Introduction
An Explicit Formula
Energy Estimates
Fundamental Solution and Energy Estimate for the
Characteristic Problem for the Wave Equation
in collaboration with Damiano Foschi
Piera Montanari
Università degli Studi di Ferrara
Journées “Equations dispersives sur les variétés”
Laboratoire de Mathématiques, Applications, et Physique Mathématique
Université d’Orléans
April 9-11, 2008
Piera Montanari
Characteristic Problem for the Wave Equation
Introduction
An Explicit Formula
Energy Estimates
Summary
Statement of the problem
References
Some remarks
Summary
1
Statement of the problem and references;
2
A Fundamental Solution in 3 dimensions;
3
Energy Estimates;
4
A local result and the extra regularity hypothesis due to the tip of
the cone.
Piera Montanari
Characteristic Problem for the Wave Equation
Summary
Statement of the problem
References
Some remarks
Introduction
An Explicit Formula
Energy Estimates
Statement of the problem
The Characteristic Problem for the Wave Equation
u = f
t ≥ 0, x ∈ Rn
u|C = ϕ
where C = {(t, x) ∈ [0, +∞) × Rn s.t. t = |x|} is the (future) light cone.
t
0
x
x1
0
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Characteristic Problem for the Wave Equation
Introduction
An Explicit Formula
Energy Estimates
Summary
Statement of the problem
References
Some remarks
It’s not a new problem...
R. Courant & D. Hilbert, M.H. Protter, E.C. Young
derive explicit formulas for the wave equation;
F. Cagnac (Problème de Cauchy sur un conöide caractéristique)
uses a parametrix to obtain the solution of a hyperbolic system of
second order differential equations;
H. Müller Zum Hagen (Characteristic initial value problem for
hyperbolic systems of second order differential equations, Ann. Inst.
Henri Poincaré, Vol.53, No.2 (1990) 159-216) solves the local
problem for a generic hyperbolic system of second order avoiding the
tip of the cone;
M. Dossa (Espaces de Sobolev non isotropes, à poids et problemes
de Cauchy quasi-linéaieres sur un conöide caractéristique, Ann. Inst.
Henri Poincaré, section A, Vol.66, No.1 (1997) 37-107) solves
locally the quasi-linear problem on the tip of the cone.
Piera Montanari
Characteristic Problem for the Wave Equation
Introduction
An Explicit Formula
Energy Estimates
Summary
Statement of the problem
References
Some remarks
...but
We want to:
understand the Müller Zum Hagen and Dossa work looking at simple
semi-linear Wave Equation, in particular Energy Estimates on a
neighborhood of the tip of the cone;
solve the characteristic problem for the semi-linear wave equation
locally and globally.
Piera Montanari
Characteristic Problem for the Wave Equation
Summary
Statement of the problem
References
Some remarks
Introduction
An Explicit Formula
Energy Estimates
The differences with respect to the Classical Cauchy
Problem
We don’t need to assign on C the derivatives of the initial datum;
There is a gap of regularity between the solution and the data.
Main difficulties:
We want to solve the problem for smooth data, but when we restrict
a smooth function to C we may get a singular function (the tip of
the cone is a geometrical singularity),
u(t, x) = t
⇒
u(|x|, x) = |x| ;
The transversal derivatives are not immediately determined from the
initial datum (they must solve a Transport equation along C).
Piera Montanari
Characteristic Problem for the Wave Equation
Introduction
An Explicit Formula
Energy Estimates
An Explicit Formula
The Fundamental Solution E+ in 3 dimensions is given by
Z
φ(|x|, x)
1
dx , φ ∈ D(R4 ).
hE+ , φi =
8π
|x|
Let χ ∈ C ∞ (R) equal to 0 on (−∞, 0] and 1 on (1, +∞), set
t − |x|
uε (t, x) = χ
u(t, x) .
ε
Easily we get
2
uε = χε f +
ε
1
∂t + ∂ω +
uχ0ε .
|x|
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Characteristic Problem for the Wave Equation
Introduction
An Explicit Formula
Energy Estimates
Hence, when t − |x| > ε, we have
u(t, x) = E+ ∗ uε (t, x) =
Z
t − |y | − |x − y | f (t − |y |, x − y )
1
χ
dy +
=
8π R3
ε
|y |
Z
1
1
u(t − |y |, x − y )
1 0 t − |y | − |x − y |
+
χ
∂t + ∂ω +
dy ,
8π R3 ε
ε
|x − y |
|y |
letting ε → 0, we get
the Explicit Formula
u(t, x) =
1
8π
f (t − |y |, x − y )
dy +
|y |
|y |+|x−y |≤t
Z
∇ϕ(x − y )
ϕ(x − y )
1
+
+
dσy .
8π |y |+|x−y |=t
|y |
|y ||x − y |
Z
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Characteristic Problem for the Wave Equation
Non-reduced Energy Estimate
Reduction to data
Reduced Energy Estimates
The differentiability gap
A local result
Introduction
An Explicit Formula
Energy Estimates
First Order Energy Estimate
Integrating thy identity ∂t |∂u|2 = 2 [∂t uu + ∇ · (∇u∂t u)] on the cone
t
τ
Cτ
x1
0
x0
Bτ
we immediately get the First Order Energy Estimate
Z
k∂u(t)kL2 (Bt ) ≤ k∇ϕkL2 (Bt ) +
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0
t
kf (τ )kL2 (Bτ ) dτ .
Characteristic Problem for the Wave Equation
Introduction
An Explicit Formula
Energy Estimates
Non-reduced Energy Estimate
Reduction to data
Reduced Energy Estimates
The differentiability gap
A local result
“Non-reduced” Energy Estimate
Then we easily derive the following
“Non-reduced” Energy Estimate
X
k∂ α u(t)kL2 (Bt )
. (1 + t)
hP
|α|≤s−1
k∇∂ α u|C kL2 (Bt ) +
|α|≤s
Z
t
+

X
0 |α|≤s−1
k∂ α f (τ )kL2 (Bτ ) dτ 
where the term k∇∂ α u|C kL2 (Bt ) still have to be reduced to the initial
data ϕ and f .
Piera Montanari
Characteristic Problem for the Wave Equation
Introduction
An Explicit Formula
Energy Estimates
Non-reduced Energy Estimate
Reduction to data
Reduced Energy Estimates
The differentiability gap
A local result
Characteristic coordinates
We introduce (ξ, η, ω) as new variables according to the relations

 ξ = t + |x|
x = ξ−η
ω
2
η = t − |x| .
⇐⇒
ξ+η
 ω= x
t= 2
|x|
The equation u = f becomes
Lv := ∂ξ ∂η v −
n−1
1
∆ n−1 v = g .
(∂ξ v − ∂η v ) −
2(ξ − η)
(ξ − η)2 S
The cone C becomes the hyperplane η = 0 in the (ξ, η, ω) space.
The derivative ∂ξ is tangential to C , so ∂ξk v (ξ, 0, ω) = ∂ξk ϕ(ξ, ω); on the
other hand the non-tangential derivative ∂η must be reduced.
Piera Montanari
Characteristic Problem for the Wave Equation
Non-reduced Energy Estimate
Reduction to data
Reduced Energy Estimates
The differentiability gap
A local result
Introduction
An Explicit Formula
Energy Estimates
Reduction to data
We set
X` (ξ, ω) := ∂η` v (η, 0, ω) ,
then X1 solves Lv = g on η = 0, that is
X10 (ξ) +
n−1
1
n−1
X1 (ξ) =
∂ξ ϕ + 2 ∆S n−1 ϕ + g (ξ, 0) .
2ξ
2ξ
ξ
In general, all the functions X` fulfill on η = 0 a
Transport Equation
X`0 (ξ) +
n−1
X` (ξ) = G` (ξ)
2ξ
⇔
n−1
n−1
∂ξ X` (ξ)ξ 2 = G` (ξ)ξ 2 ,
where the datum G` is given and ω is fixed. Moreover
Z
− n−1
+
− n−1
2
2
X` = ◦(ξ
), when ξ → 0
⇒ X` (ξ) = ξ
ξ
G` (ξ 0 )ξ 0
0
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Characteristic Problem for the Wave Equation
n−1
2
dξ 0 .
Introduction
An Explicit Formula
Energy Estimates
In general, if X` = ◦(ξ −
∂ξm X` (ξ)
n−1
2 +m
L2 (Sξ )
.
) when ξ → 0+ , we have
m Z
X
k=0
where
G` .
X
h+k=`+1
Non-reduced Energy Estimate
Reduction to data
Reduced Energy Estimates
The differentiability gap
A local result
ξ
0
1
∂ k G` (ξ 0 )
ξ 0m−k ξ
dξ 0 .
1
Xk + lower order terms .
ξh
Inductively, we can estimate the term ∂ξm X` (ξ)
k∂ωm X` (ξ)kL2 (Sξ ) ,
L2 (Sξ0 )
L2 (Sξ )
, and similarly
by terms involving only the initial data and their
derivatives.
Note that
X
k∇∂ α u|C kL2 (Bt ) .
|α|≤s−1
X
m
∂+
X`+1
L2 (Bt )
+ lower order terms .
m+`+1=s
Piera Montanari
Characteristic Problem for the Wave Equation
Introduction
An Explicit Formula
Energy Estimates
Non-reduced Energy Estimate
Reduction to data
Reduced Energy Estimates
The differentiability gap
A local result
Reduced Energy Estimates
Finally, we get
Reduced Energy Estimates
n
sup t − 2 ku(t)kH s (Bt ) .
t∈[0,T ]
. (1 + T )
" s−1
X
k=0
sup t −
n−1
2 +(s−1)−k
t∈[0,T ]
kϕ(t)kH 2(s−1)−k+1 (St ) +

2(s−1)−k−1
+ sup t −
t∈[0,T ]
n−1
2 +(s−1)−k
X
j=0
j
∂−
f (t)
H 2(s−1)−k−j−1 (St )
+
#
+ sup t
t∈[0,T ]
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− n2
kf (t)kH s−1 (Bt ) .
Characteristic Problem for the Wave Equation
Introduction
An Explicit Formula
Energy Estimates
Non-reduced Energy Estimate
Reduction to data
Reduced Energy Estimates
The differentiability gap
A local result
Hypothesis on initial data
For deriving the Energy Estimate in H s we need at least
2s − 1 derivatives for ϕ, 2s − 3 derivatives for f , in other words
2s−1
2s−3
ϕ ∈ Hloc
(Rn ), f ∈ Hloc
(Rn+1 );
ϕ and f must satisfy
n−1 n−1
ϕ(x) = ◦ x − 2 +s
and f (t, x) = ◦ (t, x)− 2 +s−2 ;
s
then we have u ∈ Hloc
(Rn+1 ).
Idea:
If ϕ and f which admit Taylor expansion of order − n−1
2 + s and
− n−1
+
s
−
2
respectively,
we
can
split
the
initial
data
ϕ = ϕp + ϕr ,
2
f = fp + fr and solve
up = fp
ur = fr
(Decomp. Harmonic Pol.)
(Energy)
up |C = ϕp
ur |C = ϕr
Piera Montanari
Characteristic Problem for the Wave Equation
Non-reduced Energy Estimate
Reduction to data
Reduced Energy Estimates
The differentiability gap
A local result
Introduction
An Explicit Formula
Energy Estimates
Is it possible to reduce the gap of regularity?
Consider the simple case of two characteristic hyperplanes
π± = {s± := t ± x1 = 0}. In the new variables
s± := t ± x1
,
y = x0
we have
u = 0
u|π∓ = ϕ
⇒
∂+ ∂− u = ∆y u
.
u(s+ , 0, y ) = u(0, s− , y ) = ϕ(y )
s−`− 12
s
`
If u ∈ Hloc
, the Trace Theorem gives U` := (∂+
u)|π+ ∈ Hloc
moreover U` (0, y ) = 0.
It’s easy to see that U1 solves on π+ the equation
∂+ U1 = ∆y ϕ
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⇒
U1 = s+ ∆ϕ.
Characteristic Problem for the Wave Equation
(π+ ),
Non-reduced Energy Estimate
Reduction to data
Reduced Energy Estimates
The differentiability gap
A local result
Introduction
An Explicit Formula
Energy Estimates
Generalizing, we have
∂+ U` = ∆y U`−1
hence
s−`− 12
∆` ϕ ∈ Hloc
`
s+
∆` ϕ.
`!
⇒
U` =
⇒
ϕ ∈ Hloc
s+`− 21
.
When ` = s − 1 the previous computation gives
2s− 23
ϕ 6∈ Hloc
⇒
s
u 6∈ Hloc
,
on the other hand by Energy Estimates we have
2s−1
ϕ ∈ Hloc
⇒
s
u ∈ Hloc
.
The s − 1 differentiability gap cannot be reduced by more than
derivatives.
Piera Montanari
Characteristic Problem for the Wave Equation
1
2
Introduction
An Explicit Formula
Energy Estimates
Non-reduced Energy Estimate
Reduction to data
Reduced Energy Estimates
The differentiability gap
A local result
Theorem:
If the data ϕ and f are such that
ϕ = ϕp + ϕr
f = fp + fr
n−1 2s−3
2s−1
(Rn+1 ),
where ϕr ∈ Hloc
(Rn ), ϕr (x) = ◦ x − 2 +s , fr ∈ Hloc
n−1
fr (t, x) = ◦ (t, x)− 2 +s−2 and ϕp , fp are polynomials of degree at
n−1
most − n−1
2 + s and − 2 + s − 2 respectively, then
u = f
t ≥ 0, x ∈ Rn
u|C = ϕ
has a unique local solution
u ∈ H s (Rn+1 ) .
Piera Montanari
Characteristic Problem for the Wave Equation
Introduction
An Explicit Formula
Energy Estimates
Non-reduced Energy Estimate
Reduction to data
Reduced Energy Estimates
The differentiability gap
A local result
Final Remarks
Remarks:
For some couple of n and s no hypothesis on the Taylor expansion of
the initial data are required.
We improve the Dossa result concerning this particular hypothesis
on the initial data.
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Characteristic Problem for the Wave Equation