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# Harmoni ity up to rearrangement and isothermal gas

-

Description

Harmoni ity up to rearrangement and isothermal gas dynami s Yann Brenier Abstra t. The on ept of harmoni (or wave) maps up to rearrangement is introdu ed. A relation is established with the smooth solutions of the isother- mal irrotational invis id gas dynami s equations. On introduit la notion d'appli ation harmonique a rearrangement pres. On etablit un lien ave les solutions regulieres des equations de rivant la dy- namique isotherme des gaz sans vis osite ni tourbillon. 1. Review of some lassi al on epts 1.1. Harmoni (or wave) maps. Let S > 0, T > 0, U =?0; T [?0; S[ and D = T d = (R=Z) d the at torus. A map (t; s) 2 U ! X(t; s) 2 D is usually alled a harmoni map (resp. a wave map) if it is a riti al point of the fun tional Z U 1 2 (j t X(t; s)j 2 + j s X(t; s)j 2 )dtds;(1.1) where = 1 (resp. = 1), with respe t to perturbations with ompa t support in U .

• gas dynami

• isothermal irrotational

• alled minimizing harmoni

• fun tions

• vlasov-poisson equations

• ribing isothermal

• vibrating string

• invis id

Subjects

Informations

X
Harmonicit
w
y
D
up
(resp.
to
map
rearrangemen
Compact
t
strings.
and
.
isothermal
+
gas

manifolds.
Y
manifolds
ann
35J
Brenier
1
Abstra
p

norm
The
a

In
of
minimizes
harmonic
R
(or
e)
w
fairly
a
b
v
in
e)
Classic
maps
Harmonicit
up
the
to
where
rearrangemen
with
t
supp
is
j
in
.)
tro
homogeneous

A
=
relation

is
minimizing
established
v
with
b
the

smo
(or
oth

solutions
=
of
t
the
b
isother-
w
mal
or
irrotational
space
in
1991
viscid
58E30
gas
ds

equations.
supp
On
ST
in
dtds;
tro
=
duit
=
la
ect
notion
with
d'application
in
harmonique
j

the
a
R
r
means

es
earrangemen
(resp.
t
e)
pr
tt

ss
es.
:
On
harmonic

1,
etablit
is
un
if
lien
as
a
X
v
along
ec
is
les
A
solutions
the
r
a

a
eguli
is

general
eres

des
d

but
equations
our
d
manifolds

ecriv

an
er,
t
b
la

dy-
the
namique
d
isotherme

des
Subje
gaz
Primary
sans
;
viscosit
Key

phr
e
,
ni
t,
tourbillon.
ork
1.
b
Review
pro
of
T
some
)

(1.1)

1.1.
1
Harmonic

(or
1),
w
resp
a
to
v
erturbations
e)

maps.
ort
Let
U
S
(Here
>
:
0,
denotes
T

>
on
0,
d
U
This
=]0
that
;
solv
T
the
[
Laplace

w

v
;
equation
S

[
X
and

D
X
=
0
T
(1.2)
d
the
=

(
=
R
a
=
X
Z

)
harmonic
d
it
the
(1.1)
at
its
torus.
alue
A
j
map
U
(
the
t;
oundary
s
xed.
)
emark.
2
natural
U
for
!
\target"
X
of
(
harmonic
t;
w
s
v
)
map
2
the
D
of
is
Riemannian
usually
The

D
a
T
harmonic
is
map
trivial
(resp.
Æ
a
for
w
discussion.
a
Riemmanian
v
without
e
oundary
map)
also
if
e
it
Ho
is
ev
a

with
p
oundaries
oin
non
t
manifolds,
of
particular
the

functional
R
Z
,
U
ould
1
Æ
2
Mathematics
(

ation.
j
58E20

76N
t

X
35Q35.
(
wor
t;
and
s
ases.
)
y
j
gas
2
rearrangemen
+
vibrating
j
W

partly
s
orted
X
y
(
austrian
t;

s
AR
)
(FWF-TEC-Y-137).
j
2
2
s;
Y
s
ANN
the
BRENIER

1.2.
an
La
u
ws,
)
rearrangemen
olution
ts,
h
Moser's
of
lemma.
1
Let
temp
(
time
A;
::::;
da
expresses
)
(1.7)
b
)
e
p
a

probabilit
;
y
Isothermal
space

(t
b
ypically
p
A
0,
=
R
[0
v
;
notations
1]
the
or
u
A
usually
=
u
T
the
d
eld
equipp
is
ed

with

the
v
Leb

esgue
+
measure
An
da
in
).
The
F
mo
or
(ph
a
3)
measurable
y
function
0,
a
)
2
s;
A
elo
!
),
X
s
(
x
a
are
)
(where
2

D
)
,

w
+
e
;
dene
of
the
tin
\la
(
w"

of
equiv
X
momen
to
if
b
t
e
0

y
(
for
x
tro
)
;
=
as
Z
system
A
s
Æ

(
0
x
u:
X
r
(
s
a
tary
))
pro
da;
8.8
(1.3)
gas
whic
olution
h
viscid
is
in
a
R
probabilit
d
y
2
measure
describ
on
a
D

,
)
more
pressure
precisely
s;
dened
0,
b
eld
y
)
Z
a
D
y
h
t;
(
alued
x
,
)
for
d
ariable
(
the
x
These
)

=
wing
Z
e
A
for
h
1
(
x
X
:
(
inner
a
R
))
s
da;
:
(1.4)
=
for
whic
all

h
and
2
\the
C
y
(
u
D
r
).
r
(An
0
usual
h
denomination
t
for
ation

The
is

\push-forw
temp
ard"
a
of
s;
da

b
the
y
en
X
=
.)
;
W

e
0.
no

w

sa
these
y
e
that

t
erb
w
space
o
(

h
u:
measurable
p
functions
:u
X
(1.10)
and
+
Y
)
are

equal
=
up
on
to
.
rearrangemen
elemen
t
pro
if
is
they
vided
ha

v
1.3.
e
irrotational
the

same
ev
la
of
w,
in
namely
gas
Z
ving
A
the
Æ
space
(
d
x
ysically
Y
=
(
;
a
;
))
is
da
ed
=
y
Z
densit
A
eld
Æ
(
(
x
x
>
X
a
(
eld
a
(
))
x
da:
>
(1.5)
a
Let
erature
us
(
quote
x
a
>
v
and
ery
v
useful

result,