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Estimates
Energy hierarchy
An Overview of Recent Progress on
Shock Formation in Three Spatial
Dimensions
Jared Speck
with G. Holzegel, S. Klainerman, & W. Wong
Massachusetts Institute of Technology
Intro
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Main results
Geometric formulation
Estimates
Energy hierarchy
Maximal developments
Theorem (Classic, Sbierski, Wong)
Consider the initial value problem
m φ = N (φ, ∂φ, ∂ 2 φ) := Nonlinearity,
φ|Σ0 = φ̊,
∂t φ|Σ0 = φ̊0 ,
on R1+3 , where m = diag(−1, 1, 1, 1), m := −∂t2 + ∆, Σ0
is a spacelike hypersurface, and (φ̊, φ̊0 ) are smooth. Then
∃! maximal development.
What is the nature of the maximal development?
Global existence?
Singularities?
Cauchy horizon?
Intro
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Maximal developments
Theorem (Classic, Sbierski, Wong)
Consider the initial value problem
m φ = N (φ, ∂φ, ∂ 2 φ) := Nonlinearity,
φ|Σ0 = φ̊,
∂t φ|Σ0 = φ̊0 ,
on R1+3 , where m = diag(−1, 1, 1, 1), m := −∂t2 + ∆, Σ0
is a spacelike hypersurface, and (φ̊, φ̊0 ) are smooth. Then
∃! maximal development.
What is the nature of the maximal development?
Global existence?
Singularities?
Cauchy horizon?
Intro
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Estimates
Energy hierarchy
A maximal development with a singular
boundary
boundary of maximal development
6
t
Σ0 = spacelike
Intro
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Estimates
Energy hierarchy
A model problem in 1D
Burgers’ equation for Ψ(t, x):
∂t Ψ + Ψ∂x Ψ = 0
˚
= data size;
data := Ψ(0, x)
It is famously known that shock singularities form at
TShock = O(˚
−1 ); Ψ remains bounded but |∂x Ψ| → ∞
A much less appreciated fact is that:
The singularity is renormalizable. That is, the solution
is globally smooth relative to good coordinates.
Intro
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Estimates
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A model problem in 1D
Burgers’ equation for Ψ(t, x):
∂t Ψ + Ψ∂x Ψ = 0
˚
= data size;
data := Ψ(0, x)
It is famously known that shock singularities form at
TShock = O(˚
−1 ); Ψ remains bounded but |∂x Ψ| → ∞
A much less appreciated fact is that:
The singularity is renormalizable. That is, the solution
is globally smooth relative to good coordinates.
Intro
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Main results
Geometric formulation
Estimates
Energy hierarchy
A model problem in 1D
Burgers’ equation for Ψ(t, x):
∂t Ψ + Ψ∂x Ψ = 0
˚
= data size;
data := Ψ(0, x)
It is famously known that shock singularities form at
TShock = O(˚
−1 ); Ψ remains bounded but |∂x Ψ| → ∞
A much less appreciated fact is that:
The singularity is renormalizable. That is, the solution
is globally smooth relative to good coordinates.
Intro
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Estimates
Energy hierarchy
“Hiding” the singularity
Characteristics: x = x(t; u) solves
d
x(t; u) = Ψ(t, x(t; u)),
dt
x(0; u) = u
In characteristic coordinates (t, u), Burgers’ equation is
∂
Ψ(t, u) = 0
∂t
Thus,
Ψ(t, u) = F (u)
Intro
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Estimates
Energy hierarchy
“Hiding” the singularity
Characteristics: x = x(t; u) solves
d
x(t; u) = Ψ(t, x(t; u)),
dt
x(0; u) = u
In characteristic coordinates (t, u), Burgers’ equation is
∂
Ψ(t, u) = 0
∂t
Thus,
Ψ(t, u) = F (u)
Intro
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Estimates
Shocks in Burgers’ equation solutions
TShock
u ≡ const
↓
µ := 1/∂x u = 0
↓
= O(˚
−1 )
˚
= data size
t
x
Energy hierarchy
Intro
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Estimates
Energy hierarchy
The inverse foliation density µ
Set
µ :=
1
∂x u
Evolution equation for µ:
∂
∂
µ(t, u) =
Ψ(t, u) = F 0 (u)
∂t
∂u
CHOV relation =⇒ ∂x Ψ blows up when µ → 0 :
∂x Ψ =
1 ∂
1
Ψ = F0
µ ∂u
µ
Intro
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Energy hierarchy
The inverse foliation density µ
Set
µ :=
1
∂x u
Evolution equation for µ:
∂
∂
µ(t, u) =
Ψ(t, u) = F 0 (u)
∂t
∂u
CHOV relation =⇒ ∂x Ψ blows up when µ → 0 :
∂x Ψ =
1 ∂
1
Ψ = F0
µ ∂u
µ
Intro
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Geometric formulation
Estimates
Energy hierarchy
The inverse foliation density µ
Set
µ :=
1
∂x u
Evolution equation for µ:
∂
∂
µ(t, u) =
Ψ(t, u) = F 0 (u)
∂t
∂u
CHOV relation =⇒ ∂x Ψ blows up when µ → 0 :
∂x Ψ =
1 ∂
1
Ψ = F0
µ ∂u
µ
Intro
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Estimates
The 1D theory
There are a huge number of related results 1D :
Challis (1848)
···
Zabusky (1962)
Lax (1964)
Keller and Ting (1966)
John (1974)
Klainerman and Majda (1980)
Majda (1984)
···
Energy hierarchy
Intro
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Estimates
Energy hierarchy
Non-constructive breakdown in 3D
Theorem (John 1980)
In R1+3 , consider nontrivial initially smooth solutions to
(
−(∂t φ)2 ,
m φ =
−(∂t φ)∂t2 φ
φ|t=0 = φ̊,
,
∂t φ|t=0 = φ̊0 ,
where m = diag(−1, 1, 1, 1) and m := −∂t2 + ∆. Then φ
breaks down for some unknown reason.
A different kind of non-constructive blow-up proof for
the Euler equations was provided by Sideris and later
by Guo+Tahvildar-Zadeh.
Intro
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Non-constructive breakdown in 3D
Theorem (John 1980)
In R1+3 , consider nontrivial initially smooth solutions to
(
−(∂t φ)2 ,
m φ =
−(∂t φ)∂t2 φ
φ|t=0 = φ̊,
,
∂t φ|t=0 = φ̊0 ,
where m = diag(−1, 1, 1, 1) and m := −∂t2 + ∆. Then φ
breaks down for some unknown reason.
A different kind of non-constructive blow-up proof for
the Euler equations was provided by Sideris and later
by Guo+Tahvildar-Zadeh.
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Null condition in 3D
In the previous examples, S. Klainerman’s classic null
condition fails; there are quadratic terms proportional to
(∇LFlat φ)2 ,
∇LFlat φ · ∇2LFlat φ
t
t−
LF lat = ∂t − ∂r
r=
LF lat = ∂t + ∂r
co
ns t
angular directions
Intro
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Relevance of null condition
Theorem (Christodoulou, Klainerman 1984)
For data of small size ˚
, solutions to
m φ = Q1 (∂φ, ∂φ) + Q2 (∂ 2 φ, ∂φ)
exist globally when the Qi verify the classic null condition.
Example: Q1 (∂φ, ∂φ) = (∂t φ)2 − (∂r φ)2 = ∇LFlat φ · ∇LFlat φ
Key ingredient: linear solutions m φ = 0 disperse:
|∇LFlat φ| . ˚
1
,
1+t
|∇LFlat φ|, |∇
/ φ| . ˚
1
(1 + t)2
Intro
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Estimates
Energy hierarchy
Relevance of null condition
Theorem (Christodoulou, Klainerman 1984)
For data of small size ˚
, solutions to
m φ = Q1 (∂φ, ∂φ) + Q2 (∂ 2 φ, ∂φ)
exist globally when the Qi verify the classic null condition.
Example: Q1 (∂φ, ∂φ) = (∂t φ)2 − (∂r φ)2 = ∇LFlat φ · ∇LFlat φ
Key ingredient: linear solutions m φ = 0 disperse:
|∇LFlat φ| . ˚
1
,
1+t
|∇LFlat φ|, |∇
/ φ| . ˚
1
(1 + t)2
Intro
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Estimates
Energy hierarchy
John’s proof of spherically symmetric blow-up
Theorem (John 1985)
In 3D, consider nontrivial spherically symmetric data of
small size ˚
for
−∂t2 φ + (1 + ∂t φ)∆φ = 0
Then ∇L ∂r φ blows up at TLifespan = O(exp(c˚
−1 )). φ and
its first rectangular derivatives remain bounded.
Blow-up follows from a Riccati inequality for w ∼ r ∇L ∂r φ
along characteristics:
1
d
w ≥c
w 2 + Error
dt
1+t
Intro
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Energy hierarchy
John’s proof of spherically symmetric blow-up
Theorem (John 1985)
In 3D, consider nontrivial spherically symmetric data of
small size ˚
for
−∂t2 φ + (1 + ∂t φ)∆φ = 0
Then ∇L ∂r φ blows up at TLifespan = O(exp(c˚
−1 )). φ and
its first rectangular derivatives remain bounded.
Blow-up follows from a Riccati inequality for w ∼ r ∇L ∂r φ
along characteristics:
1
d
w ≥c
w 2 + Error
dt
1+t
Intro
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Energy hierarchy
Main difficulties away from symmetry
Away from symmetry, the term Error in
d
1
w ≥c
w 2 + Error
dt
1+t
depends on the angular Laplacian ∆
/ φ.
Angular derivatives can only be controlled with
energy estimates.
Suitable energy estimates showing that ∇
/ φ = Error
are very hard to derive near the blow-up points.
The Riccati inequality approach is not precise
enough to work.
Intro
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Energy hierarchy
Main difficulties away from symmetry
Away from symmetry, the term Error in
d
1
w ≥c
w 2 + Error
dt
1+t
depends on the angular Laplacian ∆
/ φ.
Angular derivatives can only be controlled with
energy estimates.
Suitable energy estimates showing that ∇
/ φ = Error
are very hard to derive near the blow-up points.
The Riccati inequality approach is not precise
enough to work.
Intro
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Alinhac’s shock formation results
Theorem (Alinhac 1990s-early 2000s)
In 3D, consider nontrivial data of small size ˚
for
(g −1 )αβ (∂φ)∂α ∂β φ = 0,
gαβ (∂φ) = mαβ +
(Einstein summation)
Pert
(∂φ),
gαβ
Pert
(∂φ = 0) = 0
gαβ
Assume that the null condition fails. Then under a
non-degeneracy assumption on the data:
∃ exactly one blow-up point at TShock = O(exp(c˚
−1 )).
Singularity can be partially renormalized.
Blow-up is tied to intersecting characteristics.
Extensions by Huicheng, Bingbing, Witt
Intro
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Estimates
Energy hierarchy
Alinhac’s shock formation results
Theorem (Alinhac 1990s-early 2000s)
In 3D, consider nontrivial data of small size ˚
for
(g −1 )αβ (∂φ)∂α ∂β φ = 0,
gαβ (∂φ) = mαβ +
(Einstein summation)
Pert
(∂φ),
gαβ
Pert
(∂φ = 0) = 0
gαβ
Assume that the null condition fails. Then under a
non-degeneracy assumption on the data:
∃ exactly one blow-up point at TShock = O(exp(c˚
−1 )).
Singularity can be partially renormalized.
Blow-up is tied to intersecting characteristics.
Extensions by Huicheng, Bingbing, Witt
Intro
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Main results
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Estimates
Energy hierarchy
The region studied by Alinhac
St,0
Σut
St,u
6
t
S0,u
Σu0
S0,0 nontrivial data
Cut
C0t
trivial data
φ≡0
Intro
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Main results
Geometric formulation
Estimates
Energy hierarchy
Christodoulou’s shock formation results in 3D
In his famous book of 2007, Christodoulou (and with
Miao in 2013) proved a related small-data blow-up
result. His proof yielded valuable new information that
is inaccessible via Alinhac’s approach.
He studied perturbations of the solution φ ≡ t to all
Euler-Lagrange PDEs from irrotational fluid
mechanics.
Example equations are the following with s > 1:
∂α
∂σs
∂(∂α φ)
= 0,
σ := −(m−1 )κλ ∂κ φ∂λ φ
Singularities form ⇐⇒ characteristics intersect.
To generate shocks, he made assumptions on
Z
Z
r {(∂t φ − 1) − c∂r φ} dσ +
2(∂t φ − 1) − c∂r φ d 3 x
S0,u
Σu0
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Christodoulou’s shock formation results in 3D
In his famous book of 2007, Christodoulou (and with
Miao in 2013) proved a related small-data blow-up
result. His proof yielded valuable new information that
is inaccessible via Alinhac’s approach.
He studied perturbations of the solution φ ≡ t to all
Euler-Lagrange PDEs from irrotational fluid
mechanics.
Example equations are the following with s > 1:
∂α
∂σs
∂(∂α φ)
= 0,
σ := −(m−1 )κλ ∂κ φ∂λ φ
Singularities form ⇐⇒ characteristics intersect.
To generate shocks, he made assumptions on
Z
Z
r {(∂t φ − 1) − c∂r φ} dσ +
2(∂t φ − 1) − c∂r φ d 3 x
S0,u
Σu0
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Christodoulou’s shock formation results in 3D
In his famous book of 2007, Christodoulou (and with
Miao in 2013) proved a related small-data blow-up
result. His proof yielded valuable new information that
is inaccessible via Alinhac’s approach.
He studied perturbations of the solution φ ≡ t to all
Euler-Lagrange PDEs from irrotational fluid
mechanics.
Example equations are the following with s > 1:
∂α
∂σs
∂(∂α φ)
= 0,
σ := −(m−1 )κλ ∂κ φ∂λ φ
Singularities form ⇐⇒ characteristics intersect.
To generate shocks, he made assumptions on
Z
Z
r {(∂t φ − 1) − c∂r φ} dσ +
2(∂t φ − 1) − c∂r φ d 3 x
S0,u
Σu0
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Christodoulou’s shock formation results in 3D
In his famous book of 2007, Christodoulou (and with
Miao in 2013) proved a related small-data blow-up
result. His proof yielded valuable new information that
is inaccessible via Alinhac’s approach.
He studied perturbations of the solution φ ≡ t to all
Euler-Lagrange PDEs from irrotational fluid
mechanics.
Example equations are the following with s > 1:
∂α
∂σs
∂(∂α φ)
= 0,
σ := −(m−1 )κλ ∂κ φ∂λ φ
Singularities form ⇐⇒ characteristics intersect.
To generate shocks, he made assumptions on
Z
Z
r {(∂t φ − 1) − c∂r φ} dσ +
2(∂t φ − 1) − c∂r φ d 3 x
S0,u
Σu0
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Christodoulou’s shock formation results in 3D
In his famous book of 2007, Christodoulou (and with
Miao in 2013) proved a related small-data blow-up
result. His proof yielded valuable new information that
is inaccessible via Alinhac’s approach.
He studied perturbations of the solution φ ≡ t to all
Euler-Lagrange PDEs from irrotational fluid
mechanics.
Example equations are the following with s > 1:
∂α
∂σs
∂(∂α φ)
= 0,
σ := −(m−1 )κλ ∂κ φ∂λ φ
Singularities form ⇐⇒ characteristics intersect.
To generate shocks, he made assumptions on
Z
Z
r {(∂t φ − 1) − c∂r φ} dσ +
2(∂t φ − 1) − c∂r φ d 3 x
S0,u
Σu0
Intro
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New information in Christodoulou’s proof
singular
H
C
regular
H
∂− H
Figure: The maximal development
Intro
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New shock formation results - Part I
Theorem (G.H.,S.K.,J.S.,W.W.)
Consider the initial value problem
g(Ψ) Ψ := (g −1 )αβ (Ψ)∂α ∂β Ψ − Γα (Ψ, ∂Ψ)∂α Ψ = 0,
Ψ|t=0 = Ψ̊,
∂t Ψ|t=0 = Ψ̊0
where (Ψ̊, Ψ̊0 ) are compactly supported and of small size
˚
. Assume that
Pert
Pert
gαβ (Ψ) = mαβ + gαβ
(Ψ), gαβ
(0) = 0
Classic null condition fails
We provide a complete description of the dynamics.
Singularity ⇐⇒ characteristics intersect.
Rescaling (Ψ̊, Ψ̊0 ) → λ(Ψ̊, Ψ̊0 ) for small λ always leads
to a shock at TShock ∼ exp(c˚
−1 ).
Intro
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New shock formation results - Part I
Theorem (G.H.,S.K.,J.S.,W.W.)
Consider the initial value problem
g(Ψ) Ψ := (g −1 )αβ (Ψ)∂α ∂β Ψ − Γα (Ψ, ∂Ψ)∂α Ψ = 0,
Ψ|t=0 = Ψ̊,
∂t Ψ|t=0 = Ψ̊0
where (Ψ̊, Ψ̊0 ) are compactly supported and of small size
˚
. Assume that
Pert
Pert
gαβ (Ψ) = mαβ + gαβ
(Ψ), gαβ
(0) = 0
Classic null condition fails
We provide a complete description of the dynamics.
Singularity ⇐⇒ characteristics intersect.
Rescaling (Ψ̊, Ψ̊0 ) → λ(Ψ̊, Ψ̊0 ) for small λ always leads
to a shock at TShock ∼ exp(c˚
−1 ).
Intro
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New shock formation results - Part II
Corollary (G.H.,S.K.,J.S.,W.W.)
The same result holds for Alinhac’s equations
(g −1 )αβ (∂φ)∂α ∂β φ = 0,
Pert
gαβ = mαβ + gαβ
(∂φ),
Pert
gαβ
(∂φ = 0) = 0
Remark
Miao-Yu recently
obtained
2 a related large-data blow-up
2
result for − 1 + (∂t φ) ∂t φ + ∆φ = 0
Intro
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New shock formation results - Part II
Corollary (G.H.,S.K.,J.S.,W.W.)
The same result holds for Alinhac’s equations
(g −1 )αβ (∂φ)∂α ∂β φ = 0,
Pert
gαβ = mαβ + gαβ
(∂φ),
Pert
gαβ
(∂φ = 0) = 0
Remark
Miao-Yu recently
obtained
2 a related large-data blow-up
2
result for − 1 + (∂t φ) ∂t φ + ∆φ = 0
Intro
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The three main features of the proof
The singularity is renormalizable, just like in Burgers’
equation!
Unlike in 1D, basic existence theory requires
estimates for higher derivatives.
Moreover, the first statement is false for the high
derivatives: They are not renormalizable! This leads
to severe complications!
We must construct a sharp geometric framework to see
the renormalizability and to overcome the difficulties.
Intro
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The three main features of the proof
The singularity is renormalizable, just like in Burgers’
equation!
Unlike in 1D, basic existence theory requires
estimates for higher derivatives.
Moreover, the first statement is false for the high
derivatives: They are not renormalizable! This leads
to severe complications!
We must construct a sharp geometric framework to see
the renormalizability and to overcome the difficulties.
Intro
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The three main features of the proof
The singularity is renormalizable, just like in Burgers’
equation!
Unlike in 1D, basic existence theory requires
estimates for higher derivatives.
Moreover, the first statement is false for the high
derivatives: They are not renormalizable! This leads
to severe complications!
We must construct a sharp geometric framework to see
the renormalizability and to overcome the difficulties.
Intro
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Main results
Geometric formulation
Estimates
Energy hierarchy
The three main features of the proof
The singularity is renormalizable, just like in Burgers’
equation!
Unlike in 1D, basic existence theory requires
estimates for higher derivatives.
Moreover, the first statement is false for the high
derivatives: They are not renormalizable! This leads
to severe complications!
We must construct a sharp geometric framework to see
the renormalizability and to overcome the difficulties.
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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 delicate local and global
results for wave equations, especially Einstein’s
equations (starting with the
Christodoulou-Klainerman proof of the stability of
Minkowski spacetime)
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Fundamental geometric constructions
St,u
6
t
C0t
Cut
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
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Estimates
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
∇
/ := (true) angular derivatives
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
Energy hierarchy
Intro
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Estimates
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
∇
/ := (true) angular derivatives
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
Energy hierarchy
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Behavior of lower-order derivatives
•˚
= 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
Captures the renormalizability of the singularity
Is essential for controlling error terms
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Behavior of lower-order derivatives
•˚
= 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
Captures the renormalizability of the singularity
Is essential for controlling error terms
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Behavior of lower-order derivatives
•˚
= 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
Captures the renormalizability of the singularity
Is essential for controlling error terms
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Strong 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
• 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
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Strong 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
• 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
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Heuristics behind µ → 0
Lµ(t, u, ϑ) ∼ R̆Ψ(t, u, ϑ)
From R̆Ψ(t, u, ϑ) ∼
1
f (u, ϑ),
1+t Data
Lµ(t, u, ϑ) ∼
Integrating along L =
∂
∂t
we obtain:
1
fData (u, ϑ)
1+t
and using µ|t=0 ∼ 1, we conclude:
µ(t, u, ϑ) ∼ 1 + ln(1 + t)fData (u, ϑ)
Hence, µ → 0 at time TShock
∼ exp | min
u,ϑ
1
fData (u, ϑ)
|
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Heuristics behind µ → 0
Lµ(t, u, ϑ) ∼ R̆Ψ(t, u, ϑ)
From R̆Ψ(t, u, ϑ) ∼
1
f (u, ϑ),
1+t Data
Lµ(t, u, ϑ) ∼
Integrating along L =
∂
∂t
we obtain:
1
fData (u, ϑ)
1+t
and using µ|t=0 ∼ 1, we conclude:
µ(t, u, ϑ) ∼ 1 + ln(1 + t)fData (u, ϑ)
Hence, µ → 0 at time TShock
∼ exp | min
u,ϑ
1
fData (u, ϑ)
|
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Heuristics behind µ → 0
Lµ(t, u, ϑ) ∼ R̆Ψ(t, u, ϑ)
From R̆Ψ(t, u, ϑ) ∼
1
f (u, ϑ),
1+t Data
Lµ(t, u, ϑ) ∼
Integrating along L =
∂
∂t
we obtain:
1
fData (u, ϑ)
1+t
and using µ|t=0 ∼ 1, we conclude:
µ(t, u, ϑ) ∼ 1 + ln(1 + t)fData (u, ϑ)
Hence, µ → 0 at time TShock
∼ exp | min
u,ϑ
1
fData (u, ϑ)
|
Intro
Previous work
Main results
Geometric formulation
Estimates
Justifying the heuristics
We need to show that
Error terms are small
Solution survives until shock
Big difficulty: only L2 regularity is propagated at top
order.
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
Energy hierarchy
Intro
Previous work
Main results
Geometric formulation
Estimates
Justifying the heuristics
We need to show that
Error terms are small
Solution survives until shock
Big difficulty: only L2 regularity is propagated at top
order.
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
Energy hierarchy
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Main a priori L2 estimate hierarchy (hard!)
Let ˚
:= data size, µ? (t, u) := min{1, minΣut µ}, EM (t, u) :=
2
L (Σut )−based energy of M derivatives of Ψ. Then,
ignoring ln t factors at top-order, we have:
E24 (t, u) . ˚
2 µ−17.5
(t, u)
?
E23 (t, u) . ˚
2 µ−15.5
(t, u)
?
···
E16 (t, u) . ˚
2 µ−1.5
(t, u)
?
2
E15 (t, u) . ˚
···
E0 (t, u) . ˚
2
We recover C k bounds for Ψ via Sobolev
We recover C k bounds for µ, Li from the eikonal
equation
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Main a priori L2 estimate hierarchy (hard!)
Let ˚
:= data size, µ? (t, u) := min{1, minΣut µ}, EM (t, u) :=
2
L (Σut )−based energy of M derivatives of Ψ. Then,
ignoring ln t factors at top-order, we have:
E24 (t, u) . ˚
2 µ−17.5
(t, u)
?
E23 (t, u) . ˚
2 µ−15.5
(t, u)
?
···
E16 (t, u) . ˚
2 µ−1.5
(t, u)
?
2
E15 (t, u) . ˚
···
E0 (t, u) . ˚
2
We recover C k bounds for Ψ via Sobolev
We recover C k bounds for µ, Li from the eikonal
equation
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Energy estimate (divergence theorem) region
Σut
Mt,u
Cut t
C0 Ψ ≡ 0
Σu0
E0 (t,u)
z
Z
}|
{
z
Z
F0 (t,u)
}|
{
P1 [∂Ψ, ∂Ψ] +
P2 [∂Ψ, ∂Ψ]
Cut
Z
Z
=
P1 [∂Ψ, ∂Ψ] +
Q[∂Ψ, ∂Ψ]
Σut
Σu0
+
Z
Mt,u
Mt,u
0
o
z }| { n
(g(Ψ) Ψ) (1 + µ)LΨ + 2R̆Ψ
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Energies and fluxes feature µ weights
E0 (t, u) ∼
F0 (t, u) ∼
Z
Σut
µ|LΨ|2 + |R̆Ψ|2 + µ|∇
/ Ψ|2 ,
Cut
|LΨ|2 + µ|∇
/ Ψ|2
Z
Similarly,
E1 (t, u) ∼
Z
Σut
µ|LZ Ψ|2 + |R̆Z Ψ|2 + µ|∇
/ Z Ψ|2 ,
etc., where the Z are vectorfields depending on ∂u, Ψ, ∂Ψ
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Energies and fluxes feature µ weights
E0 (t, u) ∼
F0 (t, u) ∼
Z
Σut
µ|LΨ|2 + |R̆Ψ|2 + µ|∇
/ Ψ|2 ,
Cut
|LΨ|2 + µ|∇
/ Ψ|2
Z
Similarly,
E1 (t, u) ∼
Z
Σut
µ|LZ Ψ|2 + |R̆Z Ψ|2 + µ|∇
/ Z Ψ|2 ,
etc., where the Z are vectorfields depending on ∂u, Ψ, ∂Ψ
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Top-order degeneracy
Why is the top-order estimate so degenerate?
Because the best a priori inequality we are able to obtain is:
2
E24 (t, u) ≤ ˚
+B
B ∼ 10
Z
t
t 0 =0
!
∂
sup ln µ E24 (t 0 , u) dt 0 + · · ·
∂t
Σt 0
The key difficult factor supΣt 0 ∂t∂ ln µ is connected to
avoiding derivative loss in u.
Caricature estimate: if µ were a function of only t, then
Gronwall =⇒ E24 (t, u) . ˚
2 µ−B + · · ·
The correct Gronwall-type estimate is hard to prove
because µ = µ(t, u, ϑ)
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Top-order degeneracy
Why is the top-order estimate so degenerate?
Because the best a priori inequality we are able to obtain is:
2
E24 (t, u) ≤ ˚
+B
B ∼ 10
Z
t
t 0 =0
!
∂
sup ln µ E24 (t 0 , u) dt 0 + · · ·
∂t
Σt 0
The key difficult factor supΣt 0 ∂t∂ ln µ is connected to
avoiding derivative loss in u.
Caricature estimate: if µ were a function of only t, then
Gronwall =⇒ E24 (t, u) . ˚
2 µ−B + · · ·
The correct Gronwall-type estimate is hard to prove
because µ = µ(t, u, ϑ)
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Top-order degeneracy
Why is the top-order estimate so degenerate?
Because the best a priori inequality we are able to obtain is:
2
E24 (t, u) ≤ ˚
+B
B ∼ 10
Z
t
t 0 =0
!
∂
sup ln µ E24 (t 0 , u) dt 0 + · · ·
∂t
Σt 0
The key difficult factor supΣt 0 ∂t∂ ln µ is connected to
avoiding derivative loss in u.
Caricature estimate: if µ were a function of only t, then
Gronwall =⇒ E24 (t, u) . ˚
2 µ−B + · · ·
The correct Gronwall-type estimate is hard to prove
because µ = µ(t, u, ϑ)
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Open problems
Important open problems: proving shock formation for
systems with multiple characteristics
Figure: System with two characteristics
Cosmology (Euler-Einstein)
Magnetohydrodynamics
Elasticity
Nonlinear electromagnetism
Crystal optics
Intro
Previous work
Main results
Geometric formulation
Estimates
Energy hierarchy
Latest preprints of the book and survey article are
available at:
http://math.mit.edu/ jspeck/publications.html