Ammari H., Benkirane A., Touzani A. (editors) Recent Developments in Nonlinear Analysis
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where
ρ
0
(t) =
−~ν
2
e
iνt
ν − 2ω
0
−
e
iνt
ν + 2ω
0
−
i~ω
0
16
e
2iωt
ω − ω
0
−
e
−2iωt
ω + ω
0
e
−2iω
0
t
(234)
η
2
(t) =
ω
0
sin 2ωt
4ω
(235)
3.4.1. First order solutions
From Eq. (19), with the initial condition
(1)
V
I
(t
0
= 1, we have
(1a)
U(t) =
1 −
iµ
~
e
−iω
0
ta
+
a
e
H
I
e
−iω
0
ta
+
a
e
iH
0
t/~
=
(1a)
T (t)e
−i
(1a)
Rt/~
(236)
The first order ameliorated Floquet operators are given by
(1a)
R = H
0
(237)
(1a)
T (t) = 1 −iµ
ρ
1
(t)a
2
+ ρ
∗
1
(t)a
+2
+ η
2
(t)
aa
+
+
1
2
(238)
where
ρ
1
(t) = e
2iω
0
t
ρ
0
(t)
~
(239)
We notice that, to first order approximation, the operator
(1a)
R is equal
to the unperturbed Hamiltonian, and the
(1a)
T (t) operator is unitary and
periodic with the same period of H
1
(t). From Eqs. (6) and (8) and using
Eqs. (237) and (238), the first order Floquet states are such as
|
(1a)
ψ
n
(t)i = e
−i(n+1/2)ω
0
t
1 − iµη
2
(t)
n +
1
2
|ni − iµ
h
p
n(n−)ρ
1
(t)|n − 2i
(240)
+
p
(n + 1)(n + 2)ρ
∗
1
(t)|n + 2i
i
The expectation values of Q, P , Q
2
and P
2
are deduced and one gets
hQi
n,n
= hP i
n,n
= 0 (241)
h
(1a)
Q
2
i
n,n
=
~
2β
0
(2n + 1)
n
1 + µξ
1
(t) + µ
2
ξ
2+
(t)
o
(242)
h
(1a)
P
2
i
n,n
=
~β
0
2
(2n + 1)
n
1 −µξ
1
(t) + µ
2
ξ
2−
(t)
o
(243)
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where
ξ
1
(t) = 4iIm
ρ
1
(t)
(244)
ξ
2±
(t) =
n+
1
2
2
η
2
2
(t)+2(n
2
+n+5)|ρ
1
(t)|
2
±2(n
2
+n+1)η
2
(t)Re(ρ
1
(t))
(245)
The first order uncertainty relation in Floquet-states
(1a)
|ψ
n
(t)i is obtained
in the following form
(1a)
(∆q∆p)
n,n
=
~
2
(2n + 1)
1 + µ
2
F
1a
(n, t)
1
2
(246)
F
1a
(n, t) = 2
n+
1
2
2
η
2
2
(t)+4(n
2
+n+5)Re
2
(ρ
1
(t))−4(n
2
+n+1)Im
2
(ρ
1
(t))
(247)
The Heisenberg uncertainty condition ∆q∆p ≥
~
2
is satisfied for all time t.
3.4.2. Second order solutions
As
H
I
(t) = 0 and S
I
(t) = 0 , the differential equation (21) may be written
in the following form
i~
dΓ
dt
=
iµ
2
2~
Z
I
(t)Γ(t) (248)
where
Γ(t) = e
−iµ
2
λ
mΩ
H
0
t
~
(249)
and
λ
mΩ
=
4ν
2
ν
2
− 4ω
2
0
+
ω
2
0
16(ω
2
− ω
2
0
)
(250)
Then, the second order evolution operator (Eq. (20)) is obtained in the
following form
(2)
U(t) =
1 −
iµ
~
e
−i
H
0
t
~
e
H
I
e
i
H
0
t
~
e
−i(1+µ
2
λ
mΩ
)
H
0
t
~
(251)
The corresponding Floquet operators are
(2)
R = (1 + µ
2
λ
mΩ
)H
0
(252)
(2)
T (t) =
(1a)
T (t) (253)
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322
Determination of the ameliorated second order solution (Eq. (24)) necessi-
tates the calculation of the operators
e
Z
I
(t),
e
H
2
I
(t) such as
e
Z
I
(t) = ~
2
h
ϕ
0
(t)
2a
+
a + 1
+ ϕ
0
2
(t)a
2
− ϕ
0
∗
2
(t)a
+2
i
(254)
with
ϕ
0
(t) = −i~
2
ω
0
(
2ν sin 2νt
ν
2
− 4ω
2
0
+
ω
2
0
sin 4ωt
2
6
ω(ω
2
− ω
2
0
)
+
ν
4(ν
2
− 4ω
2
0
)(ω
2
− ω
2
0
)
×
(255)
"
(ν − 2ω)(νω + 2ω
2
0
) cos(ν + 2ω)t
ν + 2ω
−
(ν + 2ω)(νω − 2ω
2
0
) cos(ν − 2ω)t
ν − 2ω
#)
ϕ
0
2
(t) =
~
2
ω
0
8ω
e
−2iω
0
t
(
ω
0
8
"
ω
0
e
−4iνt
(ω + ω
0
)(2ω + ω
0
)
−
ω
0
e
4iνt
(ω − ω
0
)(2ω − ω
0
)
+
2ω
ω
2
− ω
2
0
#
(256)
+iν
"
(ν + 2(ω − ω
0
))e
i(ν−2ω)t
(ν − 2ω
9
)(ν − 2(ω + ω
0
))
−
(ν − 2(ω + ω
0
))e
i(ν+2ω)t
(ν − 2ω
9
)(ν + 2(ω − ω
0
))
+
(ν + 2(ω + ω
0
))e
−i(ν−2ω)t
(ν + 2ω
9
)(ν − 2(ω − ω
0
))
−
(ν − 2(ω − ω
0
))e
−i(ν+2ω)t
(ν + 2ω
9
)(ν + 2(ω + ω
0
))
#)
and
e
H
2
I
= ρ
2
0
(t)a
4
+ ρ
∗2
0
(t)a
+4
+ |ρ
0
(t)|
2
(a
2
a
+2
+ a
+2
a
2
) (257)
The operator A
2
(t) given by Eq. (25) becomes
A
2
(t) =
1
2~
2
e
Z
I
(t) −
e
H
2
I
(t)
(258)
So, the ameliorated second order evolution operator is given by
(2a)
U(t) =
1+e
−i
H
0
t
~
−
iµ
~
e
H
I
+µ
2
A
2
(t)
e
i
H
0
t
~
e
−i(1+µ
2
λ
mΩ
)
H
0
t
~
(259)
One may deduce the following Floquet operators
(2a)
R =
(2)
R (260)
(2a)
T (t) =
(2)
T (t) + µ
2
A
0
2
(t) (261)
with
A
0
2
(t) = e
−i
H
0
t
~
A
2
(t)e
i
H
0
t
~
(262)
December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent
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then
(2a)
T (t) = 1 − iµ
ρ
1
a
2
+ ρ
∗
1
a
2
+ η
2
a
+
a +
1
2
+
µ
2
2
× (263)
2ϕ
0
~
2
a
+
a+
1
2
−2|ρ
1
|
2
a
+
a+
1
2
2
+
3
4
+ϕ
2
a
2
−ϕ
∗
2
a
+2
−ρ
2
1
a
4
−ρ
∗2
1
a
+4
where
ϕ
2
(t) = e
2iω
0
t
ϕ
0
2
(t)
~
2
(264)
According to Eq. (6), the second order Floquet states are
|
(2a)
ψ
n
(t)i =
(2a)
T (t)|
(2a)
φ
n
(t)i (265)
where |
(2a)
φ
n
(t)i , the eigenstates of operator
(2a)
R , have the following
form
|
(2a)
φ
n
(t)i = e
−i(n+
1
2
)(1+µ
2
λ
mΩ
)ω
0
t
|ni (266)
By substituting Eq. (263) into Eq. (265), the construction of the ameliorated
second order Floquet states gives
|
(2a)
ψ
n
(t)i = e
−i
λ
mΩ
t
~
{D
n
0
(t)|ni + D
n
−2
(t)|n − 2i+ D
n
2
(t)|n + 2i
+D
n
−4
(t)|n −4i + D
n
4
(t)|n + 4i}
(267)
D
n
0
(t) = 1−iµ
n+
1
2
η
2
(t)+µ
2
−(n
2
+n+1)|ρ
1
(t)|
2
+(2n+1)
ϕ
0
t
2~
2
(268)
D
n
−2
(t) =
p
n(n − 1)
n
− iµρ
1
(t) + µ
2
ϕ
2
(t)
2
o
(269)
D
n
2
(t) =
p
(n + 1)(n + 2)
n
− iµρ
∗
1
(t) − µ
2
ϕ
∗
2
(t)
2
o
(270)
D
n
−4
(t) =
−µ
2
p
n(n −1)(n − 2)(n −3)ρ
2
1
(t) (271)
D
n
4
(t) =
−µ
2
p
(n + 1)(n + 2)(n + 3)(n + 4)ρ
∗2
1
(t) (272)
The expectation values of q , p, q
2
and p
2
in the |
(2a)
ψ
n
(t) states are such
as
(2a)
hqi
n,n
=
(2a)
hpi
n,n
= 0 (273)
December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent
324
h
(2a)
q
2
i
n,n
=
~
2β
(2n+1){1+µg
n
1
(t)+µ
2
g
n
2+
(t)+µ
3
g
n
3+
(t)+µ
4
g
n
4+
(t)} (274)
h
(2a)
p
2
i
n,n
=
~β
2
(2n+1){1−µg
n
1
(t)+µ
2
g
n
2−
(t)+µ
3
g
n
3−
(t)+µ
4
g
n
4−
(t)} (275)
where
β(t) = ω
0
m(t) (276)
g
n
1
(t) = 4iIm(ρ
1
(t)) (277)
g
n
2±
(t) = 8|ρ
1
|
2
+
n +
1
2
η
2
2
∓ 2Re(ϕ
2
) ± 2(n
2
+ n + 1)η
2
Re(ρ
1
) (278)
g
n
3±
(t) = i(2n + 1)
2
η
2
ϕ
0
2~
+ 2i(n
2
+ n + 5)
h
Im(ρ
∗
1
ϕ
2
) ±2Im(η
1
)|η
1
|
2
i
(279)
±2i(n
2
+ n + 1)
ϕ
0
Re(ρ
1
)
~2
+
η
2
Im(ϕ
2
)
2
g
n
4±
(t) =
n+
1
2
2
|ϕ
0
|
2
~
2
N
1
(n, 6)
4
+(n
2
+n+1)
2
|ρ
1
|
4
+
(n
2
+ n + 5)
2
|ϕ
2
|
2
(280)
∓(n
2
+ n + 1)
ϕ
0
Im(ϕ
2
)
~
2
± 2(3n
2
+ 3n + 7)Re(ϕ
2
)Re
2
(ρ
1
)
±2(n
2
+ n + 5)Re(ϕ
2
)Im
2
(ρ
1
) ∓ 8(n
2
+ n + 3)Im(ϕ
2
)Re(ρ
1
)Im(ρ
1
)
and
N
1
(n, k) =
(n + 4)!
n!
+
n!
(n −4)!
+ 2
k
(n
2
+ n + 3) (281)
Then, the second order uncertainty relation in Floquet-states
(2a)
|ψ
n
(t)i
has the following form
(2a)
(∆q∆p)
n,n
=
~
2
(2n+1)
1+µ
2
J
n
2
(ν, t)+µ
3
J
n
3
(ν, t)+µ
4
J
n
4
(ν, t)
1
2
(282)
where
J
n
2
(ν, t) = (2n + 1)
2
η
2
2
2
+ 16Re
2
(ρ
1
(t)) (283)
J
n
3
(ν, t) =
i
~
(2n + 1)
2
η
2
ϕ
0
+ 4i(n
2
+ n + 5)Re(ρ
1
)Im(ϕ
2
) (284)
December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent
325
−4i(n
2
+ n + 1)Im(ρ
1
)[Re(ϕ
2
) + 4η
2
Re(ρ
1
)]
J
n
4
(ν, t) = (2n + 1)
2
|ϕ
0
|
2
2~
2
+ 4|ρ
1
|
2
η
2
2
+
n +
1
2
2
η
4
2
4
+ (n
2
+ n + 5)|ϕ
2
|
2
(285)
+
N
1
(n, 6)
2
+ 2(n
2
+ n + 1)
2
+ 64
Re
4
(ρ
1
) − 4Re
2
(ϕ
2
)
+
N
1
(n, 6)
2
+ 2(n
2
+ n + 1)
2
− 32(n
2
+ n + 3)
Im
4
(ρ
1
)
−(N
1
(n, 6) −4(n
2
+ n + 1)(n
2
+ n − 7))Re
2
(ρ
1
)Im
2
(ρ
1
)
+8(n
2
+ n + 1)
2ϕ
0
Re(ρ
1
)Im(ρ
1
)
~
2
+ η
2
Im(ρ
1
)Im(ϕ
2
)
−4(n
2
+ n + 1)[(n
2
+ n + 1)η
2
2
Re
2
(ρ
1
) + 2η
2
Re(ρ
1
)Re(ϕ
2
)]
4. Conclusion
When the time variation of the hamiltonian of a quantum system is peri-
odic, the Floquet theory is one of the most useful approaches which permits
resolution of the Schr¨odinger equation. Therefore, in this paper, we have
treated the Floquet decomposition, solution of the evolution equation, by
using the resonating averages method from the first order in µ to the second
ameliorated order. The approach was applied to the forced harmonic oscilla-
tor, to the harmonic oscillator with a time-dependent mass and a constant
frequency, to the harmonic oscillator with a constant mass and a time-
dependent frequency and to the harmonic oscillator with time-dependent
mass and frequency. We have determined the approached forms of the evolu-
tion operator up to second ameliorated order in µ . Then, we have calculated
the Floquet operators, Floquet states and verified that the uncertainty prin-
ciple of Heisenberg is satisfied. Comparisons of our results, corresponding
to each case of the above mentioned quantum systems, showed that they
are identical to the results published by other authors which have used dif-
ferent methods,
3
,
59
-,
1218
This perfect concordance showed the efficiency of
our approach and the validity of the method.
December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent
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December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent
327
Renormalized solutions of nonlinear degenerated parabolic
problems: Existence and uniqueness
Y. Akdim, J. Bennouna, M. Mekkour, and M. Rhoudaf
D´epartement de Math´ematiques Facult´e des Sciences Dhar-Mahraz F`es, Morocco
In this paper, we study the existence and uniqueness of renormalized solutions
for the nonlinear degenerated parabolic problem
∂u
∂t
− div(a(x, t, Du)) = f in Ω × [0, T ],
u(x, 0) = u
0
in Ω
u = 0 in ∂Ω × [0, T ]
where a : Ω× [0, T ] × R
N
→ R
N
is a Carath´eodory function satisfying the coer-
civity condition, the general growth condition and only the large monotonicity.
The second term f belongs to L
1
(Q) and u
0
∈ L
1
(Ω).
Keywords: Weighted Sobolev spaces; Truncations; Renormalized solutions.
1. Introduction
Let Ω be a bounded open set of R
N
, p be a real number such that 2 <
p < ∞, Q = Ω × [0, T ] and w = {w
i
(x), 0 ≤ i ≤ N} be a vector of weight
functions (i.e., every component w
i
(x) is a measurable function which is
positive a.e. in Ω) satisfying some integrability conditions. The objective of
this paper is to study the following problem, in the weighted Sobolev space,
∂u
∂t
+ Au = f in Ω ×[0, T ],
u = 0 on ∂Ω×]0, T [,
u(x, 0) = u
0
on Ω
(1)
where A = −div(a(x, t, Du)) and a : Ω × [0, T ] × R
N
→ R
N
is a
Carath´eodory function, satisfying the coercivity condition
a(x, t, ξ).ξ ≥ α
N
X
i=1
w
i
|ξ
i
|
p
,
December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent
328
the general growth condition
|a
i
(x, t, ξ)| ≤ βw
1/p
i
(x)[k(x, t) +
N
X
j=1
w
1/p
0
j
(x)|ξ
j
|
p−1
],
and only the large monotonicity,
[a(x, t, ξ) − a(x, t, η)](ξ − η) ≥ 0 for all (ξ, η) ∈ R
N
× R
N
,
f belongs to L
1
(Q).
The difficulties that arise in problem 1 are due to the following facts:
the data f belongs to L
1
(Q) and a is monotone (but not necessarily strictly
monotone).
In the classical Sobolev space, existence of a weak solution has been
established by L. Boccardo and T. Gallouet,
2
and by D.Blanchard
3
when a
is strictly monotone. We consider here renormalized solutions, for reason,
this definition allows us to prove the uniqueness of a renormalized solution
for problem 1, in contrast to the framework of weak solution where this
question is still open. The notion of renormalized solution was introduced
by J. Diperna and P.L. Lions
8
for the study of the Boltzmann equation.
For the parabolic equation (1) the existence and uniqueness of a renor-
malized solution has been proved by D. Blanchard and F. Murat
4
in the
classical Sobolev space, they only supposed the large monotonicity. By L.
Aharouch et al in
1
in the weighted Sobolev space with f belongs to the
dual space and a is strictly monotone.
It is our purpose in this paper to generalize the result of
4
and prove
the existence and uniqueness of renormalized solution for the problem (1.1)
in the setting of the weighted Sobolev spaces. The proof uses techniques
different from that given in [,
34
].
2. Preliminaries and basic assumptions
Let Ω be a bounded open set of R
N
, p be a real number such that 2 < p < ∞
and w = {w
i
(x), 0 ≤ i ≤ N } be a vector of weight functions. Further, we
suppose in all our considerations that , there exit
r
0
> max(N, p) such that w
r
0
r
0
−p
i
∈ L
1
loc
(Ω), (2)
w
−1
p−1
i
∈ L
1
loc
(Ω), for any 0 ≤ i ≤ N. (3)
W
1,p
(Ω, w) is the space of all real-valued functions u ∈ L
p
(Ω, w
0
) such that
the derivatives in the sense of distributions
∂u
∂x
i
∈ L
p
(Ω, w
i
).
December 28, 2009 12:15 WSPC - Proceedings Trim Size: 9in x 6in recent
329
Which is a Banach space under the norm
kuk
1,p,w
=
h
Z
Ω
|u(x)|
p
w
0
(x) dx +
N
X
i=1
Z
Ω
|
∂u(x)
∂x
i
|
p
w
i
(x) dx
i
1/p
. (4)
The condition (3) implies that C
∞
0
(Ω) is a space of W
1,p
(Ω, w) and conse-
quently, we can introduce the subspace V = W
1,p
0
(Ω, w) of W
1,p
(Ω, w) as
the closure of C
∞
0
(Ω) with respect to the norm (4). Moreover, condition (3)
implies that W
1,p
(Ω, w) as well as W
1,p
0
(Ω, w) are reflexive Banach spaces.
Assumption (H1)
For 2 ≤ p < ∞, we assume that the expression
k|u|k
V
=
N
X
i=1
Z
Ω
|
∂u(x)
∂x
i
|
p
w
i
(x) dx
1/p
(5)
is a norm defined on V which equivalent to the norm (4), and there exist a
weight function σ on Ω such that,
σ ∈ L
1
(Ω) and σ
−1
∈ L
1
(Ω).
We assume also the Hardy inequality,
Z
Ω
|u(x)|
q
σ dx
1/q
≤ c
N
X
i=1
Z
Ω
|
∂u(x)
∂x
i
|
p
w
i
(x) dx
1/p
, (6)
holds for every u ∈ V with a constant c > 0 independent of u, and moreover,
the imbedding
W
1, p
(Ω, w) ,→,→ L
p
(Ω, σ), (7)
expressed by the inequality (6) is compact. Note that (V, k|.|k
V
) is a uni-
formly convex (and thus reflexive) Banach space.
Assumption (H2)
|a
i
(x, t, ξ)| ≤ βw
1/p
i
(x)[k(t, x) +
N
X
j=1
w
1/p
0
j
(x)|ξ
j
|
p−1
] f or 1 ≤ i ≤ N, (8)
[a(x, t, ξ) −a(x, t, η)](ξ − η) ≥ 0 for all (ξ, η) ∈ R
N
× R
N
, (9)
a(x, t, ξ).ξ ≥ α
N
X
i=1
w
i
|ξ
i
|
p
, (10)
a(x, t, 0) = 0 (11)