STRAUSS EXPONENT AND SMALL DATA GLOBAL EXISTENCE:
STRICHARTZ ESTIMATE FOR NON-ADMISSBLE PAIR
TADAHIRO OH
1. Introduction
Consider the nonlinear Schrödinger equation (NLS) with a power-type nonlinearity:
(
iut + ∆u = Np (u),
(x, t) ∈ Rd × R,
(1.1)
u t=0 = u0 ,
where the nonlinearity Np (u) = Np (u, u) is homogeneous of order p.
Local well-posedness of (1.1) easily follows from the Strichartz estimates, which we state
here for convenience. A pair (q, r) of exponents are called Schrödinger admissible if 2 ≤
q, r, ≤ ∞,
d
2 d
+ = ,
(1.2)
q
r
2
and (q, r, d) 6= (2, ∞, 2). Then, we have the following Strichartz estimates:
Lemma 1.1 (Strichartz estimate). Let (q, r) and (e
q , re) be admissible pairs. Then, we have
(a) homogeneous Strichartz estimate:
kS(t)u0 kLqt Lrx (R×Rd ) . ku0 kL2 (Rd )
(b) dual homogeneous Strichartz estimate:
ˆ
S(−t0 )F (t0 )dt0 R
L2 (Rd )
(1.3)
. kF kLqe0 Lre0 (R×Rd )
t
(c) nonhomogeneous (retarded) Strichartz estimate:
ˆ t
S(t − t0 )F (t0 )dt0 q r
. kF kLqe0 Lre0 (R×Rd ) .
d
t0
Lt Lx (R×R )
(1.4)
x
t
(1.5)
x
The contraction argument with the Strichartz estimates yields small data global existence 1
2
in the scaling critical regularity sc = d2 − p−1
; if ku0 kH sc is sufficiently small, then the
corresponding solution u exists globally in time.
In establishing global well-posedness, there are basically two kinds of results available; (i)
global existence by conservation laws and (ii) small data global existence. When Np (u) =
±|u|p−1 u, (1.1) enjoys the conservation of the L2 -norm and the Hamiltonian. In particular,
in the defocusing case, i.e. Np (u) = |u|p−1 u, we obtain a uniform control of the H 1 -norm of
a solution. This allows us to iterate the local argument and construct global-in-tim solutions
4
in the energy-subcritical regime: 1 < p < 1 + d−2
. In the focusing case, global existence
4
holds for p < 1 + d . In general, some finite-time blowup solutions are known, but global
1. For some global existence results mentioned in this note, scattering results are also known. However,
we will not dicuss them. Note also that below the short range exponent, i.e. p ≤ 1 + d2 , it is known that
scattering does not always hold.
1
2
TADAHIRO OH
existence still holds provided that the H 1 -norm (or L2 -norm if p = 1 + d4 ) are sufficiently
small. In the energy-critical case, the situation is more subtle. Nonetheless, global existence
holds in the defocusing case (and in the focusing case under some conditions.) Finally, no
global result is known in the energy-supercritical case. The uniform bound on the H 1 -norm
of a solution does not help us iterate the local argument in H s , s > 1.
In the following, we discuss small data global existence above the Strauss exponent pS
given by
√
d + 2 + d2 + 12d + 4
.
(1.6)
pS =
2d
Here, pS appears as the unique positive root of
dp2 − (d + 2)p − 2 = 0.
(1.7)
Note that pS lies between the short range exponent ps = 1 + d2 and the scaling L2 -critical
exponent pL2 = 1 + d4 .
In this note, we discuss the following small data global existence result above the Strauss
exponent:
4
Theorem 1.2. Let pS < p < 1 + d−2
and a > 0 be as in (2.9) below. Then, there exists
it∆
ε > 0 such that if u0 satisfies ke u0 kLa ([0,∞);Lp+1
≤ ε, then there exists a global solution
)
x
t
d
u to (1.1) with initial condition u0 . Moreover, u ∈ Lat ([0, ∞); Lp+1
x (R )).
Remark 1.3. If u0 ∈ H 1 , then it follows from Sobolev embedding ku0 kLp+1 . ku0 kH 1
that there exists ε0 > 0 such that ku0 kH 1 ≤ ε0 implies global existence. Moreover, u ∈
Lqte([0, ∞); Wx1,er (Rd )) for any Schrödinger admissible pair (e
q , re).
p+1
If u0 ∈ L2 ∩ L p with pS < p ≤ 1 + d4 , then the conclusion of Theorem 1.2 also holds,
0
i.e. that there exists ε0 > 0 such that ku0 k
p+1 ≤ ε implies global existence.
L2 ∩L
p
There are time-decay and scattering results as well, but we will not discuss them here.
Theorem 1.2 holds regardless of the precise form of the nonlinearity Np (u). Below the
Strauss exponent, however, the situation is quite different. On the one hand, GermainMasmoudi-Shatah [2] proved small data global existence (and scattering) for 2-d quadratic
NLS, i.e. p = 1 + d2 if the nonlinearity is given by N2 (u) = αu2 + βu2 . On the other hand,
Ikeda-Wakasugi [3] showed that for the nonlinearity Np (u) = λ|u|p with 1 < p ≤ 1 + d2 ,
there is a finite-time blowup and hence there is no global solution, provided
ˆ
ˆ
Re λ · Im
u0 (x) dx < 0, or Im λ · Re
u0 (x) dx > 0.
(1.8)
Rd
Rd
In particular, there is no global well-posedness even for small data. Note that this result
applies to 2-d quadratic NLS with N2 (u) = λ|u|2 .
Lastly, we point out that the proof of Theorem 1.2 is based on the dispersive estimate
(see (2.1) below). Such a dispersive estimate does not hold in the periodic setting. Indeed,
the recent result by Oh [4] shows a finite-time blowup for NLS on Td with the nonlinearity
Np (u) = λ|u|p , p > 1, as long as the condition (1.8) is satisfied. Hence, a general small
data global existence result as in Theorem 1.2 does not hold on Td .
SMALL DATA GLOBAL EXISTENCE
3
2. Proof of Theorem 1.2
The proof of Theorem 1.2 is based on the argument in Section 6.3 of Cazenave [1]. Recall
the following dispersive estimate:
d
. |t|− 2 ku0 kL1x ,
kS(t)u0 kL∞
x
t 6= 0.
(2.1)
r0
By interpolating (2.1) with kS(t)u0 kL2x = ku0 kL2x , we obtain the L -Lr estimate:
d
d
kS(t)u0 kLrx . |t|−( 2 − r ) ku0 kLr0 ,
x
t 6= 0,
(2.2)
for 1 ≤ r0 ≤ r ≤ ∞, where p0 is the Hölder conjugate of r.
Write (1.1) in the Duhamel formulation:
ˆ t
S(t − t0 )Np (u)(t0 )dt0 ,
u(t) = Γ(t)u := S(t)u0 +
(2.3)
0
where S(t) = eit∆ denotes the linear semigroup. The goal is to construct a solution to (2.3)
by the contraction mapping principle.
First, we consider the nonlinear part:
ˆ t
Φ(t)(u) :=
S(t − t0 )Np (u)(t0 )dt0 .
(2.4)
0
With r = p + 1 in (1.2), the Schrödinger admissible exponent q is given by see that
2
d
d
q = , where β := −
> 0.
(2.5)
β
2 p+1
By (2.2) and Hardy-Littlewood-Sobolev inequality, we have
ˆ t
d
d
0 −( 2 − p+1 )
0
0
kΦ(t)(u)kLa Lp+1
.
|t
−
t
|
kN
(u)(t
)k
p+1 dt
p
a
p
T x
Lx
0
. kNp (u)k
p+1
LbT Lx p
where LaT := La ([0, T ]) and
1+
,
LT
(2.6)
1
1
=β+ .
a
b
(2.7)
pb = a.
(2.8)
Now, we impose the following condition
Namely, we set
p−1
2(p − 1)(p + 1)
=
(2.9)
1−β
2p + 2 − dp + d
4
The denominator in (2.9) is positive if and only if p < 1 + d−2
; the Cauchy problem (1.1)
is energy-subcritical. With this choice of a and b, (2.6) yields
a=
kΦ(t)(u)kLa Lp+1
. kukLa Lp+1
.
x
x
T
(2.10)
T
Note that in applying Hardy-Littlewood-Sobolev inequality in (2.6), we need 1 < a, b, β1 <
4
∞. On the one hand, by the condition β < 1, we obtain p < 1 + d−2
. On the other hand,
by (2.8), we have p < a < ∞. With (2.9), this implies that
dp2 − (d + 2)p − 2 > 0
Hence, we must have pS < p < 1 +
4
d−2 .
=⇒
p > pS .
4
TADAHIRO OH
Once we establish the nonlinear estimate (2.10), the standard argument shows that Γ(t)
is a contraction on the ball
≤ Cε}
Bε = {u ∈ LaT Lp+1
: kukLa Lp+1
x
x
T
for any T > 0.
In the remaining part of this note, we discuss the results in Remark 1.3. First, we
estimate the linear part in the LaT Lp+1
x -norm. First, consider the case t ≥ 1. Here, we
impose the condition:
2a > q.
(2.11)
In particular, we have βa > 1 from (2.5) and (2.11). Then, by (2.2) with (2.5), we have
−β
.
|t|
ku
k
. ku0 k p+1 ,
(2.12)
p+1 kS(t)u0 kLp+1
0
a
a
x
p
p
Lt ([1,T ])
Lt ([1,T ])
Lx
Lx
where the implicit constant is independent of T . Next, we consider the case t < 1. In this
case, we impose the condition
q ≥ a.
(2.13)
Then, by the Strichartz estimate (1.3), we have
kS(t)u0 kLa ([0,1))Lp+1
≤ kS(t)u0 kLq ([0,1))Lp+1
. ku0 kL2x .
x
x
t
(2.14)
t
From (2.12) and (2.14), we conclude that
kS(t)u0 kLa ([0,1))Lp+1
≤ kS(t)u0 kLq ([0,1))Lp+1
. ku0 k
x
x
t
t
p+1
L2x ∩Lx p
.
(2.15)
Let us now consider the conditions (2.11) and (2.13). From (2.5) and (2.9), we see that
the condition (2.11) is equivalent to p > pS . Similarly, it follows from (2.5) and (2.9), we
see that the condition (2.13) is equivalent to p ≤ 1 + d4 .
Next, we consider the case u0 ∈ H 1 . Then, for t ≤ 1, by Sobolev embedding, we have
kS(t)u0 kLa ([0,1);Lp+1
. kS(t)u0 kLat ([0,1);Hx1 ) . ku0 kH 1 .
)
x
(2.16)
t
Note that in applying Sobolev embedding, we used the fact that p < 1 +
we can proceed as in (2.12) along with Sobolev inequality.
kS(t)u0 kLa ([1,∞);Lp+1
. ku0 k
)
x
p+1
Lx p
t
. ku0 kH 1 .
4
d−2 .
For t ≥ 1,
(2.17)
Hence, from (2.16) and (2.17), we conclude that ku0 kH 1 ≤ ε0 implies Theorem 1.2. Let
r = p + 1 and q as in (2.5). Then, we have pr0 = p and q10 = 1q + p−1
a , where a is as in (2.9).
Hence, given any Schrödinger admissible pair (e
q , re), from (1.5), we have
r
kΦ(t)(u)kLqeLWx1.er . kNp (u)kLq0 W 1,r0 ≤ kukp−1
La Lrx kukLq W 1,
t
t
x
t
t
x
(2.18)
First, by (2.18) with kukLat Lrx . ε, we can control kukLq W 1,r , i.e. (e
q , re) = (q, r). This
x
t
provides a control on kukLqeW 1,er for any Schrödinger admissible pair (e
q , re).
t
x
SMALL DATA GLOBAL EXISTENCE
Appendix A. Local well-posedness in high regularity setting: s >
5
d
2
In this appendix, we prove local well-posedness of (1.1) in the high regularity setting
s > d2 via Sobolev embedding: kukL∞ (M ) . kukH s (M ) , where M = Rd or Td . In particular,
we do not use Strichartz estimates. In the following, we consider Np (u) = |u|p , p ≥ 2.
First, we consider the case when p is an even integer. Let CT H s = C([−T, T ]; H s (M )).
Then, from (2.3), we have
kΓukCT H s ≤ ku0 kH s + T kukpCT H s ,
(A.1)
p−1
p−1
kΓu − ΓvkCT H s ≤ +T kukC
s + kvkC H s ku − vkCT H s .
TH
T
(A.2)
Here, we used the algebra property of H s , s > d2 along with the fact that we have an
p
p
algebraic nonlinearity |u|p = u 2 u 2 when p is an even integer. Then, by setting R = 2ku0 kH s
and T = T (R) sufficiently small, namely 2T Rp−1 < 1, we see that Γ is a contraction on the
ball BR of radius R in C([−T, T ]; H s ). Note that one also has unconditional uniqueness
in this case. 2 Here, unconditional uniqueness means uniqueness in C([−T, T ]; H s ) without
intersecting any auxiliary function spaces. For examples, if we use Strichartz estimates to
prove local well-posedness, then we have conditional uniqueness only in C([−T, T ]; H s ) ∩
LqT Lrx in such a case. A separate argument is required to prove unconditional uniqueness.
See Kato [5]
Next, we consider the case p > 2. In this case, the nonlinearity |u|p is no longer algebraic
p
and a care must be taken. With F (z) = |z|p = (zz) 2 , we have
∂z F (z) = p2 |z|p−2 z
and ∂z F (z) = p2 |z|p−2 z.
(A.3)
Then, we have
|u(x)|p − |u(y)|p = F (u(x)) − F (u(y))
ˆ 1
=
∂z F u(y) + θ(u(x) − u(y)) (u(x) − u(y))
0
+ ∂z F u(y) + θ(u(x) − u(y)) (u(x) − u(y))dθ.
In particular, from (A.3) and (A.4), we have
|u(x)|p − |u(y)|p . (|u(x)|p−1 + |u(y)|p−1 ) |u(x) − u(y)|.
(A.4)
(A.5)
Then, from (A.5)
ˆ
ˆ
1
|u(x)|p − |u(y)|p 2
2
dxdy
d+2s
|x − y|
M M
ˆ ˆ
1
2
p−1 |u(x) − u(y)|2
. |u|
dxdy
∞
L
d+2s
M M |x − y|
p
|u| s =
Ḣ
p
= kukp−1
L∞ kukḢ s . kukH s .
We also have |u|p L2 = kukpL2p . kukpH s from interpolation of L2p with L2 and L∞ and
Sobolev embedding. Hence, we obtain
p
|u| s . kukp s .
(A.6)
H
H
Then, (A.1) follows from (A.6) in this case.
2. Since we just constructed a contraction on BR , one still needs to eliminate the possibility of a solution
u with kukCT H s > R and u|t=0 = u0 to prove unconditional uniqueness.
6
TADAHIRO OH
Next, we estimate the difference Γu − Γv. By proceeding as before, we have
|u|p − |v|p = F (u) − F (v)
ˆ 1
∂z F v + θ(u − v) (u − v) + ∂z F v + θ(u(−v) (u − v)dθ.
=
(A.7)
0
From (A.3), (A.6) (with p − 1 instead of p), we have
∂z F v + θ(u − v) s . |v + θ(u − v)|p−1 Hs
H
p−1
kukH
s
kvkp−1
Hs
.
+
(A.8)
A similar estimate holds for ∂z F v + θ(u(−v) . Then, from (A.7) and (A.8), we have
p
p−1 |u| − |v|p s . kukp−1
s
(A.9)
H s + kvkH s ku − vkH .
H
Hence, (A.2) follows in this case as well. The rest of the argument is the same as before
and thus we omit details. Clearly, this argument fails for p < 2 precisely at (A.8) (since we
need p − 1 > 1 ). Nonetheless, (A.1) still holds even when p < 2 and one can show existence
of solutions by parabolic regularization and the energy estimate (A.1).
References
[1] T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10. New York
University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society,
Providence, RI, 2003. xiv+323 pp.
[2] P. Germain, N. Masmoudi, J. Shatah, Global solutions for 2D quadratic Schrödinger equations, J. Math.
Pures Appl. (2011), doi:10.1016/j.matpur.2011.09.008.
[3] M. Ikeda, Y. Wakasugi, Nonexistence of a non-trivial global weak solution for the nonlinear Schrödinger
equation with a nongauge invariant power nonlinearity, arXiv:1111.0178v2 [math.AP].
[4] T. Oh, A blowup result for the periodic NLS without gauge invariance, to appear in C. R. Math. Acad.
Sci. Paris.
[5] T. Kato, On nonlinear Schrödinger equations. II. H s -solutions and unconditional well-posedness, J.
Anal. Math. 67 (1995), 281–306.
Tadahiro Oh, Department of Mathematics, Princeton University, Fine Hall, Washington
Rd., Princeton, NJ 08544-1000, USA
E-mail address: hirooh@math.princeton.edu