Unique Continuation from Infinity for Waves, and Applications Arick - - PowerPoint PPT Presentation

Unique Continuation from Infinity for Waves, and Applications

Arick Shao

Queen Mary University of London

Cardiff Analysis Seminar 23 January, 2017

Arick Shao (QMUL) Unique Continuation 1 / 30

Background on Wave Equations

Wave Equations

The (free) wave equation on flat spacetime: φ := (−∂2

t + ∆x)φ = 0,

φ : Rt × Rn

x → R.

Generalisations: linear and nonlinear waves, systems, geometric waves.

Examples of hyperbolic PDEs:

Finite speed of propagation, dispersion, conservation, formation of singularities.

Waves found in many equations of physics:

Electromagnetism, gravitation, fluids.

Arick Shao (QMUL) Unique Continuation 2 / 30

Background on Wave Equations

Well-Posedness

Natural question—Cauchy problem: φ = 0, φ|t=0 = φ0, ∂tφ|t=0 = φ1.

t x t=0

φ=φ0 ∂t φ=φ1 φ=? φ=?

The Cauchy problem is well-posed.

1

The solution exists.

2

The solution is unique.

3

The solution is stable. (Depends continuously on initial data)

Given the current state of a system, we can predict the future.

Arick Shao (QMUL) Unique Continuation 3 / 30

Background on Wave Equations

Radiation

x t Propagation of waves (R1+1).

Regular solutions of φ = 0:

Propagate at fixed, finite speed. Decay in space and time at known rates.

Can make sense of “asymptotics at infinity”:

Leading order coefficient: radiation field.

Physics: “what is seen by faraway observer”

Electromagnetic, gravitational radiation.

Arick Shao (QMUL) Unique Continuation 4 / 30

Background on Wave Equations

Minkowski Geometry

Theme: Geometric viewpoint for studying wave equations.

Robust techniques applicable to many curved backgrounds. Applications to problems in relativity.

Natural setting: Minkowski spacetime (R1+n, m).

Minkowski metric: m := −dt2 + d(x1)2 + · · · + d(xn)2. Setting of special relativity.

= mαβ∇αβ is m-trace of the Hessian.

Natural second-order PDO associated with Minkowski geometry. Analogous to ∆ in Euclidean geometry.

Arick Shao (QMUL) Unique Continuation 5 / 30

Background on Wave Equations

Infinity

R T R1+n I+ I− R × Sn, mod Sn−1. R T R1+n I+ I− Previous picture, projected.

Infinity visualised via Penrose compactification.

Conformal transformation m → Ω2m. (R1+n, Ω2m) isometrically embeds into relatively compact region in R × Sn. Minkowski analogue of stereographic projection.

Infinity realised as boundary of shaded region.

Future/past null infinity I±: Null geodesics (bicharacteristics of ) terminate here. Radiation field manifested at I±. Here, useful for pictures.

Arick Shao (QMUL) Unique Continuation 6 / 30

The Main Problem

Main Questions

Question (I) Can one reconstruct a wave from its values at infinity?

In particular, are nonradiating waves trivial?

Question (II) If so, can results be further generalised?

Linear and nonlinear waves. Geometric waves (on curved backgrounds).

Question (III) Can techniques behind (I) and (II) be applied to other problems?

Arick Shao (QMUL) Unique Continuation 7 / 30

The Main Problem

Scattering Theory

There are classical scattering results for (I). (Friedlander) φ = 0 on R1+n

Initial data (t = 0) ⇔ radiation field (I+). ⇒ Solve Cauchy problem. ⇒ Solve backwards from radiation field.

Generalisations:

Product manifolds R × X. Special nonlinear waves. Special black hole spacetimes.

t=0 I+ Red: solve forward from t = 0. Purple: solve backward from I+. Arick Shao (QMUL) Unique Continuation 8 / 30

The Main Problem

Beyond Scattering

What about other wave equations?

General linear and nonlinear waves, e.g., ( + ∇X + V )φ = 0. Geometric waves on curved backgrounds, e.g., gφ = g αβ∇2

αβφ = . . . .

Can we only impose data on part of I+ and I−?

Can waves be defined only locally near I±?

These problems tend to be ill-posed:

Cannot solve the wave equation. But, can ask if solutions are unique.

I+ I−

φ,∇φ=0 φ,∇φ=0 φ=0?

An ill-posed question. Arick Shao (QMUL) Unique Continuation 9 / 30

The Main Problem

The Result Near Infinity

Theorem (Alexakis–Schlue–S., 2015)

Assume φ is a solution, near I±, of φ + ∇Xφ + V φ = 0, where X, V decay sufficiently toward I±. If φ, ∇φ vanish to ∞-order on ( 1

2 + ε)I±, then φ = 0 in the

interior near ( 1

2 + ε)I±. I+ I−

φ,∇φ=0 φ,∇φ=0 φ=0

Uniqueness on shaded region, with data on ( 1

2 + ε)I±.

• Remark. The ∞-order vanishing is optimal.

Counterexamples if φ vanishes only to finite order. n = 3: φ = ∇k

xr −1. Arick Shao (QMUL) Unique Continuation 10 / 30

Geometric Unique Continuation

Unique Continuation

Formulate main question as unique continuation (UC) problem. Problem (Unique Continuation) Suppose:

φ solves (g + ∇X + V )φ = 0. φ, dφ vanish on a hypersurface Σ.

Must φ vanish on one side of Σ? In particular, we are interested in Σ ⊆ I±.

Σ

g φ+...=0 φ,dφ=0 φ=0?

Unique continuation problem. Arick Shao (QMUL) Unique Continuation 11 / 30

Geometric Unique Continuation

The Classical Theory

Ancient theory: analytic PDE, noncharacteristic Σ.

(Cauchy–Kovalevskaya) Existence, uniqueness of analytic solutions. (Holmgren, F. John) Solution unique even in nonanalytic classes.

Classical theory for non-analytic equations (H¨

• rmander):

Crucial point: pseudoconvexity of Σ. Pseudoconvexity ⇒ Carleman estimates ⇒ UC

Classical results are purely local.

Applies to small neighbourhoods of P ∈ Σ.

Arick Shao (QMUL) Unique Continuation 12 / 30

Geometric Unique Continuation

The Geometric Approach

Σ := {f = 0} is pseudoconvex (wrt g and f ) ⇔ ∇2f (X, X) < 0 on Σ, if g(X, X) = Xf = 0.

−f convex in tangent null (bicharacteristic) directions.

Null curve hitting Σ tangentially lies in {f < 0} nearby.

Relativity: bending of light.

Σ f >0 f <0 null geodesic P Σ pseudoconvex at P.

• Note. Pseudoconvexity is conformally invariant.

Sensible to take Σ ⊆ I±.

Arick Shao (QMUL) Unique Continuation 13 / 30

Geometric Unique Continuation

Zero Pseudoconvexity

We next consider the (non-classical) borderline case. Σ is zero pseudoconvex ⇔ Σ is ruled by null geodesics.

Also conformally invariant. Bad news: I± is zero pseudoconvex, not pseudoconvex.

Possible loss of local UC in zero pseudoconvex settings:

(Alinhac–Baouendi) Counterexample to local UC when Σ = {xn = 0} ⊆ R1+n. (Kenig–Ruiz–Sogge) Global UC from all of Σ. Main result: Semi-global UC (from “large enough” part of I±).

Arick Shao (QMUL) Unique Continuation 14 / 30

Uniqueness Near Infinity

Finding Pseudoconvexity

We now return to the main problem.

• Q1. Can we find pseudoconvexity near I±?

Consider hyperboloids in R1+n

Blue level sets of r 2 − t2. These are only zero pseudoconvex.

• Idea. Take instead (1

2 + ε)I±”.

Consider red “warped hyperboloids”. These are (inward) pseudoconvex.

ι+ ι− ι0 I+ I− Blue: level hyperboloids of f . Red: pseudoconvex hyperboloids. Arick Shao (QMUL) Unique Continuation 15 / 30

Uniqueness Near Infinity

Strong Unique Continuation

• Q2. Where do we see the ∞-order vanishing?

By another conformal transformation:

UC from ( 1

2 + ε)I± ⇔ UC from cone r 2 − t2 = 0.

(But with “warped” geometry.)

Strong UC for elliptic equations:

UC from a single point (say r 2 = 0). Requires ∞-order vanishing (in r).

ι0 f = c −UV = c I+ I− ¯ I

Warped inversion of Minkowski. (Figure by V. Schlue.)

UC from r2 − t2 = 0: hyperbolic analogue of strong UC.

Arick Shao (QMUL) Unique Continuation 16 / 30

Uniqueness Near Infinity

Carleman Estimates

Q1 + Q2 + hard work ⇒ Carleman estimates.

Weighted spacetime integral inequalities. Standard tool for proving UC. Technical difficulties due to degenerate pseudoconvexity.

Carleman estimates + standard argument ⇒ main UC result.

• Remark. Carleman estimates useful in many other PDE problems.

Inverse problems. Controllability of PDEs.

Arick Shao (QMUL) Unique Continuation 17 / 30

Uniqueness Near Infinity

Geometric Robustness

Carleman estimate proved using geometric computations.

Main result is geometrically robust—extend to many curved backgrounds. Asymptotically flat spacetimes: those with “similar structure of infinity”.

Theorem (Alexakis–Schlue–S., 2015)

The main result extends to a large class of (both stationary and dynamic) asymptotically flat spacetimes, including:

1

Perturbations of Minkowski spacetimes.

2

Schwarzschild and Kerr spacetimes, and perturbations. For (2), result can be localised near εI±.

I+ I−

φ,∇φ=0

UC from εI± to shaded region. Arick Shao (QMUL) Unique Continuation 18 / 30

Global Uniqueness

Toward Finite-Order Vanishing

Let us return to Minkowski spacetime (R1+n, m)

• Question. Can we remove ∞-order vanishing condition?
• Recall. Counterexamples to finite-order vanishing:

∇k

xr−1

(n = 3).

These counterexamples blow up at r = 0. Same property holds for other dimensions.

• Idea. Impose global regularity for φ.

Eliminates above counterexamples.

Arick Shao (QMUL) Unique Continuation 19 / 30

Global Uniqueness

The Global Linear Theorem

Theorem (Alexakis–S., 2015)

Assume φ is a regular solution in D of φ + V φ = 0, V L∞ ≤ C. If φ, ∇φ vanish to order A at 1

2I±, with A depending on C,

then φ = 0 everywhere on D.

• Remark. φ = 0 on D ⇒ φ = 0 on R1+n.

I+ I−

φ,∇φ=0 φ,∇φ=0

D

φ=0

UC from 1

2 I± to shaded region.

• Remark. The L∞-assumption on V is necessary.

Otherwise, one can construct counterexamples.

Arick Shao (QMUL) Unique Continuation 20 / 30

Global Uniqueness

Carleman to Uniqueness

• Q. Why was ∞-order vanishing needed?

f =f0 Ω

ι0 I+ I−

f =+∞ f =+∞ f =0 f =0

Consider the Carleman estimate in this setting:

• Ω

f 2aφ2 a−1

• Ω

f 2a|φ|2 + . . . .

Carleman weight f 2a, with f = r 2 − t2. f ր ∞ on 1

2I±.

Need φ vanishing at 1

2I±.

For Carleman ⇒ UC argument, one takes a ր ∞.

Thus, need ∞-order vanishing of φ at 1

2I±. Arick Shao (QMUL) Unique Continuation 21 / 30

Global Uniqueness

A Global Carleman Weight

• Q. Why do we need a ր ∞?

Comes from cutoff function to eliminate φ at f = f0.

f =f0 Ω

ι0 I+ I−

f =+∞ f =+∞ f =0 f =0

• Idea. Avoid cutoff function ⇒ avoid taking a ր ∞.

Weight f 2a vanishes on null cone f = 0. If Carleman estimate can be pushed to f = 0, then there is no contribution there, even without cutoff function.

This can indeed be done:

Very careful construction of a refined Carleman weight.

Arick Shao (QMUL) Unique Continuation 22 / 30

Global Uniqueness

Further Improvements

• Question. Can we, in some cases, remove L∞-assumption for V ?

Yes, if V satisfies certain monotonicity properties.

• Idea. Derive Carleman estimate for + V , rather than .

More generally, can prove nonlinear Carleman estimates.

• Idea. Work with nonlinear operator φ → V ,pφ := φ + V · |φ|p−1φ.

Works if V has good monotonicity properties (depending on p).

• Examples. φ ± |φ|p−1φ = 0.

Standard focusing and defocusing nonlinear waves (NLW). Common model nonlinear equations in dispersive PDEs.

Arick Shao (QMUL) Unique Continuation 23 / 30

Global Uniqueness

The Global Nonlinear Theorem

Theorem (Alexakis–S., 2015)

Suppose φ is a regular solution in D of φ + |φ|p−1φ = 0, 1 ≤ p < 1 + 4 n − 1, φ − |φ|p−1φ = 0, p ≥ 1 + 4 n − 1. If φ, ∇φ vanishes to order δ at 1

2I± for any δ > 0, then φ = 0 on D.

The result generalises to some equations φ + V |φ|p−1φ = 0, if V satisfies a monotonicity property (depending on p).

Arick Shao (QMUL) Unique Continuation 24 / 30

Singularity Formation

Singularity Formation

Consider the focusing subconformal NLW: φ + |φ|p−1φ = 0, 1 < p < 1 + 4 n − 1.

Solutions can blow up in finite time. We essentially know the blow-up rate near singularity.

(0,0)

t=−2 t=−1.2 t=−0.6 t=−0.3

Past null cone of a singular point.

Theorem (Merle–Zaag, 2005)

Suppose a solution φ blows up at (0, 0) ∈ R1+n: If (0, 0) is noncharacteristic, then ∃ ε > 0 such that ∀ 0 < t ≪ 1, ε ≤ t

2 p−1 − n 2 φ(−t)L2(B(0,t)) + t 2 p−1 − n 2 +1∇t,xφ(−t)L2(B(0,t)).

Moreover, given any σ ∈ (0, 1), we have that ∀ 0 < t ≪ 1, t

2 p−1 − n 2 φ(−t)L2(B(0,σt)) + t 2 p−1 − n 2 +1∇t,xφ(−t)L2(B(0,σt)) ≤ Kσ.

Arick Shao (QMUL) Unique Continuation 25 / 30

Singularity Formation

A Singularity Result

(Merle–Zaag) gives blow-up rate of H1-norm near singularity.

But, no information about profile of blowup for n > 1.

Theorem (Alexakis–S., 2014)

Suppose φ ∈ C 2 is as before, with blow-up at (0, 0). If lim sup

t∗ր0

|t∗|2−n+

4 p−1

• {σ0|t∗|<r<σ1|t∗|}

(|∇t,xφ|2 + t−2

∗ φ2)|t=t∗ < δ,

then lim sup

t∗ր0

|t∗|1−n+

4 p−1

• {|t|≃|t∗|, r<σ0|t∗|}

(|∇t,xφ|2 + t−2

∗ φ2)|t=t∗ δ. (0,0) Arick Shao (QMUL) Unique Continuation 26 / 30

Singularity Formation

Interpretations

The theorem gives a hint of the profile near the singularity. Near the blow-up point at (0, 0):

The H1-norm of φ cannot concentrate mostly in a past time cone of 0 (e.g., the red region). Significant action must occur near the past null cone of 0 (e.g., the blue region).

The result rules out certain classes of profiles:

Uses novel tools (nonlinear Carleman estimates) not previously applied in this area.

(0,0) Past null cone of a blowup point of φ. Arick Shao (QMUL) Unique Continuation 27 / 30

Singularity Formation

Carleman Estimates Revisited

• Idea. Nonlinear Carleman estimates still hold on bounded U ⊆ D.

Contain extra boundary terms on ∂U... ... but not on f = 0.

C: past timecone from blow-up point.

Nonlinear Carleman estimate yields, for (a),

• shaded

w1|φ|p+1

• red

G(φ, dφ). Applying above twice yields, for (b),

• shaded

|φ|p+1

• red

G(φ, dφ).

(a)

C (0,0) Q

(b)

C (0,0) Arick Shao (QMUL) Unique Continuation 28 / 30

Epilogue

Other Applications

Geometric UC results have applications to relativity.

(Alexakis–Schlue) Nonexistence of time-periodic vacuum spacetimes. Key step: UC results of (Alexakis–Schlue–S.).

Analogous UC results on asymptotically Anti-de Sitter spacetimes.

(Holzegel–S.) UC for waves from conformal infinity. In progress. Rigidity results, possible connections to holography.

In progress. Refined Carleman estimates on bounded U ⊆ D:

Applications to controllability and inverse problems?

Arick Shao (QMUL) Unique Continuation 29 / 30

Epilogue

The End

Thank you for your attention!

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