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# Inviscid Limits of the Complex Ginzburg-Landau Equation

Inviscid Limits of the Complex Ginzburg-Landau Equation
Departamento de Matematica Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain, and Laboratoire Dieudonne, Universite de Nice, BP 71, F { 06108 Nice Cedex 2, France, e-mail: phbe@math.unice.fr Fachbereich Mathematik, Technische Universitat Berlin, Stra e des 17. Juni 136, D { 10623 Berlin, Germany, e-mail: jungel@math.tu-berlin.de, and Laboratoire Dieudonne, Universite de Nice, BP 71, F { 06108 Nice Cedex 2, France

Philippe Bechouche

Ansgar Jungel

Abstract
In the inviscid limit the generalized complex Ginzburg-Landau equation reduces to the nonlinear Schrodinger equation. This limit is proved rigorously with H 1 data in the whole space for the Cauchy problem and in the torus with periodic boundary conditions. The results are valid for nonlinearities with subcritical or critical growth exponent at the level of H 1 in any spatial dimension. Furthermore, optimal convergence rates are proved. The proofs are based on estimates of the Schrodinger energy functional and on Gagliardo-Nirenberg inequalities.

Keywords. Generalized complex Ginzburg-Landau equation, nonlinear Schrodinger
equation, energy estimates, optimal rate of convergence, strong solutions.

1991 Mathematics Subject Classi cation. 35K55, 35Q55, 81Q05. Acknowledgments. The authors acknowledge support from the DAAD-PROCOPE Program, the DAAD `Acciones Integradas' Program, and from the TMR Project \Asymptotic Methods in Kinetic Theory", grant number ERB-FMBX-CT97-0157. The second author is partly supported by the DFG Projects, grant numbers MA 1662/2-2 and 1-3, and by the Gerhard-Hess Program of the DFG.

1

1 Introduction
The description of spatial pattern formation or chaotic dynamics in continuum systems, in particular uid dynamical systems, is a challenging task in theoretical physics and applied mathematics. Due to the complexity of the corresponding nonlinear evolution equations, simpler model equations for which the mathematical issues can be solved with greater success, have been derived. The complex Ginzburg-Landau equation is one of these equations. It models the evolution of the amplitude of perturbations to steady-state solutions at the onset of instability 20, 27]. Originally discovered by Ginzburg and Landau for a phase transition in superconductivity 15], this equation was subsequently derived for instability waves in hydrodynamics such as the nonlinear growth of Rayleigh-Benard convective rolls 29, 30], the appearance of Taylor vortices in the Couette ow between counterrotating cylinders 33], and the development of Tollmien-Schlichting waves in plane Poiseuille ows 1]. The equation is also used to model the transition to turbulence in chemical reactions 16, 17, 21] or to describe pattern formation 6, 28]. The generalized complex Ginzburg-Landau (CGL) equation for the function u(x; t) is given by

@t u = (a + i ) u + Ru ? (b + i )f (u) in ; t > 0; (1.1) u(x; 0) = u0(x) for x 2 : (1.2) We consider this equation either in the whole space = R d or in the torus T = (R =2 Z)d . In the latter case we prescribe periodic boundary conditions. The parameters a, b, , , R are real with R 0, a > 0 and b > 0. Without loss of generality, we take > 0. The interaction term f (u) is a nonlinear function of u, a typical form of which is f (u) = ujuj2 ; u 2 C ; (1.3) for some > 0. Usually, the CGL equation with cubic nonlinearity = 1 has been considered in physics. The Eq. (1.1) reduces to the nonlinear Schrodinger (NLS) equation for a = b = 0 @t u = i u + Ru ? i f (u) in ; t > 0; (1.4) (more commonly with R = 0) and to the nonlinear heat equation with dissipative nonlinearity for = = 0, so that one may expect for it a mixture of the properties that occur in those two cases. Now, the self-focusing mechanism and its resulting nite-time blow-up for solutions of the NLS equation (with R = 0) is well understood both physically and mathematically 23, 26, 31]. More precisely, necessary conditions for the possible formation of nite-time singularities are < 0 and d=2. In this case, the interaction is attractive, and when d=2 this attraction is strong enough to lead to singularities. When > 0, the interaction is repulsive, and there is no tendency for the particles to concentrate. Therefore, it is of particular mathematical interest to ask how \far" is the solution of the CGL equation (1.1) from the solution of the NLS equation (1.4), in order to
2

relate the results for the NLS equation to the behavior of the solution to the CGL equation. In this paper we answer this question. More precisely, we prove rigorously the inviscid limit a ! 0, b ! 0 in the Eq. (1.1) and give optimal convergence rates for the di erence of the solutions to Eqs. (1.1) and (1.4). Optimal convergence rates for the di erence of the solutions to Eqs. (1.1) and (1.4) have also been given by Wu in 35], however, his estimates do not allow the limit a ! 0, b ! 0, except in the case of linear interactions. The results of this paper are valid for any nonlinearity satisfying some structure conditions (see below). Another paper which relates the CGL equation to the NLS equation is 22]. The authors show that so-called traveling holes of the CGL equation are singular perturbations of nonlinear Schrodinger dark solitons. Furthermore, Levermore studied recently the inviscid limit in terms of wavenumber modes 24]. In order to describe the main results of this paper we suppose that

> 0; R 0; 2 R ; d 1; (1.5) and, for the sake of presentation, we assume that the nonlinearity f (u) is given by (1.3). In Section 2 we present results for more general nonlinearities. We show the inviscid limit for nonlinearities with subcritical growth exponent at the level of H 1 when 0 and at the level of L2 when < 0. Here, the term \subcritical at the s " indicates that < 2=(d ? 2s). More precisely, we assume that level of H if 0 : < 2=(d ? 2) ( < 1 if d 2); (1.6) if < 0 : (i) < 2=d; or (ii) = 2=d and ku0kL is small enough. (1.7) Let " = (a; b), let the initial datum satisfy u0 2 H 1( ) \ L2 +2 ( ), T > 0, and let u" be the solution to the CGL equation (1.1){(1.2) in (0; T ) (see Section 2 for a review on existence results). Furthermore, when < 0, let (ba?d =2 ) be bounded as " ! 0.
0 < a < a0 ; 0 < b < b0 ;
2

Theorem 1.1 Let the above assumptions hold. Then there exists a subsequence of (u") (not relabeled) such that, as " ! 0, u" converges to a function u which solves the NLS equation (1.4), (1.2). The convergence is strong in the space L1 (0; T ; Lr ( )), loc where r < d=(d ? 2), and weak- in the spaces L1 (0; T ; H 1( )) and W 1;1(0; T ; H ?1( )). For the convergence rates, let u" be the solution to the CGL equation (1.1), (1.2) with initial datum u"0 and let u be the solution to the NLS equation (1.4), (1.2) with initial datum u0. We assume that u"0, u" 2 H n( ) with n > d=2 and ku"0 ? u0kL = O(a) + O(b) as " ! 0. Theorem 1.2 Let the above assumptions hold. Then there exists a constant T > 0 independent of a; b such that for all 0 t < T ,
2

k(u" ? u)(t)kL = O(a) + O(b)
2

as " ! 0:

Moreover, we can take T = T if d = 1.

3

The growth condition for the nonlinearity in the above results is sharp. Indeed, when = R d and u0 2 H 1( ), there exists a global in time weak solution to the NLS equation if < 2=(d ? 2) and the conditions (1.6){(1.7) hold. Moreover, if 2=d then there exist solutions that blow up in nite time (see 4]). Theorems 1.1 and 1.2 are corollaries from the more general results of Section 2 (see Theorems 2.1, 2.2, 2.4, and 2.5). In the following we explain our method of proof of the inviscid limits. The basic idea is to treat the CGL equation as a perturbation of the NLS equation. Now, the NLS ow (with R = 0) conserves the mass M and the energy E , which are de ned by

M (t) =
and F is given by

Z

E (t) = 2

Z

ju(t)j2 dx; jru(t)j2 dx +
2

Z

(1.8)

F (ju(t)j2) dx;

(1.9)

1 F (s) = + 1 s +1 ; s 0: One may expect that in the case of the CGL equation, these quantities are also bounded in terms of M (0), E (0), and Rt (if R > 0). This is indeed true. Let u be a weak solution to the CGL equation. A simple computation shows (see Section 3) @t M 2RM; @t E 2R( + 1)E; for all t > 0, which implies that M (t) M (0)eRt ; E (t) E (0)e2R( +1)t : When 0 we immediately obtain a H 1 bound for u(t) uniformly in a; b. In the case < 0 we need to control the integral

ju(t)j2 +2dx 2( + 1) with the H 1 norm of u(t). This is achieved through the Gagliardo-Nirenberg inequality (see Lemma 2.6)
C ( ; d)kukH kuk1? ; L where the interpolation parameter satis es = d=(2 +2). Notice that 0 < < 1 since 0 < < 2=(d ? 2). This inequality gives, for 0 t T ,

?2

Z

F (ju(t)j2)dx = ?

Z

kukL

2 +2

1

2

E (t) + 2 M (t) = 2 ku(t)k2 + 2 ku(t)k2 +2 H L (2 +2) (2 +2)(1? ) 2 0 : 2 ku(t)kH + 2 C ku(t)kH ku(t)kL 4
1 2 +2 1 1 2

Because ku(t)kL is uniformly bounded, this estimate provides an upper bound for ku(t)kH uniformly in a; b, whenever the exponent of the H 1 norm of u(t) in the second term of the right-hand side is less than in the rst term, which is the case if and only if < 2=d. For the limit case = 2=d, the above estimate gives a uniform bound for ku(t)kH provided that ku(t)kL or, equivalently, ku0kL is small enough. It is clear that the uniform H 1 bound for u(t) is the key estimate in the proof of Theorem 1.1. In a similar way, one can obtain uniform H n bounds for u(t) (n > 1) by estimating the quantity @t kuk2 n (see Sections 5 and 7). These bounds can be only H derived for small enough t > 0, since the use of the Gagliardo-Nirenberg inequality leads to inequalities of the type @t kuk2 n C1 + C2 kuk2+n ; H H with positive = ( ; d). For the proof of Theorem 1.2 we consider the equation satis ed by the di erence u" ? u, where u" with " = (a; b) solves the CGL equation and u solves the NLS equation. In order to estimate the term f (u") ? f (u) we need uniform L1 bounds for the solutions which are obtained via the Sobolev embedding theorem from the uniform H n bounds with n > d=2. This paper is organized as follows. In Section 2 we recall some existence results for the CGL equation and state our main theorems. Section 3 is concerned with the uniform H 1 estimate for the solutions of the CGL equation. The rst convergence result is proved in Section 4. A uniform bound in H 2 is derived in Section 5. Finally, the proof of the optimal convergence rate is presented in Section 6 (for at most three spatial dimensions) and Section 7 (for any spatial dimension).
2 1 1 2 2

2 Preliminaries and main results
The (global in time) existence and uniqueness of solutions to the generalized CGL equation (1.1){(1.2) is shown by Ginibre and Velo for = R d 12, 13, 14] and by Doering, Gibbon and Levermore for = Td 7] (also see the review 25]) under some assumptions on the nonlinearity. In particular, assume the following: Let = R d or = Td (d 1), u0 2 H 1( ) \ L2 +2 ( ), T > 0, (2.1) 0 < < 2=(d ? 2) ( < 1 if d 2), and, in the case < 0, let either (i) < 2=d, or (ii) = 2=d and ku0kL be small enough. We impose the following assumption on the non-linearity f (u): f (u) = ug(juj2); u 2 C ; (2.2) where the function g(s) satis es: g 2 C 1( 0; 1); 0; 1)) and there exist ; ; ?; ?0 > 0 such that for all s 0 (2.3) s g(s) ?(1 + s ), 0 sg0(s) ?0(1 + s ).
2

5

For instance, the functions g(s) = s ; g(s) = s + log(1 + s), or g(s) = ses=(1 + es) satisfy the condition (2.3). In order to de ne the mass (1.8) and energy (1.9) we introduce Z s F (s) = g( )d ; s 0: Under the above assumptions there exists a unique solution u to (1.1){(1.2) satisfying u 2 H 1(0; T ; L2( )) \ C 0( 0; T ]; H 1( )) \ L2 (0; T ; H 2( )): This follows from the a priori estimates of Section 3 and the compactness arguments of 7, 12]. Furthermore, if additionally u0 2 H n( ) (n 2 N ), then the solution u satis es u 2 H 1(0; T ; H n?1( )) \ C 0 ( 0; T ]; H n( )) \ L2 (0; T ; H n+1( )): This regularity result can be achieved by applying, for instance, Theorems II.3.2 and II.3.3 in 34, Ch. I.3.3]. Concerning the existence and uniqueness of solutions to the NLS equation (1.4), (1.2) we refer to 4, 5, 11] for = R d and to 2, 3, 18] for = Td (see also 19] and the reviews in 9, 10]). Our rst main result is concerned with the convergence of the solution u" of the CGL equation with initial datum u0 and parameter " = (a; b) to a solution u of the NLS equation. In the case < 0 we need an additional technical assumption on the parameters a and b: Let (ba?d =2 ) be bounded as (a; b) ! 0: (2.4) Theorem 2.1 Let the assumptions (1.5), (2.1){(2.4) hold. Furthermore, let = Td . Let u" be the solution to the CGL equation (1.1){(1.2). Then there exists a subsequence of u" (not relabeled) such that, as " ! 0, u" * u in L1 (0; T ; H 1( )) weakly- ; @t u" * @t u in L1 (0; T ; H ?1( )) weakly- ; u" ! u in L1 (0; T ; Lr ( )) strongly; loc f (u") ! f (u) in L1 (0; T ; Lq ( )) strongly; loc where r < 2d=(d ? 2) (r < 1 if d 2), q = 2d=(d + 2), and u 2 L1 (0; T ; H 1( )) \ W 1;1(0; T ; H ?1( )) \ L(2 +1)q ( (0; T )) solves the NLS equation (1.4), (1.2). Theorem 2.2 Let the assumptions of Theorem 2.1 hold, but assume that = R d and that g(s) ?s for 0. Then the same conclusion as in Theorem 2.1 holds. The proof of this result is based on a uniform bound for u" in L1(0; T ; H 1( )), which allows to bound the term f (u"). In one space dimension, the Sobolev embedding H 1( ) ,! L1( ) therefore provides a uniform bound for u" in L1 ( (0; T )) p p from which we can conclude a O( a) + O( b) convergence rate for the di erence u" ? u in L1(0; T ; L2( )) under the condition that the initial data satis es p (2.5) ku"0 ? u0kL = O(pa) + O( b) as a; b ! 0:
2

0

6

Theorem 2.3 Let d = 1 and assume that the hypotheses (1.5), (2.1){(2.5) hold. Then we have for all 0 t T , p k(u" ? u)(t)kL = O(pa) + O( b) as a; b ! 0:
2

More precisely, there exist two constants M1 (T ); M2(T ) > 0 depending on the data but not on a; b such that for all 0 t T , p k(u" ? u)(t)kL ku"0 ? u0kL + M1(T )pa + M2 (T ) b: In order to get the optimal convergence rate O(a)+ O(b) for (u" ? u)(t) in L2 ( ), we have to impose stronger assumptions on the nonlinearity g. Suppose that g 2 C 2 ( 0; 1); 0; 1)) and there exist 1=2; ?1; ?01 > 0 such that for all s 0 (2.6) 0 sg0(s) ?1 (s1=2 + s ); 0 s2g00(s) ?01(s1=2 + s ). It can be easily checked that the condition on g0 in (2.6) implies that 0 sg0(s) ?2 (1 + s ) for all s 0; for some ?2 > 0. Thus the condition on g0 in (2.6) is stronger than the corresponding condition in (2.3). Theorem 2.4 Let 1 d 3 and u"0; u0 2 H 2( ) and assume the conditions (1.5), (2.1){(2.6). Then, if ku"0 ? u0 kL = O(a) + O(b), there exists a constant T > 0 depending on the data but not on a; b such that for all 0 t < T , k(u" ? u)(t)kL = O(a) + O(b) as a; b ! 0: Moreover, we can take T = T if d = 1. More precisely, there exist two constants M3 (t); M4(t) > 0 which are independent of a; b such that for all 0 t < T , k(u" ? u)(t)kL ku"0 ? u0kL + M3 (t)a + M4 (t)b: The constant T > 0 depends on d; ; R; ?1; jb0 + i j; C4; C6 and k u0kL (see (5.7)). See (5.3) and (5.6) for the de nitions of C4 and C6. Our last result is concerned with the optimal convergence rate in any space dimension. In order to avoid cumbersome assumptions on the function g, we suppose that g(s) = s for some > 0: (2.7) Theorem 2.5 Let d 1, u"0; u0 2 H n( ) with n 2 N , n > d=2, and suppose that (1.5), (2.1){(2.4) and (2.7) hold. Then, if ku"0 ? u0kL = O(a) + O(b), there exists a constant T > 0 depending on the data but not on a; b such that for all 0 t < T , k(u" ? u)(t)kL = O(a) + O(b) as a; b ! 0:
2 2 2 2 2 2 2 2 2

7

In the proofs of the main theorems we employ the Gagliardo-Nirenberg inequality (see, e.g., 4, p. 21] or 8, p. 242]): Lemma 2.6 Let = R d , or = Td , or let R d be a bounded domain with 0;1 , and d; m 1. Furthermore, let 1 p; q; r 1; j 2 N f0g with j < m @ 2C and 2 j=m; 1] such that 1 = j + 1 ? m + (1 ? ) 1 ; p d r d q provided that m ? j ? d=r is not a nonnegative integer (else take = j=m). Then there exists G = G(d; m; j; ; q; r) such that for all u 2 W m;r ( ) \ Lq ( )
X

j j=j

kD ukLp GkukW m;r kuk1? : Lq
C0(kvkL + k vkL )
2 2

Frequently, we use the inequality

kvkH

2

(2.8)

which is valid for all v 2 H 2( ) with = R d or = Td .

3 A uniform bound in H 1
We show that the mass M and the energy E (see (1.8), (1.9), respektively) of the solution u to the CGL equation (1.1){(1.2) is bounded uniformly in a; b. In the following we assume that (1.5), (2.2){(2.3) hold.

Lemma 3.1 Let u0 2 L2 ( ). For all 0 t T , it holds
M (t) K0(T );
where K0 (T ) = e2RT ku0k2 . L
2

Proof. The proof of this lemma is standard. Indeed, multiply the equation (1.1) by u , integrate over , integrate by parts and take the real part to get 1 @ ku(t)k2 + akruk2 + b Z juj2g(juj2)dx = Rkuk2 : L L L 2 t An application of Gronwall's lemma gives the assertion. Lemma 3.2 Let u0 2 H 1( )\L2 +2 ( ); 0, and 2=(d?2) ( < 1 if d 2). Then, for all 0 t T , E (t) K1(T ); where K1 (T ) = (E (0) + R?TK0(T ))e2R max(1;( +1)? ? )T :
2 2 2 1

8

Proof. For every solution u to (1.1), (1.2), we get

E (t) ? E (0) = Re

Z tZ

= ?a

0 Z

(? u + f (u))@tu dxd

t
0 Z

k
t

uk2 2 d L

? b

Z t

0

kf (u)k2 d L
2

+ R

? Re ? Re
= ?a + R

kruk2 2 d L 0 Z tZ
0

+ RRe

Z tZ

0

f (u)u dxd

0 Z tZ

(a ? i )rf (u) ru dxd
Z tZ

Z t

(b ? i )rf (u) ru dxd

0 Z

k
t

uk2 2 d L
2

? b

0

kruk2 d + R L
Z tZ

0Z Z t 0

juj2g(juj2) dxd juj2g(juj2) dxd
(3.1)

? (a + b )Re
Z tZ

The fourth term on the right-hand side of (3.1) are estimated as follows. We use (2.3) to obtain

0

rf (u) ru dxd :

R

0

juj2g(juj2) dxd
Z tZ Z t

R? R?

0

(juj2 + juj2 +2) dxd + R?
?1 (

For the last integral we use g0(s) 0 and Re((ru

0

kuk2 2 d L
Z tZ

+ 1)

Z tZ

0 )2u2)

F (juj2) dxd :

jruj2juj2:

?(a + b )Re

= ?(a + b )

0Z Z t h 0

rf (u) ru dxd
g(juj2)jruj2 + g0(juj2)juj2jruj2
i

+ g0(juj2)Re((ru )2u2) 0: Therefore, we get from (3.1)

?(a + b )

Z tZ

dxd

0

g(juj2)jruj2 dxd
Z t

E (t)

E (0) + R

Z t

0

kruk2 2 d L
9

+ R?

0

kuk2 d L
2

+ R?

?1 (

+ 1)

Z tZ

0

F (juj2) dxd
?1 (

E (0) + R?K0(T )T + 2R max(1; ? E (t) K1(T ):
The lemma is proved.

+ 1))

Z t

0

E( ) d ;

using Lemma 3.1. From Gronwall's lemma, we conclude

Remark 3.3 The condition (2.3) on g0 can be weakened. Indeed, instead of assu-

ming s g0(s) 0 for s 0 it su ces to assume that 1 sg0(s) ? 2 g(s) for s 0: (3.2) In order to see this we have to estimate the last term on the right-hand side of (3.1):

?(a + b )Re

Z tZ

(a + b ) (a + b ) = 0:

0 Z tZ
Z tZ

rf (u) ru dxd jruj2 ? g(juj2) + juj2(jg0(juj2)j ? g0(juj2)) jruj2 ?g(juj2) + g(juj2)] dxd
h i

0

0

In the case < 0 we need more restrictive conditions on : Lemma 3.4 Let u0 2 H 1( ) \ L2 +2 ( ); < 0, and either (i) = 2=d and ku0kL is small enough. Then, for all 0 t T ,
2

< 2=d, or (ii)

kru(t)k2 L

2

K2 (T );

where K2 (T ) depends on the given data, on T and on ba?d =2 . We use the Gagliardo-Nirenberg inequality (see Lemma 2.6) in some special cases: There exist G1; G2; G3; G4 > 0 such that for all u 2 H 2( )

kukL kukL krukL kukL

2 +2

1 G1 kukH kukL? ;
1 1 2 1

2 +2

1 G2 kukH kukL? ;
2 2 2 2

4 +2

1 G3 kukH kukL? ;
3 2 2 3

2 +2

1 G4 kukH kukL? ;
4 2 2 4

= d 2 (0; 1); 2 +2 d 2 0; 1 i; 2= 4 +4 2 d 2 (0; 1); 3= 4 +2 (2 + d) + 2 2 h0; 1 : 4= 4 +4 2
1

(3.3) (3.4) (3.5) (3.6)

10

These inequalities are valid for all

2=(d ? 2) if d > 2 and < 1 if d 2.

Proof of Lemma 3.4. Again we start from the relation (3.1). Since < 0, we get

kru(t)k2 2 L 2

k uk2 d L 0 Z kru0k2 + j 2 j F (ju(t)j2) dx L 2
+a
2 2

Z t

+ j jb

Z tZ

0

juj2g(juj2)2 dxd
Z tZ

+ R

Z t

0

kruk2 d L
2

(3.7)

? (a + b )Re

0

rf (u) ru dxd :

We have to estimate the second, third and fth term on the right-hand side. For the second term we use the hypothesis (2.3), the Gagliardo-Nirenberg inequality (3.3) and Lemma 3.1:

H L 2 +2 1 j j ?K (T ) + j j?G2 +2 (ku(t)kd + kru(t)kd )ku(t)k2 +2?d : 1 L L L 2 0 +1 In the last step we have used the inequality

j j Z F (ju(t)j2 dx 2 j j ? Z (ju(t)j2 + ( + 1)?1ju(t)j2 +2) dx 2 j j ?K (T ) + j j? G2 +2 ku(t)k(2 +2) ku(t)k(2 +2)(1?
2
0
1 1 2 2 2

1

)

2

(x + y)(2 since d

+2)

1

= (x + y)d

2(xd + yd ) for x; y 0; (3.8)

j j Z F (ju(t)j2) dx 2

2. Again using Lemma 3.1 we get

Now we have to distinguish the cases d < 2 and d = 2. Let rst d < 2. Then, by Young's inequality, j j?G2 +2 kru(t)kd ku(t)k2 +2?d 1 L L +1 2 +2 " d2 j j?G11 kru(t)k2 L + d =(2?d ) 2 ? d j j?G2 +2 1 ku(t)k(4 +4?2d )=(2?d ) + 1 L " 2 +1 11
2 2 2 2

j j ?K (T ) + j j?G2 +2 K (T ) +1 1 2 0 +1 0 2 +2 2 + j j?G1 kru(t)kd ku(t)kL +2?d : L +1
2 2

and taking " = ( + 1)=(2d j j?G2 +2), we conclude 1 where

j j Z F (ju(t)j2) dx C + kru(t)k2 ; 1 L 2 4
2

(3.9)

2 +2 (3.10) C1 def j 2 j ?K0(T ) + j j?G11 K0 (T ) +1 = + 2d j j?G2 +2 d =(2?d ) (2 ? d )j j?G2 +2 K (T )(2 +2?d )=(2?d ) : 1 1 + 0 ( + 1) 2 +2 In the case d = 2 we choose ku0kL small enough such that j j?G2 +2 e2 RT ku k2 1 : 0 L +1 4 Hence, by (3.8), j j ?K (T ) + j j?G2 +2 K (T ) +1 + kru(t)k2 j j Z F (ju(t)j2) dx 1 L 2 2 0 +1 0 4 C1 + 4 kru(t)k2 ; L
2 2 2 2

and C1 > 0 is given by (3.10). The third term on the right-hand side of (3.7) is estimated by employing the assumption (2.3), the Gagliardo-Nirenberg inequality (3.5), and the relation (2.8)

bj j

Z tZ

0

juj2g(juj2)2 dxd j?2
Z tZ

2bj 2bj

0

(juj2 + juj4 +2) dxd

Again, we have to distinguish the cases d < 2 and d = 2. If d < 2 we use Young's inequality Z t d Z t k uk2 d d kuk4 +2?d d "2 k ukL L L 0 0 Z 1 d =(2?d ) 2 ? d t kuk(8 +4?2d )=(2?d ) d + " L 2 0
2 2 2 2

0 Z t 2TK (T ) + 4bj j?2 (G C 3 )4 +2 2bj j? 0 (kukd 2 + k 3 0 L 0 2bj j?2TK0 (T ) + 4bj j?2(G3C0 3 )4 +2 TK0(T )2 +1 Z t 2 (G C 3 )4 +2 4 + 4bj j? 3 0 k ukd 2 kukL2+2?d d : L 0

j?2TK

0 (T ) + 2bj

j?2G4 +2 3

Z t

(4 kukH +2) kuk(4 +2)(1? ) d L
2 3 2 3

4 ukd )kukL +2?d d L
2 2

(3.11)

12

and choose to get where

"= a b bj j
Z tZ

4j j?2d (G3C0 )4
3

+2
Z t

0

juj2g(juj2)2 dxd

C2 + a2
3

0

k uk2 d ; L
2

(3.12)

C2 def 2b0 j j?2TK0 (T ) + 4b0 j j?2(G3 C0 )4 +2 TK0(T )2 +1 = + 2(ba?d =2 )2=(2?d ) j j?2(G3C0 )4 +2 (2 ? d ) (3.13) ? ?1 d =(2?d ) TK0 (T )(4 +2?d )=(2?d ) : 4 j j?2d (G3C0 )4 +2 In the case d = 2 choose ku0kL such that a: 4 4j j?2(G3 C0 )4 +2 e(4 +2?d )RT ku0kL +2?d 2b Clearly, we have to assume that the quotient b=a remains bounded as a; b ! 0. Thus, we conclude from (3.11)
3 3 2 3 2

bj j

Z tZ

0

juj2g(juj2)2 dxd j?2TK
0 (T ) + 4bj Z t k uk2 2 d L 0

2bj

j?2(G
;

3 C0

3

)4 +2 TK

0

(T )2 +1 +

and C2 is given by (3.13). It remains to estimate the last term on the right-hand side of (3.7):

C2 + a2

a 2

Z t

0

k uk2 d L
2

(3.14)

?(a + b )Re
aj j aj
= aj
0

Z tZ

0 Z tZ

rf (u) ru dxd
Z tZ

g(juj2)jruj2 + 2g0(juj2)juj2jruj2] dxd
0 Z 0

j(? + 2?0) j(? + 2?0)

(1 + juj2 )jruj2 dxd + aj

(3.15)
Z tZ

t

kruk2 2 d L

j(? + 2?0)

The last integral is estimated by employing the Holder inequality and the GagliardoNirenberg inequalities (3.4) and (3.6):
Z tZ

0

juj2 jruj2 dxd :

0

juj2

jruj2 dxd

Z t

0 G2 2

kuk2 kruk2 L L
2 +2

2 +2

d
2 2

G2 4

Z t

0

1+ 1+ kukH kukL d ;
2 1

13

where
1 2

def

d 2 (0; 1]; 2 def = 2 (1 ? 2 ) + 2(1 ? 4 ) ? 1 = (4 ? d) 2 (?1; 3]; 2
= 2
2+2 4?1=

since imply

2=d. The inequalities (2.8) and (x + y)1+
1

2(x1+ + y1+ ) for all x; y 0
1 1

aj

j(? + 2?0)

Z tZ

Z t 1+ 1+ 1+ 2 2aj (kukL2 1 + k ukL2 1 )kukL2 2 G4 0 2aj j(? + 2?0)G2 G2TK0(T ) +1 2 4 Z t 1+ 1+ 0 )G2 G2 + 2aj j(? + 2? 2 4 k ukL2 1 kukL2 2 d : 0

0

juj2 jruj2 dxd

j(? + 2?0)G2

2

d
(3.16)

If < 2=d then 1+
Z t

1

< 2, and we can apply Young's inequality to the last integral:
2

0

k

1+ ukL2

1

1+ kukL2

d

and taking we obtain from (3.16)

"1 + 2 + 1 " "1 + 2 + 1 "

1

Z t

0 (1+ 1 1
Z t

k uk2 d L )=(1? ) 1 ?
2 1 2

2

1

Z t

0

(2+2 kukL
2

1

)=(1? 1 )

d

0 (1+ 1

k uk2 d L )=(1? ) 1 ? 1 TK0(T )(1+ 2
1

1

)=(1? 1 ) ;

" = 2j j(1 + )(? + 2?0 )G2 G2 ; 1 2 4
Z tZ

aj
where

j(? + 2?0)

0

juj2

jruj2 dxd

C3 + a2

Z t

0

k uk2 d ; L
2

C3 def a0 j j(? + 2?0)G2 G2T 2K0 (T ) +1 = 1 4 + (1 ? 1)"?(1+ )=(1? ) K0 (T )(1+
1 1

1

)=(1? 1 )

:

(3.17)

14

In the case = 2=d we get 1 +

= 2. Choose ku0kL small enough such that a Z t k uk2 d : 1+ kuk1+ d 0 )G2 G2 2aj j(? + 2? 2 4 k ukL L L 2 0 0 Hence, the estimate (3.15) becomes
1 Z t
2 2 1 2 2 2

a Z t k uk2 d : a0 j + C3 + 2 L 0 0 Now we insert the inequalities (3.9), (3.12), and (3.18) into (3.7) to get

?(a + b )Re

Z tZ

0

j(? + 2?0)
22

Z t

rf (u) ru dxd kruk2 2 d L

2

(3.18)

4 kru(t)kL

2 2 kru0kL + C1 + C2 + C3 Z t 0 )) kruk2 d : + ( R + a0 j j(? + 2? L
2

0

2

Notice that C2 depends on ba?d =2 . An application of Gronwall's inequality yields kru(t)k2 K2(T ); 0 t T; L
2

where

K2 (T ) def =
This proves the lemma.

2kru0k2 + 4 ?1 (C1 + C2 + C3) L
2

(3.19)

exp 4T (R + ?1 a0 j j(? + 2?0)) : The Lemmas 3.1, 3.2 and 3.4 yield a uniform H 1-bound for the solutions to (1.1),(1.2), provided that ba?d =2 remains bounded as a; b ! 0 in the case < 0: Corollary 3.5 Let (2.1){(2.3) hold. In the case < 0 we also have to require that ba?d =2 remains bounded as a; b ! 0. Then there exists a constant K3(T ) > 0, only depending on the data and on T , but not on a; b, such that for every solution u to (1.1),(1.2) it holds

K3 (T ) for all 0 t T: Remark 3.6 The constant K3(T ) is de ned by K (T ) + K (T ) if 0 K3(T ) = K0(T ) + K1 (T ) if < 0: 0 2 The precise values of K0(T ) and K1 (T ) are given in Lemma 3.1, 3.2, respectively. The constant K2 (T ) is de ned in (3.19), with constants C1 , C2, C3 which are introduced in (3.10), (3.13) and (3.17), respectively.
1

ku(t)k2 H

15

Remark 3.7 The above corollary also holds true if

inequalities in condition (2.3) is replaced by 1 ? 2 g(s) s g0(s) ?0 (1 + s ) for s 0 (see Remark 3.3).

0 and the second chain of

Let u" be a solution to (1.1){(1.2). Let p = 2d=(d ? 2) (p < 1 if d 2) and let q = 2d=(d + 2) be the conjugate exponent to p. Then, the condition < 2=(d ? 2) ( < 1 if d 2) implies (2 + 1)q < p, showing that the embedding H 1( ) ,! L(2 +1)q ( ) is continuous. In order to give uniform bounds for f (u") we have to consider the cases = R d and = Td separately. First let = Td . From (2.3) we get, for all 0 t T ,

4 Proof of Theorems 2.1 and 2.2

kf (u"(t))kLq

?

Z

ju"(t)jq (1 + ju"(t)j2 )q dx
(2 +1) 2 1

1=q

(2 ? ku"(t)kq q + ku"(t)kL +1)qq L ? c ku"(t)kL + ku"(t)k2 +1 ; H

1=q

where here and in the following we denote by c, ci positive constants independent of " = (a; b) and t. In the last step we have used that q < 2 and Lq ( ) ,! L2 ( ) since is bounded. Now let = R d . Then the above argument does not apply since the injection Lq ( ) ,! L2 ( ) is generally not valid. Here we need the condition g(s) ?s . Then Applying Corollary 3.5, we conclude that (f (u")) is uniformly bounded in L1(0; T ; Lq ( )). Hence

kf (u"(t))kLq ?ku"(t)k2 +1 L

(2 +1)

q

?ku"(t)k2 +1 : H
1

c(k u"(t)kH ? + kf (u"(t))kH ? ) c(ku"kH + kf (u"(t))kLq ) c: This shows that the sequence (u") is uniformly bounded in L1(0; T ; H 1( )) and W 1;1(0; T ; H ?1( )). From the above estimates we get the following weak convergence results:

k@t u"(t)kH ?

1

1

1

1

u" * u @t u" * @t u f (u") * f

in L1(0; T ; H 1( )) weakly- ; in L1(0; T ; H ?1( )) weakly- ; in L1(0; T ; Lq ( )) weakly- : 16

It remains to identify the limit f . For this, let ! be an arbitrary bounded 1 (0; T ; H 1 (! )) and W 1;1 (0; T ; H ?1(! )). open set. Then (u") is also bounded in L By Aubin's lemma 32], there exists a subsequence, still denoted by (u"), such that for all r < p u" ! u in L1 (0; T ; Lr (!)) strongly; where u 2 C 0( 0; T ]; Lr (!)). Applying the mean value theorem to both Re(f ) and Im(f ), we get for all u; v 2 C , (u ? v)(g(jw j2) + jw j2 g0(jw j2)) + (u ? v) w2 g0(jw j2) d ; 0 (4.1) where w = u + (1 ? )v. Thus, using Holder's inequality and condition (2.3), for all 0 s; t T , kf (u"(t)) ? f (u(t))kLq (!) (? + 2?0) (? + 2?0)
Z Z Z

f (u) ? f (v) =

Z 1 ?

! !

ju"

(t) ? u(t)jq
?

Z 1

0

(1 + j u"(t) + (1 ? )u(t)j2 )d
1=q

q

dx

1=q

ju"(t) ? u(t)jq 1 + max(1; 22 ?1)(ju"(t)j2 + ju(t)j2 ) q dx
Z

1=q

c1

!

ju"(t) ? u(t)jq (1 + ju"(t)j2 + ju(t)j2 )q dx

c2 ku"(t) ? u(t)kLq (!) + c2 ju"(t) ? u(t)jq (ju"(t)j2 q + ju(t)j2 q )dx ! c2 ku"?t) ? u(t)kLq (!) ( 2 + c2 ku"(t)kL q + ku(t)k2 q (!) ; q ku" (t) ? u(t)kL L where c1 = (? + 2?0) max(1; 22 ?1) and c2 = 31?1=q c1. Therefore, by Corollary 3.5, kf (u"(t)) ? f (u(t))kLq (!) 2 c2ku"(t) ? u(t)kLq (!) + 2c2 sup ku"(t)kH ( ) ku"(t) ? u(t)kL q (!) 0<t<T c2ku"(t) ? u(t)kLq (!) + cku"(t) ? u(t)kLr (!) : To see the last step observe that, since (2 + 1)q < p, there exists r < p such that (2 + 1)q r. The strong convergence of u" in L1(0; T ; Lr (!)) and hence in L1(0; T ; Lq (!)) now implies that f (u") ! f (u) in L1(0; T ; Lq (!)) strongly. Therefore, we get f (u) = f in ! (0; T ) for every bounded ! , hence almost everywhere in (0; T ). Now, the limit in the equation satis ed for u" can be performed and we see that u solves the nonlinear Schrodinger equation (1.4). Finally, notice that the initial value is satis ed in the sense of H ?1( ).
(2 +1) (2 +1) (2 +1) 1 (2 +1)

1=q

17

5 A uniform bound in H 2( )
In this section we derive a uniform bound in H 2( ) for the solution u of the CGL equation for space dimensions d 3. First we prove the existence of a uniform bound in one space dimension. Lemma 5.1 Let the conditions (2.1){(2.2), (2.4), (2.6) hold. Furthermore, let d = 1 and u0 2 H 2( ). Then there exists a constant K4(T ) > 0, only depending on the data and on T , but not on a; b such that k@xxu(t)k2 K4 (T ) for all 0 t T: L Proof. The rst part of the proof is valid for any space dimension. We impose the restriction d = 1 later for the estimation of the L1 bound of u. Taking the Laplacian of Eq. (1.1) we obtain
2

@t u = (a + i ) 2 u + R u ? (b + i ) g(juj2) u + 2g0(juj2)ru rjuj2 + ug00(juj2)(rjuj2)2 + ug0(juj2) juj2 : Multiplying this equation by u , taking the real part and integrating over yields 1 @ k uk2 = ?akr uk2 + Rk uk2 ? b Z g(juj2)j uj2dx L L L 2 t Z ? 2Re (b + i )g0(juj2)ru rjuj2 u dx
2 2 2

? Re (b + i )ug00(juj2)(rjuj2)2 u dx ? Re (b + i )ug0(juj2) juj2 u dx:
Z Z

Z

(5.1)

The fourth term on the right-hand side can be estimated by using (2.6) and the inequality jrjuj2j 2jrujjuj:

?2Re (b + i )g0(juj2)ru rjuj2 u dx
4?1jb + i j (1 + juj2 ?1)jruj2j ujdx
2 ? 4?1jb + i j(1 + kukL1 1)kruk2 k ukL : L The fth term on the right-hand side of (5.1) is treated similarly
4 2

Z

?Re (b + i )ug00(juj2)(rjuj2)2 u dx
4?0 jb + i
1

Z

j (1 + juj2 ?1)jruj2j ujdx 2 ? 4?01jb + i j(1 + kukL1 1)kruk2 k ukL : L
4 2

Z

18

In the last term of (5.1) we use the inequality j juj2j 2j ujjuj + jruj2 to get

?Re (b + i )ug0(juj2) juj2 u dx
2?1jb + i j (1 + juj2 ?1)(2j ujjuj + jruj2)j ujdx
2 ? 2?1jb + i j(1 + kukL1 1)(2k ukL kukL1 + kruk2 )k ukL : L
2 4 2

Z

Z

Hence, we obtain from Eq. (5.1)

@t k uk2 L

2

2 ? 2Rk uk2 + 4?1jb + i j(1 + kukL1 1 )kukL1 k uk2 L L 2 ? + 8(2?1 + ?01 )jb + i j(1 + kukL1 1 )kruk2 k ukL : L
2 4 2

2

The following Gagliardo-Nirenberg inequality holds for d 4:

krukL kruk2 L
4

4

G5krukd=4kruk1?d=4 : H L
1 2 1 2

By employing Lemma 3.2 and Corollary 3.5 this gives

G2krukd=2kruk2?d=2 5 H L 2 (kruk2 + k uk2 )d=4 kruk2?d=2 G5 L L L 2 K 1?d=4 max(1; K d=4 )(1 + k ukd=2 ); G5 3 3 L
2 2 2 2

and we can continue to estimate

@t k uk2 L
where we have set

2

2 ? 2R + 4?1jb0 + i j(1 + kukL1 1 )kukL1 ]k uk2 L 2 1 1 )(1 + k ukd=2 )k uk ; ? + C4(1 + kukL L L
2 2

2

(5.2) (5.3)

C4 def 8(2?1 + ?01)jb0 + i jG2K31?d=4 max(1; K3d=4 ): = 5

In order to bound the L1 norm of u, we have to impose the condition d = 1. Then it holds kukL1 Gkuk1=2kuk1=2 (5.4) H L p p if = T. Thus, by Lemma 3.1 and Corollary with G = 2 if = R and G = 1= 3.5,
1 2

kukL1 (4K0K3 )1=4 ;
and using the elementary inequalities

k@xxukL

2

k@xxuk3=2 L
2

1 (1 + k@ uk2 ); xx L 2 1 (1 + 3k@ uk2 ); xx L 4
2 2

19

we obtain from (5.2)

@t k@xxuk2 L

2

where

2R + 4?1jb + i j(1 + (4K0K3 ) =2?1=4 )(4K0K3)1=4 ]k@xxuk2 L =2?1=4 ) 3 + 5 k@ uk2 + C4(1 + (4K0K3 ) 4 4 xx L 3 C (1 + (4K K ) =2?1=4 ) + C k@ uk2 ; 0 3 5 xx L 4 4
2 2

2

By Gronwall's lemma, we conclude that

C5 def 2R + 4?1jb + i j(1 + (4K0K3) =2?1=4 )(4K0K3 )1=4 = + 5 C4 (1 + (4K0 K3) =2?1=4 ): 4

k@xxu(t)k2 L
where

2

K4(T ) for all 0 t T;
2 5

3 K4(T ) def k@xxu0k2 + 4 C4(1 + (4K0K3) =2?1=4 )eC T : = L This nishes the proof. In two and three space dimensions we have the following result: Lemma 5.2 Let the conditions (2.1){(2.2), (2.4), (2.6) hold. Furthermore, let 1 d 3 and u0 2 H 2( ). Then there exist constants K5(t), T > 0 depending on the data but not on a; b such that

k u(t)k2 L

2

K5(t) for all 0 t < T :

Remark 5.3 The constant K5(t) satis es limt!T K5 (t) = +1.
Proof. We start from the inequality (5.2) which holds for 1 d 4. For space dimensions d 2, we cannot use the H 1 norm for estimating the L1 norm of u as in the proof of Lemma 5.1. Instead we employ the Gagliardo-Nirenberg inequality

kukL1 G6 kukd=4kuk1?d=4; H L
2 2

(5.5)

which holds for d 3. Thus, using the inequality (2.8) we get

kukL1
where we have set

G6 C0d=4 (K0d=8 + k ukd=4)K01=2?d=8 L d=4 ); C6(1 + k ukL
2 2

C6 def G6C0d=4 K01=2?d=8 max(1; K0d=8 ): =
20

(5.6)

The above estimate together with the inequality (5.2) yields

@t k uk2 L

2

2 2Rk uk2 + 4?1jb0 + i jC6(1 + C6 ?1 (1 + k ukd=4 )2 ?1 ) L L d=4 )k uk2 (1 + k ukL L 2 ?1 (1 + k ukd=4 )2 ?1 )(1 + k ukd=2 )k uk : + C4(1 + C6 L L L
2 2 2 2 2 2 2 2

Since d < 4 the largest exponent of k ukL is 2 + d=2, and therefore, there exist two constants L1; L2 > 0 only depending on d; ; R; ?1; jb0 + i j; C4, and C6 such that @t k uk2 L1 + L2k uk2+ d=2 : L L From the Gronwall-type inequality of the appendix we conclude that
2 2

k u(t)k2 L
where

2

K5(t) for all 0 t < T ; dL t 4 2
)?4= d

K5(t) def =
and

(

k u0k2 + L
2

L1 4=(4+ d) ? d=4 ? L2
2

? L1 L
:

4=(4+ d)

2

L 4 T = dL k u0k2 + L1 L 2 2 This proves the lemma.
def

4=(4+ d) ? d=4

(5.7)

Proof of Theorem 2.3. The di erence w = u" ? u satis es the equation

6 Proof of Theorems 2.3 and 2.4

@t w = i w ? i (f (u") ? f (u)) + Rw + a u" ? bf (u"):
Multiplying this equation by w , integrating over , taking the real part and nally, using the formula (4.1) gives 1 @ kwk2 = 2 t L
2

Im

Z Z 1 Z0

w2 g0(jw j2)(w )2d dx + Rkwk2 L u"w dx ? bRe
Z

2

+ aRe

f (u")w dx;

where w = u" + (1 ? )u. Therefore, by assumption (2.3),

@t kwk2 2 L

2j

j?0

Z

(1 + max(1; 22 ?1)(ju"j2 + juj2 ))jwj2dx
Z

+ 2Rkwk2 2 + 2a L

u"w dx

21

+ 2b? (1 + ju"j2 )ju"jjwjdx 2j j?0(1 + max(1; 22 ?1 )(ku"k2 1 + kuk2 1 ))kwk2 L L L Z u"w dx + 2Rkwk2 + 2a L
2 2

Z

+ 2b?(1 + ku"k2 1 )ku"kL kwkL : L
2 2

get

Now, let d = 1. Then, using the inequality (5.4) and observing that G

p

(6.1)

2, we

ku"k2 1 L kuk2 1 L kwkL kwkH
we obtain from (6.1)
2 1

G2 (K0K3)1=2 G2 (K0K3)1=2

(4K0 K3)1=2 ; (4K0 K3)1=2 ;
1 2K0 =2 ; 1 2K3 =2 ;

where we also have used Lemma 3.1 and Corollary 3.5. Hence, since

ku"kL + kukL ku"kH + kukH
2 1

2 1

@t kwk2 L

2

2j j?0(1 + max(2; 22 )(4K0K3) =2 )kwk2 + 2Rkwk2 L L + 2ak@xu"kL k@xwkL + 4b?(1 + (4K0 K3) =2 )K0 C7kwk2 + a 4K3 + b 4?K0(1 + (4K0K3 )1=2 ); L
2 2 2 2

2

where we have set

C7 def 2R + 2j j?0(1 + max(2; 4 )(4K0K3) =2 ): =
Applying Gronwall's lemma, we conclude that

(6.2)
7

kw(t)k2 L

2

kw(0)k2 + a 4K3 eC T + b 4?K0 (1 + (4K0 K3) =2 )eC T ; L
2 7

which proves the theorem.
Proof of Theorem 2.4. We start from the inequality (6.1) which is valid for any d 1. In order to bound the L1 norm of u" and u we use the Gagliardo-Nirenberg inequality (5.5), the inequality (2.8) and Lemmas 5.1 and 5.2:

ku"kL1
where

G6ku"kd=4 ku"k1?d=4 H L d=4 (ku kd=4 + k u kd=4 )ku k1?d=4 G6C0 " L " L " L C8 ;
2 2 2 2 2

C8 def G6 C0d=4 K0(4?d)=8 (K0d=8 + max(K4; K5)d=8 ): =
22

(6.3)

Moreover,

Then the estimate (6.1) yields 2 2R + 2j j?0(1 + max(2; 4 )C8 )]kwk2 @t kwk2 L L 2 )K 1=2 kwk + 2ak u"kL kwkL + 2b?(1 + C8 0 L 2 + 2 a max(K ; K )1=2 + b?(1 + C 2 )K 1=2 ]kwk ; 2C9kwkL 4 5 L 8 0 where C9 def R + j j?0(1 + max(2; 4 )C82 ): = (6.4) Therefore, @t kwkL C9kwkL + a max(K4; K5)1=2 + b ?(1 + C82 )K01=2 ; and we obtain from Gronwall's lemma kw(0)kL + a max(K4; K5)1=2 eC t kw(t)kL 1 2 + b ?(1 + C8 )K0 =2 eC t ; for all 0 t < T (and 0 t T if d = 1). This nishes the proof.
2 2 2 2 2 2 2 2 2 2 2 9 9

kukL1 C8:

7 Proof of Theorem 2.5
First we prove the following uniform bound in H n( ) for the solution u to (1.1){ (1.2). Lemma 7.1 Under the assumptions of Theorem 2.5 there exist constants T > 0 and K6 (t) > 0 independent of a; b such that ku(t)kH n K6 (t) for all 0 t < T : Proof. Let D u denote the partial derivative of u according to the multi-index . Applying D with j j = n to the CGL equation (1.1), multiplying this equation by D u , taking the real part and integrating over yields

@t

XZ

j j=n

jD

uj2dx

= ?2a

X Z X

j j=n+1 j j=n

jD

uj2dx + 2R
Z

XZ

j j=n

jD uj2dx

? 2Re
Using the notations

(b + i )

D (juj2 u)D u dx:

rnu = (D uZ)j j=n; X jD uj2dx; krnuk2 = L
2

j j=n

23

we obtain

@t krnuk2 2 L

2Rkrnuk2 2 + 2jb0 + i L

j

Z

rn(juj2 u) rnu dx :

(7.1)

From 25, p. 183] we get the existence of a constant c(d) > 0 only depending on the space dimension, such that the last term of the above inequality is majorized by 2jb0 + i jc(d)krnuk2 kuk2 1 : L L
2

The L1 norm of u is estimated by using the Gagliardo-Nirenberg inequality

kukL1 G7kukH n kuk1? ; L
2

(7.2)

with = d=2n < 1, so that we conclude from (7.1)

@t krnuk2 L

2

2Rkrnuk2 + 2G2 c(d)jb0 + i jK0 (1? ) kuk2 n krnuk2 : 7 H L L
2 2 2 2

e There exists a constant C0 > 0 such that e kukH n C0(kukL + krnukL ): Hence, introducing y(t) = krnu(t)k2 , this gives L
2

@t y (2R + C10 K0 )y + C10y1+ ;
where
e C10 def 2C72 c(d)jb0 + i jK0 (1? ) C02 max(1; 22 ?1): = Employing Holder's inequality, there exist two constants D1, D2 > 0 such that

@t y D1 + D2 y1+ :
Lemma 8.1 of the Appendix implies that for 0 t < T ,

D1 y(t) K6(t) = y(0) + D2 where T > 0 is given by
def

1=(1+ ) ?

? D2 t
1=(1+ )

?1=

D ? D1
2

1=(1+ )

;

T =
The lemma is proved.

def

D D2 y(0) + D1 2

?1

:

Proof of Theorem 2.5. The proof is very similar to the proof of Theorem 2.4. We only have to use the Gagliardo-Nirenberg inequality (7.2) instead of (5.5) and to apply Lemma 7.1 in order to obtain a uniform L1 bound on u".

24

8 Appendix

Lemma 8.1 Let the function y : (0; T ) ! R be absolutely continuous and satisfy
the following di erential inequality:

y0 Ay + B
"

for t > 0;

with > 1, A; B > 0 and y(0) > 0. Then there exists a constant T > 0 such that for all 0 t < T ,

y(t)

y(0) + B A

1=

1?

+ (1 ? )At

#?1=( ?1)

? B A
:

1=

;

where T > 0 is given by

T = ( ? 1)A y(0) + (B=A)1=
(y + C ) the di erential inequality This inequality can be integrated:
?

?1 ?1

Proof. Via the change of variable z = y + C , where C = (B=A)1= , we get from

y +C
0:
?1=( ?1)

z0 ? Az

z(t)

z(0)1? + (1 ? )At

;

for all 0 t < T . The lemma follows.

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L L ;T B

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