Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 170, pp. 1–34.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
GLOBAL SOLUTIONS FOR 3D NONLOCAL GROSS-PITAEVSKII
EQUATIONS WITH ROUGH DATA
HARTMUT PECHER
Abstract. We study the Cauchy problem for the Gross-Pitaevskii equation
with a nonlocal interaction potential of Hartree type in three space dimensions. If the potential is even and positive definite or a positive function and
its Fourier transform decays sufficiently rapidly the problem is shown to be
globally well-posed for large rough data which not necessarily have finite energy and also in a situation where the energy functional is not positive definite.
The proof uses a suitable modification of the I-method.
1. Introduction and main results
We consider the Cauchy problem for the Gross-Pitaevskii equation with nonlocal
nonlinearity in three space dimensions:
∂v
− ∆v = v(W ∗ (1 − |v|2 ))
(1.1)
i
∂t
v(x, 0) = v0 (x) ,
(1.2)
under the condition
v → 1 as |x| → +∞ ,
where v : R1+3 → C.
More generally one could also consider the condition
|v| → 1
as |x| → +∞ ,
(1.3)
(1.4)
but for simplicity we restrict ourselves to (1.3). This problem was introduced by
Gross [9] and Pitaevskii [19] for modeling the kinetic of a weakly interacting Bose
gas. Here W describes the interaction between bosons. The original equation reads
as follows
Z
~2
∂ψ
i
(x, t) +
∆ψ(x, t) = ψ(x, t)
V (x − y)|ψ(y, t)|2 dy
∂t
2m
n
R
and is equivalent modulo normalizations to equation (1.1), provided W ∗ 1 is welldefined and positive, which in the cases we consider (under the assumptions (A1)
and either (A2) or (A3) below) is obviously true. For a detailed derivation of our
problem from the original Gross-Pitaevskii form we refer to [17].
2000 Mathematics Subject Classification. 35Q55, 35B60, 37L50.
Key words and phrases. Gross-Pitaevskii equation; global well-posedness;
Fourier restriction norm method.
c
2012
Texas State University - San Marcos.
Submitted July 6, 2012. Published October 4, 2012.
1
2
H. PECHER
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The most studied case is W = δ (= Dirac function), which occurs in theoretical physics e.g. in superfluidity, nonlinear optics and Bose-Einstein condensation
[5],[12],[13],[16]. For further references we also refer to the introduction of [17].
Using the energy conservation law which in the case W = δ is
Z
1
E(v(t)) = (|∇v(x, t)|2 + (|v(x, t)|2 − 1)2 )dx = E(v0 ) ,
2
it was shown by Bethuel and Saut [2], Appendix A, that the problem is globally
well-posed for data of the form v0 ∈ 1 + H 1 (R3 ). Gérard [7] proved the same
result for data in the larger energy space in two and three space dimensions. Gallo
[6] generalized these results to a class of local nonlinearities for data with finite
energy and space dimension n ≤ 4. Global well-posedness for the Gross-Pitaevskii
equation in the case n = 4 in the (critical) energy space was proven by Killip, Oh,
Pocopnicu and Visan [15], a case which was not considered in Gallo’s paper. The
author [18] showed that global well-posedness holds true even for data with less
regularity, namely v0 = 1 + u0 , where u0 ∈ H s (R3 ) for 5/6 < s < 1. To prove
this result one uses Bourgain type spaces and the so-called I-method (or method
of almost conservation laws), which was introduced by Colliander, Keel, Staffilani,
Takaoka and Tao [4] and successfully applied to various problems.
We now want to study the problem for two types of nonlocal nonlinearities.
Nonlocal nonlinearities were as mentioned above already introduced by Gross and
Pitaevskii. In the case of three space dimensions Shchesnovich and Kraenkel [21]
c
consider W (x) = 4π12 |x| exp(− |x|
) for > 0 with Fourier transform W (ξ) =
1
1+2 |ξ|2 . The case W = χ{|x|≤a} (χA = characteristic function of the set A) was used
in the study of supersolids [1, 11, 20]. These examples are included in the class of
nonlocal nonlinearities with suitable mapping properties and positivity conditions
on W considered by de Laire [17] such that the Cauchy problem (1.1),(1.2),(1.4) is
globally well-posed in the space φ + H 1 (Rn ), where φ has finite energy and fulfills
suitable boundedness assumptions, in particular |φ(x)| → 1 as |x| → ∞.
Our aim is to give similar results for less regular data. From now on we consider
the case of three space dimensions and make the following assumptinos:
General Assumption on W :
c (ξ)| . hξi−2 for all ξ ∈ R3 and
(A1) W : R3 → R is even, W ∈ L1 (R3 ), |W
either
c (ξ) ≥ 0 for all ξ ∈ R3 or
(A2) W
(A3) W (x) ≥ 0 ll x ∈ R3 .
c is real-valued and even, if W has the same properties.
Let us remark, that W
We have especially the following two examples in mind, which we mentioned
above:
1
c (ξ) = 1 2 , so that (A1),(A2) and (A3)
Case A: W (x) = 4π|x|
e−|x| . We have W
1+|ξ|
are satisfied.
c (ξ) =
Case B: W = χ{|x|≤a} . Obviously (A3) is satisfied. We also have W
− 23
− 23 3
a |ξ| J 2 (2πa|ξ|), where J 23 is the Bessel function of the first kind of order 32 ,
which has the properties J 32 (|ξ|) ∼ |ξ|3/2 as |ξ| 1 and J 23 (|ξ|) . |ξ|11/2 as |ξ| 1.
Thus (A1) is also satisfied.
For simplicity we assume φ ≡ 1 and consider (1.1),(1.2),(1.3). As usual we
transform the problem (1.1),(1.2),(1.3) by setting u = v − 1 into the equivalent
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NONLOCAL GROSS-PITAEVSKII EQUATION
3
form
∂u
− ∆u + F (u) = 0
∂t
u(x, 0) = u0 (x)
(1.6)
F (u) = (1 + u)(W ∗ (|u|2 + 2 Re u))
(1.7)
i
(1.5)
where
under the condition
u → 0 as |x| → +∞ .
Assuming W to be real-valued and even the conserved energy is given by
Z
Z
1
2
(W ∗ (|u|2 + 2 Re u))(|u|2 + 2 Re u)dx .
E(u(t)) = |∇u(t)| dx +
2
(1.8)
(1.9)
We remark that no L2 -conservation law holds.
Under our hypothesis on W de Laire’s results [17] especially imply that the
Cauchy problem (1.5),(1.6),(1.7),(1.8) is globally well-posed in C 0 (R, H 1 (R3 )) for
data u0 ∈ H 1 (R3 ). We now show that this problem for data u0 ∈ H s (R3 ) is
globally well-posed in C 0 (R, H s (R3 )), i.e. (1.1),(1.2),(1.3) for v0 ∈ 1 + H s (R3 ), if
1/2 < s < 1 by application of the I-method. As usual the energy conservation
law is not directly applicable for H s -data with s < 1. However there is an almost
conservation law for the modified energy E(Iu), which is well defined for u ∈ H s
(see the definition of I below). If we assume (A1) and (A2), this leads to an apriori bound of k∇Iu(t)kL2 , if s is close enough to 1, namely s > 1/2, because the
energy functional is positive definite, a property which is usually assumed when
the I-method is applied. If we assume (A1) and (A3) however it is not obvious
that the H 1 -norm of the solution can be controlled by the energy, because it is not
definite. Nevertheless it is possible to modify the I-method in this case suitably, but
the argument to get the required bound for k∇Iu(t)kL2 is more involved. Once a
bound for k∇Iu(t)kL2 is achieved we can also deduce an a-priori bound for ku(t)kL2 ,
which together gives an a-priori bound for ku(t)kH s .
The main results (cf. the definition of the X s,b -spaces below) are summarized in
the following three theorems:
Theorem 1.1 (Unconditional uniqueness). Assume (A1) and moreover (A2) or
(A3), u0 ∈ L2 (R3 ). The Cauchy problem (1.5),(1.6),(1.7) has at most one solution
u ∈ C 0 ([0, T ], L2 (R3 )) for any T > 0.
Theorem 1.2 (Local well-posedness). Assume (A1) and moreover (A2) or (A3),
s ≥ 0 and u0 ∈ H s (R3 ). Then the Cauchy problem (1.5),(1.6),(1.7) has a unique
−
1
4
−
local solution u ∈ X s, 2 + [0, δ], where δ can be chosen as δ ∼ ku0 kH s2s+1 . This
solution belongs to C 0 ([0, δ], H s (R3 )) and is also unique in this space.
Theorem 1.3 (Global well-posedness). Assume (A1) and moreover (A2) or (A3),
T > 0, s > 1/2 and u0 ∈ H s (R3 ). Then the Cauchy problem (1.5),(1.6),(1.7) has a
1
unique global solution u ∈ X s, 2 + [0, T ]. This solution belongs to C 0 ([0, T ], H s (R3 ))
and is also unique in this space.
We use the following notation and well-known facts: the multiplier I = IN is for
given s < 1 and N ≥ 1 defined by
b
Id
N f (ξ) := mN (ξ)f (ξ) ,
4
H. PECHER
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where b denotes the Fourier transform with respect to the space variables. Here
mN (ξ) is a smooth, radially symmetric, nonincreasing function of |ξ| with
(
1
|ξ| ≤ N
mN (ξ) =
N 1−s
)
|ξ| ≥ 2N.
( |ξ|
We remark that I : H s → H 1 is a smoothing operator, so that especially E(Iu) is
well-defined for u ∈ H s (R3 ). This follows from W ∈ L1 , Young’s inequality and
Sobolev’s embedding H 1 (R3 ) ⊂ L4 (R3 ).
We use the Bourgain type function space X m,b belonging to the Schrödinger
equation iut − ∆u = 0, which is defined as follows: let b or F denote the Fourier
transform with respect to space and time and F −1 its inverse. X m,b is the completion of S(R × R3 ) with respect to
kf kX m,b = khξim hτ ib F(e−it∆ f (x, t))kL2ξ,τ = khξim hτ + |ξ|2 ib fb(ξ, τ )kL2ξ,τ ,
For a given time interval I we define
kf kX m,b (I) := inf kgkX m,b .
g|I =f
We recall the following facts about the solutions u of the inhomogeneous linear
Schrödinger equation (see e.g. [8])
iut − ∆u = F ,
0
u(0) = f .
(1.10)
0
For b + 1 ≥ b ≥ 0 ≥ b > −1/2 and T ≤ 1 we have
0
kukX m,b [0,T ] . kf kH m + T 1+b −b kF kX m,b0 [0,T ] .
For 1/2 > b > b0 ≥ 0 or 0 ≥ b > b0 > −1/2:
0
kf kX m,b0 [0,T ] . T b−b kf kX m,b [0,T ]
(see e.g. [10, Lemma 1.10]).
Fundamental are the following Strichartz type estimates for the solution u of
(1.10) in three space dimensions (see [3, 14]):
kukLq (I,Lr (R3 )) . kf kL2 (R3 ) + kF kLq̃0 (I,Lr̃0 (R3 ))
with implicit constant independent of the interval I ⊂ R for all pairs (q, r), (q̃, r̃)
3
3
= 34 , 1q̃ + 2r̃
= 43 , where 1q̃ + q̃10 = 1 and 1r̃ + r̃10 = 1.
with q, r, q̃, r̃ ≥ 2 and 1q + 2r
This implies
kψkLq (I,Lr (R3 )) . kψk 0, 12 + .
X
(I)
For real numbers a we denote by a+, a + +, a− and a − − the numbers a + ,
a + 2, a − and a − 2, respectively, where > 0 is sufficiently small. We also use
the notation hxi := (1 + |x|2 )1/2 for x ∈ R3 .
The paper is organized as follows: in section 1 we prove the uniqueness result
Theorem 1.1 and two versions of a local well-posedness result for (1.5),(1.6),(1.7),
1
namely u ∈ X s, 2 + [0, δ] for data u0 ∈ H s with s ≥ 0 (Theorem 1.2), and a modi1
fication where ∇Iu ∈ X 0, 2 + [0, δ] for data ∇Iu0 ∈ L2 (Proposition 2.1), which is
necessary in order to combine it with an almost conservation law for the modified
energy E(Iu). In section 2 we use these local results and bounds for the modified
energy given in section 3 in order to get the main theorem (Theorem 1.3). Under
the assumptions (A1) and (A2) it is namely shown that the bounds for the modified energy immediately give a polynomial bound for k∇Iu(t)kL2 , which can be
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NONLOCAL GROSS-PITAEVSKII EQUATION
5
shown to imply a uniform exponential bound for ku(t)kL2 , and as a consequence
for ku(t)kH s , which in view of the local well-posedness result suffices to get a global
solution. Under the assumptions (A1) and (A3) we cannot immediately get a bound
for k∇Iu(t)kL2 from the bound for the modified energy, but first we have to show
an (exponential) bound for kIu(t)kL2 , which together with an energy bound gives
the desired (exponential) bound for k∇Iu(t)kL2 and after that as in the previous
d
E(Iu) for any
case the global well-posedness result. In section 3 we first calculate dt
solution of the equation (1.5) and estimate the time integrated terms which appear
d
in dt
E(Iu), which is the most complicated part. In section 2 these estimates are
shown to control the modified energy E(Iu) uniformly on arbitrary time intervals
[0, T ], provided s > 1/2.
2. Uniqueness and local well-posedness
Proof of Theorem 1.1. Let u, v ∈ C 0 ([0, T ], L2 )) be two solutions. Using Strichartz
type estimates in order to control
2
ku − vkL2t L6x + ku − vkL∞
t Lx
we have to estimate the various terms of F (u) − F (v). By (A1) and the HausdorffYoung inequality we have W ∈ Lp for 1 ≤ p < 3 so that by Young’s inequality we
obtain
ku − vkL2t L6x
. kW ∗ |u|2 kL2+ L3−
k(W ∗ |u|2 )(u − v)kL1+ L2−
x
x
t
t
kuk2L4+ L2 ku − vkL2t L6x
. kW kL3−
x
x
t
.T
1
2−
kuk2L∞
2 ku
t Lx
− vkL2t L6x ,
kukL2+ L2
. kW ∗ (|u|2 − |v|2 )kL2 L∞−
k(W ∗ (|u|2 − |v|2 ))vkL1+ L2−
x
x
t
t
x
t
k|u|2 − |v|2 kL2 L3/2 kukL2+ L2
. kW kL3−
x
t
.T
1
2−
x
t
x
2 + kvkL∞ L2 )ku − vkL2 L6 kukL∞ L2 ,
(kukL∞
x
x
t
t
t Lx
t x
2
k(W ∗ Re u)(u − v)kL1t L2x . kW ∗ Re ukL1t L∞
ku − vkL∞
t Lx
x
2 ku − vkL∞ L2 ,
. kW kL2 T kukL∞
x
t Lx
t
2 .
kW ∗ Re(u − v)kL1t L2x . kW kL1 T ku − vkL∞
t Lx
Similarly the remaining terms can be estimated. Therefore, choosing T small
enough, we obtain u ≡ v.
Next we prove the local well-posedness results.
Proof of Theorem 1.2. We want to apply the contraction mapping principle in the
1
Bourgain type space X s, 2 + [0, δ]. We have to estimate
k(W ∗ (|u|2 + 2 Re u))(1 + u)k
1
X s,− 2 ++
l
.
l
We denote by D the operator with symbol |ξ| and similarly by hDil the operator
1
with symbol hξil . To estimate the cubic term by δ 2 +s− kuk3 s, 1 + we have to show
X
2
(ignoring complex conjugates, which play no role for the calculations):
Z
3
Y
1
hDis (hDi−2 (u1 u2 )u3 )ψ dx dt . δ 2 +s−
kui k s, 12 + kψk 0, 12 −− .
X
i=1
X
6
H. PECHER
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This is sufficient, because we can assume without loss of generality that the Fourier
b τ ) are nonnegative, so that using the fundamental
transforms u
bi (ξi , τi ) and ψ(ξ,
−2
c
assumption |W (ξ)| . hξi it is possible to replace here and in similar situations in
the following the convolution with W by application of hDi−2 . Using the Leibniz
rule for fractional derivatives we reduce to the estimates (assuming without loss of
generality |ξ2 | ≥ |ξ1 |):
1
khDi−2 (u1 hDis u2 )u3 ψkL1xt . δ 2 +s−
3
Y
kui k
1
X s, 2 +
kψk
1
X 0, 2 −−
i=1
and
−2
khDi
s
(u1 u2 )hDi u3 ψk
L1xt
.δ
1
2 +s−
3
Y
kui k
1
X s, 2 +
kψk
1
X 0, 2 −−
.
i=1
We obtain
khDi−2 (u1 hDis u2 )u3 ψkL1xt . khDi−2 (u1 hDis u2 )u3 kL1+ L2 kψkL∞− L2
x
t
x
t
and
khDi−2 (u1 hDis u2 )u3 kL1+ L2 . khDi−2 (u1 hDis u2 )kL1+ Lp̂x ku3 kL∞ Lq̂x
t
x
t
t
s
. ku1 hDis u2 kL1+ Lr̂ ku3 kL∞
t Hx
x
t
2 ku3 kL∞ H s
. ku1 kL1+ Lv̂ khDis u2 kL∞
x
t
t Lx
x
t
. ku1 kL1+ H s,t̂ ku2 k
1
X s, 2 +
x
t
ku3 k
1
. δ s+ 2 − ku1 kLŵ H s,t̂ ku2 k
x
t
.δ
s+ 12 −
3
Y
kui k
1
X s, 2 +
1
X s, 2 +
1
X s, 2 +
ku3 k
1
X s, 2 +
.
i=1
Here we set
1
q̂
=
1
2
− 3s ,
1
t̂
1
p̂
s 1
2
s
3 , r̂ = 3 + 3 ,
Lv̂x , and finally ŵ1
=
1
v̂
so that Hxs ⊂ Lq̂x , and
=
1
6
+ 3s ,
= 61 + 32 s , so that Hxs,t̂ ⊂
= 12 − s, so that ŵ1 + 23t̂ = 43 , which
allows to apply Strichartz’ estimate in the last line. Similarly we obtain
2
khDi−2 (u1 u2 )hDis u3 kL1+ L2 . khDi−2 (u1 u2 )kL1+ L∞ khDis u3 kL∞
t Lx
t
x
x
t
. ku1 u2 k
3+
2
L1+
t Lx
ku3 k
1
X s, 2 +
. ku1 kL1+ Lv̂ ku2 kL∞ Lq̂x ku3 k
t
x
1
X s, 2 +
t
1
s ku3 k
. δ s+ 2 − ku1 kLŵ H s,t̂ ku2 kL∞
t Hx
t
1
. δ s+ 2 −
3
Y
kui k
1
X s, 2 +
x
1
X s, 2 +
.
i=1
Here 1q̂ = 12 − 3s , v̂1 = 61 + 3s −, 1t̂ = 16 + 32 s−, ŵ1 = 12 − s+, so that ŵ1 + 23t̂ = 34 allows
to apply Strichartz’ estimate again.
The quadratic terms are handled as follows (assuming again |ξ2 | ≥ |ξ1 |):
khDi−2+s (u1 u2 )kL1+ L2 . khDis (u1 u2 )kL1+ L1 . ku1 hDis u2 kL1+ L1
t
x
t
x
t
. ku1 kL2+ L2 khDis u2 kL2t L2x . δ 1− ku1 k
t
x
x
1
X s, 2 +
ku2 k
1
X s, 2 +
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NONLOCAL GROSS-PITAEVSKII EQUATION
7
and
khDis ((hDi−2 u1 )u2 )kL1+ L2
x
t
−2+s
. k(hDi
u1 )u2 kL1+ L2 + khDi−2 u1 hDis u2 kL1+ L2
t
x
t
x
−2
2 + khDi
. khDi−2+s u1 kL1+ L∞ ku2 kL∞
u1 kL∞
khDis u2 kL1+ L2
t,x
t Lx
x
t
t
x
1−
s
2 ku2 kL∞ L2 + δ
2 khDi u2 kL∞ L2
. δ 1− khDis u1 kL∞
ku1 kL∞
x
x
t Lx
t
t Lx
t
. δ 1− ku1 k
1
X s, 2 +
ku2 k
1
X s, 2 +
.
Similar estimates hold for the difference F (u) − F (v), so that a standard Picard
1
iteration under the conditions δ s+ 2 − ku0 k2H s . 1 and δ 1− ku0 kH s . 1 shows the
1
existence of a unique solution in X s, 2 + [0, δ] ⊂ C 0 ([0, δ], H s ). It is also unique is
this latter space by Theorem 1.1.
Remark: This Theorem shows that in order to get a global solution it is sufficient
to give an a-priori bound of ku(t)kH s .
We next prove a modified local well-posedness result involving the operator I
(recall that I depends on s and N ).
Proposition 2.1. Assume s ≥ 0 and ∇Iu0 ∈ L2 . Then the Cauchy problem
(1.5),(1.6),(1.7) (after application of
√ I) has a unique local solution u with ∇Iu ∈
1
≤ M k∇Iu0 kL2 , where M ≥ 1 is independent of
X 0, 2 + [0, δ] and k∇Iuk 0, 12 +
X
[0,δ]
u0 , and δ ≤ 1 can be chosen such that
1
(δ 2 − N −2 + δ 1− )k∇Iu0 k2L2 ∼ 1 .
Proof. The cubic term in the nonlinearity will be estimated as follows (dropping
[0, δ] from the notation):
k∇I(W ∗ (u1 u2 )u3 )k
This follows from
Z
A :=
δ
Z
M (ξ1 , ξ2 , ξ3 )
0
∗
1
2−
.(
1
X 0,− 2 ++
3
Y
.(
1
3
Y
δ 2−
1−
+
δ
)
k∇Iui k 0, 12 + .
X
N2
i=1
b 4 , t) dξ1 dξ2 dξ3 dξ4 dt
u
bi (ξi , t)ψ(ξ
i=1
3
Y
δ
+ δ 1− )
kui k 0, 12 + kψk 0, 21 + ,
X
X
N2
i=1
where
|ξ1 + ξ2 + ξ3 |
m(ξ1 + ξ2 + ξ3 )
·
,
m(ξ1 )m(ξ2 )m(ξ3 ) hξ1 + ξ2 i2 |ξ1 ||ξ2 ||ξ3 |
P4
and * denotes integration over the region { i=1 ξi = 0}. We assume here and
in the following again without loss of generality that the Fourier transforms are
nonnegative, and also without loss of generality that |ξ1 | ≥ |ξ2 |. We again used the
c (ξ)| . hξi−2 .
property |W
We make a case by case analysis depending on the relative size of the frequencies.
M (ξ1 , ξ2 , ξ3 ) :=
8
H. PECHER
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Case 1: |ξ1 | ≥ |ξ2 | ≥ |ξ3 | & N .
1a. |ξ1 + ξ2 + ξ3 | ≥ N . We obtain
M (ξ1 , ξ2 , ξ3 ) .
3
Y
N 1−s
|ξ1 + ξ2 + ξ3 |
|ξi | 1−s
)
(
1−s
N
|ξ1 + ξ2 + ξ3 |
|ξ1 ||ξ2 ||ξ3 |hξ1 + ξ2 i2
i=1
.
|ξ1 |s
1
. 2
.
N hξ1 + ξ2 i2
N 2(1−s) |ξ1 |s |ξ2 |s |ξ3 |s hξ1 + ξ2 i2
1
This implies by Sobolev’s embedding and Strichartz’ estimates:
1
3 ku3 kL2 L6 kψkL2 L2
khDi−2 (u1 u2 )kL∞
t Lx
t x
t x
N2
1
1
−
1 ku3 kL2 L6 δ 2
. 2 ku1 u2 kL∞
kψkL∞− L2
t Lx
t x
x
t
N
3
1 1 Y
kui k 0, 12 + kψk 0, 12 −− .
. 2 δ 2−
X
X
N
i=1
A.
1b. |ξ1 + ξ2 + ξ3 | ≤ N . We have
M (ξ1 , ξ2 , ξ3 ) .
.
3
Y
|ξi | 1−s
N
(
)
2
N
|ξ
||ξ
||ξ
1 2 3 |hξ1 + ξ2 i
i=1
N
N 3(1−s) |ξ1 |s |ξ2 |s |ξ3 |s hξ1
+ ξ2
i2
.
1
+ ξ2 i2
N 2 hξ1
as in case 1a.
Case 2: |ξ3 | ≥ |ξ1 | ≥ |ξ2 |. This case can be treated similarly as case 1.
Case 3: |ξ1 | ≥ |ξ2 | & N ≥ |ξ3 |.
3a. |ξ1 + ξ2 + ξ3 | ≥ N . We obtain
|ξ1 + ξ2 + ξ3 |
|ξ1 |1−s |ξ2 |1−s
N 1−s
N 1−s N 1−s |ξ1 + ξ2 + ξ3 |1−s |ξ1 ||ξ2 ||ξ3 |hξ1 + ξ2 i2
|ξ1 |s
1
.
.
,
|ξ1 |s |ξ2 |s |ξ3 |N 1−s hξ + ξ2 i2
N |ξ3 |hξ1 + ξ2 i2
M (ξ1 , ξ2 , ξ3 ) .
so that
1
−1
3 kD
6 kψkL1 L2
khDi−2 (u1 u2 )kL∞
u3 kL∞
t Lx
t Lx
t x
N
1
1 ku3 kL∞ L2 kψk ∞− 2
. δ 1− ku1 u2 kL∞
Lt Lx
x
t Lx
t
N
3
Y
1
. δ 1−
kui k 0, 12 + kψk 0, 12 −− .
X
X
N
i=1
A.
3b. |ξ1 + ξ2 + ξ3 | ≤ N . Similarly we obtain
M (ξ1 , ξ2 , ξ3 ) .
|ξ1 |1−s |ξ2 |1−s
N
1
.
N 1−s N 1−s |ξ1 ||ξ2 ||ξ3 |hξ1 + ξ2 i2
N |ξ3 |hξ1 + ξ2 i2
as in case 3a.
Case 4: |ξ1 |, |ξ3 | & N & |ξ2 |. Similarly as in case 3 we obtain
M (ξ1 , ξ2 , ξ3 ) .
1
,
N |ξ2 |hξ1 + ξ2 i2
EJDE-2012/??
NONLOCAL GROSS-PITAEVSKII EQUATION
9
so that by Sobolev’s embedding and Strichartz’ estimates
1
2 kψk 1+ 2
khDi−2 (u1 D−1 u2 )kL∞− L∞ ku3 kL∞
Lt Lx
t Lx
x
t
N
1
2 kψk ∞− 2
. δ 1− ku1 D−1 u2 k ∞− 32 + ku3 kL∞
Lt Lx
t Lx
Lt Lx
N
1
6 ku3 kL∞ L2 kψk ∞− 2
. δ 1− ku1 kL∞− L2+
kD−1 u2 kL∞
Lt Lx
x
t Lx
t
x
t
N
3
Y
1
kui k 0, 21 + kψk 0, 12 −− .
. δ 1−
X
X
N
i=1
A.
Case 5: |ξ3 | & N |ξ1 | ≥ |ξ2 |. We have
N 1−s
|ξ1 + ξ2 + ξ3 |
|ξ3 |1−s
1−s
1−s
N
|ξ1 + ξ2 + ξ3 |
|ξ1 ||ξ2 ||ξ3 |hξ1 + ξ2 i2
1
,
.
|ξ1 ||ξ2 |hξ1 + ξ2 i2
M (ξ1 , ξ2 , ξ3 ) .
leading to
∞ ku3 kL∞ L2 kψk 1 2
A . khDi−2 (D−1 u1 D−1 u2 )kL∞
Lt Lx
t Lx
x
t
3 ku3 kL∞ L2 kψk ∞− 2
. δ 1− kD−1 u1 D−1 u2 kL∞
L
L
x
t
t Lx
x
t
−1
6 ku3 kL∞ L2 kψk ∞− 2
6 kD
u2 kL∞
. δ 1− kD−1 u1 kL∞
L
L
x
t
t Lx
t Lx
t
. δ 1−
3
Y
kui k
1
X 0, 2 +
kψk
1
X 0, 2 −−
x
.
i=1
Case 6: |ξ1 | & N |ξ2 |, |ξ3 |. Similarly as in case 5 we obtain
M (ξ1 , ξ2 , ξ3 ) .
1
,
|ξ2 ||ξ3 |hξ1 + ξ2 i2
which implies
−1
6 kψk ∞− 2
3 kD
u3 kL∞
A . δ 1− khDi−2 (u1 D−1 u2 )kL∞
L
L
t Lx
t Lx
t
x
2 kψk ∞− 2
. δ 1− ku1 D−1 u2 kL∞− L3/2 ku3 kL∞
L
L
t Lx
x
t
.δ
1−
. δ 1−
2 kD
ku1 kL∞
t Lx
3
Y
kui k
1
−1
X 0, 2 +
t
x
6 ku3 kL∞ L2 kψk ∞− 2
u2 kL∞
L
L
x
t Lx
t
t
kψk
1
X 0, 2 −−
x
.
i=1
Case 7: N |ξ1 |, |ξ2 |, |ξ3 |. We easily obtain
M (ξ1 , ξ2 , ξ3 ) .
1
1
|ξ1 + ξ2 + ξ3 |
.
or .
,
|ξ1 ||ξ2 ||ξ3 |hξ1 + ξ2 i2
|ξ2 ||ξ3 |hξ1 + ξ2 i2
|ξ1 ||ξ2 |hξ1 + ξ2 i2
which can be handled like case 6 or case 5. This completes the claimed estimate
for the cubic term.
Next we consider the quadratic terms in the nonlinearity. They turn out to be
less critical. First we prove the estimate
1
k∇IW ∗ (u1 u2 )k
1
X 0,− 2 ++
.(
δ 2−
+ δ 1− )k∇Iu1 k 0, 12 + k∇Iu2 k 0, 21 + .
X
X
N
10
H. PECHER
EJDE-2012/??
This follows from
δ
Z
Z
B :=
M (ξ1 , ξ2 )
∗
0
b 3 , t) dξ1 dξ2 dξ3 dt
u
bi (ξi , t)ψ(ξ
i=1
2
Y
1
2−
.(
2
Y
δ
+ δ 1− )
kui k 0, 12 + kψk 0, 12 + ,
X
X
N
i=1
where
m(ξ1 + ξ2 )
|ξ1 + ξ2 |
·
,
m(ξ1 )m(ξ2 ) hξ1 + ξ2 i2 |ξ1 ||ξ2 |
P3
and * denotes integration over the region { i=1 ξi = 0}. We assume without loss
of generality |ξ1 | ≥ |ξ2 |.
Case 1: |ξ1 | ≥ |ξ2 | ≥ N .
1a. |ξ1 + ξ2 | ≥ N . We obtain
M (ξ1 , ξ2 ) :=
N 1−s
|ξ1 + ξ2 |
|ξ1 |1−s |ξ2 |1−s
1−s
1−s
N
N
|ξ1 + ξ2 |1−s |ξ1 ||ξ2 |hξ1 + ξ2 i2
|ξ1 + ξ2 |s
1
.
.
,
s
s
1−s
2
|ξ1 | |ξ2 | N
hξ1 + ξ2 i
N hξ1 + ξ2 i2
M (ξ1 , ξ2 ) .
so that by Strichartz’ estimates
1
B . khDi−2 (u1 u2 )kL2t L2x kψkL2t L2x
N
1
. ku1 u2 kL2 L3/2 kψkL2t L2x
t x
N
1 1−
2 ku2 kL2 L6 kψk ∞− 2
. δ 2 ku1 kL∞
Lt Lx
t Lx
t x
N
1 1
. δ 2 − ku1 k 0, 21 + ku2 k 0, 12 + kψk 0, 21 −− .
X
X
X
N
1b. |ξ1 + ξ2 | ≤ N . This case is similar to case 1a.
Case 2: N & |ξ2 | and |ξ1 | |ξ2 |. One has
M (ξ1 , ξ2 ) .
1
|ξ1 + ξ2 |
,
.
hξ1 + ξ2 i2 |ξ1 ||ξ2 |
|ξ2 |hξ1 + ξ2 i2
so that
B . khDi−2 (u1 D−1 u2 )kL2t L2x kψkL2t L2x
. ku1 D−1 u2 kL2 L3/2 kψkL2t L2x
x
t
−1
2 kD
6 kψk ∞− 2
. δ 1− ku1 kL∞
u2 kL∞
L
L
t Lx
t Lx
x
t
.δ
1−
ku1 k
1
X 0, 2 +
ku2 k
1
X 0, 2 +
kψk
1
X 0, 2 −−
.
Case 3: N ≥ |ξ1 | ∼ |ξ2 |. This case can be handled like case 2.
Finally we show
k∇I((W ∗ u1 )u2 )k
1
X 0,− 2 ++
. δ 1− k∇Iu1 k
1
X 0, 2 +
k∇Iu2 k
Here B is as in case 2 with
M (ξ1 , ξ2 ) :=
m(ξ1 + ξ2 )
|ξ1 + ξ2 |
·
.
m(ξ1 )m(ξ2 ) hξ1 i2 |ξ1 ||ξ2 |
1
X 0, 2 +
.
EJDE-2012/??
NONLOCAL GROSS-PITAEVSKII EQUATION
11
Because the estimates are similar to the previous case we only consider the most
critical low frequency cases.
Case 1: |ξ1 | |ξ2 | and N ≥ |ξ1 | (or N |ξ1 | ∼ |ξ2 |). The estimate
1
hξ1 i2 |ξ1 |
M (ξ1 , ξ2 ) .
implies
B . khDi−2 D−1 u1 kL∞
ku2 kL2t L2x kψkL2t L2x
x,t
2 ku2 kL2 L2 kψkL2 L2
. ku1 kL∞
t Lx
t x
t x
. δ 1− ku1 k
1
X 0, 2 +
ku2 k
1
X 0, 2 +
kψk
1
X 0, 2 −−
.
Case 2: |ξ1 | |ξ2 | and N ≥ |ξ2 |. The estimate
1
hξ1 i2 |ξ2 |
M (ξ1 , ξ2 ) .
implies
−1
3 kD
u2 kL2t L6x kψkL2t L2x
B . khDi−2 u1 kL∞
t Lx
2 ku2 kL∞ L2 kψk ∞− 2
. δ 1− ku1 kL∞
L
L
x
t
t Lx
t
. δ 1− ku1 k
1
X 0, 2 +
ku2 k
1
X 0, 2 +
x
kψk
1
X 0, 2 −−
.
We remark that similar estimates can be given for the difference terms in order
to use Banach’s fixed point theorem. In order to get a contraction we have to fulfill
the estimates
1
(δ 2 − N −2 + δ 1− )k∇Iu0 k2L2 1
1
and (δ 2 − N −1 + δ 1− )k∇Iu0 kL2 1 .
The latter requirement is weaker, so that the claimed result follows.
Remark: We want to iterate this local existence theorem with time steps of equal
length until we reach a given (large) time T . To achieve this we need to control
k∇Iu(t)kL2 ≤ c(T ) ∀ 0 ≤ t ≤ T .
(2.1)
s
This will be shown under the assumption u0 ∈ H with s > 1/2.
3. Proof of Theorem 1.3
In this section we first show that the bound (2.1) implies global well-posedness
and after that we derive such a bound from the estimates for the modified energy
E(Iu) in the next section.
Proof of Theorem 1.3. So let us assume for the moment that (2.1) holds. This
means that on any existence interval [0, T ] we have an a-priori bound (for fixed N )
of
k∇Iu(t)kL2 ∼ k|ξ|b
u(ξ, t)kL2 ({|ξ|≤N }) + k|ξ|s u
b(ξ, t)kL2 ({|ξ|≥N }) N 1−s ,
(3.1)
especially
k|ξ|s u
b(ξ, t)kL2 ({|ξ|≥1}) ≤ c(T ) .
(3.2)
If we can show that this implies an a-priori bound for ku(t)kL2 , which is done in the
following lemma, we immediately get an a-priori bound for ku(t)kH s , 0 ≤ t ≤ T ,
1
thus a unique global solution in X s, 2 + [0, T ] ⊂ C 0 ([0, T ], H s ) for any T using our
12
H. PECHER
EJDE-2012/??
local well-posedness result (Theorem 1.2), which is also unique in this latter space
by Theorem 1.1.
Lemma 3.1. Assume (2.1) and s ≥ 1/2. On any existence interval [0, T ] of the
1
solution u ∈ X s, 2 + [0, T ] we have ku(t)kL2 ≤ c(T ).
Proof. We decompose u
b=u
c1 +c
u2 smoothly with supp u
c1 ⊂ {|ξ| ≤ 2} and supp u
c2 ⊂
{|ξ| ≥ 1}. Then we have by Gagliardo-Nirenberg
1/2
1/2
kukL3 ≤ ku1 kL3 + ku2 kL3 . k∇u1 kL2 ku1 kL2 + k|D|1/2 u2 kL2
1/2
1
1/2
1−
1
. k∇u1 kL2 ku1 kL2 + k|D|s u2 kL2s2 ku2 kL2 2s
2
2
2
. k∇u1 k2L2 + ku1 kL3 2 + ku2 kL3 2 + k|D|s u2 kL3−2s
,
2
so that by (2.1),(3.1) and (3.2) we obtain on [0, T ]:
6
ku(t)k3L3 . k|ξ|c
u1 (ξ, t)k6L2 + ku1 (t)k2L2 + ku2 (t)k2L2 + k|ξ|s u
c2 (ξ, t)kL3−2s
2
≤ c0 (T )(ku(t)k2L2 + 1) .
Multiplying the differential equation (1.5) with iu and taking the real part we obtain
by Young’s inequality, because W ∈ L1 is real-valued:
Z
Z
d
2
2
ku(t)kL2 = (W ∗ (|u| + 2 Re u))udx + Re i (W ∗ (|u|2 + 2 Re u))|u|2 dx
dt
Z
Z
. |W ∗ (|u|2 )||u|dx + 2 |W ∗ Re u||u|dx
. kW ∗ |u|2 kL3/2 kukL3 + kuk2L2
. kuk3L3 + kuk2L2
≤ c0 (T )(ku(t)k2L2 + 1) ,
so that Gronwall’s lemma gives
0
ku(t)k2L2 + 1 ≤ (ku0 k2L2 + 1)ec (T )T
on [0, T ].
We recall our aim to give an a-priori bound of k∇Iu(t)kL2 (cf. (2.1)) on [0, T ]
for an arbitrarily given T . We want to show this in the rest of this section as a
consequence of Proposition 2.1 and the estimates for the modified energy which we
give in the next section.
Let N ≥ 1 be a number to be specified later and s > 1/2. Let data u0 ∈ H s be
given. Then we have
k∇Iu0 k2L2 . k|ξ|c
u0 (ξ)k2L2 ({|ξ|≤N }) + kN 1−s |ξ|s u
c0 (ξ)k2L2 ({|ξ|≥N })
. kN 1−s |ξ|s u
c0 (ξ)k2L2 (R3 ) = N 2(1−s) ku0 k2Ḣ s . N 2(1−s) .
(3.3)
(3.4)
This implies an estimate for |E(Iu0 )| as follows: we have by Young’s inequality,
using W ∈ L1 :
Z
| (W ∗ (|Iu0 |2 + 2 Re Iu0 ))(|Iu0 |2 + 2 Re Iu0 )dx| . kIu0 k4L4 + kIu0 k3L3 + kIu0 k2L2 .
Now by Sobolev’s embedding
kIu0 k4L4
EJDE-2012/??
NONLOCAL GROSS-PITAEVSKII EQUATION
. kIu0 k4
Ḣ
.
3
3
4
b0 k4L2 ({|ξ|≤N }) + k
. k|ξ| 4 u
kb
u0 k4L2 ({|ξ|≤1})
13
N 1−s 3
b0 k4L2 ({|ξ|≥N })
|ξ| 4 u
|ξ|1−s
3
+ k|ξ| 4 −s |ξ|s u
b0 k4L2 ({1≤|ξ|≤N }) + N 3−4s k|ξ|s u
b0 k4L2 ({|ξ|≥N })
. ku0 k4L2 + hN i3−4s ku0 k4Ḣ s . N 2(1−s) ku0 k4H s ,
using in the last line the assumption s ≥ 1/2. Moreover
kIu0 k3L3 . kIu0 k3Ḣ 1/2 ≤ ku0 k3Ḣ 1/2 ≤ ku0 k3H s ,
kIu0 k2L2 ≤ ku0 k2L2 ,
so that
|E(Iu0 )| ≤ c0 N 2(1−s) .
The local existence theorem (Prop. 2.1) implies that there exists a solution u on
some time interval [0, δ] with
k∇Iuk2
1
X 0, 2 + [0,δ]
≤ M k∇Iu0 k2L2 ≤ c0 M N 2(1−s)+2
(3.5)
under the assumption k∇Iu0 k2L2 ≤ c0 N 2(1−s+) , where ≥ 0 and
1
.
N 2(1−s+)
Now we use the results of the next section. We have the following estimate
δ∼
(3.6)
|E(Iu(δ)) − E(Iu0 )|
δ 1−
1
δ 1/2
δ 1/2
4
+
+
)k∇Iuk
+
k∇Iuk3 0, 1 +
1+
0,
X 2 [0,δ]
X 2 [0,δ]
N 2−
N 1−
N 3−
N 2−
1
δ
δ
+ ( 4− + 2− )k∇Iuk6 0, 1 +
+ 2− k∇Iuk5 0, 1 +
.
X 2 [0,δ]
X 2 [0,δ]
N
N
N
If we use (3.5) and (3.6) we easily see that the decisive term is
.(
(3.7)
N 6(1−s+) 1−
δ 1−
δ
N 4(1−s+)
k∇Iuk4 0, 1 +
+
)δ .
+ 2− k∇Iuk6 0, 1 +
.(
1−
1−
X 2 [0,δ]
X 2 [0,δ]
N
N
N
N 2−
This is the bound for the increment of the modified energy from time 0 to time δ.
Similarly we obtain the same bound for the increment from time t = kδ to time
t = (k + 1)δ for 0 ≤ k ≤ T /δ, k ∈ N, provided we have a uniform bound
k∇Iu(kδ)k2L2 ≤ 2c0 N 2(1−s+) ,
(3.8)
which implies
k∇Iuk2
1
X 0, 2 + [kδ,(k+1)δ]
≤ 2c0 M N 2(1−s+)
(3.9)
by the local existence theorem. The number of iteration steps to reach the given
time T is T /δ, so that the increment of the energy from time t = 0 to time t =
(k + 1)δ, 0 ≤ k ≤ Tδ , is bounded by
N 6(1−s+) 1−
T N 4(1−s+)
(
+
)δ ≤ c0 N 2(1−s)
δ
N 1−
N 2−
independent of k, if T N 1− N 4(1−s+) N 2(1−s) and T N −2+ N 6(1−s+) N 2(1−s) .
These conditions are fulfilled for N sufficiently large, if
s>
s−
1
+ 2 ⇐⇒ <
2
2
1
2
,
(3.10)
14
H. PECHER
EJDE-2012/??
as one easily calculates. Choosing sufficiently small this condition is fulfilled under
our assumption s > 1/2. We recall again that we used (3.8). We arrive at
T
, k fixed.
(3.11)
δ
Now we consider the cases where either (A1) and (A2) or else (A1) and (A3) hold
separately.
c (ξ) > 0, which immediately implies that the
If (A1) and (A2) hold we have W
energy functional is positive definite, both terms in (1.9) are namely nonnegative,
so that one gets
k∇Iu(t)k2L2 ≤ E(I(u(t)) ≤ 2c0 N 2(1−s)
|E(Iu(t))| ≤ 2c0 N 2(1−s)
∀t ∈ [0, (k + 1)δ] , 0 ≤ k ≤
for 0 ≤ t ≤ (k + 1)δ and 0 ≤ k < Tδ , where c0 is independent of k and where we can
choose = 0. Remark that on the right-hand side the same constant 2c0 appears
as in (3.8). Thus step by step after ∼ Tδ steps we obtain the desired a-priori bound
k∇Iu(t)kL2 ≤ c(T ) ∀0 ≤ t ≤ T .
Thus we are done in this case (modulo the results of the next section).
If (A1) and (A3) hold, the energy functional is not necessarily positive definite
and it is more difficult to obtain a bound for k∇Iu(t)kL2 from energy bounds.
We follow the computations of de Laire [17] in this case and obtain
Z
1
(W ∗ (|Iu|2 + 2 Re Iu))(|Iu|2 + 2 Re Iu)dx
E(Iu) = k∇Iuk2L2 +
2
(3.12)
2
˜
˜
˜
= k∇IukL2 + I1 (Iu) + I2 (Iu) + I3 (Iu) ,
where
Z
˜
I1 (Iu) := (W ∗ Re Iu) Re Iudx
Z
1
I˜2 (Iu) :=
(W ∗ |Iu|2 )(|(Iu)1 |2 + 4 Re(Iu)1 )dx
2
Z
1
I˜3 (Iu) :=
(W ∗ |Iu|2 )(|(Iu)2 |2 + 4 Re(Iu)2 )dx .
2
Here
(Iu)1 := Iuχ{|Iu|≤5} ,
(Iu)2 := Iuχ{|Iu|>5} .
We used that W is even which implies
Z
Z
2
(W ∗ |Iu| ) Re Iu dx = (W ∗ Re Iu)|Iu|2 dx .
Using W ∈ L1 (R3 ) we easily see that
|I˜1 (Iu)| + |I˜2 (Iu)| . kIuk2L2 .
Moreover using (A3) we obtain
Z
1
˜
(W ∗ |Iu|2 )(|(Iu)2 |2 − 4|(Iu)2 |)dx
I3 (Iu) ≥
2
Z
1
=
(W ∗ |Iu|2 )|(Iu)2 |(|(Iu)2 | − 4)dx
2
Z
1
(W ∗ |Iu|2 )|(Iu)2 |dx =: J3 (Iu) ≥ 0 .
≥
2
(3.13)
(3.14)
(3.15)
EJDE-2012/??
NONLOCAL GROSS-PITAEVSKII EQUATION
15
This implies by (3.12)
k∇Iuk2L2 + I˜3 (Iu) ≤ |E(Iu)| + akIuk2L2 .
(3.16)
kIuk2L2
In order to estimate
we apply I to the differential equation (1.5), multiply
with iIu and take the real part. This leads to
d
kIuk2L2 = ImhF (Iu) − IF (u), Iui − ImhF (Iu), Iui .
dt
Let us consider the first term on the right-hand side. We obtain
(3.17)
ImhF (Iu) − IF (u), Iui = Imh(W ∗ |Iu|2 )Iu − I((W ∗ |u|2 )u), Iui
+ 2Imh(W ∗ (Re Iu))Iu − I((W ∗ Re u)u), Iui
+ Imh(W ∗ |Iu|2 ) − I(W ∗ |u|2 ), Iui
+ 2ImhW ∗ (Re Iu) − I(W ∗ Re u), Iui
= ImhI(W ∗ |u|2 ) − (W ∗ |Iu|2 ), Iui .
Now we claim
Z (k+1)δ
hI(W ∗ |u|2 ) − (W ∗ |Iu|2 ), Iuidt . N −2 δk∇Iuk3
1
X 0, 2 + [kδ,(k+1)δ]
kδ
Using |W̃ (ξ)| .
1
hξi2
and defining
M (ξ1 , ξ2 ) :=
1
|m(ξ1 + ξ2 ) − m(ξ1 )m(ξ2 )|
·
m(ξ1 )m(ξ2 )
hξ1 + ξ2 i2 |ξ1 ||ξ2 ||ξ3 |
we have to show
Z (k+1)δ Z
3
3
Y
Y
A :=
M (ξ1 , ξ2 )
u
bi (ξi , t) dξ1 dξ2 dξ3 dt . N −2 δ
kui k3
kδ
. (3.18)
∗
i=1
i=1
P3
where * denotes integration over {
ality |ξ1 | ≥ |ξ2 |.
Case 1: |ξ1 | ≥ |ξ2 | & N : We have
M (ξ1 , ξ2 ) . (
i=1 ξi
1
X 0, 2 +
,
= 0}. We assume without loss of gener-
1
|ξ1 | 1/2 |ξ2 | 1/2
) (
)
.
N
N
hξ1 + ξ2 i2 |ξ1 ||ξ2 |
Thus
A . N −2 ku1 kL2x,t ku2 kL2x,t khDi−2 D−1 u3 kL∞
x,t
−1
2 ku2 kL∞ L2 kD
6
. N −2 δku1 kL∞
u3 kL∞
x
t Lx
t
t Lx
. N −2 δ
3
Y
kui k
1
X 0, 2 +
.
i=1
Case 2: |ξ1 | & N |ξ2 | (=⇒ |ξ3 | ∼ |ξ1 | & N ). By the mean value theorem we
obtain
1
|ξ2 |
.
M (ξ1 , ξ2 ) .
2
|ξ1 | hξ1 + ξ2 i |ξ1 ||ξ2 ||ξ3 |
Thus
−3
2 . N
1 ku3 kL∞ L2
A . N −3 khDi−2 (u1 u2 )kL1t L2x ku3 kL∞
δku1 u2 kL∞
x
t Lx
t Lx
t
16
H. PECHER
EJDE-2012/??
. N −3 δ
3
Y
kui k
1
X 0, 2 +
,
i=1
which completes the proof of (3.18).
Next we estimate the last term in (3.17). We have
Z
|ImhF (Iu), Iui| = |Im (W ∗ (|Iu|2 + 2 Re Iu))(1 + Iu)I ūdx|
Z
= |Im (W ∗ (|Iu|2 + 2 Re Iu))I ūdx|
Z
Z
Z
≤ 2 |W ∗ Re Iu||Iu|dx + (W ∗ |Iu|2 )|(Iu)1 |dx + (W ∗ |Iu|2 )|(Iu)2 |dx
. kIuk2L2 + J3 (Iu) . kIuk2L2 + I˜3 (Iu) . kIuk2L2 + |E(Iu)|
by (3.13) and (3.16).
From (3.17) we conclude for kδ ≤ t ≤ (k + 1)δ:
kIu(t)k2L2 − kIu(kδ)k2L2
. c1 N
−2
δk∇Iuk3 0, 1 +
X 2 [kδ,(k+1)δ]
Z
(k+1)δ
Z
t
kIu(s)k2L2 ds) .
|E(Iu(s))|ds +
+
kδ
kδ
Now we have
c1 (N −2 δk∇Iuk3
1
X 0, 2 + [kδ,(k+1)δ]
≤ c1 (2c0 )3/2 N −2 δN 3(1−s)+3 ≤ c2 δN 2(1−s) ,
provided (3.8) holds (and therefore (3.9)) and is sufficiently small. Using the
uniform energy bound (3.11) we obtain for t ∈ [kδ, (k + 1)δ]:
Z t
kIu(t)k2L2 ≤ kIu(kδ)k2L2 + c2 δN 2(1−s) + 2c0 δN 2(1−s) + c1
kIu(s)k2L2 ds .
kδ
Gronwall’s lemma implies
sup
kδ≤t≤(k+1)δ
kIu(t)k2L2 ≤ (kIu(kδ)k2L2 + c3 δN 2(1−s) )ec1 δ
under our assumptions (3.11)
|E(Iu(t))| ≤ 2c0 N 2(1−s)
on
[0, (k + 1)δ]
and (3.8)
k∇Iu(kδ)k2L2 ≤ 2c0 N 2(1−s+) .
Here c1 and c3 are independent of k. Using the bound for kIu(kδ)k2L2 this implies
sup
kδ≤t≤(k+1)δ
kIu(t)k2L2 ≤ [(kIu((k − 1)δ)k2L2 + c3 δN 2(1−s) )ec1 δ + c3 δN 2(1−s) ]ec1 δ
= kIu((k − 1)δ)k2L2 e2c1 δ + c3 δN 2(1−s) (e2c1 δ + ec1 δ ) .
Iterating this procedure after k ≤ T /δ steps we arrive at
T
sup
kδ≤t≤(k+1)δ
kIu(t)k2L2
≤
kIu0 k2L2
e
T
δ
c1 δ
+ c3 δN
2(1−s)
δ
X
l=0
≤ ku0 k2L2 ec1 T + c4 N 2(1−s) ec1 T
c0
≤ N 2(1−s+)
a
(ec1 δ )l
EJDE-2012/??
NONLOCAL GROSS-PITAEVSKII EQUATION
17
choosing N so large that ec1 T N with a small > 0, which fulfills (3.10), and
N also so large, that ku0 k2L2 N 2(1−s) . We used
T
δ
X
l=0
T
(ec1 δ ) δ − 1
ec1 T
) =
.
.
ec1 δ − 1
δ
c1 δ l
(e
This bound for kIu(t)kL2 for t ∈ [kδ, (k + 1)δ] implies by (3.16),(3.8),(3.11):
k∇Iu(t)k2L2 ≤ |E(Iu(t))| + akIu(t)k2L2
≤ 2c0 N 2(1−s) + c0 N 2(1−s+) ≤ 2c0 N 2(1−s+)
for t ∈ (kδ, (k + 1)δ] (and choosing N so large, that N 2 ≥ 2), the same bound
which we had for t = kδ (cf. (3.8)). By iteration we thus get
sup k∇Iu(t)k2L2 ≤ 2c0 N 2(1−s+) =: c(T ) .
0≤t≤T
This completes the proof of the a-priori bound for k∇Iu(t)kL2 for the problem
under the assumptions (A1) and (A3), so that now (2.1) holds in both cases. Thus
the global well-posedness result is proven (modulo the results in the next section).
4. Estimates for the modified energy
In order to estimate the increment of the modified energy E(Iu(t)) of a solution
u of the Cauchy problem (1.5),(1.6),(1.7) from time t0 to time t0 + δ, say t0 = 0 for
ease of notation, we have to control its time derivative. We calculate
d
E(Iu) = 2 Reh−∆Iu, Iut i
dt
Z
1
+
(W ∗ (IuI ūt + Iut I ū + 2 Re Iut ))(|Iu|2 + 2 Re Iu)dx
2
Z
1
(W ∗ (|Iu|2 + 2 Re Iu))(IuI ūt + I ūIut + 2 Re Iut )dx
+
2
= 2 Reh−∆Iu, Iut i + 2 Reh(W ∗ (|Iu|2 + 2 Re Iu))(1 + Iu), Iut i ,
where we used that W is even, so that the second and third term coincide. Now
−∆Iu = −iIut − IF (u) ,
so that
d
E(Iu) = 2 RehF (Iu) − IF (u), Iut i
dt
= 2 Im(h∇(F (Iu) − IF (u)), ∇Iui − hF (Iu) − IF (u), IF (u)i)
and
d
E(Iu)| ≤ 2(|h∇(F (Iu) − IF (u)), ∇Iui| + |hF (Iu) − IF (u), IF (u)i|) (4.1)
dt
with (cf. (1.7))
F (u) = (1 + u)(W ∗ (|u|2 + 2 Re u)) .
This especially shows the standard energy conservation law (setting I = id).
The estimates which now follow are given in terms of bounds of Fourier transforms of the corresponding functions. The only property of W which we use is the
c (ξ)| . hξi−2 , so that both cases, namely assuming either (A1) and (A2)
bound |W
or else (A1) and (A3) can be handled in the same way. The most critical cases
|
18
H. PECHER
EJDE-2012/??
are the terms of fourth and third order of the first term on the right-hand side of
(4.1) and the term of sixth order of the second term. In fact we shall refer to the
c = 1, in our
estimates in the case of the local Gross-Pitaevskii equation, where W
earlier paper [18] for the remaining terms of lower order on the right-hand side of
(4.1).
We start with the terms of highest order in the first term. Taking again the time
interval [0, δ] instead of [kδ, (k + 1)δ] just for the ease of notation we claim
Z
δ
h∇((W ∗ |Iu|2 )Iu − I((W ∗ |u|2 )u)), ∇Iuidt
0
δ 1−
. ( 1− +
N
Here and in the following we
variables ξi , where |ξi | ∼ Ni
(4.2)
δ 1/2
4
)k∇Iuk 0, 1 +
.
X 2 [0,δ]
N 2−
use dyadic decompositions with respect to the space
with Ni = 2ki , ki ∈ Z. In order to sum the dyadic
1∧N 0+
parts at the end we always need a convergence generating factor N 0+min , where
max
Nmin and Nmax is the smallest and the largest of the numbers Ni , respectively.
Nmax ≥ N (≥ 1) can be assumed in all cases, because otherwise our multiplier M
is identically zero. We have to take care of low frequencies especially, because we
need an estimate in terms of ∇Iu. Assuming without loss of generality that the
Fourier transforms are nonnegative we have to show:
Z δZ
4
4
Y
δ 1−
δ 1/2 Y
A :=
M (ξ1 , ξ2 , ξ3 )
ubi (ξ, t) dξ1 . . . dξ4 dt . ( 1− + 2− )
kui k 0, 12 +
,
X
[0,δ]
N
N
0
∗
i=1
i=1
P
where * always denotes integration over { ξi = 0}, and
M (ξ1 , ξ2 , ξ3 ) :=
|ξ1 + ξ2 + ξ3 |
|m(ξ1 + ξ2 + ξ3 ) − m(ξ1 )m(ξ2 )m(ξ3 )|
·
.
m(ξ1 )m(ξ2 )m(ξ3 )
hξ1 + ξ2 i2 |ξ1 ||ξ2 ||ξ3 |
Case 1: N3 N1 ≥ N2 . In this case we have N3 ∼ N4 & N , N2 = Nmin and
N3 ∼ Nmax . By the mean value theorem we obtain
N1
1
M (ξ1 , ξ2 , ξ3 ) .
·
N3 N1 |ξ2 |hξ1 + ξ2 i2
and by Hölder’s inequality, Sobolev’s embedding and Strichartz’ estimate we obtain
1
A.
ku3 kL2+ L2+
ku4 kL2t L2x
khDi−2 (D−1 u2 u1 )kL∞− L∞−
x
x
t
t
N3
δ 1−
2
.
kD−1 u2 u1 k ∞− 23 + ku3 kL∞ L2+
ku4 kL∞
t Lx
x
t
Lt Lx
N3
δ 1−
2
.
kD−1 u2 kL∞ L6+
ku1 kL∞− L2+
ku3 kL∞ L2+
ku4 kL∞
t Lx
x
x
x
t
t
t
N3
4
0+ Y
δ 1− (1 ∧ Nmin
)
.
kui k 0, 12 + ,
0+
X
Nmax
N 1− i=1
0+
2 and ku3 k ∞ 2+ . N
2.
using kD−1 u2 kL∞ L6+
. N20+ ku2 kL∞
3 ku3 kL∞
Lt Lx
t Lx
t Lx
x
t
Case 2: N1 N2 , N3 and N1 & N . We have similarly as in case 1:
1
N3
·
M (ξ1 , ξ2 , ξ3 ) .
N1 |ξ2 |N3 hξ1 + ξ2 i2
and get the same estimate as in case 1 interchanging the roles of N1 and N3 .
EJDE-2012/??
NONLOCAL GROSS-PITAEVSKII EQUATION
19
Case 3: N1 ∼ N3 (=⇒ N1 , N3 & N ). In this case we obtain
1
N1 1 − N3 1 − N1 1 −
)2 ( )2 h i2
N
N
N
|ξ2 ||ξ3 |hξ1 + ξ2 i2
1
1
N2 1
. 0+ 1− h i 2 −
.
N
N2 hξ1 + ξ2 i2
N3 N
M (ξ1 , ξ2 , ξ3 ) . (
a. N2 & N . We have
A.
1
2 ku4 k 2 6+
ku3 kL∞
khDi−2 (u1 u2 )kL2 L3−
Lt Lx
t Lx
t x
N30+ N 2−
.
δ 1/2 N40+
2 ku4 k 0, 1 +
ku3 kL∞
ku1 u2 kL∞ L1+
t Lx
x
X 2
t
N30+ N 2−
.
4
0+ Y
δ 1/2 (1 ∧ Nmin
)
kui k 0, 12 + ,
0+
X
Nmax
N 2− i=1
. N40+ ku4 kL2t L6x . ku4 k 0, 21 + by Strichartz’ estimate.
using ku4 kL2 L6+
X
t x
b. N2 N . In this case we obtain
1
2 ku4 k ∞ 2+
A . 0+ 1− khDi−2 (u1 D−1 u2 )kL1 L∞−
ku3 kL∞
Lt Lx
t Lx
t x
Nmax N
δ
2 ku4 k ∞ 2+
. 0+ 1− ku1 D−1 u2 k ∞ 23 + ku3 kL∞
Lt Lx
t Lx
Lt Lx
Nmax N
δ
−1
2 ku4 k ∞ 2+
2 kD
ku3 kL∞
u2 kL∞ L6+
. 0+ 1− ku1 kL∞
Lt Lx
t Lx
t Lx
x
t
Nmax N
4
0+ Y
δ(1 ∧ Nmin
)
.
kui k 0, 12 + .
0+
X
Nmax
N 1− i=1
Case 4: N1 ∼ N2 & N3 (=⇒ N1 , N2 & N ). In this case we obtain
N1 1 − N2 1 − N3 1 −
1
)2 ( )2 h i2
N
N
N
|ξ1 ||ξ3 |hξ1 + ξ2 i2
1
N3 1
1
.
. 0+ 1− h i 2 −
N
N3 hξ1 + ξ2 i2
Nmax N
M (ξ1 , ξ2 , ξ3 ) . (
The case N3 & N is handled like case 3a, whereas in the case N3 N we obtain
1
A . 0+ 1− khDi−2 (u1 u2 )kL1 L3−
kD−1 u3 kL∞ L6+
ku4 kL∞ L2+
x
x
t x
t
t
Nmax N
0+
δ(1 ∧ Nmin
)
1 ku3 kL∞ L2 ku4 kL∞ L2
.
ku1 u2 kL∞
0+
x
x
t Lx
t
t
Nmax N 1−
4
0+ Y
δ(1 ∧ Nmin
)
.
kui k 0, 12 + .
0+
X
Nmax
N 1− i=1
Case 5: N2 ∼ N3 (=⇒ N1 & N2 ∼ N3 ). If N1 N2 we are in the situation of
case 2, otherwise N1 ∼ N2 ∼ N3 & N , so that
N1 1 N2 1 N3 1
1
M (ξ1 , ξ2 , ξ3 ) . ( ) 2 − ( ) 2 − ( ) 2 −
N
N
N
N2 N3 hξ1 + ξ2 i2
1
1
. 0+ 2−
2
N
hξ
Nmax N
2 1 + ξ2 i
20
H. PECHER
EJDE-2012/??
as in case 3a.
Dyadic summation gives estimate (4.2).
We next consider the cubic part of the first term on the right-hand side of (4.1).
We claim
Z
δ
δ 1/2
h∇(W ∗ (|Iu|2 ) − I(W ∗ |u|2 )), ∇Iuidt . 2− k∇Iuk3 0, 1 +
.
(4.3)
X 2 [0,δ]
N
0
We have to show
Z δZ
3
3
Y
δ 1/2 Y
kui k 0, 12 +
B :=
M (ξ1 , ξ2 )
ubi (ξi , t) dξ1 dξ2 dξ3 dt . 2−
X
[0,δ]
N i=1
0
∗
i=1
with
|ξ1 + ξ2 |
|m(ξ1 + ξ2 ) − m(ξ1 )m(ξ2 )|
·
.
m(ξ1 )m(ξ2 )
hξ1 + ξ2 i2 |ξ1 ||ξ2 |
Case 1: N1 ≥ N2 & N . We have
M (ξ1 , ξ2 ) :=
M (ξ1 , ξ2 ) . (
1
N1 1/2 N2 1/2
) ( )
,
N
N
N1 N2 hξ1 + ξ2 i
so that by Strichartz’ estimate:
1
ku3 kL∞ L2+
B . 0+ 2− khDi−1 (u1 u2 )kL1 L2−
x
t
t x
Nmax N
1
. 0+ 2− ku1 u2 k 1 32 + ku3 kL∞ L2+
x
t
Lt Lx
Nmax N
δ 1/2
2 ku2 k 2 6+ ku3 k ∞ 2+
. 0+ 2− ku1 kL∞
Lt Lx
Lt Lx
t Lx
Nmax N
3
0+ Y
δ 1/2 (1 ∧ Nmin
)
.
kui k 0, 12 + .
0+
X
Nmax
N 2− i=1
Case 2: N1 ≥ N N2 . Similarly, by the mean value theorem we obtain
1
1
N2
. 0+ 2−
M (ξ1 , ξ2 ) .
N1 N1 N2 hξ1 + ξ2 i
Nmax N hξ1 + ξ2 i
leading to the same bound as in case 1, thus (4.3) is proven.
Concerning the second cubic term we claim
Z δ
δ
|
h∇((W ∗ Iu)Iu − I((W ∗ u)u)), ∇Iuidt| . 1− k∇Iuk3 0, 1 +
.
X 2 [0,δ]
N
0
We again have to consider a term like B but with
M (ξ1 , ξ2 ) :=
|m(ξ1 + ξ2 ) − m(ξ1 )m(ξ2 )|
|ξ1 + ξ2 |
·
.
m(ξ1 )m(ξ2 )
hξ1 i2 |ξ1 ||ξ2 |
We concentrate on the more difficult case N2 ≥ N1 and have to consider
Case 1: N2 ∼ N1 & N . We have
M (ξ1 , ξ2 ) . (
N1 1 − N2 1 − 1
1
)2 ( )2
. 0+ 3− .
N
N
N13
Nmax N1
Thus
B.
δ
2 ku3 k ∞ 2+
ku1 kL∞ L∞−
ku2 kL∞
Lt Lx
0+
t Lx
x
t
Nmax
N13−
(4.4)
EJDE-2012/??
NONLOCAL GROSS-PITAEVSKII EQUATION
.
21
3
0+
δ(1 ∧ Nmin
) 3/2 Y
2
N
kui kL∞
1
0+
t Lx
Nmax
N13−
i=1
3
.
0+ Y
δ(1 ∧ Nmin
)
0+
Nmax
N
3
2−
kui k
1
X 0, 2 +
.
i=1
Case 2: N2 N1 (=⇒ N2 & N ). By the mean value theorem we obtain
M (ξ1 , ξ2 ) .
N1
1
1
=
,
N2 hξ1 i2 N1
N2 hξ1 i2
so that
B.
.
δ
∞ ku2 kL∞ L2 ku3 kL∞ L2
khDi−2 u1 kL∞
0+
t Lx
x
x
t
t
Nmax N 1−
3
0+ Y
)
δ(1 ∧ Nmin
2 .
kui kL∞
0+
t Lx
Nmax N 1− i=1
Thus (4.4) follows.
We now have to consider the sixth order term on the right-hand side of (4.1).
Our aim is to show the following estimate:
Z
δ
h(W ∗ |Iu|2 )Iu − I((W ∗ |u|2 )u), I((W ∗ |u|2 )u)idt
0
(4.5)
δ
1
6
. ( 2− + 4− )k∇Iuk 0, 1 + .
X 2
N
N
We have to show
Z δZ
6
Y
C :=
M (ξ1 , . . . , ξ6 )
ubi (ξi , t) dξ1 . . . dξ6 dt
0
.
∗
i=1
6
Y
1 δ
+ 4−
kui k 0, 12 +
2−
[0,δ]
X
N
N
i=1
with
M (ξ1 , . . . ., ξ6 )
:=
6
Y
|m(ξ1 + ξ2 + ξ3 ) − m(ξ1 )m(ξ2 )m(ξ3 )|
m(ξ4 + ξ5 + ξ6 )
·
·
|ξi |−1 .
m(ξ1 )m(ξ2 )m(ξ3 )hξ1 + ξ2 i2
m(ξ4 )m(ξ5 )m(ξ6 )hξ4 + ξ5 i2 i=1
We assume without loss of generality N1 ≥ N2 and N4 ≥ N5 .
Case 1: N N4 ≥ N5 and N N6 .
a. N1 ≥ N2 & N & N3 . In this case we obtain
N2
1
N1
khDi−2 (u1 u2 )k 1 23 − kD−1 u3 kL∞ L6+
C . ( )1/2 ( )1/2
x
t
Lt Lx
N
N
N1 N2
−2
−1
−1
−1
∞ kD
× khDi (D u4 D u5 )kL∞
u6 kL∞ L6+
t Lx
x
t
δ
−1
1 kD
. 0+ 2− ku1 u2 kL∞
u3 kL∞ L6+
kD−1 u4 D−1 u5 kL∞ L3+
kD−1 u6 kL∞ L6+
t Lx
x
x
x
t
t
t
Nmax N
δ
−1
−1
2 ku2 kL∞ L2 kD
6 kD
u3 kL∞ L6+
kD−1 u4 kL∞
u5 kL∞ L6+
. 0+ 2− ku1 kL∞
x
t Lx
t
t Lx
x
x
t
t
Nmax N
× kD−1 u6 kL∞ L6+
x
t
22
H. PECHER
EJDE-2012/??
6
.
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 21 + .
0+
2−
X
Nmax N
i=1
b. N1 , N3 & N N2 . The estimate
M (ξ1 , . . . , ξ6 ) . (
1
N1 1/2 N3 1/2
) ( )
2
2
N
N
N1 hξ4 + ξ5 i |ξ1 ||ξ2 |N3 |ξ4 ||ξ5 ||ξ6 |
implies
C.
δ
−1
6 kD
2
kD−1 u1 kL∞
ku3 kL∞
u2 kL∞ L6+
0+
t Lx
t Lx
x
t
Nmax N 3−
kD−1 u6 kL∞ L6+
× khDi−2 (D−1 u4 D−1 u5 )kL∞ L∞−
x
x
t
t
δ
−1
−1
2 kD
2 kD
. 0+ 3− ku1 kL∞
ku3 kL∞
u2 kL∞ L6+
u4 D−1 u5 kL∞ L3+
t Lx
t Lx
x
x
t
t
Nmax N
−1
× kD u6 kL∞ L6+
x
t
.
6
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 21 +
0+
X
Nmax
N 3− i=1
.
c. N1 , N2 , N3 & N .
In this case we obtain
N1
N2
N3
1
2
ku3 kL∞
khDi−2 (u1 u2 )kL1 L3−
C . ( )1/2 ( )1/2 ( )1/2
t Lx
t x
N
N
N
N1 N2 N3
−1
∞ kD
× khDi−2 (D−1 u4 D−1 u5 )kL∞
u6 kL∞ L6+
t Lx
x
t
δ
−1
1 ku3 kL∞ L2 kD
. 0+ 3− ku1 u2 kL∞
kD−1 u6 kL∞ L6+
u4 D−1 u5 kL∞ L3+
x
t
t Lx
x
x
t
t
Nmax N
6
0+ Y
δ(1 ∧ Nmin
)
.
kui k 0, 12 + .
0+
X
Nmax N 3− i=1
Case 2: N N4 ≥ N5 and N6 & N .
a. N3 & N & N1 ≥ N2 . By the mean value theorem we have
C.
δ 1−
N1
2
·
khDi−2 (u1 D−1 u2 )kL∞− L∞ ku3 kL∞
t Lx
x
t
N3 N1 N3 N6
∞ ku6 kL∞ L2
× khDi−2 (D−1 u4 D−1 u5 )kL∞
t Lx
x
t
δ 1−
.
.
ku1 D
0+
Nmax
N 3−
1−
−1
u2 k
3+
Lx2
L∞−
t
−1
2 kD
2
ku3 kL∞
u4 D−1 u5 kL∞ L3+
ku6 kL∞
t Lx
t Lx
x
t
δ
−1
2 kD
6
ku1 kL∞− L2+
kD−1 u2 kL∞ L6+
ku3 kL∞
u4 kL∞
0+
t Lx
t Lx
x
x
t
t
Nmax
N 3−
2
× kD−1 u5 kL∞ L6+
ku6 kL∞
t Lx
x
t
.
δ
6
0+ Y
(1 ∧ Nmin
)
kui k 0, 21 +
0+
X
Nmax
N 3− i=1
1−
.
b. N1 & N N2 ≥ N3 (or N1 & N N3 ≥ N2 by exchanging u2 and u3 ).
Similarly as in a. we use the mean value theorem and obtain
C.
N2
δ
−1
6 ku2 kL∞ L2 kD
·
kD−1 u1 kL∞
u3 kL∞ L6+
x
t Lx
t
x
t
N1 N12 N2
EJDE-2012/??
NONLOCAL GROSS-PITAEVSKII EQUATION
23
6
kD−1 u6 kL∞
× khDi−2 (D−1 u4 D−1 u5 )kL∞ L∞−
t Lx
x
t
δ
−1
2 ku2 kL∞ L2 kD
2
. 3 ku1 kL∞
ku6 kL∞
kD−1 u4 D−1 u5 kL∞ L3+
u3 kL∞ L6+
x
t Lx
t Lx
t
x
x
t
t
N1
6
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 12 + .
.
0+
X
Nmax
N 3− i=1
c. N1 ≥ N2 & N N3 . We obtain
N1 1 − N2 1 −
1
khDi−2 (u1 u2 )kL2t L3x kD−1 u3 kL∞ L6+
)2 ( )2
x
t
N
N
N1 N2 N6
−2
−1
−1
2
ku6 kL∞
× khDi (D u4 D u5 )kL2 L∞−
t Lx
x
C.(
t
δ
−1
1 kD
2
. 0+ 3− ku1 u2 kL∞
u3 kL∞ L6+
kD−1 u4 D−1 u5 kL∞ L3+
ku6 kL∞
t Lx
t Lx
x
x
t
t
Nmax N
6
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 12 + .
.
0+
X
Nmax
N 3− i=1
d. N1 ≥ N2 ≥ N3 & N or N1 ≥ N3 ≥ N2 & N . This case can be handled similarly
1
as case c. with an additional factor ( NN3 ) 2 − .
e. N3 & N1 & N & N2 (or N1 ≥ N3 & N & N2 by exchanging the roles of u1 and
u3 ). We obtain
δ
N3 1 − N1 1 −
∞ ku3 kL∞ L2
)2 ( )2
khDi−2 (u1 D−1 u2 )kL∞
t Lx
x
t
N
N
N1 N3 N6
−2
−1
−1
∞ ku6 kL∞ L2
× khDi (D u4 D u5 )kL∞
t Lx
x
t
C.(
δ
−1
2
2 kD
ku6 kL∞
u4 D−1 u5 kL∞ L3+
ku1 D−1 u2 k ∞ 23 + ku3 kL∞
0+
t Lx
t Lx
x
t
Lt Lx
Nmax
N 3−
δ
−1
−1
−1
6 kD
2 kD
2 kD
. 0+ 3− ku1 kL∞
u5 kL∞ L6+
u4 kL∞
ku3 kL∞
u2 kL∞ L6+
t Lx
t Lx
t Lx
x
x
t
t
Nmax N
2
× ku6 kL∞
t Lx
.
.
6
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 12 + .
0+
X
Nmax
N 3− i=1
f. N3 ≥ N1 ≥ N2 & N (or N1 ≥ N3 ≥ N2 & N ). This case can be treated as case
1
a. with an additional factor ( NN2 ) 2 − .
Case 3: N4 ≥ N5 & N .
a. N1 , N2 , N3 . N and N6 ≤ N . We obtain
M (ξ1 , . . . , ξ6 ) .
1
hξ1 +
ξ2 i2 hξ4
+ ξ5
i2 N
N4 1 − N5 1 −
)2 ( )2 ,
N
4 N5 |ξ1 ||ξ2 ||ξ3 ||ξ6 | N
(
so that
C.
δ
khDi−2 (D−1 u1 D−1 u2 )kL∞ L∞−
kD−1 u3 kL∞ L6+
0+
x
x
t
t
Nmax
N 2−
−2
−1
× khDi (u4 u5 )kL∞ L3/2 kD u6 kL∞ L6+
x
t
x
t
δ
−1
1 kD
kD−1 u3 kL∞ L6+
ku4 u5 kL∞
u6 kL∞ L6+
. 0+ 2− kD−1 u1 D−1 u2 kL∞ L3+
t Lx
x
x
x
t
t
t
Nmax N
24
H. PECHER
EJDE-2012/??
6
.
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 21 + .
0+
2−
X
Nmax N
i=1
b. N1 , N2 , N3 . N and N6 ≥ N . We argue similarly as in case a with an additional
1
factor ( NN6 ) 2 − and get
C.
.
δ
kD−1 u3 kL∞ L6+
khDi−2 (D−1 u1 D−1 u2 )kL∞ L∞−
0+
x
x
t
t
Nmax
N 3−
3 ku6 kL∞ L2
× khDi−2 (u4 u5 )kL∞
x
t Lx
t
6
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 12 + .
0+
X
Nmax
N 3− i=1
6 we
c. N3 & N and N1 , N2 , N6 . N . Replacing kD−1 u3 kL∞ L6+
by kD−1 u3 kL∞
t Lx
x
t
obtain the same result as in case a.
d. N3 & N , N1 , N2 . N and N6 & N . The additional factor ( NN6 )1/2 leads to the
bound
δ
−1
∞ kD
6
u3 kL∞
C . 0+ 3− khDi−2 (D−1 u1 D−1 u2 )kL∞
t Lx
t Lx
Nmax N
3 ku6 kL∞ L2
× khDi−2 (u4 u5 )kL∞
x
t
t Lx
.
6
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 12 + .
0+
X
Nmax
N 3− i=1
e. N1 ≥ N2 & N and N3 , N6 . N . We obtain
1
M (ξ1 , . . . , ξ6 ) .
1
1
1
( NN1 ) 2 − ( NN2 ) 2 − ( NN4 ) 2 − ( NN5 ) 2 −
,
hξ1 + ξ2 i2 hξ4 + ξ5 i2 N1 N2 N4 N5 |ξ3 ||ξ6 |
so that
C.
.
.
δ
kD−1 u3 kL∞ L6+
khDi−2 (u1 u2 )kL∞ L3−
0+
x
x
t
t
Nmax N 4−
−1
−2
3 kD
u6 kL∞ L6+
× khDi (u4 u5 )kL∞
t Lx
x
t
δ
1 ku3 kL∞ L2 ku4 u5 kL∞ L1 ku6 kL∞ L2
ku1 u2 kL∞
0+
x
x
x
t
t
t
t Lx
Nmax
N 4−
6
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 12 + .
0+
X
Nmax N 4− i=1
f. N1 ≥ N2 & N , N3 . N and N6 & N . The additional factor ( NN6 )1/2 leads to
C.
δ
khDi−2 (u1 u2 )kL∞ L3−
kD−1 u3 kL∞ L6+
0+
x
x
t
t
Nmax N 4−
−2
−1
2
× khDi (u4 D u5 )kL∞ L∞−
ku6 kL∞
t Lx
x
t
δ
−1
1 ku3 kL∞ L2 ku4 kL∞ L2 kD
6 ku6 kL∞ L2
u5 kL∞
. 0+ 4− ku1 u2 kL∞
x
x
x
t Lx
t
t
t Lx
t
Nmax N
6
0+ Y
δ(1 ∧ Nmin
)
.
kui k 0, 21 + .
0+
X
Nmax
N 4− i=1
EJDE-2012/??
NONLOCAL GROSS-PITAEVSKII EQUATION
25
g. N1 ≥ N2 & N , N6 . N and N3 & N . This case can be treated as case f. with
u3 and u6 exchanged.
h. N1 ≥ N2 & N and N3 , N6 & N (=⇒ Nmin & N ). We obtain
M (ξ1 , . . . , ξ6 ) .
6
6
Y
Ni 1 Y −1
1
,
Ni
( )2−
2
N
hξ1 + ξ2 i hξ4 + ξ5 i2
i=1
i=1
so that
C.
1
3 ku3 kL2 L6
khDi−2 (u1 u2 )kL∞
0+
t Lx
t x
Nmax N 6−
−2
3 ku6 kL2 L6
× khDi (u4 u5 )kL∞
t Lx
t x
1
1 ku3 k 0, 1 + ku4 u5 kL∞ L1 ku6 k 0, 1 +
ku1 u2 kL∞
0+
x
t Lx
t
X 2
X 2
Nmax
N 6−
6
Y
1
. 0+ 6−
kui k 0, 12 + .
X
Nmax N i=1
.
i. N1 & N N2 and N3 , N6 . N . We obtain
1
1
1
1
( N1 ) 2 − ( NN2 ) 2 − ( NN4 ) 2 − ( NN5 ) 2 −
,
M (ξ1 , . . . , ξ6 ) . N
N12 hξ4 + ξ5 i2 N1 N2 |ξ3 |N4 N5 |ξ6 |
so that
C.
1
kD−1 u3 kL∞ L6+
ku1 kL2t L6x ku2 kL2 L6+
0+
x
t
t x
Nmax N 6−
−1
−2
kD u6 kL∞ L6+
× khDi (u4 u5 )kL∞ L3−
x
x
t
t
.
6
0+ Y
1 ∧ Nmin
kui k 0, 12 +
0+
X
Nmax
N 6− i=1
.
j. N1 & N N2 and N3 , N6 & N . We obtain
1
M (ξ1 , . . . , ξ6 ) .
1
1
1
1
( NN1 ) 2 − ( NN3 ) 2 − ( NN4 ) 2 − ( NN5 ) 2 − ( NN6 ) 2 −
,
N12 hξ4 + ξ5 i2 |ξ1 ||ξ2 |N3 N4 N5 N6
thus using N1 . max(N3 , N4 , N5 , N6 ):
C.
1
−1
6 kD
kD−1 u1 kL∞
u2 kL2 L6+
ku3 kL2t L6x
0+
t Lx
t x
Nmax
N 6−
× khDi−2 (u4 u5 )kL∞ L3−
ku6 kL2 L6+
x
x
t
.
6
0+ Y
1 ∧ Nmin
kui k 0, 12 +
0+
X
Nmax
N 6− i=1
t
.
k. N1 & N N2 , N3 . N and N6 . N . This case can be handled like j. without
1
the factor ( NN6 ) 2 − by exchanging u1 and u6 .
l. N1 & N N2 , N3 . N and N6 & N . The mean value theorem gives the bound
1
C.
1
1
N2 ( NN4 ) 2 − ( NN5 ) 2 − ( NN6 ) 2 −
−1
6 ku2 k 2 6+ kD
kD−1 u1 kL∞
u3 kL∞ L6+
Lt Lx
t Lx
x
t
N1
N12 N2 N4 N5 N6
× khDi−2 (u4 u5 )kL∞ L3−
ku6 kL2t L6x
x
t
26
H. PECHER
.
EJDE-2012/??
6
0+ Y
1 ∧ Nmin
kui k 0, 21 + .
0+
X
Nmax
N 6− i=1
Case 4: N4 ≥ N N5 , N6 .
a. N3 & N & N1 ≥ N2 . The mean value theorem gives
δ
N1
6 ku3 kL∞ L2
khDi−2 (u1 D−1 u2 )kL∞
C.
x
t Lx
t
N3 N1 N3 N4
−1
−2
−1
kD u6 kL∞ L6+
× khDi (u4 D u5 )kL∞ L6−
x
x
t
t
δ
2
. 0+ 3− ku1 D−1 u2 k ∞ 32 + ku3 kL∞
t Lx
Lt Lx
Nmax N
× ku4 D−1 u5 k ∞ 23 + kD−1 u6 kL∞ L6+
x
t
Lt Lx
.
.
δ
−1
2
2 kD
ku3 kL∞
u2 kL∞ L6+
ku1 kL∞
0+
t Lx
t Lx
x
t
Nmax N 3−
−1
2 kD
kD−1 u6 kL∞ L6+
u5 kL∞ L6+
× ku4 kL∞
t Lx
x
x
t
t
6
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 12 + .
0+
X
Nmax
N 3− i=1
b. N1 ≥ N N2 ≥ N3 . The mean value theorem implies
1
N2
·
ku1 kL2t L6x ku2 kL2t L6x kD−1 u3 kL∞ L6+
C.
x
t
N1 N12 N1 N2
kD−1 u6 kL∞ L6+
× khDi−2 (D−1 u4 D−1 u5 )kL∞ L3−
x
x
t
t
.
6
0+ Y
1 ∧ Nmin
kui k 0, 12 +
0+
X
Nmax
N 4− i=1
.
c. N1 ≥ N N3 ≥ N2 . This case can be treated like case b. with u2 and u3
exchanged.
d. N3 ≥ N1 & N N2 . We obtain
N3
1
N1
ku3 kL2t L6x
C . ( )1/2 ( )1/2 2
ku1 kL2t L6x kD−1 u2 kL∞ L6+
x
t
N
N
N1 N1 N3
× khDi−2 (D−1 u4 D−1 u5 )kL∞ L3−
kD−1 u6 kL∞ L6+
x
x
t
t
.
6
0+ Y
1 ∧ Nmin
kui k 0, 12 +
0+
X
Nmax
N 4− i=1
.
e. N1 ≥ N2 & N & N3 . We obtain in this case
1
M (ξ1 , . . . , ξ6 ) .
1
( NN1 ) 2 − ( NN2 ) 2 −
,
hξ1 + ξ2 i2 N42 N1 N2 |ξ3 ||ξ4 ||ξ5 ||ξ6 |
so that
C.
δ
khDi−2 (u1 u2 )kL∞ L3−
kD−1 u3 kL∞ L6+
0+
x
x
t
t
Nmax N 4−
−1
−1
−1
6 kD
× kD u4 kL∞
u5 kL∞ L6+
kD u6 kL∞ L6+
t Lx
x
x
t
.
6
0+
Y
δ(1 ∧ Nmin
)
∞ L1
2
ku
u
k
kui kL∞
1
2
L
0+
x
t Lx
t
Nmax
N 4−
i=3
t
EJDE-2012/??
NONLOCAL GROSS-PITAEVSKII EQUATION
27
6
.
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 12 + .
0+
4−
X
Nmax N
i=1
f. N1 ≥ N3 & N & N2 . We obtain
C.(
N1 1 − N3 1 −
δ
2
ku3 kL∞
)2 ( )2
khDi−2 (u1 D−1 u2 )kL∞ L∞−
2
t Lx
x
t
N
N
N4 N1 N3
−1
6 kD
× kD−1 u4 kL∞
kD−1 u6 kL∞ L6+
u5 kL∞ L6+
t Lx
x
x
t
t
δ
2
. 0+ 4− ku1 D−1 u2 k ∞ 23 + ku3 kL∞
t Lx
Lt Lx
Nmax N
−1
6 kD
× kD−1 u4 kL∞
kD−1 u6 kL∞ L6+
u5 kL∞ L6+
t Lx
x
x
t
t
δ
−1
2 kD
2
. 0+ 4− ku1 kL∞
u2 kL∞ L6+
ku3 kL∞
t Lx
t Lx
x
t
Nmax N
−1
6 kD
kD−1 u6 kL∞ L6+
u5 kL∞ L6+
× kD−1 u4 kL∞
t Lx
x
x
t
.
6
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 12 +
0+
X
Nmax
N 4− i=1
t
.
g. N1 ≥ N2 & N and N3 & N . This case can be treated as case f. with an
1
additional factor ( NN2 ) 2 − .
Case 5: N4 , N6 ≥ N N5 .
a. N3 & N & N1 ≥ N2 . This case can be treated as case 2a, because the additional
1
1
factor ( NN4 ) 2 − ( NN6 ) 2 − is harmless, when one uses N4 . max(N3 , N6 ).
b. N1 & N N2 ≥ N3 (or N1 & N N3 ≥ N2 by exchanging u2 and u3 ). We
have by the mean value theorem
C.
1
N2 N4 1 − N6 1 −
ku2 kL2t L6x kD−1 u3 kL∞ L6+
ku1 kL∞ L2+
( )2 ( )2
x
x
t
t
N1 N
N
N12 N1 N2 N4 N6
ku6 kL2t L6x
× khDi−2 (u4 D−1 u5 )kL∞ L∞−
x
t
1
−1
2 ku2 k 0, 1 + kD
u3 kL∞ L6+
. 0+ 6− ku1 kL∞
t Lx
x
X 2
t
Nmax N
−1
× ku4 D u5 k ∞ 23 + ku6 k 0, 12 +
Lt Lx
.
X
1
−1
−1
2 ku2 k 0, 1 + kD
2 kD
u3 kL∞ L6+
ku4 kL∞
ku1 kL∞
u5 kL∞ L6+
0+
t Lx
t Lx
x
x
X 2
t
t
Nmax
N 6−
× ku6 k 0, 21 +
X
6
1 ∧ N 0+ Y
kui k 0, 21 + .
. 0+ min
X
Nmax N 6− i=1
c. N1 ≥ N2 & N N3 . This case is treated like case 2c, because the additional
1
1
factor ( NN4 ) 2 − ( NN6 ) 2 − can be handled using N4 . max(N1 , N6 ).
d. N3 ≥ N1 & N N2 or N1 ≥ N3 & N ≥ N2 . This case can be handled like case
2e, using N4 . max(N1 , N3 , N6 ).
e. N1 ≥ N2 & N and N3 & N . This case is also treated like case 2e, because
1
1
1
the additional factor ( NN2 ) 2 − ( NN4 ) 2 − ( NN6 ) 2 − is acceptable, using N2 ≤ N1 and
N4 . max(N1 , N3 , N6 ).
28
H. PECHER
EJDE-2012/??
Case 6: N4 , N5 , N6 & N .
a. N3 & N N1 , N2 . The mean value theorem allows to estimate
1
M (ξ1 , . . . , ξ6 ) .
1
1
( NN4 ) 2 − ( NN5 ) 2 − ( NN6 ) 2 −
1
N1
·
. 0+ 4− ,
N3 hξ1 + ξ2 i2 hξ4 + ξ5 i2 N1 |ξ2 ||ξ3 |N4 N5 N6
Nmax N
thus
C.
δ
−1
∞ kD
6
khDi−2 (u1 D−1 u2 )kL∞
u3 kL∞
0+
t Lx
t Lx
Nmax
N 4−
3 ku6 kL∞ L2
× khDi−2 (u4 u5 )kL∞
x
t Lx
t
δ
2 ku4 u5 kL∞ L1 ku6 kL∞ L2
ku1 D−1 u2 k ∞ 23 + ku3 kL∞
0+
x
x
t Lx
t
t
Lt Lx
Nmax
N 4−
δ
−1
2 kD
2 ku4 kL∞ L2 ku5 kL∞ L2 ku6 kL∞ L2
. 0+ 4− ku1 kL∞
u2 kL∞ L6+
ku3 kL∞
x
x
x
t Lx
t Lx
t
t
t
x
t
Nmax N
6
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 21 + .
.
0+
X
Nmax N 4− i=1
.
b. N1 , N2 , N3 & N and without loss of generality N1 ≥ N2 and N4 ≥ N5 . We have
C.
6
Y
δ 1−
Ni 1
2
khDi−2 (u1 D−1 u2 )kL∞− L∞ ku3 kL∞
( )2−
t Lx
x
t
N
N
N
N
N
1
3
4
6
i=1
2
× khDi−2 (u4 D−1 u5 )kL∞− L∞ ku6 kL∞
t Lx
t
x
δ 1−
−1
2 ku4 D
2
. 0+ 4− ku1 D−1 u2 k ∞− 23 + ku3 kL∞
u5 k ∞− 23 + ku6 kL∞
t Lx
t Lx
Lt Lx
Lt Lx
Nmax N
δ 1−
−1
6
2 ku4 k ∞− 2+ kD
u5 kL∞
ku3 kL∞
kD−1 u2 kL∞ L6+
. 0+ 4− ku1 kL∞− L2+
Lt Lx
t Lx
t Lx
x
x
t
t
Nmax N
2
× ku6 kL∞
t Lx
6
.
0+ Y
δ 1− (1 ∧ Nmin
)
kui k 0, 21 + .
0+
4−
X
Nmax N
i=1
c. N1 , N3 & N & N2 . This case can be treated as case b. without the factor
1
( NN2 ) 2 − .
d. N1 ≥ N2 & N & N3 and without loss of generality N4 ≥ N5 . We obtain
Y Ni 1
δ
−1
3 kD
C.
( )2−
khDi−2 (u1 u2 )kL∞
u3 kL∞ L6+
t Lx
x
t
N
N1 N2 N4 N6
i6=3
6 ku6 kL∞ L2
× khDi−2 (u4 D−1 u5 )kL∞
x
t Lx
t
0+
δ(1 ∧ Nmin
)
−1
1 ku3 kL∞ L2 ku4 D
2
ku1 u2 kL∞
u5 kL∞ L3/2 ku6 kL∞
0+
x
t Lx
t
t Lx
4−
x
t
Nmax N
0+
δ(1 ∧ Nmin
)
−1
2 ku2 kL∞ L2 ku3 kL∞ L2 ku4 kL∞ L2 kD
6
ku1 kL∞
u5 kL∞
.
0+
x
x
x
t Lx
t
t
t
t Lx
Nmax
N 4−
2
× ku6 kL∞
t Lx
.
.
6
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 12 + .
0+
X
Nmax
N 4− i=1
EJDE-2012/??
NONLOCAL GROSS-PITAEVSKII EQUATION
29
This completes the proof of (4.5).
We now start to consider the fifth order terms and claim
Z
δ
δ
h(W ∗ |Iu|2 )Iu − I((W ∗ |u|2 )u), I(W ∗ |u|2 )idt . 2− k∇Iuk5 0, 1 + . (4.6)
X 2
N
0
We have to show
Z δZ
5
5
Y
δ Y
D :=
M (ξ1 , . . . , ξ5 )
ubi (ξi , t) dξ1 . . . dξ5 dt . 2−
kui k 0, 12 +
[0,δ]
X
N i=1
0
∗
i=1
with
M (ξ1 , . . . ., ξ5 ) :=
|m(ξ1 + ξ2 + ξ3 ) − m(ξ1 )m(ξ2 )m(ξ3 )|
m(ξ1 )m(ξ2 )m(ξ3 )hξ1 + ξ2 i2
×
5
Y
m(ξ4 + ξ5 )
·
|ξi |−1 .
m(ξ4 )m(ξ5 )hξ4 + ξ5 i2 i=1
We assume without loss of generality N1 ≥ N2 and N4 ≥ N5 .
Case 1: N N4 ≥ N5 .
a. N1 ≥ N2 & N & N3 . In this case we obtain
N2
1
N1
khDi−2 (u1 u2 )k 1 56 kD−1 u3 kL∞ L6+
D . ( )1/2 ( )1/2
x
t
Lt Lx
N
N
N1 N2
−2
−1
−1
× khDi (D u4 D u5 )kL∞ L∞−
x
t
δ
−1
1 kD
kD−1 u4 D−1 u5 kL∞ L3+
u3 kL∞ L6+
. 0+ 2− ku1 u2 kL∞
t Lx
x
x
t
t
Nmax N
5
0+ Y
δ(1 ∧ Nmin
)
.
kui k 0, 12 + .
0+
X
Nmax N 2− i=1
(4.7)
b. N1 , N3 & N N2 . We have
N3
δ
N1
2 ku3 kL∞ L2
khDi−2 (u1 D−1 u2 )kL∞
D . ( )1/2 ( )1/2
x
t
t Lx
N
N
N1 N3
−2
−1
−1
∞
× khDi (D u4 D u5 )kL∞
t Lx
.
.
δ
−1
2 kD
ku1 D−1 u2 k ∞ 23 + ku3 kL∞
u4 D−1 u5 kL∞ L3+
0+
t Lx
x
t
Lt Lx
Nmax N 2−
5
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 12 + .
0+
X
Nmax N 2− i=1
c. N1 , N2 , N3 & N . This leads to
N1
N2
N3
δ
2 ku3 kL∞ L2
D . ( )1/2 ( )1/2 ( )1/2
khDi−2 (u1 u2 )kL∞
x
t Lx
t
N
N
N
N1 N2 N3
∞
× khDi−2 (D−1 u4 D−1 u5 )kL∞
t Lx
5
.
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 12 + .
0+
3−
X
Nmax N
i=1
1
1
Case 2: N4 ≥ N5 & N . The additional factor ( NN4 ) 2 − ( NN5 ) 2 − can be compensated
by replacing the last factor in (4.7) by
−1/2
N4
−1/2
N5
−1/2
kD−1/2 u4 D−1/2 u5 kL∞ L3/2 . N4
t
x
−1/2
N5
2 ku5 kL∞ L2
ku4 kL∞
x
t Lx
t
30
H. PECHER
EJDE-2012/??
leading to even an improved bound.
Case 3: N4 ≥ N ≥ N5 . We argue as before replacing the last factor in (4.7) by
−1/2
N4
−1/2
. N4
kD−1/2 u4 D−1 u5 kL∞ L2+
x
t
−1
2 kD
ku4 kL∞
.
u5 kL∞ L6+
t Lx
x
t
This proves (4.6).
Next we claim
Z δ
h(W ∗ |Iu|2 )Iu − I((W ∗ |u|2 )u), I((W ∗ Re u)u)idt .
0
δ
k∇Iuk5 0, 1 + . (4.8)
X 2
N 2−
We again consider D with
M (ξ1 , . . . ., ξ5 )
:=
5
Y
|m(ξ1 + ξ2 + ξ3 ) − m(ξ1 )m(ξ2 )m(ξ3 )|
m(ξ4 + ξ5 )
·
·
|ξi |−1 .
m(ξ1 )m(ξ2 )m(ξ3 )hξ1 + ξ2 i2
m(ξ4 )m(ξ5 )hξ4 i2 i=1
We argue similarly as in the previous case and consider only the more difficult case
N5 ≥ N4 .
Case 1: N N5 ≥ N4 . The last term in (4.7) with a suitable change of the
Hölder exponent in the first factor can be replaced by
−2 −1
−1
∞ kD
6
6 . khDi
D u4 kL∞
u5 kL∞
khDi−2 D−1 u4 D−1 u5 kL∞
t Lx
t Lx
t Lx
2
ku5 kL∞
. kD−1 u4 kL∞ L6+
t Lx
x
t
leading to the same bound.
1
1
Case 2: N5 ≥ N4 & N . The additional factor ( NN4 ) 2 − ( NN5 ) 2 − is compensated by
replacing the last factor in (4.7) by
−1/2
N4
−1/2
N5
−1/2
. N4
−1/2
. N4
3
khDi−2 D−1/2 u4 D−1/2 u5 kL∞
t Lx
−1/2
N5
−1/2
N5
−1/2
3
3 kD
u5 kL∞
kD−1/2 u4 kL∞
t Lx
t Lx
2 ku5 kL∞ L2 .
ku4 kL∞
x
t Lx
t
Case 3: N5 ≥ N ≥ N4 . Replace the last factor in (4.7) by
6
kD−1 u5 kL∞
. kD−1 u4 kL∞ L6+
khDi−2 D−1 u4 D−1 u5 kL∞ L3+
t Lx
x
x
t
t
2 ku5 kL∞ L2 .
. N40+ ku4 kL∞
x
t Lx
t
Thus (4.8) is proven.
The next claim is
Z
δ
h(W ∗ |Iu|2 ) − I(W ∗ |u|2 ), I((W ∗ |u|2 )u)idt .
0
δ
k∇Iuk5 0, 1 + .
X 2
N 2−
We again consider D with
M (ξ1 , . . . ., ξ5 )
:=
5
Y
m(ξ3 + ξ4 + ξ5 )
|m(ξ1 + ξ2 ) − m(ξ1 )m(ξ2 )|
|ξi |−1 .
·
·
m(ξ1 )m(ξ2 )hξ1 + ξ2 i2
m(ξ3 )m(ξ4 )m(ξ5 )hξ3 + ξ4 i2 i=1
We assume without loss of generality N2 ≥ N1 and N3 ≥ N4 .
Case 1: N N3 , N4 , N5 (=⇒ N1 ∼ N2 & N ). In this case we obtain
D.(
N1 1/2 N2 1/2 δ
) ( )
khDi−2 (u1 u2 )k ∞ 56
Lt Lx
N
N
N1 N2
(4.9)
EJDE-2012/??
NONLOCAL GROSS-PITAEVSKII EQUATION
31
kD−1 u5 kL∞ L6+
× khDi−2 (D−1 u3 D−1 u4 )kL∞ L∞−
x
x
t
t
δ
−1
1 kD
kD−1 u5 kL∞ L6+
u3 D−1 u4 kL∞ L3+
. 0+ 2− ku1 u2 kL∞
t Lx
x
x
t
t
Nmax N
5
0+ Y
δ(1 ∧ Nmin
)
.
kui k 0, 12 + .
0+
X
Nmax N 2− i=1
Case 2: N3 ≥ N N4 , N5 .
a. N3 ≥ N N1 . We obtain
δ
−2
2 khDi
khDi−2 (D−1 u1 u2 )kL∞
kD−1 u5 kL∞ L6+
(u3 D−1 u4 )kL∞ L3−
t Lx
x
x
t
t
N2 N3
δ
kD−1 u1 u2 k ∞ 23 + ku3 D−1 u4 k ∞ 23 + kD−1 u5 kL∞ L6+
.
x
t
Lt Lx
Lt Lx
N2 N3
5
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 21 + .
.
0+
X
Nmax
N 2− i=1
D.
1
1
b. N2 ≥ N1 & N . We obtain an additional factor ( NN1 ) 2 − ( NN2 ) 2 − , which is
acceptable, when one estimates as in a.
Case 3: N5 ≥ N N3 , N4 (=⇒ N2 & N ).
a. N1 ≤ N . We obtain
δ
−2
∞ ku5 kL∞ L2
2 khDi
(D−1 u3 D−1 u4 )kL∞
khDi−2 (D−1 u1 u2 )kL∞
t Lx
x
t
t Lx
N2 N5
δ
2
.
ku5 kL∞
kD−1 u1 u2 k ∞ 23 + kD−1 u3 D−1 u4 kL∞ L3+
t Lx
x
t
Lt Lx
N2 N5
5
0+ Y
δ(1 ∧ Nmin
)
.
kui k 0, 21 + .
0+
X
Nmax
N 2− i=1
D.
1
1
b. N1 ≥ N . We obtain an additional factor ( NN1 ) 2 − ( NN2 ) 2 − , which can be compensated as in a.
Case 4: N3 , N4 & N & N5 .
a. N2 ≥ N N1 . We obtain
δ
N3 1/2 N4 1/2
−2
2 khDi
) ( )
khDi−2 (D−1 u1 u2 )kL∞
(u3 u4 )kL∞ L3−
t Lx
x
t
N
N
N2 N3 N4
× kD−1 u5 kL∞ L6+
x
D.(
t
N3
N4
δ
−1
1 kD
. ( )1/2 ( )1/2
kD−1 u1 u2 k ∞ 23 + ku3 u4 kL∞
u5 kL∞ L6+
t Lx
x
t
Lt Lx
N
N
N2 N3 N4
5
0+ Y
δ(1 ∧ Nmin
)
.
kui k 0, 21 + .
0+
X
Nmax
N 3− i=1
1
1
b. N1 ≥ N . We obtain an additional factor ( NN1 ) 2 − ( NN2 ) 2 − , which can be compensated by replacing the term kD−1 u1 u2 k ∞ 32 + in a. by
Lt Lx
1 .
kD−1 u1 u2 kL∞
t Lx
1
2 ku2 kL∞ L2 .
ku1 kL∞
x
t Lx
t
N1
32
H. PECHER
EJDE-2012/??
Case 5: N3 , N5 ≥ N ≥ N4 .
a. N2 ≥ N ≥ N1 . We have
N3 1/2 N5 1/2 δ
) ( )
khDi−2 (D−1 u1 D−1 u2 )kL∞ L3+
x
t
N
N
N3 N5
−2
−1
2
ku5 kL∞
× khDi (u3 D u4 )kL∞ L6−
t Lx
x
D.(
t
δ
2
ku3 D−1 u4 k ∞ 23 + ku5 kL∞
. 0+ 2− kD−1 u1 D−1 u2 kL∞ L3+
t Lx
x
t
Lt Lx
Nmax N
5
0+ Y
δ(1 ∧ Nmin
)
.
kui k 0, 12 + .
0+
X
Nmax
N 2− i=1
1
1
b. N2 ≥ N1 ≥ N . We obtain an additional factor ( NN1 ) 2 − ( NN2 ) 2 − , which can be
compensated by replacing the term kD−1 u1 D−1 u2 kL∞ L3+
in a. by
x
t
kD−1 u1 D−1 u2 kL∞ L1+
x
t
1
2 ku2 kL∞ L2 .
. 1− 1− ku1 kL∞
x
t
t Lx
N1 N2
Case 6: N3 , N4 , N5 ≥ N .
a. N2 ≥ N ≥ N1 . We have
δ
N3 1/2 N4 1/2 N5 1/2
6
) ( ) ( )
khDi−2 (D−1 u1 D−1 u2 )kL∞
t Lx
N
N
N
N3 N4 N5
3 ku5 kL∞ L2
× khDi−2 (u3 u4 )kL∞
x
t
t Lx
D.(
.
.
δ
kD
0+
Nmax
N 3−
−1
1 ku5 kL∞ L2
ku3 u4 kL∞
u1 D−1 u2 kL∞ L3+
x
t
t Lx
x
t
5
Y
0+
δ(1 ∧ Nmin
)
kui k 0, 12 + .
0+
X
Nmax N 3− i=1
1
1
b. N2 ≥ N1 ≥ N (⇒ Nmin & N ). We obtain an additional factor ( NN1 ) 2 − ( NN2 ) 2 − .
( NN1 )1− leading to
D.
.
.
δ
−2
3 ku5 kL∞ L2
6 khDi
(u3 u4 )kL∞
khDi−2 (u1 D−1 u2 )kL∞
0+
x
t
t Lx
t Lx
Nmax N 4−
δ
1 ku5 kL∞ L2
ku1 D−1 u2 kL∞ L3/2 ku3 u4 kL∞
0+
x
t Lx
t
x
t
Nmax N 4−
5
0+ Y
δ(1 ∧ Nmin
)
kui k 0, 21 + .
0+
X
Nmax N 4− i=1
which completes the proof of (4.9).
Next we want to prove the following estimate:
Z δ
h(W ∗Re Iu)Iu−I((W ∗Re u)u), I((W ∗|u|2 )u)idt .
0
δ
k∇Iuk5 0, 1 + . (4.10)
X 2
N 2−
We again consider D with
M (ξ1 , . . . ., ξ5 )
:=
5
Y
m(ξ3 + ξ4 + ξ5 )
|m(ξ1 + ξ2 ) − m(ξ1 )m(ξ2 )|
·
·
|ξi |−1 ,
m(ξ1 )m(ξ2 )hξ1 i2
m(ξ3 )m(ξ4 )m(ξ5 )hξ3 + ξ4 i2 i=1
EJDE-2012/??
NONLOCAL GROSS-PITAEVSKII EQUATION
33
and treat only the more difficult case N2 ≥ N1 , and assume without loss of generality N3 ≥ N4 . We consider the same cases as for (4.9).
In case 1 we replace khDi−2 (u1 u2 )k ∞ 56 by
Lt Lx
khDi−2 u1 u2 k
2 ku2 kL∞ L2 .
. ku1 kL∞
x
t Lx
t
6
5
L∞
t Lx
2 by
In the cases 2, 3 and 4a we replace khDi−2 (D−1 u1 u2 )kL∞
t Lx
−1
2 . kD
2 .
khDi−2 D−1 u1 u2 kL∞
ku2 kL∞
u1 kL∞ L6+
t Lx
t Lx
x
t
In case 4b. estimate
2 . ku1 kL∞ L2 ku2 kL∞ L2 .
khDi−2 D−1 u1 u2 kL∞
x
x
t Lx
t
t
Case 5a is essentially unchanged, whereas in Case 5b replace
khDi−2 (u3 D−1 u4 )kL∞ L6−
khDi−2 (D−1 u1 D−1 u2 )kL∞ L3+
x
x
t
t
by
khDi−2 (u3 D−1 u4 )kL∞ L∞−
khDi−2 D−1 u1 D−1 u2 kL∞ L2+
x
x
t
t
−1
2 kD
ku3 D−1 u4 k
u2 kL∞ L2+
. kD−1 u1 kL∞
t Lx
x
3+
2
L∞
t Lx
t
1
−1
2 ku2 kL∞ L2 ku3 kL∞ L2 kD
.
u4 kL∞ L6+
ku1 kL∞
x
x
t
t
t Lx
x
t
N1 N21−
.
In case 6a estimate
−2 −1
−1
∞ kD
6
6 . khDi
D u1 kL∞
u2 kL∞
khDi−2 D−1 u1 D−1 u2 kL∞
t Lx
t Lx
t Lx
2 .
ku2 kL∞
. kD−1 u1 kL∞ L6+
t Lx
x
t
Similarly case 6b can be handled, so that (4.10) is complete.
The forth and third order terms in |hF (Iu) − IF (u), IF (u)i| turn out to be less
critical. We omit any detailed calculations here and just refer to the recent paper
of the author [18], where the following estimates were given even under the weaker
c (ξ)| . 1 (compared to the property |W
c (ξ)| . hξi−2 which we have in
assumption |W
the present study). We have to remark that the assumption s ≥ 43 in that paper is
not really necessary for these forth and third order terms, but could be replaced by
1
Ni 12 −
i 4
s > 1/2, because the factors ( N
without
N ) can everywhere be replaced by ( N )
significance for the results. We have ([18], section 4.6, 4.7 and 4.8):
δ
Z
Z
Z
0
δ
0
|hI(u3 ) − (Iu)3 , Iui|dt . N −3+ k∇Iuk4
1
X 0, 2 + [0,δ]
0
δ
|hI(u2 ) − (Iu)2 , (Iu)2 i|dt . N −3+ k∇Iuk4
1
,
X 0, 2 + [0,δ]
,
5
|hI(u2 ) − (Iu)2 , Iui|dt . N − 2 + δ 1/2 k∇Iuk3
1
X 0, 2 + [0,δ]
Summarizing all our estimates in this chapter we arrive at (3.7).
.
34
H. PECHER
EJDE-2012/??
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Hartmut Pecher
Fachbereich Mathematik und Naturwissenschaften, Bergische Universität Wuppertal,
Gaußstr. 20, 42097 Wuppertal, Germany
E-mail address: pecher@math.uni-wuppertal.de