Hydrostatic flows with convex velocity profiles pdf file See final version in Nonlinearity

Yann Brenier

Description

HOMOGENEOUS HYDROSTATIC FLOWS WITH CONVEX VELOCITY PROFILES Yann Brenier Abstra t We onsider the Euler equations of an in ompressible homogeneous uid in a thin two-dimensional layer 1 < x < +1, 0 < z < , with slip boundary onditions at z = 0, z = and periodi boundary onditions in x. After res aling the verti al variable and letting go to zero, we get the following hydrostati limit of the Euler equations t u + u x u + w z u + x p = 0; (1) x u + z w = 0; z p = 0; (2) supplemented by slip boundary onditions at z = 0 and z = 1 and periodi boundary onditions in x. We show that the orresponding initial-value problem is lo ally, but generally not globally, solvable in the lass of smooth solutions with stri tly onvex horizontal velo ity proles, with onstant slopes at z = 0 and z = 1. Resume On onsidere les equations d'Euler d'un uide in ompressible ho- mogene se mouvant dans une ou he min e 1 < x < +1, 0 < z < , glissant sur les bords z = 0, z = et periodique en x.

• orresponding


• equations


• verti al


• lagrangian sheets


• slip boundary


• horizontal variable


• equations d'euler


• boundary ondi


• ave onditions de glissement en z


• semi-lagrangian equations

Sujets

• Équation
• Équations d'Euler

Informations

• Publié par: pefav
• 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