the water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf ·...
TRANSCRIPT
![Page 1: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/1.jpg)
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
![Page 2: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/2.jpg)
Introduction Formulation The Model Problem The Full Problem
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
![Page 3: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/3.jpg)
Introduction Formulation The Model Problem The Full Problem
Goals
Inspiration?
Figure: Author: Mila,http://home.comcast.net/∼milazinkova/Fogshadow.html
![Page 4: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/4.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
![Page 5: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/5.jpg)
Introduction Formulation The Model Problem The Full Problem
Goals
An old topic
![Page 6: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/6.jpg)
Introduction Formulation The Model Problem The Full Problem
Goals
Goals of this talk
Describe water-waves problem
![Page 7: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/7.jpg)
Introduction Formulation The Model Problem The Full Problem
Goals
Goals of this talk
Describe water-waves problem
Brief historical account
![Page 8: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/8.jpg)
Introduction Formulation The Model Problem The Full Problem
Goals
Goals of this talk
Describe water-waves problem
Brief historical account
Recast problem as a dispersive equation
![Page 9: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/9.jpg)
Introduction Formulation The Model Problem The Full Problem
Goals
Goals of this talk
Describe water-waves problem
Brief historical account
Recast problem as a dispersive equation
Dispersive equation can be approached by energy methods andFourier transform methods
![Page 10: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/10.jpg)
Introduction Formulation The Model Problem The Full Problem
Goals
Goals of this talk
Describe water-waves problem
Brief historical account
Recast problem as a dispersive equation
Dispersive equation can be approached by energy methods andFourier transform methods
Statement of results
![Page 11: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/11.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves
The Problem
Incompressible Euler equations in 2d:{∂t u + (u · ∇)u = −∇p + (0,−g),
∇ · u = 0.
![Page 12: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/12.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves
The Problem
Incompressible Euler equations in 2d:{∂t u + (u · ∇)u = −∇p + (0,−g),
∇ · u = 0.
u(x , y , t) is velocity field, p(x , y , t) is pressure, g ≥ 0 is gravity
![Page 13: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/13.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves
The Problem
Incompressible Euler equations in 2d:{∂t u + (u · ∇)u = −∇p + (0,−g),
∇ · u = 0.
u(x , y , t) is velocity field, p(x , y , t) is pressure, g ≥ 0 is gravity
Assume irrotational:∇× u = 0
![Page 14: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/14.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves
The Problem
Incompressible Euler equations in 2d:{∂t u + (u · ∇)u = −∇p + (0,−g),
∇ · u = 0.
u(x , y , t) is velocity field, p(x , y , t) is pressure, g ≥ 0 is gravity
Assume irrotational:∇× u = 0
All the “action” occurs on the surface
![Page 15: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/15.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves
The Picture
air
fluid
x
y
y = η(t , x)
Figure: The “free boundary” setup
![Page 16: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/16.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves
Boundary Conditions I
Assume interface moves with the particle velocity:
u · n = 0, n normal vector
![Page 17: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/17.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves
Boundary Conditions I
Assume interface moves with the particle velocity:
u · n = 0, n normal vector
Assume pressure “jump” at interface is proportional to meancurvature:
[p] = Sκ, κ curvature
![Page 18: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/18.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves
Boundary Conditions I
Assume interface moves with the particle velocity:
u · n = 0, n normal vector
Assume pressure “jump” at interface is proportional to meancurvature:
[p] = Sκ, κ curvature
S ≥ 0 is surface tension.
![Page 19: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/19.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves
Boundary Conditions II
Assume not much happens at large depths:
u → 0 as |(x , y)| → ∞
![Page 20: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/20.jpg)
Introduction Formulation The Model Problem The Full Problem
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 | → ∞
![Page 21: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/21.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves
The guiding principle
Water waves are affected by two things, gravity and surface tension.
![Page 22: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/22.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves
The guiding principle
Water waves are affected by two things, gravity and surface tension.From a distance, you don’t see the surface tension
![Page 23: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/23.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves
The guiding principle
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.
![Page 24: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/24.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves
The guiding principle
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?
![Page 25: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/25.jpg)
Introduction Formulation The Model Problem The Full Problem
History
A little history I
Early history is patchy:
Euler (1757,1761) hydrodynamic equations
![Page 26: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/26.jpg)
Introduction Formulation The Model Problem The Full Problem
History
A little history I
Early history is patchy:
Euler (1757,1761) hydrodynamic equations
Laplace (1776) static surface tension
![Page 27: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/27.jpg)
Introduction Formulation The Model Problem The Full Problem
History
A little history I
Early history is patchy:
Euler (1757,1761) hydrodynamic equations
Laplace (1776) static surface tension
Lagrange (1781, 1786) velocity potential
![Page 28: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/28.jpg)
Introduction Formulation The Model Problem The Full Problem
History
A little history I
Early history is patchy:
Euler (1757,1761) hydrodynamic equations
Laplace (1776) static surface tension
Lagrange (1781, 1786) velocity potential
Cauchy (1815) and Poisson (1816) studied as an initial valueproblem
![Page 29: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/29.jpg)
Introduction Formulation The Model Problem The Full Problem
History
A little history I
Early history is patchy:
Euler (1757,1761) hydrodynamic equations
Laplace (1776) static surface tension
Lagrange (1781, 1786) velocity potential
Cauchy (1815) and Poisson (1816) studied as an initial valueproblem
Weber-Weber (1825) Careful experimental data
![Page 30: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/30.jpg)
Introduction Formulation The Model Problem The Full Problem
History
A little history I
Early history is patchy:
Euler (1757,1761) hydrodynamic equations
Laplace (1776) static surface tension
Lagrange (1781, 1786) velocity potential
Cauchy (1815) and Poisson (1816) studied as an initial valueproblem
Weber-Weber (1825) Careful experimental data
Airy (1841) linearized gravity waves, ...
![Page 31: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/31.jpg)
Introduction Formulation The Model Problem The Full Problem
History
A little history II
More recently, the water-waves problem has been studied as an initialvalue problem
![Page 32: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/32.jpg)
Introduction Formulation The Model Problem The Full 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?
![Page 33: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/33.jpg)
Introduction Formulation The Model Problem The Full 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:
![Page 34: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/34.jpg)
Introduction Formulation The Model Problem The Full 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:
Existence of solution on some time interval (local or global intime)?
![Page 35: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/35.jpg)
Introduction Formulation The Model Problem The Full 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:
Existence of solution on some time interval (local or global intime)?
Uniqueness of solution on the interval of existence?
![Page 36: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/36.jpg)
Introduction Formulation The Model Problem The Full 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:
Existence of solution on some time interval (local or global intime)?
Uniqueness of solution on the interval of existence?
Continuous dependence on initial conditions?
![Page 37: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/37.jpg)
Introduction Formulation The Model Problem The Full 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:
Existence of solution on some time interval (local or global intime)?
Uniqueness of solution on the interval of existence?
Continuous dependence on initial conditions?
In addition, we may ask “what does the solution look like”?
![Page 38: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/38.jpg)
Introduction Formulation The Model Problem The Full 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:
Existence of solution on some time interval (local or global intime)?
Uniqueness of solution on the interval of existence?
Continuous dependence on initial conditions?
In addition, we may ask “what does the solution look like”? Properties:
![Page 39: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/39.jpg)
Introduction Formulation The Model Problem The Full 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:
Existence of solution on some time interval (local or global intime)?
Uniqueness of solution on the interval of existence?
Continuous dependence on initial conditions?
In addition, we may ask “what does the solution look like”? Properties:
How regular is the solution compared to initial data?
![Page 40: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/40.jpg)
Introduction Formulation The Model Problem The Full 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:
Existence of solution on some time interval (local or global intime)?
Uniqueness of solution on the interval of existence?
Continuous dependence on initial conditions?
In addition, we may ask “what does the solution look like”? Properties:
How regular is the solution compared to initial data?
If the solution exists for long times, what are the long-timeasymptotics?
![Page 41: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/41.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
![Page 42: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/42.jpg)
Introduction Formulation The Model Problem The Full Problem
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.S = 0 (gravity waves) hard!
![Page 43: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/43.jpg)
Introduction Formulation The Model Problem The Full Problem
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.S = 0 (gravity waves) hard!
small data in Sobolev spaces: Nalimov (’74), Yosihara (’82,’83),Craig (’85); analytic category: Kano-Nishida (’79),Sulem-Sulem-Bardos-Frisch (’81), many more...
![Page 44: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/44.jpg)
Introduction Formulation The Model Problem The Full Problem
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.S = 0 (gravity waves) hard!
small data in Sobolev spaces: Nalimov (’74), Yosihara (’82,’83),Craig (’85); analytic category: Kano-Nishida (’79),Sulem-Sulem-Bardos-Frisch (’81), many more...
Beale-Hou-Lowengrub (’93): linear problem well-posed ⇐⇒∇p · n < 0 (Taylor-Young inequality)
![Page 45: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/45.jpg)
Introduction Formulation The Model Problem The Full Problem
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.S = 0 (gravity waves) hard!
small data in Sobolev spaces: Nalimov (’74), Yosihara (’82,’83),Craig (’85); analytic category: Kano-Nishida (’79),Sulem-Sulem-Bardos-Frisch (’81), many more...
Beale-Hou-Lowengrub (’93): linear problem well-posed ⇐⇒∇p · n < 0 (Taylor-Young inequality)
Wu (’97): interface nonself-intersecting =⇒ Taylor-Younginequality holds
![Page 46: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/46.jpg)
Introduction Formulation The Model Problem The Full Problem
History
A little history IV
What about global/almost global well-posedness?
![Page 47: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/47.jpg)
Introduction Formulation The Model Problem The Full Problem
History
A little history IV
What about global/almost global well-posedness? Obviously muchharder! Not many results so far.
![Page 48: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/48.jpg)
Introduction Formulation The Model Problem The Full Problem
History
A little history IV
What about global/almost global well-posedness? Obviously muchharder! Not many results so far.
Wu (’09) almost global-well-posedness for 2d gravity waves
![Page 49: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/49.jpg)
Introduction Formulation The Model Problem The Full Problem
History
A little history IV
What about global/almost global well-posedness? Obviously muchharder! Not many results so far.
Wu (’09) almost global-well-posedness for 2d gravity waves
Germain-Masmoudi-Shatah (’09) and Wu (’09)global-well-posedness for 3d gravity waves
![Page 50: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/50.jpg)
Introduction Formulation The Model Problem The Full Problem
History
A little history V
S > 0 (our case) Taylor-Young inequality always holds
![Page 51: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/51.jpg)
Introduction Formulation The Model Problem The Full Problem
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?
![Page 52: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/52.jpg)
Introduction Formulation The Model Problem The Full Problem
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?
Spirn-Wright (’09) Dispersion for linearized water-waves problemwith surface tension
![Page 53: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/53.jpg)
Introduction Formulation The Model Problem The Full Problem
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?
Spirn-Wright (’09) Dispersion for linearized water-waves problemwith surface tension
C-Hur-Staffilani (’08-’09) 2d local-well-posedness, parametrixconstruction, Strichartz estimates, local smoothing
![Page 54: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/54.jpg)
Introduction Formulation The Model Problem The Full Problem
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?
Spirn-Wright (’09) Dispersion for linearized water-waves problemwith surface tension
C-Hur-Staffilani (’08-’09) 2d local-well-posedness, parametrixconstruction, Strichartz estimates, local smoothing
Alazard-Burq-Zuily (’09) nd irregular bottomlocal-well-posedness, 2d local smoothing
![Page 55: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/55.jpg)
Introduction Formulation The Model Problem The Full Problem
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?
Spirn-Wright (’09) Dispersion for linearized water-waves problemwith surface tension
C-Hur-Staffilani (’08-’09) 2d local-well-posedness, parametrixconstruction, Strichartz estimates, local smoothing
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)
![Page 56: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/56.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
![Page 57: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/57.jpg)
Introduction Formulation The Model Problem The Full Problem
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 plane-wave is of the form ei(λt+xξ) (Fourier decomposition).
![Page 58: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/58.jpg)
Introduction Formulation The Model Problem The Full Problem
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 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.
![Page 59: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/59.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
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,
![Page 60: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/60.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
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,
so λ(ξ) = ξ2.
![Page 61: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/61.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
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,
so λ(ξ) = ξ2.The phase function is ϕ = tξ2 + xξ, so the exponential oscillates a lotunless ∂ξϕ = 0, or the phase is stationary.
![Page 62: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/62.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
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,
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.
![Page 63: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/63.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Transport equation
The transport equation{
(∂t − ∂x)u = 0,
u(0, x) = u0(x),
is not dispersive.
![Page 64: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/64.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Transport equation
The 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.
![Page 65: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/65.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Transport equation
The 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.
![Page 66: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/66.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Other dispersive equations
The wave equation (dimension n ≥ 2) (∂2t − ∆)u = 0 is degenerately
dispersive
![Page 67: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/67.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Other dispersive equations
The wave equation (dimension n ≥ 2) (∂2t − ∆)u = 0 is degenerately
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
![Page 68: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/68.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Other dispersive equations
The wave equation (dimension n ≥ 2) (∂2t − ∆)u = 0 is degenerately
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
but stationary phase implies ±t ξ|ξ| + x = 0, so propagation depends
on direction but not magnitude of momentum.
![Page 69: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/69.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Other dispersive equations
The wave equation (dimension n ≥ 2) (∂2t − ∆)u = 0 is degenerately
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
but stationary phase implies ±t ξ|ξ| + x = 0, so propagation depends
on direction but not magnitude of momentum.
KdV equation is dispersive
![Page 70: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/70.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Other dispersive equations
The wave equation (dimension n ≥ 2) (∂2t − ∆)u = 0 is degenerately
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
but stationary phase implies ±t ξ|ξ| + x = 0, so propagation depends
on direction but not magnitude of momentum.
KdV equation is dispersive
Benjamin-Ono equation is dispersive
![Page 71: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/71.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Other dispersive equations
The wave equation (dimension n ≥ 2) (∂2t − ∆)u = 0 is degenerately
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
but stationary phase implies ±t ξ|ξ| + x = 0, so propagation depends
on direction but not magnitude of momentum.
KdV equation is dispersive
Benjamin-Ono equation is dispersive
Heat equation is dissipative but not dispersive
![Page 72: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/72.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Dispersive properties I
What do we gain from dispersion?
![Page 73: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/73.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Dispersive properties I
What do we gain from dispersion? Two very robust tools are energyestimates and dispersion estimates.
![Page 74: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/74.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Dispersive properties I
What do we gain from dispersion? Two very robust tools are energyestimates and dispersion estimates.Energy estimates:
![Page 75: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/75.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Dispersive properties I
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.
![Page 76: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/76.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Dispersive properties I
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 wave equation,
ddt
(‖∂x u(t , ·)‖2L2 + ‖∂tu(t , ·)‖2
L2) = 0.
![Page 77: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/77.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Dispersive properties II
A common dispersion estimate is a time dependent L1 → L∞
estimate.
![Page 78: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/78.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Dispersive properties II
A common dispersion estimate is a time dependent L1 → L∞
estimate.For Schrödinger equation, phase ϕ = tξ2 + xξ.
![Page 79: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/79.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersive Equations
Dispersive properties II
A common dispersion estimate is a time dependent L1 → L∞
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 .
![Page 80: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/80.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
![Page 81: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/81.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves formulation
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
2The Birkhoff-Rott integral arises as the interfacial limit of the Biot-Savart law torecover the velocity from the vorticity distribution.
![Page 82: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/82.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves formulation
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
Renormalize arclength so |(x , y)α| = 1
2The Birkhoff-Rott integral arises as the interfacial limit of the Biot-Savart law torecover the velocity from the vorticity distribution.
![Page 83: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/83.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves formulation
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
Renormalize arclength so |(x , y)α| = 1 T is determined
2The Birkhoff-Rott integral arises as the interfacial limit of the Biot-Savart law torecover the velocity from the vorticity distribution.
![Page 84: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/84.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves formulation
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
Renormalize arclength so |(x , y)α| = 1 T is determined
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).
2The Birkhoff-Rott integral arises as the interfacial limit of the Biot-Savart law torecover the velocity from the vorticity distribution.
![Page 85: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/85.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves formulation
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
Renormalize arclength so |(x , y)α| = 1 T is determined
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
2The Birkhoff-Rott integral arises as the interfacial limit of the Biot-Savart law torecover the velocity from the vorticity distribution.
![Page 86: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/86.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves formulation
New Formulation II
Introduce modified tangential velocity u (in terms of γ) and tangentialangle θ,
![Page 87: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/87.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves formulation
New Formulation II
Introduce modified tangential velocity u (in terms of γ) and tangentialangle θ, use Hilbert transform to approximate Birkhoff-Rott integral
![Page 88: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/88.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves formulation
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).
![Page 89: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/89.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves formulation
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).
The rj are “nicer” than the explicit terms
![Page 90: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/90.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves formulation
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).
The rj are “nicer” than the explicit terms
H is Hilbert transform: Hf (ξ) = −isgn(ξ)f (ξ).
![Page 91: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/91.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves formulation
New Formulation III
Differentiate utt and back substitute
![Page 92: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/92.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves formulation
New Formulation III
Differentiate utt and back substitute second equivalent system:
![Page 93: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/93.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves formulation
New Formulation III
Differentiate utt and back substitute second equivalent system:
∂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)
![Page 94: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/94.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves formulation
New Formulation III
Differentiate utt and back substitute second equivalent system:
∂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.
![Page 95: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/95.jpg)
Introduction Formulation The Model Problem The Full Problem
Water-waves formulation
New Formulation III
Differentiate utt and back substitute second equivalent system:
∂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).
![Page 96: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/96.jpg)
Introduction Formulation The Model Problem The Full Problem
Energy estimates
A Dispersive Equation!
Model case: take g = 0, S/2 = 1, all nonlinear terms zero:
![Page 97: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/97.jpg)
Introduction Formulation The Model Problem The Full Problem
Energy estimates
A Dispersive Equation!
Model case: take g = 0, S/2 = 1, all nonlinear terms zero:
∂2t u − H∂3
αu = 0
u(0, α) = u0(α),
ut(0, α) = u1(α)
(3.1)
![Page 98: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/98.jpg)
Introduction Formulation The Model Problem The Full Problem
Energy estimates
A Dispersive Equation!
Model case: take g = 0, S/2 = 1, all nonlinear terms zero:
∂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
![Page 99: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/99.jpg)
Introduction Formulation The Model Problem The Full Problem
Energy estimates
A Dispersive Equation!
Model case: take g = 0, S/2 = 1, all nonlinear terms zero:
∂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.
![Page 100: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/100.jpg)
Introduction Formulation The Model Problem The Full Problem
Energy estimates
Energy conservation
Energy conservation comes from multiplying equation by u andintegrating by parts:
E(t) :=
∫(|ut |
2 − uH∂3αu)dα
![Page 101: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/101.jpg)
Introduction Formulation The Model Problem The Full Problem
Energy estimates
Energy conservation
Energy conservation comes from multiplying equation by u andintegrating by parts:
E(t) :=
∫(|ut |
2 − uH∂3αu)dα
satisfiesE ′ = 0
![Page 102: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/102.jpg)
Introduction Formulation The Model Problem The Full Problem
Energy estimates
Energy conservation
Energy conservation comes from multiplying equation by u andintegrating by parts:
E(t) :=
∫(|ut |
2 − uH∂3αu)dα
satisfiesE ′ = 0
E controls 1 time derivative and 3/2 space derivative of u inL∞
t L2α
![Page 103: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/103.jpg)
Introduction Formulation The Model Problem The Full Problem
Energy estimates
Energy conservation
Energy conservation comes from multiplying equation by u andintegrating by parts:
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
![Page 104: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/104.jpg)
Introduction Formulation The Model Problem The Full Problem
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
![Page 105: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/105.jpg)
Introduction Formulation The Model Problem The Full Problem
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
If u0 and u1 have compact support in ξ > 0, compute dispersionestimate via stationary phase: We write u = u+ + u− with
![Page 106: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/106.jpg)
Introduction Formulation The Model Problem The Full Problem
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
If u0 and u1 have compact support in ξ > 0, compute dispersionestimate via stationary phase: We write u = u+ + u− with
u±(t , α) =
∫K±
0 (t , α, β)u0(β) + K±1 (t , α, β)u1(β)dβ.
![Page 107: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/107.jpg)
Introduction Formulation The Model Problem The Full Problem
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
If u0 and u1 have compact support in ξ > 0, compute dispersionestimate via stationary phase: We write u = u+ + u− with
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.
![Page 108: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/108.jpg)
Introduction Formulation The Model Problem The Full Problem
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
If u0 and u1 have compact support in ξ > 0, compute dispersionestimate via stationary phase: We write u = u+ + u− with
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.
![Page 109: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/109.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersion Estimate
Dispersion estimate II
Consider ϕ = ϕ+. Phase is stationary when ϕξ(t , α, β, ξc) = 0, or
![Page 110: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/110.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersion Estimate
Dispersion estimate II
Consider ϕ = ϕ+. Phase is stationary when ϕξ(t , α, β, ξc) = 0, or
∂ξϕ =32
tξ1/2c + α− β = 0, or ξc =
49
(β − α
t
)2
.
![Page 111: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/111.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersion Estimate
Dispersion estimate II
Consider ϕ = ϕ+. Phase is stationary when ϕξ(t , α, β, ξc) = 0, or
∂ξϕ =32
tξ1/2c + α− β = 0, or ξc =
49
(β − α
t
)2
.
Applying the method of stationary phase,
![Page 112: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/112.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersion Estimate
Dispersion estimate II
Consider ϕ = ϕ+. Phase is stationary when ϕξ(t , α, β, ξc) = 0, or
∂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.
![Page 113: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/113.jpg)
Introduction Formulation The Model Problem The Full Problem
Dispersion Estimate
Dispersion estimate II
Consider ϕ = ϕ+. Phase is stationary when ϕξ(t , α, β, ξc) = 0, or
∂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
.
![Page 114: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/114.jpg)
Introduction Formulation The Model Problem The Full Problem
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
![Page 115: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/115.jpg)
Introduction Formulation The Model Problem The Full Problem
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
Of course the full, nonlinear problem is much harder...
![Page 116: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/116.jpg)
Introduction Formulation The Model Problem The Full Problem
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
Of course the full, nonlinear problem is much harder...
Conjecture
The estimate (3.2) holds for the nonlinear problem and is sharp.
![Page 117: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/117.jpg)
Introduction Formulation The Model Problem The Full Problem
Outline of full problem
We want to prove Strichartz estimates for the nonlinear problem
Prove local well-posedness and regularity of solution
![Page 118: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/118.jpg)
Introduction Formulation The Model Problem The Full Problem
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
![Page 119: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/119.jpg)
Introduction Formulation The Model Problem The Full Problem
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
![Page 120: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/120.jpg)
Introduction Formulation The Model Problem The Full Problem
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
![Page 121: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/121.jpg)
Introduction Formulation The Model Problem The Full Problem
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
Conclude Strichartz estimates.
![Page 122: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/122.jpg)
Introduction Formulation The Model Problem The Full Problem
Energy estimates and LWP
Local Well-posedness
Extend energy estimate to quasilinear case.
![Page 123: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/123.jpg)
Introduction Formulation The Model Problem The Full Problem
Energy estimates and LWP
Local Well-posedness
Extend energy estimate to quasilinear case.Introduce nonlinear “material derivative” instead of ∂t u:
v = ∂tu + u∂αu.
Rewrite energy in terms of v .
![Page 124: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/124.jpg)
Introduction Formulation The Model Problem The Full Problem
Energy estimates and LWP
Local Well-posedness
Extend energy estimate to quasilinear case.Introduce nonlinear “material derivative” instead of ∂t u:
v = ∂tu + u∂αu.
Rewrite energy in terms of v . A difficult integration by parts usingsymmetry yields similar energy estimates to model problem,
![Page 125: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/125.jpg)
Introduction Formulation The Model Problem The Full Problem
Energy estimates and LWP
Local Well-posedness
Extend energy estimate to quasilinear case.Introduce nonlinear “material derivative” instead of ∂t u:
v = ∂tu + u∂αu.
Rewrite energy in terms of v . A difficult integration by parts usingsymmetry yields similar energy estimates to model problem, at leastfor short times.
![Page 126: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/126.jpg)
Introduction Formulation The Model Problem The Full Problem
Energy estimates and LWP
Local Well-posedness
Extend energy estimate to quasilinear case.Introduce nonlinear “material derivative” instead of ∂t u:
v = ∂tu + u∂αu.
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).
![Page 127: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/127.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
![Page 128: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/128.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
We want to solve the homogeneous equation Pu = 0
![Page 129: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/129.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
We want to solve the homogeneous equation Pu = 0Make WKB type ansatz:
![Page 130: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/130.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
We want to solve the homogeneous equation Pu = 0Make WKB type ansatz:
u =
∫e−iβξ
(eiϕ+
f +(β) + eiϕ−
f−(β))
dβdξ,
![Page 131: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/131.jpg)
Introduction Formulation The Model Problem The Full Problem
Microlocal Dispersion
Dyadic-frequency parametrix II
ϕ± functions of (t , α, ξ)
![Page 132: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/132.jpg)
Introduction Formulation The Model Problem The Full Problem
Microlocal Dispersion
Dyadic-frequency parametrix II
ϕ± functions of (t , α, ξ)
ϕ±|t=0 = αξ (Fourier inversion)
![Page 133: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/133.jpg)
Introduction Formulation The Model Problem The Full Problem
Microlocal Dispersion
Dyadic-frequency parametrix II
ϕ± functions of (t , α, ξ)
ϕ±|t=0 = αξ (Fourier inversion)
f± chosen to satisfy initial conditions
![Page 134: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/134.jpg)
Introduction Formulation The Model Problem The Full Problem
Microlocal Dispersion
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±):
![Page 135: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/135.jpg)
Introduction Formulation The Model Problem The Full Problem
Microlocal Dispersion
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)
![Page 136: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/136.jpg)
Introduction Formulation The Model Problem The Full Problem
Microlocal Dispersion
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 is a perturbation of 0-coefficient part (αξ ± t |ξ|3/2)
ϕ± = αξ ± |ξ|3/2(t + θ±(t , α, ξ))
![Page 137: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/137.jpg)
Introduction Formulation The Model Problem The Full Problem
Microlocal Dispersion
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 is a perturbation of 0-coefficient part (αξ ± t |ξ|3/2)
ϕ± = αξ ± |ξ|3/2(t + θ±(t , α, ξ))
θ± only exists on timescales t ∼ |ξ|−1/2.
![Page 138: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/138.jpg)
Introduction Formulation The Model Problem The Full Problem
Microlocal Dispersion
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 is a perturbation of 0-coefficient part (αξ ± t |ξ|3/2)
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).
![Page 139: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/139.jpg)
Introduction Formulation The Model Problem The Full Problem
Microlocal Dispersion
Dispersion Estimate I
Estimate integral kernel in dyadic frequency regions
![Page 140: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/140.jpg)
Introduction Formulation The Model Problem The Full Problem
Microlocal Dispersion
Dispersion Estimate I
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,
![Page 141: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/141.jpg)
Introduction Formulation The Model Problem The Full Problem
Microlocal Dispersion
Dispersion Estimate I
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,
K±0 (t , α, β) =
∫eiϕj,±(t,α,β,ξ)ψ(2−jξ)dξ
![Page 142: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/142.jpg)
Introduction Formulation The Model Problem The Full Problem
Microlocal Dispersion
Dispersion Estimate I
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,
K±0 (t , α, β) =
∫eiϕj,±(t,α,β,ξ)ψ(2−jξ)dξ
with suppψ ∼ 1 and
ϕj,± = (α− β)ξ ± |ξ|3/2(t + θj,±)
(K±1 similarly..)
![Page 143: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/143.jpg)
Introduction Formulation The Model Problem The Full Problem
Microlocal Dispersion
Dispersion Estimate I
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,
K±0 (t , α, β) =
∫eiϕj,±(t,α,β,ξ)ψ(2−jξ)dξ
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
![Page 144: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/144.jpg)
Introduction Formulation The Model Problem The Full Problem
Microlocal Dispersion
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
![Page 145: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/145.jpg)
Introduction Formulation The Model Problem The Full Problem
Microlocal Dispersion
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
This does not work for a fixed time scale!
![Page 146: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/146.jpg)
Introduction Formulation The Model Problem The Full Problem
Microlocal Dispersion
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
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
![Page 147: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/147.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
![Page 148: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/148.jpg)
Introduction Formulation The Model Problem The Full Problem
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.Summing over frequencies using almost orthogonality ofLittlewood-Paley decomposition for q <∞ we get the followingTheorem.
![Page 149: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/149.jpg)
Introduction Formulation The Model Problem The Full Problem
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.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.
![Page 150: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/150.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
![Page 151: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/151.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
![Page 152: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/152.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
![Page 153: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/153.jpg)
Introduction Formulation The Model Problem The Full Problem
Main Results
Perspectives II
We compare 1/p derivative loss estimates:
![Page 154: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/154.jpg)
Introduction Formulation The Model Problem The Full Problem
Main Results
Perspectives II
We compare 1/p derivative loss estimates:
Scaling suggests 1/p derivitive loss if
52p
+1q
=12
![Page 155: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/155.jpg)
Introduction Formulation The Model Problem The Full Problem
Main Results
Perspectives II
We compare 1/p derivative loss estimates:
Scaling suggests 1/p derivitive loss if
52p
+1q
=12
Energy estimates (L2α × H−3/2
α 7→ L∞T L2
α) plus Hölder’s inequalityin α
![Page 156: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/156.jpg)
Introduction Formulation The Model Problem The Full Problem
Main Results
Perspectives II
We compare 1/p derivative loss estimates:
Scaling suggests 1/p derivitive loss if
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
![Page 157: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/157.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
![Page 158: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/158.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
![Page 159: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/159.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
If the fluids are ideal, the system of PDEs can be reduced to asystem on the interface.
![Page 160: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/160.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
If the fluids are ideal, the system of PDEs can be reduced to asystem on the interface.
If the surface tension S > 0, this system can be reduced to asingle nonlinear dispersive PDE.
![Page 161: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/161.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
If the fluids are ideal, the system of PDEs can be reduced to asystem on the interface.
If the surface tension S > 0, this system can be reduced to asingle nonlinear dispersive PDE.
This PDE is approachable by Fourier analysis techniques.
![Page 162: The water-waves problem with surface tensionhans.unc.edu/files/2011/11/water-wave-colloquium.pdf · 2011-11-03 · Introduction Formulation The Model Problem The Full Problem The](https://reader033.vdocuments.us/reader033/viewer/2022042102/5e7fa63b92f5725cfe7ec573/html5/thumbnails/162.jpg)
Introduction Formulation The Model Problem The Full Problem
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.
If the fluids are ideal, the system of PDEs can be reduced to asystem on the interface.
If the surface tension S > 0, this system can be reduced to asingle nonlinear dispersive PDE.
This PDE is approachable by Fourier analysis techniques.
Solutions exhibit improved regularity, expressed as integrabilityestimates called Strichartz estimates.