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On global solutions to a defocusing semi linear wave

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16 pages
On global solutions to a defocusing semi-linear wave equation Isabelle Gallagher C.N.R.S. U.M.R. 8628, Departement de Mathematiques Universite Paris Sud, 91405 Orsay Cedex, France Fabrice Planchon C.N.R.S. U.M.R. 7598, Laboratoire d'Analyse Numerique Universite Paris 6, 75252 Paris Cedex 05, France Abstract We prove that the 3D cubic defocusing semi-linear wave equation is globally well-posed for data in the Sobolev space _ H s where s > 3 4 This result was obtained in [11] following Bour- gain's method ([3]). We present here a dierent and somewhat simpler argument, inspired by previous work on the Navier-Stokes equations ([4, 7]). 1 Introduction and main theorem We consider the equation @ 2 t + 3 = 0 in R R 3 (; @ t ) jt=0 = ( 0 ; 1 ); (1.1) where is real valued. This equation is sub-critical with respect to the H 1 norm, and, since the nonlinearity is defocusing, local well-posedness in H 1 extends to global well-posedness using the conservation of the Hamiltonian

  • global solution

  • sobolev space

  • schrodinger equation

  • linear wave

  • navier stokes equations

  • solution can

  • defocusing semi-linear


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sum
us
t
mak
Theorem
e
x;
a
Definition
remark
f
concerning
,
the
(
L
.
4
0
norms.
v
F
temp
or
e
an
b
y
f


>
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then
w
~
e
p
ha
exp
v
then
e
3
k
v

def
0
q
(
whic

)

1
)
(
k
x
L
2
4
f
=
p

x;
3
inhomogeneous
4
L
k
k

0
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q
k
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2
and
<
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to

s
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only
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ards

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k
b
_
.
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the
s
only
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p;
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i.e.
3
q
2
,
k
q

,
0
t
k
j
_
denote
H
e
s
k
:
)
(2.1)
(
It
and,
follo

ws
the
that
Cauc
as
x;
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L
on
2
as
t
s
(
>
k
3
p
4
2
,
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the
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norm
p
of
2

If
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p
can
b
b
homogeneous
e
_
made
(
arbitrarily
if
small
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compared
m
to
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as
H
distribution,
s

norm
2
b

y
L
rescaling
to
the
Z
data.
,
In
v
particular
hartz
since
w
k
in
u
([9,
0
p;
k
(
L
q
4
admissible
.
h
k
+

1
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>
k
similarly
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p;
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.
w
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e
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conclude
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that
o
the
at
quan
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Then
u
i!
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p
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)
b
;
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u
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to
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with
u
y
0
u
k
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2
t
_
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j
1
p
,
(
and
k
w
0
e
~
assume
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this
(
to
k
b
(
e
)
the
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case
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for
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rest
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of
(
the
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pro
where
of;
the
that
t
will
5
b
;
e
).
useful
2
to
s
estimate
3
the
,
Hamiltonian
f
of
elongs
u
the
in
Beso
Section
space
2.3.
B
Finally
p;q
,
R
w
)
e
and
recall
if
some
partial
denitions
P
and
m
prop
j
erties
con
of
erges
the
w
w
f
a
a
v
ered
e
and
equation
sequence
whic
j
h
=
will
j
b
k
e
j
of
k
use
p
later.
elongs
First
`
w
(
e
)
de

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