Overview FLRW Geometric Energies Recent Research
Geometric Methods in Hyperbolic PDEs
Jared Speck jspeck@math.princeton.edu
Department of Mathematics
Princeton University
January 24, 2011
Unifying mathematical themes
Many physical phenomena are modeled by systems of hyperbolic PDEs .
Keywords associated with hyperbolic PDEs:
Initial value problem formulation (the “evolution problem”)
Finite speed of propagation
Energy estimates
Goal: Show how geometrically motived techniques can help us extract information about solutions.
Unifying mathematical themes
Many physical phenomena are modeled by systems of hyperbolic PDEs .
Keywords associated with hyperbolic PDEs:
Initial value problem formulation (the “evolution problem”)
Finite speed of propagation
Energy estimates
Goal: Show how geometrically motived techniques can help us extract information about solutions.
The Einstein equations
M def
= 1 + 3 dimensional manifold
Ric
µν
−
1
2
Rg
µν
= T
Matter equations = 0
µν g
µν
Ric
= Lorentzian metric of signature ( − , + , + , +)
µν
= Ric
µν
( g , ∂ g , ∂
2 g ) = Ricci tensor
R = ( g
− 1
)
αβ
Ric
αβ
= scalar curvature
T
µν
= energy-momentum tensor of the matter
Convention: X
ν def
= ( g
− 1
)
να
X
α
, etc.
Euler-Einstein equations with Λ > 0
Ric
µν
−
1
2
Λ > 0 ∼ “Dark Energy” z }| {
Rg
µν
+ Λ g
µν
∇
α
αµ
Euler equations
↔
Cosmology
= T
µν
= 0
T
µν
= ( ρ + p ) u
µ u
ν g
αβ u
α u
β
= − 1 u 0
>
+ pg
0 (future-directed)
µν p = c
2 s
ρ = equation of state ; c s
= speed of sound
Stability assumption : 0 < c s
< 1 / 3
The Friedmann-Lemaître-Robertson-Walker (FLRW) family on ( −∞ , ∞ ) ×
T
3
Background solution ansatz : e = p ( t ) , e µ
= ( 1 , 0 , 0 , 0 ) , e = − dt
2
+ a
2
( t )
X
( dx i
)
2 i = 1
3 ( 1 + c
2 s
) ≡ ¯ = const
˙ = a s
Λ
3
+ const
3 a 3 ( 1 + c
2 s
)
| {z }
Friedmann’s equation
, a ( 0 ) = 1
Can show: a ( t ) ∼ e
Ht
, H = p
Λ / 3 = Hubble’s constant
Accelerated expansion of the universe
S. Weinberg: “Almost all of modern cosmology is based on this Robertson-Walker metric.” e = − dt
2
+ a
2
( t )
X
( dx i
)
2 i = 1 a ( t ) ∼ e
Ht
Notation : Ω( t ) def
= ln a ( t )
Ω( t ) ∼ Ht
Side remark: Λ = 0 = ⇒ a ( t ) ∼ t C
, C = t = 0 ∼ “Big Bang”
2
3 ( 1 + c
2 s
)
;
Stability
S. Hawking: “We now have a good idea of [the universe’s] behavior at late times: the universe will continue to expand at an ever-increasing rate.”
Fundamental question
What happens if we slightly perturb the initial conditions of the special FLRW solution e , e , e ?
Trustworthiness of FLRW solution ⇐⇒ its salient features are stable under small perturbations of the initial conditions
(I. Rodnianski & J.S., 2009; J.S., 2010: Nonlinear stability holds!)
Stability
S. Hawking: “We now have a good idea of [the universe’s] behavior at late times: the universe will continue to expand at an ever-increasing rate.”
Fundamental question
What happens if we slightly perturb the initial conditions of the special FLRW solution e , e , e ?
Trustworthiness of FLRW solution ⇐⇒ its salient features are stable under small perturbations of the initial conditions
(I. Rodnianski & J.S., 2009; J.S., 2010: Nonlinear stability holds!)
Stability
S. Hawking: “We now have a good idea of [the universe’s] behavior at late times: the universe will continue to expand at an ever-increasing rate.”
Fundamental question
What happens if we slightly perturb the initial conditions of the special FLRW solution e , e , e ?
Trustworthiness of FLRW solution ⇐⇒ its salient features are stable under small perturbations of the initial conditions
(I. Rodnianski & J.S., 2009; J.S., 2010: Nonlinear stability holds!)
Stability
S. Hawking: “We now have a good idea of [the universe’s] behavior at late times: the universe will continue to expand at an ever-increasing rate.”
Fundamental question
What happens if we slightly perturb the initial conditions of the special FLRW solution e , e , e ?
Trustworthiness of FLRW solution ⇐⇒ its salient features are stable under small perturbations of the initial conditions
(I. Rodnianski & J.S., 2009; J.S., 2010: Nonlinear stability holds!)
The Cauchy problem for the Einstein equations
Fundamental facts:
Local existence (Choquet-Bruhat, 1952)
Constraint equations
Elliptic
Ric
µν
− 1
2
Rg
µν
+ Λ g
µν
= T
µν
Evolution equations
Hyperbolic ∼ wave equations
Data: • 3 − manifold Σ , • Riemannian metric ˚ ,
• second fundamental form
˚
, • matter data
(Choquet-Bruhat & Geroch, 1969) Each data set launches a unique
∗
“maximal solution:” the Maximal Globally Hyperbolic
Development of the data (MGHD)
In other words · · ·
Rephrasing of fundamental question: What is the nature of the MGHD of data near that of the special FLRW solution e , e , e ?
Continuation principle (to avoid blow-up)
Lemma (heuristic)
If a singularity forms at time T max
, then lim
T → T
− max sup
0 ≤ t ≤ T
S ( t ) = ∞
S ( t ) = suitably-defined Sobolev-type (i.e., L 2 x
− type) norm
Moral conclusion: To avoid blow-up, show that S ( t ) remains finite.
Continuation principle (to avoid blow-up)
Lemma (heuristic)
If a singularity forms at time T max
, then lim
T → T
− max sup
0 ≤ t ≤ T
S ( t ) = ∞
S ( t ) = suitably-defined Sobolev-type (i.e., L 2 x
− type) norm
Moral conclusion: To avoid blow-up, show that S ( t ) remains finite.
Fundamental result with moral implications
Theorem (Littman, 1963)
Assume that − ∂ t
2
φ + ∂ i
2
φ = 0 in m ≥ 2 spatial dimensions.
i = 1
If and only if p = 2 , we have k∇ φ ( t ) k
L p x
≤ f ( t ) k∇ φ ( 0 ) k
L p x
, f = independent of φ
Moral conclusion: L 2 x
− based estimates ( energy estimates ) play an indispensable role in hyperbolic PDE.
Fundamental result with moral implications
Theorem (Littman, 1963)
Assume that − ∂ t
2
φ + ∂ i
2
φ = 0 in m ≥ 2 spatial dimensions.
i = 1
If and only if p = 2 , we have k∇ φ ( t ) k
L p x
≤ f ( t ) k∇ φ ( 0 ) k
L p x
, f = independent of φ
Moral conclusion: L 2 x
− based estimates ( energy estimates ) play an indispensable role in hyperbolic PDE.
Norms 0
Ω ∼ Ht , H
< def
= q p
1
Λ / 3
, h jk
= e
− 2 Ω g jk
, P = e 3 ( 1 + c 2 s
)Ω p ,
S g
00
+ 1 ; N
= e q Ω k ∂ t g
00 k
H N
+ e
( q − 1 )Ω k ∂ g
00 k
H N
+ e q Ω k g
00
+ 1 k
H N
,
S g
0 ∗
; N
= j = 1 e
( q − 1 )Ω k ∂ t g
0 j k
H N
+ e
( q − 2 )Ω k ∂ g
0 j k
H N
+ e
( q − 1 )Ω k g
0 j k
H N
,
S h
∗∗
; N
= e q Ω k ∂ t h jk k
H N
U
N j , k = 1
= e
( 1 + q )Ω v t
X k u j k 2
H N
,
+ e
( q − 1 )Ω k ∂ h jk k
H N j = 1
+ k ∂ h jk k
H N − 1
S
P − ¯ , u ∗
; N
S
N
= e
Ω k u j k
H N
+ k P − ¯ k
H N
,
= S
| g
00
+ 1 ; N
+ S g
0 ∗
; N
+ S h
{z
; N
+ U
N − 1
+ S
P − ¯ , u
∗
;
}
Measures the deviation from the FLRW solution
.
,
The FLRW stability theorem
Theorem (I. Rodnianski & J.S. 2009, J.S. 2010)
Assume that N ≥ 3 and 0 < c 2 s
< 1 / 3 .
Then there exist constants
S
N
( 0 ) ≤ C
0
− 1
> 0 and C > 1 such that for all ≤
0
, if
, then there exists a future global causal geodesically complete solution to the Euler-Einstein system with Λ > 0 .
Furthermore,
S
N
( t ) ≤ ( ⇐ continuation principle ) holds for all t ≥ 0 .
(i.e., the MGHD is singularity-free in the future direction)
• Previous work: Anderson, Brauer/Rendall/Reula, Friedrich,
Ringström , Isenberg, · · ·
Strategy of proof: derive norm inequalities
Lemma (J.S. 2010)
S
2
P − p , u ∗ ; N
( t ) ≤ S
2
P − ¯ , u ∗ ; N
( t
1
) + C
S
2 g
00
+ 1 ; N
( t ) ≤ S
2 g
00
+ 1 ; N
( t
1
) +
Z t
1
Z t t
S
U
2
N − 1
( t ) ≤ U
2
N − 1
( t
1
) +
S
2 g
0 ∗
; N
( t ) ≤ S
2 g
0 ∗
; N
( t
1
) +
2 h
∗∗
; N
( t ) ≤ S
2 h
∗∗
; N
( t
1
) +
Z t
τ = t
1
Z t
1 t t
1 z
2 ( 3 c
2 s
Z
− t
1
4 t
}|
0
{
− 1 + q ) e
( 1 + q )Ω
U
2
N − 1 e
− qH τ
S
2
N qHS
2 g
00
− 4 qHS
2 g
0 ∗
; N d
+ 1 ; N
+
He
− qH τ
S
2 h
∗∗
; N
C
+
τ,
+ Ce
− qH τ
S h
∗∗
Ce
; N
S g
0 ∗
; N
− qH τ
S
N
S
N
+ C S
N
U
N − 1 d τ,
S
+ g
00
Ce
S h
∗∗
; N
+ 1 ; N d
− qH τ
τ, d τ,
S
N
S g
0 ∗
; N d τ,
S
N
H def
= def
=
U p
N − 1
Λ / 3
+ S
P − ¯ , u ∗ ; N
+ S g
00
+ 1 ; N
+ S g
0 ∗
; N
+ S h
∗∗
; N
Geometric energy methods
Goal: geo-analytic structures
↔ useful coercive quantities
Ex: L = − 1
2
( g
− 1
)
αβ ∇
α
φ ∇
β
φ
L 0 s Euler-Lagrange eqn : ( g
− 1
)
αβ ∇
α
∇
β
φ = I
T
µν def
= 2
∂ L
∂ g
µν
+ ( g
− 1
)
µν
L
= ( g
− 1
)
µα
( g
− 1
)
νβ
∇
α
φ ∇
β
φ −
1
2
( g
− 1
)
µν
( g
− 1
)
αβ
∇
α
φ ∇
β
φ
Lemma encodes the
}|
∇
α
T
αµ
= −I ( g
− 1
{
)
µα
∇
α
φ
Vectorfield method of (John), Klainerman ,
Christodoulou
X
ν
( X )
J
= “multiplier vectorfield”
µ
[ ∇ φ, ∇ φ ] =
( X )
J
µ
= T
µ
ν
X
ν
= “compatible current”
∇
µ
( X ) J
µ
= −I X
α ∇
α
φ + T
µ
ν
∇
µ
X
ν
The divergence theorem
ξ
Σ t
Ω
ξ
Σ
0
Integral identity one for each X !
Z
“energy” z }| {
( X )
J
α
ξ
α
Σ t
=
Z
Σ
0
( X )
J
α
ξ
α
+
ZZ
Ω
−I X
α ∇
α
φ + T
ξ = oriented unit normal to Σ
µ
ν
∇
µ
X
ν
,
Coerciveness example for g = diag ( − 1 , 1 , 1 , 1 )
( X ) J
Σ
ξ t
µ
= { ( τ, x ) | τ = t }
= ( − 1 , 0 , 0 , 0 )
X = ( r − t ) 2 L + ( r + t ) 2
L = ∂ t
− ∂ r
, L = ∂ t
+ ∂ r
L , conformal Killing (Morawetz)
µ
ξ
µ
= T
µ
ν
ξ
µ
X
ν
=
1
|
2 n
( t − r )
2
( ∇
L
φ )
2
+ ( t + r )
2
( ∇
L
φ )
2
+ ( t
2
+ r
2
{z positivity ∼ dominant energy condition
) | 6∇ φ | 2 o
}
Morawetz-type coercive energy estimate (with | φ | 2 “correction”)
Z
Σ t Z
.
n
( t − r )
2 n r
2 |∇
L
|
Σ
0
|∇
L
φ | 2
φ | 2
+ r
+ (
2 t
|∇
L
+
φ | r
{z
2
)
2 |∇
L
φ | 2
+
“Data” r
2 | 6∇
+ (
φ | 2 t
2
+ |
+
φ | r
2
2
) | 6∇ φ | 2 o dx
}
+
ZZ
I n
( r − t )
2 ∇
L
φ + ( r + t )
2 ∇
L
φ o dxdt
+ | φ | 2 o dx
Ω
Null condition t t
− r
= co nst
L = ∂ t
− ∂ r
L = ∂ t
+ ∂ r angular directions g = diag ( − 1 , 1 , 1 , 1 ) g
αβ
L
α
L
β
= 0 = g
αβ
L
α g
αβ
L
α
L
β
= − 2
L
β
“null vectorfields”
( g
− 1
)
αβ ∇
α
φ ∇
β
φ = − ∇
L
φ ∇
L
φ + | 6∇ φ | 2
∼ “null condition”
Geometric methods: recent applications
Proofs of local well-posedness: Wong, Smulevici , J.S.
Global existence and almost global existence for nonlinear wave and wave-like equations: Katayama, Klainerman,
Lindblad, Morawetz, Sideris, Metcalfe/Nakamura/Sogge,
Keel/Smith/Sogge, Chae/Huh, Sterbenz , Yang , J.S.
The analysis of singular limits in hyperbolic PDEs: J.S.,
J.S./ Strain
Geometric continuation principles: Klainerman/Rodnianski,
Kommemi, Shao, Wang
Wave maps: Ringström
Geometric methods: recent applications
The decay of waves on black hole backgrounds & the stability of the Kerr black hole family:
Dafermos/Rodnianski, Baskin, Blue, Andersson/ Blue,
Soffer/ Blue, Holzegel, Schlue, Aretakis,
Finster/Kamran/Smoller/Yau, Luk , Tataru/ Tohaneanu
Black hole uniqueness (in smooth class): Klainerman,
Ionescu, Chrusciel, Alexakis, Nguyen, Wong
The evolutionary formation of trapped surfaces for the vacuum Einstein equations: Christodoulou,
Klainerman/Rodnianski , Yu
The formation of shocks for the 3 − dimensional Euler equations: Christodoulou
The proof of Strichartz-type inequalities for quasilinear wave equations: Klainerman
Regular hyperbolicity à la Christodoulou
φ = ( φ
1
, · · · , φ n
) :
R m →
R n h
αβ
AB
∂
α assume :
∂
β h
φ
B
αβ
AB
= I
A
= h
βα
BA assume : “hyperbolicity”
Theorem (J.S. & W. Wong, in preparation)
Under the above conditions, one can completely identify all of the ξ
µ
, X
ν that lead to a coercive (in the integrated sense) integral identity. Counterexamples if h
αβ
AB
= h
βα
BA
.
Canonical stress:
Q
µ
ν def
= h
( µλ )
AB
∂
λ
φ
B
∂
ν
φ
A − 1
2
δ
µ
ν h
( κλ )
AB
∂
κ
φ
A
∂
λ
φ
B ∼ T
µ
ν
Geometry of the equations
X
J p
I ∗ p
ξ
(a) C ∗ p
C ∗ p def
= Characteristic subset of T
∗ p
R m def
= { ζ | det h
αβ
AB
ζ
α
ζ
β
= 0 }
C p def
= Characteristic subset of T p R m
; is the “dual”
(b) C p
The Einstein-nonlinear electromagnetic equations
M =
R
1 + 3
Ric
µν
−
1
2
Rg
µν
( d F )
λµν
= T
µν
= 0
( d M )
λµν
= 0
In Maxwell’s theory:
•M
( Maxwell )
µν
=
?
F
µν
• T
( Maxwell )
µν
= F
µ
κ F
νκ
− 1
4 g
µν
F
κλ
F κλ
For many other Lagrangian theories:
•M
µν
= M
( Maxwell )
µν
• T
µν
= T
( Maxwell )
µν
+
+ quadratic “error” terms cubic “error” terms
The stability of Minkowski spacetime
Theorem (J.S. 2010; upcoming PNAS article)
The Minkowski spacetime solution e = diag ( − 1 , 1 , 1 , 1 ) , globally stable. The result holds for a large family of e
= 0 electromagnetic models that reduce to Maxwell’s theory in is the weak-field limit.
Proof: vectorfield method + regular hyperbolicity
Stability: Christodoulou-Klainerman, Lindblad-Rodnianski,
Bieri , Zipser, Choquet-Bruhat/Chrusciel/ Loizelet
Electromagnetism: Bialynicki-Birula, Boillat, Kiessling,
Tahvildar-Zadeh, Carley , · · · study e.g. the Maxwell-Born-Infeld model ( ↔ minimal surface equation)
Above results ∼ the alternate electromagnetic models are physically reasonable
Future research directions
MGHD - AIM workshop on shocks (with S. Klainerman, G.
Holzegel, W. Wong, P. Yu )
Sub-exponential expansion rates
←→
Wave equations on expanding Lorentzian manifolds
Alternative matter model: the relativistic Boltzmann equation (with R. Strain)
Future research directions continued
Singularity formation in general relativity
Regular hyperbolicity (with W. Wong)
• classification of integral identities for nonlinear electromagnetism
Great open problem
Maximal Globally Hyperbolic Developments
• Weak Cosmic Censorship Hypothesis
(R. Penrose, 1969)
I
+
Singularity r
=
0 r =
2
M
Black hole
Σ r
=
2
M
I
+
Far-away observer
D
I
−
I
− r
=
0
Figure: Schwarzschild solution of mass M (M. Dafermos)
Wanted list
Wanted: work on solving the Einstein constraint equations for some of the matter models:
• Euler-Einstein with Λ > 0
• Einstein-nonlinear electromagnetic
• Einstein-Boltzmann
Wanted: numerical evidence for the cases c
2 s
≥ 1 / 3
Wanted: blow-up results away from the small-data regime
Thank you
The “Divergence Problem”
Electric field surrounding an electron: E
( Maxwell )
R
= 4 π
R
∞ r = 0 r
1
2 dr = ∞ e
( Maxwell )
=
R
3
= −
E
( Maxwell )
| 2 dVol
| r | 2
Lorentz force: F
( Lorentz )
=
(at the electron’s location) q E
( Maxwell )
= Undefined
Analogous difficulties appear in QED
M. Born: “The attempts to combine Maxwell’s equations with the quantum theory (...) have not succeeded. One can see that the failure does not lie on the side of the quantum theory, but on the side of the field equations, which do not account for the existence of a radius of the electron (or its finite energy=mass).”
The “Divergence Problem”
Electric field surrounding an electron: E
( Maxwell )
R
= 4 π
R
∞ r = 0 r
1
2 dr = ∞ e
( Maxwell )
=
R
3
= −
E
( Maxwell )
| 2 dVol
| r | 2
Lorentz force: F
( Lorentz )
=
(at the electron’s location) q E
( Maxwell )
= Undefined
Analogous difficulties appear in QED
M. Born: “The attempts to combine Maxwell’s equations with the quantum theory (...) have not succeeded. One can see that the failure does not lie on the side of the quantum theory, but on the side of the field equations, which do not account for the existence of a radius of the electron (or its finite energy=mass).”
