Cauchy problem for the nonlinear Schrödinger equation coupled with the Maxwell equation

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Cauchy problem for the nonlinear Schrödinger equation coupled with the Maxwell equation

Mathieu Colin, Tatsuya Watanabe

To cite this version:

Mathieu Colin, Tatsuya Watanabe. Cauchy problem for the nonlinear Schrödinger equation coupled

with the Maxwell equation. 2018. �hal-01928694�

Cauchy problem for the nonlinear Schr¨ odinger equation coupled with the Maxwell equation

Mathieu Colin

and Tatsuya Watanabe

∗ INRIA CARDAMOM, 200 Avenue de la Vieille Tour, 33405 Talence, Cedex-France

Bordeaux INP, UMR 5251, F-33400,Talence, France mcolin@math.u-bordeaux1.fr

∗∗ Department of Mathematics, Faculty of Science, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto-City, 603-8555, Japan

tatsuw@cc.kyoto-su.ac.jp

In this paper, we study the nonlinear Schr¨odinger equation coupled with the Maxwell equation. Using energy methods, we obtain a local existence result for the Cauchy problem.

Key words: Schr¨odinger-Maxwell system, Cauchy problem, symmet- ric hyperbolic system, energy method

2010 Mathematics Subject Classification. 35L45, 35Q60, 35L70

1 Introduction

In this paper, we consider the following nonlinear Schr¨ odinger equation cou- pled with Maxwell equation stated in R

× R

:

iψ

+ ∆ψ = eφψ + e

|A|

ψ + 2ie∇ψ · A + ieψ divA − g(|ψ|

)ψ, (1.1) A

− ∆A = e Im( ¯ ψ∇ψ) − e

|ψ|

A − ∇φ

− ∇ divA, (1.2)

−∆φ = e

2 |ψ|

+ divA

, (1.3)

where ψ : R

× R

→ C , A : R

× R

→ R

, φ : R

× R

→ R , e ∈ R

and i denotes the unit complex number, that is, i

= −1. In this setting,

ψ is an electrically charged field and (φ, A) represents a gauge potential of

an electromagnetic field. System (1.1)-(1.3) describes the interaction of this

Schr¨ odinger wave function ψ with the Maxwell gauge potential. The constant

e represents the strength of the interaction. For more details and physical

backgrounds, we refer to [15].

Since we are interested in the Cauchy Problem, let us consider the fol- lowing set of initial data:

ψ(0, x) = ψ

(x), A(0, x) = A

(x), A

(0, x) = A

(x), (1.4) where the regularity of each functions is given in Theorem 1.1. It is known that System (1.1)-(1.3) has a so-called gauge ambiguity. Namely if (ψ, A, φ) is a solution of (1.1)-(1.3), then (exp(ieχ)ψ, A +∇χ, φ − χ

) is also a solution of (1.1)-(1.3) for any smooth function χ : R

× R

→ R . To push out this ambiguity, we adopt in the sequel the Coulomb gauge:

divA = 0, (1.5)

which is propagated by the set of Equations (1.1)-(1.3). Indeed, if initially divA(0, ·) = divA

(0, ·) = 0,

then (1.5) holds for all t > 0. (See e.g. [12] for the proof.) In this setting, the last Equation (1.3) can be solved explicitly and the solution is given by

φ = e

2 (−∆)

|ψ|

, which imposes that

φ(0, x) = e

2 (−∆)

|ψ

(x)|

. From (1.5), we also observe that (1.1) can be written as

iψ

+ L

ψ − V (x)ψ + g(|ψ|

)ψ = 0, (1.6) where V is the non-local potential: V (x) =

(−∆)

|ψ|

and L

is the magnetic Schr¨ odinger operator which is defined by A = (A

, A

, A

) and

L

ψ :=

3

X

j=1

∂

∂x

− ieA

(x)

ψ = ∆ψ − 2ie∇ψ · A − e

|A|

ψ. (1.7) In this context, the two conserved quantities of the Schr¨ odinger-Maxwell system are the charge Q and the energy E:

Q(ψ) = Z

R³

|ψ|

dx, (1.8)

E(ψ, A, φ) = 1 2

Z

R³

|∇ψ − ieAψ|

+ |∇A|

+ |∂

A|

dx + e

4 Z

R³

φ|ψ|

dx − Z

R³

G(|ψ|

) dx, (1.9) where G(t) = R

0

g(s)ds. To prove that (1.8) is formally conserved, one has to multiply Equation (1.1) by ψ, integrate over R

and take the imaginary part of the resulting equation. In a similar way, the conservation of (1.9) can be proved by multiplying (1.1)-(1.3) by ∂

ψ, ∂

A and ∂

φ respectively . This conserved quantities play a fundamental role if one wants to investigate the stability properties of such system, which is one of our main motivations.

Indeed, in a previous paper [13], we have showed that for small e > 0, System (1.1)-(1.3) admits a unique orbitally stable ground state of the form:

(ψ

, A

, φ

) :=

exp(iωt)u

, 0, e

2 (−∆)

|u

|

. (1.10)

In order to investigate the stability of such standing waves (ψ

, A

, φ

), it is necessary to prove that the Cauchy Problem (1.1)-(1.3) is almost locally well-posed around (ψ

, A

, φ

).

In a previous paper [12], we have proved the local existence of solutions for the nonlinear Klein-Gordon-Maxwell system in Sobolev spaces of high regu- larity. The method was to convert the Klein-Gordon-Maxwell system into a symmetric hyperbolic system and apply the standard energy estimate. Al- though our Schr¨ odinger-Maxwell system (1.1)-(1.3) looks similar, especially Equation (1.2) is completely the same, the usual reduction tools does not lead us to a symmetric hyperbolic system, which causes the necessity of a new strategy.

Let us also introduce results concerning the solvability of the Cauchy problem related to (1.1)-(1.3). In [3], [20], the linear Schr¨ odinger equation (g ≡ 0) coupled with the Maxwell equations has been studied. Using the Strichartz estimate, the authors obtained the global well-posedness in the energy space. However, it is not clear that their argument can be applied to the nonlinear case g 6≡ 0. We also mention the paper [19], where the Cauchy problem of the Schr¨ odinger-Maxwell system in the Lorentz gauge has been studied by using the energy method. On the other hand, a huge attention has been paid in the magnetic Schr¨ odinger equation (1.6). Especially in [18], the local well-posedness for (1.6) in the energy space has been established in the case V ≡ 0. However, in this situation, the magnetic potential A is given and was assumed to be C

, which cannot be expected a priori in our case.

We also refer to [14] for the Strichartz estimate for the magnetic Schr¨ odinger

operator (1.7) in the case A ∈ L

( R

).

We mention that if we look for the standing wave (1.10), we are led to the following non-local elliptic problem:

−∆u + ωu + e

8π|x| ∗ |u|

u = g(|u|

)u in R

, (1.11) which is referred as the Schr¨ odinger-Poisson(-Slater) equation. The existence of ground states of (1.11) as well as their orbital stability have been widely studied (see [2], [5], [4], [7], [17] and references therein). Finally, the orbital stability of standing waves for the magnetic Schr¨ odinger equation (1.6) has been considered in [8], [16]. Our study on the solvability of the Cauchy prob- lem for (1.1)-(1.3) and the result established in [13] enable us to generalize these previous results to the full Schr¨ odinger-Maxwell system.

Before stating the main result of this paper, we introduce the following notations. As usual, L

( R

) denotes the usual Lebesgue space:

L

( R

) =

u ∈ S

( R

) ; kuk

< +∞ , where

kuk

= Z

R³

|u(x)|

dx

if 1 ≤ p < +∞

and

kuk

= ess sup

|u(x)| ; x ∈ R

. We define the Sobolev space H

( R

) as follows:

H

( R

) =

u ∈ S

( R

) ; kuk

= Z

R³

(1 + |ξ|

)

|F (u)(ξ)|

dξ < +∞

, where F(u)(ξ) is the Fourier transform of u. We also introduce the homoge- neous Sobolev space ˙ H

( R

) as being the completion of C

( R

) for the norm u → k|ξ|F(u)(ξ)k

. Recall that the space ˙ H

( R

) is continuously em- bedded into L

( R

). Finally let C(I, E) be the space of continuous functions from an interval I of R to a Banach space E. For 1 ≤ j ≤ 3, we set ∂

=

j

and ∂

=

. For k ∈ N

, k = (k

, k

, k

), we denote D

u = ∂

∂

∂

u and for a non-negative integer s, D

denotes the set of all partial space derivatives of order s. Different 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 respectively.

We assume that g satisfies

g ∈ C

( R , R ) and g(0) = 0, (1.12)

for some m ∈ N with m ≥ 2, so that the function W : C → C defined by W (u) := g(|u|

)u satisfies W ∈ C

( C , C ), W (0) = W

(0) = 0. Some typical examples of the nonlinear term g are the power nonlinearity g(s) =

±s

with [p] ≥ 2m + 3 ([p] denotes the integer part of p), or the cubic- quintic nonlinearity g(s) = s − λs

for λ > 0, which frequently appears in the study of solitons in physical literatures. (See [22] for example.) In this setting, we prove the following result.

Theorem 1.1. Let s be any integer larger than

and assume that ψ

∈ H

( R

, C ), A

∈ H

( R

, R

), A

∈ H

( R

, R

) with divA

= 0, divA

= 0 and g satisfies (1.12). Then there exist T

> 0 and a unique solution (ψ, A, φ) to System (1.1)-(1.3) satisfying the initial condition (1.4) such that

ψ ∈ C([0, T

]; H

( R

)) ∩ C

([0, T

]; H

( R

)), A ∈ C([0, T

]; H

( R

)) ∩ C

([0, T

]; H

( R

)), φ ∈ C([0, T

]; ˙ H

( R

) ∩ L

( R

)), ∇φ ∈ C([0, T

]; H

( R

)), φ

∈ C([0, T

]; ˙ H

( R

) ∩ L

( R

)), ∇φ

∈ C([0, T

]; H

( R

)).

The proof of Theorem 1.1 is based on energy estimates and particularly, on the strategies developed in [9] and [10]. Note also that to overcome the loss of derivatives embedded in Equation (1.2), we use the original idea of Ozawa and Tsutsumi presented in [21].

The paper is organized as follows. In Section 2, we transform System (1.1)-(1.3) into a system to which we can apply the usual energy method.

Section 3 is devoted to the proof of Theorem 1.1.

2 Transformation of the equations

In this section, we transform the original System (1.1)-(1.3) into a new sym- metric system to which we can apply an energy method. In order to overcome the loss of derivatives contained in Equations (1.1)-(1.2), we introduce the following new unknowns (see [21]):

Ψ = ∂

ψ and Φ = ∂

φ.

Let us first derive equations for Ψ and Φ. Differentiating Equation (1.1) with respect to t, one obtains

iΨ

+ ∆Ψ =eΦψ + eφΨ + e

|A|

Ψ + 2e

ψ A · A

+ 2ie∇Ψ · A + 2ie∇ψ · A

− g

(|ψ|

)(|ψ|

Ψ + ψ

Ψ) − g(|ψ|

)Ψ.

Taking advantage of the new unknown Ψ, we also transform Equation (1.1) into an elliptic version

iΨ + ∆ψ = eφψ + e

|A|

ψ + 2ie∇ψ · A − g(|ψ|

)ψ.

Moreover, we derive an equation for Φ by applying ∂

on Equation (1.3):

−∆Φ = e

2 ψΨ + Ψψ .

Next, in order to ensure the Coulomb condition on A for all t > 0, we introduce the projection operator P on divergence free vector fields :

P :

L

( R

)

−→

L

( R

)

A 7−→ P A = − ∆

rot rot A,

so that if div A = 0, then P A = A. Thus applying P on Equation (1.2), we derive

A

− ∆A = P

e Im( ¯ ψ∇ψ) − e

|ψ|

A − ∇Φ

. (2.1)

Note that any solution to (2.1) satisfying

div A(0, ·) = 0 and div A

(0, ·) = 0, obviously satisfies

div A(t, ·) = 0 for all t > 0.

At this step, we have transformed System (1.1)-(1.3) into

iΨ + ∆ψ =eφψ + e

|A|

ψ + 2ie∇ψ · A − g(|ψ|

)ψ, (2.2) iΨ

+ ∆Ψ =eΦψ + eφΨ + e

|A|

Ψ + 2e

ψA · A

+ 2ie∇Ψ · A

+ 2ie∇ψ · A

− g

(|ψ|

)(|ψ|

Ψ + ψ

Ψ) − g(|ψ|

)Ψ, (2.3) A

− ∆A = P

e Im( ¯ ψ∇ψ) − e

|ψ|

A − ∇Φ

, (2.4)

−∆φ = e

2 |ψ|

, (2.5)

−∆Φ = e

2 ψΨ + Ψψ

. (2.6)

In order to take advantage of elliptic regularity properties, we transform Equations (2.2) by adding −αψ (α > 0 will be chosen in Lemma 3.3 below) to both sides of the equation to obtain:

(−∆ + α)ψ = iΨ − eφψ − e

|A|

ψ − 2ie∇ψ · A + g(|ψ|

)ψ + αψ. (2.7)

For simplicity, introduce U = (φ, Φ) and rewrite Equations (2.5) and (2.6) as

−∆ U = F

(ψ, Ψ), (2.8)

where

F

(ψ, Ψ) = e 2

|ψ|

ψΨ + Ψψ

. Equation (2.3) is then transformed into

i∂

Ψ + ∆Ψ = 2ie∇Ψ · A + 2ie∇ψ · A

+ F

(U , ψ, Ψ, A, A

), (2.9) where

F

(U , ψ, Ψ, A, A

) =eΦψ + eφΨ + e

|A|

Ψ + 2e

ψA · A

− g

(|ψ|

)(|ψ|

Ψ + ψ

Ψ) − g(|ψ|

)Ψ.

It is then necessary to work with A

as new unknown. We recall first that A = (a

, a

, a

). To properly write the equations on A and A

, for j = 1, 2, 3, k = 1, 2, 3 and ` = 1, 2, 3, we introduce

p

= ∂

a

, q

= ∆a

, r

= ∂

a

,

λ

= ∂

∆

∂

a

,

µ

= ∂

λ

= ∂

∂

∆

∂

a

, ν

= ∆λ

= ∂

∂

a

,

τ

= ∂

λ

= ∂

∆

∂

a

, and set A = (A

, A

, A

) with A

: R

× R

→ R

and

A

=

(a

, p

, q

, r

, λ

, µ

, ν

, τ

).

We also need to give some details on the projection operator P . For that purpose, we introduce the Riesz transform R

from L

( R

) to L

( R

) which is given by

R

= ∂

(−∆)

for j = 1, 2, 3.

Then, P can be rewritten as P = ( P

)

where

P

= δ

+ R

R

.

Now we compute the equations for each components of A

. First by the definitions of A

, one finds that

∂

a

= ∆∆

∂

a

=

3

X

k=1

∂

(∂

∆

∂

a

) =

3

X

k=1

∂

λ

,

∂

p

= ∂

∂

a

= ∆(∂

∆

∂

a

) = ∆λ

=

3

X

`=1

∂

(∂

λ

) =

3

X

`=1

∂

µ

,

∂

q

= ∂

∆a

=

3

X

k=1

∂

∆(∂

∆

∂

a

) =

3

X

k=1

∂

∆λ

=

3

X

k=1

∂

ν

,

∂

r

= ∂

a

= ∆∆

∂

a

=

3

X

k=1

∂

∂

(∂

∆

∂

a

)

=

3

X

k=1

∂

∂

λ

=

3

X

k=1

∂

τ

. Next from Equation (2.4), we have

∂

a

= ∆a

+

3

X

m=1

P

e Im( ¯ ψ∂

ψ) − e

|ψ|

a

− ∂

Φ , which provides

∂

λ

= ∂

∂

∆

∂

a

= ∂

∆

∂

a

= ∂

∆

∆a

+

3

X

m=1

P

e Im( ¯ ψ∂

ψ) − e

|ψ|

a

− ∂

Φ

= ∂

a

+ h

(ψ, A),

∂

µ

= ∂

∂

∂

∆

∂

a

= ∂

∂

∆

(∂

a

)

= ∂

∂

∆

∆a

+

3

X

m=1

P

e Im( ¯ ψ∂

ψ) − e

|ψ|

a

− ∂

Φ

= ∂

p

+ h

(ψ, A),

∂

ν

= ∂

∂

a

= ∂

∆a

+

3

X

m=1

P

e Im( ¯ ψ∂

ψ) − e

|ψ |

a

− ∂

Φ

= ∂

q

+ h

(ψ, A),

where h

, h

, h

are non-local functions defined as follows:

h

(ψ, A) = ∂

∆

3

X

m=1

P

eIm(ψ∂

ψ) − e

|ψ|

a

− ∂

Φ ,

h

(ψ, A) = ∂

∂

∆

3

X

m=1

P

eIm(ψ∂

ψ) − e

|ψ|

a

− ∂

Φ ,

h

(ψ, A) = ∂

3

X

m=1

P

eIm(ψ∂

ψ) − e

|ψ|

a

− ∂

Φ . Finally one has

∂

τ

= ∂

∆

∂

(∂

a

)

= ∂

∆

∂

∆a

+

3

X

m=1

P

e Im( ¯ ψ∂

ψ) − e

|ψ|

a

− ∂

Φ . Computing separately each term of the right-hand side of the previous equa- tion, we obtain

∂

(ψ∂

ψ) = Ψ∂

ψ + ψ∂

Ψ,

∂

(|ψ|

a

) = (Ψψ + ψΨ)a

+ |ψ|

r

. Moreover from (2.3) and (2.6), one finds that

∂

(∂

Φ) = ∂

e

2 ∂

(−∆)

(ψΨ + Ψψ)

= e

2 ∂

(−∆)

(2|Ψ|

+ ψ∂

Ψ + ψ∂

Ψ)

= e∂

(−∆)

|Ψ|

+ e∂

(−∆)

Im(i∂

Ψψ)

= e∂

(−∆)

n

|Ψ|

+ Im

− ψ∆Ψ + eΦ|ψ|

+ eφψΨ + e

|A|

ψ Ψ + 2e

|ψ |

A · A

+ 2ieψ∇Ψ · A + 2ieψ∇ψ · A

− g

(|ψ|

)|ψ|

(ψΨ + ψΨ) − g(|ψ|

)ψΨ o , from which we conclude that

∂

τ

= ∂

r

+ h

(ψ, Ψ, A, R),

where R = (r

, r

, r

) and h

(ψ, Ψ, A, R)

= ∂

∆

3

X

m=1

P

eIm(Ψ∂

ψ + ψ∂

Ψ) − e

(|ψ|

r

+ (Ψψ + ψΨ)a

)

− ∂

∆

3

X

m=1

P

h

e∂

(−∆)

n

|Ψ|

+ Im

− ψ∆Ψ + eΦ|ψ|

+ eφψΨ + e

|A|

ψΨ + 2e

|ψ|

A · R + 2ieψ∇Ψ · A + 2ieψ∇ψ · R

− g

(|ψ |

)|ψ|

(ψΨ + ψΨ) − g(|ψ|

)ψΨ oi .

The equation on A

can be written as a symmetric system of the form

∂

A

+ M

(∇)A

+ H

(ψ, Ψ, A, R) = 0 (j = 1, 2, 3), where H

=

(0, 0, 0, 0, h

, h

, h

, h

), M

(∇) =

3

X

k=1

M f

∂

are 24 × 24 symmetric matrices. Recalling that A

= (a

, p

, q

, r

, λ

, µ

, ν

, τ

), where a

, q

, r

are scalar functions, p

, λ

, ν

and τ

are functions with values in R

and µ

is a function with values in R

, M

can be simply written by blocks:

M

(∇) =

























0 0 0 0 ∇· 0 0 0

0 0 0 0 0 ∇· 0 0

0 0 0 0 0 0 ∇· 0

0 0 0 0 0 0 0 ∇·

∇ 0 0 0 0 0 0 0

0 N (∇) 0 0 0 0 0 0

0 0 ∇ 0 0 0 0 0

0 0 0 ∇ 0 0 0 0























 ,

with

N (∇) =





∇ 0 0

0 ∇ 0

0 0 ∇



 .

Note that M f

are 24 × 24 symmetric matrices whose components are all constants.

Thus from (2.7), (2.8) and (2.9), we have transformed Equations (1.1)-

(1.3) into the following system:

−∆U = F

(ψ, Ψ), (2.10)

(−∆ + α)ψ + 2ie∇ψ · A = iΨ − eφψ − e

|A|

ψ + g(|ψ|

)ψ + αψ, (2.11) i∂

Ψ + ∆Ψ − 2ie∇Ψ · A = 2ie∇ψ · R + F

(U , ψ, Ψ, A, R), (2.12) 0 = ∂

A

+ M

(∇)A

+ H

(ψ, Ψ, A, R). (2.13)

3 Solvability of the Cauchy Problem

The aim of this section is to prove Theorem 1.1. To this end, we use a fix-point argument on a suitable version of System (2.10)-(2.13). In this procedure, the necessary estimates follow from the application of the usual energy methods.

For s ∈ N with s >

, take an initial data ψ

∈ H

( R

, C ),

A

= (a

, a

, a

) ∈ H

( R

, R

), and

A

= (r

, r

, r

) ∈ H

( R

, R

), satisfying

divA

= 0, divA

= 0.

Let us define Ψ

∈ H

( R

, C ) by

Ψ

= i ∆ψ

− eφ

ψ

− e

|A

|

ψ

− 2ie∇ψ

· A

− g(|ψ

|

)ψ

), (3.1) Φ

= e

2 (−∆)

(ψ

Ψ

+ Ψ

ψ

), where φ

=

(−∆)

|ψ

|

. We also put

A

= (A

, A

, A

) ∈ H

( R

), for i, j, k, l = 1, 2, 3,

A

=

(a

, p

, q

, r

, λ

, µ

, ν

, τ

), p

= ∂

a

, q

= ∆a

, λ

= ∂

∆

r

,

µ

= ∂

∂

∆

r

, ν

= ∂

r

and

τ

= ∂

a

+∂

∆

3

X

m=1

P

e Im(ψ

∂

ψ

)−e

|ψ

|

a

−∂

Φ

. We introduce R = 2 kψ

k

+ kΨ

k

+ kA

k

and let B(R) be the ball of radius R in C [0, T ]; (H

( R

)

for T > 0.

We prove the existence of a solution (U , ψ, Ψ, A

) of (2.10)-(2.13) by the following procedure. Take (Ψ, A) ∈ B (R) with divA = 0 arbitrarily and construct new functions Q and B = (B

, B

, B

) as follows.

First we define ψ ∈ C([0, T ]; H

( R

, C )) by ψ(t, x) := ψ

(x) +

Z

Ψ(s, x) ds. (3.2)

Then by the construction of ψ, one finds that, for T small enough, kψk

≤ R.

Next let U ∈ C([0, T ]; ˙ H

( R

)) be a solution to

−∆U = F

(ψ, Ψ). (3.3)

We note that U ∈ C([0, T ]; L

( R

)) and ∇U ∈ C([0, T ]; H

( R

)). (See Lemma 3.2 below.) Next we introduce the solution χ ∈ C([0, T ]; H

( R

, C )) of the following elliptic equation:

(−∆ + α)χ + 2ie∇χ · A = iΨ − eφψ − e

|A|

ψ + g(|ψ|

)ψ + αψ. (3.4) We now consider a linearized version of (2.12)-(2.13). We take (Q, B) ∈ H

( R

) × H

( R

) solutions to

i∂

Q + ∆Q − 2ie∇Q · A = 2ie∇χ · R + F

(U , χ, Ψ, A, R),

Q(0, x) = Ψ

, (3.5)

∂

B

+ M

(∇)B

+ H

(χ, Ψ, A, R) = 0,

B

(0, x) = A

. (3.6)

Let

S : (Ψ, A) 7−→ (Q, B).

Our strategy consists in showing that S is a contraction mapping on B(R), provided that T > 0 is sufficiently small and to prove that χ = ψ, from which we obtain the existence of a solution (U , ψ, Ψ, A

) of (2.10)-(2.13) and complete the proof of Theorem 1.1.

The proof is divided into 6 steps. We first recall the following classical

lemma. (See e.g. [1, Proposition 2.1.1, p. 98] for the proof.)

Lemma 3.1. Let u, v ∈ L

( R

)∩H

( R

) for s ∈ N . Then for all (m

, m

) ∈ N

× N

with |m

| + |m

| = s, one has

kD

uD

vk

≤ C kuk

kvk

+ kvk

kuk

.

• Step 1: Solving the elliptic equation (3.3).

Lemma 3.2. There exists a unique solution U ∈ C([0, T ]; ˙ H

( R

)) of (3.3).

Moreover, U = (φ, Φ) satisfies the following estimates.

k∇φk

≤ C

(R), kφk

≤ C

(R), (3.7) k∇Φk

≤ C

(R), kΦk

≤ C

(R), (3.8) where C

, C

, C

and C

are positive constants depending only on R.

Proof. First we note that the bilinear form a(u, v) :=

Z

R³

∇u · ∇v dx

is continuous and elliptic on ˙ H

( R

, R ) × H ˙

( R

, R ). Moreover since ψ ∈ H

( R

) and Ψ ∈ H

( R

), a direct computation gives

|ψ|

= kψk

L¹²⁵

≤ Ckψk

, kψΨ + Ψψk

L⁶⁵

≤ Ckψk

kΨk

≤ Ckψk

kΨk

. Then by the Sobolev embedding L

( R

) , → H ˙

( R

)

and the Lax-Milgram theorem, we deduce that there exists a unique solution U ∈ C([0, T ]; ˙ H

( R

)) of (3.3).

Next for 0 ≤ k ≤ s, we apply D

to the first line of (3.3), multiply the resulting equation by D

φ and make an integration by parts to obtain

k∇(D

φ)k

= e 2 Z

R³

D

|ψ|

D

φ dx

≤ C Z

R³

D

|ψ|

|D

φ| dx.

Using the Leibniz rule, Lemma 3.1 and the Schwarz inequality, one has k∇φ(t, ·)k

≤ Ckψ(t, ·)k

for all t ∈ [0, T ]. (3.9) Summing inequalities (3.9) from k = 0 to s and recalling the fact that kψk

≤ R, we obtain

k∇φk

≤ C

(R),

where C

(R) is a constant depending only on R.

Finally, the Sobolev embedding W

( R

) , → L

( R

) provides that kφ(t, ·)k

≤ C

3

X

k=1

k∂

φ(t, ·)k

+ kφ(t, ·)k

!

≤ C

3

X

k=1

k∇(∂

φ)(t, ·)k

+ k∇φ(t, ·)k

! ,

from which we deduce that there exists a constant C

(R) depending only on R such that

kφk

≤ C

(R),

which ends the proof of (3.7). The proof of estimates (3.8) is similar and we omit the details.

• Step 2: Solving the elliptic equation (3.4).

Lemma 3.3. Suppose that A ∈ H

( R

, R

), s >

and divA = 0. Then for sufficiently large α > 0, the bilinear form

b(u, v) :=

Z

R³

(∇u · ∇v + αuv + 2ie∇u · Av) dx is hermitian, continuous and elliptic on H

( R

, C ) × H

( R

, C ).

As a consequence, there exists a unique solution χ(t, ·) ∈ H

( R

, C ) to (3.4) and there exists a constant C

(R) such that

kχk

≤ C

(R).

Proof. First we note that b is hermitian by the condition divA = 0. Indeed, one has

2ie Z

R³

(∇u · A)v dx = −2ie Z

R³

divAuv dx − 2ie Z

R³

(∇v · A)u dx

= 2ie Z

R³

(∇v · A)u dx,

from which it follows directly that b(u, v) = b(v, u). The continuity is a direct consequence of the Cauchy-Schwarz inequality and the fact that A ∈ H

( R

) , → L

( R

). Finally for all u ∈ H

( R

, C ), we have

2ie Z

R³

(∇u · A)u dx

≤ 2ekAk

k∇uk

kuk

≤ 1

2 k∇uk

+ 2e

kAk

kuk

≤ 1

2 k∇uk

+ Ce

kAk

kuk

.

Taking α ≥ 2Ce

kAk

, one gets b(u, u) ≥ 1

2 k∇uk

+ α 2 kuk

. This shows that b is elliptic on H

( R

, C ) × H

( R

, C ).

Now since ψ, Ψ ∈ H

( R

) and φ ∈ H ˙

( R

) ∩ L

( R

), it is obvious that iΨ − eφψ − e

|A|

ψ − g(|ψ|

)ψ + αψ belongs to L

( R

) , → (H

( R

))

. Then the Lax-Milgram theorem ensures the existence of a unique solution χ to (3.4) in H

( R

). Using the elliptic regularity theory and recalling that

(ψ, Ψ) ∈ C([0, T ], H

( R

))

, φ ∈ C(([0, T ]; ˙ H

( R

) ∩ L

( R

)),

∇φ ∈ C([0, T ]; H

( R

)), A ∈ C([0, T ], H

( R

)), one gets

kχk

≤ C

(R),

where C

(R) is a constant depending only on R. This ends the proof of Lemma 3.3.

• Step 3: Solving the Schr¨ odinger equation (3.5).

For convenience, we introduce the real form of Equation (3.5). Denote R = (R

, R

) = (Re Q, Im Q) and write

∂

R + J ∆R − 2e

3

X

j=1

K

(A)∂

R = L

(∇χ, R) + L

(U , χ, Ψ, A, R), (3.10) R(0, x) = Re Ψ

(x), Im Ψ

(x)

, where

J =

0 1

−1 0

, K

(A) =

a

0 0 a

, L

(∇χ, R) =

Im (2ie∇χ · R)

−Re (2ie∇χ · R)

, L

(U , χ, Ψ, A, R) =

Im F

(U , χ, Ψ, A, R)

−Re F

(U , χ, Ψ, A, R)

.

Now for ε > 0, we consider a long-wave type regularization of (3.10) (see [11]):

∂

(1 − ε∆)R

+ J ∆R

− 2e

3

X

j=1

K

(A)∂

R

= L

+ L

, (3.11)

with R

(0) = (1 − ε∆)

(Re Ψ

, Im Ψ

). Since Equation (3.11) is linear and contains differential operator in space of at most zero order, one can show that there exists a unique solution R

∈ C([0, T ]; H

( R

)) to Equation (3.11).

Furthermore, we have the following estimate.

Lemma 3.4. Let R

be the unique solution of Equation (3.11). Then there exist constants C

(R), C

(R) independent of ε such that

kR

k

≤ e

kΨ

k

+ e

− 1

.

Proof. We first begin with the L

-estimate. We multiply (3.11) by R

and integrate over R

. Since J is skew-symmetric, one obtains

∂

∂t 1

2 Z

R³

|R

|

+ ε|∇R

|

dx

= 2e Z

R³ 3

X

j=1

K

(A)∂

R

· R

dx + Z

R³

L

· R

dx + Z

R³

L

· R

dx. (3.12) For j = 1, 2, 3, we have from k∂

a

k

≤ R that

Z

R³

K

(A)∂

R

· R

dx

= 1 2

Z

R³

a

∂

|R

|

dx

= 1 2

Z

R³

∂

a

|R

|

dx

≤ 1

2 k∂

a

k

kR

k

≤ C(R)kR

k

. (3.13) Since Ψ, R ∈ H

and A ∈ H

, using Lemmas 3.2-3.3, one can also compute as follows :

Z

R³

L

(∇χ, R) · R

dx

≤ C(R)kR

k

,

Z

R³

L

(U , χ, Ψ, A, R) · R

dx

≤ C(R)kR

k

. (3.14) Collecting (3.12)-(3.14), we derive

∂

∂t kR

k

≤ C(R)kR

k

+ C(R).

By the Gronwall inequality and from

kR

(0, ·)k

= k(1 − ε∆)

(Re Ψ

, Im Ψ

)k

≤ kΨ

k

,