Rearrangement of functions and steady vortex rings
Dirar REBAH
Labo. AMNEDP, USTHB, Algeria
february 2nd, 2009
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
1 / 16
In R3 , We consider axisymmetric motion without swirl of an
incompressible inviscid ‡uid with unit density ideal ‡uid ‡ow ;
Incompressibility guarantees the existence of a Stokes steam function
Ψ:Π!R
that is symmetric about z axis with respect to the cylindrical
coordinates (r , θ, z ) 2 R3 for the ‡ow, which means that the velocity
~V is given as
~V = ( 1 ∂Ψ , 0, 1 ∂Ψ )
r ∂z
r ∂r
where
Π = fr > 0, θ = 0 and z 2 Rg
is the half-plane in R3 .
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
2 / 16
The vorticity ~ω is then given in terms of Stokes stream function by
~ω = r
where
L=
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
~V = (0, r LΨ, 0),
1 ∂ 1 ∂
1 ∂2
(
)+ 2 2
r ∂r r ∂r
r ∂z
CIMPA EGYPT ’09
.
february 2nd, 2009
3 / 16
The vorticity ~ω is then given in terms of Stokes stream function by
~ω = r
where
L=
~V = (0, r LΨ, 0),
1 ∂ 1 ∂
1 ∂2
(
)+ 2 2
r ∂r r ∂r
r ∂z
.
The region where
ω = r LΨ 6= 0
is called the vortex core.
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
3 / 16
The vorticity ~ω is then given in terms of Stokes stream function by
~ω = r
where
L=
~V = (0, r LΨ, 0),
1 ∂ 1 ∂
1 ∂2
(
)+ 2 2
r ∂r r ∂r
r ∂z
.
The region where
ω = r LΨ 6= 0
is called the vortex core.
If the ‡ow is in steady state, then the Stokes stream function
satis…es the equation
LΨ = φ Ψ,
(1)
where φ : R ! R is an unknown function (H. Lamb [6].)
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
3 / 16
If the Stokes stream function Ψ approaches at in…nity λ2 r 2 which
represents a uniform ‡ow of velocity λ in the negative z-direction,
where λ is a positive constant then steady vortex ring is governed by
LΨ = φ(Ψ
λ 2
r )
2
in Π,
(2)
where, φ : R ! R is a non-negative increasing function.
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
4 / 16
If the Stokes stream function Ψ approaches at in…nity λ2 r 2 which
represents a uniform ‡ow of velocity λ in the negative z-direction,
where λ is a positive constant then steady vortex ring is governed by
LΨ = φ(Ψ
λ 2
r )
2
in Π,
(2)
where, φ : R ! R is a non-negative increasing function.
Thus, by (1), Ψ
‡ow.
λ 2
2r
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
is the Stokes stream function for the uniform
CIMPA EGYPT ’09
february 2nd, 2009
4 / 16
If the Stokes stream function Ψ approaches at in…nity λ2 r 2 which
represents a uniform ‡ow of velocity λ in the negative z-direction,
where λ is a positive constant then steady vortex ring is governed by
LΨ = φ(Ψ
λ 2
r )
2
in Π,
(2)
where, φ : R ! R is a non-negative increasing function.
Thus, by (1), Ψ
‡ow.
λ 2
2r
is the Stokes stream function for the uniform
The existence of steady vortex rings in a uniform ‡ow, is to show that
(2) has a solution with respect to the boundary conditions
8
< Ψ(0, z ) = 0,
Ψ(r , z ) ! 0 as r 2 + z 2 ! ∞.
(BC)
:
jrΨj ! 0 as r 2 + z 2 ! ∞.
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
4 / 16
B. Benjamin [1] was seeking a solution for the equation (2) complying
with the conditions (BC) and for which
1
The function ζ := ω/r is a rearrangement of a prescribed function
ζ0.
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
5 / 16
B. Benjamin [1] was seeking a solution for the equation (2) complying
with the conditions (BC) and for which
1
The function ζ := ω/r is a rearrangement of a prescribed function
ζ0.
2
A value is prescribed for either the speed λ at in…nity, or the impulse
I2 in z-direction, which is given in terms of ζ as
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
I2 (ζ ) =
1
2
Z
r 2 ζ.
Π
CIMPA EGYPT ’09
february 2nd, 2009
5 / 16
B. Benjamin [1] was seeking a solution for the equation (2) complying
with the conditions (BC) and for which
1
The function ζ := ω/r is a rearrangement of a prescribed function
ζ0.
2
A value is prescribed for either the speed λ at in…nity, or the impulse
I2 in z-direction, which is given in terms of ζ as
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
I2 (ζ ) =
1
2
Z
r 2 ζ.
Π
CIMPA EGYPT ’09
february 2nd, 2009
5 / 16
B. Benjamin [1] was seeking a solution for the equation (2) complying
with the conditions (BC) and for which
1
The function ζ := ω/r is a rearrangement of a prescribed function
ζ0.
2
A value is prescribed for either the speed λ at in…nity, or the impulse
I2 in z-direction, which is given in terms of ζ as
I2 (ζ ) =
1
2
Z
r 2 ζ.
Π
One of the facts motivating this approach I2 (ζ ) and the volume of the sets
fζ αg (for each α > 0) are preserved in axisymmetric motion of an
ideal ‡uid in R3 , so I2 (ζ ) and the prescribed function ζ 0 are physically
meaningful.
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
5 / 16
To show the existence, Benjamin proposed a method based on solving
one of two variational problems
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
max (E (ζ )
ζ 2R(ζ 0 )
CIMPA EGYPT ’09
λI2 (ζ )),
(3)
february 2nd, 2009
6 / 16
To show the existence, Benjamin proposed a method based on solving
one of two variational problems
max (E (ζ )
ζ 2R(ζ 0 )
λI2 (ζ )),
(3)
E (ζ )
(4)
or
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
max
ζ 2R(ζ 0 ),I2 (ζ )=I
CIMPA EGYPT ’09
february 2nd, 2009
6 / 16
To show the existence, Benjamin proposed a method based on solving
one of two variational problems
max (E (ζ )
ζ 2R(ζ 0 )
λI2 (ζ )),
(3)
E (ζ )
(4)
or
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
max
ζ 2R(ζ 0 ),I2 (ζ )=I
CIMPA EGYPT ’09
february 2nd, 2009
6 / 16
To show the existence, Benjamin proposed a method based on solving
one of two variational problems
max (E (ζ )
ζ 2R(ζ 0 )
λI2 (ζ )),
(3)
E (ζ )
(4)
or
max
ζ 2R(ζ 0 ),I2 (ζ )=I
where E (ζ ) is the kinetic energy, I > 0 and R(ζ 0 ) is the set of all
rearrangements of a prescribed function ζ 0 0.
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
6 / 16
Here, we are concerned with Poiseuille ‡ow which means, the Stokes
stream function
Ψ : Π ! R approaches
λ 4
r ,
4
given a velocity λr 2 in the z direction.
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
7 / 16
Here, we are concerned with Poiseuille ‡ow which means, the Stokes
stream function
Ψ : Π ! R approaches
λ 4
r ,
4
given a velocity λr 2 in the z direction.
Steady vortex rings in Poiseuille ‡ow is then governed by
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
LΨ = φ(Ψ
λ 4
r ) in Π
4
CIMPA EGYPT ’09
(5)
february 2nd, 2009
7 / 16
Here, we are concerned with Poiseuille ‡ow which means, the Stokes
stream function
Ψ : Π ! R approaches
λ 4
r ,
4
given a velocity λr 2 in the z direction.
Steady vortex rings in Poiseuille ‡ow is then governed by
LΨ = φ(Ψ
Thus, by (1), Ψ
‡ow.
λ 4
4r
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
λ 4
r ) in Π
4
(5)
is Stokes stream function for the Poiseuille
CIMPA EGYPT ’09
february 2nd, 2009
7 / 16
Here, we are concerned with Poiseuille ‡ow which means, the Stokes
stream function
Ψ : Π ! R approaches
λ 4
r ,
4
given a velocity λr 2 in the z direction.
Steady vortex rings in Poiseuille ‡ow is then governed by
LΨ = φ(Ψ
Thus, by (1), Ψ
‡ow.
λ 4
4r
λ 4
r ) in Π
4
(5)
is Stokes stream function for the Poiseuille
According to Benjamin’s approach, the existence of steady vortex
rings is to show that (5)
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
7 / 16
Here, we are concerned with Poiseuille ‡ow which means, the Stokes
stream function
Ψ : Π ! R approaches
λ 4
r ,
4
given a velocity λr 2 in the z direction.
Steady vortex rings in Poiseuille ‡ow is then governed by
LΨ = φ(Ψ
Thus, by (1), Ψ
‡ow.
λ 4
4r
λ 4
r ) in Π
4
(5)
is Stokes stream function for the Poiseuille
According to Benjamin’s approach, the existence of steady vortex
rings is to show that (5)
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
7 / 16
Here, we are concerned with Poiseuille ‡ow which means, the Stokes
stream function
Ψ : Π ! R approaches
λ 4
r ,
4
given a velocity λr 2 in the z direction.
Steady vortex rings in Poiseuille ‡ow is then governed by
LΨ = φ(Ψ
Thus, by (1), Ψ
‡ow.
λ 4
4r
λ 4
r ) in Π
4
(5)
is Stokes stream function for the Poiseuille
According to Benjamin’s approach, the existence of steady vortex
rings is to show that (5) has a solution with respect to boundary
conditions (BC ),
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
7 / 16
Here, we are concerned with Poiseuille ‡ow which means, the Stokes
stream function
Ψ : Π ! R approaches
λ 4
r ,
4
given a velocity λr 2 in the z direction.
Steady vortex rings in Poiseuille ‡ow is then governed by
LΨ = φ(Ψ
Thus, by (1), Ψ
‡ow.
λ 4
4r
λ 4
r ) in Π
4
(5)
is Stokes stream function for the Poiseuille
According to Benjamin’s approach, the existence of steady vortex
rings is to show that (5) has a solution with respect to boundary
conditions (BC ),for which, the function ζ = ω/r is a rearrangement
of a prescribed function.
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
7 / 16
Let ν be a measure on Π having density 2πr with respect to the
2-dimensional Lebesgue measure µ2 on Π.
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
8 / 16
Let ν be a measure on Π having density 2πr with respect to the
2-dimensional Lebesgue measure µ2 on Π.
Now for all (r , z ) and (r 0 , z 0 ) in Π, let G (r , r 0 , z, z 0 ) denotes Green’s
function for the operator L with homogeneous Dirichlet boundary
conditions on Π.
For ζ 2 Lp (Π, ν), where p > 52 , the function K ζ that de…ned by
K ζ (r , z ) : =
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
Z
G (r , r 0 , z, z 0 )ζ (r 0 , z 0 )2πr 0 dr 0 dz 0
Π
CIMPA EGYPT ’09
february 2nd, 2009
8 / 16
Let ν be a measure on Π having density 2πr with respect to the
2-dimensional Lebesgue measure µ2 on Π.
Now for all (r , z ) and (r 0 , z 0 ) in Π, let G (r , r 0 , z, z 0 ) denotes Green’s
function for the operator L with homogeneous Dirichlet boundary
conditions on Π.
For ζ 2 Lp (Π, ν), where p > 52 , the function K ζ that de…ned by
K ζ (r , z ) : =
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
Z
G (r , r 0 , z, z 0 )ζ (r 0 , z 0 )2πr 0 dr 0 dz 0
Π
CIMPA EGYPT ’09
february 2nd, 2009
8 / 16
Let ν be a measure on Π having density 2πr with respect to the
2-dimensional Lebesgue measure µ2 on Π.
Now for all (r , z ) and (r 0 , z 0 ) in Π, let G (r , r 0 , z, z 0 ) denotes Green’s
function for the operator L with homogeneous Dirichlet boundary
conditions on Π.
For ζ 2 Lp (Π, ν), where p > 52 , the function K ζ that de…ned by
K ζ (r , z ) : =
Z
G (r , r 0 , z, z 0 )ζ (r 0 , z 0 )2πr 0 dr 0 dz 0
Π
is the weak solution in distribution sense for the problem
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
LΨ = ζ in Π
CIMPA EGYPT ’09
february 2nd, 2009
8 / 16
Let Ω be a bounded domain in Π and H the Hilbert space de…ned by
the completion of D(Ω) with scalar product
hu, v i =
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
Z
Ω
1
ru.rvd ν.
r2
CIMPA EGYPT ’09
february 2nd, 2009
9 / 16
Let Ω be a bounded domain in Π and H the Hilbert space de…ned by
the completion of D(Ω) with scalar product
hu, v i =
Z
Ω
1
ru.rvd ν.
r2
Burton [2] showed that the operator K :
Lp (Π, ν) ! Lq (Π, ν)
ζ 7! Ψ
has the following properties :
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
9 / 16
Let Ω be a bounded domain in Π and H the Hilbert space de…ned by
the completion of D(Ω) with scalar product
hu, v i =
Z
Ω
1
ru.rvd ν.
r2
Burton [2] showed that the operator K :
Lp (Π, ν) ! Lq (Π, ν)
ζ 7! Ψ
has the following properties :
1
For ζ 2 Lp (Ω, ν) there is a unique K ζ 2 H that is a weak solution of
LΨ = ζ in Ω.
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
9 / 16
Let Ω be a bounded domain in Π and H the Hilbert space de…ned by
the completion of D(Ω) with scalar product
hu, v i =
Z
Ω
1
ru.rvd ν.
r2
Burton [2] showed that the operator K :
Lp (Π, ν) ! Lq (Π, ν)
ζ 7! Ψ
has the following properties :
1
2
For ζ 2 Lp (Ω, ν) there is a unique K ζ 2 H that is a weak solution of
LΨ = ζ in Ω.
K : Lp (Ω, ν) ! H is a bounded linear operator
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
9 / 16
Let Ω be a bounded domain in Π and H the Hilbert space de…ned by
the completion of D(Ω) with scalar product
hu, v i =
Z
Ω
1
ru.rvd ν.
r2
Burton [2] showed that the operator K :
Lp (Π, ν) ! Lq (Π, ν)
ζ 7! Ψ
has the following properties :
1
2
3
For ζ 2 Lp (Ω, ν) there is a unique K ζ 2 H that is a weak solution of
LΨ = ζ in Ω.
K : Lp (Ω, ν) ! H is a bounded linear operator
K : Lp (Ω, ν) ! Lq (Ω, ν) is symmetric, strictly positive and compact
operator
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
9 / 16
Let Ω be a bounded domain in Π and H the Hilbert space de…ned by
the completion of D(Ω) with scalar product
hu, v i =
Z
Ω
1
ru.rvd ν.
r2
Burton [2] showed that the operator K :
Lp (Π, ν) ! Lq (Π, ν)
ζ 7! Ψ
has the following properties :
1
2
3
4
For ζ 2 Lp (Ω, ν) there is a unique K ζ 2 H that is a weak solution of
LΨ = ζ in Ω.
K : Lp (Ω, ν) ! H is a bounded linear operator
K : Lp (Ω, ν) ! Lq (Ω, ν) is symmetric, strictly positive and compact
operator
2 ( Ω ).
If ζ 2 Lp (Ω, ν) then K ζ 2 Wloc
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
9 / 16
The kinetic energy E and the generalised impulse I4 are given in terms of
ζ 2 Lp (Π, ν) by
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
10 / 16
The kinetic energy E and the generalised impulse I4 are given in terms of
ζ 2 Lp (Π, ν) by
E (ζ ) =
1
2
I4 (ζ ) =
1
4
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
Z
ζ (r , z )K ζ (r , z )d ν,
Π
Z
r 4 ζ (r , z )d ν.
Π
CIMPA EGYPT ’09
february 2nd, 2009
10 / 16
The kinetic energy E and the generalised impulse I4 are given in terms of
ζ 2 Lp (Π, ν) by
E (ζ ) =
1
2
I4 (ζ ) =
1
4
Z
ζ (r , z )K ζ (r , z )d ν,
Π
Z
r 4 ζ (r , z )d ν.
Π
Any measurable function ζ on Π will be called Steiner-symmetric if ζ
satis…es
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
10 / 16
The kinetic energy E and the generalised impulse I4 are given in terms of
ζ 2 Lp (Π, ν) by
E (ζ ) =
1
2
I4 (ζ ) =
1
4
Z
ζ (r , z )K ζ (r , z )d ν,
Π
Z
r 4 ζ (r , z )d ν.
Π
Any measurable function ζ on Π will be called Steiner-symmetric if ζ
satis…es
r > 0, 0 z 0 z ) ζ (r , z 0 ) ζ (r , z ) 0
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
10 / 16
The kinetic energy E and the generalised impulse I4 are given in terms of
ζ 2 Lp (Π, ν) by
E (ζ ) =
1
2
I4 (ζ ) =
1
4
Z
ζ (r , z )K ζ (r , z )d ν,
Π
Z
r 4 ζ (r , z )d ν.
Π
Any measurable function ζ on Π will be called Steiner-symmetric if ζ
satis…es
r > 0, 0 z 0 z ) ζ (r , z 0 ) ζ (r , z ) 0
and
z 2 R, r > 0 ) ζ (r ,
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
z ) = ζ (r , z ).
CIMPA EGYPT ’09
february 2nd, 2009
10 / 16
Let ζ 1 and ζ 2 be two measurable non-negative functions on Π.
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
11 / 16
Let ζ 1 and ζ 2 be two measurable non-negative functions on Π.
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
11 / 16
Let ζ 1 and ζ 2 be two measurable non-negative functions on Π. We
say that ζ 1 is rearrangement of ζ 2 on Π with respect ν measure, or
ζ 2 is a rearrangement of ζ 1 if
ν(f(r , z ) 2 Πjζ 1 (r , z )
k g)
= ν(f(r , z ) 2 Πjζ 2 (r , z )
k g)
for all k > 0.
Hence if ζ 1 2 Lp (Π, ν) for some p
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
1, then
CIMPA EGYPT ’09
february 2nd, 2009
11 / 16
Let ζ 1 and ζ 2 be two measurable non-negative functions on Π. We
say that ζ 1 is rearrangement of ζ 2 on Π with respect ν measure, or
ζ 2 is a rearrangement of ζ 1 if
ν(f(r , z ) 2 Πjζ 1 (r , z )
k g)
= ν(f(r , z ) 2 Πjζ 2 (r , z )
k g)
for all k > 0.
Hence if ζ 1 2 Lp (Π, ν) for some p
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
1, then
CIMPA EGYPT ’09
february 2nd, 2009
11 / 16
Let ζ 1 and ζ 2 be two measurable non-negative functions on Π. We
say that ζ 1 is rearrangement of ζ 2 on Π with respect ν measure, or
ζ 2 is a rearrangement of ζ 1 if
ν(f(r , z ) 2 Πjζ 1 (r , z )
k g)
= ν(f(r , z ) 2 Πjζ 2 (r , z )
k g)
for all k > 0.
Hence if ζ 1 2 Lp (Π, ν) for some p
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
1, then it follows that
k ζ 1 kp = k ζ 2 kp .
CIMPA EGYPT ’09
february 2nd, 2009
11 / 16
If ζ 0 a non-negative measurable function on Π, then
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
12 / 16
If ζ 0 a non-negative measurable function on Π, then the decreasing
rearrangement ζ 0 of ζ can be de…ned as the essentially unique
non-negative decreasing function on (0, ∞) that
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
12 / 16
If ζ 0 a non-negative measurable function on Π, then the decreasing
rearrangement ζ 0 of ζ can be de…ned as the essentially unique
non-negative decreasing function on (0, ∞) thatsatis…es
ν(f(r , z ) 2 Πjζ 0 (r , z )
kg
= µ1 ft 2 [0, ∞)jζ 0 (t )
k g)
for all k > 0, where
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
12 / 16
If ζ 0 a non-negative measurable function on Π, then the decreasing
rearrangement ζ 0 of ζ can be de…ned as the essentially unique
non-negative decreasing function on (0, ∞) thatsatis…es
ν(f(r , z ) 2 Πjζ 0 (r , z )
kg
= µ1 ft 2 [0, ∞)jζ 0 (t )
k g)
for all k > 0, whereµ1 is the 1-dimension Lebesgue measure.
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
12 / 16
We use R(ζ 0 ) to denote the set of all rearrangements of ζ 0 on Π;
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
13 / 16
We use R(ζ 0 ) to denote the set of all rearrangements of ζ 0 on Π;
R(ζ 0 ) can contain some functions denoted by ζ s , that are
Steiner-symmetric.
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
13 / 16
We use R(ζ 0 ) to denote the set of all rearrangements of ζ 0 on Π;
R(ζ 0 ) can contain some functions denoted by ζ s , that are
Steiner-symmetric.
With the same function ζ 0 we de…ne the sets W (ζ 0 ) and RC(ζ 0 ) as
follows:
RC(ζ 0 ) = fζ
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
0 ; Πjζ = ζ 0 1[0,β] , β
CIMPA EGYPT ’09
0g,
february 2nd, 2009
13 / 16
We use R(ζ 0 ) to denote the set of all rearrangements of ζ 0 on Π;
R(ζ 0 ) can contain some functions denoted by ζ s , that are
Steiner-symmetric.
With the same function ζ 0 we de…ne the sets W (ζ 0 ) and RC(ζ 0 ) as
follows:
RC(ζ 0 ) = fζ
W (ζ 0 ) = fζ
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
0 ; Πjζ = ζ 0 1[0,β] , β
0; Πj8k > 0
Z
(ζ
Π
CIMPA EGYPT ’09
k )+
Z
Π
0g,
(ζ 0
k ) + g.
february 2nd, 2009
13 / 16
The set W (ζ 0 ) is called the weak closure of the set R(ζ 0 ) in
Lp (Π, ν) and RC(ζ 0 ) is the set of all rearrangements of
curtailments of ζ 0 .
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
14 / 16
The set W (ζ 0 ) is called the weak closure of the set R(ζ 0 ) in
Lp (Π, ν) and RC(ζ 0 ) is the set of all rearrangements of
curtailments of ζ 0 .
Following Burton [3] and Douglas [4, 5] we have
R(ζ 0 )
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
RC(ζ 0 )
CIMPA EGYPT ’09
W ( ζ 0 ).
february 2nd, 2009
14 / 16
The set W (ζ 0 ) is called the weak closure of the set R(ζ 0 ) in
Lp (Π, ν) and RC(ζ 0 ) is the set of all rearrangements of
curtailments of ζ 0 .
Following Burton [3] and Douglas [4, 5] we have
R(ζ 0 )
RC(ζ 0 )
W ( ζ 0 ).
In the case of a non-negative function ζ 0 2 Lp (U, ν), where
1 < p < ∞ and U Π having in…nite measure, Douglas proved that
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
14 / 16
The set W (ζ 0 ) is called the weak closure of the set R(ζ 0 ) in
Lp (Π, ν) and RC(ζ 0 ) is the set of all rearrangements of
curtailments of ζ 0 .
Following Burton [3] and Douglas [4, 5] we have
R(ζ 0 )
RC(ζ 0 )
W ( ζ 0 ).
In the case of a non-negative function ζ 0 2 Lp (U, ν), where
1 < p < ∞ and U Π having in…nite measure, Douglas proved that
1
W (ζ 0 ) is convex and weakly sequentially compact, and
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
14 / 16
The set W (ζ 0 ) is called the weak closure of the set R(ζ 0 ) in
Lp (Π, ν) and RC(ζ 0 ) is the set of all rearrangements of
curtailments of ζ 0 .
Following Burton [3] and Douglas [4, 5] we have
R(ζ 0 )
RC(ζ 0 )
W ( ζ 0 ).
In the case of a non-negative function ζ 0 2 Lp (U, ν), where
1 < p < ∞ and U Π having in…nite measure, Douglas proved that
1
2
W (ζ 0 ) is convex and weakly sequentially compact, and
The set of extreme points of W (ζ 0 ) is the set RC(ζ 0 ).
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
14 / 16
Now, with all these notations, our main result is given as follows
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
15 / 16
Now, with all these notations, our main result is given as follows:
Theorem
1
the functional E attains a maximum value subject to
ζ 2 W (ζ 0 ) and I4 (ζ ) = I .
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
15 / 16
Now, with all these notations, our main result is given as follows:
Theorem
1
2
the functional E attains a maximum value subject to
ζ 2 W (ζ 0 ) and I4 (ζ ) = I .
every maximiser is an element of the set RC(ζ 0 ).
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
15 / 16
Now, with all these notations, our main result is given as follows:
Theorem
1
2
3
the functional E attains a maximum value subject to
ζ 2 W (ζ 0 ) and I4 (ζ ) = I .
every maximiser is an element of the set RC(ζ 0 ).
For any maximiser ζ, there exist a positive number λ and an
increasing function φ such that the function ψ := K ζ, λ and φ satisfy
the equation
λ 4
r )
LΨ = φ (Ψ
4
almost everywhere in Π.
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
15 / 16
Now, with all these notations, our main result is given as follows:
Theorem
1
2
3
4
the functional E attains a maximum value subject to
ζ 2 W (ζ 0 ) and I4 (ζ ) = I .
every maximiser is an element of the set RC(ζ 0 ).
For any maximiser ζ, there exist a positive number λ and an
increasing function φ such that the function ψ := K ζ, λ and φ satisfy
the equation
λ 4
r )
LΨ = φ (Ψ
4
almost everywhere in Π.
If ζ is a maximiser of E , then there exists I > 0 such that ζ 2 R(ζ 0 )
for all I > I .
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
15 / 16
Therefore, we proved the existence of a steady vortex ring in Poiseuille
‡ow. In order to prove this theorem, we need to use the same strategy
that used by Rebah in [8].
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
16 / 16
1
2
3
4
5
6
7
8
T. B. Benjamin. The alliance of practical and analytical insight into
the nonlinear problems of ‡uid mechanics. Applications of methods of
functional analysis to problems in mechanics. Lecture notes in
mathematics 503, 8-29. Springer-Verlag, 1976.
G. R. Burton. Rearrangements of functions, maximization of convex
functionals and vortex rings. Math. Ann., 276, 225-253, 1987.
G. R. Burton. Vortex-rings of prescribed impulse. Math. Proc.
Cambridge Philos. Soc. vol. 134, p.515-528, 2003
R. J. Douglas. Rearrangements and nonlinear analysis of vortices.
PhD thesis, University of Bath 1992.
R. J. Douglas. Rearrangements of function on unbounded domains.
Proc Royal Soc. Edinb. 124A, 621-644, 1994.
H. Lamb. Hydrodynamics, London. Cambridge University Press. 6th
edn, 1932.
D. Rebah. A steady vortex ring in Poiseuille ‡ow and rearrangements
of function. Proc Royal Soc. London A. 462, 1235-1253, 2006.
D. Rebah. A constrained variational problem for an existence theorem
of a steady vortex pair in two phase shear ‡ow. Journal of Nonlinear
Analysis, NA, (Online 3 Nov 2006).
D. Rebah (Labo. AMNEDP, USTHB, Algeria)
CIMPA EGYPT ’09
february 2nd, 2009
16 / 16