Nonlinear dispersive equations with random initial c ata
by
ARCHIVES
MASSACHUSETTS INSTITUTE
OF TECHNOLOLGY
Dana Sydney Mendelson
JUN 3 0 2015
B.Sc., McGill University (2010)
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
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Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
@ Dana Sydney Mendelson, MMXV. All rights reserved.
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A u th o r . .... ..... ......................... ........... ......................
Department of Mathematics
May 1, 2015
Certified by.
Signature redacted ..............................
Gigliola Staffilani
/ Abby Rockefeller Mauz6 Professor of Mathematics
Thesis Supervisor
Signature redacted
A ccepted by . .
..............................
Alexei Borodin
Chairman, Department Committee on Graduate Theses
Nonlinear dispersive equations with random initial data
by
Dana Sydney Mendelson
Submitted to the Department of Mathematics
on May 1, 2015, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
In the first part of this thesis we consider the defocusing nonlinear wave equation of power-type on
R 3 . We establish an almost sure global existence result with respect to a suitable randomization of
the initial data. In particular, this provides examples of initial data of supercritical regularity which
lead to global solutions. The proof is based upon Bourgain's high-low frequency decomposition
and improved averaging effects for the free evolution of the randomized initial data.
In the second part of this thesis, we consider the periodic defocusing cubic nonlinear KleinGordon equation in three dimensions in the symplectic phase space H2 (T3 ) x H--i 3 ). This space
is at the critical regularity for this equation, and in this setting there is no global well-posedness
nor any uniform control on the local time of existence for arbitrary initial data. We prove several
non-squeezing results: a local in time result and a conditional result which states that uniform
bounds on the Strichartz norms of solutions for initial data in bounded subsets of the phase space
implies global-in-time non-squeezing. As a consequence of the conditional result, we conclude nonsqueezing for certain subsets of the phase space and, in particular, we obtain deterministic small
data non-squeezing for long times.
To prove non-squeezing, we employ a combination of probabilistic and deterministic techniques.
Analogously to the work of Burq and Tzvetkov, we first define a set of full measure with respect to a
suitable randomization of the initial data on which the flow of this equation is globally defined. The
proofs then rely on several approximation results for the flow, one which uses probabilistic estimates
for the nonlinear component of the flow map and deterministic stability theory, and another which
uses multilinear estimates in adapted function spaces built on UP and VP spaces. We prove
non-squeezing using a combination of these approximation results, Gromov's finite dimensional
non-squeezing theorem and the infinite dimensional symplectic capacity defined by Kuksin.
Thesis Supervisor: Gigliola Staffilani
Title: Abby Rockefeller Mauz6 Professor of Mathematics
Acknowledgments
I would like to thank my advisor Gigliola Staffilani, who has made this thesis possible and given
me so much of her time over these past five years. She has taught me an incredible amount about
mathematics and being a mathematician, all with an immense amount of kindness and patience.
I owe much gratitude to Andrea Nahmod for many mathematical conversations and plentiful
sound advice, and to Michael Eichmair for support and guidance during some stressful times. I
would also like to thank David Jerison and Jared Speck for serving on my thesis committee, and
Jonas Liihrmann for our extremely enjoyable collaboration (which has, in particular, given rise to
the contents of Chapter 2 of this thesis).
My time at MIT would have been nowhere near as enjoyable without the friendship of my
fellow classmates, particularly Michael Andrews, Nate Bottman, Saul Glasman, Jiayong Li and
Alex Moll, and I am grateful to them for making these past years at the math department as
colorful as they have been.
I would like to thank my family and friends for their support, including my grandparents for
having set the best possible example for us all. My parents, Beverley and Morton, have given me
so much throughout my life. I would like to specifically thank them for the confidence they have
always had in me, and for believing that I was capable of finishing this thesis, even at times when
I did not.
Finally, thank you, Michael, for being my family and my home.
Table of Contents
Introduction
.
. . . .
12
1.1.2
Symplectic non-squeezing for the cubic NLKG . . . . . .
15
.
Random data Cauchy theory on Euclidean space
Notation and conventions
. . . . . . . . . . . . . . . . . . . . .
21
1.3
Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . .
23
.
.
1.2
25
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2
Deterministic and Probabilistic estimates
. . . . . . . . . . . .
31
2.2.1
Large deviation estimate . . . . . . . . . . . . . . . . . .
32
2.2.2
Averaging effects for the randomized initial data
. . . .
33
2.3
Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . .
36
2.4
Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . .
44
2.5
Appendix A: Probabilistic estimates
46
.
.
.
.
.
.
Random data Cauchy theory for NLW of power-type on R3
.
. . . . . . . . . . . . . . .
49
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . .
49
3.2
Set-up . . . . . . . . . . . . . . . . . . . .
51
3.3
Strichartz estimates
. . . . . . . . . . . .
54
3.4
Multilinear estimates . . . . . . . . . . . .
58
3.5
Stability theory in adapted function spaces
64
3.6
A low frequency equation
. . . . . . . . .
70
3.7
Proof of Theorem 3.1 . . . . . . . . . . . .
73
.
.
.
An approximation result for the cubic NLKG in the critical space
Symplectic non-squeezing for the cubic NLKG on
In 3
75
75
4.1.1
Overview of Proof
. . . . . . . . . . .
78
4.1.2
Conditional non-squeezing . . . . . . .
85
4.1.3
Probabilistic non-squeezing
. . . . . .
86
4.1.4
Organization of Chapter . . . . . . . .
87
.
.
.
Introduction . . . . . . . . . . . . . . . . . . .
.
4.1
.
4
9
1.1.1
.
3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
2
O verview
.
1.1
9
.
1
6
. . . . . . . . . . .
89
. . . . . . . . . . .
90
.
.
An infinite dimensional symplectic capacity . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
Deterministic preliminaries
. . . . . . . . . . . . . .
. . . . . . . . . . .
90
4.3.2
Probabilistic preliminaries . . . . . . . . . . . . . . .
. . . . . . . . . . .
91
. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . .
92
A . . . . . . . . . . . .
. . . . . . . . . . .
95
Probabilistic bounds for the nonlinear component of the flow
. . . . . . . . . . .
96
4.5.1
Boundedness of the flow map . . . . . . . . . . . . .
. . . . . . . . . . .
98
4.5.2
Boundedness of the flow with truncated nonlinearity
. . . . . . . . . . . 102
4.5.3
Continuity estimates for the flow map
Well-posedness theory
.
.
.
4.3.1
.
Prelim inaries
4.4.1
4.5
87
.
4.4
. . . . . . . . . . .
.
4.2.1
4.3
. . . . . . . . . . . . . . . . . . .
Symplectic Hilbert spaces
Definition and properties of
.
4.2
. . . . . . . . . . . 102
4.6
Probabilistic approximation of the flow of the NLKG . . . .
. . . . . . . . . . . 105
4.7
Approximation of the flow on open sets
. . . . . . . . . . .
. . . . . . . . . . . 109
4.7.1
Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 109
4.7.2
Proof of Theorem 4.8 . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 111
4.7.3
Proof of Theorem 4.7. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 113
Appendix A: Stability Arguments . . . . . . . . . . . . . . .
. . . . . . . . . . . 115
4.8.1
Stability theory for NLKG . . . . . . . . . . . . . . .
. . . . . . . . . . . 116
4.8.2
Stability theory for the truncated NLKG . . . . . . .
. . . . . . . . . . . 119
Appendix B: Probabilistic bounds for the cubic NLKG . . .
. . . . . . . . . . . 122
4.9
.
.
.
.
.
.
.
.
.
4.8
.
. . . . . . . .
A Facts from Harmonic analysis
125
Strichartz Estimates
. . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 125
A.2
Adapted Function spaces . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 127
A.2.1
Xsb spaces
. . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 127
A.2.2
UP and VP Function spaces
. .
. . . . . . . . . . . . . . . . . . . . . . . . 128
.
.
.
.
A.1
133
Bibliography
7
8
Chapter 1
Introduction
1.1
Overview
Nonlinear dispersive Partial Differential Equations (PDEs) model wave propagation phenomena
in many physical systems.
For the last several decades, the study of dispersive equations has
focused on questions about the existence and uniqueness of solutions, their asymptotic behaviour,
and singularity formation.
Many of these equations enjoy a scaling symmetry which gives rise
to the notion of criticality and relatedly, that of subcritical and supercritical equations.
While
there is a well established local theory in the subcritical or critical regimes, there are ill-posedness
results, for example [17], [39] and [31], which show that one cannot expect local well-posedness for
all initial data at supercritical regularities. Moreover, in cases where the critical regularity does
not correspond to a conserved quantity, global deterministic results still do not even reach critical
regularities.
In recent years, probabilistic tools have been extremely useful in obtaining almost sure wellposedness theorems in super-critical regimes, as well as in closing the gap between the scaling
prediction and existing deterministic results. One fruitful vein of research has been the study of
invariant Gibbs measures for Hamiltonian PDEs. Such measures have been studied in the work
of Zhidkov [73, 74, 75], Lebowitz, Rose and Speer [40] and subsequently by many others.
In
[6, 8], working with the Gibbs measure introduced in [40], Bourgain proved the existence of a welldefined Hamiltonian flow on the support of this measure for the nonlinear Schrddinger equation
in one and two spatial dimensions. Bourgain then used the invariance of this measure to prove
almost sure global well-posedness for these equations, for supercritical initial data. In [14, 15],
9
Burq and Tzvetkov consider the cubic nonlinear wave equation on a three-dimensional compact
manifold. They construct large sets of initial data of supercritical regularities that give rise to
local solutions, using a randomization procedure which relies on expansion of the initial data with
respect to an orthonormal basis of eigenfunctions of the Laplacian. Together with invariant measure
considerations, they prove almost sure global existence for the cubic nonlinear wave equation on
the three-dimensional unit ball. Many further results in this direction have been obtained in recent
years, see [5], [9], [15], [69], [70], [68], [51], [46], [47], [58], [11], [71] and references therein.
While it is desirable to establish the existence of an invariant Gibbs measure, in many situations
there are serious technical difficulties associated with defining such a measure. Most notably, in
dimension d > 3 it is only possible to define a Gibbs measure for initial data in very rough Sobolev
spaces.
In such spaces, the multilinear analysis necessary for well-posedness arguments is not
available.
In the absence of an invariant measure, other approaches have been developed to prove almost
sure global existence for supercritical equations via a suitable randomization of the initial data.
Energy estimates are one such approach, which was used, for instance, by Nahmod, PavloviV and
Staffilani [49] in the context of the periodic Navier-Stokes equation in two and three dimensions
and by Burq and Tzvetkov [16] for the three-dimensional periodic defocusing cubic nonlinear
wave equation. Another approach was employed by Colliander and Oh in [19], where they adapt
Bourgain's high-low frequency decomposition [101 to prove almost sure global existence of solutions
to the one-dimensional periodic defocusing cubic nonlinear Schr6dinger equation below L 2 (T).
In this thesis, we pursue the study of dispersive equations via probabilistic techniques, particularly in the absence of an invariant measure. We will focus mainly on two specific equations: the
defocusing nonlinear wave equation with power type nonlinearity
(NLW),
utt-Au
juj
(u, ut)ItO
=
1 u=0,
u:Rx(R 3 -. R
(fo, fi) C Hs(R 3 ) x Hs-1 (R 3 )(
and the defocusing cubic nonlinear Klein-Gordon equation
Utt-AU +
c
(NLKG)
+
3u=0,
u:IRx T
3
-+ R
(u, tu) J=O = (uo, ui) E Hs(T 3 ) x Hs-33
10
(1.2)
where m is the mass, c is the speed of light and h is Planck's constant. Here HS denotes the usual
inhomogenous Sobolev space. We will normalize m = c = h = 1.
The nonlinear wave equation enjoys the scaling symmetry
u(t, x)
-+
u\(t, x) := A2 /(p-)u(At, Ax)
and the scale invariant critical space corresponds to the homogeneous Sobolev space at regularity
sc :=
-
. Additionally, (1.1) has a conserved Hamiltonian
HNLW(U(t))
=
2
IV XU2 +
10tU2 +
2
alternatively called the energy, which controls the k
1
p
ulp+ 1
1
+I
Sobolev norm of solutions.
We will use
the terminology subcritical (respectively critical or supercritical) to refer to regularities above
(respectively at or below) the scale invariant Sobolev space
H'c.
The Hamiltonian is also called
the energy, and occasionally we will refer to the equation (NLW), as energy subcritical (respectively
critical and supercritical) if one is interested in the Cauchy problem (1.1) with p < 5 (respectively
p = 5 and p > 5) since the scale invariant Sobolev space for p = 5 is at regularity s, = 1, which is
controlled by the energy functional.
There is no scaling symmetry for the nonlinear Klein-Gordon equation due to the presence of the
mass term. However, since ill-posedness and blowup are normally associated with high frequencies
or short time scales, the nonlinear term dominates the mass term and one can still regard s, as
the critical regularity for this equation. The Hamiltonian for the nonlinear Klein-Gordon equation
has the form
i p+f
HNLKG(U(t))
JV
ul 2 + C
J
U12 +
J
tu12
+
l
The presence of the L 2 norm provides control over low frequencies of solutions, which makes the
nonlinear Klein-Gordon equation somewhat less singular than the nonlinear wave equation. On
Euclidean space, a useful heuristic is that the Klein-Gordon behaves like the Schrbdinger equation
at frequencies
< c where c is the speed of light, and the wave equation at frequencies > c. In
certain cases, this heuristic can be made rigorous, and one can prove that in the non-relativistic
limit as c -+ oc one recovers the nonlinear Schrbdinger equation from (1.2), see for instance [43].
11
Solutions to both the linear and nonlinear Klein-Gordon equations exhibit finite speed of propagation, which mean that information can only propagate at the speed of light. Because of this,
there is usually no difference between compact and non-compact settings if one localizes in time.
In the sequel, we will always consider solutions to the Klein-Gordon equation on bounded time
intervals and hence we will often make use of this fact and carry over Strichartz estimates from
the Euclidean to the periodic setting.
1.1.1
Random data Cauchy theory on Euclidean space
In the first part of this thesis, we will focus on the study of the nonlinear wave equation with
power type nonlinearities on Euclidean space, without any radial symmetry assumption on the
initial data. This is joint work with J. Liihrmann, which has appeared in [42]. Many previous
results on Euclidean space have involved first considering a related equation in a setting where
an orthonormal basis of eigenfunctions of the Laplacian exists. This orthonormal basis is used to
randomize the initial data for the related equation and an appropriate transform is then used to
map solutions of the related equation to solutions of the original equation. Instead of using such
transformations, we randomize functions directly on Euclidean space via a unit-scale decomposition
in frequency space. More precisely, we fix a nonzero Q E
C (R3 ) with supp b C B(O, 2), define
the Fourier projection operator
Pkf( )
Let {hk,
lk}keZ3
( - k)f((),
k E Z3
denote a sequence of iid, mean-zero Gaussian random variables on a probability
space (Q, T, P) and consider the randomized initial data
(1.3)
fW = (f,f'):= (>E hk(w)Pkfo, E lk(W)Pf).
kEZ
3
kEZ
3
We crucially exploit that these unit projections satisfy a unit-scale Bernstein inequality. In Chapter
2, we prove the following theorem for energy subcritical nonlinearities.
Theorem 2.1 (Liihrmann-Mendelson, [42]). Let 3 < p < 5 and let
f = (fo, fl) E HI(R3 ) x H- 1((R 3 ) and let fW =
(fo, fw)
p
3
2-1p3
< s < 1.
Fix
be the randomized initial data defined in
(1.3), and u" the associated free evolution. For almost every w E Q there exists a unique global
12
solution
(u, ut)
E
(u', atu') + C(R; Hx(R 3 ) x L2 (R3))
to the Cauchy problem for the nonlinear wave equation (NLW)p
-utt
+ Au = JuIP-lu on R x R
(U, ut) It= = (0,
A).
S
1.0-
0.9-
0.8
0.7-
0.6
3.5
Figure 1.1: The dashed line is the critical regularity s,
for the exponent s in Theorem 2.1.
Remark 1. For 1(7
+ v ')
5.0
4.5
4.0
= 2
22
p-1
The solid line is the threshold
~ 3.89 < p < 5, the range of allowable regularity exponents s in
Theorem 2.1 contains supercriticalexponents; see Figure 1.1. To the best of the authors'knowledge,
for 4 < p < 5, Theorem 2.1 is the first result which establishes the existence of large sets of initial
data of supercriticalregularity which lead to global solutions to (NLW)P.
The proof of Theorem 2.1 combines a probabilistic local existence argument with Bourgain's
high-low frequency decomposition [10], an approach introduced by Colliander and Oh [19, Theorem
2] to show almost sure global existence of the one-dimensional periodic defocusing cubic nonlinear
Schr6dinger equation below L2 (T). The high-low method was previously used by Kenig, Ponce
and Vega [34, Theorem 1.2] to prove global well-posedness of (1.1) for 2 < p < 5 for initial data
in a range of sub-critical spaces below the energy space (see also [1], [24] and [60] for p = 3). We
will discuss this further in Chapter 2.
13
When p = 5, the nonlinear wave equation (NLW), is energy critical and the high-low argument,
being an energy subcritical technique, no longer works. However, a similar argument to the one
used to prove the key averaging effects for Theorem 2.1 yields bounds on the L5L1 0 norm of the
free evolution u'. Together with straightforward contraction and bootstrap arguments we obtain
the following theorem.
xH-
1
(R3
)
Theorem 2.2 (Liihrmann-Mendelson, [42]). Let 2 < s < 1. Fix f = (fo, fi) E Hx(R 3 )
and let fW = (fo', fl') be the randomized initial data defined in (1.3), and uw the associated free
evolution. There exists Qf C Q with
P(QfI);1 - C exp (-C/llf12
for some absolute constants C, c > 0, such that for every w E Qf there exists a unique global
solution
(u, ut) E (uf, atuw) + C(R; Q(R 3) x L (R3))
to the quintic nonlinear wave equation
+ Au
=
u1 4 u on
(u, ut)Ito
=
(fU, ffl.
R
x
R3 ,
(1.4)
Moreover, (1.4) scatters in the sense that for every w c Qf there exist (v 1 ,v 2 )
c H(R3 ) x L.(R 3
)
S-utt
such that the free evolution v(t) = cos(t|V|)v 1 + sinti) v 2 satisfies
(u(t)
-
uy(t)
-
v(t), atu(t) - 9t u"(t) - atv(t)) |Hk(R3)xLl(R3)
-+ 0 as t
-
oo.
In [54], Pocovnicu proves almost sure global well-posedness for the energy critical defocusing
nonlinear wave equation on Rd for d = 4, 5. Her arguments are based on a probabilistic perturbation
theory which relies on the fact that the cubic nature of the nonlinearity she considers allows one to
treat the difference equation for the nonlinear component of the solution perturbatively and one
can use the energy to obtain the necessary bounds for the perturbation argument. Recently, Oh
and Pocovnicu have obtained control of an energy functional in [52] for the quintic equation, which
similarly yields global existence of solutions when combined with the probabilistic perturbation
theory. Such methods only yield a mild form of uniqueness in the energy critical setting.
14
1.1.2
Symplectic non-squeezing for the cubic NLKG
In the second part of the thesis, we study symplectic non-squeezing for the periodic defocusing
cubic nonlinear Klein-Gordon equation
utt - Au+u+u
(u, tu)
0
3
=0,
u:lRx T 3 -+ R
1./2 (T 3 ),
= (uo, ui) E H1(T3) x H-21(T3
where H2 (T) is the usual inhomogeneous Sobolev space.
Local strong solutions to (1.5) can be constructed by adapting the arguments from [41] to the
compact setting. Due to the critical nature of this problem, however, the local time of existence
for solutions depends not only on the norm of the initial data but also on its profile. Moreover,
as the critical regularity for this equation does not correspond to a conserved quantity, global
well-posedness for this equation remains open. The best known results on Euclidean space are
for subcritical regularities s > 3/4, which was proved by Miao, Zhang and Fang [45] by working
in Besov spaces and adapting Bourgain's high-low argument [10] and arguments from [34]. Once
again, these results can be adapted to the periodic setting. We recall that the L' Strichartz norm
controls the global well-posedness for (1.5) and we have the standard finite time blow-up criterion,
namely if T. denotes the maximal time of existence for a solution u to (1.5) then
T* < 00
==
IIUI|LG([OT,)xT3)
=
oo.
(1.6)
We are interested in studying symplectic non-squeezing for (1.5), which is formally an infinite
dimensional Hamiltonian system, as we will see in Section 4.2. In the finite dimensional setting,
Gromov's celebrated non-squeezing theorem states that there is no symplectic embedding of a ball
into a cylinder unless the radius of the ball is less than or equal to that of the cylinder. Smooth
finite-dimensional Hamiltonian flows are particular examples of symplectomorphisms and hence,
they exhibit non-squeezing by Gromov's theorem.
One may wonder which parts of this theory, if any, carry over to infinite dimensions. There is a
natural symplectic structure on Sobolev spaces, and for infinite dimensional Hamiltonian equations,
there are specific regularities where this structure is compatible with the flow of the equation,
and Hamiltonian PDEs can be realized, at least formally, as a symplectic flow on these infinite
dimensional phase spaces. Beyond the question of merely generalizing Gromov's theorem to certain
15
infinite dimensional cases, symplectic non-squeezing for Hamiltonian equations is connected to the
problem of weak turbulence, which examines, for example, whether the energy of a given solution
concentrates at high frequencies over time, see Section 4.2 for a discussion of this interpretation.
Gromov proved his non-squeezing theorem by showing that a certain symplectic capacity,
called the Darboux width, is invariant under the flow of a symplectomorphism via a sophisticated
analysis using pseudoholomorphic curves. Symplectic capacities, which we define for symplectic
Hilbert spaces in Subsection 4.2.1, are an important invariant of symplectic flows. Subsequent
to Gromov's proof, there were other (comparable but not necessarily equivalent) definitions of
symplectic capacities, introduced by Ekeland-Hofer, Viterbo, and Hofer-Zehnder among others.
We refer the reader [30] and references therein for a more thorough account of these developments.
The study of infinite dimensional symplectic capacities and non-squeezing for nonlinear Hamiltonian PDEs was initiated by Kuksin in [38]. There, he extended the definition of the HoferZehnder capacity to infinite dimensional Hilbert spaces and proved the invariance of this capacity
under the flow of certain Hamiltonian equations with flow maps of the form
(t) = linear operator + compact smooth operator.
(1.7)
This infinite dimensional symplectic capacity inherits the finite dimensional normalization
cap(Br(u*)) = cap(Cr(z; ko))
=
rr2
where Br(U*) is the infinite dimensional ball centered at u, in the Hilbert space, and Cr(z; ko), the
infinite dimensional cylinder, see (1.14) for the precise definition of the cylinder in our context. The
proof of this normalization in infinite dimensions is an adaptation of the original proof by Hofer
and Zehnder which can be found in [30], see [38] for details of the infinite dimensional argument.
Consequently, if a flow map D preserves capacities, one can conclude that squeezing is impossible,
namely
'1D(t)(BR(U*))
9 Cr(z; ko)
if R < r.
Several examples of nonlinear Klein-Gordon equations with weak nonlinearities can readily be
shown to be of the form (1.7), see [38]. Symplectic non-squeezing was later proved for certain
subcritical nonlinear Klein-Gordon equations in [7] using Kuksin's framework, see also [591. Bourgain later extended these results to the cubic NLS in dimension one in [4], where the flow is not
16
a compact perturbation of the linear flow. There, the argument follows from approximating the
full equation by a finite dimensional flow and applying Gromov's finite dimensional non-squeezing
result to this approximate flow. Symplectic non-squeezing was also proven for the KdV [20]. In
this situation, there is a lack of smoothing estimates in the symplectic space which would allow
the infinite dimensional KdV flow to be easily approximated by a finite-dimensional Hamiltonian
flow. To resolve this issue, the authors of [20] invert the Miura transform to work on the level of
the modified KdV equation, for which stronger estimates can be established.
As we are interested in studying non-squeezing for an equation at the critical regularity, we face
substantial difficulties if we try to naively adapt the previous approaches to this setting. Additionally, there is no uniform control on the local time of existence and as the critical regularity is not
controlled by a conserved quantity, the global well-posedness of (1.5) remains open. Ultimately,
however, we are able to circumvent these difficulties, using a combination of probabilistic and
deterministic techniques, which we combine to obtain several deterministic non-squeezing results.
The first result, which we prove in Chapter 4 is a local-in-time non-squeezing theorem.
Theorem 4.1. Let (D denote the flow of the cubic nonlinear Klein-Gordon equation (1.5).
R > 0, ko E Z3, z E C, and u* E H 1/ 2(T 3 ).
Fix
For all 0 < q < R, there exists N = N(7,u, R, ko)
and o- = o-(r, N, u*) > 0 such that for all 0 < t < o-,
4(t)(UNBR(u*))
V Cr(z; ko)
for r < R - 7.
In the statement of this theorem, HN is a projection onto frequencies
Ikl
(1.8)
N, see (1.13) for its
precise definition.
Remark 1.1. The parameter q which appears in Theorem
4.1 corresponds to the control we can
obtain over the radius of the cylinder. If we demand better control over the radius, this theorem
only holds for shorter time scales. See also Remark
4.4
for a discussion of the dependence on u*.
Remark 1.2. To prove this theorem, we combine a probabilisticapproximation argument with the
available stability theory to prove Theorem 4.1. As we have no control on the local time of existence
in the critical space, a priori we cannot ensure the flow map 4(t) is well-defined on any infinite set
for any positive time. We fix the projection in (1.8) at frequency N so that we have enough control
to define the flow map D(t) for t E [0,o-].
17
In order to state our global-in-time results, we need to introduce the following nonlinear Klein-
(UN)tt
--
-+UN ~+
AUN
PN(PNUN)3 =
(uN, atUN) t=0 = (uou, ul) E
where PN
= P<N
3(.
T3
,uR
-
Gordon equation with truncated nonlinearity
:/T
denotes the smooth projection operator defined in (1.12). We obtain the following
global-in-time non-squeezing result.
Theorem 4.2. Let c
denote the flow of the cubic nonlinear Klein-Gordon equation (1.5).
R,T > 0, ko E Z3, z E C, and u,
E W1/2(
3).
Fix
Suppose there exists some K > 0 such that for all
(uo,u1) E BR(u*), the corresponding solutions u to (1.5) and uN to (1.9) satisfy
Nx
,([0,T)xT3)
< K.
(1.10)
Then
1(T)(BR(u*))
Z
Cr(z;ko)
for r < R.
Moreover, if BR(u*) c BPO for some sufficiently small po(T) > 0, then non-squeezing holds without
any additional assumptions on the initial data.
Remark 1.3. For small data, the additional assumptions on the Strichartz norm of solutions
automatically hold by the standard arguments, see Lemma 4.31. By (1.6), the assumption (1.10)
implies that in particular, the corresponding solutions u to (1.5) and uN to (1.9) exist on [0,T).
We will frequently be implicitly making use of this fact throughout this thesis.
Remark 1.4. This theorem implies, in particular, that if one can define a global flow for the nonlinear Klein-Gordon and prove uniform Strichartz bounds for solutions to both (1.5) and (1.9) for
initial data in bounded subsets, then one obtains the full, deterministic statement of non-squeezing
theorem for this equation. In practice a global well-posedness result for the full equation will typically also yield estimates for the equation with truncated nonlinearity, so while the requirement for
bounds on the solution to the equation with truncated nonlinearity may seem artificial, we do not
believe it is too restrictive a requirement.
The main tool in the proof of Theorem 4.2, is an approximation result which we prove in
Chapter 3. It says that in the critical space, solutions to (1.5) are stable at low frequencies under
18
high-frequency perturbations to the initial data. The proof uses the UP and VP function spaces,
whose definition and properties we record in Appendix A. These spaces were applied for the first
time by Koch and Tataru in [37]. They have previously been used in the context of critical problems
by Hadac, Herr and Koch [28] for the KP-II equation, and by Herr, Tataru and Tzvetkov [29] for
the quintic nonlinear Schr6dinger equation on T3, as well as by Nahmod and Staffilani [48] for
probabilistic well-posedness of the quintic nonlinear Schrbdinger equation on T3 below the energy
space. See [36] or [28] and references therein for a more complete overview of these function spaces.
Theorem 3.1. Let 4 denote the flow of the cubic nonlinear Klein-Gordon equation (1.5).
Let
T > 0 and 1 < N' < N,. Let (uo,ui), (i&,i 1 ) E BR C ?-t1/ 2 (T 3 ) be such that P<N.(uO,ul) =
P N. (iO, il 1 ), and suppose there exists some K > 0 such that corresponding solutions u and i! to
(1.5) satisfy
3
Li,([O,T]XT3) + PLiILt([O,T]xA )
K
IIP NI (1(t)(uO, U 1 )
-
((t)(iiO, il)) 1L
41/2([lT)xT)
(
)
Then for sufficiently large N. depending on R, T and K,
with implicit constant depending on R, T, K.
We prove Theorem 3.1 by demonstrating that under the above assumptions, the low frequency
component u1o satisfies a perturbed cubic Klein-Gordon equation given by
Eujo + uio = PjoF(uO, u1o, ujo) + err,
where err is an error term which we can control by the well-posedness theory.
Non-squeezing and Liouville's theorem
We will now provide a heuristic interpretation of non-squeezing for Hamiltonian systems.
We
consider the simplest example of a symplectic phase space, given by the vector space (R 2,, WO)
representing the positions and momenta of n particles, with coordinates (xi, yi) corresponding to
the i-th particle, symplectic form wo to be given by
wo = dx1 A dy 1 +...+ dx, A dyn.
19
Given a smooth Hamiltonian H: R2 d -+ R, Hamilton's equations of motion can be written as
.x=
Xii
.
OH
---
OH
yi = ax
or equivalently, as the flow induced by the vector field XH defined implicitly by wo(XH,*)
-dH.
Liouville's well-known theorem in classical mechanics states that Hamiltonian flows preserve
volume. Thus Liouville's theorem exerts some control over of how much uncertainty can arise in
such a system. Namely, if you start with initial conditions in a certain region in phase space, you
will not be able to determine the state of the system with any greater accuracy (in terms of phase
space volume) after it undergoes a Hamiltonian evolution, nor will you lose accuracy completely.
It is thus natural to ask whether one will ever be able to determine the location of one particle
very precisely, if one is willing to give up accuracy for the remaining particles. Say we start with
initial data in a ball in the phase space, and after some time we wish to know the position and
momentum of the k-th particle very precisely for some 1 < k < n. Geometrically this corresponds
to starting with initial conditions in a ball or radius R in phase space and embedding this ball into
a very thin cylinder in the k-the coordinate plane, defined by
C,.(z; k) := {(x, y) c R 2n : (xk - zo) 2 - (k -- zl)
2
<r2},
for z = (zo, zi) c C. Recalling that Hamiltonian equations are a particular example of symplectic
flows, Gromov's non-squeezing theorem says this is impossible unless R < r.
Non-squeezing and transfer of energy to higher Fourier modes
Finally, we explain briefly the heuristics of how non-squeezing relates to the transfer of energy to
higher Fourier modes for Hamiltonian equations. For a solution u to a Hamiltonian equation with
Fourier series
u(x, t) =
'k(t)e, Uk(t)
c C.
3
kEZ
The energy transition problem investigates whether
ik (t)| decays with t for some fixed k.
To
reformulate this question, let 'W denote the phase space for the given equation, then this becomes
20
a question about whether for some ball B C N, for k E Z 3 and for 6 < 1 one has
Iuk(t) !5 0,
t=tk>l
(1.11)
for solutions with u(O) E B. If the equation we consider has a conserved energy which controls
a given Sobolev norm, as is the case for the nonlinear Klein-Gordon equation, we can interpret
(1.11) as a statement that the higher Sobolev norms of a solution grow in time. This is due to the
fact that if there is decay of certain Fourier modes, but a certain Sobolev norm remains bounded,
then some other higher Sobolev norm must compensate by growing, since higher Sobolev norms
weight higher Fourier modes more heavily.
Now let <b denote the flow map of the Hamiltonian
equation. We can rewrite the condition from (1.11) as the requirement that <D embeds the ball B
into a thin cylinder,
4D(t)(B) C {u E N : Iuk(t)I
<
6}.
If the ball has radius R > 0, then non-squeezing says this is impossible unless R < 6. Thus,
non-squeezing implies a negative answer to the question of the uniform decay of Fourier modes.
1.2
Notation and conventions
We let C > 0 denote a constant that depends only on fixed parameters and whose value may
change from line to line. Where relevant, we will explicitly record the dependence of C on these
fixed parameters. We write X < Y to denote X < CY and similarly X > Y. If there exists some
small constant c > 0 such that X < cY, then we write X < Y and similarly for X
> Y. For some
constant a we write a+ as a shorthand for a+ e, for e > 0 some arbitrarily small, fixed parameter.
Let A
=
T or R. Throughout we will be using space-time norms
1U11L qL;;(RxAd) :=-
t X
(fR (fAd
I~')1)")1
We use the usual convention of L2 normalized Fourier transform with the notation
f^(n) = (27r)-
j
e-if
(x)dx
and
(
:= 216+(.
V)
We make use of the Japanese bracket
21
f^(6) = (27r)-
e-i"f(x)dx.
As mentioned above, H4(A 3 ) denotes the
usual inhomogeneous Sobolev space endowed with the norm
IIUI H:(Rd)
)'
= 11(
and
L2(Rd)
11UlU1Hs(rd) -
11(n)7(n) 112 (Zd)
while H#(R3 ) denotes the corresponding homogeneous spaces with 1 j or Inj in place of the Japanese
bracket. We use the notation
Ws(A 3 ) := Hs(A 3) x Hs-'(A3),
where 7s(A 3 ) is endowed with the euclidean norm on the product. We fix a nonzero @
with supp
4
C
C B(-2, 2), and define the Littlewood-Paley projection operators
N-1 )f^( ),
N)
PNf (
and note the PN form a partition of unity.
decomposition and take N C
2
N
C 22
In most cases we will be using an inhomogeneous
N, a dyadic integer. We use the notation UN = PNU and we define
P<N u:=
E
P>N
UM,
=
1 - P<N-
IMIN
For more properties of these operators, see Appendix A in [66]. On the torus, PN will similarly
denote a smooth projection operator, defined by
P-N(u)(r) =
(-N
2
A)(u)(x) = -(0) + E
p
()
i(ri(n)ein")
(1.12)
N
nEZ
for 0 a smooth cut-off as above. We will need the sharp Fourier projection operators
IIHK(uo,
(
3
|kIKK
o(k)eik-x,
(1.13)
Ui(k)eik-x)
E
|k| K
We define the ball of radius R centered at u, in
W1/
BR(U,) = {u E W1/2 :
2
by
11u -
u*I1,
1/2
< R}
and we use the shorthand BR := BR(O). We define the cylinder in
22
1/
2
(T 3 )
of radius r > 0 in the
k-th frequency, centered at z = (zo, zi) C C by
C,(z; ko)
{
(ui, U2) E ?j1/2(T3)
: (ko)IU1 (ko) -- zo
2
+ (ko) 1!IZ2 (ko)
-
z11 2 < r
,
(1.14)
for real-valued Fourier coefficients fi(k). Since we only consider non-squeezing for the phase space
of real-valued functions u E
1/ 2 (T 3 ), we can always identify such a function with real-valued
Fourier coefficients.
1.3
Outline of Thesis
In Chapter 2 we study the random data Cauchy problem for the nonlinear wave equation with
power type nonlinearity on R3 . These results are joint work with Jonas Liihrmann and have
appeared in [42]. In Chapter 3, we prove an approximation result for the cubic nonlinear KleinGordon equation in the critical space. In Chapter 4, we prove symplectic non-squeezing for the
cubic nonlinear Klein-Gordon equation on T3 . Many of the results from Chapter 4 have appeared
in the preprint [44].
Finally we record some background material from Harmonic analysis in
Appendix A.
The author was supported in part by the U.S. National Science Foundation grant DMS-1068815
and by the NSERC Postgraduate Scholarships Program
23
24
Chapter 2
Random data Cauchy theory for NLW
of power-type on R3
2.1
Introduction
We consider the Cauchy problem for the defocusing nonlinear wave equation
{-utt
(2.1)
R x 3R,
julP fi)u EonH.(R
) x H - (R3),
(u, ut)It=o
+ Au = (fo,
where 3 < p < 5 and H,(R 3 ) is the usual inhomogeneous Sobolev space. The main result of this
chapter establishes the almost sure existence of global solutions to (2.1) with respect to a suitable
randomization of initial data in Hj(R3 ) x Hi-1R 3 ) for 3 <; p < 5 and p
P-3 < s <
1. In
particular, for 1 (7 + V 7 3) < p < 5, this yields examples of initial data of super-critical regularity
for which solutions to (2.1) exist globally in time. See Remark 7 for a discussion of uniqueness of
these solutions.
A systematic investigation of the local well-posedness of (2.1) for initial data in homogeneous
Sobolev spaces is undertaken by Lindblad and Sogge in [41], where local strong solutions to (2.1)
are constructed for s ;> j - T
3
using Strichartz estimates for the wave equation. When s <
this is the super-critical regime and the well-posedness arguments based on Strichartz
estimates break down.
Recently, several methods have emerged proving ill-posedness for (2.1)
below the critical scaling regularity (see Lebeau [39], Christ-Colliander-Tao [17] and Ibrahim-
25
Majdoub-Masmoudi [31]).
In this chapter we study almost sure global existence for nonlinear wave equations on Euclidean
space without any radial symmetry assumption on the initial data. Previous results on Euclidean
space have involved first considering a related equation in a setting where an orthonormal basis
of eigenfunctions of the Laplacian exists. This orthonormal basis is used to randomize the initial
data for the related equation and an appropriate transform is then used to map solutions of the
related equation to solutions of the original equation. Burq, Thomann and Tzvetkov [12, Theorem
1.2] prove almost sure global existence and scattering for the defocusing nonlinear Schrbdinger
equation on R by first treating a one-dimensional nonlinear Schrddinger equation with a harmonic
potential and then invoking the lens transform [12, (10.2)]. Subsequently, similar approaches were
used by Deng [22, Theorem 1.2], Poiret [56], [55] and Poiret, Robert and Thomann [57, Theorem
1.3] to study the defocusing nonlinear Schr6dinger equation on Rd for d > 2. In the context of
the wave equation, Suzzoni [64], [21, Theorem 1.2] first considers a nonlinear wave equation on
the three-dimensional sphere and then uses the Penrose transform [64, (3)] to obtain almost sure
global existence and scattering for (2.1) for 3 < p < 4.
Instead of using such transforms, we will randomize functions directly on Euclidean space via
a unit-scale decomposition in frequency space. More precisely, let p E Co(R 3 ) be a real-valued,
smooth, non-increasing function such that 0 < p < 1 and
For every k E Z3 set pk( )
=
1
for
< 1,
0
for
> 2.
p( - k) and define
ZlEZ3
Then Ok is smooth with support contained in
3
{{
E R3
3
(()
I_
kj
K
2}. Note that EkEza
3
for all ( E R . For f E L2(R ) define the function Pkf :R -4 C by
(Pkf)(X) = F-' (@k(()fQ()) (X) for x E R 3
26
kQ)
=
1
If
f
E Hs(R3 ) for some s E R, then Pkf E H,(s(R 3) and
lfIH-(R3)
~
||P:
(
f =keZ3
kf in Hj(3) with
H.(R3))/
3
kEZ
In the proof of the almost sure existence of global solutions to (2.1), we will crucially exploit that
these projections satisfy a unit-scale Bernstein inequality, namely that for all 2 < p :5 P2
there exists C
C(pi, p2) > 0 such that for all
E L (
IIPkf 1L2(R3)
(2.2)
ClPkfIILP1(R3).
f
3
<
oo
) and for all k E
L
For a formal statement of this inequality, see Lemma 2.3. Let now {(hk,
1k)}keZ3
be a sequence
of independent, zero-mean, real-valued random variables on a probability space (Q, A, P) with
distributions /k
E I and for all k E Z3,
and vk. Assume that there exists c > 0 such that
(2.3)
e? dpUk(<)
<ec
2
for all
R
and similarly for Vk. The assumption (2.3) is satisfied, for example, by standard Gaussian random
variables, standard Bernoulli random variables, or any random variables with compactly supported
distributions. For a given
f=
(fo, fi)
C H((R 3 )
x
H- 1 (lR 3 ) we define its randomization by
hk(w)Pkfo, E
fW = (fYOJfA) := (
lk(w)Pkfl.
(2.4)
kEZ 3
kEZ3
The quantity EkeZ3 hk(W)Pkfo is understood as the Cauchy limit in L 2 (Q; Hx(R 3 )) of the sequence
(ElkI
N hk(w)Pkfo) NEN
and similarly for EkEZ3 lk(W)Pkfl. Let
Wsin(tJV
df = cos(t|VI)fow
+
I)
VfW
(2.5)
be the free wave evolution of the initial data fW defined in (2.4).
In the case of random variables such that there exists c > 0 for which their distributions satisfy
sup3
kEZ
([-CC C]) < 1,
one can show that if f does not belong to Hx+e(R3 ) x H-+e((R3 ), then the probability that fW
27
belongs to H,+e(R3 ) x H,-+e(R 3 ) is zero, see Lemma 2.12. Thus, our randomization procedure
does not regularize at the level of Sobolev spaces.
A similar randomization procedure on Euclidean space was used by Zhang and Fang in [72,
(1.12)] to study random data local existence and small data global existence questions for the
generalized incompressible Navier-Stokes equation on
Rd
for d > 3.
We are now in a position to state our results.
Theorem 2.1. Let 3 < p < 5 and let
p3 + 5p 2 - 1lp - 3 <
9p 2 -6p- 3
Let
f
= (fo, f1) C Hs(R 3 ) x Hs-1 (R 3 ).
<
1.
Let { (hk, ik)}keZ3 be a sequence of independent, zero-mean
value, real-valued random variables on a probability space (Q, A, P) with distributions yA
and
Vk.
Assume that there exists c > 0 such that
]0
e
dyuk(x)
< ec
2
for all y E R and for all k E Z3
and similarly for Vk. Let fW = (foJ, fl') be the associated randomized initial data as defined in (2.4)
and let u' be the associatedfree evolution as defined in (2.5). For almost every w G Q there exists
a unique global solution
(u, ut) C (u, atu') + C(R; Hx(R 3 ) x L2 (R 3 ))
to the nonlinear wave equation
-utt + Au = JuIP-lu on R x R 3
6
(2.6)
(ulut)It=o =
(fYof').
Here, uniqueness only holds in a mild sense; see Remark 7.
The proof of Theorem 2.1 combines a probabilistic local existence argument with Bourgain's
high-low frequency decomposition [10], an approach introduced by Colliander and Oh [19, Theorem
2] to show almost sure global existence of the one-dimensional periodic defocusing cubic nonlinear
Schr6dinger equation below L 2 (T). The high-low method was previously used by Kenig, Ponce
and Vega [34, Theorem 1.2] to prove global well-posedness of (2.1) for 2 < p < 5 for initial data
28
in a range of sub-critical spaces below the energy space (see also [1], [24] and [60] for p = 3). We
adopt arguments from [34].
Theorem 2.1 permits initial data at lower regularities than the deterministic result [34, Theorem
1.2]. This is mainly due to so-called averaging effects for the free evolution of randomized initial
data (see Lemma 2.7 below), which are proven by combining the unit-scale Bernstein estimate
(2.2) and Strichartz estimates for the wave equation on R . In particular, we use here that the
randomization is performed directly on Euclidean space.
Remark 2. For3 < p < 4, random data Cauchy theory for the nonlinearwave equation (2.1) on R3
has been addressed by Suzzoni /64, 21! using different approaches than in the proof of Theorem 2.1.
In
[64, Theorem 2/, using methods from [14, 15], almost sure global existence and scatteringfor
(2.1) for 3 < p < 4 is established for radially symmetric initial data in a class of spaces of supercritical regularity related to the Penrose transform. For p = 3, almost sure global existence and
scatteringfor (2.1) is proven in [21, Theorem 1.2/, using methods from [16/. In both these cases,
the spaces for the initial data do not coincide with Hx(R3 ) x Hx- 1 (R 3 ).
We do not address the
question of scattering of the constructed solutions in Theorem 1.1. The main difficulty is that the
high-low method does not yield bounds on any global space-time norm of the nonlinear component
of the solution u.
Remark 3. During the final revision of this article, the preprints [2, 3/ by Binyi-Oh-Pocovnicu
and [54/ by Pocovnicu appeared which use a similar randomizationprocedure. In [2, 3] almost sure
well-posedness results are establishedfor the cubic nonlinearSchrddinger equation on Rd for d > 3.
In [54] almost sure global well-posedness is proven for the energy-criticaldefocusing nonlinear wave
equation on Rd for d = 4, 5.
Remark 4. For 1(7 + vf7 3)
~ 3.89 < p < 5, the range of allowable regularity exponents s
in Theorem 2.1 contains super-critical exponents; see Figure 2.1.
knowledge, for 4
To the best of the authors'
; p < 5, Theorem 2.1 is the first result which establishes the existence of large
sets of initial data of super-critical regularity which lead to global solutions to (2.1).
Remark 5. For the specific case p
=
3, our randomizationprocedure allows us to straightforwardly
adapt the proofs in [16/ to the Euclidean setting. In particular, by modifying the proof of the averaging effects of Corollary A.4 of [16, we can use the arguments of Proposition2.1 and Proposition
29
S
1.0
0.9
0.8
0.75
0.61
3.5
4.5
4.0
Figure 2.1: The dashed line is the critical regularity s, =
for the exponent s in Theorem 2.1.
5
0
. The solid line is the threshold
-
2.2 of /16] to prove that if 0 < s < 1 and f = (fo, fi) C H,(R') x H
1
(R'), then for almost every
w E Q there exists a unique global solution
(u, ut) E (u, atU') + C(R; Hx(R 3 ) x L 2((R 3 ))
to the defocusing cubic nonlinearwave equation on R 3 with initial data fW. This is an improvement
over the corresponding range
< s < 1 in Theorem 2.1. In contrast, these arguments which proceed
via energy estimates cannot be applied directly in the case 3 < p < 5. In such an argument, one
would need bounds on IIu|Pu-lUiLL2(RXg
=
HuH2PL2
but one no longer has the necessary
Sobolev embedding to close the argument.
When p = 5, the nonlinear wave equation (2.1) is energy-critical and the high-low argument no
longer works. However, by establishing probabilistic a priori estimates on the L5L((R x R 3 ) norm
of the free evolution u'f, we obtain the following probabilistic small data global existence result.
Theorem 2.2. Let 2 < s < 1 and f = (fo,fi) C Hx(R 3 ) x
x- 1 (iR 3 ).
Let {(hk,lk)}kCz3
be
a sequence of independent, zero-mean value, real-valued random variables on a probability space
(Q, A, P) with distributionspk and vk. Assume that there exists c > 0 such that
]
eyxdyuk(x)
< e
for all -y
R and for all k G Z3
and similarlyfor uk. Let fW = (fo, f1') be the associated randomized initial data as defined in (2.4)
30
and let u' be the associatedfree evolution as defined in (2.5). There exists Qf C Q with
IP(Qf) > 1 - Ce
(2.7)
xH
for some absolute constants C, c > 0, such that for every w E Qj there exists a unique global
solution
(u, ut)
E
(u- , tu ') + C(R; H (R 3 ) x L (R3))
to the quintic nonlinear wave equation
-utt + Au = ju1 4 u on R x R3(2.8
(fO', fl).
(
(u, ut)It=o =
Moreover, we have scattering in the sense that for every w C Qf there exist (vI, v2)
L2(R3) such that the free evolution v(t)
SW(u(t) -
UY (t) - v(t),
atu(t) -
=
+
cos(t|VI)v|
OtUY(t) -
"
c H]R3 ) x
v2 satisfies
atv(t)) ||i(R3)xL2(R3)
---+ 0 as t
-+
00.
Remark 6. The statement is only meaningful if
S< C
1f1H,'xkx l
log (C)/
)1/21
which reflects that Theorem 2.2 is a small data result.
2.2
Deterministic and Probabilistic estimates
The following unit-scale Bernstein estimate is a crucial ingredient in the proofs of the probabilistic
a priori estimates on the randomized initial data in Section 2.2.2.
Its advantage compared to
the ordinary Bernstein estimate for the dyadic Littlewood-Paley decomposition is that there is no
derivative loss.
Lemma 2.3 ("Unit-scale Bernstein estimate"). Let 2 < pi 5 P2 < oc.
C
C(p1, p2) > 0 such that for all
There exists a constant
f C LX(R 3 ) and for all k c
||Pkf IILP2(R3) :5 Cl|pkf 1ILP1(R3).
31
(2.9)
Proof. Let 7 E C
(R3) be such that 0 < 7 < 1 with 77() = 1 for
= 0 for
1 < 2 and 7()
> 3.
For k E Z3 define
k
From Young's inequality with 1+ P2 =
I|PkfflLP2(R3) =(-1(-F kf)H 2(
<
2.2.1
q
E R3
( - k) for
+ P1
1 we then obtain
1LT =(rik(kf)HL2(Ra) =fk * (Pkf)||L2 ()
|6kINLk( R3)PfIILg1(R3)
=
7LX(R3)EPkfHLP(R3).
Large deviation estimate
Most of these
We will record a basic probabilistic result about the randomization procedure.
estimates are consequences of the classical estimates of Paley-Zygmund for random Fourier series
on the torus. These estimates were used heavily in the works of Burq and Tzvetkov, see especially
[131 for proofs.
Proposition 2.4 (Large deviation estimate; Lemma 3.1 in [14]). Let {hn}'
1
complex-valued independent random variables with associated distributions{Ipn}
'
be a sequence of
on a probability
space (Q, A, P). Assume that the distributions satisfy the property that there exists c > 0 such that
ec
e'xdynu(x)|
2
for all y c R and for all n E N.
Then there exists c > 0 such that for every A > 0 and every sequence {cn}%1
00
IF(W :
CA
cnhn(w)I >
2
(N) of complex
2
A) < 2e En Cn
.
numbers,
c
n=1
As a consequence there exists C > 0 such that for every p > 2 and every {cn}
cnhn(w)LP (
CP
n=1
(
1
E
f2(N),
ICn2)1/2
n=1
Remark 2.1. Gaussian iid or Bernoulli random variables satisfy the exponential moment assumption in the statement of Proposition2.4.
32
2.2.2
Averaging effects for the randomized initial data
In the proof of Theorem 2.1 we decompose the randomized initial data into a low frequency and a
high frequency component. More precisely, for N E N define
f0,N =
and f\>N
=
-
E
hk(w)Pkfo
. Similarly, define f 1,N and f'>N. The next two lemmata establish
probabilistic a priori estimates on the low frequency component. See also Proposition 4.4 in [14].
7Vs(R 3 ).
Lemma 2.5. Let 3 < p < 5. Let s > 0 and f = (fo, fl) C
AK,N
E Q : 1fO' $-Nl,+1
For any K >0 and NE N, let
(2.10)
< K}.
/)
Then there exist constants C = C(p) > 0 and c = c(p) > 0 such that
P(ACN
(2.11)
(3)
-cK4/(p+1)
for every K > 0 and N E N.
Proof. For every p ;> p + 1 and every N E N, we bound
IkIN hk(w)Pkfo
|f06N LP(Q;LP+1(R3))
k
S
hk(w)PkfOjLP(0)
Ikl N
<p-
LG+1 (R3)
pk fo(X)|12)1/2L+(R)
(
r-_
k
|k| N
k
(2.12)
(2.13)
LPX+(R3) 3/
I
/
Nkf01
$
(foI
IPk
(R 3))/
(2.14)
IkI N
VV
4fo|L(R3).
Using p > p+ 1, we switched the order of integration in (2.12) and used the large deviation estimate
from Proposition 2.4 in (2.13). We then used the unit-scale Bernstein estimate (2.9) in (2.14). The
33
claim follows from Lemma 2.10.
Lemma 2.6. Let s E R and f = (fo, fi) E 7s(R 3 ). For any K > 0 and N
{
BK,N :=
Then there exist constants C
Q : Hf0 >NlH,(R3) + I1fl >NflHs-1(R3)
C N, let
(2.15)
K}.
C(s) > 0 and c - c(s) > 0 such that
P(Byc,N) K C-cK 2 / f'Is(R 3
)
(2.16)
for every K > 0 and N C N.
Proof. For every p > 2 and every N E N, using the large deviation estimate Proposition 2.4 we
have
-
A)s/ 2 f0
>k hk(w)((1
N LP(Q;L(R3))
1
(
((1
-
A)ls/ 2 pkf)
-
L(Q;L2(R 3))
A)/2Pk f)(X) 2 1/2
LI)
LXR3
Jkl<N
(1- A)s/2pkfo2(R3) 1/2
Ik|<N
V IIfo IH(R3).
Similarly, we obtain
(1
-
A)(s-l)/
2
fwN
(;L(R3))
<L V
fH7(R3)F:
The assertion then follows from Lemma 2.10.
Analogously to (2.5), define the free evolution of the high frequency component of the randomized initial data by
Uf,>N = cos(tV7I)fO>N +
1tV1 fW>N-
In the next lemma we prove probabilistic a priori estimates which exhibit a decay in N on certain
space-time norms of the free evolution Uf,>N once one restricts to suitable subsets of the probability
space Q.
The decay is ultimately the reason why we obtain an improved range of regularity
34
exponents s in Theorem 2.1 compared to [34, Theorem 1.2] and results from the use of the unitscale Bernstein estimate (2.9) and the Strichartz estimate (A.1).
Lemma 2.7. Let T > 0 and 3 < p < 5.
Let 0 < s < 1 and 0 < e < min(.,(1 -
i)).
Let
f = (fo fl) E7t(R 3 ). ForanyK>0 andN>3, let
DK,N,E :=
{w E Q
: Ns-2
U>NL
L$([,T]xR3)
(2.17)
K}.
Then there exist constants C = C(s, p, e) > 0 and c = c(s, p, e) > 0 such that
2
3)2
,NCe-cK
E Ce(02(II
,N,)
P(DP(D
(2.18)
for every K > 0 and N > 3.
Proof. Set r(-) = 1_22c.
Then the exponent pair
(}, r(e)) is wave-admissible and Strichartz-
admissible at regularity -y = 2e. For every p > max( , 2p) and every N > 3, we now estimate
1f,>N
(2
cos(tiV i)Pkfo
~ ~(S
.19)
1
jk>N
sin(tVi)
1/2
2
+ Vli
kfl L
P([O,T]xR3)
L
|kI>N
S ( cos(tIVI)Pfo
/eL([O,TXR3)
(2 .20)
1/2
k|>N
+
S
\/ip(
2
sin(t1V 1)
lv
|kI>N
L
1 L
I
S
EjPkf1
+
fN12
IkP>
1/2
,E([ T xR3)
(2.21)
126-1(R3))/
jk>N
|k >N
1N-(s-2E)( E
kIp
31/2
+
N(s-2e)
(
1)/2
([0,T]xR3))
-(--2.)
H
(R3)
+
1
(R3)) 1/2
2
N-(s- e)
Ik
>1
/2
(2.22)
|kl>N
|k >N
Usn p p-N -m(w2e) Ifs
|k>Pkf
IkI>N
IkI>N
/
LP(OTXR
ILP(Q;L 'LL
th (R3).
Using p ;> max( , 2p), we switched the order of integration and used the large deviation esti-
35
mate Proposition 2.4 in (2.19). The assumption
E 7(R3 ) together with Plancherel's theorem
f
guarantees that
(cos(tiV)Pkfo)(x) 2) 1/ 2 <:00
(S
kEZ
and
(sin((x
3
kEZ
2 1/2
3
<00
for almost every x C R 3 and for every t E [0, T], allowing us to apply Proposition 2.4. We use the
unit-scale Bernstein estimate (2.9) in (2.20), noting that r(F) < 2p, and then apply the Strichartz
estimates (A.1) at regularity ^y = 2e in (2.21).
In (2.22) we may estimate flPkf1i||ts-1(R3)
IIPkf1|H;-(R3) uniformly for all Jkl > N > 3 even though s - 1 < 0, since F(Pkfi)( ) = 0 for
1 1 < 1 for all
Ikl
> N > 3 due to the support properties of the unit-scale projections. The claim
0
then follows from Lemma 2.10.
2.3
Proof of Theorem 2.1
This section is devoted to the proof of the following proposition, which immediately implies Theorem 2.1.
Proposition 2.8. Let T > 0. Let 3 < p < 5 and
p 3 + 5p 2 - 11p - 3
9P2 - 6p - 3
Fix f
=
(fo, fl)
C H((RI) x Hs- 1 (R3 ).
Let fW
=
(fo,
fl")
be the associated randomized initial
data as defined in (2.4) and u' the corresponding free evolution as defined in (2.5).
Then there
exists QT C Q with P(QT) = 1 such that for every w G QT there exists a unique solution
(u, ut) C (uf , atuW) + C([0, T]; Hx(R3 ) x L (R3 ))
(2.23)
to the nonlinear wave equation
+ Au = luIP-lu on [0, T]
x
R3
(
-utt
(2.24)
-
(UI ut) It=0 = (U0, fI A).
Here, uniqueness only holds in a mild sense; see Remark 7.
Proof of Theorem 2.1. We only present the argument to construct solutions that exist for all pos36
Define Tj = j for
itive times since the argument for negative times is similar.
E =
F1o-i Qr 3.
j C N and set
By Proposition 2.8, we have that P(E) = 1 and for every w C E, we have global
E
existence for (2.6) on the time interval [0, oc).
In the proof of Proposition 2.8 we will repeatedly use the following probabilistic low regularity
local well-posedness result whose proof we defer to the end of this section.
It is here that we
from Lemma 2.7. We introduce
invoke the crucial averaging effects for the free evolution u,
the notation
2p
q( p)
=
5 - p
2
p-3
Lemma 2.9. Let 3 < p < 5 and 0 < s < 1. Let 0 < e < min(!, !(1 - -)) and K > 0 be fixed. For
N > 3 and 0 < c < 1 set Ti
=
c(KNl-s)(P-1 )/a(P). Let v : [0, T1I x R3 -
C satisfy
(2.25)
CKN1-s
5[,T]x3
||V||L q()
and let w E Q be such that
uf,>N
/EL 2P([OT]xR3)
(2.26)
< KN-s+2e.
For 0 < c < 1 sufficiently small (independent of the size of K and N) and N = N(K) sufficiently
large, there exists a unique solution
(P) , X3) xC([, Ti];L (3
)
(@,iit)
c C([0, Ti]; fI
(R3))
x
to the nonlinear wave equation
-Vtt
+ Aw
()
@t)It=o
=
v+
u,>N
u>N +
IvIP 1 V on [0, Tfl x R 3
(2.27)
(0, 0),
satisfying
I~(T1I~fl(Ra) + !it(TiLl(3) +Nt(T04L2(R3) <
We now present the proof of Proposition 2.8.
37
(pKpN
2+(1-s)p-1.
(2.28)
Proof of Proposition 2.8. The bulk of the proof is devoted to the construction of subsets
for every K E N such that for every w G
QK,T
QK,T
C Q
there exists a unique solution of the form (2.23) to
(2.24) and such that
P(Q'K,T)
We then set
QT = UK' 1 QK,T
Ce
(2.29)
KjS(g3
and conclude from (2.29) that P(QT)
=
1, which completes the proof
of Proposition 2.8.
In what follows, let K E N be fixed. Let N = N(K, T) c N be sufficiently large, to be fixed
later in the proof. We also make use of a fixed small parameter 0 < E < min(!, 1(1 - -)) whose
value depends on p and s, but is independent of K and N, and is specified further below. We
define
QK,T = AK,N
n BK,N n DK,N,E,
where these sets are as in (2.10), (2.15) and (2.17). The estimate (2.29) then follows from (2.11),
(2.16) and (2.18).
From now on we only consider w c
QK,T.
It suffices to show that there exists a unique solution
(U, Ut) E (U0 , tU ) + C([0, T]; Hx (R 3 )
x L
(R3)
since by a persistence of regularity argument, one has u E uc + C([0, T]; Hx(R 3 )).
We first construct a solution u() = v() + wl) to (2.24) on a small time interval [0, T] with
frequency initial data
{
+ Av
-o
)
t(),
7 41))1t~=
v() solves the following nonlinear wave equation with low
-
P-1 v(1) on [0, T1] x R 3 ,
(fU0'N,
(230)
'<N)
-
0 < T < 1 to be fixed later and where
Note that the initial data (fO"<N, f "<N) lies in H (R 3 ) x L2(R 3 ), since by (2.15)
|fO', N11kt(R3) +
OH(R3) + I1fl NflH,-'I(R3)) ;< KN 1-8. (2.31)
<
fl NIIL2(R3) ;< N1->(jfO'KN
Thus, by the deterministic global existence theory [25, Proposition 3.2] there exists a unique
solution (v( 8 , v(1)) E C([0, Ti]; H (R3 ) x L2(R 3 )) to (2.30). Moreover, we have energy conservation
38
since ||f0W<N LP1(R) < K2 by (2.10). Hence, for all t E
[0, Ti]
EHv 1 )t) 21.R3
v1)(t)ji(R 3 )+ |v(1)(t)j2
~I~f~NllR)--~~~NIa
1
1
1Iv'(~l
+
|()()L
(R3)
(2.32)
1
H
<(KN1~8 )2.
We note that the exponent pair (q(p), 2p) is Strichartz-admissible at regularity y =
1. Using
Strichartz estimates (A.1) and (2.31), we find
HV(
I1I)(0) +HHt( (R3) + 1 + V LL ([O,TIx R3)
||
<KN-s + TOR)|
1
~
Hence, choosing Ti
=
c(KN 1-s
.
3)
lv(')IL
L,(P)L
L([0,T,] xR3)*
with 0 < c < 1 sufficiently small (independently of the size
)a()
of K and N), we obtain
Hv()
L2p([OT1 ] x
lL
Next, we consider the nonlinear wave equation that
{-w
+ Aw) = IV' + W
=0= (f
(01), w 0)|)
>N
P
;
<3)
(2.33)
KN'-s.
-)
- v) must satisfy, namely
(v() + WU ) - lo
v-1 on [0, T1] x R3,
1> N
We look for a solution of the form
Uf,>N
+
W(=
where @(1) solves the following initial value problem on [0, T] x R3
_
(
-
V( 1 ) + Uf,>N + "
(f,
>N
(2.34)
(P))It=O = (0, 0).
( ()
Using (2.33) and the averaging effects (2.17) for the free evolution u',>N, Lemma 2.9 yields a
unique solution
((I M
)E
C([O, Ti];k (R 3 ))
(P)L 2P ([0, T1) x R 3)
39
X
C ([0, T1 ]; L 2(R33))
to (2.34) provided 0 < c < 1 in the definition of T1 is chosen sufficiently small (independently of
the size of K and N) and N is chosen sufficiently large. Moreover, we have
10"i)1
,<
(R3)
+ 1l ii5 1)(T1)HL2(R3) +
T
KP2e+(1-s)p-1.
In the next step we build a solution U( 2 )
-
V(2) +w(
2
( T1
LP+1(g3)
I ILp+'R3)(2.35)
W' (TOI
to (2.24) on the time interval [T1, 2T1]. As
before, we would like to construct V(2 using the deterministic global existence theory at the energy
level and construct W(2 through the probabilistic local well-posedness result from Lemma 2.9. To
this end, we note that w(l is comprised of the free evolution uf,>N, which is at low regularity, and
the nonlinear component 0( 1), which lies in the energy space by (2.35). As initial data for V(2 at
time T we therefore take the sum of v( 1 )(T1) and @(7)(T 1), and consider
{
2
(2 ) +
V
(v( 2), V( 2 )t=
tt - V 2 on [Ti, 2T ] x
V 2)l
(V()(T1) + iZ (T1), v(T 1 ) +
(2.36
(2.36)
-V
1
-
(T)).
Once again, by the deterministic global theory, this initial value problem has a unique solution
C([T1, 2T1 ]; ftx'(R 3)
C
(V (2, Vt()
x
L 2(R 3)
Moreover, we obtain bounds on the L qL'([T1, 2T] x R 3 )-norm of V(2 as in (2.33). Using these
bounds and the averaging effects (2.17), we apply Lemma 2.9 to solve the difference equation for
-
{
V2 on [T1 , 2T1 ] x R 3
,
W(2 = U(
W
(2)+AW( 2 ) = IV 2)+W(2 )P- 1 (V( 2 ) W( 2 )) _ JV2
2
(w( ),
tt
2
t( )) t=T1
1
= (u, >N (T1),
(2.37)2)
(2.37)
OtuW,>n (T1))-
Note that Lemma 2.9 can also be applied on the time interval [T1 , 2T1] by time translation. We
therefore find a solution W(
(,
t )
= Uf,>N +
C Q(T1,I 2T1]; k,(R
( 2 ) to (2.37) with
)) nt L(P)Lx ([T1,I 2T1 ] x R) x C ([T1, 2T1 ]; LX (R3
In order to obtain a solution u to (2.24) on the whole time interval [0, T], we iterate this
procedure on consecutive intervals for
F- 1 times.
40
At every step we redistribute the data as in
(2.36). To make the process uniform and thus reach the time T, we have to take into account that
this redistribution increases the energy at each step. To estimate this growth, we invoke energy
conservation for the solution to (2.30). We have
1
E(v(1 )(T) + @U(1)(T)) = E(v(1 )(0)) + (E(v)(T1) + @(
5)(T
1 )) - E(O)(T1))I
and
E (v (1 (T1) + @(1 (T1)) - E (v (1) (To))
$ Ilv I (1)(TO)I I (R3)11@(1)(T1)I1ft.(R3) +|jV(1)(T 1)|IL2(R3) 1 P(Ti) Ll(R3)
+ v(
1
)ig~+i
$ E(v(1) (0))i
@(l(1)
l(
1
(T1)Htl (R3) +
(2.38)
(T))
)(T1)|LP+1(R3) + E(i
1 @(P)(T)11L2(R3))
|iii1)(T1)|L+1(R3 ) + E(@1(Ti)).
+ E(v(1)(0))
The term E(v(1)(0)) 4 i|i(
(R3)
1
)(T1)ILP+1(R3) gives the largest contribution to the energy increment
(2.38). In light of (2.32) and (2.35), we must ensure that
(KN'-8)2
Inserting Ti
c(KNi-)
=
P-
Tc(
For any s >
(),
3
2
(KN1-8 )2.
KPN2e+(1-s)p-1)
this is equivalent to
2
9P -,,P-3
K(5p)(p++e
P +
qp
+1T
2
(p-1)
5-, N
3
p +5 2_11-3-s
5
9P 2-6p-3) +E(___(p1)+22
( -p (pl)p
e
(
-)+2)
(2.39)
1.
3--3, we can make the exponent on N in (2.39) negative by fixing E > 0
sufficiently small at the beginning (depending on the values of s and p, which are fixed during
the course of the argument). Hence, taking N = N(K, T) sufficiently large we can ensure that
E
condition (2.39) is satisfied. This completes the proof of Proposition 2.8.
We now present the proof of Lemma 2.9.
Proof of Lemma 2.9. In this proof we are working on [0, T] x R 3 and will omit this notation. For
iiq E L"LP define the map
= -
sin((t -s)VI)
V+
f>N
41
+
+Uf,>N
+ @-
(s) ds.
Using the general inequality
IIziI'-'z -
- z21
z2
$ zi
Z 2 1-) for any z1,z2 E C
I- +
- z 2 1(Iz 1
and the Strichartz estimates (A.1) at regularity -y = 1 we bound
IID(@)IIgfto
+ Iltb(
|)IIL-'L +
2p(
)I LP
t
fN.d
1
@-
N+@
-II-V
+ VV1
L>
|,U>N
LL
2
1 +I:sg.)
fy,
u
+
T1
ILL 2
+ luw,>NI
2
t
x
+.)IP-d)
L
L
lU>N
LL
x
t
LL
Te+I:sig(22PI
21 +||||||P
< IV+ W1
-
'
)
| f,>
-1( >N
= JI +II + I(IV + UI>N +
V
1
L
(2.40)
L1L2
t
x
L1Lp
I+II1+III+IV +V.
We now estimate the terms I
-
V separately.
Term 1: By H6lder's inequality in time and (2.26), we obtain
P)II K
2
-P)E(KN
-
>NIPL2<T1
t
x
Term III: Using (2.25) and (2.26), we find
III,>NIIVPWL
fK
t
Uf,>NPLPIVIpL,
< Tl/PIuwN2
T1
, T
<
f,>
j
KPN1
42
IVW>IL~L
H 2
Lt~L
1
t()
Term IV: By (2.25) we have
|G| |P-
LtL X <PT P pi( P)( P-1)/ P hV|It
XL
T~ac
AIq(PL
&p
X
t
L)L
KN1-s)P-1.
Term V: Using (2.26) we bound
-
<
pT-
I1UI,>N L /eLNpTa)p1wl
-
-
< 1
p-~
LL
K-s+2
+ IT(+ai)1LPL
L2p
2,1
*+IN~) UNl1F P1 l@I'PL
Collecting terms we obtain
TqG())(p
1
L L
(KN-s
+ T1
2
e)
t aX
(P1p
KII-
P2
1
q
It follows that by choosing 0 < c < 1 sufficiently small (independently of K and N) and N
sufficiently large, '1 maps a ball B of radius R(K, N) > 0 with respect to the L 4LS4-norm into
itself, where
KPN 2 +(-Ls)P-
R(K, N ) , T
In a similar vein, we show that
(2.42)
is a contraction on B with respect to the
has a unique fixed0point
< c )<E B1 and
sufficiently small and N sufficiently large. Thus,
-
for
o
3 is the unique solution to (2.27).
In order to obtain (2.28) it remains to estimate the LP+ 1(R3)-norm of @ at time T1 . By Sobolev
43
embedding and Minkowski's integral inequality we find
||"'(T1)I|I|P+1(R3)
I (TO1) HH1(W3)
T
sin((Ti - s
s)IVI) (V +
sin((T
fTi
(
T1
-
< (1 + T1 ) |V +
<
(
s)
U,>N +
U>N
f,> NNLf,>X
ivPv)(s)
P
ds Hi(
v)()
+ if~(
+
+ Uw>N +
+ U"I
+f,,>
~i3
Iv
<N
v()
3
()
L2
(Ko +
+'I-V
V + uW>
,
)
1
L
(1R3
)
JT
dR)s
)
-
<
+ "0i - II-V'2
where we used that T < 1. From (2.40), (2.41) and (2.42) we infer
I|(T1)IILP+1(R3) < R(K, N) < T,
KPN 2 e+(-s)P-1
This completes the proof of Lemma 2.9.
E
Finally, we address the uniqueness statements in Theorem 2.1 and Proposition 2.8.
Remark 7. Analogously to [19, Remark 1.2], uniqueness for the "low frequency part" (v(j), v
in the j-th step holds in the space C([(j - 1)T1, jT1]; Hxj(R3 )
x
L (R 3 )). However, uniqueness for
the "high frequency part" (w ),w (j)) in the j-th step only holds in the ball centered at u'>N(t) of
small radius in Lq()Lx
2.4
([(j - 1)T,jT1]
x R
Proof of Theorem 2.2
In this section we prove the almost sure global existence result for the quintic nonlinear wave
equation from Theorem 2.2.
Proof of Theorem 2.2. We first derive probabilistic a priori estimates on the L5L' 0 (R x R 3 ) norm
of the free evolution u" and then use these to construct global solutions to (2.8) through a suitable
fixed point argument.
44
Since we have by assumption that 25 <
s < 1, there exists -3
< r < 10 such that the exponent
pair (5, r) is wave-admissible and Strichartz-admissible at regularity s. Similarly to the proof of
Lemma 2.7, using the large deviation estimate Proposition 2.4, the unit-scale Bernstein estimate
(2.9) and the Strichartz estimate (A.1) at regularity s, we obtain for any p > 10
ULkLZ
(LL5 Lx(RRxR3))
CO
~
2V~
Lkcos(tV)Pfo
12
)1/2
+V
(RxR3))
kEZ
kZ3
CO~,V~kfI2
cor|V)ko
t
XZ
V5(IfoOIIH-(R3)
VtX
2
1/x
kjj2L5L(RxR3)) 1/2
sin(tjVI)
+
kEZ
1/2
P|I
si(tIVI)
V
kEZ3
Ikflf
< 1~l(RP3)) 1/2
3
kEZ
3
kEZ3
1/2
Id(x3
kEZ3
v(E
IPkfo s(R ) ) 1/2 -
Z
si
P
afi L L-(RxR3))
3
+ lfI l ft'~1(R3))-
Then Lemma 2.10 implies that there exist absolute constants C, c > 0 such that
: Iuy'LiLo(RxR3) >K) Ce
MK2
(2.43)
P(LC Q
I
for every K > 0.
We now look for global solutions of the form u = u' + w to (2.8), where w has to satisfy the
nonlinear wave equation
{-Wtt
+ w| 4 (ul + w) on R x R 3
+u7
+ Aw =
(2.44)
(w,wt) Ito= (0, 0).
It is straightforward to see that there exists e > 0 such that if
t~~~ si((
<D(w)(t)
=
sin((t
0
-
I|ulIIL5Lio(RxR3)
e, then the map
-- sIV
s)IV1) (1u + w1 4 (u,,+ w))(s) ds
VI
is a contraction on a ball of radius e with respect to the L5L' 0 (1R x R3 ) norm. Its unique fixed
point
(w, wt) E C(R; x (R 3 ))
L L'
(IR x 1R3 ) x C(JR; L 2(R 3 ))
is the global solution to (2.44). The probability estimate (2.7) on the event Qf in the statement of
Theorem 2.2 follows immediately from (2.43), while the scattering statement follows readily using
45
that
fuy + WI5 ILiL2(RxR3)
2.5
Appendix A: Probabilistic estimates
<
oo for every w E
[l
Qj.
We record several useful probabilistic facts. We begin with the following Lemma which is a variant
of Lemma 4.5 in [69], and is often used to prove bounds on the probability of various events.
Lemma 2.10. Let Ff be a real valued measurable function on a probability space (Q, ?). Suppose
that there exists a > 0, N > 0, k E N* and C > 0 such that for every p > ro one has
Then, there exists J and C1 depending on C and ro but independent of f such that for A > 0
P(E,f) :=P(w Ez- Q
2
Ff(w)I > A) < Cie_
'S.
Proof. Note that by Chebychev's inequality,
P(EA,f) -< A~P(Cl~f 1-Lfp)',
for all p > ro,
so for c > 0 to be specified later we consider the quantity e3
W
f P(Ef ) and we will show that
it is bounded by a constant C1. We expand
=
7
E!
2
1f12
ro
A2
(6
n=1
We start with the first term: when
)P(E,f)+
li h
(6
n=ro
A<eCrT
we bound P(EAJ)
A2 )n
P(E\,f).
-H
1 and let 6
=
(C2 roe2 - 1
.
0
P(EAf)
e
We obtain
ro
I(
A2
n=1
Il12
ro
n
P(EA,f)
Z=
ri=1
I
(6
ro
2
If 11
46
I
\2
n ! Ilf112.e2C2ro
HS
n=1
n
ro
n=1
n!
When eCV'Th !
2
(eC) we bound
Ii!weK set p = [N%'] which is > ro and with 6= (eC)-
A
(
n=1
n!
A2
11
\n
11|2
-4S=
\2
i(
ro
P
___-
eCA
A2
S
n
2e2C2
Cl|f 11.
A
ec
|
R
n=1n
1112
22
ro
\2
n
-H
n1n!
which can be bounded by a constant. For the second term, we observe that
00 1 (
6'f ~ 5 )
P (E , f)
< "0
( 6
A )
A --2n
Using Stirling's approximation and choosing 6 < (2eC2 )-'
(nv= )2 n5n- ( 26C
2
)n
.
(f|
C_f
1-h
P(EA )i
o
,
n
yields the result.
We now turn to the proof of the fact that our randomization procedure does not regularize at
the level of Sobolev spaces. We recall the definition of our randomization set-up.
Definition 2.11 (HS randomization). Let {hk}kEz3 be a sequence of independent, 0 mean value,
real-valuedrandom variables with associated distributions{ p }kez3 on a probability space (Q, A, P).
Assume that there existc c > 0 such that
e'
e~xd pk(x)|
for all y c R and for all k z Z3.
(2.45)
For f E Hs(R 3 ) we define
hk(w)(Pkf)(X) for every x E R 3 .
fW(x) :=
kEZ
(2.46)
3
Condition (2.45) ensures that (2.46) defines a measurable map w F-+ fW from (Q, A) to HS(R3)
and that fW
E L 2 (Q; HS(R 3 )).
The following lemma is a variant of Lemma B.1 in [14] and
demonstrates that the randomization procedure does not regularize functions at the level of Sobolev
spaces.
Lemma 2.12. Let f =
Hs+E(R 3 ).
ZkeZ3
Let (hk(w))kebz2
Pkf E H'(R 3 ) be such that for some e > 0, f does not belong in
be a sequence of independent random variables with distributions pk
47
such that there exists c > 0 satisfying
sup3 Pk([-c, c]) < 1
kEZ
Consider fW as defined in (2.46). Then the probability that fW belongs to HS+E(R 3 ) is zero.
Proof. Let c, 6 > 0 be such that Pk([-c, c])
e
H s+E(R3)
dI
j
I
1kZ3
kEZZ
j
-
(1 - 6) and we compute
E112
~X2II 11pk
e
+IE (R3 )HfdP(
2
1pkf 1Is12(R3
- cE(
3
) I
kEZ
c2 IPkf
kEZ
k2 e
HE(R3
|x>c
kEZ3
s+E(R 3 J).
)
-(1-6+(
Sf1
W
3
We know that E I|PkfI 211(23)
=
oo, and hence EkeZ3 1
2
e-
1PkII11s+ (3))
=
oo. Thus, since
1 - x < e x for all x,
e-c
kC7Z
2
k1
1Ipkf 11HS+E2R3)
3
exp
e
-c2
lpk
s
H1
g3
3
kEZ
=
exp( -6 E
kEZ3
=0.
48
(1
-
eC 2IPkf1s+1(23)
Chapter 3
An approximation result for the cubic
NLKG in the critical space
3.1
Introduction
In this chapter, we study the defocusing cubic nonlinear Klein-Gordon equation
utt - Au+u+u 3
= 0,
u:R X T3 -+ R
(u,&tu)|s. = (uo, ui) E H2(T3) x H-21 (T3
7 1/ 2 (T 3 ),
where H2 (T 3 ) is the usual inhomogeneous Sobolev space. If we consider the general power-type
nonlinear Klein-Gordon equation
Utt - AU + U +
then s
:
-
p-
IuI
1u = 0,
defines the critical regularity, hence we are interested in studying the cubic
nonlinear Klein-Gordon equation at the critical regularity for the cubic nonlinear Klein-Gordon in
dimension three.
The purpose of this chapter is to prove that solutions of (3.1) are stable at low frequencies
under high-frequency perturbations to the initial data.
While this type of argument is often
seen at subcritical regularities, there are additional difficulties at the critical regularity since the
decay one usually needs for such arguments is not available in this setting. Nonetheless, one can
manufacture some decay using refined bilinear Strichartz estimates which, roughly speaking, show
49
that for M < N, dyadic frequencies,
Iei(
L
N e$M
<
M||ONH
for PNqN
L2 II)ML2
=
ON, PMOM =hM.
This corresponds to one half derivative loss on each function. Thus, if one can control the frequency
separation between functions in certain key multilinear estimates, the bilinear Strichartz estimate
demonstrates that it's possible to regain some decay even in the critical setting. Our main result
is the following.
Theorem 3.1. Let D denote the flow of the cubic nonlinear Klein-Gordon equation (3.1).
T > 0 and 1 < N' <
N,. Let (uo,u1),( io,i)
C BR C HI/ 2([
3
Let
) be such that P N.(uO,u1) =
P<N.(i0 0 , ii1 ), and suppose there exists some K > 0 such that corresponding solutions u and ii to
(3.1) satisfy
|UflLU([0,TXTJ3)
+
1IL4 ([O,T]x A 3) < K.
Then for sufficiently large N. depending on R, T and K,
((N' t)(uO, u) - 4D(t)(o, il 1 )) 11L'i/2 (0,T)xT3)
<
log N)
with implicit constant depending on R, T, K.
Remark 3.1. The same result holds in the Euclidean setting with almost no modifications to the
arguments. We restrict the statements to the periodic case, however, for simplicity of exposition.
Moreover, in the Euclidean setting we can eliminate the dependence on time in the implicit constants.
We prove Theorem 3.1 by demonstrating that under the above assumptions, the low frequency
component of the solutions, ujo = P<Mu for some M C N satisfies a perturbed cubic Klein-Gordon
equation given by
ELujo + ujo = P MF(uo, ujo, u1o) + err.
(3.2)
where err is an error term which we can control by the well-posedness theory. Such an expression
will allow us to prove Theorem 3.1 by a stability type argument.
We will work with the UP and VP function spaces. These spaces have previously been used
in the context of critical problems by Hadac, Herr and Koch [28] for the KP-II equation, and by
50
Herr, Tataru and Tzvetkov [29] for the quintic nonlinear Schrddinger equation on T3 . See [36] or
(28] and references therein for a more complete overview of these function spaces. We record the
basic definitions and properties of these spaces in Appendix A.2.2.
The key benefit of these function spaces is that they recover the endpoint embeddings which
fail in X',2 but still enable us to exploit the same type of multilinear estimate machinery, and are
thus are a suitable setting for critical problems. Unfortunately, however, although one can treat
small data theory in these spaces, they miss out on a key property of the Strichartz spaces which
is exploited in the stability theory, namely that if a space-time norm is finite on a time interval,
then one can isolate subintervals where the norm is small.
To overcome this difficulty, we introduce a weaker norm (3.6) which recovers the necessary
properties in order to prove stability, but which still controls the well-posedness theory.
This
approach was used by Ionescu and Pausader in [32] to prove global well-posedness of the energy
critical nonlinear Schrbdinger equation on T3 , although the norm we introduce is slightly different
than the one used in that work.
We refer the reader to Chapter 4 for an application of Theorem 3.1 to the symplectic nonsqueezing of the cubic nonlinear Klein-Gordon equation (3.1).
3.2
Set-up
In the sequel, we let (V) be the operator with symbol V1+
II2 and we let F(u)
u 3 . We rewrite
(3.1) as a system of first order equations by factoring
att - A + 1 = ((V) +
iAt)((V)
- A).
_
U
then u
=
u+ + u- and the functions u
(
t)U
-F(u+
=
((VTFi(t)
2(V)
'
For a sufficiently regular solution u to the NLKG (3.1) we can define
solve the equations
+ u-)
2(V)
1()
(uo Tz
2
51
(3.3)
(U17)
We set
) ds
ei(t-S)(v' )
S
and we obtain a Duhamel's formula for solutions of (3.3) given by
U
ui -4 il
= eC
(F (u)).
This formulation is equivalent to (3.1) and since u = u+ + u~, we can reconstruct a solutions to
(3.1) from this system and bounds for u
imply the same bounds for u. We will not be using the
so we will often drop the notation where it has no impact on the argument.
specific structure of u
Before proceeding, we recall the definition of the UP and VP spaces. Consider partitions given
< to < t 2 < ... tK <
by a strictly increasing finite sequence -oo
convention v(tK) := 0 for all functions v : R -+ H.
0-
If tK = co we use the
We will usually be working on bounded
intervals I C R. For some additional details about these function spaces, see Appendix A.2.2. In
what follows, B will denote an arbitrary Banach space.
Definition 3.2 (UP spaces). Let 1 < p < oo. Consider a partition {to,...
B
1
with Zfi-
|1 pL2
,tKj
and let (Vk)k=O
C
= 1. We define a UP atom to be a function
K
a
>1[tk-1tk)Vk-1
E=
k=1
and we define the atomic space UP(R, B) to be the set of all functions u: R
-
13 such that
00
U
= E
ja
j=1
for a9 UP atoms, and {A}
ull
c
0(C), endowed with the norm
:= inf
E
j=1
jAjj, U = E Ajaj : a1 is a UP atom}
j=1
Definition 3.3 (VP spaces).- Let I < p < oc.
We define VP(R, B) as the space of all
52
functions,
v : R -+ B, such that the norm
1Ki/
IIvIIVP(R,B)
=
||v(ti) - v(ti.i)IIiL
sup
partitions
< 00
with the convention v(oo) = 0.
We let V-(R, B) denote the subspace of all functions satisfying limte,-o v(t)
=
0 and V/c(IR, B)
denote the subspace of all right continuous functions in V-(R, B), both endowed with norm defined
above.
We will establish a local well-posedness theory for (3.3) via a contraction mapping argument
IuIIx- = (S(k)'sjlU(k)|2)
,
in the adapted functions space
lIf JuJ
=
||eFit(V)fI U2L2(RxT3).
(3.4)
k
Similarly we define
IIly =
112
)2_911
(k)|
((k2sU'
1/2
2
2
rIl,2
=|e'FitV
If V,2,L 2 (RxT 3 ).
(3.5)
k
c,. We define
We will simplify the notation and let V :=
and
XS=X'xX.
YS=YxYS
endowed with the obvious norm. These spaces are at a slightly finer scale than the spaces used in
[28], and one should view these as analogous to the spaces used by Herr, Tataru and Tzvetkov in
[29] for the quintic nonlinear Schrddinger equation on T 3 . This choice of scale does not affect the
multilinear estimates, as we only need the weaker Strichartz estimates at dyadic scales, however it
allows us to make use of the following important orthogonality property.
Corollary 3.4 (Corollary 2.9, [29]). Let Z3
2 ||PCk
V::2,
-
U Ck be a partition of Z3 or Rk. Then
(IxT)
k
53
)1/2 $llg~1
We will work with the restriction spaces
X'(I) := {u E C(1; HS(T 3 ) x H"(rIF3 )) I(t) = u(t) for t
c
I,
e Xs}
endowed with the norm
Ijujjxs(I) = inf{1ilixs Ii : i(t) = u(t) for t C I},
and similarly for Ys(I). In our estimates we will often implicitly multiply functions by a sharp
time cut-off. When I = [0, T) we will use the notation X'. Perhaps most importantly, bounds for
solutions of (3.3) in these spaces implies the same bounds in L 7LR
for solutions of (3.1), which
was precisely the endpoint embedding we were missing in the X'12 spaces. Finally, for I C J, we
have the embedding X'(I) " X 8 (J) which can be seen via extension by zero.
As mentioned above, we will need a weaker norm which controls the local well-posedness theory
for the equation in order to prove the necessary stability theory. This norm is similar to the norm
introduced in [32, Section 2] for the same purpose and should be thought of as the appropriate
substitute for the L4
Strichartz norm. We define the Z(I) norm by
7
fIz(I)=
Jic,
For u = (u+,
sup _1PNf 1(Jx)
Jjj<1 (N
Lt;(
1/2
)1/2
(3.6)
u-), we obtain as a consequence of Corollary 3.4 and the Strichartz estimates in
Corollary 3.6 below that
jIUlIL4(Ix3) < jjuIZ(I) < tIU'1y/2(I) < IIUI1/2(I),
hence this indeed defines a weaker norm. This norm only plays a role in the stability theory and
is not necessary to prove the low-frequency component satisfies the perturbed equation (3.2).
3.3
Strichartz estimates
We record some Strichartz estimates for the Klein-Gordon equation in the UP and VP spaces.
By finite speed of propagation, the Strichartz estimates for the torus are the same as those in
Euclidean space, provided one localizes in time.
54
Lemma 3.5 (Strichartz). Let 2 < q, r < oo with - +
3
LqL;([o,TIxT3)
<
= 1 - s
Then
CTIPIIHS.
We also obtain the following formulation of Strichartz estimates in UP spaces. The reasoning
is identical to the proof of Corollary 2.21 in [28].
Corollary 3.6. Let 1 < p < 4 and 2 < q
(i) IUNL4 ,([0,T]xT)
(ii) ||UNK1L
(iii)~~~
[
xT
'.O~,([0,T] XT3)
4. Let T > 0, and let uN
= PNUN.
Then,
CTN/2IIUNIIU
N
6ONe|u
1UN
Proof. The first claim follows from Lemma 3.5 and the transfer principle, Proposition A.13. The
second claim follows from (A.3) and the fact that u agrees with its right-continuous variant almost
everywhere. The final claim follows from interpolation with the trivial L 2, estimate, since we are
E
on bounded time intervals.
We have the following refined bilinear Strichartz estimate for the Klein-Gordon equation. Such
estimates were treated in full for wave equations in [231.
For a proof of this statement for the
Klein-Gordon equation on Euclidean space, see [61, Appendix A] which adapts the geometric
proofs from [621 for the nonlinear wave equation, see also [35]. One can also adapt the proof from
[62] to the compact setting to prove Proposition 3.7 directly, replacing the volume estimates with
fairly straightforward lattice counting.
)
Proposition 3.7. Fix T > 0 and let 0, M, N be dyadic numbers and OM, ON functions in L 2 (T 3
localized at frequencies M, N respectively. Define um = e -lit(VV/M
and vM = e
2it(V)
N
Denote
L = min(0, M, N), and H = max(O, M, N). Then
I2 CT
I~P(UMN)IL',
L||$MI
Sl
L2(T3)
1NIIL2(T3)
if M<N
1 1
H L'I OMIIL2 (T3)1
N IL2 (T3)
if
M ~ N.
In order to prove the multilinear estimates required for the stability theory, we will make use
of the following refined Strichartz estimate which is proved using Proposition 3.7.
55
Proposition 3.8 (Proposition 10, [61]). Let n > 3 and let M < N and let
support in a ball of radius
-
UM,N
have Fourier
M centered at frequency N. Then
HUM,NlIL
([0,T)xT 3 )
i CT N1/4MnY HIuM,NflU.
In particularif UN = PNUN then the result holds with M = N.
By the transfer principle, an orthogonality argument and Corollary 3.6, we obtain the following
UP version of the above estimates.
Proposition 3.9 (Proposition 7, [61]). Fix T > 0 and let L, respectively H, denote the highest
and lowest frequencies of M, N, 0. Let um C U 1 , UN E U1 2 . Then
||Po(uMvN)IIL2'
if
|VN Iu2
L||umI U2
M< N
CT
H L2UMIU4 1 IIVN U4
Remark 3.2. One can convert these estimates to bounds in V
2
if M ~ N.
with a logarithmic loss in the first
estimate, and no loss in the second estimate, see PropositionA.14.
The following lemma follows immediately from the atomic structure of U 2 . See, for instance,
[61, (17)] for the computation.
Lemma 3.10 (Linear solutions lie in XS). Let s > 0, 0 < T < oo and uo G HS(T 3 ).
Then
lle*uit(U IX -([O,T)) < ||UO I H'(T3).
We have the following duality estimate. The proof of this result is a straightforward adaptation
of the proof of Proposition 2.11 in [29].
L' Hs([0, T) x TV) we have
Proposition 3.11. Let s > 0 and T > 0. For f
||I (f)|x. <
sup
j
Proof. We let (an)neZ3 E f 2 (Z 3 ) be with I1(an) ep
f x6.
a (n)" X[O,T) e W-I )(n)
nEZ3
f
56
=
T
f
(t, xv(t, x)dtdx
1 such that
We use the definition of U1, and we use duality from Theorem A.9 and Proposition A.11 to
estimate each piece by
___
f
eis(n)vn(s)d
f
)
U
X[o,T) e Fis(n)>dt
for a sequence vn E V2 with |IVnIIV2
=
2
2(n)n
f(s)(n)
<
1 supported on [0, T). We then define
(an (n) S-1 eit()
V(t, X)
Vn M)
eix,
nEZ3
and we observe that v E Y.1 s([0,T)) with ||vj|yIi-. < 1. Hence
L j ()()t)(n) + e,
II(f)IIx
f
f
nEZ
3
and the claim follows by dominated convergence theorem and Plancherel.
L
Finally, the following proposition demonstrates that a priori bounds on the Strichartz norm
control the norm of solutions in the adapted function spaces (3.4).
Proposition 3.12. Let u be a solution to the cubic nonlinear Klein-Gordon equation for initial
data (uo, ul) E BR C 7 1/ 2 which satisfies
|IU||gL4
,,rfl
,g3)
(3.7)
< K.
Then ||u|lX|/2 < C(K, R).
Proof. Let u solve the cubic nonlinear Klein-Gordon equation (3.3) and suppose that u satisfies
the uniform bound (3.7). Fix T > 0 then by Lemma 3.10 we estimate
|I|U
1/2
<
11 (Uo, uI) 11,/2
+II(F)
TX/2,
F(u) = u
We expand the nonlinear term and we deal with I+ as the other term is handled analogously. By
57
Proposition 3.11 and H6lder's inequality
|I+(F)|11/2
+
IT
sup
vEY
2
vY
2 :||V||
vE Y
2:|vI|| 1/ 2 =1 N
=
: |vI
2=1
T3
1
//
sup
<
F(t, x)v(t, x)dtdx
1/2=1 N
sup
PNF - PNv(t,x)dtdx
T3
|PNF||L4/3||PNV(t,X)L4.
By complex interpolation, we have the dual square function type inequality
)1/2
N
Applying Cauchy-Schwarz, and noting that Remark A.4 applies to 1+, we use Corollary 3.6 part
(ii) and Corollary 3.4, to obtain
sup
VEY 1/2
:||t 11I
Nt
/2=
(z
<1f1' LX
||PNf 11L4/3
/2V71,1/2
IfHL4/3
E
which yields the result.
3.4
HU13 4
Multilinear estimates
Multilinear estimates for nonlinear Klein-Gordon equations in the form we need for well-posedness
were proven in [61], where more general non-linearities were treated, but we state only the estimates
we require. The statements are slightly modified for our setting and we include the proofs for
completeness. Ultimately, however, we will need slightly stronger estimates for the stability theory,
which we prove in Proposition 3.15.
Theorem 3.13 (Theorem 3, [61]). Suppose that the signs
i (i = 0, 1, 2, 3) are arbitrary and
H ~ H'. Then
2
Li<H
2
1/2
_=i )H
=1
58
and
S
IlWIly/2= 1 L<H L 1 <H
3
2
1
Proof. We estimate
1H
IiUL2 VHIILt~
,SHjfUL1H UHIiILZ
2
rJ~iULjuH/vHdxdt
2 ~
2
< I (
L 1 L 2 |UL 1 IIV
12
IIUL2 V2 11UH' I V,
H IIV2,
where we used the improved bilinear estimates from Proposition 3.9 and Remark A.6 in each term.
Summing over Li < H, we use Cauchy-Schwarz on the terms
L
which yield the i
=
H-|ULIj
=VL?
||UL IIV2
1, 2 factors in (3.8). Ultimately, it suffices to bound
H-1+26(
Z
L
H- 1+2
L1--26L'-26 1/2
L 1 <H L 2
L1
L1
<4 1,
H
and since 1 - 26 > 0 for 6 sufficiently small, this yields (3.8).
The second estimate is treated similarly, with the roles of
UL
and VL swapped. That is, we
2
bound
UL 1 UH'UHVLdxdt
$
IIUL1 UH IiL2
)1L
".1
iUH'VLiIL2
jL1L
||UL1 I JH
2
1 H IV
1 LIV0
Collecting terms, summing in L 1 , L < H, and applying Cauchy-Schwarz yields
(3.9) < H2 ( E
E
L1-26L
26) 1/2
L,<HL1,<H
as(rHrH')||Uqr1|y d.
2|U
H ||UH'VLI,
as required.
59
|I
IIUH
||/2
2
2UHv
We will now review the well-posedness theory for (3.3). We need to estimate the cubic nonlinearity F(u) = u3 for u = u+
+ u-. Hence we can decompose
F(u) =
uY,) u() u(k)
{i,j,k}E{1,2,3}
for u() = u . Thus, it suffices to estimate these eight cubic terms, which we do in the sequel. Due
to finite speed of propagation, the arguments from [61] apply if one allows implicit constants to
depend on the time interval.
Theorem 3.14 (Theorem 4, [61]). Fix T > 0. There exists a constant C depending only on T > 0
such that
3
||I4 U(1)
5u(2) 1u(3)) 11,/2 < CH
(3.10)
JI|USi)11/2.
i=1
In particular, since X
-
Y', this implies
3
|(U (1),
(2),
(3)) lIX /2 < C
110 JJX1||2.
Remark 3.3. The proof works similarly for any s > 1/2 with the obvious modifications, however
we omit this generalizationfor simplicity of presentation. The wellposedness of (3.3) follows from
these estimates via a straightforwardcontraction mapping argument.
Proof. We only treat I+ as I- follows analogously, and T > 0 will be fixed throughout. In the
usual manner, we take extensions of the u(') to R, which we still denote by u),
and (3.10) follows
by taking infimums over all such extensions. We also suppress the notation
on each function.
+
()
U(2)
(3)
2
sup
E
1WII,1/ 2=1 No N 3 +N 2 +N 1 =No
+
uN uN
3
2uNvNodxdt
.
We do not repeat these considerations. By Proposition 3.11, we estimate
The convolution requirement, implies that the right-hand side above vanishes unless Ni ~ N for
some i
# j, and hence we may assume without loss of generality that
N3 < A
2
;< N 1,
N ~ max{No, N 2 }.
60
In the case that No ~ N1 , we use (3.8), and we bound
z
UN3 UN2
NUNoVNdxdt
N 3 +N 2 +N1=No
1/2
3
el2
V
No~N1 i=2
Ni,<NO
:
By Corollary 3.4 and Cauchy-Schwarz in No
-
I+(U(1), I (2) 1 ())
N1
,
ll
()11,y
2
11U(2
11
2|11U3
2
1
+
)| 11
2.
3
When N 2 ~ N 1 , we use (3.9) and we can similarly bound
+ 1, U(2) U(3)
sup
X1/2
+
I1wII 1/ 2=/ N 3 +N 2 +N1=No
<
ff UN 3 UN2 UN
J U31l y1/2 E
(N1N2)1/2
3NI~N2
1 vNO
dxdt
IIUN, II ,2 1 UN2 11 V,2,'
Once again, we obtain the desired bound using Corollary 3.4 and Cauchy-Schwarz.
E
Finally, we need to following refinement to Theorem 3.14 which says that we can replace some
of the factors with the mixed norm
I1u
t
Z (I)
t3/4
=
t1/4
IIU'flZ(I)HU I1
1/2
,
-
N|N
2i NO
J|UN1 JV2 INo J0
for Z the norm defined in (3.6), and we set Z' = Z+ x Z_.
As in the proof of Theorem 3.14, we need to consider cubic expressions in u+ and u-. It will
be clear from the proof that we could have, instead, relied only on bounding u
+ u- in the Z(I)
norm, which is more consistent with the L'X Strichartz norm from the standard well-posedness
theory. Ultimately, however, such considerations do not affect our arguments given Proposition
3.12 which allows us to control the Z(I) norm of solutions to (3.3) via Strichartz bounds.
Proposition 3.15. Under the above assumptions,
(U(1), U (2) 1 (3)) X1/2(j) <
||U9y,1/2(I)|
E
{i,j,k} E {1,2,3}
61
Iz{(I)l|Uk})e|z{(I).
Proof. We fix T > 0 and we only treat the 1+ term. By Proposition 3.11, we estimate
(2)
(3)
suSUP
E
UN
UN
UN, VNo dxdt
liw||1 1/2 = N3+N 2+N1=No
1/2
+
The convolution requirement implies that the right-hand side above vanishes unless Ni ~ Nj for
some i # j, and hence we may without loss of generality assume that
Ni ~ max{No, N2 }.
N3 < N 2 < N1 ,
In the case that No
-
N1 , we apply H6lder's inequality and obtain
<
UNUN 2UNVNodxdt
2
3 UNL.UN
2VNoL 2UN
N 3 +N 2 +N 1 =No
N 3 +N 2 +N1=No
Let us consider the first term. Let C be a cube of size N 3 centered in frequency space at do E
with 1 oI
-
Z3
N and let PC denote the (sharp) Fourier projection onto this cube. Since the spatial
Fourier support of
(PCUN 1 )UN 3
is contained in a fixed dilate of C,
I| I (PC
N1 IIL2
UN
3U
SN
UN,
C
L2
and hence by Husder's inequality and Proposition 3.8 on the term with PCUN, we can bound
S N3 UN
2
1
Thus for fixed N1, we obtain
3
N
S
(N3N1)
L
L(IXT3)
(N34N
1
( 1PCUN1
C
y21-
C1N
1L
IN3UN11 L x
N3,<N1
IIuN3
U1PCUN1
UN3 1L x(IxT3)
y2 1
12
C
N3<N1
N,
L1,(I xT 3)
(N1||PCU
C
N3<N1
62
N1
V:Ly3
hence by resumming over C and using Cauchy-Schwarz in N3 , we can bound
<
1
<
IUN3UN1I L2
N3<Ni
N3 <Nl
(N3)
4,.,,
4 1JUN, 11
X73)
JUNi
11
/2
)1/2
N3
which yields
IJUN3UN
1IL2
<
t ,x
IIU1IIZ11UNjJJy1/2-
N 3<N1
We perform the same analysis on the second term.
By Cauchy-Schwarz in No ~ NI and sym-
metrizing we obtain
II(U('), U (2) ,U(3))1
X1/2(I
U(i)
1,/2(j) I U(
JZ(j) J U(k) JZ(I)-
{i,j,k} E {1,2,3}
Using the definition of the Z'(I) norm, and combining this bound with the estimates from Theorem
3.14 yields the result in this case.
The case when N1 ~ N 2 and No < N2 requires a bit more care. We estimate
S
UN3UN2
_J
N 3 +N 2 +N1=No
S
UNvNodxdt
IUN 3 UN1 1IL
N2 VNO
L-
N 3 +N 2 N1 =No
As above, for the first term, we obtain
S
I UN3UN 1 IL2
IIUN1
UN31L t1 (IVxT3)
N,
N3<Nl
1yl/2
N1jJPCjUNj 112V2
:L
(N3 4
S
N3,<N1
KN,)
IUN3
ILt,X(IxT3).
For the second term we use H6lder's inequality to bound
2UN2NO
I IUN2VNo L
=
,
IIUN2 VNO 1IL2
No<N 2
No<N 2
63
I
) 1/2
and we use Propositions 3.9 and 3.8 to bound this by
38
VNO
UN2 N
/ 1VN
3/4
(N) 1 /4
(N)
3
No$N2
2
0
0
(N2)1/8IiN2 / HIUN 2 |3 4 N 0 2||vNO V2
<
No<N 2
We have split this term using H61der's inequality in order to gain some term which enables us to
sum in No without loss. Using that
(N)'!-
N1/2||VNOHV2 ;<
HWIIy/2
No<N2
and
(N3)
1UN3
HL4(IxT 3 )
$U3HZ(I),
N3<N1
we are left with
1UN 1 Ijyl/2
E
(N2) 1 /81UN 2 1
IIUN2 1L4
L
N3~N2
and we once again conclude by Cauchy-Schwarz with 1 +
3.5
3
+ 1
=
1 and Corollary 3.4.
E3
Stability theory in adapted function spaces
In this section, we prove the necessary stability theory for the nonlinear Klein-Gordon equation
in the adapted function spaces. As discussed in the introduction, the key difficulty in proving a
satisfactory stability theory in this setting is that even if X 1/ 2 (I) is bounded, we cannot isolate a
small interval on which the norm is small. Ultimately, however, using the intermediate Z'(I) norm
we are able to recover the desired stability theory. We record the following results which are, for
the most part, straightforward adaptations of the analogous results for the nonlinear Schr6dinger
equation from [32, Section 3].
Proposition 3.16. Suppose that R > 0 is fixed and let uO
Then there exists 6O = 60 (R) > 0 such that if
|le it(V)uo|z (I) < 6
64
=
(u+,u-) with ||u I|H1/2(IxJF) < R.
for some 6 < o, on some interval I with 0 E I and |I < 1, then there exists a strong solution to
(3.3) in X 1/ 2 (I) with initial data u(0) = uo and
X1
|1u, - e* (uJ
2
(3.11)
6 .
Remark 3.4. The choice 5/3 is arbitraryand in fact the statement holds for any a with 1 < a < 2,
and we merely require a > 1 for our applications. This proposition can be thought of as an version
of a small data result in the adapted function spaces which does not require that the initial data be
small in the H1/ 2 norm.
Proof. The statement about existence follows from a standard fixed point argument. Indeed, Let
R, a > 0 and consider
S
=
{u E X 1/ 2 (I) : I|uIIXl/2(I) 5 4R, I|uIlz'(I)
< 2a}
and the mapping
(u) = e t(u
t iI (F(u)).
By Proposition 3.15,
IP(u)lz,(I)
I)
Zo H1/2
+CRa2 < R +CRa 2
I e it(V)u I|z,(I) + CRa2 < 6 + CRa2
,
|I'V'(u)||X/2
and similarly for the difference expression. Choosing a = 26 for 0 < J < 60 and 60
enough so that 4CR6 < 1, we find that D = D+
=
JO(R) small
+ 4- possesses a unique fixed point u in S.
Finally, to prove (3.11), we obtain by another application of Proposition 3.15 that
Iu-e
it(V)u~~a
2 1
,
j6 2
hence taking JO even smaller if necessary, we obtain the statement.
In light of the fact that the Z' norm controls the Lt, norm of solutions, the standard blow-up
criterion in Strichartz spaces together with Proposition 3.12 imply that this weaker norm controls
the global existence theory.
65
For simplicity, we define the norm
jhIjN
(I)
e it(V)
fa
(s) ds
2(V)
X 1 2 (I)
(312)
The following is the main result of this section.
Proposition 3.17. Let I C R a compact time interval and to C I. Let v be a solution defined on
I x T 3 of the Cauchy problem
v t - Av + v + F(v) = e
(, Itv) It,
=
(vo, vi) c
3,
h1/2
and identify the solution v with (v+, v-). Suppose that
I|V'IlZ(I) + IV'IILH1/2(IXT3) < K.
Let (u, atu)I _
=
(uo, uI) C 711/ 2 (T 3 ) and suppose we have the smallness condition
11vO
for some 0 < e <
(3.13)
-
uO, Vi
- ul)IK1/2(T3)
+ H1eIN (I)
El where El < 1 is a small constant E1
=
(3-14)
E < El
E1(K, I) > 0. Then there exists a
unique solution (u, tu) to the cubic nonlinear Klein-Gordon equation on I x T3 with initial data
(uo,ul) at time to and C = C(K, I) > 1 which satisfies 1|v - u|IX1/2(I)
Ce.
Remark 3.5. In particular, (3.13) holds if we have X 1/ 2 (I) bounds on the solution v, and consequently, by Proposition 3.12, if we have Lt'(I x T ) bounds. Additionally, the computations in
Proposition 3.12 also imply that L' (I x T ) bounds on the error imply the N (I) bounds on the
error in (3.14). Hence Proposition 3.17 can be seen as a refined version of the long-time stability
theory from Appendix
4.8.
We include a proof of this fact for completeness, although it follows almost identically to the
corresponding statement for the NLS in [32, Section 3]. The main idea is to mimic the proof of
the standard Strichartz space stability, exploiting the extra properties of the Z'(I) norm. Roughly
speaking, we will work on small intervals where
IMv
66
z'(Ik) is sufficiently small and then, in spirit, the
computations which yield the standard stability theory yield the result. We only have to check that
at each step we can guarantee that the assumptions still hold, namely that the difference between
the solutions remains sufficiently small. This is possible since the number of steps depends only
on K and el, hence we can iterate such an argument to cover the whole interval in order to obtain
the result.
Proof. Without loss of generality, we may assume 11 5 1. As in the proof of Proposition 3.16,
there exists some J 1(K) such that if for some J D to,
IIei(*-tO)(V)V(to)I|zI(J) + leIIN (J) 5 6 1,
then there exists a unique solution v to (3.3) on J and
IIv
v(to)I|
t
- e *i(tto)(V)v(to)|X/ 2 (J)
je +
2
.15)
Ile||N (J).
Next we claim that there exists el = Ei(K) such that if for some Ik = (tk, tk+1) it holds that
l|ellN (Ik)
IIVIIZ(Ik) < 6
and
61
(3.16)
61
then
le
i(t-tk)(V)v
|v ||z'(Ik)
Indeed, we let h(s) := ||e
such that h(s) 5
|e
'itt
6
1/
2
C(1 + K)(c + |le||N (Ik)) 4
(tk)IIz (Ik)
i(ttk)(V)v
C(1 + K)(e + e
(tk)lz+(tk,tk+s).
3N.(Ik))-
Let Jk = [tk, t') C Ik be the largest interval
, for 6 1(K) as above. Then by Duhamel's formula
)(V)v (tk) I Z(tk,tk+S) <
Kv
e
IIZ(tk,tk+s) +
IV
-
esittk)(V)V (tk) I
+ h(s)3 + 2|leIIN (Ik).
By definition,
h(s)
(3.17)
<5 ||e3i(t-t1)(V)V
(tk)d
4
Z(tk,tk+s)
67
e
V
X
2(tk,tk+s)
2(tk,tk+s)
hence by (3.15), the boundedness of the free evolution in X/2 and (3.13),
(e + h(s)3 + 2|1e|N,(I,)) K4
h(s)
C(1 + K)(e +
+ C(1 + K)h(s)
Ie|N (I))/
and we can conclude the claim provided el is chosen sufficiently small. Let now Ik be an interval
such that
|e i(t-tk)(V)v (tk)IIz
(jk)+
VIVZ'(I)
< E < CO
IeIN (Ik)
it holds by the above considerations that |IV
< K
IIX12
(3.18)
C,
+ 1. Fix such an interval and let u be
a solution to (3.3) defined on an interval Ju 3 tk with
II(u(tk)
Set
#=
u - v, then
#
{vt
-
v(tk),Otu(tk)
atv(tk)IIR1/2
-
< CO.
solves the difference equation
(#,
- AO +
tk) ttk
and we can identify 0 with
+ (v + 0) 3 - v 3 - e = 0,
= (U(tk) -
9
v(tk),
tu(tk)
(+, #~) as before. Let
u:Rx T3 -+ R
-
&tv(tk))
Jk = [tk, tk
c
H1/2
+ s] n
'k
3
n
Ju be the maximal
interval such that
1
~
Zg(Jk)
1JCeJ
Such an interval exists and is non-empty since s
H-
-
10(K +1)'
I|zejto4to4s) is a continuous function which
vanishes at s = 0. Similarly to the argument used in the standard Strichartz stability theory, by
Proposition 3.15 we can then estimate
fle
<
i(t~tk)(V)(u
(tk)
-
V (tk))I1xi/2
+ II(V + 03
-
V3 N (Jk) + fleHN
Il(u (tk) - V (tk),&tu (tk) - atv (tk)hK1/2 + C(-oI|#||/2(
68
+
le
(Jk)
lN (Jk))-
(3.19)
Thus, if 60 is sufficiently small,
IiV'liZ'(J) 5 ClI10I11,/2(J)
and in fact Jk = Ifn
5 8C00,
(3.20)
J,. Moreover, u can be extended to all of Ik by the remark after Proposition
3.16, and further (3.19) and (3.20) hold on all of Ik. We conclude the proof by splitting I into
subintervals such that
llV'HZ(Ik)
lie1IN (Ik) < KE2-
62,
On each interval (3.16) holds, hence by (3.17), (3.18) also holds, and as above we conclude that
(3.19) and (3.20) hold. This concludes the proof of the proposition.
IZ
Remark 3.6. If we consider the nonlinear Klein-Gordon equation with truncated nonlinearity,
uN + uN +P<N(
(uN)tt -
NuN)
(uN, OtuN) t=0 = (uO, ul) E W1/2
3
0,
3 _R
uIRx
3
then all the estimates from Sections 3.4 and 3.5 go through with constants uniform in the truncation
parameter. Indeed, set G(u)
F(PNu) and with VN
=
||P NG(u)IN(I)
sup
=
P<NVN, we can estimate
fJ PNG(u)VN
=
IVN Iyl/2
Sup
G(u)vN
Ny1/2
and all of the previous multilinear estimates go through as before. Moreover, to repeat the stability
argument above, we only need
IIP NvljZ(I) + IMILQ1
1/2 (IxT3)
<K.
Moreover, if one only needs to compare low frequencies, it suffices to require
IIP N(vO
-
u0,vi -
u1)Ii-1/2(73) + HP NeIIN(I)
6 <
This statement should be compared with Remark 4.20 and Lemma 4.35.
69
El.
3.6
A low frequency equation
The next propositions should be compared with Proposition 5.1 in [20]. As we are at the critical
level, we cannot hope to achieve the gain of derivatives for the Strichartz estimates, as was obtained
in Theorem 4.3 in [20].
On the other hand, we can still obtain some decay by exploiting the
improved Strichartz estimates from Proposition 3.9 provided we are able to create a scale separation
between low and high-frequencies.
In the sequel, where we denote errors of a given order, this
is always understood to be in the N(I) norm defined in (3.12).
In the sequel we will write
F(u, v, w) = uvw as a means to record to various combinations in the nonlinearity.
Proposition 3.18. Let R, T > 0 and let u be a solution to the cubic nonlinear Klein-Gordon
c
equation for initial data (uo, u1) E BR
W1/2
which satisfies
IUILt(([0,T)x7T3)
Let 1 < N'
< N,.
(3.21)
< K.
Then there exists M E [N', N,] such that the low frequency component
u10 = P<Mu satisfies the perturbed cubic nonlinear Klein-Gordon equation
Fujo+ujo = P<MF(uo,uj, ujo) + OK,R,T
log N'
.
(3.22)
Proof. In this proof we allow the value of the parameter 0 > 0 to change from line to line, and we
allow implicit constants to depend on the various parameters involved. Given dyadic frequencies
N', N, the frequency interval [N', N*] contains 0
1,
intervals of the form
(.IN))
[N'(log(N/N'))Q, N'(log(N*/N')),+2 ]
for a = 2k, k E N.
By definition of the Y' norm (3.5), for any subset S C Z3,
|Ps u||Yg
>u(N)
~
Since L'
sIpQNu 112
)
(QNnS#0
1/2
2
bounds imply X 1/ 2 bounds by Proposition 3.12, and X 1/ 2
Y 1/ 2, we obtain yi/
2
bounds for solutions of the nonlinear Klein-Gordon given the assumption (3.21). Consequently, we
can decompose the sum in the definition of
Ilullyl/2
must be some frequency interval where
70
into the blocks 1, and we conclude that there
H(P<N'(log(N/N))+2
(3.23)
P<N'(log(N,/N'))-) u d y1/2 , (og
-
Fix this a. We introduce the notation M:= N'(Iog(N*/N'))', and we define
U1 0 =
Let a, b, c
P<M,
Umed =(<M(log(N/N'))2
P M)U,
-
Uhi
-
P<M(log(N*/NI))2)U.
C {lo, med, hi}, then we can decompose the nonlinearity as
E
+
PG<MF(u,u, u) = P MF(uo, U 0 , u1 0 )
P
M
F (Ua, Ub, uc).
max(a,b,c) > med
By (3.23) and (3.10), any term involving Umed will be an error term.
Hence, we only need to
consider terms with Uhi and ujo components. We will use largely the same analysis used to prove
Theorem 3.14, but in this case, the dual function will always be localized to low frequencies. First
we consider
P<MF(uo,Uhi, uhi) =
: PNOP MF(uN 3, uN2 , UN 1 ),
N1 ,N 2
N 3 ,No<M
which we will treat with similar estimates to those used to prove (3.9). By the convolution requirement, the term above will vanish unless Ni ~ Nj for some i
/ j, hence we may assume without
loss of generality that
N1 ~ N2
,
N 3<N 2 < N1 ,
and we bound
i
UN3 UN2
UN3 UN1 L2 IUN1VNo
NUN1 VN dxdt
f-'
u2 V o OV20.
L2
Ns N UwNo|uN
N3No C'yAppNyNg
IaIyUN2-w iV2 IIUN3 H IIVwb
2
222
1
Applying Cauchy-Schwarz in No, N 3 < M, we obtain
sup
/2=1No,<M N 3 <M
(NN 2 )N|UN
N3 No /
v HUN3 0V
V
3 I2 INo
1VN
IUN
1 2 V12
1
N1 ,N2 ~hi
N1 ,N2 -hi
1-1/2
(N1N2 )||UN
(N1 N 2 )71
1H
I
UN2IV
V2
.
>I
1
*
$ |
03
In particular, by the restriction
N 2 , N, > M(log(N*/N')) 2
,
(3.24)
we can bound the above multilinear estimate by
SU3 11 Y 1/2
(o(N/1))-41
(N1 N2 ) 2 11UN1 V, O2, 0V
N1,N 2 ~hi
and by applying Cauchy-Schwarz in N 2 ~ N 1 , we see this term is part of the error in (3.22).
For the term with high-frequencies
P<MF(h%, Uhi, Uhi)
we can once again assume without loss of generality that N 3 < N 2
Jf
UN 3 uN 2 UN 1 vNodxdt
IUNj
IIUN 3 UN 2 ILt
<(N1 N 2
r
N N
N
1
-
N 1 . We obtain
NO 11L2
NO |UN1 V
UNIv
2 21Vv2UN
IIU
211U 31 3 I N 0
VNO V2
By Cauchy-Schwarz in No and N 3 , we obtain
sup
11WI
Y /
2=1 NO/
<
IU31||V/2
M
S
(N)6
1
N1 ,N 2 ,N 3 -hi
S
N1 ,N 2 ~hi
N
N2
No N3
N3 N0 IUN
_ (N1N2 )1 |UN 1 IIV
IIv UN2 HV11 IIUN3 IIVA HVNO 11V2
UN2 IV2
and once again by (3.24) and Cauchy-Schwarz in N 2 ~ Ni we conclude that such expressions
contribute to the error term in (3.22). Finally, we note that by the convolution requirement, the
term
P<MF(ujo,uio, Uhi)
vanishes provided log(N./N') 2 > 8, which concludes the proof.
Remark 3.7. The key difference between the proofs of Propositions 3.18 and 3.13 is that the low
frequency is always bounded by M in Propositions3.18, hence when we apply Cauchy-Schwarz in
72
the low frequency, we only lose a factor of M, instead of the next largest frequency. This fact,
together with the manufactured frequency separation, enables us to achieve the necessary decay.
3.7
Proof of Theorem 3.1
Finally we arrive at the proof of the main result of this chapter.
Proof of Theorem 3.1. Let u and U denote the solutions to (3.1) with initial data (uo, u1) and
(iEo, iEi), respectively. Let K > 0 be such that
|u|L([OTI]xT3) +
,L
X([0,T xT3)
< K.
Since ul, = P<Mulo, and similarly for i1 o, the same bounds hold for the low frequency components.
By Proposition 3.18, the low frequency component satisfies the equation
and similarly for U1c. Since (ujo -
uiio)jt
=
K,R,T((og(N*N'))-
0
)
1 ujo + ujo = P<MF(ujo,ui0 , uio) +
0, let N* be chosen sufficiently large depending on
R, K and T so that the smallness requirement of Proposition 3.17 is satisfied. Then the result
follows from the stability theory, Remark 3.5 and the fact that the bounds for the equation with
0
truncated nonlinearity are uniform in N.
73
74
Chapter 4
Symplectic non-squeezing for the cubic
NLKG on T3
4.1
Introduction
We consider the behaviour of solutions to the initial-value problem for the periodic defocusing
cubic nonlinear Klein-Gordon equation
utt - Auu+u
3
=0,
u:IRx T 3 -+ R
(u, atu)1,, = (uo, ui) E H2(T3) x H- (T3
1/2(T 3)(
where H2 (T 3 ) is the usual inhomogeneous Sobolev space. Recall that if we consider the general
power-type nonlinear Klein-Gordon equation
Uti - Au + U + lul)- 1 u = 0,
then sc :=
--
is the critical regularity for the equation. In (4.1) we have p
=
d = 3, hence we
are interested in studying the Cauchy problem (4.1) at the critical regularity.
In this chapter, we will study the qualitative behaviour of solutions to (4.1) by investigating
symplectic non-squeezing for the flow of this equation. The study of infinite dimensional symplectic
capacities and non-squeezing for nonlinear Hamiltonian PDEs was initiated by Kuksin in [38].
There, he extended the definition of the Hofer-Zehnder capacity to infinite dimensional phase
spaces and proved the invariance of this capacity under the flow of certain Hamiltonian equations
75
with flow maps of the form
<1(t) = linear operator
+ compact smooth operator.
(4.2)
This infinite dimensional symplectic capacity inherits the finite dimensional normalization
cap(B,(u*)) = cap(Cr(z; ko))
=
rr2
Br(u*)
and for z
=
(zo, zo) E C and k
{u E
C Z3,
Cr(z; ko) := {(uo, ui) c
1(/
1/2u(T3) : ||U - UK1/2 < r
,
where
the infinite dimensional cylinder
2
(T3 ) : (k) Io(k)
-
zo1 2 + (k)-
We will always be working with real-valued functions and the
1 iG1(k)
fi7(k)
- zo
2
< r2
are taken to be real-valued
Fourier coefficients. The proof of this normalization in infinite dimensions is an adaptation of the
original proof by Hofer and Zehnder which can be found in [30], see [38] for details of the infinite
dimensional argument. Consequently, if a flow map <D preserves capacities, one can conclude that
squeezing is impossible, namely
<I(t)(BR(U*))
9 Cr(z; k)
if R < r.
Several examples of nonlinear Klein-Gordon equations with weak nonlinearities can readily be
shown to be of the form (4.2), see [38].
Symplectic non-squeezing was later proved for certain
subcritical nonlinear Klein-Gordon equations in [7] using Kuksin's framework, see also [59]. Bourgain later extended these results to the cubic NLS in dimension one in [4], where the flow is not
a compact perturbation of the linear flow. There, the argument follows from approximating the
full equation by a finite dimensional flow and applying Gromov's finite dimensional non-squeezing
result to this approximate flow. Symplectic non-squeezing was also proven for the KdV [20]. In
this situation, there is a lack of smoothing estimates in the symplectic space which would allow
the infinite dimensional KdV flow to be easily approximated by a finite-dimensional Hamiltonian
flow. To resolve this issue, the authors of [20] invert the Miura transform to work on the level of
76
the modified KdV equation, for which stronger estimates can be established.
As we will see in Section 4.2, the symplectic phase space for any nonlinear Klein-Gordon equation is
?j1/2(7d)
for any dimension d > 1. In particular, for the cubic nonlinear Klein-Gordon
equation in dimension three, the symplectic phase space is at the critical regularity, which presents
some serious obstructions to using simple modifications of the existing arguments. Kuksin's approach requires some additional regularity in the compactness estimates. In light of ill-posedness
results below the critical space, for instance the results of Christ-Colliander-Tao [17], [39] or [31]
adapted to (4.1), there is no way to gain the additional regularity needed. Bourgain's argument
in [4] uses an iteration scheme in which one needs uniform control over time-steps of the iteration,
and, once again, this seems to be a genuine obstruction to applying this argument at the critical
regularity. Finally, the arguments of [20] depend heavily on the structure of the KdV equation.
Additionally, the global well-posedness of (4.1) is not know and there is no uniform control on the
local time of existence.
Ultimately, however, we are able to circumvent these difficulties, using a combination of probabilistic and deterministic techniques, which we combine to obtain several deterministic nonsqueezing results. Our first result is a local-in-time non-squeezing theorem.
Theorem 4.1. Let 'D denote the flow of the cubic nonlinear Klein-Gordon equation (4.1). Fix
R > 0, ko E Z3, z E C, and u, C 711/2(3). For all 0 < ?I < R, there exists N = N(?, u, R, ko)
o-(r,
u N, u*) > 0 such that for all 0 < t <
,
)(t) (IINBR(U*))
7 Cr(z; ko)
for r < R
-
and o-
See Remarks 1.2 and 1.1 for a discussion of this result.
In order to state our global-in-time results, we need to introduce the following nonlinear KleinGordon equation with truncated nonlinearity
AuN -- UN + PN(PNuN) 3 = 0,
(UN, OtUN) t=0 =
where PN
=
(UO, u1) E
I/
2
(T
u:IRxT 3 -+IR(
(4.3)
3
)
(uN)tt -
P<N denotes the smooth projection operator defined in (1.12). We obtain the following
global-in-time non-squeezing result.
77
Theorem 4.2. Let D denote the flow of the cubic nonlinear Klein-Gordon equation (4.1).
R,T > 0, ko E Z3, z C C, and u,
(uo,ul)
E W1 / 2 (T 3 ).
Fix
Suppose there exists some K > 0 such that for all
c BR(u,), the corresponding solutions u to (4.1) and uN to (4.3) satisfy
tflL([o,T)XT3)
+ sup
IPNuNLi,([,T)xT3)
N
K.
Then
D(T)(BR(u*))
for r < R.
9 Cr(z; ko)
In particular, if BR(u*) C Bp, for some sufficiently small po(T) > 0, then non-squeezing holds
without any additional assumptions on the initial data.
See Remarks 1.3 and 1.4 for some discussion on this result.
4.1.1
Overview of Proof
Almost sure global well-posedness
To prove Theorem 4.1, we rely on an adaptation of the almost sure global well-posedness result from
[16]. This enables us to work on a set of full measure, E, with respect to a suitable randomization
of the initial data, on which the nonlinear Klein-Gordon equation is globally well-posed. We will
show that for a certain nested sequence of subsets EX\
c E, the flow of this equation can be seen
as a compact perturbation of a linear flow in the sense used by Kuksin (4.2). We will return to
this shortly.
We will now describe the randomization procedure for the initial data.
Let {(hk, lk)}keZ3
be a sequence of zero-mean, complex-valued Gaussian random variables on a probability space
(Q, A, P) with the condition h-k = hk for all k
{ ho, Re(hk),
C Z3 and similarly for the Ik.
We assume
Im(hk)}kex are independent, zero-mean, real-valued Gaussian random variables, where
I is such that we have a disjoint union Z3 = I U (-I) U {0}, and similarly for the lk. This set-up
ensures that the randomization of real-valued functions is real-valued.
Fix (fo, fi)
C V(
3
), and define a randomization map Q x R' -
,
kE-Z3
'kEZ3
78
NS
by
We could similarly take non-Gaussian random variables which satisfy suitable boundedness conditions on their distributions. For any (fo, fi) E ',
the map (4.4) induces a probability measure
on W', given by
p(fojf)(A)=
E Q: (fo', fw) E A).
EP(w
We denote by M' the set of such measures:
M8 :=-{(fofi)
Remark 4.1. The support of any p E M
some s, > s we have that (fo, fl) 0 I'R
: (fofi) E ft}.
is contained in V- for all s E R. Furthermore, if for
then the induced measure satisfies p(ff,)(*S) = 0. In
other words, this randomization procedure does not regularize at the level of Sobolev spaces in the
sense that almost surely, the randomization of a given function is no more regular than what you
started with. Moreover, if all the Fourier coefficients of (fo, fi) are nonzero, then the support of
the corresponding measure p(fo,f1) is all of V(, that is, [p(fo,f1) charges every open set in IV with
positive measure. As a consequence, for such a measure, sets of full p measure are dense. See [141
for details.
The arguments used to prove the almost sure global well-posedness of the defocusing cubic
nonlinear wave equation by Burq and Tzvetkov [16, Theorem 2], apply to the defocusing cubic
nonlinear Klein-Gordon equation, with the slight modification that one must consider the inhomogeneous energy functional
(w)
I=
Vw12+ w2+(wt)2
2
-1w.
2
(4.5)
We denote by S(t) the free evolution for (4.1), given by
S(t)(uo, ui) = cos(t(V))uo + sin(tKV))ui,
(V)
(4.6)
and we state [16, Theorem 2] adapted to our situation. Moreover, since the Hamiltonian for the
nonlinear Klein-Gordon controls the L2 norm of solutions, we no longer need the projection away
from constants which appears in [16].
Theorem 4.3. Let M
=
T3 with the flat metric and fix p C M', 0 < s < 1. Then there exists
a full i measure set E C W's(r 3 ) such that for every (uo, ul) C E, there exists a unique global
79
solution u of the nonlinear Klein-Gordon equation
utt- Au+u+u3 =0,
(uOtu)
t=o
u:RxV -T3IR
(4.7)
= (uoul)
satisfying
(u(t), ut(t)) c (S(t)(uo, u1), 8tS (t)(uo, ui)) + C(Rt; R1(T3)).
Furthermore, if we denote by
(D(t)(uo, ui) = (u(t), atu(t))
the flow thus defined, the set E is invariant under the flow 1(t):
Vt E R.
-1)(t) (E) = E
Finally, for any e > 0, there exist C, c, 0 > 0 such that for every (uo, u1) C E there exists M
=
M(uo, u1) > 0 such that the global solution of (4.7) given by
u(t) = S(t)(uo, ul) + w(t)
satisfies
|(w (t), aw(t ))|I
C(M +
(4.8)
4t.)
and furthermore, for M as in (4.8) and each A > 0,
p((uo,ui) C E : M > A) < Ce-ce
Let us introduce precisely the subset E of full measure we will work with. Let 0 < y < . to be
fixed later and define
{:(uoul)
02 :=((O,
C
Ui) E W1/2
IS(t)(1 _ A/2 / 2 (uo,u1)26(T3) E L' c(Rt)}
IIEt(U, U1)IIL-3)
(= L' c(Rt)}
(4.9)
n 2 and let E =0 + V. Set
The set
0 specified
:=E in
f [16, Theorem 2j imposes an L3OL 6
condition on the free evolution, however, it will be more convenient to work with the above definition and it is clear that the conditions in (4.9) are stronger. This choice will not change any of
80
the arguments from [16], and as we will see, E also has full measure with respect to any A
E M'.
In particular, by Remark 4.1, E is not comprised of initial data which are smoother at the level
of Sobolev spaces. We will work on the nested subsets EA C E, which we define in a following
section. These subsets have the property that their union is all of E, and we prove in Proposition
4.22 that there exists C, c, 0 > 0 so that for any A > 0, they satisfy
p(E) > 1 - Ce~cA.
Let 1 denote the nonlinear component of the flow map for the cubic nonlinear Klein-Gordon
equation, given by
1(t)(uo, ui) =
<(t)(uo, u1)
- S(t)(uo, ul).
(4.10)
On these subsets, we are able to prove a probabilistic version of the criteria needed for Kuksin's
argument from [38], namely we prove bounds of the form
I'D(t)(uo, U1) II4S2([OTJX) <
I(uo, ui)IIMsi,
(UO, ui) E E,
(4.11)
for some si < - < s2.
Remark 4.2. In [7/, Bourgainproves the analogue of (4.11) for subcriticalnonlinearKlein-Gordon
equations via estimates in local-in-time XS,b spaces, see (4.16). The reason Bourgain's estimates
,
fail at the critical regularity is because Strichartz estimates are not available at regularitiess1 <
which one would need in order to obtain the smoothing bound (4.11). Generally speaking, XS,b
spaces are typically ill-suited for critical problems, resulting in logarithmic divergences in the nonlinear estimates, and problems due to failure of the endpoint Sobolev embedding. The UP and VP
spaces are a more suitable substitute for critical problem, and we use these to prove a conditional
approximation result in Section 3.6. Nonetheless, we choose to prove the probabilistic convergence
argument in these spaces, since they are slightly simpler to work with and they are sufficient to
exploit the improved probabilistic bounds and to obtain (4.11)
Probabilistic approximation of the flow map
Once we have established the probabilistic bounds on the nonlinear component of the flow map,
our argument has several key components.
The first is an approximation estimate for the flow
81
map on the subsets E, C E. We consider (4.3) and we observe that this equation is defined on
the whole space and it decouples into a nonlinear evolution on low-frequencies and a linear flow on
high frequencies. This equation was used in [13] and we will show in Section 4.4 that this flow is
a symplectomorphism when restricted to the finite dimensional subspaces INW1/ 2 . Consequently,
we are able to show that this equation preserves infinite dimensional capacities.
Moreover, we
will show that this equation provides a good approximation to the full nonlinear Klein-Gordon
equation. More precisely, we will prove the following proposition.
Proposition 4.4. Let oD and 4DN denote the flows of the cubic nonlinear Klein-Gordon equation
with full (4.1) and truncated nonlinearities (4.3), respectively. Then for any T > 0 and for every
(uo, ul) (E Ex n BR,
sup j]4(t)(Uo, ul) - (DN (t)(uO, u1) li/2(T3)
C(A, T, R) E1(N)
te[O,T]
with e1(N)
-4
0 as N -+ oc.
We remark that this is a global in time approximation result in the critical space, with no restriction
on the size of the initial data, which can be viewed as a deterministic statement for initial data in
certain subsets of the phase space.
Remark 4.3. The dependence of the constants on R in Proposition4.4 is somewhat artificial, and
it can be removed by proving the statements from Section 4.5 with bounds in terms of the nonlinear
components of the solutions. Since both 4) and
4
N
have the same free evolution, these bounds
would suffice to prove Proposition 4.4. We do not undertake this here, however, since we cannot
remove this dependence in our other convergence results, and thus it would not improve our main
theorems.
An important ingredient in the proof of Theorem 4.1 is the following theorem which states
that locally in time, if one restricts to initial data supported on finitely many frequencies, this
approximation still holds uniformly
Theorem 4.5. Let 4) denote the flow of the cubic nonlinear Klein-Gordon equation (4.1) and 4DN
the flow of (4.3). Fix R > 0, u. E 711/
2
and N', N C N with N' sufficiently large, depending on u*.
Then there exists a constant Eo(u*, R) > 0 such that for all 0 < e < EO, there exists o = o-(R, E, N')
82
such that for any (uo, u1) E BR(u*),
sup I|I|(t)FN'(uO,ul) - (DN(t)lN'(uO, u1) 1/2
< C(R, u,) [E 1 (N) + e]
(4.12)
te[O'a]
with
IimN+o
E1(N) = 0.
Remark 4.4. The dependence on u, in this theorem arises in terms of the minimal A such that
the intersection EA
n BR(u.) is non-empty and in terms of the frequency required so that
||UI> NU, 1H/2
< R.
In particular, by using probabilistic techniques, we obtain a uniform statement at the critical regularity for all initial data in BR(u*). We suspect that the dependence of the constants on the various
parameters is non-optimal, but we do not undertake improvements to this theorem here.
Remark 4.5. The projection operators in (4.12) allow us to gain the necessary control via the
stability theory in order to obtain convergence on all of HN'BR. It is the combination of Proposition 4.4 with the stability theory which enables us to get a deterministic non-squeezing theorem,
regardless of the fact that Proposition 4.4 is a probabilisticresult.
Remark 4.6.
While we can prove global convergence for the flow maps for initial data in E,
these sets are not open and have a rather complicated structure. This poses serious problems for
the proof of the non-squeezing theorem for several reasons, among which is the fact that it is not at
E
all clear how to compute the capacities cap(Z
n BR), or even ensure that they are positive since
the subsets EA have empty interior. In order to upgrade the approximation to open subsets, one
must localize in time or obtain a conditional result. We have no prediction at the moment for an
estimate of this capacity using Kuksin's definition. Ultimately, it seems likely that an alternative
definition of an infinite dimensional symplectic capacity might be a more suitable approach.
Remark 4.7. As an easy consequence of the type of arguments used to prove Theorem 4.5, we
obtain well-posedness for long times on open subsets, as well as topological genericity for the set
of initial data for which this equation is globally well-posed.
Namely, we can write the set of
initial data for which we can prove global well-posedness as the intersection of dense, open sets.
Moreover, these arguments can be modified and combined with Gaussian measure considerations to
obtain some qualitative information about the initial data which fail to lie in these open subsets.
83
Non-squeezing via Gromov's Theorem
As in [4] and [20], we will prove non-squeezing for the full equation by using this approximation
theorem and Gromov's finite dimensional non-squeezing theorem, which we quote here for the
symplectic space (R 2 N, WO), where wo is the standard symplectic form on R2N, see [30] for details.
Theorem 4.6 (Gromov's non-squeezing theorem, [27]). Let R and r > 0, z E C, 0 < ko < N,
and x,,
c R 2 N. Let 0 be a symplectomorphism defined on BR(X,) C R 2N, then
g
b(BR(x*))
C,.(z; ko)
for any r < R.
We would like to take N
->
oo in Theorem 4.6. To do so, we use Theorem 4.5 to obtain the
necessary uniform local control.
The proof of Theorem 4.1 follows easily once we have proven Theorem 4.5. Indeed, fix parameters R > 0, ko C Z3, z = (zO, zo) C C, u* E R1/2 and 0 < q < R. Let Eo(R, u*) be as in the
statement of Theorem 4.5, and we fix N > IkoI sufficiently large and 0 < - < eo sufficiently small
so that for ei(N) and C
=
C(R, u*) as in (4.12) we have
CE (N) < -7
and
4
Ce <
4
Let o- > 0 be such that the conclusions of Theorem 4.5 hold with N'
=
2N and Eo as above. Note
that, in particular, this choice ensures that <bN is a true symplectomorphism, which does not hold
for N' < N, see Proposition 4.21. Then for any (uo, u1) c BR(U*),
sup ||<D(t)II2N(Uo, ui) -
1
N(t)H2N(Uo, u1)a1/2 <
2
te[O,o-]
(4.13)
Let r < R - r7 and define
2
2
The proof of Proposition 4.21 demonstrates that <DN is a symplectomorphism on H2N'1/ 2 with
the symplectic structure which is compatible with that of the full flow on W1/2. Then we can find
84
initial data (vo, vi) E II2NBR(U*) C BR, such that
1/2
....----
+....
(ko) I'DN (u) o, vi) (ko) - o2+(ko)
at 4N(0') (VO, vi) (ko) - Z1
(4.14)
> r + E2.-
By triangle inequality, (4.13) and (4.14) we obtain
1/2
.------.....
a8t D(-o)(vo, vi) (ko) -z
(ko) 14)(a-)(vo, v1) (ko) - o2+ (ko)
> r + E2 -
> r,
which concludes the proof.
Remark 4.8. As can be seen in this proof, the parameter71 provides a lower bound for the accuracy
needed in the approximation. This allows us to ensure that we can obtain some sort of uniform
lower bound on the time in order to prove non-squeezing.
Remark 4.9. If we wish to use Theorem 4.2 to conclude non-squeezing for the flow in the case
that a global flow is defined, we need to be a bit careful. There are several ways to go about this,
first, we can perform the same argument taking N sufficiently large depending on the profile of u.
so as to guarantee that for some choice of E > 0
IIvU
-
ll2Nu*IK1/2
< E,
then we can find initial data (uo, ul) E U2NBR-,e(u*), and hence
11u*
- (uo,
U1) IJW1/2
;
R.
It is also the case that fU2NBR(u*) c BR, (u*) for some R 1 > 0, so we can only conclude that the
flow does not squeeze some larger ball of initial data into the cylinder.
4.1.2
Conditional non-squeezing
In Section 3.6, we proved an approximation result, conditional on uniform Strichartz bounds for
solutions. The proof is based on an approximation result for the nonlinear Klein-Gordon equation
which quantifies the principle that perturbing initial data in high-frequencies does not affect the
low frequencies of the corresponding solutions by too much. Ultimately, by a similar argument,
we obtain the following approximation result.
85
Theorem 4.7. Let D and qIN denote the flows of the cubic nonlinear Klein-Gordon equation with
full (4.1) and truncated (4.3) nonlinearities, respectively. Fix T, R > 0 and suppose there exists
some K > 0 such that for all (uo, u1) E
the corresponding solutions u to (4.1) and uN to
BR(U,),
(4.3) satisfy
IfuflLt,([O,T)xT3) + suP JPNUN1HLt,([O,T)xT3) < K
N
Then for all E > 0 and any N' E N,
sup
IIPN' (I(t)(uO, U1 )
-
4N(t)(uO, U1)) I1/2 < E
tE [0,T]
for N
=
N(N', -, R, T, K) sufficiently large.
The ideas used in this proof are similar to those used in [20] however we mention again that in
the current setting, the problem is at the critical regularity. This is also reflected in the fact that
the constants in the above theorem depend on the Strichartz norm of the solutions and not just
the Sobolev norm, as in [20, Theorem 1.3]. The point of Theorem 4.7 is that if one only needs to
compare low frequencies, as is the case when proving non-squeezing, some decay can be regained
even though we are in the critical setting.
The proof of Theorem 4.2 follows from this approximation result using similar argument to
those in the proof of Theorem 4.1, once one uses Theorem 3.1 to restrict to a finite dimensional
subspace of initial data.
We refer the reader to the proof of Theorem 1.5 in [20] for a similar
argument. Alternatively, to conclude non-squeezing from Theorem 4.7, we can use the fact that
(DN
preserves symplectic capacities, as we do for the proof of Theorem 4.8 in Section 4.7.2
4.1.3
Probabilistic non-squeezing
Finally, we present one last version of the non-squeezing theorem which is a direct application
of Theorem 4.7. In contrast to Theorem 4.1, we do not need to consider the finite dimensional
projection of the ball and the result we obtain is for large times. In exchange for this, however, we
must restrict ourselves to initial data sufficiently close to elements of EA and the control we obtain
over the diameter of the cylinder is not as good.
Theorem 4.8. Fix p E M 1 / 2 and let 4 denote the flow of the cubic nonlinear Klein-Gordon
equation (4.1). Let R,T > 0, ko E Z3, z z C, and u, E H1/ 2 (7 3 ). Then there exists 0 > 0 such
86
that for every e > 0 there exists an open set Ue with
-
A(U) > 1
and such that
<b (T)(UenBR(U*))
V C,(z; ko),
for all r > 0 with irr2 < cap(Ue n BR(u*)).
Remark 4.10. The capacity cap(UefnBR(u*)) is positive since it is the capacity of an open set. In
practice, the sets U, will be constructed by taking the p-fattening of the subsets EA, see Section 4.7
for details. At the moment, we do not have a better bound for this capacity than the trivial bound
for open sets. In the case that the flow can be defined on the interval [0, T] for all initial data in
BR(u*). The proof of Theorem 4.8 is the only place where we need to use the infinite dimensional
symplectic capacity. As a result, the critical stability theory alone is insufficient because the trivial
lower bound for cap(Ue n BR(u*)) depends on E which is the parameter which yields the control for
long-time approximations in the stability theory.
4.1.4
Organization of Chapter
In Section 4.2, we provide some background on Symplectic Hilbert spaces and the relation of nonsqueezing to the energy transition problem and we introduce the capacity we will work with. In
Section 4.3, we collect some deterministic and probabilistic facts. In Section 4.4, we prove local
and global properties of solutions to the full equation (4.1) and similarly for the equation with
truncated nonlinearity (4.3). In Section 4.5, we prove the boundedness assumptions on the flow
maps of these equations. In Section 4.6, we prove Proposition 4.4. Finally in Section 4.7, we prove
Theorem 4.5 and Theorem 4.8. Throughout this chapter, we will use the notation PN =
4.2
P<N-
Symplectic Hilbert spaces
We begin with some background on symplectic Hilbert spaces. We follow the exposition in [38, 20].
Consider a Hilbert space W with scalar product (-, -) and a symplectic form wo on X. Let J be an
almost complex structure on W which is compatible with the Hilbert space inner product, that is,
a bounded self-adjoint operator with J 2
=
-1
such that for all u, v E W, wo(U, v) = (u, Jv).
87
Definition 4.9. We say the pair (W, wo) is the symplectic phase space for a PDE with Hamiltonian
H[u(t)] if the PDE can be written as
= JVH[u(t)].
=(t)
Here, V is the usual gradient with respect to the Hilbert space inner product, defined by
d
.
(v, VH[u]) = -H[u + cv]
wo(v, i(t)) = wo(v, JVH[u(t)])
-
((Ui, U2), (VI, V2)).! :=
2
For (u,
Ut) =
H[u+ Ev]
A) 1/ 2 and consider the Hilbert space W1/ 2 (T 3 ) with the usual scalar product
U1
- (V) v1
+
/
Let (V) := (1
-(v, VH[u]) =
.
Definition 4.9 is equivalent to the condition
T3
U2
(07)
V2
(ui, u 2 ) we can rewrite (4.1) as the system of first order equations
(Ui)t
= U2
(4.15)
-
(U2)t = -(1
A)u
-
(ui)3.
Define the skew symmetric linear operator
J : I/
2
(
3
3
) _ -U/2
then J is an almost complex structure on
wi (u,v)
W 1 / 2 (7 3 )
/:
(v1, v2),
0
(V)-1
- (V)
0
)I
compatible with the symplectic form
U - V2
U2 - V,
-
T3
2
that is, setting u := (ui, u 2) and v =
i
T3
we have w (u, v) = (u, Jv).. Then we can write
2
ii = JVH(u) for the Hamiltonian
Ju212
H(u) = 2
J
Vu1i 2
uil2
+
I
+ 1JuiK.
In particular, up to modifying the Hamiltonian, these computations holds for all nonlinearities and
in all dimensions.
88
4.2.1
An infinite dimensional symplectic capacity
Kuksin's construction of a symplectic capacity for an open set 0 is based on finite dimensional
approximations of this set. It is an infinite dimensional analogue of the Hofer-Zehnder capacity
[30].
Before defining this capacity, we first recall the definition of a symplectic capacity on a
symplectic phase space (7H, w).
Definition 4.10. A symplectic capacity on (1, w) is a function cap defined on open subsets 0 C W
which takes values in [0, oo] and has the following properties:
C7.
1) Translationalinvariant: cap(0) = cap(0 +
2) Monotonicity: cap(01) ;> cap(0 2) if 01
) for
W
D 02.
3) 2-homogeneity: cap(TO) = r2 cap(O).
4) Non-triviality: 0 < cap(0) < oo if 0 ) 0 is bounded.
In finite dimensions, the symplectic capacity is an important symplectic invariant, namely for
a given symplectomorphism, p, and an open subset, 0 C 7-, we have that cap(o(O)) = cap(O).
In [38], Kuksin shows that the infinite dimensional symplectic capacity he constructs is invariant
under the flow of certain nonlinear dispersive Hamiltonian equations. We will now define Kuksin's
infinite dimensional capacity. For a given Darboux 1 basis of 7-, let
-N
denote the span of the first
N basis vectors. Similarly, we use the notation ON for any subset 0 projected onto these basis
vectors. We collect a few definitions.
Definition 4.11 (Admissible function). Consider a smooth function
f
E C'(0) and let m > 0.
The function f is called m-admissible if
i) 0 < f < m everywhere.
ii) f = 0 in a nonempty subdomain of 0.
iii) fao =_m and the set
{f < m} is bounded and the distance from this set to 90 is d(f) > 0.
Definition 4.12 (Fast function). Let fN := f (N
vector field UfN, that is, for z,v G
and consider the corresponding Hamiltonian
(N, we have
W(UfN (z), v) = VJN(z)v.
la Darboux basis of 'R is a basis (ui,...,vi,...) such that w(ui, v) = 6ii.
89
A periodic trajectory of UfN is called fast if it is not a stationary point and its period T satisfies
T < 1. An admissible function f is called fast if there exists No(f) such that for all N > No, the
vector field UfN has a fast trajectory.
Remark 4.11. In light of the fact that J2 = -I, we also have the representation
UfN(Z)
=
JVfN(z).
With these definitions, we are now ready to state the definition of Kuksin's infinite dimensional
capacity.
Definition 4.13. For an open, nonempty domain 0 C '7, its capacity cap(0) equals
cap(0) = inf{m. Ieach m-admissible function with m > m. is fast}.
In [38], it is shown that this definition satisfies the axioms of a capacity, that is the criteria
of Definition 4.10, and while the construction of this capacity depends on the choice of Darboux
basis, if one chooses another basis which is quadratically close 2 to the first, then the capacity does
not change.
4.3
4.3.1
Preliminaries
Deterministic preliminaries
We recall the definition of the Xs,b spaces with norm
I|u|XS,,b(RXT3) =
11(n)'(1-T
-
(n))b t(n, ) 1L 2 f 2 .
(4.16)
1LUIJXs,b,6 = inf{1PiIxS,b(RXT3) : i[-J,6] = U
.
We will also work with the local-in-time restriction spaces X,,b, 6 , which are defined by the norm
We refer the reader to Appendix A for more details.
We will record some facts about the projection operator (1.12). In [13], these operators were
used to define an approximating equation for the cubic nonlinear wave equation and we similarly use
2
that is, some other Darboux basis
f } such that >j,
90
11
-
V)112 <0
them to define the equation with truncated nonlinearity (4.3). We use this smoothed projection
instead of the standard truncation because we will need to exploit the fact that this family of
operators has uniform LP bounds.
Lemma 4.14. Let M be a compact Riemannian manifold and let A be the Laplace Beltrami
operator on M. Let 1 < p < oo. Then PN =
(-N-2 A) : LP(K1 ) -+ LP(Ki) is continuous and
there exists C > 0 such that for every N > 1,
IIPNIILP-+LP
Moreover, for all f E LP(KI), PNAf ->
f in LP as n
-+
C.
o.
l
Proof. See [63, Theorem 2.1].
Finally, we will need the following identity to prove the symplectic properties for the truncated
nonlinear Klein-Gordon equation.
Lemma 4.15. Let K be large enough so that HKPN = PN, then
pN [(PNu) 3]
4.3.2
IJKV
(PNU) 3 pNV-
Probabilistic preliminaries
We will record some of the basic probabilistic results about the randomization procedure. Most of
these estimates are consequences of the classical estimates of Paley-Zygmund for random Fourier
series on the torus. These estimates were used heavily in the works of Burq and Tzvetkov, see
especially [13] for proofs.
The large deviation estimate from Proposition 2.4 is the key component in the proof of the
following corollary, which states that the free evolution of randomized initial data satisfies almost
surely better integrability properties. There is the minor modification in the following that we
are dealing with complex random variables {hk} which satisfy the symmetry condition hk = h-k.
Given that the functions we are randomizing are real-valued, and thus have Fourier coefficients
which satisfy an analogous symmetry condition, the arguments for the following results go through
unchanged. Recall in the sequel that S(t) denotes the free evolution for the nonlinear Klein-Gordon
equation, defined in (4.6).
91
Corollary 4.16 (Corollary A.5, [16]). Fix p E MS and suppose /p is induced via the map (4.4)
for (fo,f1) C 7-.
Let 2
pi < oo, 2 < p
2
< oo and 6 > 1 +- and 0 < o< s Then there exist
constants C,c > 0 such that for every A > 0,
p({(o,
i)
WS: |(t)S(t)(UO, IUOI I L
Remark 4.12. We can include the endpoint P2 =
2(RxT3)
00
> Al) < CC
2
if we restrict to bounding 0 < o- < s in the
statement above, or equivalently to bounding.
(t)3 (1
-
A),/ 2 S(t)(uo, ul) 11LT'LP2
for any 0 < - < s. By Sobolev embedding, we are also be able to include the endpoint p1
= 00
in
this case. We use this in Section 4.9, see [13, Lemma 2.21.
Given these large deviation estimates, the following result, which enables us to construct the
subset E is a simple corollary.
Corollary 4.17. Let T > 0 fixed, /
for any 0 < y <
C M and 2 < p < oo. Then there exists C = C(T) such that
1 < 1/2,
p({(VO,IV) E
7'
SM)(UO IU
: 11
1 L1 W-',P2 (RxIT3) > Al ) < Ce-c2
2Hn
Additionally, Corollaries 4.16 and 4.17 imply that the set of initial data which satisfies good
local LP bounds has full p measure.
Corollary 4.18. Fix /p C MS and let 2 < p < 00 and 0 <
H1(1
- A),/ 2 S(t)(uo, ul)IlP(, 3) E L' c(Rt),
< s. Then for a set of full p measure,
||S(t)(uo, ul) HL c(T3)
C
In particular, for E as defined in (4.9), we have p( E) = 1.
4.4
Well-posedness theory
We record some global bounds on the solution to the cubic nonlinear Klein-Gordon (4.1). The only
new component in this statement is the bounds on the L4 norm of the solution, which follows by
92
a similar argument to the proof of [16, Proposition 4.1]. We include the proofs of these statements
in Appendix 4.9.
Proposition 4.19. Let 0 < s < 1 and let p E Ms.
Then for any e > 0, there exist C,c,0 > 0
such that for every (uo, u1) E E, there exists M = M(uo, u1 ) > 0 such that the global solution u to
the cubic nonlinear Klein-Gordon equation (4.1) satisfies
U(t) = S(t)(uo, u1) + w(t)
)
II(w(t), Otw(t))IIN1 < C(M + ItI)1 +,
Iu(t)IL4(T3) C(M + It
and furthermore p((uo, ui) E E : M > A) < Ce-cA6
We now turn to studying the global well-posedness and symplectic properties of the approxLet PN be the smooth projection operator defined in (1.12).
imating equation (4.3).
{
PN, this equation is equivalent to the uncoupled system
sufficiently large so that HKPN
+ (1
ttllKuN
-
A)HKUN + PN[(PNUN)
tKUN)I t=0
(HKuN,
H;>K(uN)
=
For K
=
3
= 0,
(t,x) E R
x T3
(4.17)
(HKuO,IKul)
S(t)(;>KUO, H;>Kul)
which we can write as a first order system as in (4.15).
Remark 4.13. As remarked in the introduction, (4.17) is a nonlinearflow for low frequencies and
a decoupled linear evolution for high frequencies. In particular, the solution uN is supported on all
frequencies. We will nonetheless call this the truncatedflow for simplicity even though this defines
a flow on the whole space.
Global well-posedness for (4.17) follows from local well-posedness by observing that the linear
evolution is globally defined and the energy functional
PN(uN))4 (4.18)
HN(HK(UN, (UN)t))
=
jVxKUN 12 + (IKuN ) 2
2 73
(HK(UN )t)
2
+
(
_
4
"3
is well defined and conserved under the flow of (4.17) for bounded frequency components. Note
that the bounds on the solution depend on the energy of the initial data and consequently are not
93
uniform in the truncation parameter. Nonetheless, Burq-Tzvetkov [13] proved that if one restricts
to initial data (uo, ul) G E, then the nonlinear components of the solutions to the cubic nonlinear
wave equation satisfy uniform bounds. As was the case in Theorem 4.3, the proof of these uniform
bounds follow for the nonlinear Klein-Gordon equation from the arguments in [13] with only minor
modifications.
Proposition 4.20 (Proposition 3.1, [13]). Let 0 < s < 1 and let p
C M'. Then for any E > 0,
there exist C, c, 0 > 0 such that for every (uO, u1) E E, there exists M = M(uo, u1) > 0 such that
the family of global solutions (UN)NEN to (4.17) satisfies
UN(t)
=
St)(uO, u1) + WN(t)
+
||(WNO), 0tWN t)jR1 <_ C(M + t&
IIPNuN(t)K4(T3)
C(M +\t|
and furthermore p((uo, ul) E E : M > A) < Ce-A.
The truncated flow maps also preserve symplectic capacities. This is an easy consequence of
the fact that, when restricted to bounded frequencies, these maps are finite dimensional symplectomorphisms.
Proposition 4.21. The flow maps
o c
1/2
(N(t)
preserve symplectic capacities cap(O) for any domain
3
Proof. Let K be large enough so that I7 KPN = PN and consider the Hamiltonian (4.18). For
(VI, V 2) E HK
1/ 2(T 3 ) and (U1, U2) which solve (4.17), we have
d- HN
JIK (U 1, U2) + E(V 1, V2))
dE
+
=de 2 J3
K
fTfK U1)V1
+ EV)
2
VxHII)U1
+
I =0
+x(fKul
EV1)j 2 + (1
)
+
&Vx (HK U1) VxV1 + (IK U2)V2
-~(N
94
Ku2
+
Ev2)
U,)3pN (V1)
2
+)4
(pN(Ul
T3NJvl)
+
EVi))4]
thus by Lemma 4.15 we obtain
j
T3
=j
=
PN(PNU1) 3 HKV1
(HKU1)Vl + Vx(lKU1)VxV1 + (UKU2)V2
-
&t(I7K
2)
v1
9
-wg ((VI, V2), (tHIKU1,
t (FKul)
v2
OtHKU2))-
Thus the maps <DN are symplectomorphisms on 1IK-1/ 2 , denote this restricted map by
N. Since
the flow decouples low and high frequencies, we can write
<DN(u) =
(N(IIKu)
+ etJAJ>Ku= etJA (-tJA
N(JKU) + 11KU),
where A is the linear propagator for the Klein-Gordon equation on V1
2
. The invariance of the
symplectic capacity under the flow follows from [38, Lemma 51, since e-tJA N is also a symplectomorphism and that etJA is an isometry, hence it preserves admissible and fast functions.
4.4.1
El
Definition and properties of E\
We recall the definition of E. We let 0 < y < 1 to be fixed later and define
{(uo, ui) E j1/2 :S(t)(1 - A)y/2 (uo,u)II 6 (' 3 ) E LU)(1Rt)}
E1
E)2
Set
E
:=
1
:=
((nO U ) (E 1/2
:
n E 2 and let E = E +7H1.
|S(t)(UO,U1)||L-(T3) E L'o( t}
For (uo, ui) E E we denote by u(t) = S(t)(uo, ui) + w(t)
the global solution to (4.1). Fix e > 0 and let C > 0 be as in the statements of Proposition 4.19
and Proposition 4.20. Define the subsets
E\ :{(uo,ui) E E
I(w(t),atw(t))Ijij1 < C(A+ It)
HA
{(uo, ui) E E : I(wN (t),&twNM)I
JA
{(Uo,u) C
KA
E
{(uo, ui) E E
IUIIL4
< C(A
C(A
ItI)le},
IUNIIL < C(A+ ItI)'+',
95
and let MA be the set of (uo, u1) E E such that for -y as above we can find C
=
C(T) as in Corollary
4.17 such that
(1
-
A),/2 S(t)(uo, ui)L6([0,T]xT 3 ) < CA.
Setting
EA := EAn HAnJAnKA n MA
(4.19)
we have the following bounds for the measure of this set. The choice of -y > 0 does not affect the
following proposition.
Proposition 4.22. Fix [t E M 1 / 2 and let EA be as defined in (4.19). Then there exists C, c, 0 > 0
such that for all A > 0 we have
/I(E) > 1 - Ce-CA.
(4.20)
Proof. Suppose that p is induced by the randomization of (fo, fi) E W 1/2. By Proposition 4.19
and Proposition 4.20, there exists C, c > 0 such that
Pt(EA n Ha, n J,\ n K,) > 1 - Ce-~A,
and by Corollaries 4.16 and 4.17,
p(MA) >
1-
2
Ce-cA /I(fo,'f1)jis.
Taking intersections and using that these bounds are exponential yields (4.20).
El
Remark 4.14. In the above proof we are implicitly taking advantage of the fact that we have
exponential bounds on the measure of the sets in question.
4.5
Probabilistic bounds for the nonlinear component of the flow
In this section, we will show boundedness for the nonlinear component of the cubic nonlinear
Klein-Gordon equation on the subsets EA C E. In the sequel we let F(u) = u 3 . We will begin with
the proof of a local boundedness property for the nonlinearity F(u). The argument is based on
Strichartz estimates together with the improved averaging effects for the free evolution of initial
96
data (uo, u1)
c EA, where EA was defined in (4.19), as well as the uniform bounds on the nonlinear
component of those global solutions from Proposition 4.19 and Proposition 4.20.
for solutions to the cubic nonlinear Klein-
We wish to obtain bounds on ||F[u(t)]jj
Gordon equation (4.1) with initial data (uO, ui) E EA. We will perform the estimates with b
!+
and hence, we need to estimate the expression
[zJdin
for F(u)
=
(4.21)
wr12
IF(n
- ))2(1-b)
(n)2(1-s2)(j(
u3 and b > 1/2. If we expand F(u) for u(t)
=
S(t)(uo, ui)
+ w(t), then we need to
consider terms of the form U(I)U(2)U(3) for u(') either
(I) the free evolution S(t)(uo, ui) of initial data (uo, ui) E EA C -1 / 2 , or
(II) the nonlinear component, w(t), of a solution to (4.1) with initial data (uo, u1) E E.
We will refer to these as type (I) or type (II) functions. We define
ci (ni, ri) = (ni)" (|ri| - (nj))6|-)(ni, ri)|
then I1c11LSef(RxT3)
-
Iju(i)IXs1,b(RxT3). By duality, (4.21) can be estimated by
f
'r
+7 +7
T
where
Iv IIL2j2(RxT3)
(ni,
Ji
-r)
v(n, -r)
I (njj) 81(1i~ -r (ni))bI (n)1-S2 (ITI - (n))1T
d-r
(4.22)
< 1. We remark that this notation should not be confused with the initial
data for (4.1), and it will be clear from the context which we are considering.
We restrict the ni and n to dyadic regions Ingi ~ Ni and In
-
N. We will implicitly insert a
time cut-off with each function but we will omit the notation, since, in the usual way we can take
extensions of the ui and then take infimums. The ordering of the size of the frequencies will not
play a role in this argument. We do not repeat these considerations. Letting
UN, (ni, -ri) -
ci (ni, -rN)
VN (n, 7-
X{|nf|~N&}
97
v (n,
1-b
X{n|~N}
(r)
(4.23
we will need to estimate expressions of the form
3
(N1N 2N3 )~s
1N
Jt
1
-( -s2)
JUN, VN dx
dt.
(4.24)
We will use expressions (4.22) and (4.24) as starting points in proofs of the subsequent propositions.
In the sequel, we will always take the constant y
4.5.1
= si
in our definition of E.
Boundedness of the flow map
Proposition 4.23 (Local boundedness).
Consider the cubic nonlinear Klein-Gordon equation
(4.1). Then there exists s1 < " < s2 with s1, s2 sufficiently close to 1/2 such that for any A, R, T >
0, for every (uo, u1 ) E Ex
n BR and for any interval I C [0, T] with I| = 6, the nonlinearity
satisfies the bound
F(u)Ix
< C(A, R, T) Jc (1 +
-1
IiUlX8II+I,6Ix3)).
(4.25)
where (u, atu) is the global solution to the cubic nonlinear Klein-Gordon equation (4.1) with initial
data (uo, ul).
Proof. For any solution u(t) = S(t)(uo, ui) + w(t), our computations will yield (4.25) with the
nonlinear component of the solutions, w, on the right-hand side instead of u. Now, for any s, b E R
and any interval I C [0, T] with IIl = 6 and inf I = to, and 77(t) a Schwartz time-cutoff adapted
to that interval we claim that we it sufficed to obtain (4.25) with the nonlinear components of the
solutions on the right-hand side. Indeed,
I|7(t)wI|x.,b(IxT3) = 117(t)(u - S(t)(uo, ui)) IIXb(IxT3)
;<
<
I|n(t)S(t)(uo,ui)Iixsb(IxT3) + flr(t)uIIXs,b( 1 xT3)
1(u(to),
tu(to)) j|s + II(t)uI|x,b(IXT3).
Since the free evolution is bounded in 'Ws, we have
I(u(to),Atu(to))IRs ;< Iu(o, ui)I|H.S + j(w,tw)IL|oo1s([oTjXT3),
and since the terms on the right-hand side of (4.25) are computed with s = si < 1/2, then by the
98
choice of E,, we have that for any such I c [0, T],
$
r?(t)wjIxsb(IxT3)
lHr(t)uxIsb(IXT3)
+ C(A, R, T)
which will yield (4.25). The key point here is that on any interval [0, T], the choice of E,\ yields
uniform control on bounds of the Sobolev norm of solutions. We will not repeat these considerations.
We analyze the different combinations of ui systematically. The argument only depends on the
number of ui which are of type (I) or type (II), so we will only present one combination from each
case.
* Case (A): All u(') of type (II). Since
IIUUTaIIL4+e2 , |UN
for p =
v'||UNJ
-a)5
we use H6lder's inequality to estimate (4.22) by
(4+6-2)5
1-E2
(N 1 N2 N3 )-s1N-(l-s2)
L
3
fA
1UNj - N dxdt
3
pIUNi
< (N 1N2 N 3)-s8N-(1-s2)JIUN
IIVN I1L4-3e.
1c)5
i=1
Taking a = 1/4, we have (1 - a)5 < 4 and for
1- b >
62
sufficiently small, pa < 6. Provided
4-:6E
2(4 - 3-1)
(4.26)
-2
2'
we can bound
1
1
(1-Ti
-
(n))1b
(l
1
1
(n))1b -L+
(ITIl
-
(n))4#+ r'(I- - (ni))
and we can apply Strichartz estimates (A.2) with r = L and 01 = 14/15 for the
estimates with r = 4 - 3E1 and
02 = 46,,
4-3el
2'+
UNj
and Strichartz
for VN. By Sobolev embedding, accounting for the Nis
factors in the expressions for the UN,, we obtain
3
\2 3
2(i1.4-6c
J2IIU(i) 11 H
< (NiN 2 N 3 ) -SI N-(1
i=1
99
7
i=1
|c|(T3)
L 4
The expression on the right-hand side is summable for dyadic values of Ni and N. By the definition
of E\ (4.19), we obtain
11u(i)H11,,
(4.22) < (A + T)Z+
.
(4.27)
i=1
* Case (B): u( 1) of type (I) and U(2), U(3 ) of type (II). In this case, we use H6lder's inequality
and Strichartz estimates (A.2) with r =
18
and 0
, and provided
=
4
,
1 -b>
which holds if (4.26) holds, we can bound
1
(4.22) < (N 1 N2 N)-8N-(
< (N1N 2 N3 )-
1N
UN, UN 2 UN3
-s2)
(-s2)
|UN1 HL6||UN2
VN
dxdt
LT||UN
r (N1)-1(N2N 3 )V81 N~(-s2) |UN 1 11L6 H
2
L
HN
X, 1,b
'L
IX'lb.
Noting how we defined UN 1 , we use the fact that si < 1 and that (uo, ul) C EA, to estimate that
piece, exploiting the uniform boundedness of the frequency projections in LS spaces. Once again,
we obtain an estimate which summable for dyadic N and Ni, yielding
(4.22)
(2
< A HuI
)
Xslb, 6
iu(3)Hx. 1,b,6.
(4.28)
* Case (C): U(I), U( 2 ) of type (I) and U( 3) of type (II). In this case, we use H6lder's inequality
=2
1 - b >
and
1
3
,
and Strichartz estimates (A.2) with r = 3 and 0
which holds if (4.26) holds and estimate
(4.22) < (N 1 N 2 N)-81N-(1-s2)
J UN
1
< (N1N2N)-s1N-(1-82)||UN1
< (NiN2 )- 8 '(N 3 )3-
UN 2 UN,
VN dxdt
IL3NIJL3
L6UN IhL6|UNa
2
N-(3s2-)
100
UN 1 HL6
1UN 2 11L6
H
3 IX.1,b
which is again summable for dyadic N and Ni, yielding
(4.22) < A 2 1U(')Ixs1 ,b.
(4.29)
9 Case (D): All u(') of type (I) . In this case we estimate
(4.22) < (N1N 2 N 3 >)-1N-(l-s2) JUN1 UN 2 UN,
r< ( N1 N2 N3) - 1N -(1
,< (AiJ1
2
A
3 -*'
-52) )|
N(l-s2)
UN
11UL6 11|UN2 11L6
VN dxdt
|1UN3
11L6 11|VN 11L2
hUN1 IIL6 |UN 2 HL6 IIUN3IIL6
which is again summable for dyadic N and Ni, yielding
(4.22) < A 3.
Finally, we note that for
-
(4.30)
< -(1 - b), we have
flF(u) I1 X2-1,-(1-b),.5 6 6|jF(u)IXs2-1,-(1-b)+e,6
(4.31)
where the implicit constant depends only on e. To obtain the small time factor in the estimates,
we can perform the previous estimates replacing b with b+e on the VN factor for some small E > 0.
Provided b = !+ is chosen sufficiently close to 1 and e > 0 is sufficiently small so as to ensure
that (4.26) continues to hold for b + e, we obtain the desired estimate. We will not repeat this
consideration in the following Propositions.
Combining (4.27), (4.28), (4.29), (4.30) and (4.31) yields
F(u)|
C(A, R)6o (i
-
Remark 4.15. If we consider (uo, ul) E
,
u|
)
4
i
n BR and instead we look at solutions to the cubic
nonlinear Klein-Gordon equation with truncated nonlinearity, uN, then we obtain the same bounds
hIF~u~hI~ 2~lj+,6
||F(UN)UIXS2-1,
+,S N
C(A, R, T) Pc
s+,.')
(i +
fluNh119l4,1
7
and similarly for PNuN. Indeed, by the choice of E., the nonlinear component of the solution WN
101
will satisfy the same bounds as the nonlinear component, w, of the solution to the full equation.
Since the linear components of u and
UN
are the same, certainly we have the same bounds on
S(t)(uo, ul) and we can repeat the arguments in the previous proof to obtain (4.15).
4.5.2
Boundedness of the flow with truncated nonlinearity
We defined the truncated nonlinearity FN(uN)
= PN [(PNUN)
3
]. A direct consequence of Proposi-
tion 4.20, Proposition 4.23 and our choice of E\ is the following local boundedness result for the
truncated nonlinear Klein-Gordon equation. It is important to note that the bounds we obtain are
uniform in the truncation parameter.
Proposition 4.24 (Local boundedness for the truncated equation). Consider the cubic nonlinear
Klein-Gordon equation with truncated nonlinearity (4.3). Then there exists s1 <
sufficiently close to 1/2 such that for any A, R,T > 0, for every (uo, ul)
1 <
s2 with S1, S2
CZ E n BR and for any
(X2-1,-+,<(x
A R, T)6c 1
IIFN(UN)
where
4.5.3
(UN,
UN 11,
+
interval I C [0, T] with III = 6, the truncated nonlinearity satisfies the bound
tuN) is the global solution to (4.3) with initial data (uo, u1).
Continuity estimates for the flow map
We need the following continuity-type estimate when we compare the full nonlinearity on solutions
of the full flow to solutions of the truncated equation.
Proposition 4.25. Consider the cubic nonlinear Klein-Gordon equation (4.1) and the cubic nonlinear Klein-Gordon equation with truncated nonlinearity (4.3). Then there exists s1 < 1 < s2 with
S1, s2 sufficiently close to 1/2 such that for any A, R, T > 0, for every (uo, ul) E E\ n BR and for
any interval I c [0, T] with II| = 6, the nonlinearity satisfies the bound
< c(A, R, T)
3C
|u - uNII
6
(IxT3)
6
,(IxT3)
(1
I
,+, 6(IxT3 ) +
XUN
if4+ 6 (Ix T3)
'
|IF(u) - F(uN)XI X'~+
where (u, atu) and (UN, 0tuN) to the full and truncated equations, respectively, with data (uo, ul).
102
Proof. As in the proof of the Proposition 4.23, in light of (4.31) we will take b = 1+ for b sufficiently
close to 1 so that the time localization yields the desired
6C
2
(U)3
U
_ (uN)3J
-
factor. We first note that
UN 2
uNI (Iu2
hence these estimates are similar to those in Proposition 4.23 but we will always estimate ui =
IU - UNI
in X2'2',. More precisely, once again we estimate the expression
cl(ni, ri)
dfr
=2 N
(ni))b i21+T2+T3
(ni)fli(|l
n=nl+n2+n3
ci(ni, -r)
(Ii7 - ())
where IIvIIL2t2(RxT3) < 1 and as before the functions u or
UN
v(n, r)
(n))l-b
+ (i)
(4.32)
in the expression for c are either of
type (I) or type (II), with ui always of type (II).
We define UNj and VN as in (4.23). The key difference between this proof and the proof of
Proposition 4.23 is that in each case, we will estimate ui in XII+6 instead of using Xi,"+,6
which we use for the other functions.
* Case (A): All u(
of type (I). Once again, we recall that since
UU
for p =
(4+E2)5
| 1L4+I2 IUNI IPaI1UNi 111a) 5
we use H6lder's inequality to estimate (4.22) by
16
3
_1
UNi VN dx dt
N1 2(N2 N3 )-1Nj
3
SN1
(N1N 2 N3 )-s1N--IHUN1 IL4-ei l
H||UN IL1r |UNL(1- a)5 11VN
L4-d1
i=2
Taking a = 1/4, we have (1 - a)5 < 4 and for E2 sufficiently small, pa < 6. By Sobolev embedding
and Strichartz estimates (A.2) with r =
1
and 01 = 14/15 for the UN 2 , UN3 and by Strichartz
estimates with r = 4 - e1 and 62 = 4-2,1 for UN1 and VN we obtain
3
3
$
(NIN)(
1
4-!
ii()
)(N 2 N 3 )2_-2s1|jUN 1 IL4-Jiu
i=2
JJ(
L2
i=2
The expression on the right-hand side of the inequality is summable for dyadic values of Ni and
103
N provided el > 0. By the definition of EX (4.19), we obtain
3
<
(A+ T)12+ Iu("1)H
n ,
11 Uiu()I11
3
X7 i=2
E
The other cases follow analogously.
Finally, the last continuity type estimate we will need demonstrates that if one of the functions
in the multilinear estimates satisfies 1NU ) = 0, that is if it is only supported on high frequencies,
then one gains some additional decay in N.
Proposition 4.26. Consider the cubic nonlinear Klein-Gordon equation (4.1). Then there exists
S2 > 1 with S2 sufficiently close to 1/2 and 0 > 0 such that for any A, R, T > 0, for every
(uo, u1) E E\ n BR and for any interval I C [0, T] with jI = 6, the nonlinearity satisfies the bound
|IF(u)
- F(PNu) Xs 2 ,-
C(A R, T) c N- ( +
(IxT3)
l
where (u, otu) is the global solution to the cubic nonlinear Klein-Gordon equation (4.1) with initial
data (uo, ul).
Proof. Once again, we use the inequality.
I'- PN)u u2 +pNU 2
|F(u) - F(PNu)I
-
Let sl be as in Proposition 4.23, and we repeat those arguments but we will always estimate the
high-frequency term, (I - PN)u, in XSi,2+,, even for the linear evolution. For instance, consider
the case where all the u(') are of type (I). We once again estimate the expression
z
n=nl+n2+n3
r=rl+r2+r3
ci (ni, r)
d(n, r)
(riI - (ni)) + (
(ni)
i (d
dr
-
(())4
-)
and with the definitions for UNj and VN as in (4.23), we obtain
(4.33) < (NIN2 N 3 )~s1N-
JUNi UN 2 UN3 - VN dxdt
< (N1N2N3) 1N-I|N1UN1
N,
51+3
T1+1(N2N3)-s1NIv2
klSIv3
L3 1UN2
1L6|UN 3
fu)Hs+,6 11 UN2 1L6 11 UN
3 IL6TuU
104
L6|VN
1L3
(4.33)
which is summable for dyadic N and Ni. By Strichartz estimates, recalling that we set
U
=
(1 - PN)S(t)(O, u),
we obtain
HU 1 1X81,1+,
=|(1-PN)S(t)(uou1)fj+,
6
< N--alS(t)(uo, u)|j
1s < N-0 11
(uo, u) 11-H/2(T3),
which yields the desired estimate. The other cases follow analogously to the previous propositions,
with the modification that when we take u()
(1 - PN)u we obtain
-
which yields the result.
4.6
Probabilistic approximation of the flow of the NLKG
This section is devoted to the proof of the approximation of the flow map for the cubic nonlinear Klein-Gordon equation by the flow of the nonlinear Klein-Gordon equation with truncated
nonlinearity. We will use here the probabilistic boundedness estimates from Section 4.5.
Proposition 4.27. Let iD denote the flow of the cubic nonlinear Klein-Gordon equation (4.1), and
(N
the flow of the cubic nonlinear Klein-Gordon equation with truncated nonlinearity (4.3). Fix
R, T, A > 0. Then for every (uo, ui) E E\ n BR,
sup
te[O,T
with e 1 (N) -+ 0 as N
Ik@(t)(uo, ui)
--
--
qN(t)(uO, u1)I,/2
3
)
C(A, T, R) E1 (N)
oc.
Proof. Fix R, T, A > 0 and let E\ be as defined in (4.19).
We need to estimate the difference
( - 4N for initial data (uo, ul) E E,\ n BR. Fix such a (uo, ul) and let u(t) and uN(t) denote the
105
corresponding solutions to the full and truncated equations, respectively. By the choice of E,
I|(1
6 3
- A)S1/ 2 S(t)(uo, ui)HL6([o,T]; L (T ))
II(w,
(wN,
tw)IILoo([o,T];W 1(T3))
tWN) I LOO([O,T];7=1(T3))
< CTA
< C (A + T)1 +,
(4.34)
< C (A + T) 1+I
where, as usual, w(t) and WN(t) are the nonlinear components of the global solutions u(t) and
UN(t), respectively. Note in particular, for any subinterval I C [0, T], these bounds hold uniformly.
Furthermore, for I = 6, Proposition 4.23, and the inhomogeneous estimate for the Klein-Gordon
equation yield
|Wt|
8
,1x
,< ||F(u)H
1+,6
C(A, R, T) C (1 +),
hence if inf I = to, by Lemma A.4 we obtain
,< 1(u(to),
,6 + |WII
,
lUllX~j'16 5 |S(t)(uo, u1)llxaj
Otu(to)) 1.,si + HF(u)
_1.
By the uniform bounds (4.34) and the boundedness of the free evolution on ',
sup 11(u(to), tu(to))lIRsi < C(A, R, T),
toc[O,T]
hence
+
|UIIXSi,7+,6 5 C(A, R, T) + C(A, R, T) 6 (1
Thus, taking 6 = 6(A, R, T) > 0 sufficiently small we obtain that
I(t)(Uo, Ui)|xIi~81 73 1 C(A, T, R).
By Proposition 4.24, this argument yields the same result for
(4.35)
1
N(t) with the same choice of 6 > 0.
Note, too, that we can obtain the same bounds with s, replaced by 1. Now, define
(<0, ft) := (u - UN, atU - atUN)
106
then we have
=
(F(u) - F(UN) + F(UN) - F(PNUN) + F(PNUN) - PNF(PNUN))
-
Otto - AO + 0
Set
F(u) - F(UN),
A2
:= F(UN) - F(PNUN),
3(I
PN)F(PNUN),
-
and fix an interval I C [0, T] with III = 6. We will estimate
HAiH
'9-,+,(IxT3 )
X
+
I xT3)I + IA3l X-2 -2+'(IxT3)'
"2I
By Proposition 4.25, we can bound Al by
||A10
-,+,6
< C(A, R, T) 6C IIu
- UN 1X,+6
+
HuH 6 ,4
HuN
+
X,-+,
6
For the second term, Proposition 4.26 and Remark 4.15 yields
C(A, R, T) c N- (I + H|uN
-,
1+,
6
)
2
Finally, by Remark 4.15 we have
N- 0 |F(PNUN)IIXs 2 -1,-i+ 6
|A3jX-,i+,_
C(A,R,T)JcN-(1 + nuN
1
+,)
In the second and third terms, we used the observation that for any N E N, s E R, we have
for any v such that the right-hand side is finite. Now let I = [0, 6]. Then because u
- UN
has zero
initial data, the inhomogeneous estimate yields
1 11+I IX7
I AI 1_
_______IA
21 ,-I
II 2f7
____
107
X-
+13I
(4.36)
and together with (4.35) and the similar result for <bN, we can bound (4.36) by
111
0, (4.37)
I1+,6
, +C(A, T, R) 6c N-
C(A, T, R) 6c 1111
1, +,,5
and similarly for the time derivative component. Hence, for 6 > 0 sufficiently small,
(0,9t)
:1)5 C1 (A, R, T) E, (N),
12(
with limN+oo, 1 (N) = 0. On the next subinterval, 12
| (#,
t)L
2(I2 xT3)
=
(4.38)
[6, 26] we bound
C1 C(A,R,T)el(N) + S(t - s) [F(u) - FN(UN)L
and once again by the inhomogeneous estimate
11i''
< C1(A, R, T) ei(N) +
[F(u) - FN(uN)]
Applying the above argument, we obtain that
|(H
at,) 1L'
2(2x
T)
C 2 (A, R, T) &i(N).
(4.39)
At each stage the coefficient of the nonlinear component is independent of the step number, the
constants in (4.38) are independent of the subinterval and the bounds (4.35) are uniform, we can
choose 6 > 0 sufficiently small to obtain the analogue of (4.39) at each stage uniformly for all
subintervals. We do remark, however, that the bound that we obtain will grow with each iteration
because the constant for the initial data is compounded. Since the number of steps is controlled
by A, R and T, we obtain the desired result.
El
Remark 4.16. When the estimate is performed on the time derivative, the time localization may
increase the left-hand side of (4.37) up to a factor of
6 2--b
for b
=
as above. However, we
recall that in Section 4.5, we the exponent c which we obtain on 6 is some fixed, small constant
independent of b. By taking b sufficiently close to 1,
2' we still obtain the necessary 6 factor provided
we ensure that
1
-
Remark 4.17.
b+c>0.
The term A 2 is a key reason why this argument will not work with probabilistic
108
energy estimates alone, as in [161, say. Indeed, this term requires us to bound the nonlinearity by
a weaker norm and it does not seem possible to close the Gronwall argument if one needs to derive
a bound with respect to some norm below R1.
4.7
Approximation of the flow on open sets
The goal of this section is to prove the approximation results presented in the introduction.
4.7.1
Proof of Theorem 4.5
One key component in our argument is the critical stability theory which allows us to upgrade the
sets of large measure where the approximation holds to open sets, or at least to general initial data
in
FJ2NBR
as is required for Theorem 4.5. Stability arguments first appeared in the context of the
three-dimensional energy critical nonlinear Schr6dinger equation in [18], see also [67].
For the nonlinear Klein-Gordon equation in periodic settings, some care is required as the
Strichartz estimates need to be localized in time. Nonetheless, they follow in a similar manner
from the Strichartz estimates of Proposition A.2, and we present the proofs in Appendix 4.8. One
modification we will present are the stability arguments adapted to the nonlinear Klein-Gordon
equation with truncated nonlinearity. Importantly, we will be able to choose the small parameters
in these arguments uniformly in the truncation parameter. We recall the statement of this theorem.
Theorem 4.5. Let 4 denote the flow of the cubic nonlinear Klein-Gordon equation (4.1) and 4N
the flow of (4.3). Fix R > 0, u. E H1/
2
and N', N
C N with N' sufficiently large, depending on u*.
Then there exists a constant eo(u*, R) > 0 such that for all 0 < e < EO, there exists a = u(R, e, N')
such that for any (uo, ui)
c BR(u*),
sup ||@(t)HN'(u0,u1) - D N (t)INI(uO, U1)1Ji
/2
C(R, u*) [el(N) + e]
(4.40)
te[0,o]
with limN-+ooe1(N)
Proof of Theorem
=
0.
4.5. Fix R > 0, I = [0, o-] for some 0 < o- < I to be fixed, and let A > 0 be
sufficiently large so that we can find (vo, vi) E E,\ n BR(u*). The intersection is non-empty for
some A > 0 by density and for all N' E N it holds that PN(vO, v1)
C EA since (4.19) and (4.53)
are invariant under smooth projections. Thus, there exists some constant K 1 > 0 such that the
109
corresponding global solutions v and
to equations (4.1) and (4.3) with initial data PN(VO, v1)
VN
satisfy
Let (uO, ui)
K1
and
IIVNIIL4(IxT3) < K1
.
IIVIIL4(IxT3) -
c BR, and let N' be sufficiently large so that
I H>N' U*II'1/2
< R.
Then
|S(t)(PN'(vo, i)
- IN'(UO,
U1))
Lt,(IxT3)
Ij /4 sup IIS(t)(PN'(VO, V1) - lN'(UO, U1)II L4(T3)
tEIT
111/4
r1_l4
(N') 1/ 2 [11PN'(O, V1 ) - PNU*)IHi/2(T3) + II(PN'U*
~-
N'(UO,
Ul))IHi/2(T3)
(N')1/ R.
Let p1 = pi(K1 ) be as in the stability lemma and recall that we can choose p1 uniformly for all
o- < 1. Let 0 < Eo < pi, then setting
o- ~ (N')- 2 R
4
E0,
the smallness condition (4.45) of Lemma 4.33 is met and we conclude that for t
C I, solutions
U(t) := <(t)IIN'(UO, U1)
UN W
(D=
N W
t)NN'(VO, V1)
exist to equations (4.1) and (4.3), respectively. Moreover, we conclude from (4.46) that
jF(u)
- F(v)1L4/3(IxT3)
< C(K1 ) Eo.
Hence, by Duhamel's formula and Strichartz estimates, the nonlinear components 1 of the solutions, which was defined in (4.10), satisfy
sUP II(D(t)IIN(VO, v)
-
I(t)N'(UO,U1)I
1/2(T 3) < F(u) - F(v)I
Wt(3)IL''IX3
110
C(K1 ) o,
and similarly for
4
N. We can estimate (4.40) using the triangle inequality by
sup (D(t)71N' (uO, U1)
= sup
'DN ()N' (UO, U1 ) 1 1/23)
-
II'(t)UN'(U0, u1)
V/2e3)
(3
N(t)HN'(U0, u)
-
tEI0
< sup II(t)PN' (vo, V 1)
-
(tN'
i)
1/2(3)
+ SUP ||4(t)PN'(VO, V1) t EIo
N WP)NI(V,
+ SUP IIN(t
tE I(
-
)PN()(, VI)
and hence we obtain that for all (uo,ui) E
sup
(uo
IjID(t)llN'(U0,u1)
-
(3
X /27
N(Vo
V)lUN,
T3
< C(R, u*) [E 1 (N) + 0].
Uji/2(T /23)
(T3)
BR(U*),
(DN(t)1IN'(U0,U1)II.,//2
tE[0,0-1X
4.7.2
VJ1I)
EI
Proof of Theorem 4.8
We turn to the proof Theorem 4.8. We define the p-fattening of E,\ by
E,, :
U
Bp(u).
In this section, we will consider initial data (uo, ul) E EA,p n BR for some sufficiently small
p = p(A, T) > 0 for which we have uniform bounds for the corresponding solutions by the stability
theory.
The following theorem demonstrates that for p = p(A, T) > 0 sufficiently small, we can obtain
convergence uniformly on this open set. Our proof uses a combination of our probabilistic approximation result, Proposition 4.27, and the low-frequency stability result of Theorem 3.1. Using this
theorem, the proof of Theorem 4.8 follows similarly to that of Theorem 4.1.
Theorem 4.28. Let D denote the flow of the cubic nonlinear Klein-Gordon equation (4.1) and
(N
the flow of the cubic nonlinear Klein-Gordon equation with truncated nonlinearity (4.3). Fix
A, T, R > 0, then there exists some p1
=
p1 (A, T) > 0 sufficiently small such that for any 0 < p <
Pi, any (uo, uo) E E\,p n BR, and any e > 0 and N' E N,
sup IPN' (1(t)(UO, U 1 )
tE [0,T]
-
'N(t)(UO,
111
ul))
11,1/2
<
6
for N = N(N', e, A, R, T)
> N' sufficiently large.
Proof. Suppose that (uo, ui) c Bp(vo, vi) for some fixed (vo, vi) c E\.
Since PN.(uo, u1) E
BpPN. (vO, vj) for all N, E N, the stability theory yields solutions on [0, T] given by
U
N N(t)PN.(UO, U1)
:=
I(t)(uO, ui),
UN
i :=) (t)PN. (UO, U1),
which satisfy uniform L3([0, TI
x
=
)
UN
T3) and L 'W/
2
NN(t)(UO, U1)
([0, T] x TV) bounds depending only on A, R, T.
We apply the triangle inequality
sup IIPN' ('(t)(Uo, U1)-IN(t)(UO, Ul)) JINI/2
te[O,T]
r_
(1(t)(Uoui)
sup PjN'
-
4)(t)PN.(UO, U1)) II1/2
tE[O,T]
sup
tE [0,T]
||PN' (D(t)PN. (U0 , U)
sup
tE[O,T
'IN(t)PN. (U0 , U1 )) 131/2
-
IIPN' (bN(t)(UO, Ul)
-
'I)N(t)PN. (UO, U1))
1/2,
and we estimate each term separately. For the second term, we observe that PN, (uO, u1) is smooth,
hence PN. (uo, u1) G EA n BR for A = A(R, K, N,) by energy conservation and Sobolev embedding.
Thus Proposition 4.27 yields the bound
sup IIPN' (f(t)PN.(U 0 , Ui)
-
IN(t)PN,(UO,U1))
11,1/2
,
C(N,, R, T) Ei(N).
tE[0,T]
By Theorem 3.1 and Remark 3.6, we can bound the first and last terms by
sup
IPN'
('N(t)(UOU1) -
-
(t)PN.(UO, U1)) II,1/2 <
IN(t)PN (UO,U1))
kJ1/2
log N*N'
(log
,
sup ||PN' (1(t)(uO, U1)
t E [,T|
tE [0,T]
where the implicit constants depend on A, R, and T > 0. Thus for fixed N' E N, choosing N
sufficiently large, and subsequently N sufficiently large yields the result.
FI
Now we prove the following statement, from which we obtain Theorem 4.8 readily given the
112
bounds on the measure of the subsets EA from Proposition 4.22.
Theorem 4.29. Let 4 denote the flow of the cubic nonlinear Klein-Gordon equation (4.1). Fix
T, R > 0, ko E Z3, z E C, and u. C 7- 1/ 2 (7 3 ) and let A > 0 be such that E\ n BR(U*) / 0. Then
for all 0 < p < p1(A, T) sufficiently small,
4D(T) (EA,p n BR(U,,)) 4 Cr(z; ko)
for all r > 0 with 7rr2 < cap(E,, n BR(U*)).
Iju*11
Proof. Fix N' E N with N' > Ikol and let R1 :=
+ R. Then by Theorem 4.28, we can find
N E N sufficiently large so that for any (uo, ui) E E\,p n BR, we have
sup
IIPN' (1(t)(Uo, ul)
-
4N(t)(UO, ul))
1/2
(4.41)
< E.
tE[O,T
Since
(N
preserves capacities by Proposition 4.21, we have the equality
C((DN(t)(
,p n BR(u*))) = c(Ex,p n BR(u*))
for all t E R. Thus for z = (zo, zo) we can find some (uo, ui) E EZ,p n BR(u*) such that
..------
(ko)DN(T)(uo, uj)(ko)
-
.
+ (ko
-
1
t N(T)(uOj E)(ko)
1/2
)
>r +6
and since (uo, ui) E EZ,p n BR1 ,we conclude by the triangle inequality and (4.41) that
((ko)1i(T)(uo, ui)(ko) -
zO1 2
+ (ko)> 1ItD(T)(uo,ui)(ko) - zo 12
which completes the proof.
4.7.3
> r,
E
Proof of Theorem 4.7
The goal of this subsection is to prove the conditional global result and the small-data nonsqueezing result. The following result demonstrates that at low frequencies, the truncated flow is
a good approximated to the full equation. The proof follows from the same arguments used to
prove Theorem 3.1, which is unsurprising given that Theorem 3.1 essentially yields a decoupling
between low and high frequencies. In this setting, we do not rely on the probabilistic estimates
113
from Proposition 4.27, however, we are only able to compare the low frequency components of the
corresponding solutions. The following proposition immediately yields the large data portion of
Theorem 4.7.
Proposition 4.30. Let D and
1
N denote the flows of the cubic nonlinear Klein-Gordon equation
with full (4.1) and truncated (4.3) nonlinearities, respectively. Fix T, R > 0 and suppose there
exists some K > 0 such that for all (uo, ul) E BR(u,), the corresponding solutions u to (4.1) and
uN to (4.3) satisfy
I|ullLt,([O,T)xT3)
+ sup IjPNUNIHLt,([O,T)xT3)
K.
N
Then for all E > 0 and any N' c N, and sufficiently large N depending on R, T and K,
sup IIPN'
t e [0,T]|XN
(N1(t)(uO, ul)
-
DN(t)(uO, u1)) 11/2 ,
<
N )-0
Nlog,
with implicit constants depending on R, T, and K.
Proof. Let u and uN to the Cauchy problem (4.1) and the truncated equation (4.3) respectively
on [0, T). By Proposition 3.18, there exists some M E [N', N]
E ujo + ujo = PmF(ulo, ujo, ujo) + OK,R,T ((log(N/N')))
for ujo = PMu. By the same reasoning,
EDuN,lo + uN, lo
=
PMF(UN,lo, uN, io uN,lo) + OK,R,T ((log(N/N'))-0)
with a slightly different error term. Since u and uN have the same initial data, the arguments used
E1
to prove Theorem 3.1 yield the result.
Remark 4.18. Note that although any initial data in PN'BR gives rise to global solutions of the
relevant Cauchy problems, Proposition4.30 is insufficient to prove Theorem
4.5 since the implicit
constants would depend on the truncation parameter and thus we could not guarantee convergence
uniformly.
To conclude, we present the following lemma which yields our small data result.
Lemma 4.31. Fix T > 0 and let 4 N denote the flow of the cubic nonlinear Klein-Gordon equation
with truncated nonlinearity (4.3). There exists some sufficiently small absolute constant po
114
=
po(T)
such that for any 0 < p < po and for all (uo, ui) E BP C 71- 1/ 2 (T 3 ) there exists a unique solution
uN
DN(UO, ul)
on [0, T] which satisfies
.
|HUN IIL~,x([0,T)xT3)
Proof. Fix T > 0 By the small data theory, we know that there exists some pi(T) > 0 sufficiently
small so that for any 0 < p < pi, and for all (uo, ul) E BP, a unique solution u := D(uo, ui) exists
to the cubic nonlinear Klein-Gordon equation (4.1), which satisfies
IUItI.([O,T)xT3) I P.
Let F(u)
=
u3 , then we can expand
F(u) - PNF(PNuN)
= F(u) - PNF(u) + PNF(u) - PNF(PNu) + PNF(PNu) - PNF(PNUN),
hence by the boundedness of the smooth projections and Strichartz estimates we obtain
||u - uNIIL4 <
L
5 flufLT
+ |F(PNu) - F(PNuN)L4/ 3
, +
IOu
-
uNL
t'Xtx
t
+
LIuI,4I9
'X
-
uNIIL
t
where the implicit constants may depend on time. By taking po = po(T) > 0 smaller if necessary,
we obtain the desired result by a standard continuity argument.
4.8
E
Appendix A: Stability Arguments
This appendix is devoted the proofs of the critical stability lemmata for the cubic nonlinear KleinGordon equation. As usual, these statements are proved first for solutions which have sufficiently
small Strichartz bounds, we will call these short-time stability arguments. In order to conclude the
statement for arbitrarily large bounds, one needs to divide a given time interval into subintervals
such that the norm of the solution is sufficiently small on each subinterval, then the statement
follows from an iteration argument.
We include these proofs as we would like to make explicit
the dependence of the constants on the various parameters involved. We also point out that the
115
dependence on the time interval in the following estimates arises solely due to localizing Strichartz
estimates and thus can be taken uniformly for all I C [0, 1]. In the sequel we let F(u) := u 3 and
4.8.1
3
.
FN(UN) := PN(PNU)
Stability theory for NLKG
Lemma 4.32 (Short-time stability). Let I C R a compact time interval and to
C I. Let v be a
solution defined on I x T 3 of the Cauchy problem
vtt - Av + v + F(v) = e
= (vo, vi) E h1/2 (T3.
(V, 0to) L,_
Let (u, tu) It=
=
(uo, u1) G W 1/ 2 (T 3 ) be such that
fl(vo -
UO, vi
-
ui) I1/2(T3)
< Ri
for some R 1 > 0. Suppose also that we have the smallness conditions
IVIK4(IXT3)
H1S(t - to)(vo
-
uo, vi
-
Po
ul)L4(IX'T3) < P
(4.42)
Ile|1'qL;(X 3 < P,
for some 0 < p < po(Ri) a small constant and (q',I') a conjugate admissible pair. Then there
exists a unique solution (u(t),tu(t)) to the cubic nonlinear Klein-Gordon equation on I
x
iF 3 with
initial data (uo, ul) at time to and C = C(I) > 1 which satisfies
1K
I(V
-
u,Otu
-
&tV)11L|
ulIL4(IxT3)
CP
|IF(v) - F(u)1L4/3(IxT3)
CP
3
/2 (IxT )
+
fly -
-
u~lL r(Ix
T3)
(4.43)
(4.44)
CRi
for all admissible pairs (q, r). Furthermore, the dependence of the constant C on time arises only
from the constant in the localized Strichartz estimates.
Remark 4.19. By Strichartz estimates, assumption (4.42) is redundant if R 1
116
=
0(p).
v - u, then
Proof. Without loss of generality, let to = inf I and let q
{
problem
#
satisfies the Cauchy
Ott - AO+O+F(v) -F(+v) = e
(0, at#) I|=t
=
(vo - uo, vi - ui).
By Strichartz estimates, H6lder's inequality and the assumptions above,
V1#IHL4(IxT3)
C(H|s(t)(vo - Uo,vi - ul)IL4(IxT3) +
K
IIF( +
v) - F(v) 4/3
+
I xV))
C(2p + (Po) 2 101kIL4(IxT3) + II0 kL4(Ix T3)
hence a continuity argument yields (4.43) provided po is sufficiently small. In particular
||F(
+ v) - F(v)
4/3(IXT3)
CP
for such po. From (4.43) we have
(#l,&H) |L
3) + IIL4(IxT3)
/2(IxT3) +
" C(R1+ fF(O + v) - F(v) 114/
3)
+ IlellLq'Lf(IxT3)
" C(R1 + p + Cp),
hence we obtain (4.44) provided po = po(Ri) is chosen sufficiently small.
0
Lemma 4.33 (Long-time stability). Let I C R a compact time interval and to E I. Let v be a
{vtt
solution defined on I x V73 of the Cauchy problem
- Av + v + F(v) =e
_
= (vo,
vi) E
l1/
2
(
3
)
(V, Otv)|
Suppose that
vl|L4(IxT3) < L
for some constant L > 0. Let to
c I and let (u, tu) I=t.
II(vo -
uo, vi
= (uo, u1)
- ui)I I1/2(T3)
117
Ri
G H1/2(T3) be such that
for some R 1 > 0. Suppose also that we have the smallness conditions
IS(t -
to)(vo
-
uo, vi
-
ul)11L4(IXT3)
P
X
|1e||Lq'L i '(IxT3)<-P
for some 0 < p < p1 (R1 , I, L) a small constant and (4',f') a conjugate admissible pair. Then there
exists a unique solution (u, tu) to the cubic nonlinearKlein-Gordon equation on I x 73 with initial
C(I, L) > 1 which satisfies
data (uo,ul) at time to and C
ulIL4(IxT3)
CP
|IF(v) - F(u)1L4/3(IX T3)
Cp
1v
(v
u, &tu -
-
tO)Lt N/2(I x T3)
1lV
-
- uht (IxT3)
(4.46)
< CR 1
admissible pairs (q, r). Moreover, the dependence of the constants C and p1 on time arise solely
from the constant in the time localized Strichartz inequality.
Proof. Fix I C R and let po = po(2R1 ) > 0 be as in Lemma 4.32. This will allow for some growth
(1+ L)4 subintervals Ij
in the argument. We divide the time interval I into J
=
[tjtj+l1
such that IivlL4 (I) < po letting := u -uv, we can apply the previous lemma on the first interval,
yielding
-
u11L4(IOXT3) < Op
|IF(v) - F(u)11L4/3(IOXT3)
(v
-
u,&tv
-
tu)IIL-/ 2(IoxT3)+ 1ly
-
uliL,'-(IxT3)
CP
CR 1
.
11V
We would like to apply this argument iteratively to claim that
v
IF(v)
Il(v -
u,
Otv
-
&tu)L
-
uIIL4(IXT3) < C(j)p
- F(u)1L4/3(IjxT 3)
/2(IjXT3) + 11V
-
C(j)p
uhL-(I xT3) K C(j)R1.
In order to do this, we need to ensure that for each t3 we have
I(v(tj)
- u(tj), (tv (tj) - atu(tj))I IJ/2(T3)
<
||S(t - ti)(u(ti) - v(t))L4(jXT3)
< C(1, L) p
118
2R 1
(4.47)
(4.48)
We prove these statements by induction. For (4.47) we use Strichartz estimates and we bound
(#(tj), ato(tj)) I|W/2( 3)
1|((to), Oto(to))||7
1/2(,3)
+ ||F(#+v) - F(v) 4/3
Ie
j-1
R 1 + EC(k)p+ p
k=O
and similarly for (4.48), we have
- ti)(u(t3 ) - v(t))IIL4(I.x T'3)
< |S(t - t.)(u(to) - v(to))|L4([to,tj)XT3) +
flF(u) - F(v)
3
)
+ e
t
3
)
HS(t
j-1
,< 2 p + E
C( k )p,
k=O
so the conclusion follows by choosing p1 (R1 , I, L) sufficiently small.
4.8.2
E
Stability theory for the truncated NLKG
Lemma 4.34 (Short-time stability for the truncated equation). Let I C R a compact time interval
and to E I. Let
vN
be a solution defined on I x T3 of the Cauchy problem
{
(VN)tt
(vN, 8tvN) t=to
Let to E I and let (uN,&tuN) tt
VN + PN(PNVN)
- AVN
= (uouO
I (vo -
-
3
-
e
(vIvI) E H1/2 (3
) E H1/2(T3) be such that
Uo, vI -
u1)II1/2(T3)
R1
for some R 1 > 0. Suppose also that we have the smallness conditions
|PNVNIIL4(IxT3)
IIS(t - to)(vo - uo, Vi - ui) 1L4(IxT3)
e
119
xLL
I(IxT3)
Po
P
p,
for some 0 < p < po(Ri) a small constant and (4', i') a conjugate admissible pair. Then there
exists a unique solution
I x T
3
to the truncated cubic nonlinear Klein-Gordon equation on
(UN, &tuN)
with initial data (uo, ul) at time to and C
C(I) > 1 which satisfies
uNIL4(IxT3)
< Op
FN(uN)I L4/3(IxT3)
CP
|IVN
|IFN(VN)
II(VN
-
-
-
UN, atUN - (tUN)1 L-1/2 (Ix3) + JIVN - UNIIL
3
(Ixr
)
(4.49)
< CR1
(4.50)
for any admissible pair (q, r). Furthermore, the dependence of C on time arises only from the
constant in the localized Strichartz estimates.
Proof. Without loss of generality, let to = inf I. Let
- VN - UN,
ON
then
satisfies the Cauchy
ON
problem
(ON)tt -
AON + ON + FN(VN)
(ON, atkN)I t=0 = (Vo - uo, VI
-
FN( 5 N + vN)
=
e
ul)-
By Strichartz estimates, H6lder's inequality, the boundedness of PN and the assumptions above,
IJNlIL4(IxT3)
<
C(HS(t) (vo
< C(2p +
-
(Po)
2
uo, Vi -
u1)
L4(IX T3)
+ I FN(VN)
-
FN(ON
+ VN)
L4 (IxT3)
+ 11e L Liz '(Ix
3)
IbkNlIL4(IxT3) + IkONHL4(IxT3)),
hence a continuity argument yields (4.49) provided po is sufficiently small. Similarly
(ON, OtON) IL' W/2(IxT3) + IK/NIILqL(IxT3)) + IIqONIL4(Ix T3)
C(II(vo
-
uo,V1 - ui)IKWI/2(T3)
C(Ri + (PO) 2
IONIIL4(IxT3)
+ |HFN(VN)
-
+ II'N 1L4(IxT3)
FN(ON + VN)IlL4/3(IxT 3)
+
LeH
LfL
+ P
We conclude (4.50) by a continuity argument for po = po(Ri) > 0 sufficiently small.
Remark 4.20. If one only requires bounds on the low frequency component PN(UN
120
'(IxT3)
I
-
VN), then
from the proof of the previous Lemma, it is clear that it suffices to assume that
flS(t -
tO)PN(VO
uO, v1
-
-
u1)I L4(IxT3)
P
4'L' < P.
IlPe
t|~IIX
Li(IxT3) -'
Lemma 4.34 and the same proof as in Lemma 4.33 yields the long-time stability argument for
the truncated equation with bounds uniform in the truncation parameter.
Lemma 4.35 (Long-time stability). Let I C R a compact time interval and to E I. Let VN be a
solution defined on I x 73 of the Cauchy problem
(VN)tt
(VN,
VN + FN(VN)
AVN
-
OtVN) t=to
=
(vo, vi) c
e
=
3
H1/2
Suppose that
||PN VNHL4(IxT3)
L
for some constant L > 0. Let (uN, OtuN) t=tO = (uO, ul) E 71/ 2 (T 3 ) be such that
11(vo
-
Uo,
vi
-
R1
u1)IIy1/2(T3)
for some R 1 > 0. Suppose also that we have the smallness conditions
lS(t - to)(vo - u0, vi - ui)L4(IxT3)
P
for some 0 < p < p1 (RI, I, L) a small constant and any (4', f') a conjugate admissible pair. Then
there exists a unique solution (uN, 1tuN) to the truncated cubic nonlinear Klein-Gordon equation
on I x T 3 with initial data (uo, ul) at time to and C = C(I) > 1 which satisfies
|I(vN) 3
1
VN
-
uN, atuN -
atvN)1
L
-
1/2 (IxT 3)+
121
-
IJvN
uNIL4(IxT3)
-
(UN)
-
3
IL4/3(IxT3)
CP
OP
uNLq;(IxT3) < C R
1
.
||vN
4.9
Appendix B: Probabilistic bounds for the cubic NLKG
In this appendix, we record the proof of Proposition 4.19. The proof of Proposition 4.20 follows
almost identically. We recall the statement.
Proposition 4.20. Let 0 < s < 1 and let p E M'. Then for any e > 0, there exist C, c, 0 > 0
such that for every (uo, ui) E E, there exists M > 0 such that the family of global solution u to
cubic nonlinearKlein-Gordon equation (4.1) satisfies
u(t) = S(t)(u0, ui) + w(t)
I|(w(t),& tw(t))IIui < C(M + ltl)l
c E : M > A)
Proof of Proposition 4.19. Fix
(4.52)
C(M + j)
|Iu(t)1L4(T3)
and furthermore [((uo,ui)
(4.51)
CecA 0
p C Ms. Following the proofs of Proposition 4.1 in [16] and Lemma
2.2 from [13], fix e > 0, p > 1, p > 1/3, and p > 0 and define
UN
(VO, v1)
Cz
I:IN(vo, vi)
z
0I1 < NK-s+e
< N6}
GN
{(VO, V1) E
HN
(VO, vl) CE
KN
(VO, v1) EC
:
j(t)_
UIN (VO, 19 0L3L6(R xT3) :5 NE-s}
RN
{(vO, v) C
:
1 t)-
UIN S(t)(VO V10 0Li
INvOIIL4
:11 (t)_
ll N(vo,
V1 ))L2L-(RxT3)
<N's }
L4 (R x T3)
< N'-s
We let
EN = UNn GN
n HN n KN nRN.
(4.53)
The bounds on the measure of EN follow from Proposition 4.1 in [16] and Lemma 2.2 in [13]. We
consider the inhomogeneous energy functional (4.5)
2
Vw| 2
2(w(t))=
122
2
Fix (VO, v1) E EN and let WN denote the solution to
(WN tt
-
AWN
+ WN + (wN + S(t)rL>N(VO, Vl)) 3
then
0,
(WN, &tWN) t=,
=
11N(VO,V1)
S(WN(t))1/2 -<CN1-s+E
Since S controls the V norm, we no longer need to project away from constants to obtain (4.51).
To prove (4.52), note that by the definition of RN, we have
IlUN(t)I L4(T3)
IWN(t) HL4(T3) +
CN
+ CN-8+e
.
S (WN (t))1/4
S(t)l;>N(VO, Vl)IL4(T3)
The conclusion then follows as in Proposition 4.1 in [16].
123
E
124
Appendix A
Facts from Harmonic analysis
A.1
Strichartz Estimates
Definition A.1. A pair of real numbers (q, r), is called y admissible provided 2 < q
+oo,
2 Kr < +oo,
1 +3 =3
q
r
2
an
1
1
q
r
1
K-.
We call a pair (4', f') a conjugate admissible pair if
1
1
q
q'
11
r
for some admissible pair (4,f).
We state the following Strichartz estimates compactly for both the wave or Klein-Gordon
equations on A = R or T. These estimates are classical and due in parts to [65], [53], [26], [33].
Proposition A.2 (Strichartz estimates). Let u be a solution to the inhomogeneous equation
utt - Au + mu = F(u),
(u, atu)It= = (Uo, ui)
m = 0, 1
on I C [0, T] for T fixed. Let (q, r) be a -- admissible pair, then for m = 1
||(u, tu)IIu'
(IxA) +
UIL qLr(IxA)
.- I(UO, ul)11(Y(A) +
125
|IFIILf L(Ix A),
(A.1)
and similarly for m
=
0 with homogeneous Sobolev spaces instead of inhomogeneous ones, and
(P', 4') is a conjugate admissible pair. When A
=
T the implicit constant depends on the choice of
T but is uniform for any subinterval I C [0, T]. When A = R, the implicit constant is independent
of T and we can take T = oc
Remark A.1. In fact, a larger range of exponents are admissible for the nonlinear Klein-Gordon
equation than the range stated above, which only coincides with the admissible exponents for the
nonlinear wave equation. Since the above estimates are all we need in our arguments, we refrain
from stating the full range of Strichartz exponents for simplicity. For a full formulation, see for
instance [50], and references therein.
We use the following interpolation estimate for the Strichartz estimates for the nonlinear KleinGordon equation in Chapter 4 to estimate frequency localized functions in X,,b spaces, see [7] for
this formulation of the estimates.
S
< C(I)N1/2 (
a(n)ei(xn+t(n))
a(n) 12)
/
Proposition A.3 (Strichartz). Let I C R. There exists a constant C = C(I) > 0 such that
L 4 (Ix T 3
)
Ini~N
By Hdlder's inequality we have
5
c(n, r)
d
ei(xn+tr)
< C(I)N 1/
N
L (IxT
-nl
2
3
(
c(n, )
2
dr)
1/2
)
4
and by interpolating with Parseval'sidentity we obtain that for 2 < r < 4
InIN
-
< C(I)NO/
L'r (I x T
3
r
.
where 0 = 2 -
-rInI
ei(xn+tr)
)
c(n, 7)
fdr
126
2
Ic(n, T)12d )1/ 2
(A.2)
A.2
A.2.1
Adapted Function spaces
Xsb
spaces
For a good overview of these spaces, see Chapter 2.6 in [66].
For completeness, we recall the
definition of X,,b(R x T) spaces, with norm
-
We also work with the local-in-time restriction spaces
(n)1biU(n,
which are defined by the norm
X,,b,6,
|U|HX.,b,6 = inf{!IIx9s,b(RXT3)
1rL2e2.
-
.
||uflxsb(RxT3) = H(n)'(I-rI
We have the obvious inclusions
Xs',b' C Xs,"
for s < s' and b < b'.
We remark that these spaces are not invariant under conjugation or
modulation but they are invariant under translation.
Heuristically, these spaces measure how far a given function is from being a free solution.
Additionally, free solutions lie in
XSb,
at least when time is localized.
Lemma A.4. Let f E H' for s E R and let S(t) denote the free evolution for the Klein-Gordon
equation. Then for any Schwartz time cutoff q E S.(R),
1177(t)S(t)fixsb(RXT3) + 7I(t)tS(t)fllxs-1,b(RXT3)
c(7, b) IfI1iis(T3).
We note that it does not hold that free solutions lie in Xb globally hence these spaces are
really only suitable for local theory. An important property of these spaces is the so-called transfer
principle which allows one to convert bounds for free solutions into bounds for XSb functions
Lemma A.5 (Lemma 2.9, [66]). Let A = R or T and let L = iP(V/i) for some polynomial
P : Ad -4 R, and let s E R and let Y be a Banach space of functions on R x Ad such that
lei~eLfIly < IfI()
12od < H.(Ad)
127
for all f E H,(A) and -ro E R. Then for b >
2
IJuIly :b IlUlXsb(RxAd).
Letting Y be the Strichartz spaces from Proposition A.2, we obtain the following corollary
immediately.
Corollary A.6. Let (q, r) be Strichartz admissible pairs and let b > 1. Then
ra J IXO,b.
||U1|L L
In particular, for b > 1,
2' X,,b embeds into CtH. (for both the full and restricted spaces on
the appropriate domains). This embedding fails at the endpoint b = 1 and should be thought of
analogously to the failure of the endpoint Sobolev embedding L'
V H
. It is precisely at the
endpoint b = 1 that these spaces respect the scaling of CtHS and for critical problems where scale
invariance is an issue, one no longer has the appropriate control in order to close the contraction mapping argument.
It is possible to remedy this problem by including a Besov space type
refinement, however we will focus instead on UP and VP spaces.
A.2.2
UP and VP Function spaces
In this section, we introduce the basic facts we need about the UP and VP spaces used in Chapter 3.
We follow the exposition in [28]. Consider partitions given by a strictly increasing finite sequence
-00
< tO < t 2 < ... tK < 00-
v : R -+
If tK = oc we use the convention v(tK) := 0 for all functions
H. We usually work on bounded intervals I C R.
A step functions associated to a
partition is a function which is constant on each open sub-interval of the partition. In the sequel,
we let B denote an arbitrary Banach space.
Definition A.7 (UP spaces). Let 1 < p < oo. Consider a partition {to, ...
B with
_- 1||PkIIP2
=
1. We define a UP atom to be a function
K
a =
3
1
[tk1,)Pk-1
k=1
128
,tK}
and let (pk)K-l C
and we define the atomic space UP(R, B) to be the set of all functions u : R
-+
B such that
00
U
= E
j j
j=1
for aj UP atoms, and {Aj} E 0 (C), endowed with the norm
00
Ijullup:= inf
{E
00
Ajaj : aj is a UP atom}.
jLu, U =Z
j=1
j=1
Remark A.2. This yields a Banach space which satisfies the embeddings
UP (R, B) "- Uq (R, B) " L' (R, B)
for 1 < p < q < oo. Furthermore, every u
E
UP is right-continuous and limt,-
Definition A.8 (VP spaces). Let 1 < p < oo.
00
u(t)
=
0.
We define VP(R, 1) as the space of all B valued
functions, v, such that the norm
IVIvP(R,13)
with the convention v(oo)
=
0.
=
|Iv(ti) - v(ti_)IIp
sup
partitions
<
00
We let V_ (R, B) denote the subspace of all functions satisfying
limt-oo v(t) = 0 and we let VR(R, 1) denote the subspace of all right continuous functions in
V_(R, 1), endowing both these subspaces with the above norm.
Remark A.3. Note that for 1 < p < oo we have the embeddings
UP (R, B) " VP(R, B) " L' (R, B)
and
VP(R, B)
-+
Vq(R, B).
If further 1 < p < q < oc, then
V (R, )
o Uh(R, B).
A crucial property of the UP and VP spaces is the following duality relation.
129
(A.3)
Theorem A.9. Let 1 < p < oc and
1
P
+ P / = 1. Then
(UP(R, B))* = VP' (R, B*),
that is, there is a bounded bilinearform
T(v) := B(-, v)
T : VP'(R, B3*) -+ (UP(R, LB))*,
which is an isometric isomorphism.
Proposition A.10 (Proposition 2.9, [28]). For 1 < p < oo, let u c UP be continuous and
v, v* G VP' for 1 + P- = 1 such that v(s)
=
v*(s)= except for at most countably many points. Then
B(u, v)
Proposition A.11. Let 1 < p < oc, u
c
=
B (u, v*).
V_ (B) be absolutely continuous on compact intervals and
v E VP'(p3*) for 1 +P / =1. Then,
B(u,v) = In particularB(u, v)
=
B(u, iU) if v(t)
=
(v(t),u'(t)),3 dt.
U(t) almost everywhere. Consequently, v can be replaced
by its right-continuous version.
El
Proof. See [28]
Remark A.4 (Remark 2.11 in [28]). Let 1 < p < oc and u E UP. Then for
+
=
1 one clearly
has
Iu||up
sup
=
vEVP':
|B(u,v)|.
|v|| 1PI1
However, in light of PropositionA.10, one can restrict to taking a supremum over right-continuous
functions, which we do in the estimates in Chapter 3. The bilinear form above should be thought
of as the correspondingthe Stieltjes integral
f dg=
f (ti) (g(ti+1) - g(ti)).
130
We define variants of the UP and VP spaces adapted to the linear propagator for the KleinGordon equation e*-t(V). In the sequel we will suppress the notational dependence on I C R and
the Banach space B.
Definition A.12. Defnote by U ' the space UP adapted to the linear propagator equipped with the
norm
Sull up = Ile it(V)ullup
and similarly for the VP spaces.
Remark A.5. The space U
is again an atomic space with atoms a = eit(V)a for a UP atom,
a. Further, in the case that B = H8, we compare the above definition to that of the XS,b spaces
associated to the Klein-Gordon equation given by Iu||Xs,b
=
Ile-it(V)u HbH'
A useful feature of the UP and VP spaces is that they satisfy a transfer principle, namely one
can transfer multilinear estimates for free solutions to estimates for U
functions.
Proposition A.13 (Transfer principle, Proposition 2.19 [281). Let To
Lx x ...
x L-
L'
be
an m-linear operator. Suppose that for some 1 < q, r < oo
m
||TO(SMt)1, .
SMOt)#I)L
L
L<,
1L
i=1
then there exists an extension To : Ui x
...
x U4 -+ L L' with
|T(ui,. um)I|LqL'
<
IlUilUq
which agrees with To for almost every t C R.
Remark A.6. The transferprinciple is the key tool which allows us to use the Strichartz estimates
available for free solutions of the Klein-Gordon equation to derive bounds in UP and VP spaces.
Because of the duality relation, typically one needs to put one function in VP when performing
multilinear estimates. In these cases, the following proposition demonstrates that this is possible if
one allows for a logarithmic loss.
131
Proposition A.14.
T :
U'1
x
... Uq_
Let q1,... qm > 2 where m = 1,2 or 3 and let E be a Banach space and
-+ E be a bounded rn-linear operatorwith
m
Cl I'Uj
.
|T(ui .... Um )| E
U
i=1
Suppose further theses exists some 0 < C2 < C such that the estimate
m
||T(ui,... u m ) E
02
Jl IjUU2
hn1
holds. Then T satisfies
||T(ui,....umn)||E <C02
log C +1)
Mm
flhuJV2,
i=1
132
ujGrC,
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