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Nombre de lectures | 7 |
Langue | English |
Extrait
HOMOGENEOUS
en
HYDR
x
OST
ersit
A
TIC
et
FLO
son
WS
dans
WITH
CONVEX
h
VELOCITY
z
PR
trons
OFILES
Y
z
ann
ho-
Br
<
enier
z
!
W
u
e
+
the
Euler
temps
equations
of
don
an
v
Lab
homogeneous
1
uid
se
in
mince
a
0
thin
sur
t
=
w
x
o-dimensional
la
a
y
e
er
u
1
z
<
;
x
=
<
;
+
en
1
et
,
x
0
r
<
pas
z
<
h
,
de
with
slip
tes
b
z
oundary
um
aris
at
uide
z
=
an
0,
z
<
=
1
z
and
glissan
p
b
erio
0,
et
b
dique
oundary
P
t
in
helle
x
passage
.
limite
After
on
rescaling
the
v
u@
w
v
+
ariable
=
and
x
letting
z
;
go
=
to
v
zero,
glissemen
w
=
e
=
get
the
e
follo
Nous
wing
p
h
esoudre
ydrostatic
etit,
limit
men
of
the
le
Euler
de
equations
dans
solutions
t
u
les
+
horizon
u@
x
exes
u
p
+
tes
w
0
1.
z
d'analyse
u
erique,
+
e
F
x
d'un
p
=
mog
0
ene
;
mouv
(1)
t
une
x
he
u
1
+
x
+
z
,
w
<
=
<
0
,
;
t
les
z
ords
p
=
=
z
0
;
p
(2)
erio
supplemen
en
ted
.
b
ar
y
hangemen
slip
d'
b
ec
oundary
v
et
at
z
la
=
0
0,
and
arriv
z
aux
=
equations
1
ydrostatiques
and
t
p
+
erio
x
+
b
oundary
u
in
p
x
0
.
W
u
e
sho
w
w
0
that
the
p
0
onding
a
initial-v
ec
alue
de
problem
t
is
z
lo
0
z
,
1
but
p
generally
erio
not
globally
en
,
.
solv
mon
able
qu'on
in
eut
the
en
of
p
smo
mais
oth
globale-
solutions
t
with
g
en
eral,
v
probl
ex
eme
horizon
Cauc
tal
y
v
la
elo
des
r
y
eguli
proles,
eres
with
t
prols
t
vitesse
slop
tale
es
t
at
t
z
v
=
a
0
ec
and
en
z
=
en
1.
=
R
et
=
esum
oratoire
e
n
On
Univ
ere
P
les
6,
rance.
equations
d'EulerIn
BRENIER
;
:
e
CONVEX
after
VELOCITY
tly
PR
OFILES
℄
1
)
In
2
tro
is
b
The
r
Boussinesq
h
equations
understanding
of
z
a
)
three-dimensional
of
in
the
viscid
homogeneous
uid
equations
in
and
h
)
ydrostatic
x
balance
The
write
the
:
y
whic
t
as
u
=
+
)
(
)
u:
the
r
ariable
x
[8
)
has
u
in
+
when
w
densit
z
v
u
t
+
in
r
the
x
(
p
=
0
0
z
:
=
(3)
w
r
exactly
x
del
:u
and
+
t
Eu-
z
b
w
=
u
0
(
;
(
t;
z
(
p
t;
+
w
=
v
0
the
;
in
(4)
℄
are
t
ered.
een
+
in
(
literature
u:
degenerate
r
densit
x
whic
)
to
Then
+
w
tirely
from
z
a
y
=
absorb
0
pressure
:
e
(5)
system
In
u
these
r
equations,
+
(
u
x;
p
z
(6)
)
+
=
=
(
z
x
:
1
equations,
;
x
equations
2
ond
;
ydrostatic
z
tioned
)
h.
stands
y
for
imp
the
in
space
the
v
equations
ariable,
they
x
formally
2
the
R
ariable
2
ws
,
t;
0
!
<
x;
z
;
<
x;
1
w
b
z
eing
(8)
the
x;
v
p
z
and
ordinate,
to
r
pap
x
mainly
=
w
(
(1),
HHEs,
x
tal
1
one-dimensional,
;
[2
x
and
2
℄
)
then
is
v
the
There
horizon-
b
tal
apparen
gradien
little
t,
terest
(
the
u;
for
w
somewhat
)
=
the
(
y
u
uniform,
1
h
;
onds
u
a
2
uid.
;
the
w
y
)
b
stands
en
for
remo
the
ed
v
the
elo
y
densit
eld,
p
e
(
ed
t;
the
x;
term)
z
w
)
obtain
and
simpler
:
(
t
t;
+
x;
u:
z
x
)
u
are
w
the
z
pressure
+
and
x
the
=
densit
:
y
r
elds.
:u
T
ypical
w
b
0
oundary
p
tions
0
are
(7)
w
resulting
(
that
t;
e
x;
homogeneous
z
ydrostatic
)
(HHE),
=
0
to
at
h
z
mo
=
men
0
in
and
(c
z
4.6)
=
ma
1
pla
(slip
an
b
ortan
oundary
role
the
and
of
spatial
3D
p
ler
erio
from
h
y
in
e
x
obtained
.
rescaling
A
v
discussion
v
of
z
these
follo
equations
:
(
b
x;
e
)
found
u
in
t;
[6
z
℄
)
from
w
the
t;
Hamiltonian
z
and
!
non-linear
(
stabilit
x;
y
=
p
;
oin
p
t
t;
of
z
view.
!
(
that,
x;
when
=
the
;
Coriolis
letting
force
go
is
zero.
added,
this
the
er,
so-called
e
primitiv
address
e
t
equations
o-dimensional
widely
ersion
used
(2)
in
the
o
when
horizon
y
v
and
is
meteorology
and
[10
tro
℄
a
(see
alsoglobal
BRENIER
reform
:
Theorem
CONVEX
),
VELOCITY
that
PR
initial
OFILES
tal
particular
Z
to
of
,
solutions,
e
;
C
Lagrangian
solutions
function
and
(13)
dened
the
as
prole
follo
1
ws.
(9),
W
(2)
e
However,
assume
)
((
pro
u;
use
w
ordinate
)(
to
t;
sheets
x;
;
z
(
)
Z
;
are
p
T
(
w
t;
initial
x
x;
))
to
to
C
b
with
e
main
smo
pr
oth
al
functions,
by
sa
op
y
(0
C
solutions
1
Main
in
step
t
HHE's,
>
el
0,
the
x
this
2
is
R
W
,
e
and
(
C
y
2
is
in
:
z
)
,
0)
1-p
1)
erio
;
examples
in
C
x
solv
,
alue
for
only
whic
w
h
izon
the
(0
horizon
)
tal
e
v
e
elo
x
in
y
prole
C
z
(11).
!
is
u
The
(
for
t;
lo
x;
solvable
z
de