Electronic Journal of Differential Equations, Vol. 2009(2009), No. 82, pp. 1–9.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
ON THE CAUCHY-PROBLEM FOR GENERALIZED
KADOMTSEV-PETVIASHVILI-II EQUATIONS
AXEL GRÜNROCK
Abstract. The Cauchy-problem for the generalized Kadomtsev-PetviashviliII equation
ut + uxxx + ∂x−1 uyy = (ul )x , l ≥ 3,
is shown to be locally well-posed in almost critical anisotropic Sobolev spaces.
The proof combines local smoothing and maximal function estimates as well as
bilinear refinements of Strichartz type inequalities via multilinear interpolation
in Xs,b -spaces.
1. Introduction
Inspired by the work of Kenig and Ziesler [13, 14], we consider the Cauchy
problem
u(x, y, 0) = u0 (x, y), (x, y) ∈ R2
(1.1)
for the generalized Kadomtsev-Petviashvili-II equation (for short: gKP-II)
ut + uxxx + ∂x−1 uyy = (ul )x ,
(1.2)
where l ≥ 3 is an integer. Concerning earlier results on related problems for this
equation we refer to the works of Saut [15], Iório and Nunes [8], and Hayashi,
Naumkin, and Saut [7].
For the Cauchy data we shall assume u0 ∈ H (s) , where for s = (s1 , s2 , ε) the
Sobolev type space H (s) is defined by its norm in the following way: Let ξ :=
(k, η) ∈ R2 denote the Fourier variables corresponding to (x, y) ∈ R2 and hDx iσ1 =
F −1 hkiσ1 F, hDy iσ2 = F −1 hηiσ2 F, as well as hDx−1 Dy iσ3 = F −1 hk −1 ηiσ3 F, where
σ
F denotes the Fourier transform and hxiσ = (1 + x2 ) 2 . Setting
ku0 kσ1 ,σ2 ,σ3 := khDx iσ1 hDy iσ2 hDx−1 Dy iσ3 u0 kL2xy
we define
ku0 kH (s) := ku0 ks1 +2s2 +ε,0,0 + ku0 ks1 ,s2 ,ε .
Almost the same data spaces are considered in [13], [14], the only new element here
is the additional parameter s2 , which will play a major role only for powers l ≥ 4.
Using the contraction mapping principle we will prove a local well-posedness
result for (1.1), (1.2) with regularity assumptions on the data as weak as possible.
2000 Mathematics Subject Classification. 35Q53.
Key words and phrases. Cauchy-problem; local well-posedness; generalized KP-II equations.
c
2009
Texas State University - San Marcos.
Submitted April 9, 2009. Published July 10, 2009.
Supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 611.
1
2
A. GRÜNROCK
EJDE-2009/82
Two types of estimates will be be involved in the proof: On the one hand there is
the combination of local smoothing effect ([13], see also [10]) and maximal function
estimate (proven in [13, 14]), which has been used in [13], a strategy that had
been developed in [11] in the context of generalized KdV equations. On the other
hand we rely on Strichartz type estimates (cf. [15]) and especially on the bilinear
refinement thereof taken from [5, 6], see also [9, 16, 18, 17]. A similar bilinear
estimate involving y-derivatives (to be proven below) will serve to deal with the
hDy i containing part of the norm. To make these two elements meet, we use
Bourgain’s Fourier restriction norm method [1], especially the following function
spaces of Xs,b -type. The basic space X0,b is given as usual by the norm
kukX0,b := khτ − φ(ξ)ib FukL2ξτ ,
2
where φ(ξ) = k 3 − ηk is the phase function of the linearized KP-II equation. According to the data spaces chosen above we shall also use
kukXσ1 ,σ2 ,σ3 ;b := khDx iσ1 hDy iσ2 hDx−1 Dy iσ3 ukX0,b
as well as
kukX(s),b := kukXs1 +2s2 +ε,0,0;b + kukXs1 ,s2 ,ε;b .
Finally the time restriction norm spaces X(s),b (δ) constructed in the usual manner
will become our solution spaces. Now we can state the main result of this note.
Theorem 1.1. Let s1 > 1/2, s2 ≥
l−3
2(l−1)
and 0 < ε ≤ min (s1 , 1). Then for
s = (s1 , s2 , ε) and u0 ∈ H
there exist δ = δ(ku0 kH (s) ) > 0 and b > 21 such that
there is a unique solution u ∈ X(s),b (δ) of (1.1), (1.2). This solution is persistent
and the flow map S : u0 7→ u, H (s) → X(s),b (δ) is locally Lipschitz.
(s)
The lower bounds on s1 , s2 , and ε are optimal (except for endpoints) in the
sense that scaling considerations strongly suggest the necessity of the condition
1
2
s1 + 2s2 + ε ≥ 12 + l−3
l−1 . Moreover, for l = 3 C -illposedness is known for s1 < 2
or ε < 0, see [13, Theorem 4.1]. The affirmative result in [13] concerning the cubic
gKP-II equation, that was local well-posedness for s1 > 34 , s2 = 0, and ε > 21
is improved here by 43 derivatives. Some effort was made to keep the number of
y-derivatives as small as possible1, but we shall not attempt to give evidence for
the necessity of the individual lower bounds on s1 , s2 , and ε, respectively.
By the general arguments concerning the Fourier restriction norm method introduced in [1] and further developed in [2, 12], matters reduce to proving the following
multilinear estimate.
l−3
Theorem 1.2. Let s1 > 21 , s2 ≥ 2(l−1)
and 0 < ε ≤ min (s1 , 1). Then there exists
1
1
0
b > − 2 , such that for all b > 2 and all u1 , . . . , ul ∈ X(s),b supported in {|t| ≤ 1}
the estimate
l
l
Y
Y
k∂x
uj kX(s),b0 .
kuj kX(s),b
j=1
j=1
holds.
1For l ≥ 5 the use of local smoothing effect and maximal function estimate alone yields LWP
l−3
for s1 > l−1
, s2 = 0, and ε > 21 , which is optimal from the scaling point of view, too. This
result may be seen as essentially contained in [13, Theorem 2.1 and Lemma 3.2] plus [11, proofs
of Theorem 2.10 and Theorem 2.17]. This variant always requires 12 + derivatives in y.
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CAUCHY-PROBLEM FOR GKP-II EQUATIONS
3
To prepare for the proof of Theorem 1.2 let us first recall those estimates for
free solutions W (t)u0 of the linearized KP-II equation, which we take over from the
literature. First we have the local smoothing estimate from [13, Lemma 3.2]: For
0 ≤ λ ≤ 1,
2 . ku0 kL2
(1.3)
kDx(1−λ) (Dx−1 Dy )λ W (t)u0 kL∞
xy
x Lyt
and hence by the transfer principle [2, Lemma 2.3] for b > 21 ,
2 . kukX
kDx(1−λ) (Dx−1 Dy )λ ukL∞
.
0,b
x Lyt
(1.4)
Interpolation with the trivial case L2xyt = X0,0 and duality give for 0 ≤ θ ≤ 1,
1
1−θ
pθ = 2 ,
k[Dx(1−λ) (Dx−1 Dy )λ ]θ ukLpxθ L2yt . kukX0,θb
(1.5)
as well as
k[Dx(1−λ) (Dx−1 Dy )λ ]θ ukX0,−θb . kuk
p0
Lxθ L2yt
.
(1.6)
(For Hölder exponents p we will always have p1 + p10 = 1.) To complement the local
smoothing effect, we shall use the maximal function estimate
1
3
kW (t)u0 kL4x L∞
. khDx i 4 + hDx−1 Dy i 2 + u0 kL2xy
yT
(1.7)
due to Kenig and Ziesler [13, Theorem 2.1], which is probably the hardest part of
the whole story. The capital T here indicates, that this estimate is only valid local
in time, the +-signs at the exponents on the right denote positive numbers, which
can be made arbitrarily small at the cost of the implicit constant but independent
of other parameters. (This notation will be used repeatedly below.) The transfer
principle implies for u supported in {|t| ≤ 1}
3
1
kukL4x L∞
. khDx i 4 + hDx−1 Dy i 2 + ukX0,b ,
yt
(1.8)
where b > 21 . The Strichartz type estimate
kW (t)u0 kL4xyt . ku0 kL2xy ,
(1.9)
taken from [15, Proposition 2.3], becomes
kukL4xyt . kukX0,b ,
b>
1
.
2
(1.10)
For its bilinear refinement
−1
kuvkL2xyt . kDx1/2 ukX0,b kDx 2 vkX0,b ,
b>
1
,
2
(1.11)
and the dualized version thereof
1
,
(1.12)
2
we refer to [6, Theorem 3.3 and Proposition 3.5]. In order to estimate the Xs1 ,s2 ,ε;b0 norm of the nonlinearity the following bilinear estimate involving y-derivatives will
be useful. We introduce the bilinear pseudodifferential operator M (u, v) in terms
of its Fourier transform
Z
M\
(u, v)(ξ) :=
|k1 η − kη1 |1/2 u
b(ξ1 )b
v (ξ2 )dξ1
kDx1/2 (uv)kX0,−b . kDx1/2 ukX0,b kvkL2xyt ,
ξ=ξ1 +ξ2
b>
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A. GRÜNROCK
EJDE-2009/82
b px Lqyt by kf k bp q := kFx f k p0 q , where Fx denotes
and define the auxiliary space L
Lx L
L L
x
yt
yt
the partial Fourier transform with respect to the first space variable x only. Then
we have:
Lemma 1.3.
kM (W (t)u0 , W (t)v0 )kLb1 L2 . kDx1/2 u0 kL2xy kDx1/2 v0 kL2xy .
x
Proof.
2
(1.13)
yt
We have
\
M (W (t)u
0 , W (t)v0 )(ξ, τ )
Z
η2
η2
=
|k1 η − kη1 |1/2 δ(τ − k13 − k23 + 1 + 2 )c
u0 (ξ1 )vb0 (ξ2 )dξ1 .
k1
k2
ξ=ξ1 +ξ2
η2
η2
2
Because of k11 + k22 = ηk + k1kk2 (η1 − kk1 η)2 and with a = τ − k13 − k23 +
as g(η1 ) = k1kk2 (η1 − kk1 η)2 − a this equals
Z
|k1 η − kη1 |1/2 δ(g(η1 ))c
u0 (ξ1 )vb0 (ξ2 )dη1 dk1 .
η2
k
as well
ξ=ξ1 +ξ2
The zero’s of g are η1± =
k1
k η
±
q
k1 k 2 a
k
and for the derivative we have |g 0 (η1 )| =
2
|k1 k2 | |k1 η
− kη1 |. So we get the two contributions
Z
1
|k1 k2 |
u
c0 (k1 , η1± )vb0 (k2 , η − η1± )dk1 .
I ± (ξ, τ ) =
2 k1 +k2 =k |k1 η − kη1± |1/2
By Minkowski’s integral inequality
Z
1
±
kI (ξ, ·)kL2τ .
|k1 k2 |k|k1 η − kη1± |− 2 u
c0 (k1 , η1± )vb0 (k2 , η − η1± )kL2τ dk1 ,
k1 +k2 =k
where, with λ :=
Z
q
k1 k 2 a
k ,
the square of the last L2τ -norm equals
k1
k2
|k1 η − kη1± |−1 |c
u0 (k1 , η ± λ)vb0 (k2 , η ∓ λ)|2 dτ
k
k
Z
2
k1
k2
≤
|c
u0 (k1 , η ± λ)vb0 (k2 , η ∓ λ)|2 dλ,
|k1 k2 |
k
k
since dτ = 2 |kλk
dλ = ∓ k12k2 (k1 η − kη1± )dλ. This gives
1 k2 |
Z
Z
1/2
k1
k2
±
1/2
kI (ξ, ·)kL2τ .
|k1 k2 |
|c
u0 (k1 , η ± λ)vb0 (k2 , η ∓ λ)|2 dλ
dk1 .
k
k
k1 +k2 =k
2It is not apparent from the proof, but there is a very simple idea behind Lemma 1.3. If we take
t
3
2
the partial Fourier transform Fx W (t)u0 (k) = eitk ei k ∂y Fx u0 (k) with respect to the x-variable
only, we obtain a free solution of the linear Schrödinger equation with rescaled time variable s = kt ,
multiplied by a phase factor of size one. So any space time estimate for the Schrödinger equation
should give a corresponding estimate for linearized KP-type equations. This idea was exploitet in
[3], [4] to obtain suitable substitutes for Strichartz type estimates for semiperiodic and periodic
problems. From this point of view Lemma 1.3 corresponds to the one-dimensional estimate
1/2
kDy
2
2
(eit∂y u0 e−it∂y v0 )kL2 . ku0 kL2 kv0 kL2 .
yt
y
y
EJDE-2009/82
CAUCHY-PROBLEM FOR GKP-II EQUATIONS
5
Using Parseval and again Minkowski’s inequality for the L2η -norm we arrive at
kFx M (W (t)u0 , W (t)v0 )(k)kL2yt
Z
Z
1/2
k1
k2
.
|c
u0 (k1 , η ± λ)vb0 (k2 , η ∓ λ)|2 dλdη
|k1 k2 |1/2
dk1
k
k
k1 +k2 =k
Z
=
|k1 k2 |1/2 kc
u0 (k1 , ·)kL2η kvb0 (k2 , ·)kL2η dk1
.
k1 +k2 =k
kDx1/2 u0 kL2xy kDx1/2 v0 kL2xy
by Cauchy-Schwarz and a second application of Parseval’s identity.
Corollary 1.4. Let b > 1/2. Then
kM (u, v)kLb1 L2 . kDx1/2 ukX0,b kDx1/2 vkX0,b
x
(1.14)
yt
− 21
kDx M (u, v)kX0,−b . kukLb∞ L2 kDx1/2 vkX0,b .
x
(1.15)
yt
Proof. Lemma 1.3 implies (1.14) via the transfer principle, (1.15) is then obtained
by duality. In fact, if we fix v and consider the linear map Mv (u) := M (u, v), then
its adjoint is given by Mv∗ = Mv , and we have kvkX0,b = kvkX0,b .
Now we are prepared for the proof of the central multilinear estimate.
Proof of Theorem 1.2.
1. We use (1.6) with λ = 0, θ =
kDx1/2
l
Y
uj kX0,b0 . k
j=1
1
2
and Hölder to obtain for b0 < − 41
l
Y
j=1
uj k
4
Lx3 L2yt
. ku1 kL4x L2yt ku2 kL4x L∞
ku3 kL4x L∞
yt
yt
Y
kuj kL∞
.
xyt
j≥4
For the first factor we have by (1.5), again with λ = 0, θ = 12 ,
−1
ku1 kL4x L2yt . kDx 2 u1 kX0, 1 + ,
4
while the second and third factor are estimated by (1.8)
1
3
ku2,3 kL4x L∞
. khDx i 4 + hDx−1 Dy i 2 + u2,3 kX0,b ,
yt
where b > 12 . It is here that the time support assumption is needed. For j ≥ 4 we
use Sobolev embeddings in all variables to obtain for b > 21
1
1
kuj kL∞
. khDx i 2 + hDy i 2 + uj kX0,b .
xyt
Summarizing we have for b0 < −1/4, b > 1/2,
kDx1/2
l
Y
−1
3
1
uj kX0,b0 . kDx 2 u1 kX0,b khDx i 4 + hDx−1 Dy i 2 + u2 kX0,b
j=1
3
1
× khDx i 4 + hDx−1 Dy i 2 + u3 kX0,b
Y
1
1
khDx i 2 + hDy i 2 + uj kX0,b .
j≥4
(1.16)
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A. GRÜNROCK
EJDE-2009/82
2. Combining the dual version (1.12) of the bilinear estimate with Hölder’s
inequality, (1.11) and Sobolev embeddings we obtain for b1 < −1/2, b > 1/2,
kDx1/2
l
Y
uj kX0,b1
j=1
. kDx1/2 u3 kX0,b ku1 u2 kL2xyt
Y
kuj kL∞
xyt
(1.17)
j≥4
−1
. kDx 2 u1 kX0,b kDx1/2 u2 kX0,b kDx1/2 u3 kX0,b
Y
1
1
khDx i 2 + hDy i 2 + uj kX0,b .
j≥4
3. Bilinear interpolation involving u2 and u3 gives
kDx1/2
l
Y
−1
1
θ
θ
uj kX0,b0 . kDx 2 u1 kX0,b khDx i 2 + 4 + hDx−1 Dy i 2 + u2 kX0,b
j=1
1
θ
θ
× khDx i 2 + 4 + hDx−1 Dy i 2 + u3 kX0,b
Y
1
1
khDx i 2 + hDy i 2 + uj kX0,b ,
j≥4
where 0 < θ 1 and b0 = θb0 + (1 − θ)b1 . Now symmetrization via (l − 1)-linear
interpolation among u2 , . . . , ul yields
kDx1/2
l
Y
−1
uj kX0,b0 . kDx 2 u1 kX0,b
j=1
Y
khDx iα1 + hDy iα2 + hDx−1 Dy iα3 + uj kX0,b ,
j≥2
(1.18)
θ
l−3
θ
with b, b0 as before, α1 = 12 + 2(l−1)
, α2 = 2(l−1)
, and α3 = l−1
. Using hηi ≤
hkihk −1 ηi, we may replace α2 + by α2 in (1.18). Now, for given s1 > 12 and ε > 0
we choose θ close enough to zero, so that α1 < s1 and α3 < ε, and b0 (respectively
b1 ) close enough to − 14 (respectively to − 12 ), so that b0 > − 12 . Then, assuming by
symmetry that u1 has the largest frequency with respect to the x-variable (i. e.
|k1 | ≥ |k2,...,l |), we obtain
k∂x
l
Y
uj kXs1 +2s2 +ε,0,0;b0 . ku1 kXs1 +2s2 +ε,0,0;b
j=1
Y
kuj kXs1 ,s2 ,ε;b .
(1.19)
j≥2
Ql
4. The same upper bound holds for k∂x j=1 uj kXs1 ,s2 ,ε;b0 , if |η| ≤ |k| (or even
if |η| ≤ |k|2 ), where |k| (respectively |η|) are the frequencies in x (respectively in y)
of the whole product. In the case where |k| ≤ 1 - assuming ε ≤ min (s1 , 1) and b0
sufficiently close to − 21 , the estimate
k∂x
l
Y
j=1
uj kXs1 ,s2 ,ε;b0 .
l
Y
kuj kXs1 ,s2 ,ε;b
(1.20)
j=1
is easily derived by a combination of the standard Strichartz type estimate (1.10)
and Sobolev embeddings. So we may henceforth assume |k| ≥ 1, |η| ≥ 1, and
|k −1 η| ≥ 1.
Ql
One last simple observation concerning the estimation of k∂x j=1 uj kXs1 ,s2 ,ε;b0 :
If we assume in addition to |k1 | ≥ |k2,...,l | that u1 has a large frequency with respect
to y; i.e., |η| . |η1 |, then from (1.18) we also obtain (1.20).
EJDE-2009/82
CAUCHY-PROBLEM FOR GKP-II EQUATIONS
7
Ql
5. It remains to estimate k∂x j=1 uj kXs1 ,s2 ,ε;b0 in the case where |k1 | ≥ |k2,...,l |
(symmetry assumption as before) and |η1 | |η|. By symmetry among u2,...,l we
may assume in addition that |η2 | ≥ |η1,3,...,l | and hence that |k2 | |k|, because otherwise previous arguments apply with u1 and u2 interchanged. For this distribution
of frequencies the symbol of the Fourier multiplier M (u1 u3 · · · ul , u2 ) becomes
|kη2 − k2 η|1/2 ∼ |kη2 |1/2 & |kη|1/2 .
Now let P (u1 , . . . , ul ) denote the projection in Fourier space on {|η2 | ≥ |η1,3,...,l |} ∩
{hk2 i |k| . k1 }. Then by (1.15) we obtain for b > 12 , b1 < − 21
kDy1/2 P (u1 , . . . , ul )kX0,b1
. kDx1/2 u2 kX0,b ku1 u3 · · · ul kLb∞ L2
x
.
yt
1
kDx1/2 u2 kX0,b khDx i 2 + (u1 u3 )kL2xyt
Y
kuj kLb∞ L∞
x
yt
j≥4
1
1
. kDx0+ u1 kX0,b khDx i 2 + u2 kX0,b khDx i 2 + u3 kX0,b
Y
1
1
khDx i 2 + hDy i 2 + uj kX0,b ,
j≥4
where besides Sobolev type inequalities we have used (1.11) in the last step. Interpolation with (1.17) gives for 0 ≤ λ ≤ 1
λ
1−λ
kDx2 Dy 2 P (u1 , . . . , ul )kX0,b1
−λ
2+
. kDx
1
1
u1 kX0,b khDx i 2 + u2 kX0,b khDx i 2 + u3 kX0,b
Y
1
1
khDx i 2 + hDy i 2 + uj kX0,b .
j≥4
Yet another interpolation, now with (1.16), gives for 0 < θ 1, b0 = θb0 + (1 −
θ)b1 , sx = 12 (λ(1 − θ) + θ) and sy = 12 (1 − θ)(1 − λ)
kDxsx Dysy P (u1 , . . . , ul )kX0,b0
1
θ
θ
1
θ
θ
. kDx−sx + u1 kX0,b khDx i 2 + 4 + hDx−1 Dy i 2 + u2 kX0,b khDx i 2 + 4 + hDx−1 Dy i 2 + u3 kX0,b
Y
1
1
×
khDx i 2 + hDy i 2 + uj kX0,b .
j≥4
The next step is to equidistribute the hDy i’s on u2 , . . . , ul . Here we must be careful,
because the symmetry between u2 and u3 , . . . , ul was broken. But since u2 has the
l−3
largest y-frequency we may first shift a hDy i 2(l−1) + onto u2 and then interpolate
among u3 , . . . , ul in order to obtain
kDxsx Dysy P (u1 , . . . , ul )kX0,b0
Y
. kDx−sx + u1 kX0,b
khDx iβ1 + hDy iβ2 + hDx−1 Dy iβ3 + uj kX0,b ,
(1.21)
j≥2
l−3
where β1 = 12 + θ4 , β2 = 2(l−1)
and β3 = θ2 . Again we may replace β2 + by β2 . Now
(1.21) is applied to hDx is1 hDy is2 hDx−1 Dy iε ∂x P (u1 , . . . , ul ), where we can shift the
hDx is1 partly from the product to u1 and the hDy is2 partly to u2 . Moreover, since
|k2 | . |k| and |η| . |η2 | we have |k −1 η| . |k2−1 η2 |, so that a hDx−1 Dy iε−θ may be
8
A. GRÜNROCK
EJDE-2009/82
thrown from the product onto u2 . The result is
k∂x P (u1 , . . . , ul )kXs1 ,s2 ,ε;b0
. kDxs1 +1−θ−2sx + u1 kX0,b khDx iβ1 + hDy iβ2 +s2 −sy +θ hDx−1 Dy iε−θ+β3 + u2 kX0,b
Y
×
khDx iβ1 + hDy iβ2 hDx−1 Dy iβ3 + uj kX0,b .
j≥3
2
, s1 − 12 ) and λ such that sy = β2 +θ
Here β2 ≤ s2 and by choosing θ < min (ε, 3(l−1)
we can achieve that
• s1 + 1 − θ − 2sx < s1 + 2s2 + ε,
• β2 + s2 − sy + θ ≤ s2 ,
• ε − θ + β3 < ε,
as well as β1 < s1 , β3 < ε. This gives
Y
k∂x P (u1 , . . . , ul )kXs1 ,s2 ,ε;b0 . ku1 kXs1 +2s2 +ε,0,0;b
kuj kXs1 ,s2 ,ε;b
j≥2
as desired.
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[3] Grünrock, A., Panthee, M., Silva, J.: On KP-II type equations on cylinders. Ann. I. H.
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EJDE-2009/82
CAUCHY-PROBLEM FOR GKP-II EQUATIONS
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[18] Tzvetkov, N.: Global low regularity solutions for Kadomtsev-Petviashvili equation, Differential Integral Equations, 13 (10-12), 1289-1320, 2000
Axel Grünrock
Rheinische Friedrich-Wilhelms-Universität Bonn, Mathematisches Institut
Beringstraße 1, 53115 Bonn, Germany
E-mail address: gruenroc@math.uni-bonn.de