Energy-supercritical NLS: critical $\dot H^s$-bounds imply scattering

Abstract

We consider two classes of defocusing energy-supercritical nonlinear Schr\"odinger equations in dimensions $d\geq 5$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\in L_t^\infty \dot H^{s_c}_x$, then $u$ is global and scatters.

Full text

Page 1

arXiv:0812.2084v1 [math.AP] 11 Dec 2008
ENERGY-SUPERCRITICAL NLS:
CRITICAL
˙Hs-BOUNDS IMPLY SCATTERING
ROWAN KILLIP AND MONICA VISAN
Abstract. We consider two classes of defocusing energy-supercritical nonlinear Schr¨ odinger
equations in dimensions d ≥ 5. We prove that if the solution u is apriorily bounded in
the critical Sobolev space, that is, u ∈ L∞
t
˙Hsc
x , then u is global and scatters.
1. Introduction
We consider the initial-value problem for the defocusing nonlinear Schr¨ odinger equation
in dimension d ≥ 5,
?
u(t = 0,x) = u0(x),
iut= −∆u + F(u)
(1.1)
where the nonlinearity F(u) = |u|pu is energy-supercritical, that is, p >
The class of solutions to (1.1) is left invariant by the scaling
4
d−2.
u(t,x) ?→ λ
2
pu(λ2t,λx). (1.2)
This defines a notion of criticality. More precisely, a quick computation shows that the only
homogeneous L2
critical regularity is sc:=d
than the critical regularity sc, we call the problem subcritical/supercritical.
We consider (1.1) for initial data belonging to the critical homogeneous Sobolev space,
that is, u0∈˙Hsc
lifespan solution that is uniformly bounded (throughout its lifespan) in˙Hsc
global and scatter. We were prompted to consider this problem by a recent preprint of
Kenig and Merle [15] which proves similar results for radial solutions to the nonlinear wave
equation in R3.
Let us start by making the notion of a solution more precise.
x-based Sobolev space left invariant by the scaling is ˙Hsc
2−2
x(Rd), where the
p. If the regularity of the initial data to (1.1) is higher/lower
x(Rd), in two regimes where sc> 1 and d ≥ 5. We prove that any maximal-
x(Rd) must be
Definition 1.1 (Solution). A function u : I × Rd→ C on a non-empty time interval
0 ∈ I ⊂ R is a solution (more precisely, a strong˙Hsc
class C0
t,x
(K × Rd) for all compact K ⊂ I, and obeys the Duhamel
formula
?t
for all t ∈ I. We refer to the interval I as the lifespan of u. We say that u is a maximal-
lifespan solution if the solution cannot be extended to any strictly larger interval. We say
that u is a global solution if I = R.
x(Rd) solution) to (1.1) if it lies in the
t˙Hsc
x(K × Rd) ∩ Lp(d+2)/2
u(t) = eit∆u(0) − i
0
ei(t−s)∆F(u(s))ds(1.3)
We define the scattering size of a solution to (1.1) on a time interval I by
?
SI(u) :=
I
?
Rd|u(t,x)|
1
p(d+2)
2
dxdt.

Page 2

2ROWAN KILLIP AND MONICA VISAN
Associated to the notion of solution is a corresponding notion of blowup. As we will
see in Theorem 1.6 below, this precisely corresponds to the impossibility of continuing the
solution.
Definition 1.2 (Blowup). We say that a solution u to (1.1) blows up forward in time if
there exists a time t1∈ I such that
S[t1,supI)(u) = ∞
and that u blows up backward in time if there exists a time t1∈ I such that
S(inf I,t1](u) = ∞.
We subscribe to the following conjecture.
Conjecture 1.3. Let d ≥ 1, p ≥4
to (1.1) such that u ∈ L∞
d, and let u : I × Rd→ C be a maximal-lifespan solution
x. Then u is global and moreover,
SR(u) ≤ C??u?L∞
for some function C : [0,∞) → [0,∞).
Our primary goal in this paper is to demonstrate how techniques developed to treat
the energy-critical NLS can be applied to Conjecture 1.3 in the regime sc ≥ 1, although
some of the arguments we will use were developed first in the mass-critical setting. As we
will describe, the appearance of the L∞
t
˙Hsc
illusory the supercriticality of the equation. The famed supercriticality of Navier–Stokes is
the fact that the problem is supercritical with respect to all quantities controlled by (known)
conservation/monotonicity laws; see, for instance, the discussion in [31]. In the context of
Conjecture 1.3, the assumption that the solution is uniformly bounded in ˙Hsc
role of the missing critical conservation law. It is not surprising therefore that techniques
developed to treat problems with true critical conservation laws should be applicable in this
setting. Next we review some of this work before describing the particular contribution of
this paper.
Mass and energy are the only known coercive conserved quantities for NLS; hence, the
corresponding critical NLS equations have received the most attention. In the mass-critical
case, the critical regularity is sc = 0 (i.e. p =
invariant, that is, the conserved quantity
?
Similarly, in the energy-critical case, the critical regularity is sc= 1 (i.e. p =
and the scaling (1.2) leaves invariant the energy,
?
which is also a conserved quantity for (1.1).
In the defocusing energy-critical case, it is known that all˙H1
solutions with finite scattering size. Indeed, this was proved by Bourgain [2], Grillakis
[11], and Tao [28] for spherically-symmetric initial data, and by Colliander–Keel–Staffilani–
Takaoka–Tao [8], Ryckman–Visan [24], and Visan [37, 38] for arbitrary initial data. For
results in the focusing case see [13, 19].
Unlike for the nonlinear wave equation (NLW), all known monotonicity formulae for
NLS (that is, Morawetz-type inequalities) scale differently than the energy. Ultimately,
the ingenious induction on energy technique of Bourgain (and the concomitant identifica-
tion of bubbles) introduces a length scale to the problem which makes it possible to use
t
˙Hsc
t
˙Hsc
x
?, (1.4)
x norm on the right-hand side of (1.4) renders
x plays the
4
d) and the scaling (1.2) leaves the mass
M(u) :=
Rd|u(t,x)|2dx.
4
d−2, d ≥ 3)
E(u) :=
Rd
?|∇u(t,x)|2+
1
2+p|u(t,x)|2+p?dx,
xinitial data lead to global

Page 3

ENERGY-SUPERCRITICAL NLS: CRITICAL
˙Hs-BOUNDS IMPLY SCATTERING3
non-invariantly scaling monotonicity formulae. All subsequent work has built upon this
foundational insight.
The Lin–Strauss Morawetz inequality used by Bourgain has ˙H1/2
adapted to the spherically-symmetric problem. To treat the non-radial problem, Colliander,
Keel, Staffilani, Takaoka, and Tao introduced an interaction Morawetz inequality; this has
˙H1/4
the reader may notice, in both cases the regularity associated to the monotonicity formulae
is lower than the critical regularity of the equation.
our belief that the techniques developed for treating the energy-critical problem should be
broadly applicable to Conjecture 1.3 whenever sc≥ 1/2. In this paper, we will by no means
complete this program, but rather have chosen to present some selected results that give
the flavor of our main thesis without becoming swamped with technicalities.
We turn our attention now to the defocusing mass-critical NLS. In this case, Conjec-
ture 1.3 has been proved for spherically-symmetric L2
see [18, 21, 35]. For a proof of the corresponding conjecture in the focusing case (for
spherically-symmetric L2
xdata with mass less than that of the ground state and d ≥ 2) see
[18, 21]. At present, we do not know how to deal with the Galilean symmetry possessed by
this equation, except through suppressing it by assuming spherical symmetry. We also note
that in this case, one needs to prove additional regularity (rather than decay) to gain access
to the known monotonicity formulae.
The first instance of Conjecture 1.3 to fall at non-conserved critical regularity was the
case sc= 1/2 in dimension d = 3. This was achieved by Kenig and Merle, [14]. They used
the concentration-compactness technique in the manner they pioneered in [13] together with
the Lin–Strauss Morawetz inequality.
The present paper is motivated by a recent preprint of Kenig and Merle, [15], who consider
spherically-symmetric solutions to a class of defocusing energy-supercritical nonlinear wave
equations in three dimensions. They prove that if the solution is known to be uniformly
bounded in the critical˙Hs
x-space throughout its lifetime, then the solution must be global
and it must scatter; this is precisely the NLW analogue of Conjecture 1.3.
Earlier, we drew a parallel to the Navier–Stokes equation. The most natural analogue of
Conjecture 1.3 in that setting is to show that boundedness of a critical norm implies global
regularity. Such results are known; see [9] and the references therein.
In this paper we prove Conjecture 1.3 in several instances of defocusing energy-supercritical
nonlinear Schr¨ odinger equations in dimensions d ≥ 5 for arbitrary (not necessarily spherically-
symmetric) initial data. The approach we take is modeled after [19], which considers the
energy-critical problem in dimensions d ≥ 5 in both the defocusing and focusing cases.
We will consider two different settings. In the first one, the nonlinearity is cubic, that is,
F(u) = |u|2u, and hence the critical regularity is sc=d−2
this problem is energy-supercritical.
The second problem we consider is that of general (not necessarily polynomial) energy-
supercritical nonlinearities in dimension d ≥ 5, that is, p >
impose some additional constraints on the power p. First, we ask that the nonlinearity
obeys a certain smoothness condition; more precisely, we ask that sc < 1 + p, which is
equivalent to 2p2− p(d − 2) + 4 > 0. The role of this constraint is to allow us to take
sc-many derivatives of the nonlinearity F(u); this is important in the development of the
local theory. Moreover, in Section 6, we require that scand p obey some further constraints.
Together, these amount to
x -scaling and is best
x -scaling, which is even further from the˙H1
x-scaling of the energy-critical problem. As
This is the philosophical basis of
xinitial data in all dimensions d ≥ 2;
2. Note that in dimension d ≥ 5,
4
d−2. In this case, we also
?
1 < sc<3
2
d+2−√
for d = 5,6
1 < sc<
(d−2)2−16
4
for d ≥ 7.
(1.5)

Page 4

4 ROWAN KILLIP AND MONICA VISAN
One should not view (1.5) as a major constraint on the size of the critical regularity sc.
Indeed, we claim that an interpolation of the techniques we present to treat the two problems
outlined above can be used to treat any defocusing energy-supercritical NLS (with the
solution apriorily bounded in˙Hsc
x) in dimensions d ≥ 5, without any additional constraint
on sc if the power p is an even integer and requiring merely the smoothness condition
sc< 1+p for arbitrary powers p. However, for the sake of readability, we chose not to work
in this greater generality.
Our main results are the following:
Theorem 1.4 (Spacetime bounds – the cubic). Let d ≥ 5 and F(u) = |u|2u. Let u :
I × Rd→ C be a maximal-lifespan solution to (1.1) such that u ∈ L∞
u is global and moreover,
SR(u) ≤ C??u?
Theorem 1.5 (Spacetime bounds). Let d ≥ 5 and assume the critical regularity scsatisfies
(1.5). Let u : I ×Rd→ C be a maximal-lifespan solution to (1.1) such that u ∈ L∞
Rd). Then u is global and moreover,
SR(u) ≤ C??u?L∞
As we already mentioned, the proofs of Theorems 1.4 and 1.5 follow closely the approach
taken in [19] to study the energy-critical problem. We outline the argument in subsection 1.1
below. Our decision to work in dimensions d ≥ 5 was motivated by the fact that it allows
us to employ some techniques used in [19], in particular, the double Duhamel trick. The
natural approach in lower dimensions would be to use the frequency localized interaction
Morawetz inequality in the spirit of [8]. While subsequent developments (some of which
are reviewed in this paper) lead to simplifications of [8], this would still be a significant
undertaking and we do not pursue it here.
The arguments presented here apply mutis mutandis to the corresponding Hartree equa-
tions; indeed, the fact that the nonlinearity depends polynomially on u for that equation
means that it resembles the simpler cubic case treated here. We also believe that the argu-
ments adapt to the corresponding energy-supercritical wave equations in dimensions d ≥ 6;
however, we have not worked through the details.
To study the global theory for (1.1), we must first develop a local theory for this equation.
To this end, we revisit arguments by Cazenave and Weissler, [4], who treated the case
0 ≤ sc≤ 1, as well as more sophisticated stability results in the spirit of [23, 32].
Theorem 1.6 (Local well-posedness). Let d and scbe as in Theorem 1.4 or 1.5. Then, given
u0∈˙Hsc
to (1.1) with initial data u(t0) = u0. This solution also has the following properties:
• (Local existence) I is an open neighbourhood of t0.
• (Blowup criterion) If supI is finite, then u blows up forward in time (in the sense of
Definition 1.2). If inf I is finite, then u blows up backward in time.
• (Scattering) If supI = +∞ and u does not blow up forward in time, then u scatters
forward in time, that is, there exists a unique u+∈˙Hsc
lim
t
˙H
d−2
2
x
(I × Rd). Then
L∞
t
˙H
d−2
2
x
?.
t
˙Hsc
x(I ×
t
˙Hsc
x
?.
x(Rd) and t0∈ R, there exists a unique maximal-lifespan solution u : I × Rd→ C
x(Rd) such that
t→+∞?u(t) − eit∆u+?˙Hsc
x (Rd)= 0.(1.6)
Conversely, given u+∈˙Hsc
infinity so that (1.6) holds.
• (Small data global existence) If
u is a global solution which does not blow up either forward or backward in time. Indeed, in
this case SR(u) ?
2
.
x(Rd) there is a unique solution to (1.1) in a neighbourhood of
??|∇|scu0?2is sufficiently small (depending on d,p), then
??|∇|scu0
??p(d+2)/2

Page 5

ENERGY-SUPERCRITICAL NLS: CRITICAL
˙Hs-BOUNDS IMPLY SCATTERING5
In Section 3 we establish this theorem as a corollary of our stability results Theorems 3.3
and 3.4. These stability results are essential to the arguments we present, more specifically,
to the proof of Theorem 1.12.
1.1. Outline of the proofs of Theorems 1.4 and 1.5. We argue by contradiction. We
show that if either Theorem 1.4 or Theorem 1.5 failed, this would imply the existence of a
very special type of counterexample. Such counterexamples are then shown to have a wealth
of properties not immediately apparent from their construction, so many properties, in fact,
that they cannot exist.
While we will make some further reductions later, the main property of the special coun-
terexamples is almost periodicity modulo symmetries:
Definition 1.7 (Almost periodicity modulo symmetries). Suppose sc> 0. A solution u to
(1.1) with lifespan I is said to be almost periodic modulo symmetries if there exist functions
N : I → R+, x : I → Rd, and C : R+→ R+such that for all t ∈ I and η > 0,
?
We refer to the function N as the frequency scale function for the solution u, x the spatial
center function, and to C as the compactness modulus function.
|x−x(t)|≥C(η)/N(t)
??|∇|scu(t,x)??2dx +
?
|ξ|≥C(η)N(t)
|ξ|2sc|ˆ u(t,ξ)|2dξ ≤ η.
Remark 1.8. The parameter N(t) measures the frequency scale of the solution at time
t, while 1/N(t) measures the spatial scale. It is possible to multiply N(t) by any function
of t that is bounded both above and below, provided that we also modify the compactness
modulus function C accordingly.
Remark 1.9. When sc= 0 the equation admits a new symmetry, namely, Galilei invari-
ance. This introduces a frequency center function ξ(t) in the definition of almost periodicity
modulo symmetries; see [17, 34] for further discussion.
Remark 1.10. By the Ascoli–Arzela Theorem, a family of functions is precompact in
˙Hsc
C so that
?
for all functions f in the family. Thus, an equivalent formulation of Definition 1.7 is as
follows: u is almost periodic modulo symmetries if and only if
x(Rd) if and only if it is norm-bounded and there exists a compactness modulus function
?
|x|≥C(η)
??|∇|scf(x)??2dx +
|ξ|≥C(η)
|ξ|2sc|ˆf(ξ)|2dξ ≤ η
{u(t) : t ∈ I} ⊆ {λ
2
pf(λ(x + x0)) : λ ∈ (0,∞), x0∈ Rd, and f ∈ K}
x(Rd). for some compact subset K of˙Hsc
Remark 1.11. A further consequence of compactness modulo symmetries is the existence
of a function c : R+→ R+so that
?
for all t ∈ I and η > 0.
With these preliminaries out of the way, we can now describe the first major milestone
in the proof of Theorems 1.4 and 1.5.
|x−x(t)|≤c(η)/N(t)
??|∇|scu(t,x)??2dx +
?
|ξ|≤c(η)N(t)
|ξ|2sc|ˆ u(t,ξ)|2dξ ≤ η
Theorem 1.12 (Reduction to almost periodic solutions).
Theorem 1.5) failed. Then there exists a maximal-lifespan solution u : I × Rd→ C to (1.1)
such that u ∈ L∞
forward and backward in time. Moreover, u has minimal L∞
solutions, that is,
sup
t∈I
Suppose that Theorem 1.4 (or
t
˙Hsc
x(I ×Rd), u is almost periodic modulo symmetries, and u blows up both
t
˙Hsc
x-norm among all blowup
??|∇|scu(t)??
2≤ sup
t∈J
??|∇|scv(t)??
2