0(β) −

u1(β)

i|ξ|3/2

))dξdβ


u±(t , α) =

∫K±

0 (t , α, β)u0(β) + K±1 (t , α, β)u1(β)dβ.

K±0 (t , α, β) =

∫eiϕ±(t,ξ,α,β)

1suppu0dξ,

and similarly for K1. Here ϕ± = ±t |ξ|3/2 + (α − β)ξ.


Dispersion Estimate

Dispersion estimate II

Consider ϕ = ϕ+. Phase is stationary when ϕξ(t , α, β, ξc) = 0, or


Dispersion Estimate



∂ξϕ =32

tξ1/2c + α− β = 0, or ξc =

49

(β − α

t

)2

.


Dispersion Estimate



∂ξϕ =32

tξ1/2c + α− β = 0, or ξc =

49

(β − α

t

)2

.

Applying the method of stationary phase,


Dispersion Estimate



∂ξϕ =32

tξ1/2c + α− β = 0, or ξc =

49

(β − α

t

)2

.

Applying the method of stationary phase, we have

K0(t , α, β) = Ct−1/2 〈ξc〉1/4 + l.o.t.


Dispersion Estimate



∂ξϕ =32

tξ1/2c + α− β = 0, or ξc =

49

(β − α

t

)2

.

Applying the method of stationary phase, we have

K0(t , α, β) = Ct−1/2 〈ξc〉1/4 + l.o.t.

Recalling that on F.T. side, ξ ∼ Dα, we have (informally)

|K0(t , α, β)| ≤ Ct−1/2 〈Dα〉1/4

.


Dispersion Estimate

Dispersion implies Strichartz

Rescaling plus theorems of Ginibre-Velo (’92,’95) and Keel-Tao (’98)imply Strichartz estimates:

‖u‖Lp(T )W s−1/2p,qα

≤ C(‖u0‖Hs + ‖u1‖Hs−3/2). (3.2)

for2p

+1q

=12


Dispersion Estimate




≤ C(‖u0‖Hs + ‖u1‖Hs−3/2). (3.2)

for2p

+1q

=12

Of course the full, nonlinear problem is much harder...


Dispersion Estimate




≤ C(‖u0‖Hs + ‖u1‖Hs−3/2). (3.2)

for2p

+1q

=12

Of course the full, nonlinear problem is much harder...

Conjecture

The estimate (3.2) holds for the nonlinear problem and is sharp.


Outline of full problem

We want to prove Strichartz estimates for the nonlinear problem

Prove local well-posedness and regularity of solution





Linearize about a known solution






Construct approximate solution as oscillatory integral







Prove dispersion estimate







Prove dispersion estimate

Conclude Strichartz estimates.


Energy estimates and LWP

Local Well-posedness

Extend energy estimate to quasilinear case.




Extend energy estimate to quasilinear case.Introduce nonlinear “material derivative” instead of ∂t u:

v = ∂tu + u∂αu.

Rewrite energy in terms of v .






Rewrite energy in terms of v . A difficult integration by parts usingsymmetry yields similar energy estimates to model problem,






Rewrite energy in terms of v . A difficult integration by parts usingsymmetry yields similar energy estimates to model problem, at leastfor short times.






Rewrite energy in terms of v . A difficult integration by parts usingsymmetry yields similar energy estimates to model problem, at leastfor short times.

Theorem (LWP)

Let the surface tension coefficient S > 0 be fixed.For each s > 5/2 equation (2.1) is locally well posed in Hs × Hs−3/2

and for T > 0 sufficiently small

(u(t , ·), ut (t , ·)) ∈ C([0,T ]t ; Hs × Hs−3/2).


Microlocal Dispersion

Dyadic-frequency parametrix I

With a solution of known regularity, we linearize and try to improveregularity. Replace nonlinearities u with a known function V and lookat linear equation:

{Pu := (∂2

t − H∂3α + 2V∂t∂α + V 2∂2

α)u = R

u|t=0 = u0, ut |t=0 = u1.





{Pu := (∂2

t − H∂3α + 2V∂t∂α + V 2∂2

α)u = R

u|t=0 = u0, ut |t=0 = u1.

We want to solve the homogeneous equation Pu = 0





{Pu := (∂2

t − H∂3α + 2V∂t∂α + V 2∂2

α)u = R

u|t=0 = u0, ut |t=0 = u1.

We want to solve the homogeneous equation Pu = 0Make WKB type ansatz:





{Pu := (∂2

t − H∂3α + 2V∂t∂α + V 2∂2

α)u = R

u|t=0 = u0, ut |t=0 = u1.

We want to solve the homogeneous equation Pu = 0Make WKB type ansatz:

u =

∫e−iβξ

(eiϕ+

f +(β) + eiϕ−

f−(β))

dβdξ,



Dyadic-frequency parametrix II

ϕ± functions of (t , α, ξ)





ϕ±|t=0 = αξ (Fourier inversion)






f± chosen to satisfy initial conditions







Apply operator to ansatz to get two eikonal equations (one each for±):








ϕ±t = −V (t , α)ϕ±

α ± (ϕ±α )3/2, (4.1)








ϕ±t = −V (t , α)ϕ±

α ± (ϕ±α )3/2, (4.1)

ϕ is a perturbation of 0-coefficient part (αξ ± t |ξ|3/2)

ϕ± = αξ ± |ξ|3/2(t + θ±(t , α, ξ))








ϕ±t = −V (t , α)ϕ±

α ± (ϕ±α )3/2, (4.1)


ϕ± = αξ ± |ξ|3/2(t + θ±(t , α, ξ))

θ± only exists on timescales t ∼ |ξ|−1/2.








ϕ±t = −V (t , α)ϕ±

α ± (ϕ±α )3/2, (4.1)


ϕ± = αξ ± |ξ|3/2(t + θ±(t , α, ξ))

θ± only exists on timescales t ∼ |ξ|−1/2. partition of unity in ξ ∼ 2j

(Littlewood-Paley decomposition).



Dispersion Estimate I

Estimate integral kernel in dyadic frequency regions




Estimate integral kernel in dyadic frequency regionsWrite

u±(t , α) =

∫K±

0 (t , α, β)u0(β) + K±1 (t , α, β)u1(β)dβ,





u±(t , α) =

∫K±

0 (t , α, β)u0(β) + K±1 (t , α, β)u1(β)dβ,

K±0 (t , α, β) =

∫eiϕj,±(t,α,β,ξ)ψ(2−jξ)dξ





u±(t , α) =

∫K±

0 (t , α, β)u0(β) + K±1 (t , α, β)u1(β)dβ,

K±0 (t , α, β) =


with suppψ ∼ 1 and

ϕj,± = (α− β)ξ ± |ξ|3/2(t + θj,±)

(K±1 similarly..)





u±(t , α) =

∫K±

0 (t , α, β)u0(β) + K±1 (t , α, β)u1(β)dβ,

K±0 (t , α, β) =


with suppψ ∼ 1 and

ϕj,± = (α− β)ξ ± |ξ|3/2(t + θj,±)

(K±1 similarly..)

θ only exists for |t | ≤ 2−j/2 and |ξ| ∼ 2j



Dispersion Estimate II

Want to apply stationary phase, but ϕj,± not defined on fixed timescale.

Precise control of symbolic estimates on θ = O(t2 + t |ξ|−1/2)plus stationary phase implies

|K0| ≤ C2j/4t−1/2, |t | ≤ 2−j/2T






|K0| ≤ C2j/4t−1/2, |t | ≤ 2−j/2T

This does not work for a fixed time scale!






|K0| ≤ C2j/4t−1/2, |t | ≤ 2−j/2T

This does not work for a fixed time scale!

Rescaling plus theorems of Ginibre-Velo (’92,’95) and Keel-Tao(’98) imply

‖u‖Lp(2−j/2T )W s−1/2p,qα

≤ C(‖u0‖Hs + ‖u1‖Hs−3/2).

for2p

+1q

=12


Main Results

Strichartz Estimates

Taking the pth power and summing up over 2j/2 time slices yields amultiple of 2j/2p ∼ 〈Dα〉

1/2p on dyadic frequencies.


Main Results



1/2p on dyadic frequencies.Summing over frequencies using almost orthogonality ofLittlewood-Paley decomposition for q <∞ we get the followingTheorem.


Main Results



1/2p on dyadic frequencies.Summing over frequencies using almost orthogonality ofLittlewood-Paley decomposition for q <∞ we get the followingTheorem.

Theorem (Strichartz Estimates)

Let the surface tension coefficient S > 0 be fixed.If s ≫ 1 is sufficiently large and T > 0 is sufficiently small,

( ∫ T

0

(∫ ∞

−∞

|Ds−1/pα u(t , α)|qdα

)p/qdt)1/p

≤ C, (4.2)

holds, where2p

+1q

=12, q <∞, (4.3)

C > 0 depends only on T > 0 and the Sobolev norms of the initialdata in Hs × Hs−3/2.


Main Results

Perspectives I

Model problem:{

(∂2t − H∂3

α)U = 0U(0, α) = U0(α), Ut(0, α) = U1(α).

U satisfies scaling symmetry U(t , α) → λ1/2U(λ3/2t , λα), λ > 0.


Main Results

Perspectives I

Model problem:{

(∂2t − H∂3

α)U = 0U(0, α) = U0(α), Ut(0, α) = U1(α).

U satisfies scaling symmetry U(t , α) → λ1/2U(λ3/2t , λα), λ > 0.The estimate

‖Ds−1/2pα U‖Lp

T Lqα≤ C(‖U0‖Hs + ‖U1‖Hs−3/2)

is scale-invariant if (p, q) satisfies (4.3). For this reason, we refer tothis estimate as optimal.


Main Results

Perspectives I

Model problem:{

(∂2t − H∂3

α)U = 0U(0, α) = U0(α), Ut(0, α) = U1(α).

U satisfies scaling symmetry U(t , α) → λ1/2U(λ3/2t , λα), λ > 0.The estimate

‖Ds−1/2pα U‖Lp

T Lqα≤ C(‖U0‖Hs + ‖U1‖Hs−3/2)

is scale-invariant if (p, q) satisfies (4.3). For this reason, we refer tothis estimate as optimal.Our result has twice the loss in derivative.


Main Results

Perspectives II

We compare 1/p derivative loss estimates:


Main Results

Perspectives II


Scaling suggests 1/p derivitive loss if

52p

+1q

=12


Main Results

Perspectives II



52p

+1q

=12

Energy estimates (L2α × H−3/2

α 7→ L∞T L2

α) plus Hölder’s inequalityin α


Main Results

Perspectives II



52p

+1q

=12

Energy estimates (L2α × H−3/2

α 7→ L∞T L2

α) plus Hölder’s inequalityin α Strichartz estimates with 1/p derivative loss if

1p

+1q

=12


Main Results

Perspectives III

Hölder plus energy

1/p1/2

1/2

1

1/q

1/41/5 2/5

Main Theorem

Suggested by scaling and Sobolev

Figure: Strichartz estimates with 1 p derivative loss.


Main Results

Take home message

The water-waves problem consists of two fluids separated by aninterface and a nonlinear system of PDEs governing the fluidmotion.


Main Results

Take home message


If the fluids are ideal, the system of PDEs can be reduced to asystem on the interface.


Main Results

Take home message



If the surface tension S > 0, this system can be reduced to asingle nonlinear dispersive PDE.


Main Results

Take home message




This PDE is approachable by Fourier analysis techniques.


Main Results

Take home message




This PDE is approachable by Fourier analysis techniques.

Solutions exhibit improved regularity, expressed as integrabilityestimates called Strichartz estimates.

the water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf ·...

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