Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Shock Formation in Solutions to 3D
Wave Equations
Jared Speck
with G. Holzegel, S. Klainerman, & W. Wong
Massachusetts Institute of Technology
April 27, 2015
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Overview of prior results
In 2007 (and with Miao in 2013), Christodoulou proved
a stable shock-formation result for small-data solutions
to the 3D Euler equations in irrotational regions.
His new contribution: sharp description of the blow-up.
Sharp description is needed to set up the weak
extension problem (recently solved by
Christodoulou-Lisibach in spherical symmetry).
Alinhac had previously derived related blow-up results,
but his Nash-Moser-based approach only reveals the
first singularity point.
Before Alinhac, many researchers had used proof by
contradiction to prove blow-up (e.g. John, Sideris,
Guo/Tahvildar-Zadeh) for related equations.
Proofs of blow-up in 1D and spherical symmetry are
classical (e.g. Lax, John, Majda, Klainerman/Majda).
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Overview of prior results
In 2007 (and with Miao in 2013), Christodoulou proved
a stable shock-formation result for small-data solutions
to the 3D Euler equations in irrotational regions.
His new contribution: sharp description of the blow-up.
Sharp description is needed to set up the weak
extension problem (recently solved by
Christodoulou-Lisibach in spherical symmetry).
Alinhac had previously derived related blow-up results,
but his Nash-Moser-based approach only reveals the
first singularity point.
Before Alinhac, many researchers had used proof by
contradiction to prove blow-up (e.g. John, Sideris,
Guo/Tahvildar-Zadeh) for related equations.
Proofs of blow-up in 1D and spherical symmetry are
classical (e.g. Lax, John, Majda, Klainerman/Majda).
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Overview of prior results
In 2007 (and with Miao in 2013), Christodoulou proved
a stable shock-formation result for small-data solutions
to the 3D Euler equations in irrotational regions.
His new contribution: sharp description of the blow-up.
Sharp description is needed to set up the weak
extension problem (recently solved by
Christodoulou-Lisibach in spherical symmetry).
Alinhac had previously derived related blow-up results,
but his Nash-Moser-based approach only reveals the
first singularity point.
Before Alinhac, many researchers had used proof by
contradiction to prove blow-up (e.g. John, Sideris,
Guo/Tahvildar-Zadeh) for related equations.
Proofs of blow-up in 1D and spherical symmetry are
classical (e.g. Lax, John, Majda, Klainerman/Majda).
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Overview of prior results
In 2007 (and with Miao in 2013), Christodoulou proved
a stable shock-formation result for small-data solutions
to the 3D Euler equations in irrotational regions.
His new contribution: sharp description of the blow-up.
Sharp description is needed to set up the weak
extension problem (recently solved by
Christodoulou-Lisibach in spherical symmetry).
Alinhac had previously derived related blow-up results,
but his Nash-Moser-based approach only reveals the
first singularity point.
Before Alinhac, many researchers had used proof by
contradiction to prove blow-up (e.g. John, Sideris,
Guo/Tahvildar-Zadeh) for related equations.
Proofs of blow-up in 1D and spherical symmetry are
classical (e.g. Lax, John, Majda, Klainerman/Majda).
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Overview of prior results
In 2007 (and with Miao in 2013), Christodoulou proved
a stable shock-formation result for small-data solutions
to the 3D Euler equations in irrotational regions.
His new contribution: sharp description of the blow-up.
Sharp description is needed to set up the weak
extension problem (recently solved by
Christodoulou-Lisibach in spherical symmetry).
Alinhac had previously derived related blow-up results,
but his Nash-Moser-based approach only reveals the
first singularity point.
Before Alinhac, many researchers had used proof by
contradiction to prove blow-up (e.g. John, Sideris,
Guo/Tahvildar-Zadeh) for related equations.
Proofs of blow-up in 1D and spherical symmetry are
classical (e.g. Lax, John, Majda, Klainerman/Majda).
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
New results summary
Our results:
In 3D, we extended Christodoulou’s results to:
i) g(Ψ) Ψ = 0,
ii) (g −1 )αβ (∂Φ)∂α ∂β Φ = 0
For the equations, small-data shock formation occurs
⇐⇒ Klainerman’s null condition fails
Gave a sharp description of the singularity and the
blow-up mechanism; Ψ, ∂φ remain bounded, ∂Ψ, ∂ 2 φ
blow up
Simplified many aspects of the proof
Miao-Yu recently derived a related large-data
shock-formation result for −(1 + (∂t φ)2 )∂t2 φ + ∆φ = 0
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
New results summary
Our results:
In 3D, we extended Christodoulou’s results to:
i) g(Ψ) Ψ = 0,
ii) (g −1 )αβ (∂Φ)∂α ∂β Φ = 0
For the equations, small-data shock formation occurs
⇐⇒ Klainerman’s null condition fails
Gave a sharp description of the singularity and the
blow-up mechanism; Ψ, ∂φ remain bounded, ∂Ψ, ∂ 2 φ
blow up
Simplified many aspects of the proof
Miao-Yu recently derived a related large-data
shock-formation result for −(1 + (∂t φ)2 )∂t2 φ + ∆φ = 0
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
New results summary
Our results:
In 3D, we extended Christodoulou’s results to:
i) g(Ψ) Ψ = 0,
ii) (g −1 )αβ (∂Φ)∂α ∂β Φ = 0
For the equations, small-data shock formation occurs
⇐⇒ Klainerman’s null condition fails
Gave a sharp description of the singularity and the
blow-up mechanism; Ψ, ∂φ remain bounded, ∂Ψ, ∂ 2 φ
blow up
Simplified many aspects of the proof
Miao-Yu recently derived a related large-data
shock-formation result for −(1 + (∂t φ)2 )∂t2 φ + ∆φ = 0
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
New results summary
Our results:
In 3D, we extended Christodoulou’s results to:
i) g(Ψ) Ψ = 0,
ii) (g −1 )αβ (∂Φ)∂α ∂β Φ = 0
For the equations, small-data shock formation occurs
⇐⇒ Klainerman’s null condition fails
Gave a sharp description of the singularity and the
blow-up mechanism; Ψ, ∂φ remain bounded, ∂Ψ, ∂ 2 φ
blow up
Simplified many aspects of the proof
Miao-Yu recently derived a related large-data
shock-formation result for −(1 + (∂t φ)2 )∂t2 φ + ∆φ = 0
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
The maximal development of the data
singular
H
C
H
regular
∂− H
Figure: The maximal development
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
The covariant wave operator
{x α }α=0,1,2,3 are rectangular coordinates with x 0 = t
Pert
Pert
gαβ (Ψ) = mαβ + gαβ
(Ψ),
gαβ
(0) = 0
m = diag(−1, 1, 1, 1)
g(Ψ) Ψ := (g −1 )αβ ∂α ∂β Ψ − (g −1 )αβ (g −1 )κλ Γακβ ∂λ Ψ
Γακβ :=
1
2
{Gκβ ∂α Ψ + Gακ ∂β Ψ − Gαβ ∂κ Ψ}
Gαβ (Ψ) :=
d
g (Ψ)
dΨ αβ
Remark
Alinhac and Lindblad showed small-data global existence
for (g −1 )αβ ∂α ∂β Ψ = 0.
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
The covariant wave operator
{x α }α=0,1,2,3 are rectangular coordinates with x 0 = t
Pert
Pert
gαβ (Ψ) = mαβ + gαβ
(Ψ),
gαβ
(0) = 0
m = diag(−1, 1, 1, 1)
g(Ψ) Ψ := (g −1 )αβ ∂α ∂β Ψ − (g −1 )αβ (g −1 )κλ Γακβ ∂λ Ψ
Γακβ :=
1
2
{Gκβ ∂α Ψ + Gακ ∂β Ψ − Gαβ ∂κ Ψ}
Gαβ (Ψ) :=
d
g (Ψ)
dΨ αβ
Remark
Alinhac and Lindblad showed small-data global existence
for (g −1 )αβ ∂α ∂β Ψ = 0.
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
The covariant wave operator
{x α }α=0,1,2,3 are rectangular coordinates with x 0 = t
Pert
Pert
gαβ (Ψ) = mαβ + gαβ
(Ψ),
gαβ
(0) = 0
m = diag(−1, 1, 1, 1)
g(Ψ) Ψ := (g −1 )αβ ∂α ∂β Ψ − (g −1 )αβ (g −1 )κλ Γακβ ∂λ Ψ
Γακβ :=
1
2
{Gκβ ∂α Ψ + Gακ ∂β Ψ − Gαβ ∂κ Ψ}
Gαβ (Ψ) :=
d
g (Ψ)
dΨ αβ
Remark
Alinhac and Lindblad showed small-data global existence
for (g −1 )αβ ∂α ∂β Ψ = 0.
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
The covariant wave operator
{x α }α=0,1,2,3 are rectangular coordinates with x 0 = t
Pert
Pert
gαβ (Ψ) = mαβ + gαβ
(Ψ),
gαβ
(0) = 0
m = diag(−1, 1, 1, 1)
g(Ψ) Ψ := (g −1 )αβ ∂α ∂β Ψ − (g −1 )αβ (g −1 )κλ Γακβ ∂λ Ψ
Γακβ :=
1
2
{Gκβ ∂α Ψ + Gακ ∂β Ψ − Gαβ ∂κ Ψ}
Gαβ (Ψ) :=
d
g (Ψ)
dΨ αβ
Remark
Alinhac and Lindblad showed small-data global existence
for (g −1 )αβ ∂α ∂β Ψ = 0.
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
The dynamic region
St,0
Σut
St,u
6
Cut
t
S0,u
Σu0
S0,0 nontrivial data
C0t
trivial data
Ψ≡0
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Eikonal function
Most important proof ingredient is the eikonal function:
Du · Du := (g −1 )αβ (Ψ)∂α u∂β u = 0,
∂t u > 0,
u|t=0 = 1 − r
Truncated level sets are null cone pieces Cut
Play a critical role in many local and global results for
wave equations, especially Einstein’s equations
(starting with the Christodoulou-Klainerman proof of
the stability of Minkowski spacetime)
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Eikonal function
Most important proof ingredient is the eikonal function:
Du · Du := (g −1 )αβ (Ψ)∂α u∂β u = 0,
∂t u > 0,
u|t=0 = 1 − r
Truncated level sets are null cone pieces Cut
Play a critical role in many local and global results for
wave equations, especially Einstein’s equations
(starting with the Christodoulou-Klainerman proof of
the stability of Minkowski spacetime)
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Fundamental geometric constructions
Cut
C0t
R̆
L
X ,X
µ≈1 1 2
R̆
L
µ is small
X1 , X2
µ−1 := −Dt · Du, shock ⇐⇒ µ → 0
Lν := −µDν u, L ∼ ∂t + ∂r
R̆u = 1, g(R̆, R̆) = µ2 , shock ⇐⇒ R̆ → 0
X1 , X2 are local vectorfields on St,u = Σt ∩ Cut
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
The system of equations
g(Ψ) Ψ = 0,
1
Lµ = GLL R̆Ψ + Error(LΨ, ∇
/ Ψ, · · · ),
2
LLi = Error(LΨ, ∇
/ Ψ, · · · )
Last two equations are ∼ Du · Du = 0
∇
/ := connection of g/, the metric on the spheres St,u
L = ∂t∂ |u,ϑ
∂
| + angular error
∂u t,ϑ
d
GLL := dΨ
gαβ (Ψ)Lα Lβ
R̆ =
Classic null condition fails =⇒ GLL ∼ f (ϑ) 6= 0
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
The system of equations
g(Ψ) Ψ = 0,
1
Lµ = GLL R̆Ψ + Error(LΨ, ∇
/ Ψ, · · · ),
2
LLi = Error(LΨ, ∇
/ Ψ, · · · )
Last two equations are ∼ Du · Du = 0
∇
/ := connection of g/, the metric on the spheres St,u
L = ∂t∂ |u,ϑ
∂
| + angular error
∂u t,ϑ
d
GLL := dΨ
gαβ (Ψ)Lα Lβ
R̆ =
Classic null condition fails =⇒ GLL ∼ f (ϑ) 6= 0
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
A nonlinear analog of the null condition
Set %(t, u) := 1 − u + t ≈ 1 + t. Then µg Ψ = 0 is:
n
o
L µL(%Ψ) + 2R̆(%Ψ) = %µ∆
/ Ψ + Error(µ∂Ψ, ∇
/ µ, µ∇
/ Li )
• Key structure: Error consists of quadratic and higher
terms, where (R̆Ψ)2 , (R̆Ψ)3 , etc. are not present.
• This is a nonlinear analog of the classic null condition.
“Strong null condition”
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
A nonlinear analog of the null condition
Set %(t, u) := 1 − u + t ≈ 1 + t. Then µg Ψ = 0 is:
n
o
L µL(%Ψ) + 2R̆(%Ψ) = %µ∆
/ Ψ + Error(µ∂Ψ, ∇
/ µ, µ∇
/ Li )
• Key structure: Error consists of quadratic and higher
terms, where (R̆Ψ)2 , (R̆Ψ)3 , etc. are not present.
• This is a nonlinear analog of the classic null condition.
“Strong null condition”
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Dispersive behavior in the rescaled frame
•˚
= data size
Bootstrap “linear” behavior relative to the rescaled frame
Relative to the rescaled frame, we should see the
dispersive behavior of linear waves:
1
,
(1 + t)2
1
|Ψ|, |R̆Ψ| . ˚
1+t
|LΨ|, |∇
/ Ψ| . ˚
1
Can prove: R̆Ψ ∼ fData (u, ϑ) 1+t
, fData = O(˚
) 6≡ 0
We make related bootstrap assumptions for the
mid-order derivatives of Ψ and for µ and Li
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Dispersive behavior in the rescaled frame
•˚
= data size
Bootstrap “linear” behavior relative to the rescaled frame
Relative to the rescaled frame, we should see the
dispersive behavior of linear waves:
1
,
(1 + t)2
1
|Ψ|, |R̆Ψ| . ˚
1+t
|LΨ|, |∇
/ Ψ| . ˚
1
Can prove: R̆Ψ ∼ fData (u, ϑ) 1+t
, fData = O(˚
) 6≡ 0
We make related bootstrap assumptions for the
mid-order derivatives of Ψ and for µ and Li
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Dispersive behavior in the rescaled frame
•˚
= data size
Bootstrap “linear” behavior relative to the rescaled frame
Relative to the rescaled frame, we should see the
dispersive behavior of linear waves:
1
,
(1 + t)2
1
|Ψ|, |R̆Ψ| . ˚
1+t
|LΨ|, |∇
/ Ψ| . ˚
1
Can prove: R̆Ψ ∼ fData (u, ϑ) 1+t
, fData = O(˚
) 6≡ 0
We make related bootstrap assumptions for the
mid-order derivatives of Ψ and for µ and Li
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Heuristics behind the lower bound for |R̆Ψ|
Ignoring RHS, we integrate along L = ∂t∂ to deduce:
n
o
µL(%Ψ) + 2R̆(%Ψ) (t, u, ϑ) ∼ fData (u, ϑ)
From dispersive estimates and L% = −R̆% = 1, we obtain:
R̆Ψ(t, u, ϑ) ∼
1 1
1
fData (u, ϑ) ∼
fData (u, ϑ)
2 %(t, u)
1+t
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Heuristics behind the lower bound for |R̆Ψ|
Ignoring RHS, we integrate along L = ∂t∂ to deduce:
n
o
µL(%Ψ) + 2R̆(%Ψ) (t, u, ϑ) ∼ fData (u, ϑ)
From dispersive estimates and L% = −R̆% = 1, we obtain:
R̆Ψ(t, u, ϑ) ∼
1 1
1
fData (u, ϑ) ∼
fData (u, ϑ)
2 %(t, u)
1+t
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Heuristics behind µ → 0 and blow-up
Lµ(t, u, ϑ) ∼ h(ϑ)R̆Ψ(t, u, ϑ)
1
Using R̆Ψ ∼ fData (u, ϑ) 1+t
, we obtain
Lµ(t, u, ϑ) ∼ h(ϑ)fData (u, ϑ)
Since L =
∂
,
∂t
1
,
1+t
hf = O(˚
) 6≡ 0
we integrate dt and use µ|t=0 ≈ 1 to deduce:
µ(t, u, ϑ) ∼ 1 + h(ϑ)fData (u, ϑ) ln(1 + t)
Hence, µ → 0 at TShock ∼ exp {| minu,ϑ 1/(hfData )|}. Moreover,
R̆Ψ is non-zero where µ vanishes,
and |R̆|g = µ =⇒ |∂Ψ| blows-up like µ−1 .
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Heuristics behind µ → 0 and blow-up
Lµ(t, u, ϑ) ∼ h(ϑ)R̆Ψ(t, u, ϑ)
1
Using R̆Ψ ∼ fData (u, ϑ) 1+t
, we obtain
Lµ(t, u, ϑ) ∼ h(ϑ)fData (u, ϑ)
Since L =
∂
,
∂t
1
,
1+t
hf = O(˚
) 6≡ 0
we integrate dt and use µ|t=0 ≈ 1 to deduce:
µ(t, u, ϑ) ∼ 1 + h(ϑ)fData (u, ϑ) ln(1 + t)
Hence, µ → 0 at TShock ∼ exp {| minu,ϑ 1/(hfData )|}. Moreover,
R̆Ψ is non-zero where µ vanishes,
and |R̆|g = µ =⇒ |∂Ψ| blows-up like µ−1 .
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Heuristics behind µ → 0 and blow-up
Lµ(t, u, ϑ) ∼ h(ϑ)R̆Ψ(t, u, ϑ)
1
Using R̆Ψ ∼ fData (u, ϑ) 1+t
, we obtain
Lµ(t, u, ϑ) ∼ h(ϑ)fData (u, ϑ)
Since L =
∂
,
∂t
1
,
1+t
hf = O(˚
) 6≡ 0
we integrate dt and use µ|t=0 ≈ 1 to deduce:
µ(t, u, ϑ) ∼ 1 + h(ϑ)fData (u, ϑ) ln(1 + t)
Hence, µ → 0 at TShock ∼ exp {| minu,ϑ 1/(hfData )|}. Moreover,
R̆Ψ is non-zero where µ vanishes,
and |R̆|g = µ =⇒ |∂Ψ| blows-up like µ−1 .
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Heuristics behind µ → 0 and blow-up
Lµ(t, u, ϑ) ∼ h(ϑ)R̆Ψ(t, u, ϑ)
1
Using R̆Ψ ∼ fData (u, ϑ) 1+t
, we obtain
Lµ(t, u, ϑ) ∼ h(ϑ)fData (u, ϑ)
Since L =
∂
,
∂t
1
,
1+t
hf = O(˚
) 6≡ 0
we integrate dt and use µ|t=0 ≈ 1 to deduce:
µ(t, u, ϑ) ∼ 1 + h(ϑ)fData (u, ϑ) ln(1 + t)
Hence, µ → 0 at TShock ∼ exp {| minu,ϑ 1/(hfData )|}. Moreover,
R̆Ψ is non-zero where µ vanishes,
and |R̆|g = µ =⇒ |∂Ψ| blows-up like µ−1 .
Intro
Basic setup
Estimates
Energy hierarchy
Justifying the heuristics
We need to show that
Error terms are small
Solution survives as long as µ > 0
The only known methods involve commuting the
equations and deriving a family of L2 estimates.
Two main ingredients:
A priori L2 estimates for
Ψ, µ, Li
| {z }
hard at top order!
k
A priori C estimates for Ψ, µ, Li
Strong Null Condition
Intro
Basic setup
Estimates
Energy hierarchy
Justifying the heuristics
We need to show that
Error terms are small
Solution survives as long as µ > 0
The only known methods involve commuting the
equations and deriving a family of L2 estimates.
Two main ingredients:
A priori L2 estimates for
Ψ, µ, Li
| {z }
hard at top order!
k
A priori C estimates for Ψ, µ, Li
Strong Null Condition
Intro
Basic setup
Estimates
Energy hierarchy
Justifying the heuristics
We need to show that
Error terms are small
Solution survives as long as µ > 0
The only known methods involve commuting the
equations and deriving a family of L2 estimates.
Two main ingredients:
A priori L2 estimates for
Ψ, µ, Li
| {z }
hard at top order!
k
A priori C estimates for Ψ, µ, Li
Strong Null Condition
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Energy estimates for g(Ψ)Ψ = 0
Z
E0 (t, u) ∼
Σut
Z
F0 (t, u) ∼
µ|LΨ|2 + |R̆Ψ|2 + µ|∇
/ Ψ|2 ,
|LΨ|2 + µ|∇
/ Ψ|2
Cut
We derive energy estimates with the timelike multiplier
T := (1 + 2µ)L + 2R̆.
The implicit volume forms are bounded away from 0.
For µ small, the energies provide very weak control of
angular derivatives.
All factors of |∇
/ Ψ|2 come with a µ weight.
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Energy estimates for g(Ψ)Ψ = 0
Z
E0 (t, u) ∼
Σut
Z
F0 (t, u) ∼
µ|LΨ|2 + |R̆Ψ|2 + µ|∇
/ Ψ|2 ,
|LΨ|2 + µ|∇
/ Ψ|2
Cut
We derive energy estimates with the timelike multiplier
T := (1 + 2µ)L + 2R̆.
The implicit volume forms are bounded away from 0.
For µ small, the energies provide very weak control of
angular derivatives.
All factors of |∇
/ Ψ|2 come with a µ weight.
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Morawetz estimates for g(Ψ)Ψ = 0
Z
1
2
e
µ|(LΨ + trχΨ)|2 + µ|∇
E0 (t, u) ∼ (1 + t)
/ Ψ|2 ,
u
2
Σt
Z
1
e0 (t, u) ∼
F
(1 + t 0 )2 |(LΨ + trχΨ)|2
2
Cut
Morawetz estimates are derived with the null multiplier
e ∼ (1 + t)2 L + corrections
K
e estimate yields an integral that controls |∇
K
/ Ψ|2 near
the shock with
Z t no
Z µ degeneracy!!:
e 0 (t, u) ∼
K
s2 [Lµ]− |∇
/ Ψ|2 ds, [Lµ]− = max{0, −Lµ}
s=0
Σus
Crucial estimate:
1
µ(s, u, ϑ) < 41 =⇒ [Lµ]− (s, u, ϑ) > C (1+s) ln(e+s)
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Morawetz estimates for g(Ψ)Ψ = 0
Z
1
2
e
µ|(LΨ + trχΨ)|2 + µ|∇
E0 (t, u) ∼ (1 + t)
/ Ψ|2 ,
u
2
Σt
Z
1
e0 (t, u) ∼
F
(1 + t 0 )2 |(LΨ + trχΨ)|2
2
Cut
Morawetz estimates are derived with the null multiplier
e ∼ (1 + t)2 L + corrections
K
e estimate yields an integral that controls |∇
K
/ Ψ|2 near
the shock with
Z t no
Z µ degeneracy!!:
e 0 (t, u) ∼
K
s2 [Lµ]− |∇
/ Ψ|2 ds, [Lµ]− = max{0, −Lµ}
s=0
Σus
Crucial estimate:
1
µ(s, u, ϑ) < 41 =⇒ [Lµ]− (s, u, ϑ) > C (1+s) ln(e+s)
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Morawetz estimates for g(Ψ)Ψ = 0
Z
1
2
e
µ|(LΨ + trχΨ)|2 + µ|∇
E0 (t, u) ∼ (1 + t)
/ Ψ|2 ,
u
2
Σt
Z
1
e0 (t, u) ∼
F
(1 + t 0 )2 |(LΨ + trχΨ)|2
2
Cut
Morawetz estimates are derived with the null multiplier
e ∼ (1 + t)2 L + corrections
K
e estimate yields an integral that controls |∇
K
/ Ψ|2 near
the shock with
Z t no
Z µ degeneracy!!:
e 0 (t, u) ∼
K
s2 [Lµ]− |∇
/ Ψ|2 ds, [Lµ]− = max{0, −Lµ}
s=0
Σus
Crucial estimate:
1
µ(s, u, ϑ) < 41 =⇒ [Lµ]− (s, u, ϑ) > C (1+s) ln(e+s)
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Morawetz estimates for g(Ψ)Ψ = 0
Z
1
2
e
µ|(LΨ + trχΨ)|2 + µ|∇
E0 (t, u) ∼ (1 + t)
/ Ψ|2 ,
u
2
Σt
Z
1
e0 (t, u) ∼
F
(1 + t 0 )2 |(LΨ + trχΨ)|2
2
Cut
Morawetz estimates are derived with the null multiplier
e ∼ (1 + t)2 L + corrections
K
e estimate yields an integral that controls |∇
K
/ Ψ|2 near
the shock with
Z t no
Z µ degeneracy!!:
e 0 (t, u) ∼
K
s2 [Lµ]− |∇
/ Ψ|2 ds, [Lµ]− = max{0, −Lµ}
s=0
Σus
Crucial estimate:
1
µ(s, u, ϑ) < 41 =⇒ [Lµ]− (s, u, ϑ) > C (1+s) ln(e+s)
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Main a priori L2 estimate hierarchy
(exceptionally hard!)
Let ˚
:= data size, µ? (t, u) := min{1, minΣut µ}. Up to ln t
factors at top-order, we have
E24 (t, u) + F24 (t, u) . ˚
2 µ−17.5
(t, u)
?
2 −15.5
E23 (t, u) + F23 (t, u) . ˚
µ?
(t, u)
···
E16 (t, u) + F16 (t, u) . ˚
2 µ−1.5
(t, u)
?
E15 (t, u) + F15 (t, u) . ˚
2
···
E0 (t, u) + F0 (t, u) . ˚
2
We recover C k bootstrap assumptions for Ψ via
Sobolev
We recover C k bootstrap assumptions for µ, Li ∼ ∂u
from transport equations that lose derivatives
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Main a priori L2 estimate hierarchy
(exceptionally hard!)
Let ˚
:= data size, µ? (t, u) := min{1, minΣut µ}. Up to ln t
factors at top-order, we have
E24 (t, u) + F24 (t, u) . ˚
2 µ−17.5
(t, u)
?
2 −15.5
E23 (t, u) + F23 (t, u) . ˚
µ?
(t, u)
···
E16 (t, u) + F16 (t, u) . ˚
2 µ−1.5
(t, u)
?
E15 (t, u) + F15 (t, u) . ˚
2
···
E0 (t, u) + F0 (t, u) . ˚
2
We recover C k bootstrap assumptions for Ψ via
Sobolev
We recover C k bootstrap assumptions for µ, Li ∼ ∂u
from transport equations that lose derivatives
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Top-order degeneracy
The best top-order inequality we are able to obtain is:
Z
E24 (t, u) ≤ data + B
t
t 0 =0
!
Lµ sup E24 (t 0 , u) dt 0 + · · ·
µ
Σt 0
B ∼ 10
The key degenerate factor supΣt 0 |L ln µ| is connected to
(avoiding) derivative loss in the top derivatives of u.
Recall that L = ∂t∂ |u,ϑ
Caricature estimate: if µ were a function of only t, then
Gronwall =⇒ E24 (t, u) ≤ data × µ−B (t) + · · ·
The actual Gronwall-type estimate leads to
E24 (t, u) ≤ data × lnA (e + t)µ−B
? (t, u) and is hard to
prove because µ = µ(t, u, ϑ)
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Top-order degeneracy
The best top-order inequality we are able to obtain is:
Z
E24 (t, u) ≤ data + B
t
t 0 =0
!
Lµ sup E24 (t 0 , u) dt 0 + · · ·
µ
Σt 0
B ∼ 10
The key degenerate factor supΣt 0 |L ln µ| is connected to
(avoiding) derivative loss in the top derivatives of u.
Recall that L = ∂t∂ |u,ϑ
Caricature estimate: if µ were a function of only t, then
Gronwall =⇒ E24 (t, u) ≤ data × µ−B (t) + · · ·
The actual Gronwall-type estimate leads to
E24 (t, u) ≤ data × lnA (e + t)µ−B
? (t, u) and is hard to
prove because µ = µ(t, u, ϑ)
Intro
Basic setup
Estimates
Energy hierarchy
The tensorfield χ
χAB := g(DA L, XB ) ∼ ∇
/ ∂u,
trχ := (g/−1 )AB χAB
Strong Null Condition
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Naive energy estimates lose derivatives
Commute g Ψ = 0 with Z := {%L, R̆, O(1) , O(2) , O(3) } ∼ ∂u
Rotation vectorfields O(1) , O(2) , O(3)
O := Π
/OEuclidean ,
Π
/ := St,u projection
OEuclidean = O + small × µ−1 R̆
| {z }
Danger!!
To estimate 2 derivatives of Ψ, commute once:
g (OΨ) = (R̆Ψ)Otrχ + · · ·
∼ (µ∂Ψ)∂ 3 u + · · ·
Suggests derivative loss because
(g −1 )αβ (Ψ)∂α u∂β u = 0 =⇒ L(∂ 3 u) ∼ f (∂ 3 Ψ, · · · )
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Naive energy estimates lose derivatives
Commute g Ψ = 0 with Z := {%L, R̆, O(1) , O(2) , O(3) } ∼ ∂u
Rotation vectorfields O(1) , O(2) , O(3)
O := Π
/OEuclidean ,
Π
/ := St,u projection
OEuclidean = O + small × µ−1 R̆
| {z }
Danger!!
To estimate 2 derivatives of Ψ, commute once:
g (OΨ) = (R̆Ψ)Otrχ + · · ·
∼ (µ∂Ψ)∂ 3 u + · · ·
Suggests derivative loss because
(g −1 )αβ (Ψ)∂α u∂β u = 0 =⇒ L(∂ 3 u) ∼ f (∂ 3 Ψ, · · · )
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Naive energy estimates lose derivatives
Commute g Ψ = 0 with Z := {%L, R̆, O(1) , O(2) , O(3) } ∼ ∂u
Rotation vectorfields O(1) , O(2) , O(3)
O := Π
/OEuclidean ,
Π
/ := St,u projection
OEuclidean = O + small × µ−1 R̆
| {z }
Danger!!
To estimate 2 derivatives of Ψ, commute once:
g (OΨ) = (R̆Ψ)Otrχ + · · ·
∼ (µ∂Ψ)∂ 3 u + · · ·
Suggests derivative loss because
(g −1 )αβ (Ψ)∂α u∂β u = 0 =⇒ L(∂ 3 u) ∼ f (∂ 3 Ψ, · · · )
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Naive energy estimates lose derivatives
Commute g Ψ = 0 with Z := {%L, R̆, O(1) , O(2) , O(3) } ∼ ∂u
Rotation vectorfields O(1) , O(2) , O(3)
O := Π
/OEuclidean ,
Π
/ := St,u projection
OEuclidean = O + small × µ−1 R̆
| {z }
Danger!!
To estimate 2 derivatives of Ψ, commute once:
g (OΨ) = (R̆Ψ)Otrχ + · · ·
∼ (µ∂Ψ)∂ 3 u + · · ·
Suggests derivative loss because
(g −1 )αβ (Ψ)∂α u∂β u = 0 =⇒ L(∂ 3 u) ∼ f (∂ 3 Ψ, · · · )
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Avoiding derivative loss with Raychaudhuri
Recall that g Ψ = 0 =⇒ µ∆
/ Ψ = L(µ∂Ψ) + l.o.t.
Using curvature, we derive the key Raychaudhuri identity:
L(µtrχ) = L(µ∂Ψ) + µ∆
/ Ψ + l.o.t.
= L(µ∂Ψ) + l.o.t.
Hence, commuting with O, we derive a good eqn. for M:


:=M
z
}|
{


L µOtrχ − GLL R̆OΨ + · · · ∼ µ∂ 2 Ψ + µ (LO χ̂) ·χ̂ + · · ·
| {z }
∂3u
Identity was first used by Klainerman-Rodnianski to
prove low-regularity local well-posedness
The term LO χ̂ can be treated with elliptic estimates
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Avoiding derivative loss with Raychaudhuri
Recall that g Ψ = 0 =⇒ µ∆
/ Ψ = L(µ∂Ψ) + l.o.t.
Using curvature, we derive the key Raychaudhuri identity:
L(µtrχ) = L(µ∂Ψ) + µ∆
/ Ψ + l.o.t.
= L(µ∂Ψ) + l.o.t.
Hence, commuting with O, we derive a good eqn. for M:


:=M
z
}|
{


L µOtrχ − GLL R̆OΨ + · · · ∼ µ∂ 2 Ψ + µ (LO χ̂) ·χ̂ + · · ·
| {z }
∂3u
Identity was first used by Klainerman-Rodnianski to
prove low-regularity local well-posedness
The term LO χ̂ can be treated with elliptic estimates
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Avoiding derivative loss with Raychaudhuri
Recall that g Ψ = 0 =⇒ µ∆
/ Ψ = L(µ∂Ψ) + l.o.t.
Using curvature, we derive the key Raychaudhuri identity:
L(µtrχ) = L(µ∂Ψ) + µ∆
/ Ψ + l.o.t.
= L(µ∂Ψ) + l.o.t.
Hence, commuting with O, we derive a good eqn. for M:


:=M
z
}|
{


L µOtrχ − GLL R̆OΨ + · · · ∼ µ∂ 2 Ψ + µ (LO χ̂) ·χ̂ + · · ·
| {z }
∂3u
Identity was first used by Klainerman-Rodnianski to
prove low-regularity local well-posedness
The term LO χ̂ can be treated with elliptic estimates
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Top-order energy estimates lead to µ loss
Using M = µOtrχ − GLL R̆OΨ + · · · and Lµ = 12 GLL R̆Ψ + · · · , we
obtain
Z t Z
E[OΨ](t, u) ≤ data − 2
(R̆Ψ)(Otrχ)(R̆OΨ) dt 0 + · · ·
t 0 =0
t
Z
Σt 0
Z
≤ data − 2
t 0 =0
Z
t
Σt 0
1
GLL (R̆Ψ)(R̆OΨ)2 dt 0
µ
Z
1
(R̆Ψ)M(R̆OΨ) dt 0 + · · ·
µ
|
{z
}
very difficult; we ignore it here
Z t Z
Lµ
≤ data − 4
(R̆OΨ)2 dt 0 + · · ·
t 0 =0 Σt 0 µ
−2
t 0 =0
Σt 0
=⇒
Z
E[OΨ](t, u) ≤ data + B
t
t 0 =0
!
Lµ sup E[OΨ](t 0 , u) dt 0 + · · ·
µ
Σ
t0
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Top-order energy estimates lead to µ loss
Using M = µOtrχ − GLL R̆OΨ + · · · and Lµ = 12 GLL R̆Ψ + · · · , we
obtain
Z t Z
E[OΨ](t, u) ≤ data − 2
(R̆Ψ)(Otrχ)(R̆OΨ) dt 0 + · · ·
t 0 =0
t
Z
Σt 0
Z
≤ data − 2
t 0 =0
Z
t
Σt 0
1
GLL (R̆Ψ)(R̆OΨ)2 dt 0
µ
Z
1
(R̆Ψ)M(R̆OΨ) dt 0 + · · ·
µ
|
{z
}
very difficult; we ignore it here
Z t Z
Lµ
≤ data − 4
(R̆OΨ)2 dt 0 + · · ·
t 0 =0 Σt 0 µ
−2
t 0 =0
Σt 0
=⇒
Z
E[OΨ](t, u) ≤ data + B
t
t 0 =0
!
Lµ sup E[OΨ](t 0 , u) dt 0 + · · ·
µ
Σ
t0
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
µ−1
? improvements at the lower orders
Below top order, we can allow a loss of one derivative
in our estimate for Otrχ
All error integrals are cubic and can be bounded with a
bootstrap argument
Z t
+
0
0
1−B
Key bound:
(1 + t 0 )−(1 ) µ−B
(t, u)
? (t , u) dt . µ?
t 0 =0
=⇒ can gain powers of µ? via t decay
Proof uses a sharp version of µ? (t, u) ∼ 1 − C˚
ln t
3
Can show: EBelow−top [OΨ](t, u) . data + ˚
3
=⇒ EBelow−top [OΨ](t, u) ≤ (data + C˚
)×
−1
• Improvement in the power of µ? !!
Z
t
µ1−B
(t 0 , u) 0
?
dt
(1 + t 0 )3/2
t 0 =0
2−B
µ? (t, u)
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
µ−1
? improvements at the lower orders
Below top order, we can allow a loss of one derivative
in our estimate for Otrχ
All error integrals are cubic and can be bounded with a
bootstrap argument
Z t
+
0
0
1−B
Key bound:
(1 + t 0 )−(1 ) µ−B
(t, u)
? (t , u) dt . µ?
t 0 =0
=⇒ can gain powers of µ? via t decay
Proof uses a sharp version of µ? (t, u) ∼ 1 − C˚
ln t
3
Can show: EBelow−top [OΨ](t, u) . data + ˚
3
=⇒ EBelow−top [OΨ](t, u) ≤ (data + C˚
)×
−1
• Improvement in the power of µ? !!
Z
t
(t 0 , u) 0
µ1−B
?
dt
(1 + t 0 )3/2
t 0 =0
2−B
µ? (t, u)
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
µ−1
? improvements at the lower orders
Below top order, we can allow a loss of one derivative
in our estimate for Otrχ
All error integrals are cubic and can be bounded with a
bootstrap argument
Z t
+
0
0
1−B
Key bound:
(1 + t 0 )−(1 ) µ−B
(t, u)
? (t , u) dt . µ?
t 0 =0
=⇒ can gain powers of µ? via t decay
Proof uses a sharp version of µ? (t, u) ∼ 1 − C˚
ln t
3
Can show: EBelow−top [OΨ](t, u) . data + ˚
3
=⇒ EBelow−top [OΨ](t, u) ≤ (data + C˚
)×
−1
• Improvement in the power of µ? !!
Z
t
(t 0 , u) 0
µ1−B
?
dt
(1 + t 0 )3/2
t 0 =0
2−B
µ? (t, u)
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
µ−1
? improvements at the lower orders
Below top order, we can allow a loss of one derivative
in our estimate for Otrχ
All error integrals are cubic and can be bounded with a
bootstrap argument
Z t
+
0
0
1−B
Key bound:
(1 + t 0 )−(1 ) µ−B
(t, u)
? (t , u) dt . µ?
t 0 =0
=⇒ can gain powers of µ? via t decay
Proof uses a sharp version of µ? (t, u) ∼ 1 − C˚
ln t
3
Can show: EBelow−top [OΨ](t, u) . data + ˚
3
=⇒ EBelow−top [OΨ](t, u) ≤ (data + C˚
)×
−1
• Improvement in the power of µ? !!
Z
t
(t 0 , u) 0
µ1−B
?
dt
(1 + t 0 )3/2
t 0 =0
2−B
µ? (t, u)
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
g(Ψ)Ψ = 0 vs. (g −1)αβ (∂φ)∂α ∂β φ = 0
We commute the following equation with ∂ν :
(g −1 )αβ (∂φ)∂α ∂β φ = 0
~ := (Ψ0 , Ψ1 , Ψ2 , Ψ3 ) and obtain
We set Ψν := ∂ν φ, Ψ
~
~
µg(Ψ)
~ Ψν = µN (Ψ)(∂ Ψ, ∂Ψν ),
∂α Ψν = ∂ν Ψα
~
Remarkable fact: N verifies a strong g(Ψ)−null
condition:
~
~ ∂Ψν ) . |LΨ||
~ R̆ Ψ|
~ + µ|∇
~ 2
Ψ,
/ Ψ|
µN (Ψ)(∂
X−1
2
~X
|X
+
µX
R̆X
Ψ|
+ ···
Net effect: the difference between the system and the
scalar equation g(Ψ) Ψ = 0 is small
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
g(Ψ)Ψ = 0 vs. (g −1)αβ (∂φ)∂α ∂β φ = 0
We commute the following equation with ∂ν :
(g −1 )αβ (∂φ)∂α ∂β φ = 0
~ := (Ψ0 , Ψ1 , Ψ2 , Ψ3 ) and obtain
We set Ψν := ∂ν φ, Ψ
~
~
µg(Ψ)
~ Ψν = µN (Ψ)(∂ Ψ, ∂Ψν ),
∂α Ψν = ∂ν Ψα
~
Remarkable fact: N verifies a strong g(Ψ)−null
condition:
~
~ ∂Ψν ) . |LΨ||
~ R̆ Ψ|
~ + µ|∇
~ 2
Ψ,
/ Ψ|
µN (Ψ)(∂
X−1
2
~X
|X
+
µX
R̆X
Ψ|
+ ···
Net effect: the difference between the system and the
scalar equation g(Ψ) Ψ = 0 is small
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
g(Ψ)Ψ = 0 vs. (g −1)αβ (∂φ)∂α ∂β φ = 0
We commute the following equation with ∂ν :
(g −1 )αβ (∂φ)∂α ∂β φ = 0
~ := (Ψ0 , Ψ1 , Ψ2 , Ψ3 ) and obtain
We set Ψν := ∂ν φ, Ψ
~
~
µg(Ψ)
~ Ψν = µN (Ψ)(∂ Ψ, ∂Ψν ),
∂α Ψν = ∂ν Ψα
~
Remarkable fact: N verifies a strong g(Ψ)−null
condition:
~
~ ∂Ψν ) . |LΨ||
~ R̆ Ψ|
~ + µ|∇
~ 2
Ψ,
/ Ψ|
µN (Ψ)(∂
X−1
2
~X
|X
+
µX
R̆X
Ψ|
+ ···
Net effect: the difference between the system and the
scalar equation g(Ψ) Ψ = 0 is small
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
The state of affairs for small data
Alinhac and Lindblad proved global existence for:
(g −1 )αβ (Ψ)∂α ∂β Ψ = 0
When N verifies the strong null condition, shocks form for:
(g −1 )αβ (Ψ)∂α ∂β Ψ = −GLL (µ−1 R̆Ψ)2 + N
Not well understood:
(g −1 )αβ (Ψ)∂α ∂β Ψ = (other) × (µ−1 R̆Ψ)2 + N
Remark
(∂t Ψ)3
is not a small perturbation term near the shock.
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
The state of affairs for small data
Alinhac and Lindblad proved global existence for:
(g −1 )αβ (Ψ)∂α ∂β Ψ = 0
When N verifies the strong null condition, shocks form for:
(g −1 )αβ (Ψ)∂α ∂β Ψ = −GLL (µ−1 R̆Ψ)2 + N
Not well understood:
(g −1 )αβ (Ψ)∂α ∂β Ψ = (other) × (µ−1 R̆Ψ)2 + N
Remark
(∂t Ψ)3
is not a small perturbation term near the shock.
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
The state of affairs for small data
Alinhac and Lindblad proved global existence for:
(g −1 )αβ (Ψ)∂α ∂β Ψ = 0
When N verifies the strong null condition, shocks form for:
(g −1 )αβ (Ψ)∂α ∂β Ψ = −GLL (µ−1 R̆Ψ)2 + N
Not well understood:
(g −1 )αβ (Ψ)∂α ∂β Ψ = (other) × (µ−1 R̆Ψ)2 + N
Remark
(∂t Ψ)3
is not a small perturbation term near the shock.
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
The state of affairs for small data
Alinhac and Lindblad proved global existence for:
(g −1 )αβ (Ψ)∂α ∂β Ψ = 0
When N verifies the strong null condition, shocks form for:
(g −1 )αβ (Ψ)∂α ∂β Ψ = −GLL (µ−1 R̆Ψ)2 + N
Not well understood:
(g −1 )αβ (Ψ)∂α ∂β Ψ = (other) × (µ−1 R̆Ψ)2 + N
Remark
(∂t Ψ)3
is not a small perturbation term near the shock.
Intro
Basic setup
Estimates
Energy hierarchy
Strong Null Condition
Latest preprints of the book and survey article are
available at:
http://math.mit.edu/ jspeck/publications.html
Thank you