On a quasilinear Zakharov system
describing laser-plasma interactions
by M. Colin and T. Colin,
MAB, Université Bordeaux I,
CNRS UMR 5466 and CEA LRC MO3,
351 cours de la libération,
33405 Talence cedex, France.
[email protected]
[email protected]
Abstract : In this paper, starting from the bi-fluid Euler-Maxwell system,
we derive a complete set of Zakharov’s equations type describing laser-plasma
interactions. This system involves a quasilinear part which is not hyperbolic
and exhibits some elliptic zones. This difficulty is overcome by making a change
of unknowns that are strongly related to the dispersive part. This change of
variable is a symmetrization of the quasilinear part and is the key of this paper.
This shows that the Cauchy problem is locally well-posed.
1
Introduction.
1.1
Physical context.
The construction of always more powerful lasers allows new experiments where
hot plasma are created in which laser beams can propagate. The main goal is
to simulate nuclear fusion by inertial confinement in a laboratory. Of course,
one wants also to simulate numerically these experiences (it is much cheaper
than true experiences !). We therefore need some precise and reliable models
of laser-plasma interactions. The kinetic-type models are the more precise ones
but the cost in term of computations is exorbitant and no physically relevant
situation for nuclear fusion can be simulated in this context. Therefore, we use
a bi-fluid model for the plasma : we couple two compressible Euler systems
with Maxwell equations. Even under this form, it is not possible to perform direct computations because of high frequency motions and the small wavelength
AMS classification scheme number: 35Q55, 35Q60, 78A60
1
involved in the problem. At the beginning of the 70’s, Zakharov and its collaborators introduced the so-called Zakharov’s equation [20] in order to describe the
electronic plasma waves. Basically, the slowly varying envelope of the electric
field E = ∇ψ is coupled to the low-frequency variation of the density of the ions
δn by the following equations written in a dimensionless form :

 i∂t ∇ψ + ∆(∇ψ) = ∇∆−1 div(δn∇ψ),
(1.1)
 2
∂t δn − ∆δn = ∆(|∇ψ|2 ).
Of course, variations of this systems exist (see [17] for example). For laser
propagation, one uses the paraxial approximation and the Zakharov system
reads

 i(∂t + ∂y )E + ∆⊥ E = nE,
(1.2)
 2
(∂t − ∆⊥ )n = ∆⊥ (|E|2 ),
where ∆⊥ = ∂x2 + ∂z2 . (See [14] or [17] for a symmetric use of this kind of model
for numerical simulation). It is however not clear if system (1.2) is well-posed
or not and no results are available at the moment.
Physically, the complete situation is the following one : the laser enters
the plasma. Part of it is backscattered through a Raman-type process and a
Brillouin-type process. The Raman and Brillouin parts and the laser combine
together to create an electron-plasma wave. These four waves interact in order
to create a low-frequency variation of density of the ions which has itself an
influence on the four preceding waves. This situation can be described by two
systems of the form (1.2) and one of the form (1.1) coupled by quasilinear terms.
Such a system is used in [15] for numerical simulations.
Notations : As usual, we denote by Lp (Rd ) the Lebesgue space
n
o
0
Lp (Rd ) = u ∈ S (Rd ) / ||u||p < +∞
where
Z
||u||p =
p1
|u(x)| dx
if 1 ≤ p < +∞
p
Rd
and
||u||∞ = ess.sup |u(x)|; x ∈ Rd .
We define the Sobolev space H s (Rd ) as follows
Z
s
d
0
d
2
2 s
2
H (R ) = u ∈ S (R ) / ||u||H s (Rd ) =
(1 + |ξ| ) |b
u(ξ)| dξ < +∞
Rd
where u
b(ξ) is the Fourier transform of u. Let C(I, E) be the space of continuous
functions from an interval I of R to a Banach space E. For 1 ≤ j ≤ d, we set
2
∂u
. For k ∈ Nd , k = (k1 , ..., kd ), we denote ∂ k u = ∂xk11 ... ∂xkdd u. Different
∂xj
positive constants might be denoted by the same letter C. We also denote by
Re(u) and Im(u) the real part and the imaginary part of u.
∂xj u =
1.2
Presentation of the problem.
The system used in [15] is
2iπe2
ie
ic2
∆⊥ )A0 = −
nA0 +
(∇ · E) AR e−iθ ,
2ω0
mω0
4mω0
ic2
2iπe2
ie
(∂t + vR ∂y −
∆⊥ )AR = −
nAR +
(∇ · E ∗ ) A0 eiθ ,
2ωR
mωR
4mωR
e
4πe2
2
∇∆−1 div(nE) +
ω 2 ∇(A0 A∗R eiθ ),
(2iωpe ∂t + 3vth
∆)E =
m
2mc2 pe
!
2
ωpe
Z
2
2
2
2
2
∆
(|A0 | + |AR | ) + |E| ,
(∂t − cs ∆)n =
16πmi
c2
(∂t + v0 ∂y −
where θ = k1 y − ω1 t, t ∈ R, y ∈ R and ∆⊥ = ∂x2 + ∂z2 . For simplicity, if A and
B are two vectors of R3 , the inner product A · B is denoted by AB. In this
system, A0 denotes the envelope of the incident laser field, AR is the scattered
light, E is the electronic-plasma wave and n the variation of density of ions.
Furthermore, the vector fields A0 , AR and E are such that
A0 , AR , E : R3 −→ C3 .
At this point, we do not give the orders of the different constants. We will use
them carefully in the next section.
The authors observe some numerical instability explained by the fact that
the paraxial approximation is done on A0 and AR and not on E. This appears
through the presence of the operators
∂t + v0 ∂y −
ic2
ic2
∆⊥ and ∂t + vR ∂y −
∆⊥
2ω0
2ωR
on A0 and AR while the evolution operator on E is the complete dispersive
2
one (2iωpe ∂t + 3vth
∆). As mentioned before, no existence results are known for
(1.2). It is probably ill-posed. See however [19] for variations around this kind
of system. In order to obtain a system having better properties, we will derive a
new set of equations starting from the bi-fluid Euler-Maxwell system. This will
be done in section 2. Our system involves four Schrödinger equations coupled
by quasilinear terms and a wave equation. It reads :
2
ωpe
k0 c2
c2
k 2 c4
i ∂t +
∂y A0 +
∆A0 − 0 3 ∂y2 A0 =
< δn > (A0 + e−2ik0 y AB )
ω0
2ω0
2ω0
2ω0
e
−
(∇ · E0 ) AR e−i(k1 y−ω1 t) ,
(1.3)
2me ω0
3
2
ωpe
k0 c2
c2
k 2 c4
i ∂t −
∂y AB +
∆AB − 0 3 ∂y2 AB =
< δn > (A0 + e−2ik0 y AB )
ω0
2ω0
2ω0
2ω0
e
−
(∇ · E0 ) AR ei((2k0 −k1 )y−ω1 t) ,
(1.4)
2me ω0
2
ωpe
kR c2
c2
k 2 c4
i ∂t +
∂y AR +
∆AR − R 3 ∂y2 AR =
< δn > AR
ωR
2ωR
2ωR
2ωR
e
−
(∇ · E0∗ ) (A0 + e−2ik0 y AB )ei(k1 y−ω1 t) , (1.5)
2me ωR
i∂t E0 +
2
vth
ωpe
∆E0 =
∇∆−1 div (< δn > E0 )
2ωpe
2
eωpe
+ 2 ∇ A∗R (A0 + e−2ik0 y AB )ei(k1 y−ω1 t) ,
2c me
∂t2 − c2s ∆ < δn >=
2
ωpe
1
∆ ( |E0 |2 + 2 ( |A0 + e−2ik0 y AB |2
4πn0 mi
c
+|AR |2 .
(1.6)
(1.7)
Here A0 is the incident laser field, AB is the Brillouin component, AR is the
Raman field, E0 the electronic-plasma field and < δn > the low-frequency
variation of the density of the ions. The coefficients are described in section 2.
The complete electric field is recovered by
E=
iω0
iωR
A0 (t, X)ei(k0 y−ω0 t) +
AR (t, X)ei(kR y−ωR t)
c
c
iω0
+
AB (t, X)e−i(k0 y+ω0 t) + E0 (t, X)e−iωpe t + c.c.,
c
where (k1 , ω1 ) are defined by
k0 = kR + k1 ,
ω0 = ωR + ωpe + ω1 .
(1.8)
We are namely in a situation where we want to describe a three-waves interaction. It is easy to see that kR has to be non positive if k0 is non negative.
Therefore, the velocity of AR is non positive. System (1.3) − (1.7) differs from
that used in [15] by the presence of AB and the complete dispersion in (1.3),
(1.4), (1.5). This will play a fundamental role in the existence theory. However,
AB can be eliminated as shown in section 3.2 and the final system is using
AC = A0 + e−2ik1 y AB :
2
ωpe
k0 c2
c2
k 2 c4
i ∂t AC +
∂y AC +
∆AC − 0 3 ∂y2 AC =
< δn > AC
ω0
2ω0
2ω0
2ω0
e
−
(∇ · E0 ) AR e−i(k1 y−ω1 t) ,
(1.9)
2me ω0
4
2
ωpe
kR c2
c2
k 2 c4
i ∂t AR +
∂y AR +
∆AR − R 3 ∂y2 AR =
< δn > AR
ωR
2ωR
2ωR
2ωR
e
−
(∇ · E0∗ ) AC ei(k1 y−ω1 t) ,
(1.10)
2me ωR
i∂t E0 +
∂t2
1.3
−
c2s ∆
2
ωpe
vth
∆E0 =
∇∆−1 div (< δn > E0 )
2ωpe
2
eωpe
+ 2 ∇ A∗R AC ei(k1 y−ω1 t) ,
2c me
!
2
ωpe
1
2
2
2
∆ |E0 | + 2 |AC | + |AR |
.
< δn >=
4πn0 mi
c
(1.11)
(1.12)
Statement of the results.
The local in time Cauchy problem for the usual Zakharov equations (1.1) is
now well understood in the context of regular solutions (see [1], [12], [16] [18]
for local models, see [3] for the non-local case (1.1)). For weak solutions, one
can see [7]. For finite-time blow-up see [8, 9]. Here, we are interested in proving
local existence in time for strong solutions. We want to apply Ozawa-Tsutsumi’s
method (see [12]). The main problem is to deal with the quasilinear part of the
system :
i∂t AC = − (∇ · E0 ) AR ,
i∂t AR = − (∇ · E0∗ ) AC ,
i∂t E0 = ∇(A∗R AC ).
(1.13)
(1.14)
(1.15)
We omit here the exponentials which play no role regarding the local existence
method. Even if any regular solution of (1.13) − (1.15) satisfies the conservation
law
Z
d
(2|AC |2 + |AR |2 + |E0 |2 ) = 0,
dt
the system is not symmetric, neither hyperbolic ! It is therefore ill-posed (see
section 3.3).
However, when one adds the dispersion part, one can show that
i∂t AC + α∆AC = − (∇ · E0 ) AR ,
i∂t AR + β∆AR = − (∇ · E0∗ ) AC ,
i∂t E0 + γ∆E0 = ∇(A∗R AC ),
(1.16)
(1.17)
(1.18)
is locally well-posed. The dispersion seems thus to play a fundamental role
(see section 4). The Cauchy problem for quasilinear Schrödinger equations has
5
been studied by many authors [4], [10], [11]. We want here a proof that can
be adapted to Zakharov-type models and that involves usual energy estimates.
In fact, we show that the dispersion plays the same role than dissipation for
non hyperbolic diphasic model (see [13]). The way we proceed is the following
one : we add new unknowns to the system and we use the dispersive terms to
symmetrize the quasilinear part. This is the key of this paper.
Our main result reads (see Theorem 5.1) :
Theorem. Let (aC , aR , e) ∈ (H s+2 (Rd ))3d , n0 ∈ H s+1 (Rd ) and n1 ∈ H s (Rd )
with s > d2 +3. There exists T ∗ > 0 and a unique maximal solution (AC , AR , E, δn)
to (1.9) − (1.12) such that
(AC , AR , E) ∈ C([0, T ∗ [; H s+2 )3d ,
δn ∈ C([0, T ∗ [; H s+1 ) ∩ C 1 ([0, T ∗ [; H s )
satisfying
(AC , AR , E, δn)(0) = (aC , aR , e, n0 ),
∂t δn(0) = n1 .
It is clear that the same theorem can be proved for system (1.3) − (1.7)
The paper is organized as follows. In section 2, we carefully derive our
new model. In section 3, we discuss the algebraic structure of the system :
we introduce a dimensionless form, we eliminate the Brillouin component and
study the lack of hyperbolicity of the quasilinear part. In section 4, we solve the
Cauchy problem for (1.16) − (1.18) using the dispersive part. In section 5, we
apply the same method to (1.9)−(1.12). In section 6, we give some complements
and open problems.
Acknowledgments : this work was partially supported by ACI Jeunes
Chercheurs “solutions oscillantes d’EDP”, Ministère de la Recherche, France
and GDR EAPQ numéro 2103, CNRS.
2
The nonlinear coupled model.
In this part, we derive the nonlinear coupled model describing the interaction
of a laser beam in a plasma with the Raman and Brillouin backscattered fields.
In section 2.1, we introduce the starting point : the bi-fluid Euler-Maxwell
system. We recall the polarization conditions for the transverse and longitudinal
unknowns. In section 2.2, we obtain the amplitude equations for the different
electric fields. In section 2.3, we couple these amplitude equations with the
linearized hydrodynamical equations for the ions.
2.1
The bi-fluid Euler-Maxwell system.
The Euler-Maxwell system describes the evolution of the electromagnetic field
in a plasma. The plasma is modelized by two fluids (one for the ions, one for
6
the electrons). Therefore, we couple Maxwell system with two Euler systems.
This coupling reads
∂t B + c∇ × E = 0,
∂t E − c∇ × B = 4πe ((n0 + ne )ve − (n0 + ni )vi ) ,
(2.1)
(2.2)
(n0 + ne ) (∂t ve + ve · ∇ve ) = −
γe Te
e(n0 + ne )
1
∇ne −
(E + ve × B),
me
me
c
(2.3)
(n0 + ni ) (∂t vi + vi · ∇vi ) = −
e(n0 + ni )
1
γi Ti
∇ni +
(E + vi × B),
mi
mi
c
(2.4)
∂t ne + ∇ · ((n0 + ne )ve ) = 0,
∂t ni + ∇ · ((n0 + ni )vi ) = 0.
(2.5)
(2.6)
The unknowns are :
• E and B are respectively the electric and magnetic field.
• ve and vi denote respectively the electron’s and ion’s velocity.
• n0 is the mean density of the plasma.
• ne and ni are the variation of density respectively of electrons and ions
with respect to the mean density n0 .
The constants are :
• c is the velocity of light in the vacuum; e is the elementary electric charge.
• me and mi are respectively the electron’s and ion’s mass.
• Te and Ti are respectively the electronic and ionic temperature and γe and
γi the thermodynamic coefficient.
For a precise description of this kind of model, see classical textbooks [5]. One
of the main points is that the mass of the electrons is very small compared to
the mass of the ions : me mi . Since the Lorentz force is the same for the ions
and the electrons, the velocity of the ions will be neglectable with respect to the
velocity of the electrons. The consequence is that we neglect the contribution
of the ions in equation (2.2).
Now, we recall the polarizations for the longitudinal and transverse components of the fields in the linear regime. We begin with the linear theory of system
(2.1)−(2.6). The linearized version around (E, B, ve , vi , ne , ni ) = (0, 0, 0, 0, 0, 0)
is (neglecting the contribution of the ions in (2.2)) :
∂t B + c∇ × E = 0,
∂t E − c∇ × B = 4πene ve ,
en0
γe Te
∇ne −
E,
n0 ∂t ve = −
me
me
∂t ne + n0 ∇ · ve = 0.
7
(2.7)
(2.8)
(2.9)
(2.10)
Now writing B = Bk + B⊥ with ∇ × Bk = 0 and ∇ · B⊥ = 0 and using similar
notations for E and ve , we get
∂t Bk = 0,
∂t Ek = 4πen0 vek ,
e
γe Te
∇ne −
Ek ,
∂t vek = −
me n0
me
∂t ne + n0 ∇ · vek = 0,
(2.11)
(2.12)
∂t B⊥ + c∇ × E⊥ = 0,
∂t E⊥ − c∇ × B⊥ = 4πen0 ve⊥ ,
e
∂t ve⊥ = −
E⊥ .
me
(2.15)
(2.16)
(2.13)
(2.14)
and
(2.17)
(2.11)−(2.14) and (2.15)−(2.17) are two decoupled systems that can be treated
separately. We begin with the longitudinal part (2.11) − (2.14). Computing
∂t (2.13) and using (2.12) and (2.14) leads to
∂t2 vek =
4πe2 n0
γe Te
∇∇ · vek −
vek .
me
me
That is
2
2
[∂t2 − vth
∆ + ωpe
]vek = 0,
(2.18)
since ∇ × vek = 0 and where ωpe is the electronic plasma pulsation
s
4πe2 n0
,
ωpe =
me
and vth is the thermal velocity of electrons
r
γe Te
vth =
.
me
The dispersion relation associated to (2.18) is therefore
2
2
ω 2 = ωpe
+ k 2 vth
.
(2.19)
For the transverse part, we compute ∂t (2.16) and we use (2.15) and (2.17) to
obtain
2
∂t2 E⊥ − c2 ∆E⊥ + ωpe
E⊥ = 0,
(2.20)
since ∇ · E⊥ = 0. The dispersion relation associated to (2.20) is
2
ω 2 = ωpe
+ k 2 c2 .
8
(2.21)
For practical applications, the thermal velocity vth is always one order of magnitude (at least) smaller than c. Therefore, the status of equations (2.18) and
(2.20) are quite different. One has to think to the solution of (2.20) under the
form ei(kx−ωt) E(t, x) with ∂t E ωE and ∂x E kE while the one of (2.18)
will be e−iωpe t ve with ∂t ve ωpe ve .
We are then able to write the polarization conditions on the fields. As usual,
ne
we introduce the variation of density δne =
.
n0
• Longitudinal part : the linear equations for the longitudinal part are
(2.11) − (2.14). Using the time envelope approximation, one sets
e e)
ek , E
ek , veek , δn
(Bk , Ek , vek , δne ) = e−iωpe t (B
with
e e ) ωpe (B
e e ).
ek , E
ek , veek , δn
ek , E
ek , veek , δn
∂t (B
One obtains :
ek = 0,
B
(2.22)
iωpe e
Ek ,
4πen0
e e =− 1 ∇·E
ek .
δn
4πen0
veek = −
(2.23)
(2.24)
• Transverse part : in this case, in the physical literature, it is usual to work
with the vector potential A such that B = ∇ × A and the scalar potential ψ in
the Lorentz jauge ∂t ψ = c∇ · A. The linearized equations become :
∂t ψ = c∇ · A,
∂t A + cE = c∇ψ,
∂t E − c∇ × ∇ × A = 4πen0 ve ,
γe Te
e
∂t ve = −
∇δne −
E,
me
me
∂t δne + ∇ · ve = 0.
The transverse part is :
∂t A⊥ + cE⊥ = 0,
∂t E⊥ − c∇ × ∇ × A⊥ = 4πen0 ve⊥ ,
e
∂t ve⊥ = −
E⊥ .
me
(2.25)
(2.26)
(2.27)
e⊥ , E
e⊥ , vee⊥ )
Looking for the solution in the form (A⊥ , E⊥ , ve⊥ ) = ei(kx−ωt) (A
with
e⊥ , E
e⊥ , vee⊥ ) ω(A
e⊥ , E
e⊥ , vee⊥ ),
∂t (A
9
e⊥ , E
e⊥ , vee⊥ ) k(A
e⊥ , E
e⊥ , vee⊥ ),
∂x (A
one obtains
e⊥ = iω A
e⊥ ,
E
c
e e
vee⊥ =
A⊥ .
cme
2.2
(2.28)
(2.29)
Amplitude equations for the electric fields.
We now present a weakly nonlinear theory of the laser-plasma interaction. We
will generalized the equations used in [15]. The electromagnetic transverse part
is decomposed into three parts :
i) the incident wave with vector potential A0 ei(k0 x−ω0 t) + c.c.
ii) a Raman backscattered wave with vector potential AR ei(kR x−ωR t) + c.c.
iii) a Brillouin backscattered wave with vector potential AB ei(−k0 x−ω0 t) +c.c.
These three waves create a longitudinal electronic plasma wave at the pulsation ωpe . The corresponding electric field is E0 e−iωpe t + c.c. These four electric
fields combine in order to create a low-frequency density modulation < δn >.
Using the polarization condition (2.28) for the transverse part, we look for the
electric field in the form
E=
iω0
iωR
A0 (t, X)ei(K0 ·X−ω0 t) +
AR (t, X)ei(KR ·X−ωR t)
c
c
iω0
+
AB (t, X)e−i(K0 ·X+ω0 t) + E0 (t, X)e−iωpe t + c.c.,
c
(2.30)
with X = (x, y, z). The aim of this section is to find the envelope equations
satisfied by A0 , AR , AB and E0 . We introduce (K1 , ω1 ) satisfying
K0 = KR + K 1 ,
ω0 = ωR + ωpe + ω1 .
(2.31)
Note that (KR , ωR ) and (K0 , ω0 ) satisfy the same dispersion relation (2.21).
In order to match the notations used in [15], we take K0 = k0 (0, 1, 0) and
KR = kR (0, 1, 0). (2.30) becomes
E=
iω0
iωR
AR (t, X)ei(kR y−ωR t)
A0 (t, X)ei(k0 y−ω0 t) +
c
c
iω0
+
AB (t, X)e−i(k0 y+ω0 t) + E0 (t, X)e−iωpe t + c.c.,
c
and (2.31)
k0 = kR + k1 ,
ω0 = ωR + ωpe + ω1 .
10
(2.32)
We begin with the nonlinear Maxwell equations with the vector potential :
∂t ψ = c∇ · A,
∂t A + cE = c∇ψ,
∂t E − c∇ × ∇ × A = 4πe((n0 + ne )ve − (n0 + ni )vi ).
Combining together these equations and neglecting the effects of the ions, we
obtain
∂t2 A − c2 ∆A = −4πen0 (1 + δne )ve .
(2.33)
Using the form (2.30), the relations (2.31), the dispersion relation (2.21), the
polarization condition (2.24) and (2.29) and collecting the factor of ei(k0 y−ω0 t)
in the right-hand-side of (2.33), we get
2
∂t2 A0 − 2iω0 ∂t A0 − c2 ∆A0 − 2ik0 c2 ∂y A0 = − ωpe
< δn > (A0 + e−2ik0 y AB )
e
+
(∇ · E0 ) AR e−i(k1 y−ω1 t) .
me
We now relax the usual time envelope approximation : at the first order, one
has −2iω0 ∂t A0 − 2ik0 c2 ∂y A0 = 0, that is
∂t A0 = −
and
∂t2 A0 =
k0 c2
∂y A0
ω0
k02 c4 2
∂ A0 .
ω02 y
We therefore get
i(∂t A0 +
2
ωpe
k0 c2
c2
k 2 c4
∂y A0 ) +
∆A0 − 0 3 ∂y2 A0 =
< δn > (A0 + e−2ik0 y AB )
ω0
2ω0
2ω0
2ω0
e
−
(∇ · E0 ) AR e−i(k1 y−ω1 t) .
2me ω0
(2.34)
We make the same computation for the harmonic ei(kR y−ωR t) to obtain
i(∂t AR +
−
2
ωpe
kR c2
c2
k 2 c4
∂y AR ) +
∆AR − R 3 ∂y2 AR =
< δn > AR
ωR
2ωR
2ωR
2ωR
e
(∇ · E0∗ ) (A0 + e−2ik0 y AB )ei(k1 y−ω1 t) .
2me ωR
(2.35)
The equation for the Brillouin component e−i(k0 y+ω0 t) is similar
i(∂t AB −
2
ωpe
k0 c2
c2
k 2 c4
∂y AB )+
∆AB − 0 3 ∂y2 AB =
< δn > (AB + e2ik0 y A0 )
ω0
2ω0
2ω0
2ω0
e
−
(∇ · E0 ) AR ei((2k0 −k1 )y−ω1 t) .
(2.36)
2me ω0
11
For the electronic-plasma part, we go back to the Maxwell equations in (B, E)
(2.1) − (2.2) and we get
∂t2 E + c2 ∇ × ∇ × E = 4πe∂t ((n0 + ne )ve ).
(2.37)
We use the Euler equations (2.3) and (2.5) in order to determine the right-handside of (2.37) keeping only at most quadratic terms :
∂t2 E + c2 ∇ × ∇ × E =
e Te
4πe(−n0 ve ∇ve − γm
ne ∇δne −
e
en0
−
ve × B − ne ve ∇ · ve ).
cme
e(n0 +ne )
E
me
1
∇ · E0 thanks to the
We write E = E0 e−iωpe t and we replace δne by − 4πen
0
polarization condition (2.24) and get
2
∂t2 E0 − 2iωpe ∂t E0 + c2 ∇ × ∇ × E − vth
∇∇ · E0
Z ω2π
pe
e
e
(ve · ∇ve +
δne E +
ve × B + ve ∇ · ve )eiωpe t dt.
= −2eωpe ne
m
cm
e
e
0
The right-hand-side is computed using the polarization relations (2.28) − (2.29).
One gets
ve · ∇ve +
e
e2
e2
ve × B + ve ∇ · ve = 2 2 (A · ∇A + A∇ · A) + 2 2 A × (∇ × A)
cme
c me
c me
2
2
e
|A|
+ A∇ · A),
= 2 2 (∇
c me
2
since for all v
|v|2
+ (∇ × v) × v.
2
Moreover, at first order ∇ · A = 0. One finally gets using the time envelope
approximation
v · ∇v = ∇
i∂t E0 +
2
vth
c2
ωpe
∇∇ · E0 −
∇ × ∇ × E0 =
< δn > E0
2ωpe
2ωpe
2
eωpe
+ 2 ∇(A∗R (A0 + e−2ik0 y AB )ei(k1 y−ω1 t) ).
2c me
(2.38)
Now since vth c, we perform the limit c → +∞ in this expression. We
find ∇ × E0 = 0 and therefore E0 is a gradient. We then eliminate the term
∇ × ∇ × E0 in (2.38) by applying the divergence on it :
i∂t ∇ · E0 +
2
vth
ωpe
∆∇ · E0 =
∇ · (< δn > E0 )
2ωpe
2
eωpe
+ 2 ∆(A∗R (A0 + e−2ik0 y AB )ei(k1 y−ω1 t) ).
2c me
12
Recall now that E0 = ∇ψ and apply ∇∆−1 on this equation yields
i∂t E0 +
2
vth
ωpe
∆E0 =
∇∆−1 div(< δn > E0 )
2ωpe
2
eωpe
+ 2 ∇ A∗R (A0 + e−2ik0 y AB )ei(k1 y−ω1 t) .
2c me
(2.39)
See [6] for the justification of such an approximation in the context of nonlinear
Schrödinger equations. We still have to obtain an equation for < δn >.
2.3
Acoustic approximation for the low-frequency part.
Since ∇ · E = −e(ne − ni ) and since the electric field has a high-frequency at the
first order, one gets that electron’s low-frequency part < δn > equals ion’s lowfrequency part < δni >. We look for an equation on < δni >. The linearization
of (2.4) − (2.6) yields :
e
γi Ti
∇ < δni > +
< E >,
mi
mi
∂t < δni > +∇ · vi = 0,
∂t vi = −
where < E > is the low-frequency part of the electric field. One gets
∂t2 < δni > −
γi Ti
e
∆ < δni >= − ∇· < E > .
mi
mi
(2.40)
In order to express < E >, we divide (2.3) by n0 + ne and we take the mean
value component
2
< ve · ∇ve >= −vth
∇ < δne > −
e
e
<E>−
< ve × B >,
me
me c
that is
e
e
2
< E >= −vth
∇ < δni > − < ve · ∇ve > −
< ve × B > .
me
me c
(2.41)
The contribution of the longitudinal part to the right-hand-side of (2.41) is only
in the term < ve · ∇ve > which gives
<∇
2
ωpe
|ve |2
>=
∇|E0 |2 ,
2
2
16π e2 n20
thanks to the polarization condition (2.23) and since ∇ × E0 = 0 at the first
order.
The contribution of the transverse part is
< −ve · ∇ve −
e
e2
ve × B > = − 2 2 < A · ∇A + A × (∇ × A) >,
me c
c me
e2
= − 2 2 < ∇|A|2 >,
c me
13
thanks to the polarization condition (2.29) and B = ∇ × A. We obtain
< ve · ∇ve +
e
e2
ve × B >= − 2 2 ∇(|A0 + e−2ik0 y AB |2 + |AR |2 ).
me c
c me
Finally, the low-frequency component of E is
2
ωpe
e
2
< E > = −vth
∇ < δni > −
∇|E0 |2
me
16π 2 e2 m2e
e2
− 2 2 ∇(|A0 + e−2ik0 y AB |2 + |AR |2 ).
c me
Plugging this expression in (2.40) gives
∂t2
−
c2s ∆
where c2s =
!
2
ωpe
1
2
−2ik0 y
2
2
∆ |E0 | + 2 (|A0 + e
AB | + |AR | ) (2.42)
< δn >=
4πn0 mi
c
γi Ti
mi
+
γe Te
me
is the acoustic velocity.
Equations (2.34), (2.35), (2.36), (2.39) and (2.42) form a closed set of equations describing the laser-plasma interactions. In the next section, we investigate
the algebraic properties of this system.
3
Structural properties of the system.
This system is strongly quasilinear and we investigate its structure. In order
to make more comprehensible computations, we first introduce a dimensionless
form (3.1) − (3.5). Then we show that the quasilinear part in the equations of
A0 , AR , AB , E0 is not symmetric. Then we prove that the Brillouin component
AB can be eliminated. The system becomes (3.6)−(3.9). We finish by discussing
a resonance relation on (k1 , ω1 ) and by some remarks.
3.1
Dimensionless form.
We use
1
1
as time scale,
as space scale and introduce
ω0
k0
ei = √ωi ωpe Ai
A
with i = 0, R or B,
c α
√
√
ωpe
2me ω0 ω0 ωR ωpe
e
E=
E
with α =
,
α
ek0
k1
ω1
e
k1 = , ω
e1 =
,
k0
ω0
14
and dropping the tildes, we get the following system :
2
ωpe
k 2 c2
c2 k02
k04 c4 2
i(∂t + 0 2 ∂y ) +
∆
−
∂
< δn > (A0 + e−2iy AB )
y A0 =
2
4
ω0
2ω0
2ω0
2ω02
− (∇ · E0 ) AR e−i(k1 y−ω1 t) , (3.1)
i(∂t +
2
ωpe
c2 k02
k 2 k 2 c4
kR k0 c2
∂y )+
∆ − 0 3R ∂y2 AR =
< δn > AR
ωR ω0
2ωR ω0
2ωR ω0
2ωR ω0
− (∇ · E0∗ ) (A0 + e−2iy AB )ei(k1 y−ω1 t) ,
(3.2)
2
ωpe
c2 k02
k04 c4 2
k02 c2
∆
−
∂
A
=
< δn > (AB + e2iy A0 )
i(∂t − 2 ∂y ) +
B
ω0
2ω02
2ω04 y
2ω02
− (∇ · E0 ) AR ei((2−k1 )y−ω1 t) ,
(3.3)
i∂t E0 +
2 2
vth
k
ωpe
∆E0 =
∇∆−1 div(< δn > E0 )
2ωpe ω0
2ω0
+ ∇ A∗R (A0 + e−2iy AB )ei(k1 y−ω1 t) ,
∂t2 < δn > −
c2s k02
4me ω0 ωR
∆ < δn >=
∆
2
ω02
mi ωpe
(3.4)
ωpe
|A0 + e−2iy AB |2 + |E0 |2
ω0
ωpe
2
+
|AR | .
(3.5)
ωR
me
c2 k 2
is small, as well as s 20 . All other coefficients are of order 1
mi
ω0
2
vth
k2
except
which is small.
2ωpe ω0
Note that
3.2
Elimination of the Brillouin component.
In this section, we prove that we can in fact decrease the number of unknowns.
We introduce
AC = A0 + e−2iy AB .
Let us compute
i(∂t +
k02 c2
c2 k02
k04 c4 2
∂
)
+
∆
−
∂
AC ,
y
ω02
2ω02
2ω04 y
15
one obtains after a straightforward computation :
2
ωpe
k 2 c2
c2 k02
k04 c4 2
i(∂t + 0 2 ∂y ) +
∆
−
∂
< δn > AC
y AC =
2
4
ω0
2ω0
2ω0
2ω02
− (∇ · E0 ) AR e−i(k1 y−ω1 t) .
(3.6)
Of course (3.2), (3.4) and (3.5) become
2
2 4
ωpe
kR k0 c2
c2 k02
k02 kR
c
i(∂t +
∂y ) +
∆−
A
=
< δn > AR
R
3ω
ωR ω0
2ωR ω0
2ωR
2ωR ω0
0
− (∇ · E0∗ ) AC ei(k1 y−ω1 t) ,
i∂t E0 +
and
∂t2 −
c2s k02
∆
ω02
2 2
vth
k
ωpe
∆E0 =
∇∆−1 div(< δn > E0 )
2ωpe ω0
2ω0
+ ∇ A∗R AC ei(k1 y−ω1 t) ,
< δn >=
4me ω0 ωR
∆
2
mi ωpe
(3.7)
(3.8)
ωpe
ωpe
|AC |2 + |E0 |2 +
|AR |2 .
ω0
ωR
(3.9)
(3.6) − (3.9) form a closed set of equations that we solve in the sequel.
3.3
Study of the quasilinear part.
In this section, we consider the quasilinear part of the system (3.6) − (3.8) and
we omit the exponentials :
∂t AC = i (∇ · E0 ) AR ,
∂t AR = i (∇ · E0∗ ) AC ,
∂t E0 = −i∇(A∗R AC ).
(3.10)
(3.11)
(3.12)
Unfortunately, this system is not obviously symmetrizable as we show below for
the scalar 1-D model. Let us write AC = u1 +iu2 , AR = u3 +iu4 , E0 = u5 +iu6 .
System (3.10) − (3.12) reads :
∂t u1
∂t u2
∂t u3
∂t u4
∂t u5
∂t u6
= −∂x u6 u3 − ∂x u5 u4 ,
= ∂x u5 u3 − ∂x u6 u4 ,
= ∂x u6 u1 − ∂x u5 u2 ,
= ∂x u5 u1 + ∂x u6 u2 ,
= ∂x (u2 u3 − u1 u4 ),
= −∂x (u1 u3 + u2 u4 ),
16
and it can be rewritten in the form
∂t U = M (U)∂x U,
with U = (u1 , u2 , u3 , u4 , u5 , u6 )t and

0
0
 0
0

 0
0
M (U) = 
 0
0

 −u4 u3
−u3 −u4
0
0
0
0
u2
−u1
0
0
0
0
−u1
−u2
−u4
u3
−u2
u1
0
0
−u3
−u4
u1
u2
0
0
This matrix is obviously not symmetric and the blocks
−u2 u1
u2
M1 =
and M2 =
u1
u2
−u1
−u1
−u2




.



are such that M1t = −M2 . Therefore the Cauchy problem for (3.10) − (3.12) is
certainly ill-posed and the Cauchy problem for (3.6), (3.7), (3.8) and (3.9) can
not be handled directly by coupling Ozawa-Tsutsumi’s method with hyperbolictype energy estimates for symmetric system. We will explain in section 4 how
to overcome this difficulty by using the dispersive part.
3.4
The resonance condition (2.31).
Let us recall that the numbers k0 , kR , k1 , ω0 , ωR and ω1 satisfy
k0 = k1 + kR ,
ω0 = ωR + ωpe + ω1 .
We moreover impose (as in [15]) that ω1 = ωpe (k1 λD )2 . This corresponds to
the fact that
q
2 + v2 k2 .
ωpe + ω1 ' ωpe
th 1
Therefore, condition (2.31) is a usual 3-waves interaction between two electromagnetic waves and a plasma-electronic wave :
q
q
q
2 + c2 k 2 =
2 + c2 k 2 +
2 + v2 k2 .
k0 = kR + k1 ,
ωpe
ω
ωpe
pe
0
R
th 1
We expect a Raman growth on the AR component. It is therefore not surprising
that the quasilinear part is not hyperbolic.
3.5
A hierarchy of model.
Note that since |ω0 | > |k0 c| and |ωR | > |kR c|, the second order operators in
(3.6), (3.7) are elliptic. From now on, we will restrict ourself to the case where
this operator is
c2 k02
∆.
2ω02
17
This is the result that one obtains when making the time envelope approximation. In order to decompose the difficulty, we will consider successively more
and more complex model. The first one is the system of quasilinear Schrödinger
equations obtained by setting < δn >= 0 in (3.6), (3.7) and (3.8). We will
show how to handle the Cauchy problem in section 4. One could also use the
complete equation on E0 that is
i∂t E0 +
2 2
c2 k02
ωpe
vth
k
∇∇ · E0 −
∇ × ∇ × E0 =
(< δn > E0 )
2ωpe ω0
2ωpe ω0
2ω0
+ ∇(A∗R AC e−i(k1 y−ω1 t) ).
(3.13)
We do not discuss (3.13) in this paper. It can certainly be handled by the same
method but the proof may be more technical.
4
The Cauchy problem for the nonlinear Schrödinger
system.
Let us consider the system obtained from (3.6), (3.7), (3.8) by setting < δn >=
0. It can be rewritten in the form (omitting the exponentials that have no role
in this context)
(i(∂t + vC ∂y ) + α∆) AC = − (∇ · E0 ) AR ,
(i(∂t + vR ∂y ) + β∆) AR = − (∇ · E0∗ ) AC ,
(i∂t + γ∆) E0 = ∇(A∗R AC ),
where
vC =
α=
k02 c2
,
2ω02
k02 c2
,
ω02
β=
vR =
k02 c2
,
2ωR ω0
(4.1)
(4.2)
(4.3)
kR k0 c2
,
ωR ω0
γ=
2 2
vth
k
.
2ωpe ω0
Theorem 4.1. Let (aC , aR , e0 ) ∈ H s (Rd ) with s > d2 + 3. There exists T ∗ > 0
and a unique solution (AC , AR , E0 ) ∈ (C([0, T ∗ [; H s (Rd )))3d of (4.1), (4.2), (4.3)
satisfying (AC , AR , E0 )(0, X) = (aC , aR , e0 )(X).
Proof : A lot of work has already been done on quasilinear Schrödinger
equations. In [11], Kenig, Ponce and Vega use local smoothing effects in order
to obtain local existence. In [4] or [10], the authors use nonlinear changes of
unknowns in order to obtain local existence. Here we want a generalization of
such methods that can be adapted to Zakharov-type equations. In order to prove
Theorem 4.1, we transform system (4.1) − (4.2) into a dispersive perturbation of
a symmetric quasilinear system as follows. Let us first explain how the method
works. Consider a system of PDE’s of the form (for simplicity, we consider the
1-D case) :
∂t U + B(U)∂x U + K∂x2 U = 0,
18
(4.4)
where
U : [0, T ] × R −→ Rd ,
K is skew-adjoint and one-to-one, and
B(U) = B1 (U) + B2 (U),
where B1 (U) is symmetric and B2 (U) is skew-symmetric. Except if B2 = 0, one
cannot handle directly (4.4) by usual energy methods. One introduces V = ∂t U,
X = ∂x U and W = ∂x2 U. The equations on V, X and W read
∂t V + B(U)∂x V + K∂x2 V = semilinear terms,
(4.5)
∂t X + B(U)∂x X + K∂x2 X = semilinear terms,
(4.6)
∂t W + B(U)∂x W + K∂x2 W = semilinear terms,
(4.7)
∂t U = semilinear terms,
(4.8)
V + KW = semilinear terms,
(4.9)
and
where the semilinear terms depend on U, V, X and W. Now in (4.5), we replace
V by −KW + semilinear terms (depending only on U and X ) in B2 (U)∂x V and
in (4.7) we replace W by −K −1 V + semilinear terms (depending only on U and
X ) in B2 (U)∂x W. The new system on V and W reads
∂t V + B1 (U)∂x V − B2 (U)K∂x W + K∂x2 V = semilinear terms,
(4.10)
∂t W + B1 (U)∂x W − B2 (U)K −1 ∂x V + K∂x2 W = semilinear terms.
(4.11)
Now system (4.10) − (4.11) may be symmetrizable, depending on the properties
of B2 and K. In our case, it will possible and we explain now why.
Let us introduce the following notations (for simplicity, we denote E = E0 ):
j
j
BR
= ∂xj AR , BC
= ∂xj AC , F j = ∂xj E,
CR = ∆AR , CC = ∆AC , G = ∆E,
DR = ∂t AR , DC = ∂t AC , H = ∂t E.
We write the equations satisfied by these new unknowns. We set
U := (AC , AR , E, BC , BR , F, CC , CR , G, DC , DR , H).
We also introduce the following linear operator


i(∂t + vC ∂y ) + α∆
L =  i(∂t + vR ∂y ) + β∆  .
i∂t + γ∆
19
The system becomes


AC
e
L  AR  = f1 (U),
E
(4.12)
j 
BC
j 
L  BR
= f2j (U),
j
F
(4.13)

where Ue = (AC , AR , E, BR , BC , F ). Here f1 is a smooth function and the
f2j are non-local functions satisfying |f2j (U)| ≤ K|U|σH s for some σ, K and
j
j
s > n2 . Indeed, the f2j (U) are polynomial in AC , AR , E, BR
, BC
, F j and
2
∂xi xj (AC , AR , E) that can be expressed in terms of CC , CR , G by
∂x2i xj (AC , AR , E) = ∂x2i xj ∆−1 (CC , CR , G).
(4.14)
In order to find the equations satisfied by CC , CR and G, we need to express
the nonlinear term ∇(A∗R AC ) in (4.3) more precisely using
∇(AB) = A · ∇B + B · ∇A + A × (∇ × B) + B × (∇ × A).
(4.15)
We moreover set
mA (B) := A · ∇B + A × (∇ × B).
The system satisfied by (CC , CR , G) is therefore

 

−AR ∇ · G
CC
 + f3 (U),
−AC ∇ · G∗
L  CR  = 
∗
mA∗R (CC ) + mAC (CR
)
G
(4.16)
(4.17)
where f3 (U) is a non-local function as f2j (U). In the same way, the system for
(DC , DR , H) is


 
−AR ∇ · H
DC
 + f4 (U),
−AC ∇ · H ∗
(4.18)
L  DR  = 
∗
mA∗R (DC ) + mAC (DR )
H
where f4 (U) is a non-local function as f2j (U). As mentionned before, the quasilinear part of (4.17) and (4.18) is not symmetric and therefore usual energy
estimates are inoperative. We now prove the following proposition.
Proposition 4.1. Let b ∈ C3 be a constant. The adjoint in L2 of −b∇· is
mb∗ (·).
20
Proof : Let V and W two C ∞ vector fields in C3 . Let us compute
Z
Z
mb∗ (V ) · W ∗ = (b∗ · ∇V + b∗ × (∇ × V )) · W ∗ by (4.16),
Z
= ((b∗ · ∇)V · W ∗ + (W ∗ × b∗ ) · ∇ × V )
Z
Z
∗
∗
= (b · ∇)V · W + ∇ × (W ∗ × B ∗ ) · V
since the operator ∇× is self-adjoint. Now using the relation
∇ × (A × B) = A∇ · B − B∇ · A + B · ∇A − A · ∇B
and the fact that b is constant, one gets
∇ × (W ∗ × b∗ ) = −b∗ ∇ · W ∗ + b∗ · ∇W ∗ .
Moreover
Z
(b∗ · ∇)V · W ∗ = −
Z
V · (b∗ · ∇)W ∗ .
It follows that
Z
Z
Z
Z
mb∗ (V ) · W ∗ = − V · (b∗ · ∇)W ∗ − b∗ ∇ · W ∗ · V + b∗ · ∇W ∗ · V
Z
Z
= − b∗ ∇ · W ∗ · V = − b∇ · W · V ∗ .
This ends the proof of Proposition 4.1. In the quasilinear part of (4.17) or (4.18) the terms −AR ∇ · H in the first
equation and mA∗R (DC ) in the third one can be rewritten as



0
0 −AR ∇·
DC
 0
  DR 
0
0
∗
mAR (·) 0
0
H
where the operator


0
0 −AR ∇·
 0

0
0
∗
mAR (·) 0
0
is skew-adjoint. Therefore usual energy estimates will apply. Unfortunately,
∗
this is not the case for the other part. For the terms −AC ∇ · G∗ , mAC (CR
),
∗
∗
−AC ∇ · H and mAC (DR ) we use equations (4.2) − (4.3)
e
iDR + βCR = g1 (U),
(4.19)
i∂xj DR + β∂xj CR = g2j (U),
(4.20)
and
21
where g1 is a smooth function and the g2j are functions satisfying the same
properties as the f2j .
In the same way, one has
e
iH + γG = g3 (U),
(4.21)
i∂xj H + γ∂xj G = g4j (U).
(4.22)
and
Plugging these relationships into the bad terms of equations (4.17) and (4.18),
we obtain :


 
−AR ∇ · G
CC
iA
∗
 + f5 (U),
− γC ∇ · H
(4.23)
L  CR  = 
i
∗
∗
G
mAR (CC ) + β mAC (DR )
and

 

−AR ∇ · H
DC
 + f6 (U).
iγAC ∇ · G∗
L  DR  = 
∗
mA∗R (DC ) − iβmAC (CR
)
H
The complete

L
 L

 L
L
systems reads :


0
0


U = 


0
M (AC , AR , ∂)

Ue



(4.24)


 + F1 (U),

(4.25)
V
where
Ue = (AC , AR , E, BC , BR , F ),
V = (CC , CR , G, DC , DR , H),
and M is a 6x6 matrix of operators which value is

0
0
−W ∇·
0
 0
0
0
0

 mW ∗ (·)
0
0
0
M (V, W, ∂) = 
 0
0
0
0

 0
0
iγV ∇ · σ
0
0
−iβmV (σ·)
0
mW ∗ (·)
0
0
0
− iV
γ ∇·σ
i
m
(σ·)
0
β V
0
−W ∇·
0
0
0
0
where σ is the complex conjugation (for a vector field A in R3 , σA = A∗ ).
A straightforward change of variable enables us to restrict to the case γ =
β = 1. In this case, the operator M (V, W, ∂) with V and W fixed constants is
symmetric. (4.25) then reads
0
0
(∂t + C(∇) + iD∆) U =
U + iF1 (U),
(4.27)
0 iM (AC , AR , ∂)
22





(4.26)



where
C(∇) =
d
X
Cj ∂xj ,
j=1
corresponds to the order one terms in (4.25) and D is the diagonal dispersion
matrix. Usual energy estimates for the system (4.27) yield Theorem 4.1.
5
The Cauchy problem for the nonlinear Zakharov system.
In this section, we consider the following non-dimensional form of system (3.6)−
(3.9). We have omitted the exponentials which have no role in this context.
b2
nAC − (∇ · E) AR ,
2
bc
(i(∂t + vR ∂y ) + β∆) AR = nAR − (∇ · E ∗ ) AC ,
2
b
(i∂t + γ∆) E = nE + ∇(A∗R AC ),
2
(∂t2 − vs2 ∆)n = a∆ |E|2 + b|AC |2 + c|AR |2 ,
(i(∂t + vC ∂y ) + α∆) AC =
where
a=4
me ω0 ωR
,
2
mi ωpe
b=
ωpe
,
ω0
ωpe
,
ωR
c=
vs
(5.1)
(5.2)
(5.3)
(5.4)
cs k0
.
ω0
Here, < δn > is denoted by n for simplicity.
Theorem 5.1. Let (aC , aR , e) ∈ (H s+2 (Rd ))3d , n0 ∈ H s+1 (Rd ) and n1 ∈
H s (Rd ) with s > d2 + 3. There exists T ∗ > 0 and a unique maximal solution
(AC , AR , E, n) to (5.1) − (5.4) such that
(AC , AR , E) ∈ C([0, T ∗ [; H s+2 )3d ,
n ∈ C([0, T ∗ [; H s+1 ) ∩ C 1 ([0, T ∗ [; H s )
with the initial value
(AC , AR , E, n)(0) = (aC , aR , e, n0 ),
∂t n(0) = n1 .
The proof relies on the transformation described in section 4. In addition to
the variables of section 4, we have to use ∆n and ∂t n. We therefore write the
equations satisfied by n and ∂t n. Take
p = ∇n,
q = ∆n,
r = ∂t n,
λ = ∇∆−1 ∂t n,
µ = ∇λ,
and set
N := (n, p, q, r, λ, µ, ν, τ ),
23
ν = ∆λ,
τ = ∂t λ,
Ue := (AC , AR , E, BC , BR , F, CC , CR , G, DC , DR , H).
The equations for N are
∂t n = ∇ · λ,
e
∂t λ = ∇n + h1 (U),
∂t pj = ∇ · µj ,
e
∂t µj = ∇pj + h2j (U),
∂t q = ∇ · ν,
e U)),
e
∂t ν = ∇q + ∇(h3 (U,
∂t r = ∇ · τ,
e U)),
e
∂t τ = ∇r + ∇(h4 (U,
where h1 , h2j , h3 and h4 are smooth functions. h3 (·, ·) and h4 (·, ·) are bilinear.
The system is :
e + ∇(F3 (U,
e U))
e = 0,
∂N + A(∇)N + F2 (U)
(5.5)
where
A(∇) =
d
X
Aj ∂xj ,
j=1
the matrices Aj are symmetric, F2 is a smooth function and F3 (·, ·) a smooth
bilinear function. The equations satisfied by Ue are
0
0
e + B(N
e , U),
e (5.6)
e
e
e=
Ue + iFe1 (U)
∂t + C(∇)
+ iD∆
U
0 iM (AC , AR , ∂)
e ·) is a bilinear mapping, M (AC , AR , ∂) is the symmetric operator
where B(·,
defined by (4.26) and
d
X
ej ∂x ,
e=
C
C
j
j=1
corresponds to the order one terms in (5.1) − (5.2).
e one obtains the following system for U = Re(U),
e Im(U)
e
Now using the real form of U,
∂t U + C(∇)U + D∆U =
d
X
Aj (U)∂xj U + F1 (U) + B(N , U),
j=1
where
C(∇) =
d
X
j=1
24
Cj ∂xj ,
(5.7)
with Cj being constant symmetric matrices. D is a skew-symmetric non-singular
matrix, B(·, ·) is a (possibly non-local of order 0) bilinear mapping, for all j,
Aj (U) is symmetric and
U 7−→ Aj (U)
is linear. Moreover, system (5.5) becomes
∂t N + E(∇)N + F2 (U) + ∇(F3 (U, U)) = 0,
(5.8)
where F2 and F3 are smooth functions. We will now apply Ozawa-Tsutsumi’s
method to (5.7). Set A = ∂t U. The system satisfied by A is
0
∂t A + C(∇)A + D∆A =B(∂t N , U) + B(N , A) + F1 (U)A
+
d
X
Aj (U)∂xj A +
j=1
d
X
Aj (A)∂xj U,
(5.9)
j=1
and
A + C(∇)U + D∆U =
d
X
Aj (U)∂xj U + F1 (U) + B(N , U).
(5.10)
j=1
Let us consider an initial datum U0 ∈ H s+2 , A0 ∈ H s given by (5.10) and
N0 ∈ H s+1 . Just as in [12], we first transform (5.10) by adding −αDU on both
sides of (5.10) :
A + C(∇)U + D(−α + ∆)U =
d
X
Aj (U)∂xj U + F1 (U) + B(N , U) − αDU.
(5.11)
j=1
We now consider a linearized version of (5.9) − (5.11) and (5.8) as follows.
Take A ∈ C([0, T ]; H s ), N ∈ C([0, T ]; H s+1 ) ∩ C 1 ([0, T ]; H s ) and construct new
functions B and P by the following procedure. First construct U ∈ C([0, T ]; H s )
by
Z t
U = U0 +
A(s)ds,
(5.12)
0
and obtain V by solving
A + C(∇)V + D(−α + ∆)V =
d
X
Aj (U)∂xj V + F1 (U) + B(N , U) − αDU.(5.13)
j=1
We will obtain a solution to (5.13) for α large enough thanks to the fact that
Aj (U) are symmetric. Now take (B, P) as being the solutions to
0
∂t B + C(∇)B + D∆B =B(∂t N , U) + B(N , B) + F1 (U)B
+
d
X
Aj (B)∂xj V +
j=1
d
X
j=1
25
Aj (A)∂xj B,
(5.14)
∂t P + E(∇)P + F2 (V) + ∇(F3 (V, V)) = 0.
(5.15)
Note that the symmetry of the matrices Aj will play a crucial role in the resolution of (5.14).
Let
T : (A, N ) 7−→ (B, P).
The proof consists in showing that T is a contraction on a suitable ball of
C([0, T ]; H s ). We split the proof in 5 steps. Take
R = 2 (|U0 |H s + |N0 |H s ) ,
and consider B(R) the ball of radius R in C([0, T ]; H s (Rd ) × H s+1 (Rd )).
• step 1 : Solving the elliptic equation (5.13). Assume that (A, N ) ∈ B(R).
Then
|U|H s ≤ |U0 |H s + T R ≤ R,
provided that T is small enough. Applying D−1 on (5.13) yields
D
−1
C(∇)V + (−α + ∆)V −
d
X
D−1 Aj (U)∂xj V = D−1 F,
j=1
where
F = −A + F1 (U) + B(N , U) − αDU.
Lemma 5.1. The bilinear form
Z
a(V1 , V2 ) =
(D
−1
C(∇)V1 ·V2 −αV1 ·V2 −∇V1 ·∇V2 −
d
X
D−1 Aj (U)∂xj V1 ·V2 )
j=1
is elliptic for α large enough.
Proof : Compute
Z
d
X
I = (D−1 C(∇)V · V −
D−1 Aj (U)∂xj V · V)
j=1
≤ K (|V|L2 |∇V|L2 + |U|∞ |V|L2 |∇V|L2 ) .
Recall that
0
0
|U|∞ ≤ K |U|H s ≤ K R,
so that
00
1
I ≤ K (1 + R2 )|V|2L2 + |∇V|2L2 .
2
00
Take α = 2K (1 + R2 ). This provides
a(V, V) ≥
α 2
1
|V| 2 + |∇V|2L2 .
2 L
2
26
Then using usual arguments for elliptic equations, one obtains that there exists
a unique V ∈ H s+2 solution to (5.13) such that
|V|H s+2 ≤ C1 (R)|F |H s ,
and
|V|L∞ (0,T ;H s+2 ) ≤ C1 (R) ( |A|L∞ (0,T ;H s ) + C(|U|L∞ (0,T ;H s ) )
+|N |L∞ (0,T ;H s ) |U|L∞ (0,T ;H s ) + |U|L∞ (0,T ;H s ) ,
since
|F (U)|H s ≤ C(|U|H s ),
where x 7−→ C(x) is a continuous function. It follows that there exists C2 (R)
such that
|V|L∞ (0,T ;H s+2 ) ≤ C2 (R).
(5.16)
• step 2 : Resolution of the dispersive equation (5.14). We first use a longwave type regularization of (5.14) : take B ε solution to
0
∂t (1 − ε∆)B ε + C(∇)B ε +D∆B ε = B(∂t N , U) + B(N , B ε ) + F1 (U)B ε
+
d
X
Aj (B ε )∂xj V +
j=1
d
X
Aj (A)∂xj B ε ,
(5.17)
j=1
with B ε (0) = B0 .
There exists a unique solution
B ε ∈ C([0, T ]; H s )
to (5.17) since (5.17) is a linear, zero order (in space) equation. We will now
perform energy estimates on (5.17) and we will use several times the following
inequality :
∀ (u, v) ∈ L∞ ∩ H s , ∀ (α, β) such that |α| + |β| = s,
|∂ α u∂ β v|L2 ≤ C (|u|L∞ |v|H s + |u|H s |v|L∞ ) .
See [2]. Note that ∂ α denotes here any spatial derivative of order α.
Proposition 5.1. There exists C5 (R) > 0 such that
|Bε |L∞ (0,T ;H s ) ≤ |B0 |H s + C5 (R)T.
27
(5.18)
Proof : L2 estimate : multiply (5.17) by B ε and get
Z
1
∂t (|Bε |2 + ε|∇B ε |2 ) ≤C |∂t N |L∞ |U|L2 |Bε |L2 + |N |L∞ |Bε |2L2
2
d
X
+
|Bε |2L2 |∂xj V|L∞ + C(|U|L∞ )|Bε |2L2 )
j=1
+
d Z
X
Aj (A)∂xj B ε · B ε .
j=1
Now, since Aj (A) is symmetric, one obtains
d Z
X
d
Aj (A)∂xj B ε · B ε = −
j=1
1X
2 j=1
Z
Aj (∂xj A)B ε · B ε ,
and this last quantity is controlled by
|∂xj A|L∞ |Bε |2L2 .
Therefore, on has
Z
|Bε |2 ≤ |B0ε |2L2 + T C3 (R).
(5.19)
H s estimate : one applies ∂ s on (5.17) and takes the L2 inner product with
∂ B ε . The only non straightforward term is the quasilinear one
Z
Aj (A)∂ s ∂xj B ε · ∂ s B ε .
s
Again the symmetry of Aj gives
Z
Z
1
Aj (A)∂ s ∂xj B ε · ∂ s B ε = −
Aj (∂xj A)∂ s B ε · ∂ s B ε ,
2
which is controlled by
|∂xj A|L∞ |∂ s B ε |2L2 ≤ C|A|H s |∂ s B ε |2L2 ,
since s >
d
2
+ 1. We then obtain
1
∂t |Bε |2H s + ε|Bε |2H s+1 ≤ C3 (R)|Bε |2H s + C4 (R)|Bε |2H s+1 ,
2
from which Proposition (5.1) follows. Letting ε → 0 gives a solution
B ∈ L∞ (0, T ; H s ) ∩ C([0, T ]; H s−η )
28
∀ η > 0,
to (5.14) satisfying
|B|L∞ (0,T ;H s ) ≤ |B|H s + C5 (R)T.
(5.20)
• step 3 : Solving the “wave equation“ (5.15). Note that
|∇F3 (V, V)|L∞ (0,T ;H s+1 ) ≤ C|V|L∞ (0,T ;H s+1 ) ,
and
|V|L∞ (0,T ;H s+2 ) ≤ C2 (R),
thanks to (5.16). It follows that
|P|L∞ (0,T ;H s+1 ) ≤ |N0 |H s + C6 (R)T.
(5.21)
Inequality (5.20) and (5.21) show that :
Proposition 5.2. There exists T0 > 0 such that if T ≤ T0 , T maps B(R) into
itself.
• step 4 : T is a contraction in L∞ (0, T ; L2 ) if T is small enough. It is
obtained as usual. The crucial point is again the symmetry of Aj (U).
Proposition 5.3. There exists T1 ≤ T0 such that if T ≤ T1 , T is a contraction
mapping on B(R) in the L∞ (0, T ; L2 )-norm.
It follows that there exists a unique (A, N ) ∈ B(R) such that
T (A, N ) = (A, N ),
that is a unique solution of
∂t A + C(∇)A + D∆A =B(∂t N , U) + B(N , A) +
d
X
Aj (A)∂xj V
j=1
+
d
X
0
Aj (A)∂xj A + F1 (U)A,
j=1
∂t N + E(∇)N + F2 (V) + ∇(F3 (V, V)) = 0,
where U is given by
Z
U = U0 +
t
A(s)ds,
0
and V is the solution to the elliptic equation
A + C(∇)V + D(−α + ∆)V =
d
X
Aj (U)∂xj V + F1 (U)
j=1
+ B(N , U) − αDU.
29
(5.22)
• step 5 : Going back to the original problem. We still have to prove that
V = U which is done by making an estimate on the elliptic equation (5.22) just
as in [12]. This ends the proof of Theorem (5.1).
6
6.1
Some complements.
Invariants.
System (3.6), (3.7), (3.8) and (3.9) has some invariants that we will make more
precise now. We rewrite this system using the relations of section 4 :
b2
< δn > AC − (∇ · E0 ) AR e−iθ
2
bc
< δn > AR − (∇ · E0∗ ) AC eiθ ,
(i(∂t + vR ∂y ) + β∆) AR =
2
b
i∂t E0 + γ∆E0 = ∇∆−1 div(< δn > E0 ) + ∇ A∗R AC eiθ ,
2
∂t2 < δn > −vs2 ∆ < δn >= a∆ |E0 |2 + b|AC |2 + c|AR |2 ,
(i(∂t + vC ∂y ) + α∆) AC =
(6.1)
(6.2)
(6.3)
(6.4)
where θ = k1 y − ω1 t and
vC =
γ=
k02 c2
kR k0 c2
c2 k02
c2 k02
,
v
=
,
α
=
,
β
=
,
R
ω02
ωR ω0
2ω02
2ωR ω0
2 2
vth
k
4me ω0 ωR
ωpe
ωpe
cs k0
, a=
, b=
, c=
, vs =
.
2
2ωpe ω0
mi ωpe
ω0
ωR
ω0
Proposition 6.1. For any regular solution of (6.1) − (6.4), one has
Z
d
2|AC |2 + |AR |2 + |E0 |2 = 0.
dt
Proof : Multiply (6.1) by 2A∗C , (6.2) by A∗R , (6.3) by E0∗ , integrating over
R , summing the results and taking the imaginary part gives
Z
1
∂t
2|AC |2 + |AR |2 + |E0 |2
2
Z
= Im
−2∇ · E0 AR A∗C e−iθ − ∇ · E0∗ A∗R AC eiθ + ∇(A∗R AC eiθ )E0∗ ,
Z
= Im
2∇ · E0∗ A∗R AC eiθ − ∇ · E0∗ A∗R AC eiθ − A∗R AC ∇ · E0∗ eiθ ,
d
= 0.
Note that, this does not means that the quasilinear part is hyperbolic. In fact,
as noticed before, it is not! For some particular values of the parameters, we
have a Hamiltonian structure :
30
Proposition 6.2. Assume that ω1 = 0 (that is θ = k1 y). Then, for any regular
solution to (6.1) − (6.4) such that δn ∈ C([0, T ]; H −1 )
Z
d
1
b2
bc
α|∇AC |2 + β|∇AR |2 + γ|∇E0 |2 + δn( |AC |2 + |AR |2
dt
2
2
2
Z
2
b
bv
1
+ |E0 |2 ) + s (|δn|2 + 2 |∇∆−1 ∂t δn|2 ) − Re
∇ · E0∗ A∗R AC eiθ
2
8a
vs
Z
1
+ Im (vC ∂y AC A∗C + vR ∂y AR A∗R ) = 0
2
Proof : Multiply (6.1) by ∂t A∗C , (6.2) by ∂t A∗R and (6.3) by ∂t E0∗ , summing
the results, taking the real part and integrating over Rd leads to
Z
− Im (vC ∂y AC ∂t A∗C + vR ∂y AR ∂t A∗R )
Z
1
− ∂t (α|∇AC |2 + β|∇AR |2 + γ|∇E0 |2 )
2
Z
1
b2
bc
b
=
δn ( ∂t |AC |2 + ∂t |AR |2 + ∂t |E0 |2 )
2
2
2
2
Z
∗ −iθ
+ Re (−∇ · E0 AR ∂t AC e
− ∇ · E0∗ AC ∂t A∗R eiθ )
Z
+ Re ∇(A∗R AC eiθ )∂t E0∗ .
(6.5)
We compute each terms separately.
Z
Im (vC ∂y AC ∂t A∗C + vR ∂y AR ∂t A∗R )
Z
= −Im (vC AC ∂t ∂y A∗C + vR AR ∂t ∂y A∗R ),
Z
= Im (vC A∗C ∂t ∂y AC + vR A∗R ∂t ∂y AR ).
Therefore
Z
(vC ∂y AC ∂t A∗C + vR ∂y AR ∂t A∗R )
Z
1
= Im ∂t (vC ∂y AC A∗C + vR ∂y AR A∗R .
2
Im
(6.6)
Moreover
b2
bc
b
δn ( ∂t |AC |2 + ∂t |AR |2 + ∂t |E0 |2 )
2
2
2
Z
b2
bc
b
= ∂t δn ( |AC |2 + |AR |2 + |E0 |2 )
2
2
2
Z
b2
bc
b
− ∂t δn ( |AC |2 + |AR |2 + |E0 |2 ).
2
2
2
Z
31
(6.7)
The last term is
Z
Re (−∇ · E0 AR ∂t A∗C e−iθ − ∇ · E0∗ AC ∂t A∗R eiθ + ∇(A∗R AC eiθ )∂t E0∗ )
Z
= −Re ∂t (∇ · E0∗ A∗R AC )eiθ .
(6.8)
Now (6.6), (6.7), (6.8) in (6.5) provides
Z
1
∂t −Im
(vC ∂y AC A∗C + vR ∂y AR A∗R )
2
Z
1
−
(α|∇AC |2 + β|∇AR |2 + γ|∇E0 |2 )
2
Z
1
b2
bc
b
2
2
2
−
δn ( |AC | + |AR | + |E0 | )
2
2
2
2
Z
b2
bc
b
1
=−
∂t δn ( |AC |2 + |AR |2 + |E0 |2 )
2
2
2
2
Z
∗ ∗
iθ
− Re ∂t (∇ · E0 AR AC )e .
(6.9)
We still have to deal with the two terms in the right-hand-side of (6.9). For the
second one, recalling that θ = k1 y − ω1 t, we take ω1 = 0 (which is not a physical
case, see next section) and we get
Z
Z
∗ ∗
iθ
Re ∂t (∇ · E0 AR AC )e = ∂t Re(∇ · E0∗ A∗R AC eik1 y ).
(6.10)
For the first term, we need to write the first order equation satisfied by δn.
Introducing V = v1s ∇∆−1 ∂t δn gives
∂t δn = vs ∇ · V,
(6.11)
a
∂t V = vs ∇δn + ∇(|E0 |2 + b|AC |2 + c|AR |2 ).
vs
(6.12)
Multiply (6.11) by δn, (6.12) by V provides
Z
Z
1
a
∂t (|δn|2 + V 2 ) = −
(|E0 |2 + b|AC |2 + c|AR |2 )∇ · V,
2
vs
Z
a
= − 2 (|E0 |2 + b|AC |2 + c|AR |2 )∂t δn.
vs
Then
Z
Z
b2
bc
b
b
2
2
2
∂t δn (b|AC |2 + c|AR |2 + |E0 |2 )
∂t δn ( |AC | + |AR | + |E0 | ) =
2
2
2
2
Z
bvs2
=−
∂t (|δn|2 + V 2 ).
(6.13)
4a
32
Using (6.10) and (6.13) in (6.9) gives
Z
1
b2
bc
∂t
α|∇AC |2 + β|∇AR |2 + γ|∇E0 |2 + δn( |AC |2 + |AR |2
2
2
2
Z
2
b
bv
1
+ |E0 |2 ) + s (|δn|2 + 2 |∇∆−1 ∂t δn|2 ) − Re
∇ · E0∗ A∗R AC eiθ
2
8a
vs
Z
1
+ Im (vC ∂y AC A∗C + vR ∂AR A∗R ) = 0
2
which ends the proof of Proposition 6.2. At this point, we do not know how to use this conservation law.
6.2
Back to the resonance condition.
In order to obtain our model, we imposed the following relation (2.31) :
k0 = kR + k1 ,
ω0 = ωR + ωpe + ω1 ,
where
2
ω02 = ωpe
+ k02 c2 ,
2
2
2 2
ωR
= ωpe
+ kR
c .
In fact, in the physical literature, one finds
2
2 2
(ωpe + ω1 )2 = ωpe
+ vth
k1 .
This is the classical three waves resonance. At the first approximation, this
gives (using ω1 ωpe ) :
2 2
2ω1 ωpe ' vth
k1
=⇒
ω1 '
2 2
vth
k1
.
2ωpe
(6.14)
Now rewrite (2.38)
i∂t E0 +
2
vth
ωpe
eωpe
∆E0 =
∇∆−1 div(< δn > E0 ) + 2 ∇(A∗R A0 ei(k1 y−ω1 t) ).
2ωpe
2
2c me
The dispersion relation of the linear part is
ω=
2 2
vth
k
.
2ωpe
Equation (6.14) means that (k1 , ω1 ) satisfies the dispersion relation and therefore the term A∗R A0 ei(k1 y−ω1 t) is resonant and the process will be efficient. More
precisely, since vth = λD ωpe where λD is the Debye’s length, the equation on
E0 can be written in 1-D (neglecting δn and considering AR and A0 as given)
i∂t E0 +
λ2D ωpe 2
∂y E0 = Aei(k1 y−ω1 t) .
2
33
(6.15)
Let L be a characteristic scale in space, T in time. The dimensionless form of
(6.15) reads :
i∂t E0 +
λ2D ωpe T 2
∂y E0 = AT ei((k1 L)y−(ω1 T )t) .
2L2
Introduce k1 L = kε , ω1 T = ωε , and replace AT by A. Since ω1 = ωpe λ2D k12 and
ω1 T = ωε (with ω, k = O(1)), one gets
ωpe
ω1 T
ω
ε2 ω
λ2D T
= 2 2 = 2 = 2 ε = O(ε).
2
L
k1 L
k ε
k
The dimensionless form of (6.15) is therefore
i∂t E0 +
(ky−ωt)
εω 2
∂y E0 = Aei ε .
2
k
(6.16)
A standard WKB expansion shows that
E0 = Bei
(ky−ωt)
ε
+ O(ε),
with
ω
∂y B) = A.
k
This means a linear growth in time for B at the group velocity
hand, if the equation is
i(∂t B +
i∂t E0 +
ω
k.
On the other
0
(ky−ω t)
εω 2
i
ε
∂
E
=
Ae
,
0
y
k2
0
with ω 6= ω, then E0 will stay of size O(ε). Of course, nonlinear versions of
these results can be proved. We postpone the numerical study of this problem
to a further work.
6.3
Open problems
The existence and stability of solitary waves for these equations is open. It can
certainly be done for the coefficient corresponding to Proposition 6.2. It would
be interesting also to have some existence proofs for system (6.1) − (6.4) with
an existence time and bounds on the solutions which are independent of k1 and
ω1 in order to investigate the limit k1 → +∞ and ω1 → +∞ as in section 6.2.
Our proof is clearly not uniform with respect to these parameters.
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