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Kenji Nakanishi Wilhelm Schlag
Invariant Manifolds and Dispersive Hamiltonian Evolution Equations
Authors: Prof. Kenji Nakanishi Department of Mathematics Kyoto University 606-8502 Kyoto Japan
Prof. Wilhelm Schlag Department of Mathematics University of Chicago 5734 S. University Avenue Chicago, IL 60637-1514 USA
2010 Mathematics Subject Classification: 35L70, 35Q55 Key words: Nonlinear dispersive equations, wave, Klein–Gordon, Schrödinger equations, scattering theory, stability theory, solitons, ground states, global existence, finite time blow up, soliton resolution conjecture, hyperbolic dynamics, stable, unstable, center-stable, invariant manifolds
ISBN 978-3-03719-095-1 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.
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© 2011
European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email:
[email protected] Homepage: www.ems-ph.org
Printing and binding: Druckhaus Thomas Müntzer GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321
Contents 1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
The Klein–Gordon equation below the ground state energy 2.1 Basic existence theory . . . . . . . . . . . . . . . . . 2.2 Stationary solutions, ground state . . . . . . . . . . . 2.3 The Payne–Sattinger criterion, regions P S˙ . . . . . 2.4 Scattering in P SC . . . . . . . . . . . . . . . . . . . 2.5 Strichartz estimates for Klein–Gordon equations . . . 2.6 Summary and conclusion . . . . . . . . . . . . . . .
. . . . . . .
. . . . . . .
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17 20 29 34 37 63 77
3
Above the ground state energy I: Near Q . . . . . . . . . . . . . . . 3.1 Energy landscape . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Center, stable, and unstable manifolds in hyperbolic dynamics 3.3 Center-stable manifolds via the Lyapunov–Perron method . . . 3.4 Dispersive estimates for the perturbed linear evolution . . . . . 3.5 The center-stable manifold for the radial cubic NLS in R3 . . . 3.6 Summary and conclusion . . . . . . . . . . . . . . . . . . . .
. . . . . . .
. . . . . . .
79 80 85 108 117 130 143
4
Above the ground state energy II: Moving away from Q . . . . . . 4.1 Nonlinear distance function, eigenmode dominance, ejection 4.2 J and K0 ; K2 above the ground state energy . . . . . . . . . 4.3 The one-pass theorem . . . . . . . . . . . . . . . . . . . . . 4.4 Summary and conclusion . . . . . . . . . . . . . . . . . . .
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145 145 158 162 170
5
Above the ground state energy III: Global NLKG dynamics 5.1 Statement of the main results on global dynamics . . 5.2 The blowup/scattering dichotomy in the ejection case 5.3 Proofs of the main results . . . . . . . . . . . . . . . 5.4 Summary and conclusion . . . . . . . . . . . . . . .
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173 174 177 182 188
6
Further developments of the theory . . . . . . . 6.1 The nonradial cubic NLKG equation in R3 6.2 The one-dimensional NLKG equation . . 6.3 The cubic radial NLS equation in R3 . . . 6.4 The energy critical wave equation . . . .
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191 194 204 218 232
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1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
1
Introduction
This research monograph is intended as an introduction to, and exposition of, some of the phenomena that solutions of nonlinear dispersive focusing equations exhibit at energy levels strictly above that of the ground state soliton. It grew out of lectures that the authors have given on various aspects of their work on focusing wave equations. In particular, it is a much expanded version of the second author’s postgraduate course (Nachdiplomvorlesung), which he taught in the fall of 2010 at ETH Zürich, Switzerland. The equations which we consider in these lectures are Hamiltonian, but not completely integrable, and they all exhibit soliton-like (i.e., stationary or periodic) solutions which are unstable. Amongst those the ground state is singled out as the one of smallest energy. The aforementioned phenomena concern the transition from a region of phase space in which solutions exist globally in forward time and scatter to a free wave, to one where they blow up in finite positive time. We can describe this transition in some detail provided the energy is only slightly larger than that of the ground state. In fact, the boundary along which these open regions meet can be identified as a center-stable manifold associated with the ground state. To be more specific, consider an energy subcritical Klein–Gordon equation uR
u C u D f .u/
(1.1)
in R t Rdx with real-valued solutions. More generally, the mass term should be m2 u with m > 0, but we can set m D 1 without loss of generality. This equation should be thought of as perhaps the simplest model equation which exhibits the phenomena which we wish to describe here, but it should not be mistaken as the central object of our investigations. We begin with a fairly general discussion of (1.1). It is invariant under the full Poincaré group, i.e., under the group generated by spatial as well as temporal translations, Euclidean rotations, and Lorentz transforms. The latter are defined as the group that leave the quadratic form 2 jj2 in Rd C1 invariant (the Minkowski metric). Moreover, (1.1) is both Lagrangian as well as Hamiltonian in the following sense: at least formally, solutions to this equation are characterized as critical point of the Lagrangian, with F 0 D f , Z i h 1 1 1 L.u; u/ P D uP 2 C jruj2 C u2 F .u/ .t; x/ dtdx : d C1 2 2 2 R t;x
2
Introduction
This can be seen by integrating by parts in the integral representing L0 .E u/ D 0. The Lagrangian point of view is important with respect to conservation laws generated by one-parameter subgroups of symmetries of L (Noether’s theorem). For example, time translation invariance leads to the conservation of the energy Z h i 1 1 2 1 uP C jruj2 C u2 F .u/ .t; x/ dx E.u; u/ P D 2 2 Rd 2 invariance under spatial translations yields the conservation of the momentum P .u/ D hujrui. P Euclidean rotations are associated with the conservation of the angular momentum. The aforementioned energy subcriticality assumption which we made on (1.1) now means the following: the nonlinear term F u.t/ is strictly weaker than the H 1 part of the energy, as expressed by the Sobolev estimate. To be more specific, consider f .u/ D jujp 1 u in R3 . Then p < 5 is subcritical, whereas p D 5 is critical and p > 5 is supercritical due to the Sobolev embedding H 1 .R3 / L6 .R3 /. We shall not touch the supercritical case here at all. Even though it may seem most desirable to restrict one’s attention to classical, i.e., smooth, solutions of (1.1) this is not the case; the best notion of solution for many different reasons turns out to be that of an energy solution which is a solution which belongs to H 1 L2 for all times. To express (1.1) in Hamiltonian form, we write it as a first order system, with u dependent variable U WD uP : UP D JH U C N.U / where J D
0 1 ; 1 0
H D
C1 0 ; 0 1
! 0 N.U / D : f .u/
For simplicity, let f D 0. Then energy conservation simply means that D 0. The symplectic form associated with (1.1) can be now seen to be Z ˝ ˛ !.U; V / D J U jV D .U2 V1 U1 V2 /.x/ dx :
d hH U jU i dt
Rd
There are two main classes dividing equations of the form (1.1): the defocusing equations on the one hand, and the focusing ones on the other hand. Loosely speaking, this division can be expressed along the lines of the globalin-time existence problem for (1.1) for smooth, compactly supported data, say. Defocusing equations admit smooth solutions for all data and all times, whereas the
3
Introduction
focusing ones may exhibit finite time blowup for certain data (such as those of negative energies). To be more specific, consider monic nonlinearities f .u/ D jujp 1 u in arbitrary dimensions. Then > 0 represents the focusing case, and < 0 the defocusing one. Note that this corresponds exactly to the distinction of the energy E.E u/ being indefinite vs. positive definite, respectively. Only the focusing case will be relevant to this monograph. Of central importance to the theory of the focusing nonlinear Klein–Gordon (NLKG) equation (1.1) is the fact that they admit nonzero time-independent solutions '. Any weak H 1 solution of the semilinear elliptic PDE (1.2)
' C ' D f .'/
is such a solution. Letting the Poincaré symmetries act on ' generates a manifold of moving solutions of the following form: first, define for any .p; q/ 2 R2d Q.p; q/.x/ D Q.x q C p.hpi 1/jpj 2 p .x q//; p where hpi WD 1 C jpj2 . The traveling waves generated from ' are defined as u.t/ D ˙'.p; q.t//;
p 2 Rd ; q.t/ P D
p ; hpi
p with fixed momentum p and velocity hpi . They are solutions of (1.1). Note that jq.t/j P < 1 in agreement with the fact that the speed of light which is normalized to equal 1, acts as a barrier. Amongst all solutions of (1.2) one singles out a positive decaying one, called the ground state which we denote by Q. It is known to be unique up to translations for many different nonlinearities f .u/, and it is radial. Q is characterized as the minimizer of the stationary energy (or action) Z h i 1 1 jr'j2 C ' 2 F .'/ dx J.'/ WD 2 R3 2
subject to the constraint, with ' ¤ 0, Z K0 .'/ WD jr'j2 C ' 2 R3
f .'/' dx D 0
(1.3)
It follows that the regions P SC D f.u; u/ P j E.u; u/ P < J.Q/; K0 .u/ 0g; P S D f.u; u/ P j E.u; u/ P < J.Q/; K0 .u/ < 0g
(1.4)
4
Introduction
are invariant under the nonlinear flow in the phase space H 1 L2 where E.u; u/ P is the conserved energy for (1.1). It is a classical result of Payne and Sattinger [114] that solutions in P SC are global, whereas those in P S blow up in finite time; these results apply to both time directions, i.e., blowup occurs for both positive and negative times simultaneously, and the same is true of global existence. In particular, the stationary solution Q is unstable, see also Shatah [125] and Berestycki, Cazenave [11]. Scattering in P SC was only recently shown by Ibrahim, Masmoudi, and the first author [77] using the concentration-compactness proof method of Kenig and Merle [84]. We present these results below the energy threshold J.Q/ in Chapter 2. Starting with Chapter 3 we study solutions whose energies satisfy J.Q/ E.u; u/ P < J.Q/ C "2
(1.5)
for some small " > 0 and the special nonlinearity f .u/ D u3 (although this is out of convenience rather than necessity). It is here that one encounters the aforementioned center-stable manifolds that appear as boundaries of open blowup/global existence regions. Center/stable/unstable manifolds are well-established objects which arise in the study of the asymptotic behavior of ODEs in Rn , see for example Carr [25], Hirsch, Pugh, Shub [72], as well as Guckenheimer, Holmes [68], and Vanderbauwhede [138]. Let us recall the meaning of these manifolds: given an ODE in Rn , xP D f .x/ with f .0/ D 0, and f smooth, let A D Df .0/. Then split Rn D Xu C Xs C Xc as a direct sum into A-invariant subspaces such that all eigenvalues of A Xu lie in the right half-plane, those of A Xs lie in the left half-plane, and the eigenvalues of A Xc are all purely imaginary. An example of such a situation is given by the 7 7 matrix 2 3 1 1 0 0 0 0 0 60 1 0 0 0 0 07 6 7 60 0 7 1 0 0 0 0 6 7 6 0 1 0 07 A D 60 0 0 (1.6) 7: 60 0 0 7 1 0 0 0 6 7 40 0 0 0 0 0 15 0 0 0 0 0 0 0 The eigenvalues are f0; 1; 1; i; ig, and Xu is spanned by the first two coordinate directions, Xs by the third, and the center subspace by the final four. Note that Xc splits into a rotation in the fourth and fifth variables, but has linear growth on
Introduction
5
the subspace spanned by variables six and seven. By the center-manifold theorem, see [25], [68], [138], there exist smooth manifolds Mu ; Ms ; Mc locally around zero which are tangent to Xu ; Xs ; Xc , respectively, at x D 0, and which are transverse to each other. Moreover, Mu ; Ms ; Mc are each locally invariant under the flow (meaning that a trajectory starting on any of these, say Mc , remains on Mc as long as the trajectory itself remains in a small neighborhood of the equilibrium point). A manifold Mcs which is tangent to Xs C Xc at x D 0, of the same dimension as this tangent space, and is locally invariant under the flow is referred to as center-stable. On Ms ; Mu the solution to xP D f .x/ decreases exponentially fast as t ! 1 or t ! 1, respectively (in fact, they are characterized by this property). But on Mc the behavior can be quite complicated and that manifold is not characterized by growth conditions. Such a decomposition is relevant for several reasons. On the one hand, it reduces the dynamics in the state space to lower-dimensional subspaces which is often the only way to obtain any understanding of the flow. On the other hand, it is most relevant for bifurcation theory of ODEs which refers to situations where the vector field f .x/ depends on a parameter , see Guckenheimer, Holmes [68]. In the context of (1.1) and related Hamiltonian PDEs such as the cubic focusing nonlinear Schrödinger equation (NLS), a center-stable manifold arose in [122] from the attempt of obtaining a conditional asymptotic stability result for an unstable equation. More precisely, the second author obtained – for the cubic focusing NLS in R3 , and in a small neighborhood of the ground state soliton – a codimension one manifold with the property that any solution starting from that manifold exists globally and scatters to a (modulated) ground state soliton. The drawback of [122] lay with the topology which is not invariant under the NLS flow. But Beceanu [9] later carried out the construction in the optimal topology introducing several novel ideas, such as Strichartz estimates for linear evolution equations with small timedependent but space-independent lower order terms. This allowed him not only to obtain a similar conditional asymptotic stability result as the one in [122] (but of course without any pointwise control on the rate of convergence of various parameters, which requires a stronger topology), but also to verify the properties usually associated with the center-stable manifold such as invariance locally in time (in fact, globally in forward time, and locally in backward time). The method of proof in [122], [9] is perturbative, and is restricted to a small neighborhood of the ground states. This work left open the question as to what happens near the ground state soliton, but off the center-stable manifold. This is one of the problems we wish to address in this monograph. While there has been some heuristic and numerical work in the physics literature, see Bizo´n et al. [15], [16] and Choptuik [30], the first rigorous results on this problem were obtained in
6
Introduction
[87], [88], [109]–[111]. As an example, consider the one-dimensional Klein–Gordon equation u t t uxx C u D jujp 1 u (1.7) with p > 5. It is well-known that for this equation the solitons are given by the explicit expressions Q.x/ D ˛ cosh
1 ˇ
.ˇx/;
˛D
p C 1 p1 1 2
; ˇD
p
1 2
:
In contrast to the nonlinear Schrödinger equation, one of the advantages of (1.7) lies with the fact that under an even perturbation Q does not change; in other words, it is not modulated. In fact, the following result was obtained in [88]. Theorem 1.1. Let p > 5. There exists " > 0 such that any even real-valued data .u0 ; u1 / 2 H 1 .R/ L2 .R/ with energy E.u; u/ P < E.Q; 0/ C "2
(1.8)
have the property that the solutions u.t/ of (1.7) associated with these data exhibit exactly one scenario of the following trichotomy: ı u blows up in finite positive time ı u exists globally and scatters to zero as t ! 1 ı u exists globally and scatters to Q, i.e., there exists a free Klein–Gordon wave v.t/; v.t/ P 2 H 1 L2 with the property that u.t/; u.t/ P D .Q; 0/ C v.t/; v.t/ P C oH .1/; t ! 1: In addition, the set of even data as above splits into nine nonempty disjoint sets corresponding to all possible combinations of this trichotomy in both forward and negative times. All solutions which fall under the third alternative form the center-stable manifold. One can show that this is a C 1 (or better) manifold of codimension 1 which passes through .Q; 0/ and lies in the set described by (1.8). Figure 1.1 illustrates Theorem 1.1 for data that start off near the ground-state solution .Q; 0/. The third region is the center-stable manifold. The figure on the cover illustrates the nine sets alluded to in the final statement of the theorem. Moreover, we obtain the following characterization of the threshold solutions, i.e., those with energies E.E u/ D E.Q; 0/. Results of this type originate with the seminal work of Duyckaerts and Merle [48].
7
Introduction
I
.Q; 0/
III
II
W cs
Figure 1.1. The forward trichotomy near .Q; 0/
Corollary 1.2. The even solutions to (1.7) with energy E.E u/ D E.Q; 0/ are characterized by exactly one of the following scenarios, each of which can occur: ı they blow up in both the positive and negative time directions ı they exist globally on R and scatter as t ! ˙1 ı they are constant and equal ˙Q ı they equal one of the following solutions, for some t0 2 R: WC .t C t0 ; x/;
W .t C t0 ; x/;
WC . t C t0 ; x/;
W . t C t0 ; x/
where .W˙ .t; /; @ t W˙ .t; // approach .Q; 0/ exponentially fast in H as t ! 1, and D ˙1. In backward time, WC scatters to zero, whereas W blows up in finite time. As usual, the images of W˙ and Q form the one-dimensional stable manifold associated with .Q; 0/. The unstable manifold is obtained by time-reversal. The goal of these lectures is to prove results such as Theorem 1.1 and Corollary 1.2. More precisely, in Chapters 2–5 we restrict ourselves to the radial cubic NLKG equation in R3 and systematically develop the machinery leading to results analogous to Theorem 1.1 and Corollary 1.2 above. Loosely speaking, the argument relies on an interplay between the hyperbolic dynamics near the ground states on the one hand, and a variational analysis away
8
Introduction
from them on the other hand. While the former here means linearizing around the ground states and then performing the necessary perturbative analysis locally around the ground states, the latter refers to a type of global argument that is not at all based on linearization. More precisely, we shall use the virial identity valid for any energy solution of (1.1), E 1 d D (1.9) w uP j .x r C r x/u D K2 .u/ C error dt 2 where K2 .u/ D kruk22 34 kuk44 . The cut-off function w here is chosen in such a way that the error remains small, cf. Figure 4.6 on page 165. From variational considerations we shall be able to conclude that K2 .u/ has a definite sign away from a neighborhood of ˙Q, and by integration of (1.9) we will obtain the important no-return (or “one-pass”) theorem, see Chapter 4. This guarantees that solutions which are not trapped by ˙Q can only make one pass near the ground states which then implies that the signs of K0 .u/; K2 .u/ stabilize. The latter then allows one to conclude finite time blowup or global existence in the same way as Payne and Sattinger [114], at least for those solutions which are not trapped by the ground states. Those that are trapped are then shown to lie on the center-stable manifold whence the third alternative in Theorem 1.1. Since very little is known about solutions in the regime E.u; u/ P > J.Q/ C "2 , it seems natural to turn to numerical investigations in order to obtain some idea of the nature of the blowup/global existence dichotomy. Roland Donninger and the second author have conducted such computer experiments at the University of Chicago, see [45]. This work consists of numerical computations of radial solutions to (1.1) with f .u/ D u3 whose data belong to a two-dimensional surface (such as a planar rectangle) in the infinite dimensional phase space H WD H 1 L2 (of course the data are chosen to belong to a fine rectangular grid on that surface). Each solution is then evaluated with regard to blowup/global existence and a dot is placed on the data rectangle if global existence is observed, whereas the dot is left blank otherwise. Figure 1.2 below shows the outcome of such a computation for the data choice 2 2 u.0/; u.0/ P .r/ D Q.r/ C Ae r ; Be r with the horizontal axis being A, and the vertical being B. The central region, from which thin spikes emanate, is the set of data leading to global existence. The dropshaped region contained inside of it is the set P SC , see (1.4), whereas both the region to the far right (which meets P SC at a cusp centered at .Q; 0/ which corresponds to A D B D 0) as well as the region on the far left are P S . The region which appears blank is the one giving finite time blowup (at least numerically), and it contains P S as a subset.
Introduction
9
We refer the reader to [45] for a discussion of the numerical methods, as well as more results and figures. The appearance of the two Payne–Sattinger regions near the point .0; 0/ is reminiscent of the set 2 2 0. This is due to the fact that the energy near Q takes the form of a saddle surface, which in turn follows from the existence of negative spectrum of the linearized operator LC D C 1 3Q2 , see Section 3.1. In fact, there is a codimension-1 plane around .Q; 0/ in H such that locally around that point the energy is positive definite on this plane, whereas it is indefinite on the whole space. An important feature of the central global existence region in Figure 1.2 is the appearance of the boundary: it seems to be a smooth curve. In fact, we will prove in Chapter 3 that near .Q; 0/ in H the boundary is indeed a smooth codimension 1 manifold M with the property that solutions with data on that manifold are global and scatter to Q as t ! 1. In dynamical terms this manifold is precisely the center-stable one, which contains the 1-dimensional stable manifold. Furthermore, M is transverse to the 1-dimensional unstable manifold. The latter manifold is characterized by the property that all solutions starting on it converge to .Q; 0/ as t ! 1; in fact, this convergence is exponential. Moreover, in positive times solutions on the unique unstable manifold grow exponentially up until the time at which they leave a small neighborhood of the equilibrium .Q; 0/.
Figure 1.2. Numerically computed planar section through .Q; 0/
10
Introduction
In their ongoing numerical investigations, Donninger and the second author have found that the boundary of the forward scattering region, i.e., of SC WD f u.0/; u.0/ P j u exists for all positive times and scatters to zerog is more complicated than one might expect; more specifically, numerical experiments suggest that the boundary can display features atypical of a smooth manifold, such as many thin filaments which emerge from it, see [45] for a precise description of this phenomenon. However, the numerics in and of itself does not provide any conclusive evidence at this point that could indicate that the boundary is not a smooth manifold. The results in this book are based on the following papers: in [109], [111] the authors studied the radial as well as nonradial nonlinear cubic Klein–Gordon equation in R3 , and in [110] they treated the radial nonlinear focusing cubic Schrödinger equation in R3 . The so-called graph transform (also known as Hadamard’s method) is adapted to the nonradial NLKG equation in [112] in order to construct invariant manifolds. The point of [112] is to be able to separate the issue of the construction of the invariant manifolds from that of establishing asymptotic properties of those solutions which start on the center-stable manifold. While relatively “soft” spectral information suffices for the former, the latter – at least with current technology – depends on “hard” information such as determining the spectrum in the gap of the linearized operator and understanding the resolvent at the threshold, i.e., the edge of the continuous spectrum. To be specific, the graph transform on which [112] is based requires no knowledge of the spectrum of linearized operator in the gap, whereas the Lyapunov–Perron method in the implementation of [109], [111] does demand such spectral information. On the other hand, the graph transform does not allow any conclusions about the asymptotic stability properties for solutions belonging to the center-stable manifold. See Chapter 3 for a discussion of both methods. The energy critical wave equation in R3 and R5 was studied in [87] by J. Krieger and the authors, whereas the one-dimensional nonlinear Klein–Gordon equation was treated by these same authors in [88]. The book is organized as follows. Chapters 2 through 5 are largely devoted to the cubic focusing Klein–Gordon equation in R3 , mostly under a radial assumption (one exception being the nonradial scattering result from [77] which applies to energies below that of the ground state). The NLS equation appears in Chapter 3 where we establish the existence of center-stable manifolds for the cubic equation in R3 . The cubic NLKG equation in three dimensions turns out to be particularly well-suited for the development of the new results above the ground state energy, since it exhibits both finite propagation speed as well as a lack of symmetries (due to the mass term),
Introduction
11
at least in the radial context. The latter is convenient as it implies that stationary solutions remain static and unchanged under small perturbations (more technically speaking, one does not need any modulation parameters). In Chapter 2, we review the basics of the theory, including local and global well-posedness of the equation, as well as the Payne–Sattinger results characterizing global existence/finite time blowup below the ground state energy. Most of the effort goes into an exposition of the scattering results from [77], which rely on the Kenig–Merle method [84] and the concentration-compactness decomposition as in Bahouri, Gérard [4], and Merle, Vega [105]. The origins of this decomposition go back to the ideas introduced by Lions [98] in the elliptic context, but the adaptation to Hamiltonian evolution equations is highly nontrivial. For pedagogical reasons, we split the scattering proof into the radial and nonradial cases, respectively, with the former of course being simpler. Due to the subcriticality of the problem, the Kenig–Merle method is easier to implement for the cubic NLKG equation than for the critical wave equation as considered in [84]. In order to keep our presentation largely self-contained, Chapter 2 concludes with a presentation of the Strichartz estimates for Klein–Gordon equations. Chapter 3 initiates the discussion of the dynamics for energies which are slightly larger than the ground state energy. More specifically, we first introduce the concept of stable/unstable/center manifolds by means of the work of Bates and Jones [6], followed by a discussion of the work by the second author [122], and Beceanu [9] on the (cubic) NLKG and NLS equations. We also review some of the linear dispersive theory needed in that context. Technically speaking, [6] adapts the Hadamard method (of invariant cones) to the context of ODEs in Banach spaces, which is then applied to the NLKG equation. On the other hand, [122], as well as [9], [87], [90], [91], [109]–[111] use the Lyapunov–Perron method which allows for a detailed description of the asymptotic dynamics of solutions starting from the center-stable manifold. The “cost” of this lies with Strichartz estimates for the linearized operator which in turn rely on a careful spectral analysis of that operator. While the Hadamard method yields less information, it also requires much less information on the linearized evolution. We wish to emphasize, though, that any sophisticated spectral information such as the gap property studied in [41] is needed only for the scattering property of solutions lying in the center-stable manifold but not for the construction of the invariant manifolds themselves. See [112] for more on this matter. In Chapter 4 we present the core of the method developed in [109]. Loosely speaking, one combines the unstable hyperbolic dynamics near the ground states with the variational structure of the stationary energy functional J , as well as of functionals derived from J via the action of symmetries on the equation (for exam-
12
Introduction
ple, the Payne–Sattinger functional K0 or the virial functional K2 , see above). Of particular importance in Chapter 4 is the one-pass theorem which says that there do not exist almost homoclinic orbits connecting f.˙Q; 0/g with itself (more precisely, connecting small balls around .˙Q; 0/). The analysis of Chapters 3 and 4 then leads to a description of the global dynamics analogous to Theorem 1.1 above. This is carried out in Chapter 5, which ends with a summary of the methods developed here. In particular, we obtain the 9-set theorem for the cubic NLKG equation in R3 . We again use the Kenig–Merle method to prove scattering to zero, but the execution of this method relies crucially on the aforementioned one-pass theorem. The final Chapter 6 presents other results which are accessible (but with considerable additional work in most cases) to the ideas set forth in this book. More specifically, we consider the nonradial form of (1.1), the one-dimensional NLKG with even data, the cubic radial NLS equation in R3 , and finally the energy critical wave equation in dimensions 3 and 5. In the critical case, our results are less complete than they are in the subcritical case. In contrast to the previous chapters, the final one is purely expository and presents only select details. It is meant as an introduction to the original research presented in [87], [88], [110], [111]. There are several exercises throughout the text, most of which appear in Chapters 2 and 3. The starred ones are somewhat more involved. Some exercises ask the reader to supply technical details that were omitted from an argument. Those exercises should be considered part of the main body of the text. Exercises marked with a dagger go beyond the core material, and are thus not needed in order to follow the proofs. For the most part, those exercises tend to be quite involved as well. The questions and problems addressed in this monograph can be seen from several perspectives: (i) On the one hand, we consider dispersive Hamiltonian equations which in and of themselves constitute a vast and rapidly developing field. As far as the defocusing case is concerned, the energy subcritical as well as the energy critical equations have been studied extensively for both wave and Schrödinger equations. The earliest treatment of the global existence problem for semilinear wave equation with smooth solutions was conducted by Jörgens [80] for subcritical defocusing nonlinearities in R3 , and the global problem for the critical case (u5 nonlinearity in R3 ) was solved by Struwe [134] radially, and Grillakis [65] nonradially. The corresponding problem for the critical nonlinear Schrödinger equation in R3 was settled by Bourgain [19] radially (see also Grillakis [66]), and Colliander, Keel, Staffilani, Takaoka, Tao [37] nonradially. For more on defocusing equations, see the monographs by Bourgain [18] and Tao [136]. An alternative as well as very general approach to global existence problems was found recently by Kenig and
Introduction
13
Merle [84]. It applies to both defocusing (see [86]) as well as focusing equations, but for the latter [84] only allows energies strictly below the ground state energy. In fact, this entire monograph is devoted to the question of what happens at energies equal to or larger than the ground state energy. On the other hand, Duyckaerts and Merle [48] carried out a comprehensive analysis of the threshold behavior, i.e., at energies equal to that of the ground states for both the energy critical NLS and nonlinear wave equations. It turns out that the special threshold solutions W˙ which they found in this context are of a universal nature; for example, Duyckaerts and Roudenko [49] established their existence for the cubic (and thus energy subcritical) NLS in R3 . Furthermore, in this text we shall identify them as onedimensional stable and unstable manifolds, respectively, associated with the ground states. Even though the literature on focusing equations is generally speaking more sparse than for the defocusing ones, certain classes of equations have been studied in great detail. Especially for the L2 -critical focusing NLS equation substantial progress has been made on the very delicate blowup phenomena exhibited at and near the ground state. The L2 critical equation is special due to its invariance under the pseudo-conformal transformation, see for example [27]. Applying this class of transformations to the ground state Q gives rise to a solution blowing up in finite time, and it is unique with this property at exactly the mass of Q, see Merle [101]. Very recently Merle, Raphaël, and Szeftel [104] proved that these solutions are unstable. Prior to that, and more in the spirit of the present work, Bourgain and Wang [20] studied the conditional stability of the pseudo-conformal blowup on a submanifold of large codimension, and Krieger and the second author [89] established the existence of a codimension 1 submanifold (albeit with no regularity and in a strong topology) for which these solutions are preserved. The conjecture that the pseudo-conformal should be stable under a codimension 1 condition is due to Galina Perelman [116]. A sweeping analysis of the stable blowup regime near the ground state for the L2 -critical case was carried out by Merle and Raphaël [102] in a series of works, preceded by [116] which established the existence of the so-called log log blowup regime. In [103] Merle, Raphaël and Szeftel were able to transfer some of the techniques from the critical equation to the slightly L2 -supercritical one and established stable blowup dynamics near the ground state. The L2 -critical instability of the ground state is algebraic in nature rather than exponential, and thus very far from the considerations in this paper. We emphasize that the hyperbolic dynamics is an essential feature of the theory presented in this text. Moreover, in contrast to the aforementioned works, we do not pursue the deep question of characterizing the type of blowup when it occurs; rather, we only establish its existence via the usual
14
Introduction
convexity obstruction, see [62], [94], [114]. For a characterization of the nature of the blowup, see Merle, Zaag [106], as well as Hamza, Zaag [70]. (ii) On the other hand, one can view this text also from the viewpoint of the vast and diverse body of literature in PDEs which either construct or use (un)stable and center manifolds. Let us merely point out some relevant literature (we caution the reader, however, that the following list is by no means exhaustive). First, Ioss, Vanderbauwhede [78], Gallay [56], and Chen, Hale, Tan [29] construct center(-stable) manifolds in an “abstract” infinite-dimensional setting, as do Bates and Jones [6]. Ball [5] also has a construction which he then applies to the beam equation. The most prevalent method used in the literature appears to be that of Lyapunov–Perron, but [6] follows the so-called Hadamard approach which can sometimes be preferable, cf. [112]. An “abstract” implementation of the Lyapunov–Perron method in infinite dimensions can be found in the seminal papers by Chow, Lu [31] and Chow, Lin, Lu [32] who constructed foliations by invariant manifolds. The reader will find an exposition of both types of constructions in Chapter 3. Bates, Lu, and Zeng [7] developed the theory of center manifolds associated with invariant manifolds which are larger than isolated equilibria. See also Chow, Liu, Yi [33]. Applications of center manifolds to PDEs also abound, especially in the dissipative setting. For example, see Carr, Pego [26], Eckmann, Wayne [50], Wayne [139], Collet, Eckmann [36], Bianchini, Bressan [14], and Beck, Wayne [10], just to name a few. Less work seems to have been done on invariant manifolds for conservative equations, perhaps due to the fact that the center manifold becomes dominant in that case. Promislow [118] applies invariant manifold ideas to a dispersive equation, as do Comech, Cuccagna, Pelinovsky [38], and Weder [142]. Tsai and Yau [137] obtained conditional stability results for excited states of a nonlinear Schrödinger equation with a potential. Soffer and Weinstein [129] obtain the following long-term asymptotic result for the same type of equations studied by Tsai and Yau: either the solution approaches the ground state (the generic case), or an excited state (nongenerically). There have also been many applications of invariant manifolds to the equations of fluid dynamics. For a recent review see Wayne [140], as well as Constantin, Foias [39], Constantin, Foias, Nikolaenko, Temam [40], and Gallay, Wayne [57]. Li, McLaughlin, Shatah, and Wiggins [95] constructed homoclinic orbits for a forced-dissipative perturbation of the completely integrable cubic NLS equation on the one-dimensional torus. Their construction involved, amongst many other elements, invariant manifolds. For more background, as well as a discussion of the relevance of this work towards establishing chaotic motion for NLS, see the book by Li, Wiggins [96]. For an expert summary of [95], and an illuminating overview of much other work in this area see the introduction of the Memoirs article by Bates,
Introduction
15
Lu, and Zeng [7]. For further work on homoclinic orbits via invariant manifolds see Zeng [146] and Shatah, Zeng [127]. However, we emphasize that all aforementioned references are somewhat different from the subject matter of this monograph, as they do not exhibit the role of a center-stable manifold as the locus of transition from data leading to blowup versus data leading to global existence and scattering (in forward time, say).
2
The Klein–Gordon equation below the ground state energy
In this chapter we develop some basic tools needed for the study of the nonlinear focusing cubic Klein–Gordon (NLKG) equation in R1t R3x : (2.1) u C u u3 D 0; uŒ0 D u.0/; u.0/ P 2 H D H 1 L2 : All solutions of this equation will be real-valued. Throughout this text, we use uP and @ t u interchangeably. We begin with this specific equation for two reasons: on the one hand, it displays all the relevant features which we wish to discuss in this book; and, on the other hand, it is technically the simplest, not just due to the low subcritical power, but also since the only symmetry groups which preserve realvalued solutions of (2.1) are translations and the change of sign u 7! u. We shall see later that this rigidity considerably simplifies the stability analysis of the ground state solution, especially in the radial case. On the other hand, we shall also indicate in the final chapter that the theory as developed here extends to NLKG with more general powers and in other dimensions (especially in dimension 1), as well as to the more challenging nonlinear Schrödinger equation (NLS). We shall begin the rigorous analysis of equation (2.1) with the so-called local well-posedness theory. This refers to the fact that it admits solutions to the Cauchy problem up to some finite time (the direction of time here does not play a role since (2.1) is invariant under the time-reflection symmetry). While smooth decaying data are perhaps historically of greatest interest, we will limit our assumptions on the data to the energy class, i.e., u.0/; @ t u.0/ 2 H 1 L2 .R3 /. “Energy” here refers to the conserved quantity Z h 1 4i 1 .jruj2 C juj2 C juj P 2/ juj dx E.E u/ WD (2.2) 4 R3 2 which requires uE WD .u; @ t u/ 2 H 1 L2 in order to be well-defined. Note also that we are dealing with the energy-subcritical regime. This means that the nonlinearity in (2.2) lies below the critical one, namely u5 which is defined by the critical Sobolev embedding HP 1 .R3 / L6 .R3 /. In the following section, we prove that (2.1) admits local-in-time energy solutions which are also unique. Care needs to be taken with the definition of “solution”, and we adopt the Duhamel sense for this purpose (which is referred to as the class of
18
2 The Klein–Gordon equation below the ground state energy
“strong” solutions in [126]). These strong solutions have several natural properties, such as conservation of the energy in (2.2), preservation of regularity (if they are more regular than H 1 L2 initially, say H k H k 1 with some integer k > 1, then the solution retains this property at later times). Moreover, the solutions are guaranteed to exist for some amount of time which only depends on the size of the data in H 1 L2 , and small data lead to global-in-time solutions. In the latter case, one also has that kukL3 .RIL6x .R3 // is finite and in fact small. This latter norm is natural t from the point of view of energy estimates since they place the nonlinearity u3 in L1t L2x . The L3t L6x norm is an example of what one calls Strichartz norms, and we shall devote Section 2.5 to them. The significance of L3t L6x lies with the fact that it gives a criterion for finite time blowup, as well as scattering, see Lemma 2.2 for the precise statements. “Scattering”, or more precisely, “scattering to zero”, means that the solution becomes asymptotically free. In other words, uE D vE C oH 1 L2 .1/ as t ! C1 where . C 1/v D 0 and vE 2 H 1 L2 . As solutions of a wave equation, strong solutions of (2.1) obey the principle of finite propagation speed. This means that if uE D 0 on some ball fjx x0 j < Rg, then u.t; x/ vanishes on the cone ˚ jx
x0 j < R
t; 0 < t < min.T; R/
where .0; T / is the time-interval of existence of u. Using this principle, a standard construct gives smooth, compactly supported data which lead to finite time blowup for (2.1). Indeed, one first checks by direct verification that for every T > 0 there is a space-independent function 'T .t/
p
2.T
t/
1
(2.3)
in the sense of asymptotic equality as t ! T (without the mass term u in (2.1) the right-hand side here is an exact solution, but the mass term contributes lower orders). But then, by finite propagation speed, we may multiply 'ET .0/ by a cut-off function which is equal to 1 on a ball of radius T . This leads to smooth, compactly supported data which blowup either at time t D T , or before. It is now natural to ask how to determine from the data whether they lead to finite time blowup or global existence. In other words, is there a “criterion” which allows one to decide this dichotomy? Generally speaking, this is perhaps a bit too optimistic. However, as found by Payne and Sattinger [114] in 1975, there is such a criterion for energies below that of the ground state. The latter refers to the unique radial, positive, and stationary energy solution of (2.1), denoted by Q. We shall
2 The Klein–Gordon equation below the ground state energy
19
recall some well-known variational arguments in Section 2.2 below which guarantee the existence of Q. For the uniqueness, we refer the reader to [35]. What Payne and Sattinger found is that in the regime E.E u/ < E.Q; 0/, blowup vs. global existence is decided by the sign of the functional Z K.u/ WD .jruj2 C u2 juj4 / dx : R3
More precisely, the sign of K (where K D 0 is counted positive) is invariant under the flow of (2.1) and K positive guarantees global existence (in both time directions), whereas K negative implies finite time blowup (again in both time directions). We shall prove all of this in Section 2.3. The Payne–Sattinger criterion leaves open the possibility that u exists globally for K positive, but does not scatter to zero (such as any nonzero stationary solution). In Section 2.4 we follow Ibrahim, Masmoudi, and Nakanishi [77] to establish that solutions with K.u/ 0 (and of course E.E u/ < E.Q; 0/) do scatter to zero. The argument is based on the Kenig– Merle method [84]. Loosely speaking, the blue-print of this method rests on four pillars: (i) a small data scattering result; (ii) a concentration-compactness decomposition; (iii) a suitable perturbation theory; (iv) a rigidity argument. Part (i) is self-explanatory, and is covered by the basic existence result, see Lemma 2.2 below. Part (ii) is related to the concentration-compactness theory of Lions [98]. The specific decomposition which we use here was pioneered by Bahouri, Gérard [4], and Merle, Vega [105] for critical equations. It refers to the fact that any sequence of data .un ; uP n / which is bounded in H 1 L2 (after passing to a subsequence) can be written as the super-position of linear Klein–Gordon waves Vj;n evaluated at t D 0 with 1 j J , and a remainder nJ which is again a linear KG-wave evaluated at zero. The dependence of Vj;n on n is very specific, and is given by translation (in both time and space) of asymptotic profiles Vj . As n ! 1, it is a crucial property that the Vj;n move arbitrarily far away from each other, guaranteeing that these linear waves interact only very weakly. The remainder
nJ , on the other hand, becomes small as J ! 1 uniformly in large n, albeit not in the energy sense, but only in a weaker Strichartz sense. We emphasize that the subcritical concentration-compactness results used here are much simpler than in the critical cases of [4], [105]. In Part (iii) it is then important that the perturbation theory allows for such small perturbations (and not require that the perturbations have small energy, say). Part (iv) refers to an argument which excludes a nonzero solution whose trajectory (at least after modding out the symmetries, which here only consist
20
2 The Klein–Gordon equation below the ground state energy
of translations) is precompact in H 1 L2 . An example of such an object is given by any stationary solution, but the latter are excluded by our assumption that the energy be strictly less than E.Q; 0/. Part (iv) is the most “nonlinear” ingredient in the Kenig–Merle method, and requires the use of the virial identity, see (2.64) below. For reasons of exposition, we split the argument in Section 2.4 into the radial and nonradial cases. The latter is somewhat more involved than the former as one now also needs to control a translation parameter. We close this chapter with a derivation of Strichartz estimates for the Klein– Gordon equation in Section 2.5. For the endpoint estimate, which involves L2t , we apply the Keel, Tao theory from [82].
2.1
Basic existence theory
We begin with the following basic existence and uniqueness result for strong solutions of (2.1). First, we define what we mean by a “strong solution”. Definition 2.1. We say that u is a strong solution to equation (2.1) on the timeinterval Œ0; T / if u 2 C.Œ0; T /I H 1 / C 1 .Œ0; T /I L2 /
(2.4)
and if u satisfies the Duhamel formula sin.t!/ u1 C u.t/ D cos.t!/u0 C ! p where ! WD hri WD 1 C jr 2 j.
t
Z 0
sin..t
s/!/ !
u3 .s/ ds
(2.5)
By the imbedding H 1 ,! L6 .R3 /, the integral on the right-hand side is welldefined as it converges absolutely in L2 . In what follows, we write uŒ0 D uE .0/ D .u.0/; u.0//. P An immediate consequence of (2.5) is the basic energy estimate, where we write F .s/ instead of u3 .s/: Z t (2.6) kE u.t/kH . kuŒ0kH C kF .s/k2 ds: 0
The choice F D u3 leads to the norm kukL3 L6x which is an example of a Strichartz t norm, see Section 2.5.
21
2.1 Basic existence theory
Lemma 2.2. For any uŒ0 2 H there exists a unique strong solution u 2 C.Œ0; T /I H 1 / C 1 .Œ0; T /I L2 / for some T T0 > 0 where T0 only depends only on kuŒ0kH . The solution depends continuously on the data. ı If uŒ0 2 H 2 H 1 , then uE .t/ 2 H 2 H 1 for all 0 t < T where T is any time up to which a strong solution exists. ı The energy of such a solution, i.e., Z 1 1 4 1 2 1 juj P C jruj2 C juj2 juj dx E.u; u/ P D 2 2 4 R3 2 does not depend on time. ı If kuŒ0kH 1, then the solution exists globally in time and kukL3 .Œ0;1/;L6 .R3 // . kuŒ0kH : ı If T > 0 is the maximal forward time of existence, then T < 1 implies that kukL3 .Œ0;T /;L6 .R3 // D 1. ı If T D 1 and kukL3 .Œ0;T /;L6 .R3 // < 1, then u scatters in the following sense: there exist .uQ 0 ; uQ 1 / 2 H such that with v.t/ D S0 .t/.uQ 0 ; uQ 1 / one has u.t/; u.t/ P D v.t/; v.t/ P C oH .1/ t ! 1 where S0 .t/ is the free Klein–Gordon evolution. On the other hand, if u scatters then kukL3 .Œ0;1/;L6 .R3 // < 1. ı One has the finite speed of propagation: if uE D 0 on fjx x0 j < Rg, then u.t; x/ vanishes on the cone fjx x0 j < R t; 0 < t < min.T ; R/g. Proof. The proof is a standard contraction argument in suitable spaces. For the local existence, it suffices to use the energy spaces via the Sobolev imbedding H 1 .R3 / ,! L6 .R3 /. To be more precise, (2.6) implies kuŒtkH . kuŒ0kH C ku3 kL1 .Œ0;t/IL2 .R3 // 3 . kuŒ0kH C T kukL 1 .Œ0;t/IL6 .R3 //
. kuŒ0kH C
3 T kukL 1 .Œ0;t/IH 1 .R3 //
(2.7) ;
for any 0 t < T . We set up a contraction scheme in X WD L1 .Œ0; T /I H 1 / L1 .Œ0; T /I L2 /. Define a map ˚ on X by the Duhamel formula (2.5), i.e., Z t sin .t s/! 3 sin.t!/ u1 C u .s/ ds: ˚.u/.t/ D cos.t!/u0 C ! ! 0
22
2 The Klein–Gordon equation below the ground state energy
Denote R WD k.u0 ; u1 /kH 1 L2 . Then by the energy estimate
˚.u/ . R C T u 3 X X and for sufficiently short times T the ball of radius 2R is mapped into itself. Using the energy estimate once more one can show that ˚ is also a contraction,
˚.u/
1 ˚.u/ Q X u 2
uQ X :
Banach’s fixed point theorem then yields a unique w 2 BX .2R/ with ˚.w/ D w. With the same argument one shows that indeed the solution depends continuously on the data. This shows local well-posedness of (2.1), as well as uniqueness and continuous dependence on the data (by a small variation of the contraction argument). Differentiating the equation yields1 Z t sin .t s/! 2 sin.t!/ ru.t/ D cos.t!/ru0 C ru1 C 3 u .s/ru.s/ ds : ! ! 0 (2.8) Placing this in the energy space yields kr uE kL1 ..0;T /IH/ . k.u0 ; u1 /kH 2 H 1 C ku2 rukL1 ..0;T /IL2 / 2 E kL1 ..0;T /IH/ . k.u0 ; u1 /kH 2 H 1 C kukL 1 ..0;T /IL6 / kr u
whence by Gronwall’s inequality kE ukL1
.0;T /IH 2 H 1
on any interval .0; T / for which kuk2 1 L
<1 < 1.
.0;T /IL6 .R3 /
For the finite propagation speed, one checks that div t;x
1 2
jruj2 C juj2 C juj P 2 ; uru P D uP u3
(2.9)
where div t;x is the space-time divergence. Applying the divergence theorem to (2.9) over a truncated cone of the form fjx 1
x0 j < R
t; 0 < t < min.t0 ; R/g;
0 < t0 < T
Strictly speaking, one needs to solve (2.8) with v in place of ru and then argue that in fact v D ru.
23
2.1 Basic existence theory
yields the estimate Z jruj2 C juj2 C juj P 2 .t0 ; x/ dx Œjx x0 j
t0
Z C 0
Z
jx x0 j
ˇ 3 ˇ ˇuP u .t; x/ˇ dxdt :
Applying Cauchy–Schwarz as well as Sobolev imbedding to the nonlinear term on the right-hand side implies Z t e.t/ . e.0/ C M 2 .t/ e.s/ ds 0
where
Z
jruj2 C juj2 C juj P 2 .t; x/ dx
e.t/ WD Œjx x0 j
and M.t/ D kukL1 ..0;t/IH 1 / . If e.0/ D 0, then by Grownwall’s inequality e.t/ D 0 on 0 < t < T . Conservation of energy is first shown for sufficiently regular solutions, in fact those with data in H 2 H 1 . Since that space is dense in H, this will suffice by choosing an approximating sequence of solutions in H 2 H 1 which converges to the original H-solution in the norm of (2.4). Taking such a more regular solution in u 2 C.Œ0; T /I H 2 / C 1 .Œ0; T /I H 1 / one multiplies equation (2.1) with @ t u and integrates over space. Assume furthermore that u has compact support. By the finite speed of propagation this property is preserved. Integration by parts yields Z 1 2 1 1 1 4 d juj P C jruj2 C juj2 juj dx D 0 : dt 2 2 2 4 As already mentioned, the result extends to solutions u 2 C Œ0; T /I H 1 C 1 Œ0; T /I L2 by approximation. If kuŒ0kH 1, energy estimates alone do not suffices to establish global existence. However, by the Strichartz estimates of Section 2.5, see in particular Lemma 2.43, one obtains for any I D Œ0; T /, 3 kukL3 .I IL6 / . kuŒ0kH C kukL 3 .I IL6 / :
24
2 The Klein–Gordon equation below the ground state energy
By a continuity argument one now sees that one can take I D Œ0; 1/ and we conclude that kukL3 .Œ0;1/IL6 / < 1 as desired. If T < 1, then from (2.7) one must have k.u; u/k P L1 .Œ0;T /IH/ D 1 which implies that kukL3 .Œ0;T /;L6 .R3 // D 1. If the norm k.u; u/k P L1 .Œ0;T /IH/ does not blow up we could use short-time existence to extend the solution. For the scattering statement, set U WD uuP and rewrite (2.1) in matrix form UP D JH U C N.U /; where J D
0 1 ; 1 0
C1 0 ; 0 1
H D
0 N.U / D u3
!
whence U.t/ D e
tJH
t
Z
e .t
U.0/ C
s/JH
N U.s/ ds
0
which we want to write as e tJH UQ .0/ C oH .1/. Applying e tJH and sending t ! 1 yields Z 1 (2.10) UQ .0/ D U.0/ C e sJH N U.s/ ds : 0
Note that e tJH D
! sin.thri/ cos.thri/ hri sin.thri/hri cos.t hri/
and t
Z
e 0
sJH
N U.s/ ds D
t
Z 0
sin.shri/ 3 u .s/ ds : hri
The integral is absolutely convergent in H 1 by our assumption kukL3 .Œ0;1/IL6 .R3 // < 1. Thus UQ is well-defined as an element of H. One checks that it exhibits the desired properties.
2.1 Basic existence theory
25
Exercise 2.3. Show that S0 .t/ WD e tJH is unitary in H relative to the natural metric (2.11) k.u0 ; u1 /k2H WD khriu0 k22 C ku1 k22 : In particular, show that S0 .t/ D S0 . t/ where S0 refers to the adjoint of S0 in H relative to the scalar product induced by that metric. We remark that (2.11) defines twice the free energy. Therefore, the previous exercise amounts to nothing other than the conservation of the free energy for the free Klein–Gordon equation. We shall henceforth assume that the norm in H is the one from (2.11). Let us now precisely state the blow-up phenomenon alluded to in the introduction to this chapter. Lemma 2.4. There exist smooth compactly supported data which lead to finite time blowup for (2.1). Proof. We seek solutions of the form u.t; x/ D w.t/. This leads to the ODE w.t/ R C w.t/ D w 3 .t/ :
(2.12)
If the w term an exact solution would be p on the left-hand side were not present, then p w0 .t/ D 2.T ˙ t/ 1 for any T . We now set w0 .t/ D 2t 1 and seek a solution of (2.12) of the form w.t/ D w0 .t/.1 C .t//
(2.13)
with .t/ ! 0 as t ! 0. Inserting the ansatz (2.13) into (2.12) leads to w0 2 R C 2
wP 0 P C w0 2 .1 C / D 2 C 32 C 3 w03
or, in other words, 1 2 t .t/ R 2
t .t/ P C
1 2
t2
2 .t/ D
1 2 t C 32 .t/ C 3 .t/ : 2
(2.14)
t D 0 is a regular singular point2 of the differential operator on the left-hand side, and the roots of the indicial equation ˇ 2 3ˇ 4 D 0 are ˇ D 1; 4. Therefore, by 2
Any reader unfamiliar with Frobenius’ theory of ODEs with analytic coefficients can safely skip the remaining argument, it will not be needed again. Moreover, we will give an alternative, albeit nonconstructive, proof of the existence of blowup solutions in Corollary 2.13 below.
26
2 The Klein–Gordon equation below the ground state energy
Frobenius theory the general solutions of the homogeneous form of equation (2.14), viz. 1 2 t .t/ R t .t/ P 2.t/ D 0 2 are 1 1 X X 0 .t/ D t 4 an t n ; 1 .t/ D t 1 bm t m C c0 log t 0 .t/ nD0
mD0
where the series are convergent near t D 0 and c0 is a fixed nonzero constant. By variations of constants, the general solution to (2.14) which remains bounded near t D 0 is Z t .t/ D a0 .t/ C 0 .t/1 .s/ 0 .s/1 .t/ !.s/ f .s/ ds: 0 (2.15) 1 2 1 2 2 3 t .t/ C 3 .t/ C .t/; f0 .t/ D t f .t/ WD f0 .t/ 2 2 where a is arbitrary and ! is determined by the condition that 1 2 t P 0 .t/1 .t/ 2 In particular, !.s/ D O.s
4
0 .t/P 1 .t/ !.t/ D 1;
8 t > 0:
/ as s ! 0. A contraction argument in the space
X WD f 2 C.0; t0 / j sup t 0
2
j.t/j M g
with suitable t0 and M (depending on a) now shows that (2.15) has a unique solution in X. In conclusion, (2.12) admits a solution (in fact, infinitely many) of the form w.t/ D w0 .t/ 1 C O.t 2 / as t ! 0. To prove the lemma one now multiplies w.T t/ with T > 0 fixed (where w./ is the one we just constructed) by a smooth cutoff function to a ball of radius R > T and which is constant equal to 1 on this ball. By the property of finite propagation speed we obtain a solution with smooth, compactly supported data and which blows up at time t D T or before. The blowup solutions constructed in the previous lemma are referred to as ODEtype blowup solutions. It is a very delicate question to decide if all blowup solutions need to exhibit this kind of behavior; for a precise formulation of this question and an affirmative answer for many nonlinearities, see Merle, Zaag [106] and Hamza, Zaag [70]. We will not be addressing this question at all in this text. Of central importance to the entire book is the following forward scattering set, which simply consists of all data which lead to global existence in forward time and which scatter to zero.
27
2.1 Basic existence theory
Definition 2.5. The forward scattering set is the set ˚ SC WD .u0 ; u1 / 2 H j u.t/ WD S.t/.u0 ; u1 / exists for all times and scatters to zero, i.e., kukL3 .Œ0;1/IL6 / < 1
(2.16)
where S.t/ is the nonlinear evolution of equation (2.1). By means of Lemma 2.2 we can establish some basic properties of the forward scattering set. Note how the finiteness of L3t ..0; 1/I L6x / gives us strong control to perform simple perturbative arguments on the solutions from this set. Lemma 2.6. The forward scattering set satisfies the following properties. ı SC Bı .0/, a small ball in H, ı SC ¤ H, ı SC is unbounded, ı SC is an open set in H, ı SC is path-connected. Proof. By Lemma 2.2, SC contains a neighborhood of zero. That SC ¤ H follows from the fact that data with negative energy lead to finite time blowup, see Corollary 2.13 below. Alternatively, one can use the explicit ODE blowup construction from Lemma 2.4 above. That SC is open follows from a basic perturbative argument. Fix a given solution u D S.u0 ; u1 / with .u0 ; u1 / 2 SC . Let v WD u C w a solution with a perturbation w such that kwŒ0kH < ". Then w satisfies w C w D 3u2 w C 3uw 2 C w 3 DW F : For the Duhamel formula associated with the non-linearity F one obtains an energy inequality T
Z kwkL1 .Œ0;T /IH 1 /\L3 .Œ0;T /IL6 / . " C
0
kF .s/kL2 ds
3 . " C ku2 wkL1 L2 C kuw 2 kL1 L2 C kwkL 3 L6 : 2 2 The middle terms can be estimated by kukL 3 L6 kwkL3 L6 and kukL3 L6 kwkL3 L6 by applying Hölder’s inequality twice. A contraction scheme for infinite time T D 1 fails as kukL3 L6 might be too large. Therefore one partitions Œ0; 1/ D SJ j D1 Œtj 1 ; tj / where t0 D 0 < t1 < t2 < < tJ D 1 and J is a positive finite
28
2 The Klein–Gordon equation below the ground state energy
number such that kukL3 .Œtj
1 ;tj /IL
6/
< "0
small enough for convergence of the contraction scheme on each time interval. Then for any kvŒ0 uŒ0kH < ı0 .J / there exists a strong solution v of (2.1) with kvkL3 .Œ0;1/IL6 / < 1. So SC is open. Finally, for the path connectivity, let u and v be two solutions with data in SC . Let .u1 ; uP 1 / and .v1 ; vP 1 / be their respective scattering data in H. Connect these data by a continuous path . s ; Ps / in H, where 0 s 1. Associate to this curve solutions ws .t/ with 0 Ts < t < 1 of (2.1) such that Z 1 sin .t t 0 /! ws .t 0 /3 dt 0 : ws .t/ D S0 .t/. s ; Ps / ! t Then Ts can be chosen to depend continuously on s and set T D maxs Ts . By construction, kws kL3 .ŒT C1;1/IL6 / < 1 for each 0 s 1. One can now connect .u1 ; uP 1 / and .v1 ; vP 1 / entirely with SC by first following u.t/ up to time T C 1, then continuing along .ws .T C1/; wP s .T C1// for 0 s 1, and finally by moving along v.t/ from time T C 1 to time 0. To see that @SC is unbounded, one uses the existence of the wave operators as in the previous paragraph: to every .v0 ; v1 / 2 H 1 L2 and associated free Klein– Gordon solution .v; v/ P there exists a solution .u; u/ P of the nonlinear equation (2.1) on some interval ŒT; 1/ which approaches the free solution .v.t/; v.t// P as t ! 1. This existence is proved by a standard contraction argument. But then it follows that the energy E.u; u/ P equals the free energy of .v; v/, P whence the former as a function on SC is unbounded. In particular, SC is unbounded. Several natural questions now pose themselves: (i) What does @SC look like, is it a smooth manifold, or can it be very rough? (ii) If @SC is a smooth manifold, does it separate a region of finite time blowup from one of global existence, at least locally? (iii) What is the dynamical behavior of solutions starting on @SC ? Are there any special solutions with data on @SC ? The main goal of this book is to present some partial answers to these questions, following [109], see also [87], [88], [110], [111]. For a numerical investigation of the set SC and its boundary, see [45]. It is unclear at this point whether or not @SC remains a smooth manifold at all energies. There is some evidence in [45] that this boundary can be more complex than one might expect. In this chapter, we lay the groundwork for these investigations, by considering the easier case of solutions with energies below that of the ground state. The latter
2.2 Stationary solutions, ground state
29
refers to a special stationary solution of (2.1), see the following section. This chapter is largely based on [77], which in turn relies on the classical work of Payne and Sattinger [114], Shatah [125], and the concentration-compactness method introduced by Bahouri and Gérard [4], Merle and Vega [105], and especially Kenig and Merle [84].
2.2
Stationary solutions, ground state
A stationary, i.e., time-independent solution u.t; x/ D '.x/ of (2.1) is characterized as an H 1 weak solution of the elliptic problem in R3 ' C ' D ' 3 :
(2.17)
By standard elliptic theory, any such solution is a classical one, i.e., C 1 .R3 /. Moreover, the minimization problem ˚ 2 1 inf k'kH 1 j ' 2 H ; k'k4 D 1
(2.18)
has a radial positive solution '1 , [131], which decays exponentially and so that ' D '1 satisfies (2.17) for some > 0. Coffman [35] showed that a radial positive solution of (2.17) in H 1 is unique, see also McLeod [100]. Moreover, by [60] any positive solution of (2.17) is radial around some origin. The unique radial solution in H 1 of (2.17) is called the ground state and denoted by Q. This names derives from the fact that amongst all solutions of (2.17) the positive one minimizes the stationary energy Z 1 1 4 1 (2.19) jr'j2 C j'j2 j'j dx : J.'/ WD 2 2 4 This will follow from Lemma 2.9 below. Furthermore, (2.17) has infinitely many smooth nodal solutions, i.e., solutions which are radial and change their sign a finite number of times; in fact, for any positive integer there is one whose sign changes as often as that number, see [131], [12]. Exercise 2.7. (a) Establish the estimate j'.x/j C jxj
1
k'kH 1 ;
8 jxj 1
1 for all ' 2 Hrad .R3 /. Generalize to other dimensions, see Strauss [131].
(2.20)
30
2 The Klein–Gordon equation below the ground state energy
1 (b) Use (a) to show that the imbedding Hrad .Rd / ,! Lp .Rd / is compact for 2 < 2d p < 2 D d 2 , d 3. 1 (c) Show that the minimization problem (2.18), restricted to the radial class Hrad , has a positive solution '1 , which is a weak solution of (2.17) up to a scaling factor. Generalize to other powers in the nonlinearity as well as to all dimensions. (d) In dimension d D 1, the ground state can be stated explicitly in terms of elementary functions. Find this explicit formula, and show that apart from the ground states ˙Q there are no other nonzero solutions to the equation
' 00 C ' D j'jp
1
';
p>1
(2.21)
which belong to H 1 .R/. Hint: Multiply (2.21) by ' 0 and integrate. Then consider the phase portrait of (2.21) and find ˙Q as the two homoclinic orbits associated with .0; 0/. We will mainly be concerned with the ground state Q, but for now allow general stationary solutions in our analysis. Define Z K0 .'/ WD jr'j2 C j'j2 j'j4 dx : (2.22) The reason for the index 0 on K will become clearer later. Note that (2.17) is the same as J 0 .'/ D 0 (the Fréchet derivative), and that K0 .'/ D hJ 0 .'/j'i D @ jD0 J.e '/ which implies that K0 .'/ D 0 for any solution of (2.17). More generally, we can consider smooth 1-parameter groups of symmetries S./ and set ˇ ˇ J.S./'/ D hJ 0 .'/jS 0 .0/'i : KS .'/ WD @ ˇ D0
Of particular importance later on will be the L2 -unitary family .S./'/.x/ D 3 e 2 '.e x/ with S 0 .0/' D 12 .xr C rx/, cf. (2.64). Lemma 2.8. Every weak solution of (2.17) satisfies J 0 .'/ D 0 and K0 .'/ D 0. Moreover, for any ' 2 H 1 n f0g the function j' ./ WD J.e '/ is continuously differentiable for all 2 R, is strictly convex near D 1, and satisfies j'0 . / D 0 for a unique . Further, j' ./ decreases strictly for > with j' ./ ! 1 as ! 1.
31
2.2 Stationary solutions, ground state
j' ./
Figure 2.1. Payne–Sattinger well
Proof. With D e , one has j' ./ D a2 b4 where a; b > 0 are some constants depending on '. Clearly, j'0 ./ D 2a2 4b4 which has a unique zero . The function j' ./ has the shape of a potential well, see Figure 2.1. This feature lies at the heart of the seminal work of Payne and Sattinger [114]. To be more specific, one seeks to trap the solution in the well to the left of , which should guarantee global existence. However, in order to do this we need to find the lowest pass over the ridge defined by all possible graphs j' ./ as ' ranges over H 1 n f0g with the normalization condition D 0. ˇ In other words, we normalize @ J.e '/ˇD0 D K0 .'/ D 0. Without this requirement a comparison between different maxima defined by j' ./ is meaningless. The height of the lowest pass over the ridge is therefore given by h0 WD inffJ.'/ j ' 2 H 1 n f0g; K0 .'/ D 0g :
(2.23)
It turns out to be characterized by the ground state solution Q. That this might be expected can be seen from the Euler–Lagrange equation which a minimizer '0 ¤ 0 of (2.23) needs to satisfy, viz. J 0 .'0 / D m K00 .'0 /
32
2 The Klein–Gordon equation below the ground state energy
which implies 0 D K0 .'0 / D hJ 0 .'0 /j'0 i D mhK 0 .'0 /j'0 i D 2m.K.'0 /
k'0 k44 / D
2mk'0 k44
and thus also m D 0. But this means that J 0 .'0 / D 0, or in other words, that '0 satisfies (2.17). If '0 0, then in fact '0 is the ground state (at least up to translations). In the following lemma we establish the existence of such a minimizer and thus prove h0 D J.Q/. Lemma 2.9. Define a positive functional G0 by G0 .'/ WD J.'/
1 2 1 K0 .'/ D ' H 1 : 4 4
(2.24)
We have h0 D J.Q/ D inffJ.'/ j ' 2 H 1 n f0g; K0 .'/ D 0g D inffG0 .'/ j ' 2 H 1 n f0g; K0 .'/ 0g;
(2.25)
and these infima are achieved uniquely by the ground states ˙Q, up to translations. Proof. If K0 .'/ < 0 then K0 .e '/ D 0 for some < 0, whereas G0 .e '/ < G0 .'/. Since moreover K0 .'/ D 0 implies J.'/ D G0 .'/, the two infima are equal. Since J 0 .Q/ D 0 implies K0 .Q/ D 0, the infima are no larger than J.Q/. 1 To obtain a minimizer, let f'n gn1 Hrad n f0g be a minimizing sequence such that (the Schwarz symmetrization allows us to restrict them to the radial functions, see Exercise 2.10) K0 .'n / D 0;
J.'n / ! h0 :
Since G0 .'n / D J.'n / is bounded, 'n is bounded in H 1 . After extraction of a subsequence, it converges weakly to some '1 in H 1 , and in the strong sense in L4 , by the radial symmetry. Thus K0 .'1 / 0, J.'1 / h0 and G0 .'1 / h0 . If '1 D 0 then the strong convergence in L4 together with K0 .'n / D 0 implies that 'n ! 0 strongly in H 1 . Notice that K0 .'/ > 0 for 0 < k'kH 1 1, due to the interpolation inequality 4 3 k'kL 4 . k'kL2 kr'kL2 :
(2.26)
Hence K0 .'n / > 0 for large n, a contradiction. Thus we obtain a nonzero minimizer '1 , and G0 .'1 / D h0 . If K0 .'1 / < 0, then there is < 0 so that
33
2.2 Stationary solutions, ground state
K0 .e '1 / D 0. This implies that h0 G0 .e '1 / D e 2 G0 .'1 / < h0 which is a contradiction. Therefore K0 .'1 / D 0, which implies that 'n ! '1 strongly in H 1 , as well as J.'1 / D h0 . Next, one considers the Euler–Lagrange equation associated with the minimizer '1 . The constrained minimization implies that for some Lagrange multiplier 2 R, J 0 .'1 / D K00 .'1 /;
K0 .'1 / D 0 :
Multiplying the Euler–Lagrange equation with '1 and integrating by parts yields 0 D K0 .'1 / D hK00 .'1 /j'1 i D 2 K0 .'1 /
4 k'1 kL 4
which implies D 0, whence J 0 .'1 / D 0. Furthermore, we may assume that '1 0 by replacing 'n with j'n j. By elliptic regularity, '1 is smooth and nonnegative, whereas the maximum principle guarantees that '1 > 0. So '1 is indeed the radial ground state, i.e., '1 D Q. The only other case is '1 < 0 which then forces '1 D Q. For the uniqueness in the nonradial case, one can either use [60] to show that all positive solutions of (2.17) are radial modulo a translation, or alternatively one can use the following exercise. Figure 2.2 gives a schematic depiction of the region J.u/ < J.Q/. As we shall see later, see Section 3.1, the cuspidal appearance of this region around ˙Q derives
J WD J.u/ > J.Q/ DW J0
KD0
KD0
K<0 .J < J0 /
J < J0
J < J0 K0
Q
K<0
Q
Figure 2.2. The splitting of E.E u/ < J.Q/ by the sign of K D K0
34
2 The Klein–Gordon equation below the ground state energy
from the fact that the energy is a saddle surface near these points; more precisely, the quadratic part of the energy exhibits exactly one negative mode. Figure 2.2 relates to Lemma 2.9 as follows: the level-sets fu 2 H 1 j J.u/ D hg with h < J.Q/ do not intersect fK D 0g. And the latter intersects the level set fJ.u/ D J.Q/g in precisely two points, namely ˙Q (under a radial assumption). The reader will find detailed calculations substantiating all these claims relating to Figure 2.2 in the first section of the following chapter. The following exercise is marked with a dagger, since it is admissible on first reading to restrict to radial functions. Exercise 2.10. Provide the details of the symmetrization argument in the previous proof. You will need to use that the symmetric nonincreasing rearrangement does not increase the kinetic energy; in fact, symmetrization strictly decreases krf k2 unless f is a translate of its nonincreasing rearrangement, see Lemma 7.17 in [97]. Moreover, remove the radial assumption in Exercise 2.7 (c). A nice feature of the nonincreasing rearrangement arguments in the calculus of variations is that they automatically prove that the ground state is decreasing in the radial variable. Moreover, they apply to general nonlinearities f .Q/. Remark 2.11. An alternative approach to the existence of Q in the nonradial case, which does not rely on symmetrization, is based on the concentration-compactness method; see P. Raphaël’s lecture notes [119], as well as Tao’s book [136].
2.3
The Payne–Sattinger criterion, regions P S˙
As a corollary we now obtain the following important invariance result in the region E < J.Q/, see [114], [125]. It lies at the heart of the Payne–Sattinger theorem on the global existence/finite time blowup dichotomy for solutions of energies below the ground-state energy. Corollary 2.12. The two regions P SC WD f.u0 ; u1 / 2 H j E.u0 ; u1 / < J.Q/; K0 .u0 / 0g; P S WD f.u0 ; u1 / 2 H j E.u0 ; u1 / < J.Q/; K0 .u0 / < 0g are invariant under P 2 P SC , the flow of (2.1) in the following sense: if u.0/; u.0/ then u.t/; u.t/ P 2 P SC for as long as the solution exists, and the same holds for PS .
2.3 The Payne–Sattinger criterion, regions P S˙
35
Proof. Let u be defined on the interval I around t D 0. Since J u.t/ E u.t/; u.t/ P E.u0 ; u1 / < J.Q/ Lemma 2.9 implies the following: if K0 u.t / D 0 for some t 2 I , then u.t / D 0. However, from the Gagliardo–Nirenberg–Sobolev bound kuk44 . kruk32 kuk2 2 we conclude that K0 .u.t// ' ku.t/kH 1 0 for t near t . Thus, no solution starting from P S can end up in P SC and vice versa.
By the corollary we may abuse notation and write u.t/ 2 P S˙ , which we shall occasionally do. As discovered by Payne and Sattinger [114], the significance of P S˙ lies with the following result. We shall see in the next chapter that this then easily implies that the ground state is unstable. See Shatah [125] for a proof of instability of the ground state for more general nonlinearities, and Grillakis, Shatah, Strauss [67] for an entire “abstract” theory addressing the question of orbital stability or instability of periodic or stationary solutions of large classes of PDEs. Corollary 2.13. Solutions of (2.1) which lie in P SC exist for all times, whereas those in P S blow up in finite time (in both temporal directions). In particular, data of negative energy blow up in finite time, and SC ¤ H. Proof. If u.t/ 2 P SC , then 1 1 2 P 22 ' G0 u.t/ C kuk P : ku.t/kH P 22 E.u; u/ 1 C kuk 2 2 Therefore, Lemma 2.2 implies global existence. For the finite time blowup we use the convexity argument due to Payne and Sattinger. Suppose that u extends to all 2 t > 0 and let y.t/ WD ku.t/kL 2 . From (2.1) we have 2 yR D 2 kuk P L K0 u.t/ 2 x (2.27) 2 D 6kuk P 22 8E.u; u/ P C 2kukH 1 : Since J u.t/ E.u; u/ P D E.u0 ; u1 / < J.Q/, Lemma 2.14 below implies that K0 u.t/ > ı > 0; 8 t 0 and some fixed ı > 0. Therefore, y.t/ ! 1 as t ! 1. In view of (2.27), for large t 3 y.t/ P 2 y.t/ R 6kuk P 22 2 y.t/
36
2 The Klein–Gordon equation below the ground state energy
or @ t t .y
1 2
/D
1 y 2
5 2
3 2 yP < 0 2
y yR
which gives a contradiction. Indeed, since y.t/ ! 1, for some large t0 one has @ t .y for all t t0 . But then y
1 2
1 2
1 2
/.t/ @ t .y
/.t0 / < 0
would need to vanish at some point, which is impossible.
We close this section with some more variational results. The following lemma, together with Exercise 2.15, quantifies Lemma 2.8. We shall need these bounds later in our dynamical analysis. Lemma 2.14. If J.'/ < J.Q/ and K0 .'/ < 0, then K0 .'/ 2 J.Q/
(2.28)
J.'/ :
Proof. As before, set j' ./ D J.e '/ and let be as in Lemma 2.8. Then with D e, j'00 ./ D 2K0 .'/ 24 k'k44 2j'0 ./ which implies via integration between and 0 that j'0 .0/
j'0 . / 2 j' .0/
j' . /
which is the same as K0 .'/ 2 J.'/
J.e '/ :
Since Lemma 2.9 implies that J.e '/ J.Q/ due to K0 .e '/ D 0 and ' ¤ 0, we finally obtain (2.28). Exercise 2.15. Prove that if J.'/ < J.Q/ and K0 .'/ 0, then K0 .'/ c0 min J.Q/
2 J.'/; k'kH 1
(2.29)
for some absolute constant c0 . Generalize (2.28) and (2.29) to other powers of the nonlinearity as well as other dimensions. Hint: See the variational section in [77].
2.4 Scattering in P SC
2.4
37
Scattering in P SC
While global existence for data in P SC was shown in [114], the scattering to zero of solutions in this set was established only recently by Ibrahim, Masmoudi, Nakanishi [77], who in fact proved the following stronger result. Proposition 2.16. All solutions u.t/ of (2.1) which are associated with P SC scatter as t ! ˙1 and kukL3 L6x < 1. Moreover, there exists a function N W 0; J.Q/ ! t .0; 1/ so that P kukL3 L6x < N E.u; u/ (2.30) t for all solutions belonging to P SC . The proof of Proposition 2.16 uses the approach of Kenig–Merle [84], an outline of which we now present. It is important to note that this proof applies to the defocusing case as well. In other words, it provides an alternative approach to the proof of large data scattering for the defocusing equation which was originally based on Morawetz estimates, see Morawetz, Strauss [108], and Ginibre, Velo [61]. First, we know from Lemma 2.2 and (2.26) that P SC contains a small ball around zero for which all corresponding solutions scatter with small kukL3 L6x -norm. t Therefore, if the scattering property in P SC fails, then there exists a sequence of data ˚ 0 0 .un ; uP n / n1 P SC with corresponding global solutions un for which E.un ; uP n / % E < J.Q/;
kun kL3 .RIL6 / ! 1 :
(2.31)
Note that by construction fE un g is a bounded sequence in H. Moreover, we can assume that E > 0 is minimal with the aforementioned properties. The goal is to produce, from this sequence, a solution u 2 P SC of (2.1) which satisfies ku kL3 .RIL6 / D 1 as well as E.E u / D E . Such a solution is called a critical element and it has the additional property that at least one of the sets ˚ K˙ WD u . C x0 .t/; t/; uP . C x0 .t/; t/ j 0 ˙t < 1 (2.32) is precompact in H, where x0 .t/ is some path in R3 . In the radial case x0 .t/ 0, and in general lim sup t !˙1
jx0 .t/j 1 jt j
(2.33)
38
2 The Klein–Gordon equation below the ground state energy
by the finite speed of propagation. The ˙ here is determined by whether ku kL3 .Œ0;1/IL6x / D 1 or t
ku kL3 .. t
1;0IL6 x/
D 1:
By construction, at least one of these equalities has to hold. The final step consists in showing that K˙ compact leads to a contradiction unless u 0. But this is impossible, and therefore a sequence as in (2.31) cannot exist. The existence of the critical element u is a consequence of the minimality of E . The naive attempt of passing to a limit in un fails, as one can see from the following two scenarios, each of which presents an obstacle to compactness: (a) Each un can be arbitrarily translated; in other words, the wave moves off to infinity in space-time. (b) The un might split into individual waves which become arbitrarily separated in space-time as n ! 1; in other words, the energy splits into asymptotically independent constituents. It is simple to deal with the obstacle (a) by means of suitable translations. However, (b) is more subtle and requires the minimality of E : if (b) occurs, then asymptot.2/ ically one can split the waves un into the sum of two waves u.1/ n and un plus an .1/ .2/ error. The energies of un and un lie strictly below E and they therefore scatter with finite and uniformly controlled L3t L6x norm. But then one concludes via a perturbative argument that the same applies to un itself which is a contradiction (since u1n and u2n interact only weakly). This type or argument is reminiscent of the concentration-compactness method of Lions [98].
2.4.1 The concentration-compactness decomposition Following the seminal work of Bahouri, Gérard [4] and Merle, Vega [105], we begin with a heuristic description of the method. Henceforth we shall write uE D .u; u/. P Starting from a sequence fun .t/g of solutions to the free KG equation whose free energies E0 .E un / are uniformly bounded, one finds a sequence of free KG waves (called limiting or asymptotic profiles) v j as well as suitable “symmetry transformations” Snj which leave the free KG equation invariant, so that for all J , and up to passing to a subsequence, X un D v j ı Snj C nJ (2.34) j <J
where nJ are free waves which are “small” in a suitable sense, uniformly in n. More precisely, the Strichartz norms – but not the energy – of the nJ vanish in the limit
39
2.4 Scattering in P SC
tn1
xn1
xn2
tn3
tn1
tn2
Figure 2.3. Schematic depiction of the concentration-compactness decomposition
J ! 1. As for the symmetries Snj , one has Euclidean rotations, Lorentz transforms, j and space-time translations. Note that if (up to a subsequence) Snj ! S1 , then j j one can incorporate S1 into v . This is the case for the Euclidean rotations, but not for the other transformations as they form a noncompact group of symmetries. Moreover, as j ¤ k, the transformations Snj and Snk will diverge from each other (in precise sense, see for example (2.36)) as n ! 1. This implies in particular that the energy splits up as follows: for all J 1, X E0 .E un / D E0 .E v j / C E0 . EnJ / C o.1/; n ! 1 : j <J
The point ˚ of (2.34) 1 is therefore that it captures the failure of compactness of the sequence uE n .0/ nD1 H. Indeed, if there are two nonvanishing limiting profiles v j in (2.34), and using the aforementioned pairwise divergence of the associated sym-
40
2 The Klein–Gordon equation below the ground state energy
˚ 1 metries, one concludes that uE n .0/ nD1 cannot be relatively compact. Conversely, if there is only one nonvanishing limiting profile, and provided nJ vanishes in the energy sense, then this sequence is relatively compact. It is precisely the latter conclusion that will prove most useful in the proof of scattering. For simplicity of exposition, we first restrict ourselves to the radial case; this means that the only remaining symmetry of the KG equation is given by translations in time, which are therefore the sole obstacle to compactness. For the nonradial case see Proposition 2.24 below. The proof of Proposition 2.17 is considerably easier than the one in [4] for several reasons: (i) in contrast to the wave equation, the Klein–Gordon equation fixes a scale – in other words, there is no scaling symmetry (ii) the vanishing of the error
nk is expressed only in terms of subcritical Strichartz norms rather than the critical ones as in [4] (iii) the radial assumption ensures that the embedding H 1 ,! Lp .R3 / for 2 < p < 6 is compact (see Exercise 2.7). Proposition 2.17. Let fun g be a sequence of free radial Klein–Gordon solutions bounded in H. Then after replacing it by a subsequence, there exist a sequence of free solutions v j bounded in H, and sequences of times tnj 2 R such that for nk defined by X un .t/ D v j .t C tnj / C nk .t/ (2.35) 1j
we have for any j < k, Enk . tnj / * 0 weakly in H as n ! 1, as well as ˇ ˇ lim ˇtnj tnk ˇ D 1 n!1
(2.36)
and the errors nk vanish asymptotically in the sense that3 lim lim sup k nk k.L1 Lpx \L3 L6x /.RR3 / D 0;
k!1 n!1
t
t
8 2 < p < 6:
Finally, one has orthogonality of the free energy4 X kE un k2H D kE v j k2H C k Enk k2H C o.1/
(2.37)
(2.38)
1j
as n ! 1. 3 4
One can replace L3t L6x with many other Strichartz norms, see Section 2.5. We have chosen the one which is convenient for (2.1). Since the free energy is conserved under the free KG-flow, it does not matter at which time we evaluate the individual norms in (2.38).
41
2.4 Scattering in P SC
Proof. Since nk is bounded in Hx1 , interpolation with the Strichartz estimates of p Lemma 2.46 implies that it suffices to estimate the remainder in L1 t Lx for any fixed 2 < p < 6. Fix such a p. Let n0 WD un and k D 0. If p D 0; k WD lim sup k nk kL1 t Lx
n!1
then we are done by putting n` D nk for all ` > k. Otherwise, there exists a sequence tnk 2 R such that k nk . tnk /kLpx k =2 for large n. Since Enk . tnk / 2 H is bounded, after extracting a subsequence it converges weakly in H, and nk . tnk / converges strongly in Lpx . Let v k be the free solution given by the limit lim E k . n!1 n
tnk / D vEk .0/ :
By Sobolev kv k .0/kH 1 & k . We repeat the same procedure inductively in k 1. Let S0 .t/ denote the free Klein–Gordon propagator in H, see Exercise 2.3. If tnj tnk ! c 2 R for some j < k, then tnj ! 0; tnk D S0 tnj tnk Enk
Enk weakly in H. To see this, it suffices to show that ˛ ˝ k tnk j E ! 0; n ! 1
En E But by Exercise 2.3 one has for any Schwartz function . ˛ ˝ k ˛ ˝ tnj j S0 tnk tnj E ! 0
En . tnk / j E D Enk since S0 tnk tnj E ! S0 . c/ E strongly in L2 . Hence jtnj tnk j ! 1 as long as vEk 6D 0. Then for all j k,
EnkC1 tnj D Enk tnj vEk tnk tnj * 0 weakly in H. Indeed, if j < k then this follows from the inductive assumption, whereas for j D k it follows by construction. To prove (2.38), expand (without loss of generality at t D 0)
X
2
kE un .0/k2H D vEj .tnj / C Enk .0/ : 1j
H
The cross terms are all o.1/ as n ! 1: for k > j ¤ `, and with the scalar product in H, ˝ j j ` ` ˛ ˝ j ˛ vE tn j vE tn D vE .0/ j S0 tn` tnj vE` .0/ ! 0 ; (2.39) ˝ j j ˛ ˝ ˛ vE .tn / j Enk .0/ D vEj .0/ j Enk tnj ! 0 :
42
2 The Klein–Gordon equation below the ground state energy
E 1 ! 0 for The first line of (2.39) vanishes as n ! 1 due to kS0 .tn` tnj / k j ` any Schwartz function E since jtn tn j ! 1, see the pointwise decay estimates of Section 2.5; as usual this suffices since we can approximate vEj .0/; vE` .0/ by Schwartz functions. The second line vanishes by Enk . tnj / * 0 in H as n ! 1. Finally, one uses (2.38) to conclude that j ! 0: X X 2
2
vEj & lim sup uE n H . j /2 H n!1
j
j
uniformly in k. The final inequality follows from the radial Sobolev imbedding (in p other words, Sobolev imbedding and compactness). Hence, lim supn!1 k nk kL1 t Lx k D ! 0, as k ! 1. Later, we shall formulate and prove the non-radial version of Proposition 2.17. Figure 2.3 shows the two degrees of freedom entering into the Bahouri–Gérard decomposition for the nonradial Klein–Gordon equation: temporal and spatial translations. Exercise 2.18. Show that there exists a sequence of free KG waves fun g1 nD1 which is uniformly bounded in H 1 L2 and such that kun kL3 L6x ! 0 as n ! 1, while t the free energies are bounded away from zero. This shows that the nk above need not vanish in energy as k ! 1, even though they do so in the Strichartz sense. Proposition 2.17 is formulated for free waves, but we will of course want to apply it to solutions of the NLKG equation. To be more specific, suppose fun g1 nD1 P SC is a sequence of uniformly bounded energy. Recall that this implies that the free energies E0 .E un / are also uniformly bounded. We can therefore apply Proposition 2.17 to the sequence S0 .t/ uE n .0/ of free solutions. This yields a representation of the form X uE n .0/ D vEj .tnj / C EnJ .0/ (2.40) j <J
where v j are free KG waves. In and of itself, (2.40) is not too useful. Indeed, we would like to have a representation involving the nonlinear evolutions. One way of passing from the linear profile decomposition (2.40) to a nonlinear one is to associate with each v j a nonlinear profile U j defined as a solution of the NLKG equation and which satisfies
j j
UE .t / vEj .t j / ! 0; n ! 1 n n H
43
2.4 Scattering in P SC
j where we are assuming as we may that tnj ! t1 2 Π1; 1 as n ! 1. The existence of these U j is simple, based on the wellposedness theory from Section 2.1. It then follows that we may write, for small t , X un .t/ D U j .t C tnj / C nJ .t/ C Jn .t/ (2.41) j <J
where kE Jn .0/kH ! 0 as n ! 1. The relation (2.41) is the nonlinear profile decomposition and is of central importance. To see why, let us assume that each U j is a global NLKG solution that scatters. While the superposition principle fails for the nonlinear equation, it essentially holds for large n due to the divergence property (2.36). More precisely, due to the finiteness of the L3t L6x -norm of each U j one verifies that the sum on the right-hand side of (2.41) satisfies the NLKG equation up to an error which vanishes in the L1t L2x -norm as n ! 1. We would now like to conclude from (2.41) and the preceding considerations that un has finite L3t L6x norm uniformly in n. Clearly, this requires a perturbation theory which is capable of absorbing the errors nJ and Jn . While the latter is harmless since it is small in energy, the former is not. However, the Strichartz norms (other than the energy) of nJ are very small for large J which allows us to derive the desired conclusion by means of the perturbation theory which we shall now develop.
2.4.2 The perturbation lemma The following perturbative result will allow us to apply Proposition 2.17; we emphasize that it does not rely on any radial assumption. Note that it is designed precisely to meet the needs of Proposition 2.17. In fact, we start from a solution of (2.1) with finite L3t L6x -Strichartz norm on some time-interval I , and perturb it by a free KG solution with globally small Strichartz norm (rather than small energy, say). This of course needed in order to absorb the errors nk from Proposition 2.17. As usual S0 denotes the free KG-propagator. Lemma 2.19. There are continuous functions "0 ; C0 W .0; 1/ ! .0; 1/ such that the following holds: Let I R be an open interval (possibly unbounded), u; v 2 C.I I H 1 / \ C 1 .I I L2 / satisfying for some B > 0 kvkL3 .I IL6x / B; t
keq.u/kL1 .I IL2x / C keq.v/kL1 .I IL2x / C kw0 kL3 .I IL6x / " "0 .B/; t
t
t
44
2 The Klein–Gordon equation below the ground state energy
where eq.u/ WD u C u u3 in the sense of distributions, and w E 0 .t/ WD S0 .t t0 /.E u vE/.t0 / with t0 2 I arbitrary but fixed. Then kE u
vE
w E 0 kL1 C ku t .I IH/
vkL3 .I IL6x / C0 .B/": t
In particular, kukL3 .I IL6x / < 1. t
Proof. Let Z WD L3t L6x and w WD u
v;
e WD . C 1/.u
v/
u3 C v 3 D eq.u/
eq.v/:
There is a partition of the right half of I as follows, where ı0 > 0 is a small absolute constant which will be determined below: t0 < t1 < < tn 1; kvkZ.Ij / ı0
Ij D .tj ; tj C1 /;
.j D 0; : : : ; n
1/;
I \ .t0 ; 1/ D .t0 ; tn /;
n C.B; ı0 /:
We omit the estimate on I \. 1; t0 / since it is the same by symmetry. Let w Ej .t/ WD S0 .t tj /w.t E j / for all 0 j < n. Then Z t sin .t s/hri . C 1/.u v/.s/ ds w.t/ D w0 .t/ C hri t0 (2.42) Z t sin .t s/hri 3 3 D w0 .t/ C .e C .v C w/ v /.s/ ds hri t0 which implies that, for some absolute constant C1 1,
kw w0 kZ.I0 / . .v C w/3 v 3 C e L1 L2 .I / x 0 t 2 2 C1 ı0 C kwkZ.I0 / kwkZ.I0 / C C1 "
(2.43)
Note that kwkZ.I0 / < 1 provided I0 is a finite interval. If I0 is half-infinite, then we first need to replace it with an interval of the form Œt0 ; N /, and let N ! 1 after performing the estimates which are uniform in N . Now assume that C1 ı02 1 and fix ı0 in this fashion. By means of the continuity method (which refers to 4 using that the Z-norm is continuous in the upper endpoint of I0 ), (2.43) implies that kwkZ.I0 / 8C1 ". Furthermore, Duhamel’s formula implies that Z t1 sin .t s/hri w1 .t/ w0 .t/ D .e C .v C w/3 v 3 /.s/ ds hri t0
45
2.4 Scattering in P SC I0
t
t
3
t
2
1
t0
I1
t1
t2 t3
t4
Figure 2.4. The time decomposition
whence also Z kw1
w0 kZ.R/ .
t1
.e C .v C w/3
t0
v 3 /.s/ 2 ds
(2.44)
which is estimated as in (2.43). We conclude that kw1 kZ.R/ 8C1 ". In a similar fashion one verifies that for all 0 j < n kw
wj kZ.Ij / C kwj C1
wj kZ.R/ . k.v C w/3
v 3 C ekL1 L2x .Ij / t
C1 .ı02 C kwk2Z.Ij / /kwkZ.Ij / C C1 " (2.45) where C1 1 is as above. By induction in j one obtains that kwkZ.Ij / C kwj kZ.R/ C.j / ";
8 1 j < n:
This requires that " < "0 .n/ which can be done provided "0 .B/ is chosen small enough. Repeating the estimate (2.45) once more, but with the energy piece L1 t H included on the left-hand side, we can now bound the full Strichartz norms on w. Exercise 2.20. Justify (2.42) under the assumptions of the previous lemma. In order to complete the scattering argument, we shall need more than the functional K0 . In fact, it is natural to consider the functionals K˛;ˇ , defined by the action of symmetries on J : K˛;ˇ .'/ D @ jD0 J e ˛ '.e ˇ /
(2.46)
see Jeanjean, LeCoz [79] and [77]. Next to pure scaling, which corresponds to ˛ D 1; ˇ D 0 and gives K0 , we will use the parameters ˛ D d2 , ˇ D 1 in general dimension d . Note that the underlying symmetry is precisely the one that preserves L2 .Rd /. Its infinitesimal generator is therefore anti-symmetric (as being associated
46
2 The Klein–Gordon equation below the ground state energy
with a unitary operator) and is precisely A D 21 .x r C r x/. Thus, in R3 , K 3 ;1 .'/ WD hJ 0 .'/ j A'i 2 Z h 1 2 D @ jD0 e jr'j2 .x/ C j'j2 .x/ R3 2 Z 3 ˇˇ ˇˇ4 jr'j2 .x/ D ' .x/ dx : 4 R3
1 3 4 i e j'j .x/ dx 4
(2.47)
Following a convention from [109], we call this functional K2 . Lemma 2.21. One has ˚ J.Q/ D inf J.'/ j K2 .'/ D 0; ' 2 H 1 n f0g ˚ D inf G2 .'/ j K2 .'/ 0; ' 2 H 1 n f0g
(2.48)
1 K /.'/ D 16 kr'k22 C 21 k'k22 . The minimizers are exactly where G2 .'/ D .J 3 2 ˙Q. C x0 / where x0 2 R3 is arbitrary. The sets ˚ ' 2 H 1 j J.'/ < J.Q/; Kj .'/ 0 ; (2.49) ˚ ' 2 H 1 j J.'/ < J.Q/; Kj .'/ < 0
do not depend on the choice of j D 0 or j D 2. Moreover, if J.'/ < J.Q/ and K2 .'/ < 0, then K2 .'/ 2 J.Q/ J.'/ (2.50) and if J.'/ < J.Q/ and K2 .'/ 0, then K2 .'/ c0 min J.Q/
J.'/; kr'k22
(2.51)
for some absolute constant c0 > 0. Proof. We shall only sketch the argument, leaving the finer details to Exercise 2.22. 1 As usual, one reduces to the case of Hrad . To prove (2.48), pick a minimizing sequence 'n . Then 'n ! '1 strongly in L4 and weakly in H 1 . Due to K2 .'n / D 0 and J.'n / bounded one verifies that 'n is bounded in H 1 and moreover k'n k4 and k'n kH 1 bounded away from zero uniformly in n. Thus '1 ¤ 0, and K2 .'1 / 0. If K2 .'1 / < 0, then one obtains a contradiction via G2 , cf. the analogous argument for K0 in Lemma 2.9. Hence K2 .'1 / D 0 and '1 is a minimizer. By the Euler–Lagrange equation, J 0 .'1 / D K20 .'1 / which implies that 0 D hJ 0 .'1 / j A'1 i D K2 .'1 / D hK20 .'1 / j A'1 i D
3 k'1 k44 : 4
2.4 Scattering in P SC
47
Therefore, D 0 and J 0 .'1 / D 0. As usual, one can assume that '1 0 (replace 'n with j'n j) which then implies via elliptic regularity and the maximum principle that '1 > 0. By Coffman [35], '1 D Q and thus only ˙Q are possible minimizers. This concludes the proof of (2.48). To prove (2.49), we follow [77]. We already know that 0 is surrounded by the open set Kj .'/ 0 for both j , and Kj .'/ D 0 is prohibited except for ' D 0 by J.'/ < J.Q/, due to the above minimization property of Q. Then it is enough to show that each set KjC WD fKj 0; J < J.Q/g is contractible to f0g, which implies that it is connected and so cannot be divided by the disjoint open sets KC 2 j and K2 j D fK2 j < 0; J < J.Q/g. The contractions can be seen to be '0 WD '.x/, 3
and '2 WD 2 '.x/, respectively, for j D 0; 2 as decreases from 1 to 0. By definition @ J.'j / D Kj .'j /=, and so J.'j / decreases together with , as long as Kj .'j / > 0, which is preserved as long as J.'j / < J.Q/. Hence, 'j stays in KjC for 1 > 0. When j D 2, it converges to 0 as ! C0 only in HP 1 \ L4 , but since K˙ 2 are open in that topology this is sufficient: once '2 moves into a small ball C around 0 in K2 , one can change to the other scaling ' 7! ' to send it to 0 in H 1 , while remaining in KC 2. Finally, (2.50) and (2.51) follow by the same type of arguments which established the analogous properties of K0 , see Lemma 2.14 and Exercise 2.15. Exercise 2.22. Provide the omitted details in the proof of Lemma 2.21. In particular, prove (2.50) and (2.51). We can now establish the scattering property in P SC . We begin with the radial case.
2.4.3 Proof of scattering in the radial case Suppose the proposition fails. Then there exists a sequence fun g as in (2.31). Applying Proposition 2.17 to the linear KG evolutions of uE n .0/ yields free waves v j and times tnj as in (2.35). Let U j be the nonlinear profiles of .v j ; tnj /, i.e., those energy solutions of (2.1) which satisfy lim kE v j .t/
j t!t1
UE j .t/kH ! 0
j j where limn!1 tnj D t1 2 Π1; 1. The U j exist locally around t D t1 by the local existence and scattering theory, see Lemma 2.2. Locally around t D 0 one has
48
2 The Klein–Gordon equation below the ground state energy
the following nonlinear profile decomposition X un .t/ D U j .t C tnj / C nk .t/ C kn .t/
(2.52)
j
where kE kn .0/kH ! 0 as n ! 1. Note that X
v j .t j / 4 C o.1/ kun .0/k44 D n 4 j
as k; n ! 1 (first choose k large so that k nk kL1 L4x is small, and then n large so t
as to make all – with the possible exception of one – kv j .tnj /k4 arbitrarily small; the latter follows from the fact that either all, or all but one, jtnj j ! 1 as n ! 1). Combining this with (2.38) yields X E C o.1/ > E.E un / D E.UE j / C E Enk .0/ C o.1/; j
K un .0/ D
X
K U j .tnj / C K nk .0/ C o.1/
(2.53)
j
as k; n ! 1. Furthermore, using the notation of Lemma 2.9 and (2.38) we conclude that for large n, and with E0 denoting the free KG energy, 1 1 K un .0/ G0 un .0/ C kuP n .0/k22 J.Q/ > E C o.1/ > E uE n .0/ 4 4 1 X1 1 j j D E0 uE n .0/ D E0 UE .tn / C E0 Enk .0/ C o.1/ 2 2 2 j
as n ! 1. If follows from (2.25) that K U j .tnj / 0 and K nk .0/ 0 whence in particular also E.UE j / 0 and E Enk .0/ 0. Now suppose that either there are two non-vanishing v j , say v 1 ; v 2 , or that lim sup lim sup k Enk kH > 0 : k!1
n!1
(2.54)
Note that the left-hand side does not depend on time since nk is a free wave. In the former case, 1 1 E UE j .tnj / G0 U j .tnj / C kUP j .tnj /k22 D E0 v j .tnj / C o.1/ 4 2 1 D E0 v j .0/ C o.1/ 2
49
2.4 Scattering in P SC
as n ! 1 implies that E.UE 1 /; E.UE 2 / > 0 whereas in the case where (2.54) holds, one has E Enk .0/ > ı0 > 0 for large k and n. In conjunction with (2.53) this implies that E.UE j / < E for each j . By the minimality of E each U j is a global solution and scatters with kU j kL3 L6x < 1. t We now apply Lemma 2.19 on I D R with u D un and X v.t/ D U j .t C tnj / : (2.55) j
That keq.v/kL1 L2x is small for large n follows from (2.36). To see this, note that t
eq.v/ D . C 1/v v 3 X 3 D U j .t C tnj /
X
j
3 U j .t C tnj / :
j
The difference on the right-hand side here only consists of terms which involve at least one pair of distinct j; j 0 . But then keq.v/kL1 L2x ! 0 as n ! 1 by (2.36). In t order to apply Lemma 2.19 it is essential that
X j
lim sup U .t C tnj / L3 L6 B < 1 (2.56) n!1
t
x
j
uniformly in k, which follows from (2.36), (2.38), and Lemma 2.2. The point here is that the sum can be split into one over 1 j < j0 and another over j0 j < k. This splitting is performed in terms of the energy, with j0 being chosen such that for all k > j0 X
UE j .t j / 2 "2 lim sup n 0 H (2.57) n!1 j0 j
where "0 is fixed such that the small data result of Lemma 2.2 applies. Clearly, (2.57) follows from (2.38). Using (2.36) as well as the small data scattering theory of Section 2.1 one now obtains X
X
3
U j ./ 3 3 6 lim sup U j . C tnj / L3 L6 D L L n!1
t
j0 j
x
t
x
j0 j
C lim sup n!1
X
kUE j .tnj /k2H
j0 j
with an absolute constant C . This implies (2.56), uniformly in k.
23
(2.58)
50
2 The Klein–Gordon equation below the ground state energy
Hence one can take k and n so large that Lemma 2.19 applies to (2.52) whence lim sup kun kL3 L6x < 1 t
n!1
which is a contradiction. Thus, there can be only one nonvanishing v j , say v 1 , and lim sup k En2 kH D 0 :
(2.59)
n!1
Thus, E.UE 1 / D E and K.U 1 / 0. By the preceding, necessarily kU 1 kL3 L6x D 1 :
(2.60)
t
Therefore, U 1 DW u is the desired critical element. Suppose that (2.61)
ku kL3 .Œ0;1/IL6x / D 1 : t
Then we claim that KC is precompact with x0 0, see (2.32). If not, then there exists ı > 0 so that for some infinite sequence tn ! 1 one has
uE .tn / uE .tm / > ı; 8 n > m : (2.62) H Applying Proposition 2.17 to U 1 .tn / one concludes via the same argument as before based on the minimality of E and (2.60) that uE .tn / D VE . n / C En .0/
(2.63)
where VE , En are free Klein–Gordon waves in H, and n is some sequence in R. Moreover, k En kH ! 0 as n ! 1. If n ! 1 2 R, then (2.63) and (2.62) lead to a contradiction. If n ! 1, then kV . C n /kL3 .Œ0;1/IL6x / ! 0; t
as n ! 1
implies via the local wellposedness theory that ku . C tn /kL3 .Œ0;1/IL6x / < 1 for all t large n, which is a contradiction to (2.61). If n ! 1, then kV . C n /kL3 .. t
1;0IL6 x/
! 0;
as n ! 1
implies that ku . C tn /kL3 .. 1;0IL6x / < B < 1 for all large n where B is some t fixed constant. Passing to the limit yields a contradiction to (2.63) and (2.62) is seen to be false, concluding the proof of compactness of KC .
51
2.4 Scattering in P SC
Finally, we show that KC precompact leads to a contradiction via the following virial argument; this is the “rigidity” step from [84], cf. Part (iv) above. First, let be a compactly supported cutoff function so that D 1 on jxj 1. Then by a direct calculation one verifies that, with A D 12 .x r C r x/, Z ˛ d ˝ uP j Au D K2 .u / C O .juP j2 C jru j2 C ju j2 / dx dt R jxj>R (2.64) where K2 is as in (2.47). The O./-term here is uniformly small by the precompactness of the forward trajectory of u . The idea is now simply to integrate (2.64) from 0 to some large time t0 and to note that the resulting left-hand side is bounded by O.R/ (by energy conservation and the fact that we are in the region fK 0g which ensures that the free energy of u is uniformly bounded in time), whereas if the entire right-hand side is < ı < 0 for some fixed ı > 0, then a contradiction follows by taking t0 large. However, we cannot simply assume that K2 .u .t// obeys this uniform bound. To be more specific, K2 obeys the following bound by Lemma 2.21: K2 .u / c0 min.J.Q/
J.u /; kru k22 / ı2 kru k22
(2.65)
and some ı2 > 0. We now face the difficulty that one might have u .t1 / D 0 for some t1 which renders (2.65) void at that time; note that u .t1 / D 0 does not imply that u 0 since uP .t1 / can be large. To overcome this, we proceed as follows, see [77]. We first claim the following: for every " > 0 there exists a constant C."/ such that ku .t/k2 C."/kru .t/k2 C "kuP .t/k2 ;
8t 0 :
(2.66)
If this fails, then for some " > 0 there is a sequence tn with the property that ku .tn /k2 > nkru .tn /k2 C "kuP .tn /k2 ;
8 n 1:
(2.67)
This contradicts the precompactness of the forward trajectory of u . Next, use (2.66) with " D 12 to conclude that d hu juP i D kuP k22 C hu ju u C u3 i dt D kuP k22 kru k22 ku k22 C ku k44 1 kuP k22 C kru k22 2
52
2 The Klein–Gordon equation below the ground state energy
and thus also 1 d 2 2 kuP k22 C ku kH hu juP i : 1 C kru k2 C 4 dt
(2.68)
Due to K.u / 0, the left-hand side here is ' E.E u /. Integrating (2.68) from 0 to an arbitrary t0 > 0 yields Z t0
ru .t/ 2 dt : (2.69) t0 E.E u / . 2 0
Hence, choosing R so large in (2.64) that the energy outside the R-ball is < ı3 E.E u / SC ) and integrating (2.64) where ı3 ı2 , (which can be done by compactness of K between 0 and t0 yields Z t0 D Eˇ
1 t0 ˇ
ru .t/ 2 dt C C t0 ı3 E.E uP j .x r C r x/u 0 < ı2 u / : 2 R 2 0 Since the left-hand side here is O RE.E u / uniformly in t0 , one obtains a contradiction to (2.69) as t0 ! 1. Exercise 2.23. (a) With v defined by (2.55) prove that keq.v/kL1 L2x ! 0 as n ! 1. Moreover, t provide the remaining details in the argument that (2.56) holds uniformly in k. (b) Give a detailed proof of (2.66).
2.4.4 Proof of scattering in the nonradial case The general approach in the nonradial case is the same as in the radial setting: we argue by contradiction, choose a critical energy E and an associated critical sequence. We then apply the concentration-compactness method, but in contrast to before we now have spatial translations xnj in addition to temporal ones, see Figure 2.3. But everything else remains the same, and the combined space-time sequences .tnj ; xnj / diverge arbitrarily far away from each other for j ¤ j 0 as n ! 1. By the same concentration-compactness procedure as in the radial case we obtain a critical element u with infinite Strichartz norm, but the compactness now holds only after modding out the translation symmetry. In other words, one needs to translate u by a path x0 .t/. The latter is chosen essentially by selecting the center of mass of u .t/, see Lemma 2.25 below. By the finite propagation speed one has that x0 .t/ can grow at most linearly, but this is insufficient for the virial argument centered
2.4 Scattering in P SC
53
around (2.64). In fact, for the desired contradiction to ensue we need x0 .t/ to grow more slowly than any linear function of time. Remarkably, this is indeed the case and follows from the observation (as in [84]) that the critical element u has vanishing momentum: P .E u / D 0, see Lemma 2.28. The latter holds since otherwise a Lorentz transform will lower the energy leading to a contradiction to the minimality of E . The vanishing momentum now guarantees that x0 has the desired slow growth and we obtain a contradiction via the virial identity as before. The details are as follows. We begin with the nonradial form of the concentrationcompactness decomposition in the Klein–Gordon case. In contrast to the radial case, the embedding H 1 ,! Lp .R3 / for 2 < p < 6 is not compact, and the construction in the proof of Proposition 2.24 is therefore somewhat different from the one that we used in the proof of Proposition 2.17. An important question that arises in formulating the nonradial version of Proposition 2.17 is as follows: which symmetries Snj are relevant to the representation (2.34)? In fact, while the Snj may be a general element of the full Poincaré group, one can ignore the compact subgroups such as j the Euclidean rotations. To see this, note that one can pass to the limit Snj ! S1 , j with the latter becoming part of the limiting profiles v . Interestingly, one can also eliminate the Lorentz subgroup in this fashion. To be specific, consider the Lorentz transform L./.t; x/ D .t cosh C x1 sinh ; t sinh C x1 cosh ; x2 ; x3 / : Then as jj ! 1 one verifies that the (free) energy of any nonzero solution composed with L./ becomes unbounded. Since the concentration-compactness decomposition always remains within a bounded region in H, this implies that jj remains bounded whence the claim. In conclusion, only space-time translations remain as nontrivial noncompact Snj . Proposition 2.24. Let fun g1 nD1 be a sequence of free Klein–Gordon solutions in three dimensions satisfying sup kE un kL1 < 1: t H n
After passing to a subsequence, there exist a sequence of free solutions v j bounded in H, and sequences .tnj ; xnj / 2 R R3 such that for nk defined by X un .t; x/ D v j .t C tnj ; x C xnj / C nk .t; x/ (2.70) 1j
we have for any j < k, Enk . tnj ; xnj / * 0 in H as n ! 1, and lim jtnj tnk j C jxnj xnk j D 1 : n!1
(2.71)
54
2 The Klein–Gordon equation below the ground state energy
Moreover, the errors nk vanish asymptotically in the sense of (2.37), and one has orthogonality of the free energy as in (2.38). Proof. Interpolating with the Strichartz estimates such as L2t RI .L6x \ L9x /.R3 / p 3 reduces (2.37) to the L1 t .RI Lx .R //-norms alone. Next, we claim that it suffices to show that lim lim sup k nk k
3
2 3 L1 t .RIB1;1 .R //
k!1 n!1
D 0;
(2.72)
where for 1 q < 1 ˛ kf kBp;q D kP0 f kp C
1 X
2j˛q kPj f kpq
q1
j D1
and for q D 1 ˛ kf kBp;1 D kP0 f kp C sup 2j˛ kPj f kp :
1j <1
Here Pj are the usual Littlewood–Paley projections, see Section 2.5 and in particular (2.113) for further details. Indeed, interpolating this with the energy bound
sup lim sup
nk L1 .RIB 1 .R3 // < 1 k
t
n!1
2;2
which follows from (2.38) (and the latter not depending on (2.37) for the proof) yields
lim lim sup
nk D 0 H) lim lim sup
nk L1 .RIL3 .R3 // D 0 1 k!1 n!1
6 3 L1 t .RIB3;3 .R //
k!1 n!1
t
which is sufficient by interpolation with the energy bound. Therefore, we now turn to the proof of (2.72). Let 1 WD lim inf kun k n!1
3
2 3 L1 t .RIB1;1 .R //
If 1 D 0, then the process terminates with all vEj D 0. Thus, let 1 > 0 and pick a sequence fkn g1 nD1 of nonnegative integers so that 2
3 2 kn
1
Pk un 1 > n L t;x 2
(2.73)
where Pk are the usual Littlewood–Paley projections; more precisely, P0 is the projection onto frequencies . 1, and for k 1, Pk are the projections onto frequencies
55
2.4 Scattering in P SC
of size ' 2k . By supn;t kun .t/kH 1 < 1 it follows that fkn g is a bounded sequence (the bound depends on 1 , but this is of no consequence). So after passing to a sub1 sequence one can assume that kn D k1 is constant. Select a sequence f.tn1 ; xn1 /g1 nD1 with the property that 2
3 1 2 k1
1 ˇ ˇ ˇ.P 1 un /. t 1 ; x 1 /ˇ > ; n n k1 2
8 n:
(2.74)
Let uE n . tn1 ; xn1 / * vE1 .0/./ in H. The notation vE1 .0/ is supposed to signify that this function is viewed as being at time t D 0. Writing the projection in (2.74) as a convolution with a fixed Schwartz function and passing to the limit n ! 1 one concludes that C0 kv 1 .0/k2 > 2
3 1 2 k1
1 ˇ ˇ ˇ P 1 v 1 .0/ .0/ˇ > k1 2
(2.75)
with an absolute constant C0 . Let v 1 be the free KG wave with data vE1 .0/, and define n2 WD un v 1 . C tn1 ; C xn1 /. By construction, En2 . tn1 ; xn1 / * 0. Now repeat the construction with 2 WD lim inf k n2 k n!1
3
2 3 L1 t .RIB1;1 .R //
If 2 D 0, then the process terminates, whereas if 2 > 0, then one selects .tn2 ; xn2 / 2 2 R R3 and some integer k1 0 such that 2
3 2 2 k1
2 ˇ ˇ ˇ.P 2 2 /. t 2 ; x 2 /ˇ > n n k1 n 2
for large n. The analogue of (2.75) holds for v 2 .0/ defined by En2 . tn2 ; xn2 / * vE 2 .0/./ in H. Suppose jtn1 tn2 j C jxn1 xn2 j remains bounded as n ! 1. Then we may assume that tn1 tn2 ! and xn1 xn2 ! whence
En2 . tn2 ; xn2 / D S0 .tn1
tn2 / En2 . tn1 ; xn1 C xn1
xn2 / * 0;
n ! 1:
Thus, jtn1 tn2 j C jxn1 xn2 j ! 1 as n ! 1. Inductively, one now constructs all limiting profiles v j . As in the radial case one verifies by induction in k that for any j < k, Enk . tnj ; xnj / * 0 in H as n ! 1, and that (2.71) holds. This then can be seen to imply the orthogonality of the free energies, i.e., (2.38) which in turn ensures that, cf. (2.75), X X . j /2 . E0 .E vj / < 1 j
j
In particular, j ! 0 which concludes the argument.
56
2 The Klein–Gordon equation below the ground state energy
To prove Proposition 2.16, one again starts from a sequence as in (2.31). Applying Proposition 2.24 and using the minimality of E one argues as in the radial case that there exist .tn ; xn / 2 R R3 such that un D u . C tn ; C xn / C rn where u is a strong solution of (2.1) and krn kL1 ! 0 as n ! 1. By t H Lemma 2.19, necessarily ku kL3 L6x D 1 and u also satisfies K0 .u / 0 (by t the same proof as in the radial case). As before, we call u the critical element, and assume without loss of generality that ku kL3 ..0;1/IL6x / D 1. t Our next goal is to show that KC as defined in (2.32) is precompact for a suitable choice of x0 .t/. The following preliminary statement defines the points x0 .t/. Lemma 2.25. There exists x0 W Œ0; 1/ ! R3 so that for all " > 0 there exists R."/ 2 .0; 1/ with the property that Z jru j2 C ju j2 C juP j2 .t; x/ dx < " (2.76) Œjx x0 .t/j>R."/
for all t 0. Proof. We begin with a weaker statement, namely that for every " > 0 there exists x0;" .t/ and R."/ so that (2.76) holds. If not, then there exists " > 0 and for every n 1 some tn 0 such that Z inf jru .tn ; x/j2 C ju .tn ; x/j2 C juP .tn ; x/j2 dx > "; 8 n 1 : y2R3 Œjx yj>n
(2.77) Now apply Proposition 2.24 and use that u is a critical element to conclude that uE .tn / D VE . n ; C n / C rEn .0/
(2.78)
where V ¤ 0 is a free wave and rn is a free wave with kErn .0/kH ! 0. As in the radial compactness proof one argues that n is a bounded sequence, see (2.63). Then n ! 1 and a contradiction to (2.77) follows immediately. In order to free x0;" of its "-dependence one simply starts with " D "0 sufficiently small and then shows that, at the expense of making R."/ larger, one can take x0 D x0;"0 . To be more specific, first note that with absolute implicit constants, Z jru j2 C ju j2 C juP j2 .t; x/ dx ' E.E u /; 8 t 2 R R3
2.4 Scattering in P SC
57
since uE 2 P SC . Fix some "0 > 0 and associated R0 WD R."0 / such that for any given time t the left-hand side of (2.76) represents a small portion (less than 50%) of the total integral, with x0 .t/ WD x0;"0 .t/. Then for any 0 < " < "0 it is clear that the ball B.x0;" .t/; R."// intersects B.x0 .t/; R0 /. Thus, if we replace R."/ by 3R."/, then one can replace x0;" .t/ with x0 .t/. It is now a simple matter to establish the compactness property of KC . Corollary 2.26. The set KC defined in (2.32) with x0 .t/ as in Lemma 2.25 is precompact in H. Proof. Suppose this fails. Then there exists ı > 0 and an infinite sequence ftn gn1 such that kE u tn ; C x0 .tn / uE tm ; C x0 .tm / kH > ı; 8 n ¤ m : (2.79) Applying Proposition 2.24 and using that u is critical yields uE tn ; C x0 .tn / D VE . n ; C n / C rEn .0/ as in (2.78). As in (2.63) one checks that f n g is bounded, and using (2.76) one also sees that fn g R3 is a bounded sequence. Thus, n ! 1 and n ! 1 as n ! 1 which contradicts (2.79), concluding the proof. Exercise 2.27. (a) Provide all omitted details in the proof of Lemma 2.25 and Corollary 2.26. (b) Prove that one can choose x0 .t/ to be continuous in t 0. As in the radial case, we are now going to show that KC cannot be precompact unless u D 0, a contradiction. This is the “rigidity” step of [84]. As in the radial case, the crucial ingredient here will be the virial identity (2.64). The presence of the translation x0 .t/ of course affects the virial argument, as a nonzero momentum would render that argument invalid. Thus, we first show that the momentum vanishes. Recall that the momentum is a conserved quantity for solutions of (2.1). Lemma 2.28. For the critical element u as above one has Z P .E u / WD uP ru .t; x/ dx D 0; 8 t 0 i.e., the momentum of u vanishes.
58
2 The Klein–Gordon equation below the ground state energy
Proof. We claim first that if P .E u / ¤ 0, then a suitably chosen Lorentz transform L reduces the energy, i.e., E.E u ı L/ < E while maintaining the property K0 .u ı L/ 0. The idea behind this is simply that placing the solution in a frame that moves with the solution will reduce the kinetic energy. Since Lorentz transforms mix space and time, while the notion of the Cauchy problem is based on a specified time-direction, some care is needed in working with uE ı L. Taking these claims for granted (we shall return to them below), then by the minimality of E one concludes that u ı L exists globally and scatters. By our definition this means that ku ı LkL3 L6x < 1. By the Strichartz estimates, see t < 1 where v D u ı L (here one uses Lemma 2.46, this implies that kv k 8 L t3 L8 x
of course that v is a solution of (2.1)). Now note that
2
1 kv kL4 . v L3 1 L2 v 3 8 t;x
t
x
<1
L t3 L8 x
(2.80)
whence also ku kL4 < 1 (since j det Lj D 1). Denote by wT the free Klein– t;x Gordon solution with w E T .T / D uE .T /. Then sup kwT k
8
L t3 L8 x
T
M <1
(2.81)
< 1 (which in turn follows from K u .t/ 0 for all t). Furthersince kE u kL1 t H more, by Lemma 2.46, for all S > T , ku
wT k
8
L t3 ..T;S/IL8 x/
3 ku3 kL1 ..T;S/IL2x / . ku kL 3 ..T;S/IL6 / t
t
x
2
. ku kL4 ..T;S/IL4x / ku k
8 L t3 ..T;S/IL8 x/
t
with an absolute implicit constant. From (2.81) and (2.82) we infer that ku k
8 L t3 ..T;S/IL8 x/
M C ku kL4 ..T;S/IL4x / ku k2 8 t
L t3 ..T;S/IL8 x/
whence in particular sup ku k
S>T
8
L t3 ..T;S/IL8 x/
2M :
For the latter one takes T so large that M ku kL4 ..T;1/IL4x / < t
1 : 4
(2.82)
59
2.4 Scattering in P SC
In conclusion, ku k
8
L t3 ..0;1/IL8 x/
<1
and thus also
1 2 ku kL3 ..0;1/IL6x / . u L3 4 u 3 8 t
t;x
L t3 ..0;1/IL8 x/
<1
which is a contradiction. Therefore, P .E u / D 0 as claimed. Let us now return to the initial claim concerning the Lorentz transform L. Suppose P .E u / ¤ 0. Then without loss of generality Z P1 .E u / D uP @1 u dx ¤ 0 : We introduce the Lorentz transform L˛ .t 0 ; x10 ; x20 ; x30 / D .cosh.˛/t 0 C sinh.˛/x10 ; sinh.˛/t 0 C cosh.˛/x10 ; x20 ; x30 / and set v˛ D u ı L˛ . Then for all ˛ one calculates that, see Exercise 2.29, E.E v˛ / D E.E u / cosh.˛/ C P1 .E u / sinh.˛/ ; P1 .E v˛ / D P1 .E u / cosh.˛/ C E.E u / sinh.˛/ :
(2.83)
Hence, we can choose ˛ so small that E.E v˛ / D E.E u / C ˛P1 .E u / C O.˛ 2 / < E.E u / : As for the sign of K0 .v˛ /, note that K0 .v˛ / < 0 is impossible since it would mean that v , and therefore also u , are not global solutions. Exercise 2.29. Assuming that u has compact support on each time-slice prove (2.83). Then pass to the limit to recover the general situation. See Figure 2.5 for the geometry of the Lorentz-transformed solutions: the dashed lines, which are the planes of constant Lorentz-transformed time, must exit the double cone which is the domain of influence of a compact region on ft D 0g. See also the following lemmas for more clarification on the Cauchy problem.
Lemma 2.30. Let u be a finite energy solution of NLKG (2.1) on .T; 1/, and let u0 be a Lorentz transform of u. Then there exists T 0 2 R such that u0 extends to a finite energy solution on .T 0 ; 1/. Moreover, if ku0 kL4 .t>T 0 / < 1 then kukL4 .T;1/ < t;x t;x 1.
60
2 The Klein–Gordon equation below the ground state energy
t 0 D const
t D0 t 0 D const
Figure 2.5. The geometry of Lorentz transformed wave equations
Proof. Let 2 C01 .R3 / be the cut-offfunction as before, and let w be the solution of NLKG with w.T E / D 1 .x=S/ uE .T /. If S 1 is large enough, then w is global with kwkL4 .R1C3 / < 1, by the small data scattering theory. By the finite t;x propagation property, u.t; x/ D w.t; x/ for jxj > S C jt T j. Hence u extends to this region for t < T . The image of the region ˚ (2.84) .T; 1/ R3 [ .t; x/ j jxj > S C jt T j by any Lorentz transform contains .T 0 ; 1/ R3 for some T 0 > 0. Then the transform of u solves NLKG for t > T 0 . After a suitable rotation, we may assume that the Lorentz transform is in the form (6.15). If ku0 kL4 .t>T 0 / < 1, then t;x
1 > ku0 kL4
t;x .t>T
Since the remaining region ˚ .t; x/ j t > T; t cosh
0/
kukL4
t;x .t
:
(2.85)
Tj
(2.86)
cosh x1 sin >T 0 and t>T /
x1 sinh < T 0 ; and jxj < S C jt
is bounded in space-time, the L4t;x norm of u in that region is bounded by the Sobolev embedding. Hence kukL4 .t>T / < 1. t;x
61
2.4 Scattering in P SC
Similarly, we have a local version of the above: Lemma 2.31. Let u be a finite energy solution of NLKG (2.1) on a time interval I 3 T . Then there is an open neighborhood O of the identity in the Lorentz group, such that the transform of u by any g 2 O extends to a solution in a space-time region including a time slab which contains T . Proof. Let w be the global solution as given in the proof of the previous lemma. Then u extends to I R3 [ fjxj > S C jt T jg, which is mapped to a region containing a time slab by any Lorentz transform sufficiently close to the identity. With these technical issues involving Lorentz transformed solutions out of our way, we now return to the main argument. By the precompactness of KC , there exists for any " > 0 an R0 ."/ > 0 so that Z juP j2 C jru j2 C ju j2 C ju j4 .t; x/ dx < " E.E u / 8 t 0 : Œjx x0 .t /j>R0 ."/
(2.87) We will now set out to improve (2.33), using Lemma 2.28. In fact, we shall show that jx0 .t/ x0 .0/j grows much more slowly than t. To this end let R .x/ WD .x=R/ be a cut-off function with .x/ D 1 on jxj 1, 0 1, and .x/ D 0 for jxj 2. Then define Z XR .t/ WD R .x/xe.t; x/ dx where e is the local energy density of u , i.e., eD
1 juP j2 C jru j2 C ju j2 2
1 ju j4 4
which satisfies the local conservation law @ t e D div.uP ru /. Therefore, Z Z P XR .t/ D 1 R .x/ uP ru .t; x/ dx x uP .x/r R .x/ ru .t; x/ dx where we used that P .u / D 0 in the first integral. Hence, Z ˇ ˇ ˇXP R .t/ˇ . juP j2 C jru j2 dx Œjxj>R
(2.88)
62
2 The Klein–Gordon equation below the ground state energy
with an implicit constant that does not depend on R. The following lemma expresses the aforementioned low-velocity bound for x0 .t/, cf. [84], Lemma 5.4 and [77], Lemma 7.4. Lemma 2.32. Let R0 ."/ be as in (2.87). Provided 0 < " 1 and R R0 ."/, one has ˇ ˇ ˇx0 .t/ x0 .0/ˇ R R0 ."/ (2.89) for all 0 < t < t0 where t0 can be taken of size &
R . "
Proof. We may assume that x0 .0/ D 0 and that x0 is continuous, see Exercise 2.23. Let ˚ t0 D inf t > 0 j jx0 .t/j R R0 ."/ : Since jx0 .t/j < R
R0 ."/ for all 0 < t < t0 , it follows from (2.87) and (2.88) that ˇ ˇ ˇXP R .t/ˇ . "E.E (2.90) u /; 8 0 < t < t0 :
On the other hand, for these t, jXR .t/j jx0 .t/jE.E u /
ˇZ ˇ jx0 .t/jˇ
1
ˇ ˇ R .x/ e.t; x/ dx ˇ ˇZ ˇ ˇ ˇ x x0 .t/ R .x/e.t; x/ dx ˇ ˇ
Z jx0 .t/jE.E u /.1
jx
C "/ Œjx x0 .t/j>R0 ."/
jx0 .t/jE.E u /.1
x0 .t/j R .x/je.t; x/j dx Z R0 ."/ je.t; x/j dx
C "/
CR"E.E u / Z 1 R0 ."/ e0 .t; x/ C ju j4 .t; x/ dx 4
where e0 is the free energy density. Since Z 1 e0 .t; x/ C ju j4 .t; x/ dx D 3E.E u / 4
K0 u .t/ 3E.E u /
we obtain the lower bound jXR .t/j E.E u / jx0 .t/j.1
C "/
CR"
3R0 ."/ :
2.5 Strichartz estimates for Klein–Gordon equations
63
Sending t ! t0 implies in particular that 1 E.E u / R 2
R0 ."/ :
(2.91)
jXR .0/j . R0 ."/ C "R E.E u / :
(2.92)
jXR .t0 /j Finally,
In summary, (2.90)–(2.92) imply that 1 R 2 which implies that t0 & "
1
R0 ."/ . R0 ."/ C "R C t0 "
R as desired.
By (2.51) there exists ı2 > 0 such that K2 u .t/ ı2 kru .t/k22 ;
8 t 0:
(2.93)
With t0 defined by Lemma 2.32, we argue as in the paragraph between (2.66) and (2.69), using Lemma 2.32 instead of the radial assumption, to conclude that ˇ˝ ˛ˇˇt0 ˇˇ 1 ı2 ˇ (2.94) u / : u / & RE.E ˇ uP j .x r C r x/u ˇ ˇ & ı2 t0 E.E 0 2 " Since the left-hand side is . RE.E u /, we obtain a contradiction by taking " > 0 sufficiently small. Exercise 2.33. Verify the omitted details for (2.94).
2.5
Strichartz estimates for Klein–Gordon equations
While there are excellent references for Strichartz estimates for both the wave and the Schrödinger equation, see Shatah, Struwe [126] and Cazenave [27], respectively, the corresponding estimates for the Klein–Gordon case are still largely contained in research papers, such as Brenner [21], Pecher [115], and Ginibre, Velo [61] as well as [77]. In order to keep these lectures self-contained and for the convenience of the reader, we provide a brief account of this class of estimates. Heuristically speaking, the Klein–Gordon equation uR
u C u D 0
(2.95)
64
2 The Klein–Gordon equation below the ground state energy
displays features of both the wave and Schrödinger equations. Indeed, the dispersion relation of (2.95), viz. !./ D hi, satisfies 1 !./ D 1 C jj2 C O.jj4 / ! 0 ; 2 !./ D jj C O.jj 1 / ! 1 which are – to leading order – the dispersion relations for the Schrödinger and wave equations, respectively. Another way of exposing this mixed nature of Klein–Gordon is through complex solutions. Indeed, setting u D e it v reduces (2.95) to vR C 2i vP
v D 0 :
We commence the rigorous discussion of Strichartz estimates by deriving pointwise decay of solutions to (2.95). As in the case of the wave equation, we rely on the usual dyadic Littlewood–Paley decomposition of the frequency space. In particular, we shall use the Besov spaces Bp;2 which are defined in terms of such a dyadic decomposition, see (2.114) below. The solution of (2.95) with data u.0/ D f; u.0/ P Dg is given by sin thri (2.96) u.t/ D cos thri f C g: hri We now express the right-hand side of (2.96) via the Fourier transform on fixed frequency domains. To do so we write the trigonometric functions as exponentials. This will yield terms like e ˙it hri f D K t f with the kernel K t , defined by the formal integral Z K t .x/ WD e i.˙thiCx/ d : Rd
We split the integral on the right-hand side into dyadic shells and analyze the decay of the different contributions in time as follows. Let 0 be a standard bump function equal to one around zero, and be a bump function supported in Rd n f0g which equals one on 12 < jj < 2, say. For t 0 and 1 define Z ˙ ˚ .t; x/ D e i.˙t hiCx/ .=/ d (2.97) Rd
2.5 Strichartz estimates for Klein–Gordon equations
65
and set ˚0˙ .t; x/ D
Z Rd
e i.˙thiCx/ 0 ./ d :
(2.98)
Those expressions are of the form Z I./ D
e i ./ a./ d ;
(2.99)
Rd
with a compactly supported function a 2 C01 and smooth, real-valued phase 2 C 1 . The asymptotic behavior of the integral I./ as ! 1 is described by the method of (non)stationary phase. This refers to the distinction between having a critical point on supp.a/ or not. We begin with the easier case of non-stationary phase. The reader should note that the standard Fourier transform is of this type. Lemma 2.34. If r ¤ 0 on supp a then the integral (2.99) decays as ˇ ˇZ ˇ ˇ e i ./ a./ d ˇ C.N; a; / N ; ! 1 ˇ Rd
for arbitrary N 1. Proof. Note that the exponential e i is an eigenfunction for the operator L WD
1 r r: i jr j2
(2.100)
Therefore we can apply L to the exponential inside of (2.99) as often as we wish. The adjoint of L is i r L D r : jr j2 Then for any positive integer N , one has ˇZ ˇ ˇZ ˇ ˇ ˇ ˇ ˇ i ./ e a./ d ˇ D ˇ LN .e i ./ /a./ d ˇ ˇ Rd Rd ˇZ ˇˇ ˇ Dˇ e i ./ .L /N a./ d ˇ d Z R j.L /N a./ j d Rd
C.N; a; / as claimed.
N
66
2 The Klein–Gordon equation below the ground state energy
The situation changes when has a critical point inside supp a. In that case one no longer has arbitrary decay and it can be very difficult to determine the exact rate. However, if the critical point is nondegenerate, i.e., has a nondegenerate Hessian at that point, then there is a precise answer as we now show. Lemma 2.35. If r .0 / D 0 for some 0 2 supp a, r ¤ 0 away from 0 and the Hessian of at the stationary point 0 is non-degenerate, i.e., det D 2 .0 / ¤ 0, then for all 1, ˇ ˇZ d ˇ ˇ e i ./ a./ d ˇ C.d; a; / 2 : (2.101) ˇ Rd
In fact, ˇ h ˇ k ˇ@ e
i .0/
Z Rd
iˇ ˇ e i ./ a./ d ˇ C.d; a; ; k/
d 2
k
(2.102)
for any integer k 1. Proof. Without loss of generality we assume 0 D 0. By Taylor expansion ./ D .0/ C
˛ 1˝ 2 D .0/; C O jj3 : 2
Since D 2 .0/ is non-degenerate we have jr ./j & jj on supp a. We split the integral into two parts, Z e i ./ a./ d D I C II Rd
where the first part localizes the contribution near D 0, Z 1 I WD e i ./ a./ 0 . 2 / d Rd
and the second part restores the original integral, viz. Z 1 II D e i ./ a./ 1 0 . 2 / d : Rd
The first term has the desired decay Z ˇ ˇ ˇ 0 . 21 /ˇ d . jI j . Rd
d 2
:
67
2.5 Strichartz estimates for Klein–Gordon equations
To bound the second term, we proceed as in the proof of the nonstationary case. Thus, let L be as in (2.100) and note that ˇ r ˇ ˇ ˇ ˛ ./ˇ C˛ jj 1 j˛j ˇD jr j2 for any multi-index ˛. Thus, let N > d be an integer. Then the Leibnitz rule yields Z 1 jIIj . j.L /N 1 0 . 2 / a./j d Œjj>
.
1 2
Z1
N
N
. 2 r
N
Cr
2N
/ rd
1
dr .
d 2
1 2
as claimed. The stronger bound (2.102) follows from the observation that ˇ ˇ ˇ ./ .0/ˇ . jj2 and the same argument as before (with N > d C 2k) concludes the proof. Finer asymptotic expansions of I./ in large can be found in Hörmander [73]. As a first application of the previous stationary phase lemma we estimate ˚0˙ as in (2.98). Corollary 2.36. One has for all t 0 sup j˚0˙ .t; x/j . hti
d 2
:
x2Rd
Proof. Putting the absolute values inside, leads to the bound O.1/ uniformly in .t; x/. Thus, we can take t 1. Let ˙ .I t; x/ WD ˙hi C t 1 x . Then a critical point is characterized by @ ˙ .I t; x/ D ˙hi
1
Ct
1
xD0
which has a unique solution 0 . If j0 j 1, then 0 62 supp 0 in which case we can apply the nonstationary phase lemma, see Lemma 2.34. On the other hand, if j0 j . 1, then the Hessian D 2 ˙ .0 I t; x/ is nondegenerate, uniformly so in 0 . Therefore, Lemma 2.35 implies the desired estimate. A well-known application of Lemma 2.35 is the following representation of S d 1 .x/ for large x, where S d 1 is the surface measure of the unit sphere in Rd .
1
68
2 The Klein–Gordon equation below the ground state energy
1
This will be needed in order to bound ˚˙ . Note that the function S d 1 .x/ is smooth and bounded, so we are interested in its asymptotic behavior for large jxj, both in terms of oscillations and size. We need to understand the oscillations in order to bound derivatives of S d 1 .x/.
1
Corollary 2.37. One has the representation
1 .x/ D e
S d
1
i jxj
ijxj
!C .jxj/ C e
! .jxj/;
jxj 1
(2.103)
where !˙ are smooth and satisfy j@kr !˙ .r/j Ck r
d
1 2
k
;
8r 1
and all k 0. Proof. By definition,
1 .x/ D
S d
Z
1
Sd
1
e ix S d
1
.d /
which is rotationally invariant. Therefore we choose x D .0; : : : ; 0; jxj/ and denote the integral on the right-hand side by I.jxj/. Thus, Z I.jxj/ D e i jxjd S d 1 .d / : Sd
1
Note that d , viewed as a function on S d 1 , has exactly two critical points, namely the north and south poles .0; : : : ; 0; ˙1/ which are, moreover, nondegenerate. We therefore partition the sphere into three parts: a small neighborhood around each of these poles and the remainder. The Fourier integral I.jxj/ then splits into three integrals by means of a smooth partition of unity adapted to these three open sets, and we write accordingly X I.jxj/ D I˙ .jxj/ C Ieq .jxj/ : ˙
The latter integral has non-stationary phase and decays like jxj N for any N , see Lemma 2.34. On the other hand, the integrals I˙ exactly fit into the frame work of Lemma 2.35 which yields the desired result. Note that one can absorb the equatorial piece Ieq .jxj/ into either e ˙i jxj !˙ .jxj/ without changing the stated properties of !˙ , and (2.103) follows.
69
2.5 Strichartz estimates for Klein–Gordon equations
By means of Corollary 2.37 we can now easily estimate ˚˙ . Corollary 2.38. The kernels ˚˙ from (2.97) above satisfy the uniform estimates d
˙
˚ .t; /
1
C 2 C1 t
d 2
(2.104)
for all t > 0 and 1. Proof. Introducing polar coordinates into ˚˙ , and invoking Corollary 2.37 yields X Z ˚ .t; x/ D d e i .rIt;jxj/ !˙ rjxj .r/r d 1 dr (2.105) ˙
with ˙ .rI t; jxj/ D t 1 hri ˙ rjxj, and D ˙ a fixed sign. The phase C is easier, and we leave it to the reader. Thus, we write .rI t; jxj/ D .rI t; jxj/ and note that r @r rI t; jxj D t hri
jxj
as well as @2r .r/ ' 2 t. Consider the case t ' jxj & phase applied to (2.105) in the r-variable one obtains ˇ ˇ ˇ˚ .t; x/ˇ . d .t/ d
d
1 2
1
min.1; 2 t
1 2
1
. Then by stationary
/:
(2.106)
1
The weight .t/ 2 here derives from the decay of !˙ .rjxj/ since jxj ' t, 1 1 whereas min.1; 2 t 2 / means that one either passes absolute values inside (which gives the 1 in this minimum), or applies Lemma 2.35 leading to the other factor, i.e., @2r .r/
1 2
1
1 2
' 2 t
:
In order to apply Lemma 2.35 we are using that j@kr !˙ .rjxj/j Ck .jxj/
d
1 2
;
8k1
as guaranteed by Corollary 2.37. This is important since the proof of Lemma 2.35 requires several integrations by parts in the r-variable. If t ' jxj . 1 , then ˇ ˇ ˇ˚ .t; x/ˇ . d :
(2.107)
70
2 The Klein–Gordon equation below the ground state energy
Finally, if t 6' jxj, then j@r .rI t; jxj/j ' t C jxj whence by Lemma 2.34 ˇ ˇ ˝ ˛ ˇ˚ .t; x/ˇ . d .t C jxj/ N (2.108) for every N 1. One of the many ways to combine the bounds (2.106)–(2.108) is the universal estimate (2.104), and we are done. We now return to the free Klein–Gordon evolution and denote r U .t/ WD e it hri for 1; U0 .t/ WD e it hri 0 .r/ : These operators are L2 -bounded uniformly in . Our aim is to prove an estimate of the form kU f kLpt Lqx C hiˇ kf kL2x
(2.109)
for both D 0 and 1 where ˇ needs to be determined, depending on p; q (recall that we needed p D 3; q D 6 in Lemma 2.2, but we shall work in greater generality here). Once these bounds are obtained, we shall then sum over 2 f2j j j 0g [ f0g to obtain the desired Strichartz estimates. One elegant approach to (2.109) is to appeal to the Keel, Tao theory [82]. This merely requires inserting our pointwise estimates into their machinery and then to apply Littlewood–Paley theory to sum the resulting bounds. We find this not to be too instructive, especially for readers who have little experience with Strichartz estimates. We therefore first follow the usual Ginibre, Velo approach, see [61], which yields everything but the endpoint. The standard approach to (2.109) is the so-called T T -method. This method hinges on the following equivalencies. Lemma 2.39. Fix p; q and some constant C . Then the following are equivalent: (i) kU U F kLpt Lqx C2 kF kLp0 Lq0 , x t (ii) kU gkL2x C kgkLp0 Lq0 , x t (iii) kU f kLpt Lqx C kf kL2x where f; g; F are Schwartz functions. Proof. Estimates (ii) and (iii) are equivalent by duality. Composing these estimates implies (i). Assume (i). Then 2 kU F kL 2 D jhU F; U F ij D jhF; U U F ij x
kF kLp0 Lq0 kU U F kLpt Lqx C2 kF k2 p0 t
x
q0
L t Lx
71
2.5 Strichartz estimates for Klein–Gordon equations
and we are done. The most approachable estimate is (i). By means of it we can now convert the pointwise estimates from above into the Lpt Lqx estimates of Corollary 2.41. We require the following well-known fact about convolutions, which we state without proof (see for example [97]). Lemma 2.40 (Hardy–Littlewood–Sobolev inequality, or fractional integration). The one-dimensional convolution estimate kf
1 kLs C.r; s; ˛/kf kLrt jtj˛ t
(2.110)
holds in the range 0 < ˛ < 1, provided the scaling condition 1C
1 1 D C ˛; s r
1
is satisfied. We can now finally state and prove our first Strichartz estimates for Klein– Gordon. For all matters of interpolation in the Besov setting, we refer the reader to the standard reference Bergh, Löfström [13]. Corollary 2.41. For any d 1, 2 < p 1, 2 q 1, with has the bounds
1 p
C
d 2q
D
d , 4
one
kU f kLpt Lqx C hiˇ kf kL2x for any Schwarz function f . Here ˇ D 21 . d2 C 1/. q10
1 / q
and either D 0 or 1.
Proof. First, by Corollaries 2.36 and 2.38 and Young’s inequality d
kU f kL1 . 2 C1 t x
d 2
kf kL1x :
On the other hand, there is the obvious L2 -estimate kU f kL2x . kf kL2x and so by interpolation d
kU f kLqx . . 2 C1 t
d 2
1
/ q0
1 q
kf kLq0 : x
(2.111)
72
2 The Klein–Gordon equation below the ground state energy
We write U U F .t/ explicitly in the form .U U F /.t/ D
Z
1
e i.t
s/hri 2
1
r F .s/ ds :
The factor of 2 causes no essential difference to , so from (2.111) Z 1 1 1 d d . 2 C1 t 2 / q0 q kF .s/kLq0 ds : k.U U F /.t/kLqx . x
1
Applying Lemma 2.40 with s D p, r D p 0 , ˛ D d2 . q10 q1 /, and 1 C p1 D p10 C ˛ yields the desired conclusion. Note that this excludes the case p D 2 as it would imply that ˛ D 1. On the other hand, ˛ D 0 is allowed here since it means that q D 2, p D 1 which is of course admissible. A special case of Corollary 2.41 is the estimate, for the case of .t; x/ 2 R1C3 , 5
. C hi 9 kf k2
kU f k
18 L3t Lx5
(2.112)
for all 1 as well as D 0. The point here is that we can sum up this bound over a geometric sequence of by means of standard Littlewood–Paley theory. Since we can place f into H 1 .R3 / in the end, this frees up a derivative of order 94 on the left18 hand side. Finally, this much regularity is more than sufficient to improve L 5 .R3 / to L6 .R3 / via Sobolev imbedding (in fact, only a 31 -derivative is needed for that), whence we derive the L3t L6x .R1C3 t;x / bound required by Lemma 2.2. Let us now explain the argument in the previous paragraph in more detail. First, let 0 ./ D 1 for jj < 1 and 0 ./ D 0 for jj > 2. Further, set ./ D 0 ./ 0 .2/ and j ./ WD .2 j /. Then one has the partitions of unity X 1D 8¤0 j ./; j 2Z
as well as 1 D 0 ./ C
X
j ./;
8:
j >0
We shall only use the latter here, and define associated operators Pj f WD .
O /_ ;
jf
P0 f WD . 0 fO /_
8 j > 0;
(2.113)
2.5 Strichartz estimates for Klein–Gordon equations
73
and the Besov norms kf kBp;2 WD kP0 f kp C
X
22j kPj f kp2
12
:
(2.114)
j >0 These spaces can of course also be defined with other powers than 2, i.e., Bp;r , but we shall only use (2.114) here. Thus, square summing (2.112) yields
ke ithri f kL3 B 0
1C3 18 ;2 .R t;x / 5
t
C kf k
5
H 9 .R3 /
or equivalently, ke it hri f k
4
9 L3t B 18
5 ;2
.R1C3 t;x /
C kf kH 1 .R3 / :
(2.115)
We now wish to convert the Besov space on the left-hand side into something more useful (but strictly weaker) such as L6x .R3 /. To do so, recall that 0 Bp;2 ,! Lp for 2 p < 1 :
Indeed, by the classical Littlewood–Paley theorem,
X 21
kf kp ' kP0 f kp C jPj f j2 ; 8 1 < p < 1 ; p
j >0
see for example [121]. Further, for p 2, one has the triangle inequality
X X 12 21
kPj f kp2 : jPj f j2
p
j >0
(2.116)
(2.117)
(2.118)
j >0
Combining (2.117) with (2.118) yields (2.116) as desired. Exercise 2.42. (a) Prove (2.118), and show that it fails for p < 2. In that case, establish (2.118) with the inequality sign reversed. (b) Prove Bernstein’s inequality: if supp.fO/ E Rd where E is measurable and f is Schwartz, say, prove that 1
kf kq jEj p
1 q
kf kp ;
81pq1
(2.119)
where jEj stands for the Lebesgue measure. Hint: First handle the case q D 1; p D 2, by Plancherel and Cauchy–Schwarz, and then dualize and interpolate.
74
2 The Klein–Gordon equation below the ground state energy
(c) Using Bernstein’s inequality and (2.117) establish the embeddings 0 Bq;2 .Rd / ,! Bp;2 .Rd / ,! Lp .Rd /
for any 2 q p < 1,
1 q
1 p
, d
D
1
0. In particular,
3 B 18 .R3 / ,! L6x .R3 /; 5
;2
(2.120)
1
6 B6;2 .R3 / ,! L9x .R3 / :
(2.121)
Combining (2.115) with (2.121) now implies the desired Strichartz estimates needed in Lemma 2.2, albeit at the moment only for the homogeneous case. To be more precise, we have obtained the following with F D 0. Lemma 2.43. Any solution of u C u D F;
u.0/ D u0 ; u.0/ P D u1
in R1t R3x satisfies the estimates kukL3 L6x . ku0 kH 1 C ku1 k2 C kF kL1 L2x : t
t
(2.122)
Proof. Only the case F ¤ 0 requires proof. By linearity, we can take u0 D u1 D 0. By Exercise 2.42 it further suffices to show that kPj ukL3 L6x . kPj F kL1 L2x t
t
for any j 0. Retracing our steps, this in turn follows from kPj uk
4
9 L3t B 18
. kF kL1 L2x t
(2.123)
5 ;2
for j 0. For simplicity, we restrict ourselves to j D 0, since the other cases are the same up to the 2j -weights. We have shown, amongst other things, that 18
U0 W L2 ! L3t Lx5 ; U0 W L1t L2x ! L2 all spaces being in R3 (the second line here is just the trivial L2 bound). The composition of course yields that 18
U0 U0 W L1t L2x ! L3t Lx5 :
(2.124)
75
2.5 Strichartz estimates for Klein–Gordon equations
The left-hand side is given by the operator Z 1 U0 U0 F .t/ D e i.t 1
s/hri 2 0 ./ F .s/ ds
which is similar to the Duhamel integral, up to the fact that the latter differs by insertion of the indicator Œ0<s
0 is harmless, the other one s < t is not. Fortunately, we can appeal to the well-known Christ–Kiselev lemma, see Lemma 2.44 below.5 By means of this device, we can now conclude that the Duhamel integral satisfies the same bound as in (2.124) and we are done. We now record the Christ–Kiselev lemma from [34], and refer the reader to that reference for a proof. Note that p < q is needed as evidenced by the Hilbert transform. Lemma 2.44. Suppose T W Lp .I I X/ ! Lq .I I Y / where 1 p < q 1 and X; Y are Banach spaces. Let T be given by a kernel K, i.e., Z Tf .t/ D K.t; s/f .s/ ds I
and assume that the operator norm of T is 1. Then Z T> f .t/ WD Œs
satisfies T> W Lp .I I X/ ! Lq .I I Y / with a bound on the norm given by C.p; q/. For later purposes (the one-dimensional equation is discussed in the final chapter), we ask the reader to verify the following 1-dimensional Strichartz estimates. They follow immediately from the methods which we developed in this section. Exercise 2.45. For any Schwartz function u in R1C1 t;x derive the a prori bound kuk
1
4 L4t B1;2
C kukL1 Hx1 C kuk P L1 L2x t
t
. uŒ0
H 1 L2
5
C ku C uk
4
3
4 CL1 L2 L t3 B1;2 t x
: (2.125)
One should observe that (2.122) follows simply by integrating the bound on U0 . Nevertheless we state this lemma since it is useful when working with more complicated function spaces.
76
2 The Klein–Gordon equation below the ground state energy
By interpolation, infer the following bounds: for any 4 p 1, and any 2 q 1,
kukLpt B ˛ . uŒ0 H 1 L2 C ku C uk 4 3 (2.126) 4 CL1 L2 L t3 B1;2 t x
q;2
where ˛ D 1 p3 , q1 D q r < 1 provided ˛
1 2 1 q
2 . By the embedding p 1 , finally conclude that r
˛ Bq;2 .R/ ,! Lr .R/ for any
kukLpt Lrx . uŒ0 H 1 L2 C ku C uk
4
3
4 CL1 L2 L t3 B1;2 t x
for any 4 < p 1, 0 <
1 r
1 2
(2.127)
2 . p
To conclude this section, we now discuss the case p D 2 in Corollary 2.41 for the sake of completeness (strictly speaking, for NLKG we do not require p D 2 in what follows, but it is convenient and simplifies the proofs in the construction of the center-stable manifold; however, for NLS the Keel–Tao endpoint is crucial for the Lyapunov–Perron method in the energy space due to the presence of modulation parameters, see Chapter 3). We cannot use fractional integration for estimates in the spaces L2t Lpx as they correspond to the endpoint ˛ D 1 in Lemma 2.40. This is precisely the endpoint issue which was resolved by Keel, Tao in [82]. Here we apply 2 the endpoint Strichartz estimate of Keel–Tao [82] to U .t1C d / when 1. The re-scaling of time is explained by the fact that 2
d 2
kU .t1C d /f k1 . t
kf k1 ;
(2.128)
see Corollary 2.38. One therefore obtains from the main theorem in [82] that 1
1
kU f kL2 Lpx .Rd / C 2 C d kf k2 t
where p D
2d d 2
(2.129)
for d 3. Applying Littlewood–Paley theory as before now yields ke ithri f kL2 B 0 t
d p;2 .R /
C kf k
1C 1 d
H2
or equivalently, ke it hri f kL2 B t
p;2 .R
d/
C kf kH 1
where D 12 d1 . In summary, we obtain the following class of Strichartz estimates for the free Klein–Gordon equation.
77
2.6 Summary and conclusion
Lemma 2.46. Any solution of u C u D F;
u.0/ D u0 ; u.0/ P D u1
in R1t Rdx satisfies the estimates, with p D kukL2 B t
where D
1 2
1 1 p;2 \L t Hx
1 d
2d , d 2
d 3,
. ku0 kH 1 C ku1 k2 C kF kL2 B 0
and 0 D 1
t
p 0 ;2
(2.130)
CL1t L2 x
. 1
6 We remark that the endpoint of Lemma 2.46 in d D 3 is L2t B6;2 ,! L2t L9x , 8 3 1 6 8 3 which interpolates with L t Lx to give L t Lx .R /. We state the latter norm here explicitly, since it was used above, see (2.80). Note that for the Schrödinger case 1 , whereas for the wave equation it is the the endpoint with H 1 .R3 / data is L2t B6;2 2 P0 3 forbidden one L t B1;2 .R /. Hence, Klein–Gordon is similar to the Schrödinger equation, and better than wave, but since the curvature of the hyperboloid z D hi becomes degenerate at infinity, one loses some of the regularity here, to be precise 56 of a derivative.
Exercise 2.47. (a) Verify that (2.130) is optimal. More precisely, show that one cannot have more regularity on the left-hand side, nor replace p by any other exponent. (b) Derive other classes of Strichartz estimates than those in Lemma 2.46 by allowing for more general combinations of (2.106)–(2.108). More precisely, derive the estimates stated in Lemma 4.1 of [77].
2.6
Summary and conclusion
This chapter lays the ground work for the analysis of the cubic NLKG equation (2.1) (it applies to other powers and dimensions but we restrict ourselves to this case for simplicity) needed in this monograph. After proving the basic wellposedness theorem based on energy and Strichartz estimates (which are proved in the final section), we turn to the analysis of the ground state stationary solution and the variational structure required for this purpose. In particular, we present the basic Payne– Sattinger theory from 1975 describing the long-term qualitative behavior of solutions at energies below the ground state energy. Following Ibrahim, Masmoudi, and the first author, we then prove scattering to zero for all solutions which are known to be
78
2 The Klein–Gordon equation below the ground state energy
global from the Payne–Sattinger theory. This is done by means of the Kenig–Merle concentration-compactness method. One advantage of the subcritical NLKG equation (2.1), as compared to the energy critical equations of the original reference [84], lies with the relative simplicity of the technical details involved in executing this method. This is especially true in the radial case.
3
Above the ground state energy I: Near Q
With this chapter we begin the study of the case E.u; u/ P J.Q/. We work perE E turbatively near ˙Q where Q D .Q; 0/, and exhibit the well-known exponential instability of these equilibria. Linearizing the nonlinear equation about the ground state yields a linear operator with a negative eigenvalue which is responsible for the exponential growth. E We then ask the natural question how to stabilize the nonlinear flow near ˙Q. On the linear level this simply means projecting away the unstable mode, but on the nonlinear level it is somewhat more difficult. However, the issue at hand turns out to be a version of the well-known center-stable manifold theorem. We recall this result, and present the theory of Bates and Jones [6] which first carefully defines, and then constructs the stable, unstable, and center stable manifolds for an abstract ODE in Banach spaces. These authors use the Hadamard method [69] for this purpose, also known as the method of invariant cones or the graph transform method. In order to apply the Bates–Jones theory to the radial NLKG equation, one needs nothing more than energy estimates and Sobolev imbedding. While that can be seen as an advantage, it limits the range of powers to those no larger than 3. The extension of the Hadamard method to both higher powers as well as nonradial data is studied in [112]. Since nonradial data lead to moving solitons which generate unbounded gradient terms in the linearized equation relative to the standard energy metric, [112] differs from [6] quite substantially. Later in this section, we construct the center-stable manifolds via the Lyapunov– Perron method, see [9], [90], [122]. Implementing this method requires more information on the linearized flow as compared to the Hadamard approach. More specifically, one uses dispersive estimates (pointwise or Strichartz) for the linearized flow. Since we wish to work entirely in the energy space, we restrict ourselves to Strichartz estimates. This dispersive theory in turn rests on a full description of the spectrum of the linearized operator generated by the ground state, which is often very hard to obtain. In the one-dimensional case, see [88], [90], a purely analytical treatment is available thanks to the explicit and special form of the soliton. However, in three dimensions such explicit representations of Q are not available and we currently only have detailed information on special cases such as the cubic one, see [41], [44], and [99]. While the former two references depend on numerics, the latter is rigorous.
80
3 Above the ground state energy I: Near Q
While it requires more, the Lyapunov–Perron approach has the advantage of providing more information on the center-stable manifold, namely asymptotic stability and scattering to the ground states. Moreover, as shown in [9], [90], [122], the Lyapunov–Perron method applies equally well in the presence of symmetry parameters. In Section 3.1, we discuss the linearized operator and the energy as a saddle surface near ˙.Q; 0/. In particular, we shall see that the ground states are unstable. Section 3.2 introduces the notion of stable, unstable, and center manifolds. We then present the Bates–Jones theory for conservative equations. Section 3.3 develops the Lyapunov–Perron approach in the context of the NLKG equation, which requires the gap property of LC , see Section 3.1.2 (at least in the implementation adapted here). An overview of some rudiments of the dispersive theory is given in Section 3.4. In particular, we present Beceanu’s linear dispersive theory in the context of the NLS equation, see [8]. This in turn requires detailed spectral information on the linearized NLS matrix Hamiltonian, see [52], [99], [122]. We close this chapter with an application of this linear theory to the Lyapunov– Perron construction for the cubic radial NLS equation in R3 . The latter is essentially a restriction of the full nonradial result of Beceanu’s [9] to radial functions in the energy class.
3.1
Energy landscape
3.1.1 The linearized operator LC We now expand the energy near .Q; 0/ and exhibit the saddle structure of the energy near the ground state. This analysis involves the linearized operator LC D
C1
3Q2
(3.1)
and its spectral properties, which we discuss. Moreover, we present a quantitative form of Lemma 2.9, which will play a crucial role in the description of the global dynamics under the energy constraint (1.5). In fact, in combination with the onepass theorem and another device which we call the ejection lemma (which quantifies the dominance of the unstable dynamics for non-trapped trajectories near the ground states), it will guarantee a fixed sign for the functionals K and K2 once the solution has moved sufficiently far away from the ground states. Henceforth, we mostly restrict ourselves to radial function spaces.
81
3.1 Energy landscape
To be specific, consider solutions u D Q C v of (2.1) where v is small in a suitable sense. Then 3Q2 v D 3Qv 2 C v 3 :
(3.2)
Z ˛ 1 1 1˝ LC vjv C kvk P 22 .4Qv 3 C v 4 / dx 2 2 4 R3 ˛ 1 1˝ 3 P 22 C O kvkH D J.Q/ C LC vjv C kvk 1 ; 2 2 ˝ 0 ˛ 1 ˝ 00 ˛ 3 Ks .Q C v/ D Ks .Q/jv C Ks .Q/vjv C O kvkH 1 2
(3.3)
vR C LC v D N.v/;
N.v/ WD .v C Q/3
Q3
The energy functionals satisfy E.Q C v; v/ P D J.Q/ C
for s D 0; 2. In particular, 3 K0 .Q C v/ D 2h Q C Q 2Q3 jvi C h. C 1 6Q2 /vjvi C O kvkH 1 ˝ ˛ ˝ ˛ 3 D 2 Q3 jv C .LC 3Q2 /vjv C O kvkH 1 : (3.4) Since hLC QjQi D 2kQk44 , it follows that the energy is indefinite around .Q; 0/ which establishes the aforementioned saddle behavior of the energy near the ground state. It is now a simple matter to prove that the ground state is unstable. Corollary 3.1. For every " > 0 the ball B" .Q; 0/ in H satisfies B" .Q; 0/ \ P SC ¤ ;;
B" .Q; 0/ \ P S ¤ ;
(3.5)
where P S˙ are from the previous chapter. In other words, for every " > 0 there are two nonempty open subsets of B" .Q; 0/ of data that lead to blowup in finite positive time as well as global forward existence and scattering to zero, respectively. Proof. Insert v D "Q, vP D 0 into (3.3), (3.4) to conclude that for small " > 0 the resulting data lie in P SC , whereas for small " < 0 they lie in P S . So the theory of Chapter 2 implies the desired conclusion. Since LC has only one negative eigenvalue (as the next lemma will show), the intersections in (3.5) are approximately “cones” with the vertex at Q, see Figure 3.1 and also Figure 1.2 (p. 9). Our dynamical picture will replace those two “cones” by two “hypersurfaces”: the center-stable and center-unstable manifolds of codimension 1, which are tangent to the “cones” exactly on the two “curves”: the stable and unstable manifolds with dimension 1. See Figure 5.1 (p. 175).
82
3 Above the ground state energy I: Near Q K.u/ D 0
E.u; u t / < J.Q/
.Q; 0/ Figure 3.1. The energy functionals E and K (K0 or K2 ) near .Q; 0/
Lemma 3.2 gives a more detailed description of the spectrum of LC . Note that no statement is made about the interval .0; 1/ which appears to require much more work, see Section 3.1.2. Lemma 3.2. As an operator in L2rad , LC has only one negative eigenvalue, which is non-degenerate, and no eigenvalue at 0 or in the continuous spectrum Œ1; 1/. Proof. LC has at least one negative eigenvalue because hLC QjQi D
2kQk44 < 0;
and at most one, because hQ3 jvi D 0 H) hLC vjvi 0:
(3.6)
To see this, suppose that f 2 H 1 satisfies hQ3 jf i D 0 and hLC f jf i D 1, and let v D "Q C ıf for small "; ı 2 R. Then 4 K0 .Q C v/ D 2"kQkL ı 2 1 C h3Q2 f jf i C O."2 C "ı C ı 3 /; 4
3.1 Energy landscape
83
so there exists " D O.ı 2 / such that K0 .Q C v/ D 0. On the other hand, J.Q C v/ D J.Q/
1 2 ı C O.ı 3 / < J.Q/; 2
which contradicts the minimizing property of Lemma 2.9. Next, if 0 6D f 2 L2rad solves LC f D 0, then f ? Q3 ; Q, because LC Q D 2Q3 and LC .r@r C 1/Q D 2Q. By the Sturm oscillation theorem, f changes its sign exactly once, say at r D r0 > 0. The same applies to Q3 Q.r0 /2 Q, since Q.r/ is decreasing. Then .Q3 Q.r0 /2 Q/f is non-zero with a definite sign, contradicting f ? Q3 ; Q. The absence of embedded eigenvalues is standard, and follows also from the asymptotic equation for any radial eigenfunction, viz. LC f D 2 f H) . @2r C 1
2 /rf .r/ D 3rQ2 .r/f .r/ :
(3.7)
By the exponential decay of Q, a standard Volterra perturbative ansatz reveals that for any 1, f .r/
1 ir p2 e r
1
as r ! 1
(3.8)
whence f 62 L2 .R3 /. Hence there are neither embedded nor threshold eigenvalues. The reader may find it instructive to generalize Lemma 3.2 to other nonlinearities and dimensions. Exercise 3.3. 1 (a) Show that (3.6) can be improved to the following, provided v 2 Hrad : 2 hQ3 jvi D 0 H) hLC vjvi ' kvkH 1
(3.9)
Can this hold if we drop the radial assumption? Hint: first prove (3.9) as a lower bound with c0 kvk22 on the right-hand side for some c0 > 0. Use an argument by contradiction. (b) Denote by the L2 -normalized negative eigenmode of LC and let P? WD 1
hj
(3.10)
be the orthogonal projection perpendicular to . Prove that 2 ? 1 hLC vjvi ' kvkH 1 on P .H / :
(3.11)
84
3 Above the ground state energy I: Near Q
(c) Provide the details for (3.8). Hint: Show that (3.7) admits solutions Z 1 g.rI / D g0 .rI / C G.r; sI /V .s/g.sI / ds ; g0 .rI / D e
ir
p
r 2 1
;
V D 3Q2 : p sin .r s/ 2 1 p . 2 1
For > 1 the Green function is G.r; sI / D Find the correct expression here for D 1. (d) Show that fu 2 H 1 j K0 .u/ D 0g is a C 1 hypersurface which is bounded away from zero. Conclude from (a) that fK0 .u/ D 0g does not intersect the shaded area in Figure 3.1, which is the one where the energy is strictly less than J.Q/.
3.1.2 The gap property of LC Lemma 3.2 leaves open whether or not LC has any eigenvalues in the interval .0; 1/. While the orbital stability part of our analysis does not require any information on this issue (see Theorem 5.1 below), the construction of the center-stable manifold via the asymptotic stability approach does rely on the following gap property of LC : LC has no eigenvalue in .0; 1 and no resonance at the threshold 1.
(3.12)
For the definition of “resonance” see Section 3.4. Both parts of this statement have been verified by L. Demanet and the second author in [44] via numerics; this was done in the nonradial setting. Their approach is based on the Birman–Schwinger theorem which equates the number of eigenvalues and resonances 1 (the threshold) of LC , counted with multiplicity, to the number of eigenvalues 1 of the self-adjoint, positive, compact operator KC WD 3Q. / 1 Q, again counted with multiplicity. In fact, [44] by means of a numerical computation finds that KC has precisely four eigenvalues greater than 1 (corresponding to the negative ground state plus the zero eigenvalue of multiplicity three: LC rQ D 0), whereas the fifth largest eigenvalue was calculated to be < 0:98 with an estimated 8 to 9 digits of accuracy behind the comma (to be precise, 5 D 0:97039244 : : :). By the Birman–Schwinger theorem, this verifies (3.12). An analytical argument, which rigorously verifies the numerical findings of [44], is presented in [41]. While that argument is rather computational, it is not numerical and can be verified with some effort by “pen and paper” and a handheld calculator. The basic approach is to find a very accurate approximation to the ground state soliton, which itself is not explicitly known. The approximation is done on three intervals, namely Œ0; :9/, Œ:9; 2:5/, and Œ2:5; 1/. On the latter, one finds an
85
3.2 Center, stable, and unstable manifolds in hyperbolic dynamics r
3r
asymptotic expression for Q with leading order e r and corrections e r 3 , whereas 1 is approximated by polynomials of degree 11 with rational on the former two Q coefficients. One then proceeds to show that the true soliton lies very close to the approximate one. The spectral problem is reformulated in terms of the approximate ground state, with the verification of absence of eigenvalues amounting to checking that a Wronskian determinant does not vanish; the Wronskian here being the one between solutions on Œ0; R and ŒR; 1/ to the linear eigenvalue equation. One checks that, indeed, for the cubic nonlinearity this Wronskian does not vanish for energies in Œ0; 1, but that it can be very small. The latter is an instance of the observation from [44] that lowering the power of the nonlinearity slightly below the cubic one destroys the “gap property”. To conclude this discussion, we wish to reiterate that the difficult gap property is not needed in the construction of the invariant manifolds, but rather in order to establish the asymptotic evolution of solutions starting on the center-stable manifold; in other words, in order to prove that these solutions scatter to the ground state. For a construction of the invariant manifolds using only “soft” spectral information, see [112].
3.2
Center, stable, and unstable manifolds in hyperbolic dynamics
In this section we recall some classical results on center, stable, and unstable manifolds near an equilibrium. First, consider the ODE xP D Ax C f .x/;
f W Rd ! Rd
(3.13)
where f .0/ D 0, Df .0/ D 0, and f smooth, say. The Hartman–Grobman theorem states the following: if A has no eigenvalues on the imaginary axis, then there exists a local diffeomorphism around 0 which conjugates the nonlinear flow t with that of the linearized equation: e tA ı D ı t ;
jt j < "
The situation is very different when A does have imaginary eigenvalues. This case arises by linearizing (2.1) around the ground state. In fact, as we shall see, the center piece (i.e., the subspace of the linearized operator with spectrum on the imaginary axis) is infinite dimensional in the phase space. There are two main approaches to proving theorems involving center manifolds known to the authors. One is more geometric, involving splitting the flow along complementary subspaces corresponding to different exponential growth rates which allows for a natural construction of all invariant manifolds via a contraction on the
86
3 Above the ground state energy I: Near Q
xP D Ax C f .x/
xP D Ax
Figure 3.2. The Hartman–Grobman theorem
graphs. It is attributed to Hadamard, and was used by Bates and Jones [6] in the context of the radial nonlinear Klein–Gordon equation. The other one, called Lyapunov– Perron method, is based on finding the splitting into invariant manifolds by means of integral equations and by adjusting the initial condition of the unstable mode so that it remains small forever. A systematic exposition of this method is given in Vanderbauwhede [138], and see Ball [5] for an adaptation to infinite dimensions. The latter approach seems more powerful for the purposes of the stability analysis of (2.1), and it was used in [109], [122], [111], as well as in [9]. However, for the sake of completeness we shall begin by presenting an outline of the results in [6], but in less generality. We shall ignore the “dissipative” case (D) in [6] and restrict ourselves the the “conservative” one, which is called (C) in [6]. We remark that the dissipative case is much more stable, in fact, the center-unstable manifold in that setting is finite dimensional, in contrast to the conservative one. An early reference on dissipative equations is Keller [83]. For a comprehensive introduction to (semi)groups of operators, see [51]. Bates, Lu, Zeng [7] have obtained far-reaching generalizations of [6] that we shall not touch upon here, however.
3.2 Center, stable, and unstable manifolds in hyperbolic dynamics
87
3.2.1 Definition and construction of the manifolds following Bates and Jones Consider the equation (3.14)
uP D Au C f .u/
where u 2 X , a Banach space, and such that the following properties hold: (H1) A W X ! X is a closed, densely defined operator which generates a strongly continuous group S.t/. (H2) the spectrum .A/ can be written in the form .A/ D s [ c [ u with s D f 2 .A/ j Re < 0g ; c D f 2 .A/ j Re D 0g ; u D f 2 .A/ j Re > 0g where the associated spectral subspaces satisfy dim X u C dim X s < 1 and the evolution S.t/ restricted to X c , which we denote by S c .t/, satisfies the bound for all > 0, kS c .t/uk M./e jtj kuk;
8t:
(3.15)
(H3) The nonlinearity f .u/ is defined on all of X and is locally Lipschitz with f .0/ D 0. For all " > 0 there exists a neighborhood U D U."/ of 0 such that f has Lipschitz constant " on U."/. For the purposes of this monograph, (3.14) will be (2.1) linearized around the ground state Q. As another example, consider the finite-dimensional matrix 2 3 1 1 0 0 0 0 60 1 0 0 0 07 6 7 60 0 1 0 0 07 6 7 AD6 (3.16) 7 60 0 0 0 1 07 40 0 0 0 0 15 0 0 0 0 0 0 and let 12 , 3 , and 456 denote the projections onto the subspaces defined by the coordinates specified by the indices. Then for any ˛ < 1 and some constant M , one has ke tA 12 k Me ˛t for all t 0, ke tA 3 k Me t for all t 0, and for any ˇ > 0, ke tA 456 k Me ˇ t for all t 0.
88
3 Above the ground state energy I: Near Q
Definition 3.4. Let ˚ t denote the flow of (3.14). If V U we say that V is positively invariant relative to U iff for any v 2 V [ [ ˚s .v/ U H) ˚s .v/ V; 8 t > 0 : s2Œ0;t
s2Œ0;t
Negatively invariant relative to U is defined analogously and simply invariant relative to U means invariant in both time directions. Finally, if we can take U D X, then we drop the “relative to U ” part. In what follows c ; s ; u ; cs ; cu denote the projections onto various spectral subspaces. In the latter two cases, this means onto X cs WD X c ˚ X s and X cu WD X c ˚ X u , respectively. Definition 3.5. Given a neighborhood U of 0 we use the following terminology: ı The stable manifold is W s WD fu 2 U j ˚ t .u/ 2 U 8 t 0; ˚ t .u/ ! 0 exponentially as t ! 1g : ı The unstable manifold is W u WD fu 2 U j ˚ t .u/ 2 U 8 t 0; ˚ t .u/ ! 0 exponentially as t !
1g :
ı A Lipschitz manifold Y U is called center-stable iff (i) Y is invariant relative to U , (ii) cs .Y / contains a neighborhood of 0 in X cs , (iii) Y \ W u D f0g. We denote such a Y by W cs . Switching stable and unstable in the previous item one obtains the center-unstable manifold, and the center manifold is defined analogously but with c .Y / containing a neighborhood of X c and Y \ W u D Y \ W s D f0g. Note that reversing time exchanges the roles of the stable and unstable manifolds. The main result in [6] concerning the existence of these manifolds is the following. Theorem 3.6. Under the above assumptions on (3.14) there exists a neighborhood U of 0 in X such that W s , W u , W cs , W cu , and W c exist as Lipschitz manifolds in U and they are tangent to X s , X u , X cs , X cu , and X c , respectively. Each of these manifolds is invariant relative to U ; in fact, U can be chosen such that X s is positively invariant, and X u negatively invariant. Furthermore, the following properties hold:
3.2 Center, stable, and unstable manifolds in hyperbolic dynamics
89
(P1) W c D W cs \ W cu . If W u D f0g, then W cu D W c etc. Moreover, W cu is W cs in backward time. (P2) W cs has the following repulsion property: there exists a neighborhood V U of 0 such that if u0 62 W cs but u0 2 V , then ˚ t .u0 / leaves V in positive time. To be more specific about the nature of the manifolds, the proof shows that W s (and analogously all other manifolds) is given by a Lipschitz function hs W s .U / ! X cu whose graph is W s and so that hs .0/ D 0, Dhs .0/ D 0 where D is the derivative. As a consequence of (P2), if W u 6D f0g, then there is a solution that leaves any small neighborhood of 0 in positive time. In particular, the equilibrium 0 is unstable. A reformulation of (P2) reads as follows: if ˚ t .u0 / 2 V for all t 0, then u0 2 W cs . Uniqueness is a subtle issue for W cs and W cu since these manifolds are constructed depending on an arbitrary truncation, see [6].
Wu
Ws W
s
Wu
Figure 3.3. Stable, unstable, and center-stable manifolds
W cs
90
3 Above the ground state energy I: Near Q
Exercise 3.7. Show that the ODE in R2 xP 1 D
x13 ;
xP 2 D
x2
has a continuum of invariant curves which are tangent to x1 -axis at .0; 0/, each of which is a center manifold. However, as we shall see later, in the context of the linearization of (2.1) around the ground state one does have uniqueness since we can describe the dynamics on either side of the manifold. 3.2.2 Proof of the center manifold theorem of Bates and Jones1 Following Section 2 of [6], we now provide the essential details of the main argument. Theorem 3.6 will be a consequence of a more general global result on (3.14) under the following conditions. Since we will alter the nonlinearity by truncation, we rewrite the equation in the form uP D Au C g.u/
(3.17)
where X admits an A-invariant splitting X D X ˚ X C with closed subspaces X ˙ , dim X C < 1 and A˙ WD AjX ˙ satisfy ı A generates a C 0 -group S .t/ on X with kS .t/k Me ˛t
8 t 0:
(3.18)
8t 0
(3.19)
ı AC generates a C 0 -group S C .t/ on X C with kS C .t/k Me ˇ t
where ˇ > ˛. Note that we are not imposing any sign condition on ˛; ˇ. Finally, we demand that g satisfies the following: g.0/ D 0, and g is globally Lipschitz with a sufficiently small Lipschitz constant " > 0, depending on M; ˛; ˇ; k ˙ k where ˙ are the projections associated with the splitting of X. The reader may find it instructive to determine all possible choices of X ; X C for the examples in (3.16) and (1.6), respectively. 1
The reader can skip this section on first reading and move on to Section 3.3 without losing anything as far as later chapters are concerned.
3.2 Center, stable, and unstable manifolds in hyperbolic dynamics
91
Exercise 3.8. (a) Show that one can re-norm X ˙ so that (3.19), (3.18) hold with M D 1. Denote the new norms on X ˙ by k k˙ . Henceforth, we take j j WD k kC C k k as norm in X . Hint: Consider sup t0 e ˛t kS .t/xk on X . (b) Prove that kS C .t/k M
1
eˇ t ;
(3.20)
8 t 0:
(c) Show that the nonlinear flow ˚ t ./ associated with (3.17) is globally defined in R X. Now fix the Lipschitz constant " > 0 of g to be so small that for some one has ˇ C 2" < <
˛
2"
and define W
WD fu 2 X j e t ˚ t .u/ ! 0 as t ! 1g ;
W C WD fu 2 X j e t ˚ t .u/ ! 0 as t !
1g :
(3.21)
Note that W ˙ are invariant under the flow by definition. It is instructive to verify for the linear case, i.e., for g D 0, that W ˙ D X ˙ . The idea is of course that W cannot have any component coming from the “dominant” part of the flow which grows like e ˇ t . As we shall see, this forces it to be an “image” of X . As for W C , the point is that e t prevents any admixture of the v component to the flow, since that would lead to divergence in the limit in (3.21). The main result which we prove next is that these sets are global Lipschitz graphs. Proposition 3.9. Under the above assumptions there exist Lipschitz functions h˙ with the following properties: (a) h W X ! X C so that W D graph.h / and h .0/ D 0. (b) hC W X C ! X so that W C D graph.hC / and hC .0/ D 0. Moreover, W C \ W D f0g. Note that we do not claim that W ˙ are tangent to X ˙ , respectively. This is not possible under our assumptions, since g is not assumed to have arbitrarily small Lipschitz constants near 0. We will of course obtain Theorem 3.6 from Proposition 3.9 by localizing f near 0, and then the Lipschitz constant can be made arbitrarily small leading to the claimed tangencies in the theorem. The proof of Proposition 3.94 is divided into several steps. The main driving force is the fact that the expansion rate of S C is faster than that of S . In particular,
92
3 Above the ground state energy I: Near Q
XC
WC
W X
X
Figure 3.4. Invariance of fjwj jvjg and W C ; W
one expects that ˚ K WD .v; w/ 2 X ˚ X C D X j jvj jwj is positively invariant under ˚ t . To prove this, and even more general facts, rewrite (3.17) in the form vP D A v C g .v; w/ ; wP D AC w C g C .v; w/ : The solutions to this system are written in integral form t
Z v.t/ D S .t/v.0/ C w.t C / D S C . /w.t/ C
S .t Z0
s/g
v.s/; w.s/ ds;
t 0;
S C .
s/g C v.t C s/; w.t C s/ ds;
0
0
t
93
3.2 Center, stable, and unstable manifolds in hyperbolic dynamics
and one obtains the following estimates, see (3.19), (3.18): Z t jv.t/je ˛t jv.0/j C " jv.s/j C jw.s/j e ˛s ds; t 0; 0 Z 0 ˇ .tC / ˇt .jv.t C s/j C jw.t C s/j/e ˇ.tCs/ ds; jw.t C /je jw.t/je C"
0
t:
By means of Grownwall’s inequality we now conclude the following: if jw.s/j k1 jv.s/j for all 0 s t, then jv.t/j jv.0/j exp .˛ C ".1 C k1 //t (3.22) If jv.s/j k2 jw.s/j for all 0 s t, then jw.t/j jw.0/j exp .ˇ
".1 C k2 //t
(3.23)
see (3.20). Define ˚ K WD .v; w/ 2 X X C j jvj jwj and pick 0 < < 1 < so that ˇ ˛ ; 2CC 1 ˇ < < ".1 C / "<
".1 C
1
/
(3.24) ˛:
It is natural to call the v-component (i.e., the X -component) of the flow “subordinate” to w since it is associated with the slower rate ˛. Lemma 3.10 is a key statement for the entire Hadamard theory and should be seen in those terms. Also, note that the renorming provided by Exercise 3.8 is crucial for this lemma. Lemma 3.10. (a) For 2 Œ; , K is positively invariant. Furthermore, if .v0 ; w0 / 2 K , then (3.25) jw.t/j jw.0/j exp .ˇ ".1 C 1 / t ; 8 t 0: (b) More generally, if u2 .0/ 2 u1 .0/ C K , then the corresponding solutions satisfy u2 .t/ 2 u1 .t/ C K . Furthermore, jw1 .t/ w2 .t/j jw1 .0/ w2 .0/j exp .ˇ ".1 C 1 //t ; 8 t 0: (3.26)
94
3 Above the ground state energy I: Near Q
Proof. We only prove the weaker statement (a), which is (b) with u1 0, as the stronger one is proved in the same fashion. It suffices to consider u0 2 @K n f0g. Then jw0 j D jv0 j ¤ 0 and for any ı > 0 one has jw.s/j . C ı/jv.s/j;
jv.s/j .
ı/
1
jw.s/j
80st
where t > 0 is small. By (3.23) and (3.22), one then concludes, with ı small enough, that ˇ v.t/ ˇ ˇ v ˇ ˇ ˇ 0ˇ ˇ ˇ ˇ ˇ exp .˛ ˇ C ".2 C C ı C . ı/ 1 //t < 1 ˇ w.t/ w0 and we are done. We now turn to the proof of the proposition. To illustrate the simple ideas that drive the proof, consider first the case g D 0. Then of course W C D X C and W D X , but we need to isolate a characterization underlying this choice which is “stable” in the sense that it can be applied to g ¤ 0 as well. In view of the previous lemma, we should look for W C K1 and W X n K1 . Note that if .v; w/ 2 X n X , in other words if w ¤ 0, then .v; w/ 2 .v; 0/ C K1 . Denote by .v.t/; w.t// the (in this case linear) evolution of .v; w/ and by .v0 .t/; w0 .t// the evolution of .v; 0/. Then jw.t/ w0 .t/j jwj exp .ˇ 2"/t (3.27) for all t 0 by the previous lemma. If v.t/; w.t/ 2 X n K1 for all t 0, then (3.22) implies that jv.t/j jvj exp .˛ C 2"/t 8 t 0: In combination with (3.27) one now concludes from ˛ < ˇ that for large t necessarily jw.t/j > jv.t/j which is a contradiction. Note that of course w0 .t/ D 0 but we do not require that much. All we need to reach a contradiction is that .v0 .t/; w0 .t// remains in the complement of K1 , which is guaranteed simply by our assumption that it lies in the invariant set W . Note also that we found W “by inspection” since g D 0. In the general case, we first need to show that to given v0 ¤ 0 there exists w0 2 X nK1 so that the evolution of .v0 ; w0 / remains outside of K1 in forward time. This will be accomplished by means of a topological argument based on the Brouwer degree. As for W C one simply observes that W C would have to lie in an arbitrarily “thin” cone K (in other words 1) since the action of the flow stretches the X C component much more strongly than the X one. To be more specific, suppose
3.2 Center, stable, and unstable manifolds in hyperbolic dynamics
95
XC
X
Figure 3.5. Invariant, moving cones under ˚ t , see (b) of Lemma 3.10
.v; w/ 2 W C K1 . Since W C is invariant in time (both forwards and backwards), we can write .v; w/ D ˚n .vn ; wn / for any positive integer n and with .vn ; wn / 2 W C . But then Lemma 3.10 in combination with (3.22) implies that jwj can be made jvj C arbitrarily small by letting n ! 1. In other words, w D 0 and W X C . Proof of Proposition 3.9. To construct W , fix v0 2 X n f0g, and let ˚ B WD w0 2 X C j jw0 j jv0 j ; ˚ G t WD w0 2 B j jw.t/j jv.t/j : By Lemma 3.10, G t GsTfor 0 s t. We claim that G t ¤ ; for all t 0, which then implies that G1 WD t0 G t ¤ ;. Assuming this for now, to every v0 2 X we can then associate some w0 2 G1 . Suppose there are two distinct points w1 ; w2 2 G1 . Then Lemma 3.10 implies that .v0 ; w2 / 2 .v0 ; w1 / C K H) v.t/; w2 .t/ 2 v.t/; w1 .t/ C K ; jw2 .t/ w1 .t /j jw1 w2 j exp .ˇ ".1 C 1 //t ; 8 t 0 : By construction, jvj .t/j jwj .t/j;
j D 1; 2; 8 t 0; jvj .t/j jv0 j exp .˛ C ".1 C //t :
(3.28)
96
3 Above the ground state energy I: Near Q
Hence, 8 t 0, 2jv0 j exp .˛ C ".1 C //t jv1 .t/j C jv2 .t/j 1 jw1 .t/j C jw2 .t/j ˇ ˇ 1 ˇw2 .t/ w1 .t/ˇ 1 jw1 w2 j exp .ˇ ".1 C 1 //t : In view of (3.24) this implies w1 function h W X ! X C via
w2 D 0 as desired. Thus, G1 defines a unique (3.29)
fh .v0 /g D G1 : By (3.28), and jw.t/j jv.t/j e t ˚ t .v0 ; h .v0 // ! 0;
as t ! 1;
(3.30)
see (3.24), whence graph.h / W . If w1 ¤ h .v0 /, then .v0 ; w1 / 2 v0 ; h .v0 / C K and thus also, for all t 0, jw1 .t/
w.t/j jw1
h .v0 /j exp .ˇ
".1 C
1
//t :
Again by (3.24) this shows that graph.h / D W . The Lipschitz property of h is a simple variant of the previous arguments. In fact, suppose that for some v0 ; v1 2 X one has jh .v0 /
h .v1 /j jv0
(3.31)
v1 j :
Then for all t 0, v1 ; h .v1 / 2 v0 ; h .v0 / C K H) ˚ t v1 ; h .v1 / 2 ˚ t v0 ; h .v0 / C K which implies ˇ ˇ ˇ ˇw1 .t/ w0 .t/ˇ ˇh .v1 /
ˇ h .v0 /ˇ exp .ˇ
".1 C
1
//t ;
8 t 0:
But this contradicts (3.30) and therefore (3.31) is impossible. It remains to prove the main claim from above about G1 . Let t > 0 and define 8 ˇ ˇ ˇ ˇ <w.s/ jv0 j if ˇw.s/ˇ < ˇv.s/ˇ; jv.s/j (3.32) ˇ ˇ ˇ ˇ s .w0 / D :w.s/ jv0 j if ˇw.s/ˇ ˇv.s/ˇ: jw.s/j
97
3.2 Center, stable, and unstable manifolds in hyperbolic dynamics
Note that s is well-defined; indeed, in the first case v.s/ ¤ 0 and if w.s/ D 0 in the second case, then also v.s/ D 0 which entails that .v; w/ 0 whence v0 D 0, a contradiction. The geometry underlying (3.32) is as follows: we flow .v0 ; w0 / to v.t/; w.t/ . If v.t/; w.t/ 2 K , then project that point onto X C with C , and then intersect the segment connecting 0 2 X C to w.t/ in X C with @B. If on the other hand v.t/; w.t/ 62 K , then intersect the ray in .X ; X C / joining .0; 0/ to v.t/; w.t/ with the set fv D v0 g B. Moreover, s is continuous on Œ0; t B, and is a map from B into itself. Finally, by the positive invariance of K , we see that s takes @B to @B. We claim thatT there is w0 2 B with s .w0 / D 0. If so, then G t ¤ ; for all t 0 and G1 WD t 0 G t ¤ ; as desired. The claim is easy if dim X C D 1; indeed, in that case B is a closed interval around zero and s W B ! B is necessarily onto by the aforementioned properties and the intermediate value theorem (at s D 0 one has 0 D Id). If dim X C > 1 the surjectivity remains true but requires more topological machinery such as the Brouwer degree (which is admissible since B lies in a finite-dimensional space). In fact, deg. s ; B; 0/ is welldefined since 0 62 s .@B/ and constant by the homotopy invariance of the degree. Therefore, 1 D deg. 0 ; B; 0/ D deg. t ; B; 0/ whence t .w/ D 0 for some w 2 B (by the same argument, t .B/ D B). This concludes the discussion of W . The main idea in the construction of W C is that Lipschitz graphs lying in fjwj jvjg are compressed by ˚ t for t 0. Thus, one finds W C as a fixed point under ˚ t . Define ˚ L WD h 2 C 0 .X C ; X / j h.0/ D 0; h 2 Lip
1
(3.33)
and for any h 2 L, set Gh WD graph.h/. We endow L with the norm jh.w/j : w¤0 jwj
khkL WD sup
First, one checks the following via Lemma 3.10: for every h 2 L, and all t 0, C .˚ t .Gh // D X C ; ˚ t .Gh / D Gh t for some h t 2 L :
(3.34)
For the first property let w0 2 X C and choose R > 0 such that R exp .ˇ
".1 C
1
//s > jw0 j
80 s t :
(3.35)
98
3 Above the ground state energy I: Near Q
Set B WD fw 2 X C j jwj Rg. It follows from Lemma 3.10 and (3.35) that w0 ¤ C ı ˚s .h.w/; w/ for any w 2 @B and any 0 s t. Thus, by the homotopy invariance of the degree, deg C ı ˚ t h./; ; B; w0 D deg C ı ˚0 h./; ; B; w0 D 1 which establishes the first part of (3.34). The second part is an immediate consequence of Lemma 3.10. Indeed, if .vj .t/; wj .t// D ˚ t .h.wj /; wj / for j D 1; 2, then one needs to verify that ˇ ˇ ˇ ˇ ˇh t w1 .t/ h t w2 .t/ ˇ 1 ˇw1 .t/ w2 .t/ˇ which is the same as ˇ ˇv1 .t/
ˇ v2 .t/ˇ
1ˇ
ˇ w1 .t/
ˇ w2 .t/ˇ :
However, the latter is nothing but the moving cone invariance of Lemma 3.10. Denote the map h 7! h t by T t W L ! L. The main property of T t is the following: T t is a contraction for large t.
(3.36)
To prove it, fix t, and hj 2 L, j D 1; 2. Set gj WD T t .hj /. For any w 2 X C there exist wj 2 X C with gj .w/; w D ˚ t hj .wj /; wj ; j D 1; 2 : By Lemma 3.10, more precisely by the positive invariance of K , jwj jwj j exp .ˇ ".1 C 1 //t ; j D 1; 2 :
(3.37)
Assume that g1 .w/ ¤ g2 .w/. Then .g1 .w/; w/ 62 .g2 .w/; w/ C K implies via Lemma 3.10 that ˚s h2 .w2 /; w2 62 ˚s h1 .w1 /; w1 C K ; 8 0 s t : (3.38) By (3.22) we conclude from (3.38) that ˇ ˇ ˇ ˇ ˇg2 .w/ g1 .w/ˇ ˇh2 .w2 / h1 .w1 /ˇ exp .˛ C ".1 C //t : On the other hand, ˇ ˇ ˇ ˇh2 .w2 / h1 .w1 /ˇ ˇh2 .w2 /
ˇ ˇ ˇ h2 .w1 /ˇ C ˇh2 .w1 / h1 .w1 /ˇ ˇ ˇ 1 jw2 w1 j C ˇh2 .w1 / h1 .w1 /ˇ ˇ ˇ ˇ ˇ 1 ˇh2 .w2 / h1 .w1 /ˇ C ˇh2 .w1 / h1 .w1 /ˇ :
(3.39)
3.2 Center, stable, and unstable manifolds in hyperbolic dynamics
99
The second inequality sign here uses the Lipschitz property of h2 , and the third one uses (3.38) with s D 0. Thus, ˇ ˇ ˇ ˇ ˇh2 .w2 / h1 .w1 /ˇ .1 1 / 1 ˇh2 .w1 / h1 .w1 /ˇ which, combined with (3.39) yields ˇ ˇ ˇ ˇg2 .w/ g1 .w/ˇ . / 1 ˇh2 .w1 /
ˇ h1 .w1 /ˇ exp .˛ C ".1 C //t : (3.40)
Dividing by jwj on the left-hand side and by jw1 j on the right-hand side now implies in view of (3.37) that
T t .h2 / T t .h1 / . / 1 exp .˛ ˇ C ".2 C C 1 //t kh2 h1 kL L which for sufficiently large t gives the desired contraction property. Let ht 2 L be the unique fixed point of T t . Then T t .T ht / D T .T t ht / D T ht shows that T ht D ht . Therefore, ht does not depend on t and we denote by hC the common fixed-point of T t for all large t. To complete the proof it remains to show the following: e t ˚ t .u/ ! 0 as t !
(3.41) 1 if and only if u 2 GhC DW GC : Take u D .v; w/ 2 GC and set v.t/; w.t/ D ˚ t .u/ for t < 0. By GC K and Lemma 3.10, ˇ ˇ jwj ˇw.t/ˇ exp .ˇ ".1 C 1 //t : ˇ ˇ ˇ ˇ Together with ˇv.t/ˇ 1 ˇw.t/ˇ and the choice of , see (3.24), one obtains GC W C . For the other direction, let .v; w/ 2 W C n GC . Then by Lemma 3.10 (3.42) .v; w/ 62 hC .w/; w C K H) ˚ t .v; w/ 62 ˚ t hC .w/; w C K for all t 0. Denote v.t/; w.t/ D ˚ t .v; w/. Then (3.42) implies that ˇ ˇ ˇ ˇ 0 < ˇv hC .w/ˇ ˇv.t/ w.t/ˇ exp ˛ C ".1 C / t ; 8 t 0 : But then one obtains a contradiction to .v; w/ 2 W C , see (3.24). That W C \ W D f0g follows from the construction, since they lie in disjoint cones: W C K , and W X X C n K . Before turning to the proof of Theorem 3.6, we present some technical material related to the tangency of W u and W s to X u , and X s , respectively. Part (a) of the following exercise is an immediate consequence of Lemma 3.10, whereas (b) is evident from the definition of the differential and the choice of norm in L.
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3 Above the ground state energy I: Near Q
Exercise 3.11. (a) Prove the following: for any t > 0 and ı1 > 0 there exists ı > 0 such that if .v0 ; w0 / 2 K and jw.t/j ı then ˇ ˇ ˇ ˇ ˇv.s/ˇ C ˇw.s/ˇ ı1 for 0 s t: (b) Show that L0 WD fh 2 L j h is differentiable at 0 and Dh.0/ D 0g is a closed set in L. Lemma 3.12. Assume that g is differentiable at 0, and that Dg.0/ D 0. Then T t as defined in (3.36) satisfies T t W L0 ! L0 for all t > 0. In particular, hC as constructed in the previous proof satisfies hC 2 L0 . Proof. Fix a positive integer n and t > 0 and let 0 < " < ".n/ be so small that 1 (3.24) D n. There exist a neighborhood U of 0 such that ˇ ˇ holds with D ˇg.u/ˇ < "juj for all u 2 U . Let h 2 L0 . We can assume that U \ graph.h/ Kn . By Exercise 3.11 there exists ı > 0 such that if .v0 ; w0 / 2 K with jw.t/j ı, then .v.s/; w.s// 2 U for all 0 s t. Fix any jwj ı. Then by (3.34) there exists v 2 X with .v; w/ 2 graph.T t h/ which means that for some .v0 ; w0 / 2 graph.h/ K one has .v; w/ D v.t/; w.t/ as well as v.s/; w.s/ 2 U for all 0 s t . By the invariance of Kn relative to U , therefore ˚ .v; w/ 2 graph.T t h/ j jwj ı Kn whence
ˇ ˇ ˇ.T t h/.w/ˇ 1 sup jwj n jwjı
which implies that DT t .h/.0/ D 0. Proof of Theorem 3.6. We reduce matters to Proposition 3.9, and begin with the existence of W u and W cs locally on some neighborhood U of 0. First, multiply f .u/ by a Lipschitz cut-off function to generate the function g.u/ of sufficiently small support and small Lipschitz constant as required by Proposition 3.9. Then (H1)– (H3) guarantee that the hypotheses of Proposition 3.9 hold. The parameters ˛; ˇ are chosen such that ˚ 0 < ˛ < ˇ < inf Re z j z 2 u .A/ : Applying that proposition with X C WD X u and X WD X s ˚ X c yields Lipschitz graphs W C and W . We need to show that W u WD W C \ U and W cs WD W \ U
101
3.2 Center, stable, and unstable manifolds in hyperbolic dynamics
are the desired manifolds where U is chosen to be a sufficiently small neighborhood of 0 (which depends on the choice of cutoff above). By Lemma 3.12, W C is tangent to X C . Moreover, W C \ U W u is clear by construction. The other inclusion is more subtle, but is essentially implicit in the final step in the proof of Proposition 3.9 (dealing with the inclusion in the graph of hC ). To be more specific, note that W u does not specify the rate of exponential convergence nor the choice of norm used. Thus, one first needs to verify that W C does not depend on these choices (in other words, on the specific choice of ˛; ˇ or the norm, as well as the cut-off used to pass from f to g), at least locally near 0. This is done by the usual invariant cone arguments combined with the tangency of W C to X C at 0 that we just established. The local equality of W u with W C near the origin is then an immediate consequence. Also somewhat delicate is the tangency of W to X cs at 0. To prove it, one argues indirectly: let fun g W \ K be a sequence such that un ! 0 where 2 .0; 1/ is fixed. Furthermore, we can assume that fun g U 0 U where U 0 has the property that K \ U 0 is invariant in forward time, and Lemma 3.10 holds 2 for K while the flow remains in U 0 . This of course uses that the Lipschitz constant 2 of f can be made very small in U 0 . Now C ˚ t .un / grows exponentially inside of U due to ˇ > 0. Hence we can find tn ! 1 so that wn WD C ˚ tn .un / satisfy jwn j D ı 0 and j C ˚ t .un /j < ı 0 for 0 < t < tn for all n 1. Now let uQ n 2 W \ U be chosen so that C uQ n D wn . Then one has ˇ ˇ uQ n
ˇ ˇ ˇ ˚ tn .un /ˇ ˇ ˚ tn .un /ˇ
j uQ n j ı 0 .
1
1
/
00
DW ı > 0;
8 n 1:
Via invariant moving cone considerations one now verifies that for all 0 tn ˚ tn
.un /
2˚
Q n/ .u
C Kc ; 2
˚ tn
Q n/ .un /; ˚ .u
2 K :
Via the triangle inequality, these inclusions imply the crucial estimate j un
˚
Q n /j tn .u
4 C j ˚
Indeed, suppose jvj j jwj j for j D 1; 2 and 2 jv1 jw1
w2 j
1 jw1 j C jw2 j ; 2
jw1 j 3jw2 j;
(3.43)
Q n /j : tn .u v2 j jw1 jv1
w2 j. Then
v2 j
4 jw2 j
as desired. The right-hand side of (3.43) is bounded by 4 1 ı 0 e .ˇ 2"/tn , whereas the left-hand side is bounded from below by ı 00 e .˛C2"/tn and a contradiction ensues.
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3 Above the ground state energy I: Near Q U
c K
u C K
V W cs
Figure 3.6. Geometry in the proof of property (P2)
Therefore, W is tangent to X cs at 0. The preceding properties now imply that W \ U is a center-stable manifold in U due to the fact that W C \ W D f0g. To construct W cu one simply reverses time. For the center manifold, denote by C h W X c X s ! X u the function whose graph is W cs and by h W X c X u ! X s the function whose graph is W cu . Then for fixed x 2 X c consider the function F .y; z/ D h .x C z/; hC .x C y/ for .y; z/ 2 X s X u . Since F has Lipschitz constant < 1 there exist .y; z/ such that .y; z/ D F .y; z/. Set hc .x/ D .y; z/ and note that x; hc .x/ is Lipschitz and its graph is W c with the desired properties, locally near 0. By construction, W c D W cs \ W cu and property (P1) holds. For (P2), pick > 1 so that the moving cone property of Lemma 3.10 holds in U with K . Next, pick V such that .u C K / \ V D ; for all u 2 @U \ Kc where W cs U n K . Then for any u0 2 V n W cs find u1 2 W cs \ V (take V to be a product neighborhood, say) with u0 2 u1 C K . By the cone invariance, ˚ t .u0 / 2 ˚ t .u1 / C K provided t 0 is such that the flows remain in U . If ˚ t .u1 / remains in U for all t 0, then by the exponential growth of the w-component ˚ t .u0 / leaves V . If not, then ˚ t .u1 / exits U in Kc , which then implies that ˚ t .u0 / exits V prior to that, see Figure 3.6.
3.2.3 Applying the center manifold theorem to the NLKG equation As Bates and Jones, we now apply the preceding abstract theory to the NLKG equation. In order to verify the assumptions of Theorem 3.6, we write (3.2) as a system: ! ! ! v v 0 0 1 @t D C : (3.44) LC 0 vP 3Qv 2 C v 3 vP
3.2 Center, stable, and unstable manifolds in hyperbolic dynamics
103
ess spec
i
k
k i
ess spec Figure 3.7. Spectrum of the linear operator in (3.44) in L2rad
Setting A equal to the matrix operator on the right-hand side, we first determine the spectrum of A. Much of what is done in this section is routine. Lemma 3.13. (a) For Re z > 0 the resolvent of A on L2rad L2rad equals ! ! 2 1 f .L C z / .zf C g/ C .A z/ 1 D g f z.LC C z 2 / 1 .zf C g/ for all .f; g/ 2 L2 L2 . (b) One has .A/ D fz 2 C j z 2 2 words,
(3.45)
.LC /g as spectrum in radial L2 . In other
.A/ D f˙kg [ iŒ1; 1/ [ i. 1; 1
(3.46)
where k 2 is the negative eigenvalue of LC . Proof. Part (a) is nothing but simple algebra and (b) follows from (3.45) by inspection. We remark that (3.44) can be written in the Hamiltonian form 0 1 LC 0 VP D JHC V C N.V /; J D ; HC D : 1 0 0 1
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3 Above the ground state energy I: Near Q
Clearly, J is a symplectic matrix, and HC is selfadjoint with domain H 2 L2 . The energy conservation (for N D 0, say) is immediate in this formulation: ˛ d ˝ HC V jV D 0: dt In the nonradial setting, the zero modes are associated to the translation and Lorentz symmetries. In fact, ! ! ! ! ! rQ 0 rQ 0 rQ JHC D ; JHC J D JHC D (3.47) 0 0 0 rQ 0 where the former describes the entire kernel of A, and the latter the kernel of A2 . There are no further zero modes. We are using here that ker.LC / D spanf@j Qgj3D1 , see [143] which proves this for the case at hand (cubic nonlinearity in R3 ). That the generalized eigenspace of A has dimensions 6 reflects the fact that there a three degrees of translational freedom, both in physical space as well as in momentum (the latter determines the Lorentz transforms). Indeed, the manifold of traveling solitons can be described as follows. First, for fixed .p; q/ 2 R6 set Q.p; q/ WD Q x q C p hpi 1 jpj 2 p .x q/ ; p where hpi WD 1 C jpj2 . In terms of these solitons, the traveling waves can be described as p u.t/ D ˙Q p; q.tI p; q0 / ; p 2 R3 ; q.t P I p; q0 / D ; q.0I p; q0 / D q0 hpi with fixed momentum p and velocity
p . hpi
Then
! Q.p; q.tI p; q0 // U.t/ D q.t/ P rq Q.p; q.tI p; q0 // solves the system UP D JH0 U C N.U /;
H0 D
C1 0 0 1
;
! 0 N.u/ D : u3
Differentiating this PDE in q0 and setting q0 D 0 gives the first relation in (3.47), wheres differentiating in p and evaluating at p D 0 gives the second one. We now prove that A generates a group on H D H 1 L2 (radial). We begin by showing that A m Id generates a contraction semi-group on t 0 for some
3.2 Center, stable, and unstable manifolds in hyperbolic dynamics
105
m > 0. Clearly, A is densely defined on Dom.A/ WD H 2 L2 . By the Lumer– Phillips theorem, see [51], it suffices to show that A m Id is dissipative and that . C m/Id A has range equal to H for all > 0. Corollary 3.14. (a) There exists m > 0 such that A m Id is dissipative, i.e., one has f f .A m Id/ g ; g 0 8 .f; g/ 2 H 1 L2
(3.48)
where the scalar product is the one corresponding to the norm khrif k22 Ckgk22 . (b) The range of . C m/Id A equals H for all > 0. (c) A generates a semigroup S.t/ on t 0 with kS.t/k e mt , and S.t/ extends to a group on R with kS.t/k e mjtj for all t 2 R. Proof. Part (a) is obtained via integration by parts and Cauchy–Schwarz. Part (b) follows from (b) of Lemma 3.13, provided m > 0 is sufficiently large. Part (c) follows from (a), (b) and the Lumer–Phillips theorem, see [51], Corollary 3.18, Chapter II. The final statement is obtained by reversing time in (3.44). Next, we rewrite (3.44) in the form (3.49) XP D AX C f .X/ where X WD vvP and f .X/ D 3Qv02 Cv3 . Abusing notation, we also write f .v/ D f .X/. Using Sobolev embedding, one immediately checks that 2 2 (3.50) kf .v/ f .w/k2 . kv wkH 1 kvkH 1 C kwkH 1 C kvkH 1 C kwkH 1 for all v; w 2 H 1 with an absolute implicit constant. Moreover, the cubic equation is “critical” with regard to this property, i.e., any nonlinearity f .v/ D jvjp 1 v with p > 3 fails to satisfy a bound of the form (3.50). The only remaining condition to check is (3.15). Recall that all function spaces are radial. Lemma 3.15. Let S0 .t/ be the group generated by 0 1 A0 WD 1 0
(3.51)
on H. Then for every t 2 R, S.t/ S0 .t/ is compact. In particular, .S c .t// D 1 for every t, where is the spectral radius.
106
3 Above the ground state energy I: Near Q
Proof. By variation of constants (Duhamel) t
Z S.t/
S0 .t/ D
S.t
0
s/KS0 .s/ ds;
KD
3
0 0 Q2 0
(3.52)
1 Note that multiplication by Q2 is a compact operator from Hrad ! L2 , see Exercise 2.7. Thus, the right-hand side is compact as well. Since S0 .t/ is unitary by energy conservation for every t, it follows that S0 .t/ D ess S0 .t/ D S 1 :
By Weyl’s criterion, the essential spectra of S.t/ and S0 .t/ agree. To be more specific, we use the Weyl criterion as given in the form of [120], Vol. 4, page 111, Lemma 3 with A D S0 .t/ and B D S.t/; indeed, .A/ D S 1 has empty interior, and each component of ˚ ˚ C n .A/ D jzj < 1 [ jzj > 1 contains a point of .B/. For the unit disk one can take the point z D 0 which expresses the invertibility S.t/ 1 D S. t/, and for its complement any point z with jzj > kS.t/k. Thus, any spectrum of S.t/ outside the unit circle (or the unit disk for that matter) is necessarily discrete, i.e., eigenvalues of finite multiplicity. By [51], Chapter V, Theorem 2.6 this implies that the generator of S c .t/ which is Ac would need to have discrete spectrum in the right-half plane. But this is a contradiction, whence S c .t/ D 1 as claimed. By the previous lemma, for any t > 0, given any " > 0 one can find a positive integer n0 D n0 .t; "/ such that for all n n0 , 1
k.S c .t//n k n 1 C t " H) kS c .tn/k .1 C t "/n e "tn : This implies (3.15) and all hypotheses of Theorem 3.6 have been verified. Thus, we obtain the following, using the notation of Theorem 3.6. 1 Corollary 3.16. In a small neighborhood of .Q; 0/ in Hrad L2rad there exist onedimensional stable and unstable manifolds, as well as a center manifold of codimension two for equation (2.1). All manifolds are Lipschitz, and have X u ; X s , and X c , respectively, as tangent spaces at .Q; 0/. Moreover, properties (P1), (P2) of Theorem 3.6 hold.
3.2 Center, stable, and unstable manifolds in hyperbolic dynamics
107
Following [6], we now show that the evolution restricted to the center manifold W c is stable in the orbital sense. This follows from the observation that the energy restricted to W c acts as a Lyapunov functional for k.v; v/k P H . To be more specific, we first note the fact that 1 .X c /, which is the projection of X c onto the first component of H (i.e., H 1 ) belongs to P? .H 1 /, see (3.10). This is heuristically obvious, since otherwise W c would contain an exponentially growing solution which is impossible. One can also verify this fact via Riesz projections using the resolvent formula (3.45). The details are left to the following exercise. Exercise 3.17. Using (3.45), prove that 1 .X c / lies in the absolutely continuous spectral subspace of LC . By means of (3.3) and (3.11) one now obtains the following conclusion, which highlights the importance of the center manifold for the description of the dynamics. Corollary 3.18. Restricted to the local center manifold W c around .Q; 0/, the flow of (2.1) is orbitally stable as jtj ! 1. More precisely, there exists a neighborhood U of .Q; 0/ with the property that a solution starting in U remains in U for all times (both positive and negative) if and only if the solution intersects W c \ U (in which case it will lie entirely in W c \ U ). Proof. Simply observe that, for .v; v/ P 2 W c small, E.Q C v; v/ P
˛ 1 1˝ LC vjv C kvk P 22 C o.kvk2 / 2 2 & k.v; v/k P 2H C o.kvk2 / :
E.Q; 0/ D
(3.53)
Here we used Exercise 3.17, (3.3), and (3.11). The left-hand side is a conserved quantity whence the claim on stability (of course, one needs that W c is invariant locally around the equilibrium). If the solution does not start from W c \ U , then it either does not start from W cs \ U or W cu \ U . In the former case, Theorem 3.6 guarantees that the solution exits U (which needs to be chosen appropriately) in finite positive time and the same in the latter case in backward time. Of course, one expects the flow on W cs to be stable in forward time. We will prove this property in the stronger asymptotic stability sense in the following section.
108
3 Above the ground state energy I: Near Q
Exercise 3.19. (a) Show that Corollaries 3.16 and 3.18 remain valid for the equation in R1Cd uR
u C u
jujp
1
u D 0;
1
2 d
2
; d 3
in the energy space H 1 L2 . (b) Generalize the results of this section, including the previous part (a), to the linearization about arbitrary stationary solutions, which are not necessarily the ground state. You will find that Corollary 3.16 remains essentially the same, the only difference being the dimensions of W s and W u . For Corollary 3.18 you will need to make an assumption on the spectrum of the linearized operator; which assumption? To conclude this section, we remark that while we have followed the original reference by Bates and Jones [6] very closely, there are other ways of executing the graph transform method. For example, one can find W via a contraction argument on the level of graphs (as we did for W C ) rather than by means of the topological method used in the proof of Proposition 3.9. As far as the PDE is concerned, one also does not need to invoke semi-group theory but can rely instead on the standard wellposedness theory in the energy class as in the previous chapter. As an example of an alternate way of going about the graph transform in the context of NLKG, see [112] which goes beyond Bates and Jones in several ways. The appeal of [6], at least to the authors, lies with its geometric flavor which is easily lost in other formulations. For example, the Lyapunov–Perron method to which we now turn is less geometric but very powerful in applications. But it also comes at a higher cost as we shall see.
3.3
Center-stable manifolds via the Lyapunov–Perron method
In this section, we shall present an alternative approach to the construction of the invariant manifolds.
3.3.1 The finite-dimensional construction We begin by briefly recalling this method for the ODE (3.13) in Rd . We will also assume that kf kC 1 < " sufficiently small, which can be done by truncation as
109
3.3 Center-stable manifolds via the Lyapunov–Perron method
before. Following [138], we introduce the space of “slow growth” Y WD fy W R ! Rd j kyk WD sup e
jtj
t 2R
jy.t/j < 1g
(3.54)
where > 0 will be small. Let Rd D X u ˚ X c ˚ X s be the same type of splitting as in the previous section. By variation of constants, y.t/ D e A.t c
C s e A.t
tc /
ts /
t
Z
e A.t
y.tc / C Z
/
tc t
e A.t
y.ts / C
/
f y. / d
f y. / d
ts
C u e A.t
tu /
Z
t
e A.t
y.tu / C
/
f y. / d :
tu
Fix tc D 0, and send ts ! y 2 Y one obtains
1, tu ! 1. If > 0 is sufficiently small, then for any
Z t Z tA A.t / y.t/ D e xc C e f y. / d C 0 Z 1 e A.t / u f y. / d :
t
c
t
e A.t
/
s f y. / d
1
(3.55)
Denoting the right-hand side of (3.55) by .xc ; y/, one now seeks a fixed point in Y of this map for arbitrary but fixed xc 2 X c . We leave the construction in Rd to the patient reader as an exercise, see [138] for all details. Exercise 3.20. Let f .0/ D 0; Df .0/ D 0. Assume that kf kC 1 < " with " small, and let > 0 be small. Then prove that W X c Y ! Y , and that this map is a contraction. Denote the fixed point by y xc and define a map .xc / D y xc .0/. Show that M WD f .xc / j xc 2 X c g is a Lipschitz graph which is invariant under the flow of (3.13) and tangent to X c at the origin. Furthermore, prove that every trajectory which lies in a sufficiently small neighborhood of 0 for all times is entirely contained in M. The unstable and stable manifolds can also be obtained in this setting, as we demonstrate by means of the PDE (2.1) in the following section.
110
3 Above the ground state energy I: Near Q
3.3.2 The construction for the NLKG equation In this section, we apply the Lyapunov–Perron scheme to the PDE (2.1). Ball [5] used this scheme for the nonlinear beam equation, which is a dissipative equation. This type of equation is simpler to control than the Hamiltonian ones, since the space of all non-decaying modes is finite dimensional for dissipative equations. The second author [122] employed dispersive estimates to obtain center-stable manifolds for the cubic NLS equation in R3 , albeit in a strong topology which is not invariant under the NLS flow. Hence the manifolds obtained there are not in any reasonable sense invariant but they do give the desired “conditional asymptotic stability property” of solutions starting on these manifolds. Beceanu [9], using novel estimates on the linearized flow, was able to establish existence of these manifolds in the energy topology and he obtains true center-stable ones. At least in the radial setting, the construction of center-stable manifolds for (2.1) is much simpler than for the NLS equation studied in [9], [122]. Indeed, the only symmetries present in our case, namely the change of sign in the dependent variable and time translations, fix the ground state and the linearized operator is scalar and self-adjoint. We decompose any solution u simply by putting u.t/ D Q C v.t/;
v.t/ D .t/ C .t/;
? ;
which is obviously unique. We then obtain the equations of .; / 2 R P? .H 1 / ( R k 2 D N .v/; q (3.56)
R C ! 2 D P? N.v/ DW Nc .v/; ! WD P? LC : Nc here stands for P? N and P N.v/ DW N .v/. One natural way to proceed now is to rewrite the equation as the Hamiltonian system, see (3.44). With A equal to the linear matrix operator on the right-hand side of (3.44), we now ask the reader to verify the following properties. 0 1 Exercise 3.21. Denote J D , and set e˙ D ˙k . Show that e˙ are the 1 0 eigenvectors of A with eigenvalues ˙k, cf. (3.46). Next, prove that
P˙ WD h; Je ie˙
(3.57)
are the Riesz projections onto e˙ . By the exercise, the linear center-stable manifold is given by the condition D 0 on the data. This simply means that there is no exponentially growing PC u.0/ u.0/ P
3.3 Center-stable manifolds via the Lyapunov–Perron method
111
mode in the solution. In other words, we infer that ˝ ˛ ku.0/ C u.0/ P j D0 is the tangent plane to the putative center-stable manifold. The latter will be found as a graph over this tangent plane. To be more specific, we seek forward global solutions to (3.56) which grow at most polynomially, which is equivalent to removing the growing mode e kt . As in the finite-dimensional case, we extract the growing mode as follows: Z 1 t 1 P sinh k.t s/ N .v/.s/ ds .t/ D cosh.kt/.0/ C sinh.kt/.0/ C k k 0 Z t kt e 1P 1 ks D .0/ C .0/ C e N .v/.s/ ds C 2 k k 0 where the omitted terms are exponentially decaying. Hence, the necessary and sufficient stability condition is Z 1 P.0/ C k.0/ D (3.58) e ks N .v/.s/ ds: 0
Note that the left-hand side here vanishes exactly on the tangent plane which we found earlier. Hence, (3.58) expresses precisely that the true nonlinear center-stable manifold is obtained by higher order corrections to the linear ansatz. Under the condition (3.58), the integral equation for .t/ is reduced to Z 1 Z 1 1 1 e ks N .v/.s/ ds C e kjt sj N .v/.s/ ds; .t/ D e k t .0/ C 2k 0 2k 0 (3.59) while the integral equation for is 1 1
.t/ D cos.!t/ .0/ C sin.!t/ .0/ P C ! !
t
Z
sin.!.t
s//Nc .v/.s/ ds: (3.60)
0
The coupled equations (3.59), (3.60) can be solved by iteration, using the Strichartz estimate for e i t ! of Lemma 2.46, for any small initial data .0/; .0/ E 2 R P? .H/. The center-stable manifold is then obtained from (3.58). Recall that H is the radial energy space. We emphasize that the following result requires the gap property (3.12). However, it was shown in [112] that the construction of the manifold does not require this condition, which implies that it is really only needed in order to establish the scattering property of solutions associated with the center-stable manifold. A version of the following theorem was independently established by Stefanov and Stanislavova [130].
112
3 Above the ground state energy I: Near Q
Theorem 3.22. Assume that (3.12) holds. Then there exists > 0 small and a smooth graph M in B .Q; 0/ H (radial) so that .Q; 0/ 2 M, with tangent plane ˚ TQ M D .v0 ; v1 / 2 H j hkv0 C v1 ji D 0 (3.61) at .Q; 0/ in the sense that ˇ ˇ ˇhkv0 C v1 jiˇ . ı 2 ;
sup
80<ı<
.QCv0 ;v1 /2@Bı .Q;0/\M
Any data .u0 ; u1 / 2 M lead to global evolutions of (2.1) of the form u D Q C v where v satisfies k.v; v/k P L1 C kvkL3 ..0;1/IL6 .R3 // . t ..0;1/IH/
(3.62)
and scatters to a free Klein–Gordon solution in H, i.e., there exists a unique free Klein–Gordon solution 1 such that P j.t/j C j.t/j C k .t/ E
E1 .t/kH ! 0;
as t ! 1. In particular, we have E.E u/ D J.Q/ C k E1 k2H =2
(3.63)
Finally, any solution that remains inside B .Q; 0/ for all t 0 necessarily lies entirely on M, and M is invariant under the flow for all t 0. Proof. For the existence, we show that (3.59), (3.60) is a contraction in the norm2 k.; /kX WD kkL1 \L1 .0;1/ C k kSt.0;1/ ;
1 St WD L2t L6x \ L1 t Hx :
The Strichartz estimate for the free Klein–Gordon equation gives us, see Section 2.5, kukSt.0;T / . kE u.0/kH C k.@2t
C 1/ukL1 L2x .0;T / : t
(3.64)
By Section 3.4 below, it follows that these same estimates hold for P? LC . In fact, under the hypothesis (3.12), the operator LC satisfies the conditions of Yajima’s W k;p boundedness theorem for the wave operators, see [145]. Therefore, we can conclude that (3.64) applies to as well: 2 2 k kSt.0;T / . k .0/k E ; H C k.@ t C ! / kL1 L2 x .0;T / t
2
Strictly speaking, one does not need the L2t endpoint here but we use if for convenience.
(3.65)
113
3.3 Center-stable manifolds via the Lyapunov–Perron method
provided that .0/ E and .@2t C ! 2 / are orthogonal to . P For the solution of (3.59), with .0/ uniquely determined by (3.58), we estimate the norm by integration in t : kkL1 \L1 .0;1/ . k
1
j.0/j C k
1
kN .v/kL1 .0;1/ t
. j.0/j C kN.v/kL1 L2x .0;1/ ; t
and for the solution of (3.60) the Strichartz estimate implies k kSt.0;1/ . k .0/k E : H C kN.v/kL1 L2 x .0;1/ t
The nonlinearity N.v/ is bounded in L1t L2x by 2 kQv 2 kL1 L2x C kv 3 kL1 L2x . kvkL 2 L6 .kvkL1 L6 C kQkL1 L6 / ; x x t
t
t
t
x
t
(3.66)
where the norm of v is bounded by kvkLp L6x . kkL1 \L1 C k kLp L6x ; t
t
t
t
.1 p 1/ :
Gathering these bounds, we obtain 3 2 k.; /kX . j.0/j C k .0/k E H C k.; /kX C k.; /kX :
(3.67)
This shows that the map which takes the right-hand side of (3.59), (3.60) defined in Q / terms of .; Q with k.; /kX onto the solution .; / of this system takes a ball BC .Q; 0/ into itself, where j.0/j C k .0/k E H and C is some absolute constant. Next, one checks by the same arguments that the map is, in fact, a contraction. Thus, we obtain a unique fixed point of (3.59), (3.60) for any given small .0/; .0/ E . It is straightforward to see that u WD Q C .t/ C
.t/ solves the NLKG equation (2.1) on 0 t < 1, satisfying kE u
E H . k.; /kX . j.0/j C k .0/k Qk E H;
with smooth dependence on the data. The bound on hjuP C kui D P C k follows from (3.58) by means of the bound on v that we just proved. Moreover, the asymptotic profile of is given by Z 1 1 1
1 .t/ D cos.!t/ .0/ C sin.!t/ .0/ P C sin !.t s/ Nc .v/.s/ ds; ! ! 0
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3 Above the ground state energy I: Near Q
with the convergence property E L1 .T;1/ C k E kk t
E1 kL1 . kN.v/kL1 L2 .T;1/ ! 0; t H.T;1/ t
.T ! 1/:
The iteration can be solved with a given 1 and the equation of now reads Z 1 t (3.68) sin !.t s/ Nc .v/.s/ ds;
.t/ D 1 .t/ C ! 1 where the estimates are essentially the same. 1 .t/ can be further replaced with a free Klein–Gordon solution by the linear scattering for LC . The uniqueness part requires some more work, since a priori we do not know if the solution lies in the space X, globally in time. The R 1 obstacle here is of course that the .; /-system is non-local due to the integral 0 in the -equation, see (3.59). However, the exponentially decaying weight appearing in these integrals allows one to circumvent this difficulty very easily. Let u be a solution on Œ0; 1/ satisfying kE u
E L1 ..0;1/IH/ < Qk
Since it is bounded in the energy space, we can easily see that N .t/ is bounded. Therefore, it has to satisfy (3.58), and the reduced integral equation (3.59) as well as (3.60) for all 0 < t < 1. To see that 2 L1t .0; 1/ and 2 L2t L6x .0; 1/, consider the norm k.; /kXT WD kkL1 .0;T / C kkL1 .T;1/ C k kSt.0;T / C k E kL1 ..T;1/IH/ ; (3.69) for T > 0. The energy bound implies that k.; /kXT < 1 for all T > 0, but we require a uniform bound. From the integral equations one concludes that kkL1 .0;T /\L1 .T;1/ . kN kL1 .0;T /\L1 .T;1/ . C .kkL1 .0;T /\L1 .T;1/ C k kSt.0;T / C k k E L1 ..0;1/IH/ /; and using the Strichartz estimate, one further has k kSt.0;T / . C kkL1 .0;T / C k kSt.0;T / : Thus we obtain by Fatou k.; /kX lim inf k.; /kXT . ; T !1
and the contraction mapping principle implies the uniqueness. The invariance now follows from (3.62) and the uniqueness which we just proved.
3.3 Center-stable manifolds via the Lyapunov–Perron method
115
Exercise 3.23. Provided the details for the linear scattering alluded to above which allows one to pass from a free KG wave relative to LC to a free KG wave in the usual sense. Moreover, prove the partition of the energy as stated in (3.63). We remark that the proof also establishes the following properties of M: for any given .0/ 2 R, .0/ E 2 P? .H/ satisfying 2 2 2; P E0 WD k 2 j.0/j2 C k! .0/kL 2 C k .0/k L2
there exists a unique solution u of NLKG (2.1) on 0 t < 1 satisfying u.0/ D Q C .0/ C .0/;
P? u.0/ P D .0/; P
jhju.t/ P C ku.t/ij . E0 for all t 0, and (3.70) .0 8t < 1/: In other words, M is parametrized by .0/; .0/ E , as well as by .0/; E1 .0/ . These parametrizations are smooth in L1 .0; 1I H/. E 2 CE0 ; Qk H
kE u.t/
Exercise 3.24. Consider the equation in R1C3 with radial data uR
u C u
jujp
1
uD0
(3.71)
in the range 3 < p < 5. Assuming the gap property (3.12) for the ground state of (3.71), extend the constructions of Theorem 3.22 to this case. This exercise demonstrates one of the differences between the constructions of center-stable manifolds via the Hadamard and Lyapunov–Perron methods, respectively. While the former does not apply to nonlinearities with powers exceeding 3, at least in the original form of Bates and Jones [6], the previous exercise shows that the Lyapunov–Perron method does easily allow for such extensions due to the use of Strichartz estimates for the linearized equation, provided the gap property holds. On the other hand, see [112] for an adaptation of the graph transform method to accommodate powers exceeding 3. The strategy of [112] is to use the basic proof strategy of Hadamard (i.e., the existence of different exponential expansion rates) but not the “abstract” result of [6], and implement it on the level of the Duhamel formulation of the equation using Strichartz estimates for the free equation. In addition [112] allows for nonradial solutions which present a much more serious obstacle to the graph transform method than higher powers do.
116
3 Above the ground state energy I: Near Q
[44] shows that the gap property fails for p < 2:8 (loc. cit. contains a more precise decimal expansion of this number). Needless to say, Exercise 3.24 applies to certain powers below 3 as well, as long as the gap property remains valid. It is an interesting open problem to obtain a version of Theorem 3.22 if LC does have eigenvalues in the gap or a threshold resonance. Note that [112] does obtain such a result, but without the scattering property and an orbital, rather than an asymptotic, stability property of the center-stable manifold. We will now employ the stronger asymptotic stability statement of Theorem 3.22 in the construction of stable and unstable manifolds.
3.3.3 The stable and unstable manifolds We now construct the stable manifold as a corollary to Theorem 3.22. The unstable one is then obtained by reversing time. The main idea is that the stable manifold is uniquely characterized as that subset of the center-stable manifold (with respect to forward time, say), for which the scattering data vanish identically. This follows from the energy partition as it appears in Theorem 3.22. Corollary 3.25. Let be as in Theorem 3.22. Then any .u0 ; u1 / 2 B .Q; 0/ with the property that the associated solution u.t/ of (2.1) satisfies u.t/; u.t/ P ! .Q; 0/;
t ! C1
(3.72)
belongs to a one-dimensional smooth manifold W s in B which is tangent to the line ˚ .Q; 0/ C .; k/ j 2 R
(3.73)
at .Q; 0/. Moreover, W s n f.Q; 0/g consists of two distinct trajectories of solutions to (2.1) which approach .Q; 0/ exponentially fast. Any two solutions belonging to one of these two halves differ only by a time translation. Proof. If (3.72) holds, then from the energy splitting E.E u/ D J.Q/ C 21 k E1 k2H of Theorem 3.22 one sees that the scattering data satisfy 1 D 0. But then the solution is parametrized by .0/ alone, which determines W s as a curve. The case .0/ D 0 is the constant solution .Q; 0/, whereas .0/ > 0 and .0/ < 0 correspond to the two distinct trajectories. Time translation leaves these trajectories invariant since (3.72) is also invariant. It remains to prove that the convergence is exponentially fast. This of course follows by noting that the entire solution is controlled by .t/, which is
3.4 Dispersive estimates for the perturbed linear evolution
117
exponentially decreasing. To be more precise, instead of (3.59) and (3.60), we now have the system for v.t; x/ D .t/.x/ C .t; x/, Z 1 Z 1 1 1 ks kt e N .v/.s/ ds C e kjt sj N .v/.s/ ds .t/ D e .0/ C 2k 0 2k 0 Z 1 1 sin.!.t s//Nc .v/.s/ ds:
.t/ D ! t (3.74) The second equation here is obtained from the fact that 1 D 0. It is clear that (3.74) is contractive in the exponentially weighted space k.; /kY WD sup e kt j.t/j C k. ; /.t/k P H t0
for any small .0/, and the result follows. We note that the unstable manifold W u is tangent to ˚ .Q; 0/ C .; k/ j 2 R and therefore transverse to the center-stable one.
3.4
Dispersive estimates for the perturbed linear evolution
There are many results on dispersive estimates – in the pointwise as well as in the time-averaged, i.e., Strichartz sense – for both the linear Schrödinger and wave evolutions under perturbations. “Perturbation” here can mean a variety of things, such as perturbations of the lowest order, i.e., potentials acting as multiplication operators, or magnetic potentials which involve terms of the form iAr. We begin with the former, i.e., we replace H0 D with H D H0 C V where V is real-valued, bounded, and decays at infinity in a suitable sense (depending on the context). As usual, special care needs to be taken with the situation at zero energy; the latter is also referred to as threshold since it is the edge of the continuous spectrum. The basic distinction being between a regular threshold on the one hand, and a singular threshold on the other hand. The former refers to the case where zero energy is neither an eigenvalue nor a resonance of H D C V , whereas the latter is the opposite. These properties are most easily expressed in terms of the resolvent near zero energy. More precisely, a regular threshold is characterized as sup kw .H
Im z>0
z2/
1
Pc .H /w k2!2 < 1
(3.75)
118
3 Above the ground state energy I: Near Q
where w .x/ D hxi for some sufficiently large > 0 (and with Pc .H / being the projection onto the continuous spectrum). Equivalently, in three dimensions it is also characterized by requiring that there are no nonzero solutions to the equation H D 0 which satisfy hxi 2 L2 .R3 / for all > 21 . In fact, if there is such a 2 3 1 solution which R does not belong to L .R /, then j .x/j cjxj as jxj ! 1, with c ¤ 0, and V .x/ dx ¤ 0. Such a function is referred to as a resonance function. In one dimensions, the condition is simply 2 L1 .R/. While (3.75) is known3 to hold in general provided jzj > " > 0 with a constant C."/, the latter constant may or may not remain bounded as " ! 0. In fact, in R3 (as well as in all odd dimensions) there is a Laurent expansion near z D 0 of the form .H
z2/
1
DB
2
z
2
CB
1
z
1
C B0 C B1 z C : : : C Bk z k C Rk .z/ (3.76)
where kw Rk .z/w k2!2 Ck jzjkC1 provided is large enough depending on k. B 2 is the orthogonal projection onto the zero-energy eigenspace. Zero energy being regular, or equivalently (3.75), is the same as B 2 D B 1 D 0. The relevance of this condition for dispersive estimates can be seen from the fact that various evolutions of H appear as some form of Fourier transform of the spectral measure. More specifically, e
itH
1
Z Pc .H / D
e it E.d/Pc .H /
0
e
˙it
p
H C1
Z Pc .H / D
1
e
˙it
p
(3.77) C1
E.d/Pc .H /
0
where E./ is the spectral resolution of H and Pc is the projection onto the continuous spectral subspace of H . The first line in (3.77) is the Schrödinger evolution, whereas the second line represents Klein–Gordon evolutions associated with H . The connection with the resolvent is furnished by the well-known identity between projection-valued measures on L2 E.d/Pac .H / D Im.H
/
1
Pac .H / d;
>0
(3.78)
with Pac .H / being the projection onto the absolutely continuous subspace of H . The right-hand side of (3.78) is singular at D 0 if B 1 ¤ 0. In view of (3.77), it is therefore no surprise and also well-known that for the case where zero energy
3
This is called “limiting absorption principle”.
119
3.4 Dispersive estimates for the perturbed linear evolution
is singular, the usual dispersive estimates fail4 (in both the pointwise and Strichartz senses). There is a sizable body of literature on dispersive estimates for both wave and Schrödinger evolutions with regular threshold, the main objective being to establish the same decay properties as for the free evolutions. However, we cannot possibly review this literature here (see however [123], [124]). If the potential V exhibits sufficient decay and regularity, then Yajima [145] found a general and powerful approach based on the wave operators. Yajima’s theory allows one to deduce the desired dispersive bounds for H D H0 C V directly from those for the free equation (this applies to both pointwise as well as Strichartz estimates, as well as to different kinds of evolutions).
3.4.1 The wave operators in scattering theory The wave operators are defined as the strong limits in L2 W WD s
lim e
t !1
itH itH0
e
;
H WD
C V;
H0 D
(3.79)
which means that Wf D lim t!1 e itH e itH0 f for any f 2 L2 . It is easy to see that this limit exists in R3 provided that V 2 L2 .R3 /. Indeed, let f be a Schwartz function, say, and note that5 Z t Z t d isH isH0 i tH i tH0 e e f ds D i e isH Ve isH0 f ds : e e f f D 0 0 ds 3
Using the unitarity of the e isH evolution, as well the hsi 2 pointwise decay of the free evolution, one concludes that the integral on the right is absolutely convergent in L2 over .0; 1/ provided V 2 L2 . Finally, the Schwartz class is dense in L2 which implies that the limit exists for all f 2 L2 due to the unitarity of the evolution operators. The wave operators are isometries on L2 . Thus W W and W W are projections, the former onto Ran.W /, the latter being the identity (this is a general property of isometries; consider the right shift in `2 over the positive integers as an example). 4
5
In this regard, recall Howland’s razor [75], which states that resonances cannot be defined in terms of a single operator on an abstract Hilbert space. However, one can detect them by means of a single operator on Lp with p ¤ 2. This argument is called “Cook’s method”.
120
3 Above the ground state energy I: Near Q
By construction one has the intertwining property e isH W D W e isH0 H) E./W D WE0 ./ where E; E0 refer to the spectral resolutions of H , and H0 , respectively. Therefore, Ran.W / L2a:c: , the absolutely continuous subspace of H . In fact, it is known that for “nice” potentials (such as V D 3Q2 ) this is an equality, and that there is no singular continuous spectrum, see Agmon [1], Kuroda [93], and Reed, Simon [120], Vol. 3, for example. Hence W W D Id;
W W D Pc .H / D Id
Ppp .H /
where Ppp is the projection onto the space spanned by all eigenfunctions. The latter space is finite dimensional for “nice” potentials, for example if the Birman– Schwinger operator 1
K WD jV j 2 . /
1
1
jV j 2
is a compact operator on L2 .R3 /. In particular, f .H /Pc .H / D Wf .H0 /W
(3.80)
for any bounded continuous f . This latter identity is the key for applying Yajima’s theorem to which we now turn.
3.4.2 Yajima’s Lp bound on the wave operators Yajima showed under certain conditions on V that W is bounded on Lp .R3 / for any 1 p 1. In R3 , the special case of his theorem from [145] which we used in the proof of Theorem 3.22 reads as follows. Needless to say, the following result applies to V D 3Q2 . Theorem 3.26. Let V be real-valued and jV .x/j . hxi , where > 5. Assume furthermore that zero energy is regular for H D C V . Then the wave operator W from (3.79) is bounded on Lp .R3 / for all 1 p 1. A typical example demonstrating the applicability of Yajima’s theorem already appeared in the proof of Theorem 3.22, see how (3.64) lead to (3.65). Some version of Theorem 3.26 holds in all dimensions, as well as for W k;p spaces.
3.4 Dispersive estimates for the perturbed linear evolution
121
3.4.3 Singular thresholds Generally speaking, this case is less understood. It is definitely “non-generic” and may therefore seem somewhat exotic. However, it does arise in actual problems, such as for the energy critical wave equation in R3 , see Chapter 6, Section 6.4. There the linearized operator associated with the ground state is LC D 5W 4 with 1 W .r/ D .1 C r 2 =3/ 2 , and LC exhibits a zero energy resonance but not an eigenvalue at zero energy – at least over the radial functions; nonradially, one has root modes rW which are eigenfunctions. In this situation it is not meaningful to seek general dispersive or Strichartz estimates6 since the threshold resonance destroys the free linear decay, at least for general data. However, this failure is restricted to a finite-dimensional subspace which we can hope to isolate (this is the statement that B 1 is of finite rank). The idea is that after subtracting the linear dynamics on this subspace we should be left with an evolution which does exhibit free dispersive estimates. As an example, consider the following result from [91]. In order to apply it, one specializes to H D 5W 4 restricted to radial functions. This is legitimate, since the potential is radial. Note that W 4 decays exactly like hxi 4 . Hence it is important that we do not require too much decay in Proposition 3.27. Proposition 3.27. Assume that V is a real-valued potential such that jV .x/j . hxi where > 3 is arbitrary but fixed. If H D C V has neither a resonance nor an eigenvalue at zero, then
sin.t pH /
(3.81) Pc f . t 1 kf kW 1;1 .R3 /
p 1 H for all t > 0. Now assume that zero is a resonance but not R an eigenvalue of H . Let be the unique resonance function normalized so that V .x/ dx D 1. Then there exists a constant c0 ¤ 0 such that
sin.t pH /
Pc f c0 . ˝ /f . t 1 kf kW 1;1 .R3 / (3.82)
p 1 H for all t > 0. Note that even the regular threshold assertion (3.81) does not follow from Yajima’s theorem since that result requires more decay, see Theorem 3.26. The ap6
Note, however, that Yajima did establish Lp boundedness of the wave operators even if zero energy is 0 where p D p .d /. singular, but in a restricted range 1 < p < p < p
122
3 Above the ground state energy I: Near Q
proach to the proof of Proposition 3.27 in [91] is based on the Laurent expansion (3.76) for small energies, and a Born expansion for large energies. Note that the pointwise decay in the above proposition is that of the free wave equation in three dimensions. We remark that Proposition 3.27 was used in [91] to construct Lipschitz graphs of codimension 1 near W with the property that solutions starting from this manifold exhibit the desired forward global existence and asymptotic stability property. However, the underlying topology in this construction was much stronger than the energy space and it remains an open problem to decide whether or not such a manifold also exists in the energy topology. See Section 6.4 for more on this issue.
3.4.4 Beceanu’s linear dispersive estimate In this section7 , we present some of the linear analysis introduced by Beceanu [8], [9] in his construction of the center-stable manifolds for the cubic NLS equation in the critical topology and without any symmetry assumptions. To be more specific, consider the nonlinear equation i@ t
D j j2 ;
which admits the periodic solitons e
it˛ 2
Q C ˛ 2 Q D Q3 ;
.t; x/ 2 R1C3
(3.83)
Q.x; ˛/. Here Q.x; ˛/ solves Q.x; ˛/ D ˛Q.˛x; 1/
where Q.x; 1/ > 0 is the ground state which we encountered in Chapter 2. It has been known since the work of Berestycki, Cazenave [11], Cazenave, Lions [28], and Weinstein [143], [144] that these periodic solutions are unstable; in fact, they can blow up under arbitrarily small perturbations. This is of course in analogy with the Klein–Gordon instability, cf. Corollary 3.1. An important difference between NLKG and NLS arises at the L2 -critical powers and below: while NLS solitons become stable for L2 -subcritical nonlinearities, this is not the case for NLKG; indeed, for the latter one has the Bates–Jones theorem on the existence of unstable manifolds which implies the instability of the ground states. In [122] it was shown, relative to a suitable topology, that NLS solitons (for the cubic nonlinearity) are nevertheless conditionally asymptotically stable; this of course means that asymptotic stability is guaranteed provided the data are chosen 7
Skip on first reading; this is only relevant to the NLS equation.
3.4 Dispersive estimates for the perturbed linear evolution
123
from a small Lipschitz hyper-surface containing the 8-dimensional soliton manifold. The drawback of this result lay with the choice of topology, which contained an L1 -component and which was therefore non-invariant. Beceanu [9] then carried out the construction in the energy topology (and in fact, the weaker critical topology of (3.83)), which yielded the center-stable manifold with the desired invariance property. His construction is based on a delicate Strichartz estimate for linear operators with small time-dependent, but space-independent, coefficients which we present in this section. A significant difference between the NLS and NLKG equations lies with the need to modulate the ground state for the former, even in the radial setting whereas for NLKG this is not the case and Q remains unchanged. This refers to the fact that 2 e i t ˛ Q is multiplied by an additional phase e i .t/ and ˛ becomes time-dependent, ˛ D ˛.t/; for nonradial solutions one also encounters space and momentum translations, adding another six parameters into the equation. We now linearize (3.83) around the ground state e it Q where Q D Q.; 1/, which leads to the following matrix operator (we view all operators in this section as complex linear ones): C 1 2Q2 .; 1/ Q2 .; 1/ HD ; Q2 .; 1/ 1 C 2Q2 .; 1/ see the following section for more details on this linearization. H is conjugate to 1h 1 2 i
1 i h1 H i 1
h 0 i i L i Di i LC 0
(3.84)
where LC D
C1
3Q.; 1/2 ;
L D
C1
Q.; 1/2 :
The equality (3.84) is to be considered as one between complex linear operators. However, it is also natural to view the left-hand side as acting on vectors uu12 with u1 ; u2 real-valued. In that case the right-hand side needs to be rewritten as hL
C
0
0 i D L: L
As already noted above, the spectral properties of LC ; L and especially H are quite delicate. First, we remark that L has the same gap property, see (3.12), as LC . For this we refer the reader to [44] and [41]. The following result summarizes what can be obtained by rigorous analysis, see [52], [76], [122], combined with some numerical findings, such as [44] and [99]. In the work of Marzuola, Simpson [99]
124
3 Above the ground state energy I: Near Q
numerics is used to assist index computations of certain quadratic forms in the spirit of the virial argument of Fibich, Merle, Raphael [54]. For simplicity, we restrict ourselves to the Hilbert space8 L2rad .R3 / in Proposition 3.28. Proposition 3.28. The essential spectrum of H is . 1; 1 [ Œ1; 1/ and there are no imbedded eigenvalues or resonances in the essential spectrum, the discrete spectrum is of the form f0; i; ig where > 0 with ˙i both simple eigenvalues, the root space at 0 is of dimension two, and the thresholds ˙1 are neither eigenvalues nor resonances. In explicit form, the root space is spanned by ! ! Q @˛ Q 0 D ; 0 D (3.85) Q @˛ Q and one has H0 D 0; H2 0 D 0. Let HG˙ D iG˙ with the normalization kG˙ k2 D 1. Then the eigenfunctions G˙ are exponentially decaying and of the form G˙ D gg˙ . ˙
Proof. The description of the root space of H goes back to [143], [144]. The imaginary spectrum was identified by Grillakis [63], [64] (see also [122]), and for the exponential decay of the corresponding eigenfunctions see [76]. All these results are based on purely analytical arguments. The fact that H does not have embedded eigenvalues in the essential spectrum was shown in [99], assisted by some numerical computations. Their proof also implies that there are no non–zero eigenvalues in the gap Œ 1; 1, and that the thresholds are not resonances. The latter two facts also follow rigorously by the analytical arguments in [122] combined with the proof of the gap properties of L˙ in [41]. Furthermore, it is to be expected that by combining the analytical argument in [99] with the rigorous study of the ground state conducted in [41] one can eliminate all numerical components from [99]. This would then establish the absence of embedded eigenvalues in the essential spectrum analytically, removing all numerical components from this proof. Next, we present a result for non-selfadjoint Schrödinger evolutions which originatesTin [8] (in fact, Beceanu proves a stronger result in Lorentz spaces). Let S D p;q Lpt .RC ; Lqx / be the Strichartz space with 2 p 1, 2 q 6, and 1 0 2 3 3 C q D 2 , and let S be its dual. For the remainder of this chapter, 3 D p 0 1 8
The only change to Proposition 3.28 is that H has a root space of dimension eight rather than two.
3.4 Dispersive estimates for the perturbed linear evolution
125
i
ess spec
1
0
ess spec
1
i
Figure 3.8. Spectrum of the linearized operator as in Proposition 3.28
is the third Pauli matrix. Also, we define the matrix operators H0 ; V via C1 0 H0 D ; H D H0 C V: 0 1 Lemma 3.29. Let A; a 2 L1 .R/ be real-valued and satisfy kAk1 C kak1 < c0 for some small absolute constant c0 . The solution 2 C RI L2 .R3 / \ C 1 RI H 2 .R3 / of the problem i @ t C H C iA.t/rPc C a.t/3 Pc D F 2 S ;
.0/ D 0 2 L2 .R3 / (3.86)
where Pc is the projection corresponding to the essential spectrum of H, obeys the Strichartz estimates kPc kS . k 0 kL2 .R3 / C kF kS :
(3.87)
Furthermore, if 0 2 H 1 , then krPc kS . k 0 kH 1 .R3 / C krF kS :
(3.88)
126
3 Above the ground state energy I: Near Q
Finally, one has scattering. Suppose for simplicity that A D 0. Then there exists 1 2 H 1 such that Pc .t/ D e i3
Rt
0.
C1Ca.s// ds
1 C o.1/
(3.89)
in H 1 as t ! 1. Proof. We follow [8], and set A D 0 for simplicity. Clearly, the proof should be perturbative in a by nature, with a D 0 being the nontrivial statement that Strichartz estimates (including the endpoint) hold for the equation i@ t C H D F : However, the latter has been established by several authors, see for example [8], Theorem 1.3 and [43]. Due to the lack of any physical localization of the a.t/ term, the perturbative analysis is nontrivial. On the other hand, note that any perturbation 6 of the form a.t/ .x/ where the multiplier is bounded L6 .R3 / ! L 5 .R3 / can be taken to the right-hand side by virtue of the endpoint Strichartz estimate. To commence with the actual argument, consider the following auxiliary equation, with arbitrary but fixed ı > 0, and Pd D Id Pc : i@ t Z C HPc Z C iıPd Z C a.t/3 Pc Z D F
(3.90)
with data Z.0/ D 0 . We claim the Strichartz estimates for general data Z.0/, kZkS . kZ.0/kL2 .R3 / C kF kS :
(3.91)
Q If so, then ZQ WD Pc Z satisfies Z.0/ D Pc 0 and i@ t ZQ C HZQ C a.t/3 ZQ D Pc F C a.t/Œ3 ; Pc ZQ which is the same as the PRc projection of (3.86). Thus, Pc D ZQ and (3.91) imt plies (3.87). Let A.t/ D 0 a.s/ ds and write U.t/ D e iA.t/3 , Z.t/ D U.t/˚ . Then (3.90) becomes i@ t ˚ C U
1
.HPc C iıPd /U˚ D U
1
F C a.t/U
1
3 Pd U˚ DW F1
(3.92)
or, with ˚.0/ D Z.0/, i@ t ˚ C H0 ˚ D
U
1
.V
HPd C iıPd /U˚ C F1 :
(3.93)
127
3.4 Dispersive estimates for the perturbed linear evolution
Choose a smooth, exponentially decaying matrix potential V2 which is invertible and such that the operator V1 WD .V
1
HPd C iıPd /V2
is bounded from Lp ! Lq for any 1 p; q 1. In other words, V1 V2 D V
(3.94)
HPd C iıPd
with V1 ; V2 being bounded from Lp ! Lq for any 1 p; q 1. By Duhamel the solution to (3.93) is Z t i t H0 (3.95) ˚.t/ D e ˚.0/ i e i.t s/H0 Œ U 1 V1 V2 U˚ C F1 .s/ ds : 0
Applying U.t/ to both sides yields, since U commutes with the propagator of H0 , Z t i tH0 Z.t/ D U.t/e Z.0/ C i e i.t s/H0 U.t/U 1 .s/V1 V2 Z.s/ U.t/F1 .s/ ds: 0
(3.96)
We introduce the operators t
Z T0 F .t/ WD V2
e i.t
s/H0
V1 F .s/ ds;
e i.t
s/H0
U.t/U.s/
0
TQ0 F .t/ WD V2
t
Z
1
V1 F .s/ ds :
0
By the Strichartz estimates for the free equation, T0 ; TQ0 are bounded on L2t;x . By (3.96), Z t it H0 Q V2 Z D i T0 V2 Z C V2 U.t/e Z.0/ iV2 U.t/ e i.t s/H0 F1 .s/ ds : (3.97) 0
Suppose .Id
i TQ0 /
1
W L2t;x ! L2t;x
(3.98)
as a bounded operator. Then (3.97) implies via the endpoint Strichartz estimate, see (3.92), kV2 ZkL2 . kZ.0/k2 C kF1 kS . kZ.0/k2 C kF kS C c0 kV2 ZkL2 : t;x
t;x
128
3 Above the ground state energy I: Near Q
To pass to the final estimate we wrote Pd Z D Pd V2 1 V2 Z and used that Pd V2 1 is bounded by construction. Inserting the resulting bound on kV2 ZkL2 back into t;x (3.96) yields the desired estimate (3.91). It therefore remains to prove (3.98) which will follow from .Id
iT0 /
1
W L2t;x ! L2t;x
(3.99)
provided we can show that kT0 TQ0 k 1 in the operator norm on L2t;x . This, however, follows from the pointwise dispersive estimate on e itH0 which yields 1 5 4 ht si 4 kF .s/k2 : kV2 e i.t s/H0 U.t/U.s/ 1 1 V1 F .s/k2 . kak1 Thus, we have reduced ourselves to proving (3.99). We introduce Z t T1 F .t/ WD V2 e i.t s/HPc .t s/ıPd V1 F .s/ ds: 0
As for the meaning of T1 , first note that due to commutativity e itHPc
tıPd
D e it HPc e
tıPd
D e it H Pc C Pd e
tıPd
D e it H Pc C e
tıPd
Pd
satisfies Strichartz estimates as in (3.87), see [8] and [43]. Second, the solution to i@ t Z C HPc Z C iıPd Z D 0 can be written in two ways: Z.t/ D e itHPc
tıPd
Z.0/ ; Z t itH0 Z.t/ D e Z.0/ C i e i.t
s/H0
.V
HPd C iıPd /Z.s/ ds:
0
Thus, one further has e i t HPc
t ıPd
De
Z.0/
it H0
t
Z
e i.t
Z.0/ C i
s/H0
.V
HPd C iıPd /e isHPc
sıPd
Z.0/ ds :
0
Therefore, we conclude that Z tZ s T0 T1 F .t/ D V2 e i.t 0
s/H0
.V
HPd C iıPd /e i.s
s1 /HPc .s s1 /ıPd
0
V1 F .s1 / ds1 ds t
Z D
.e i.t
iV2 0
s1 /HPc .t s1 /ıPd
e i.t
s1 /H0
/V1 F .s1 / ds1
129
3.4 Dispersive estimates for the perturbed linear evolution
or T0 T1 C i.T1
T0 / D 0 which implies that iT0 /.Id C iT1 / D Id :
.Id On the other hand, s
Z tZ
e i.t
T1 T0 F .t/ D V2
s/HPc .t s/ıPd
.V
HPd C iıPd /e i.s
s1 /H0
0
0
V1 F .s1 / ds1 ds Z tZ
t
@s e i.t
D iV2 0
e i.s
s1 /H0
ds V1 F .s1 / ds1
s1 t
Z D
s/HPc .t s/ıPd
.e i.t
iV2
s1 /HPc .t s1 /ıPd
e i.t
s1 /H0
/V1 F .s1 / ds1
0
whence T1 T0 C i.T1
T0 / D 0 which implies that .Id C iT1 /.Id
iT0 / D Id :
These identities hold in the algebra of bounded operators on L2t;x , as justified by the endpoint Strichartz estimates. Thus (3.99) holds and (3.87) follows. For (3.88) one applies a gradient to (3.86). From (3.95), we obtain the scattering of ˚ in the following sense: ˚.t/ D e itH0 ˚1 C o.1/;
t !1
in H 1 for some ˚1 2 H 1 . Thus, Pc .t/ D Pc U.t/ e itH0 ˚1 C o.1/ D e iA.t/3 e it H0 ˚1 C o.1/;
t !1
in H 1 , as claimed. The proof with A ¤ 0 is left to the following exercise. Exercise 3.30. Adapt the previous proof to include the term iAr with A ¤ 0.
130
3 Above the ground state energy I: Near Q
3.5
The center-stable manifold for the radial cubic NLS in R3
In this section, we construct9 a center-stable manifold containing the ground state Q for the NLS equation (3.83). All function spaces will be radial. Moreover, Bı .Q/ denotes a ı-ball in the energy space centered at Q. We work with the matrix formalism as it appears in Section 3.4.4. In what follows, H.˛; / D
C ˛ 2 2Q2 .; ˛/ e 2i Q2 .; ˛/
e 2i Q2 .; ˛/ : ˛ 2 C 2Q2 .; ˛/
(3.100)
The following proposition constructs the center-stable manifold in a small neighborhood of Q. It should be compared to Definition 6.13; in fact, it provides much more detailed information than what is required by that definition. In view of the symmetries of (3.83) it is more natural to work with the “cone” of radial solitons S WD fe i Q.; ˛/ j 2 R; ˛ > 0g; rather than with a fixed soliton Q. Moreover, S˛ is the slice of S given by fixing ˛. In Remark 3.32 we therefore extend M so as to cover all of the radial soliton manifold S, and Corollary 3.33 characterizes the stable manifold, which lies in M. All of this is in analogy with the results which were obtained above for the NLKG equation. However, as we shall see, carrying out the underlying Lyapunov–Perron scheme is much more involved for the NLS equation. As we already mentioned several times before, the following proposition has its roots in the asymptotic stability analysis of solitons. For results on asymptotic stability analysis in the subcritical, and thus orbitally stable case, see Buslaev, Perelman [22], [23], Cuccagna [42], and Soffer, Weinstein [128]. See also Pillet, Wayne [117] for invariant manifolds in the context of the stability analysis of small solitons obtained by bifurcation off of linear eigenfunctions. 1 Theorem 3.31. There exists ı > 0 small and a smooth manifold M Hrad with the following properties: S \ Bı .Q/ M Bı .Q/, M divides Bı .Q/ into two connected components, and any initial data u0 2 M generates a solution of (3.83) for all t 0 of the form
u.x; t/ D e i.t/ Q x; ˛.t / C v.x; t/;
9
8t 0
Skip on first reading, this is not needed for the main theorems on the NLKG equation.
(3.101)
3.5 The center-stable manifold for the radial cubic NLS in R3
where .t/ D
Rt 0
131
˛ 2 .s/ ds C .t/, k P kL1 \L1 .0;1/ C k˛k P L1 \L1 .0;1/ . ı 2 ; supŒj˛.t/ t0
1j C j .t /j . ı :
(3.102)
The function v is small in the sense kvkL1 C kvkL2 ..0;1/IW 1;6 .R3 // . ı 1 3 t ..0;1/IH .R // t
(3.103)
and it scatters: v.t/ D e it v1 C oH 1 .1/ as t ! 1 for a unique v1 2 H 1 . M is unique in the following sense: there exists a constant C so that any u0 2 Bı .Q/ satisfies u0 2 M if and only if the solution u.t/ of (3.83) with data u0 has the property that dist.u.t/; S1 / C ı for all t 0. Proof. Inserting (3.101) into (3.83) yields ! ! v v e i@ t C H.t/ D .t/ P e .t/ v v where
e D H.t/
2Q2 .; ˛.t// e 2i.t/ Q2 .; ˛.t//
e .t; v; v/ i ˛.t/e P .t/ C N
(3.104)
e 2i.t/ Q2 .; ˛.t// C 2Q2 .; ˛.t//
(3.105)
as well as e .t/ D
! e i.t/ Q.; ˛.t// ; e i.t/ Q.; ˛.t//
! e i.t/ @˛ Q.; ˛.t// e .t/ D e i.t/ @˛ Q.; ˛.t//
(3.106)
and e .t; v; v/ D N
! 2e i.t/ Q.; ˛.t//jvj2 e i.t/ Q.; ˛.t//v 2 C jvj2 v : (3.107) 2e i.t/ Q.; ˛.t//jvj2 C e i.t/ Q.; ˛.t//v 2 jvj2 v
Next, set v.t/ D e
i0 .t/
t
Z w.t/;
˛ 2 .s/ ds:
0 .t/ D
(3.108)
0
Then (3.104) turns into i@ t W C H .t/W D .t/.t/ P
i ˛.t/.t/ P C N .t; W /
(3.109)
132
3 Above the ground state energy I: Near Q
where W D
w w
H .t/ D
, and with D .˛; /, C ˛ 2 .t/ 2Q2 .; ˛.t// e 2i .t/ Q2 .; ˛.t//
e 2i .t/ Q2 .; ˛.t// ˛ 2 .t/ C 2Q2 .; ˛.t//
(3.110)
as well as .t/ D
! e i .t/ Q.; ˛.t// ; e i .t/ Q.; ˛.t//
! e i .t/ @˛ Q.; ˛.t// .t/ D e i .t/ @˛ Q.; ˛.t//
(3.111)
and N .t; W / D
! 2e i .t/ Q.; ˛.t//jwj2 e i .t/ Q.; ˛.t//w 2 C jwj2 w : (3.112) 2e i .t/ Q.; ˛.t//jwj2 C e i .t/ Q.; ˛.t//w 2 jwj2 w
At this point we remark that all manipulations which we perform in this proof on f (3.104) and (3.109) preserve the “admissible” subspace f f j f W R3 ! Cg. This is necessary in order to return to the scalar formulation (3.101). In other words, the second row of these systems can be viewed as redundant, as it is always the complex conjugate of the first. 1 0 Let 3 D be the third Pauli matrix, and set .t/ D 3 .t/, .t/ D 0 1 3 .t/. Impose the orthogonality conditions10 ˝ ˛ ˝ ˛ W .t/; .t/ D 0; W .t /; .t/ D 0; 8t 0 : (3.113) Note that this imposes a condition on the data at t D 0. However, by the inverse function theorem there is a unique choice of ˛.0/ and .0/ in a ı-neighborhood of .1; 0/ so that (3.113) is satisfied; the needed nondegeneracy here is provided by hQj@˛ Qi ¤ 0. Since H .t/ .t/ D 0 and H .t/ .t/ D 2˛ .t/, as well as ˝ ˛ ˝ ˛ .t/; .t/ D .t/; .t/ D 0; ˝ ˛ ˝ ˛ ˝ ˛ .t/; .t/ D .t/; .t/ D 2 Qj@˛ Q ¤ 0; one obtains from (3.109) that ˝ ˛
.t/ P .t/; .t/ D ˝ ˛ i ˛.t/ P .t/; .t/ D 10
˝ ˛ i W .t/; P .t/ ˝ ˛ i W .t/; P .t/
˝ ˛ N .t; W /; .t/ ; ˝ ˛ N .t; W /; .t/ :
In this section, we write h; i for the standard inner product in L2 .R3 I C2 /.
(3.114)
3.5 The center-stable manifold for the radial cubic NLS in R3
133
The time derivatives P ; P .t/ on the right-hand side come with a small coefficient (due to the W ), and are therefore admissible in the contraction. The system (3.109), (3.113), (3.114) determines the evolution of v.t/; ˛.t/; .t/ in (3.101). It suffices for (3.113) to hold at one point, say t D 0, since it then holds for all t 0. We need to find a fixedpoint to this system consisting of a path .t/ D .˛.t/; .t// as well as a function vv , or equivalently, W satisfying the system as well as the bounds (3.102), (3.103). We begin with the stability part of the underlying contraction argument, i.e., we turn (3.102) and (3.103) into bootstrap assumptions and then recover them from this system. Thus, suppose 0 D .˛0 ; 0 / and W0 are given so that (3.102) and (3.103) hold and consider the following system of differential equations: i@ t W C H0 .t/W D .t/ P i ˛.t/ P 0 .t/ 0 .t/ C N0 .t; W0 /; ˝ ˛ ˝ ˛ ˝ ˛
.t/ P 0 .t/; 0 .t/ D i W .t/; P 0 .t/ N0 .t; W0 /; 0 .t/ ; ˝ ˛ ˝ ˛ ˝ ˛ i ˛.t/ P 0 .t/; 0 .t/ D i W .t/; P0 .t/ N0 .t; W0 /; 0 .t/ ; ˝ ˛ ˝ ˛ W .0/; 0 .0/ D 0; W .0/; 0 .0/ D 0;
(3.115)
where H0 , 0 ; 0 and N0 .t; W0 / are defined as above but relative to the given functions 0 ; W0 . The initial conditions are ˛.0/ D ˛0 .0/, .0/ D 0 .0/; in addition to the final equation in (3.115), W .0/ needs to satisfy a further codimension-1 condition which will be specified below. We begin with the ˛; P P part of (3.115). The W appearing on the right-hand side will be seen later to satisfy (3.103); for the moment, we will simply assume this bound. To be more specific, rewrite (3.102) and (3.103) in the form k k P L1 \L1 C k˛k P L1 \L1 C0 ı 2 kvkL1 1 3 C kvkL2 W 1;6 .R3 / C1 ı t H .R /
(3.116)
t
and assume that C0 C12 . Inserting these bounds in the right-hand side of (3.114) yields k k P L1 \L1 C k˛k P L1 \L1 . C0 C1 ı 3 C C12 ı 2 C0 ı 2 provided ı is small. One can thus recover (3.116). The bound on v (or W ) is more delicate. Since we are in the unstable regime, (3.109) is exponentially unstable. More precisely, write H0 .t/ D H0 C a0 .t/3 C D0 .t/
134
3 Above the ground state energy I: Near Q
with the constant coefficient operator H0 D H.˛0 .0/; 0 .0//, see (3.100), and a0 .t/ D ˛02 .t/ ˛02 .0/, as well as D0 .t/ equaling e 2i 0 .t/ Q2 ; ˛0 .t / C e 2i 0 .0/ Q2 ; ˛0 .0/ Q2 ; ˛0 .0/ 2 Q2 ; ˛0 .t / : 2 2 2i 0 .0/ 2 2i 0 .t/ 2 e
Q ; ˛0 .t /
2 Q ; ˛0 .t /
Q ; ˛0 .0/
e
2
N
Q ; ˛0 .0/
2
Note that ka0 ./k1 . ı and khxi D0 ./k1 . ı for any N provided the condition (3.102) holds. Proposition 3.28 in Section 3.4.4 details the spectral properties of H0 provided ˛.0/ D 1 and .0/ D 0. The more general case here follows by means of the rescaling f 7! ˛f .˛x/, as well as a modulation by a constant unitary matrix. Following the notation of Proposition 3.28 one writes W .t/ D C .t/GC C .t/G C W1 .t/
(3.117)
˝ ˛ where W1 .t/; 3 G˙ D 0 for all t 0. One needs to apply the aforementioned rescaling and modulationto G˙ and with the fixed parameters ˛0 .0/; 0 .0/, which means that D ˛0 .0/ , G˙ D G˙ ˛0 .0/; 0 .0/ . We remark that ˙ as defined in (3.117) are real-valued. Indeed, since kG˙ k2 D 1 and GC D ggC D G , the C Riesz projections associated with the eigenvalues ˙i can be seen to be P˙ D
h; 3 G i G˙ ; hG˙ ; 3 G i
(3.118)
where hG˙ ; 3 G i 2 i R n f0g. Therefore, C D
hW; 3 G i 2ihwjig i D 2R hG˙ ; 3 G i hG˙ ; 3 G i
and similarly for . We now rewrite the W -equation in (3.115) in the form i P C .t/GC C i P .t/G
iC .t/GC C i .t/G
C i@ t W1 .t/ C .H0 C a0 .t/3 /W1 D
D0 .t/W C .t/ P 0 .t/
a0 .t/3 GC
(3.119)
a0 .t/3 G
i ˛.t/ P 0 .t / C N0 .t; W0 / DW F .t/ :
Denote by P˙ , P0 the Riesz projections onto G˙ , and the zero root space, respectively. Note that these operators are given by integration against exponentially decaying tensor functions. Moreover, we write Pc D 1
PC
P
P0 D Preal
P0
(3.120)
3.5 The center-stable manifold for the radial cubic NLS in R3
135
for the projection onto the continuous spectrum. Applying the projections P˙ to (3.119) yields the system of ODEs P C .t/ C .t/ GC D ia0 .t/PC .3 W1 / iPC F .t/ ; (3.121) P .t/ C .t/ G D ia0 .t/P .3 W1 / iP F .t/ : For “generic” initial data C .0/ the solution C .t/ grows exponentially. However, there is a unique choice of initial condition that stabilizes C (i.e., ensures that it remains bounded) leading to the determination of the co-dimension one manifold. It is given by means of the suppose x.t/ P x.t/ D f .t/ following simple principle: with f 2 L1 .0; 1/ . Then x 2 L1 .0; 1/ iff Z 1 (3.122) 0 D x.0/ C e t f .t/ dt: 0
Thus, 1
Z 0 D C .0/GC C i
e
t
0
a0 .t/PC .3 W1 /.t/
PC F .t/ dt
is that unique choice. (3.123) has the following equivalent formulation Z 1 C .t/GC D i e .s t/ a0 .t/PC .3 W1 /.t/ PC F .t/ ds :
(3.123)
(3.124)
t
For .t/ we have the expression .t/G D e
t
t
Z .0/G C i
e
.t s/
a0 .t/P .3 W1 /.t/
0
P F .t/ ds : (3.125)
Via (3.118) one checks that ˙ as defined by these equations are real-valued. To determine the PDE for W1 D Preal W1 D Preal W , we write W1 D Pc W1 C P0 W1 D Wdisp C Wroot . Then i @ t Wdisp .t/C H0 Ca0 .t/3 Wdisp D Pc F .t/ a.t/Œ3 ; PC CP CP0 W1 : (3.126) The sought after solution ˛.t/; .t/; C .t/; .t/; Wroot .t/; Wdisp .t/
(3.127)
is now determined from the second and third equations of (3.115), from (3.124), (3.125), and (3.126). The root part is controlled by the orthogonality conditions hW; 0 i.t/ D hW; 0 i.t/ D 0;
8 t 0:
136
3 Above the ground state energy I: Near Q
The main technical ingredient for the dispersive control of (3.126) is the Strichartz estimate of Lemma 3.29. The existence of the solution (3.127) is not entirely trivial since the determining equations contain these functions linearly on the right-hand side. However, they occur with small coefficients which allows one to iterate or contract; we skip those details. The solution obeys the estimates (3.102), (3.103). While (3.102) has already been established in this fashion, (3.103) is obtained as follows. Assuming again (3.116), one concludes from (3.124) and (3.125) that k˙ kL1 \L2 . ı C .C0 C1 ı C C0 C C12 /ı 2 C1 ı provided ı is sufficiently small. Via Lemma 3.29 we conclude that kWdisp kS C1 ı where S is the Strichartz space in (3.103). Finally, now that the path ˛.t/; .t/ has been determined, as well as ˙ .t/, Wdisp .t/, the orthogonality conditions (3.113) determine Wroot which also satisfies kWroot kS C1 ı. From these estimates, we conclude (3.103) via bootstrap as claimed. The manifold is determined by (3.123) as a graph, once a fixed point .; W / D .0 ; W0 / is obtained. More precisely, for fixed ˛.0/; .0/ we prescribe initial conditions W .0/ 2 H 1 , kW .0/ kH 1 . ı for (3.115) such that P0 W .0/ D 0 as well as PC W .0/ D 0 where the projections are relative to H ˛.0/; .0/ . Such data are linearly stable. The condition (3.123) takes nonlinear corrections into account and modifies the data in the form W .0/ D W .0/ C h.0 ; W0 ; W .0/ /GC
(3.128)
ˇ ˇ where h.0 ; W0 ; W .0/ / D C .0/ is real-valued and satisfies ˇh.0 ; W0 ; W .0/ /ˇ . ı 2 . Since .0/ D 0 .0/ by construction, once we have found a fixed point, we can write h.0 ; W0 ; W .0/ / D h ˛0 .0/; 0 .0/; W .0/ where the latter is smooth in W .0/ in the sense of Fréchet derivatives. Moreover, the bound ˇ
ˇ ˇh ˛0 .0/; 0 .0/; W .0/ ˇ . W .0/ 2 1 H
(3.129)
will hold. This shows that (3.128) describes a codimension-3 manifold which is smoothly parametrized by W .0/ and tangent to the subspace of linear stability. To regain the two missing codimensions, we vary ˛0 .0/; 0 .0/ in a ı-neighborhood of .1; 0/. In other words, we let the dilation and modulation symmetries act on the codimension-3 manifold. Since these symmetries act transversely on the manifold
3.5 The center-stable manifold for the radial cubic NLS in R3
137
(for the same reason that allowed us to enforce (3.113) at t D 0 by modifying the data), we obtain a smooth codimension-1 manifold which will be parametrized by ˛0 .0/; 0 .0/; W .0/ 2 .1 ı; 1 C ı/ . ı; ı/ Bı where Bı is a ı-ball in H 1 . This is then the sought after M. Thus, one needs to find a fixed point for the system (3.115) via a contraction argument. The contraction argument is slightly delicate as it involves solving this system with two different but nearby given paths j0 , j D 0; 1 which therefore define different Hamiltonians via (3.110), and therefore also different orthogonality conditions (3.113). Note that phases of the form t˛ 2 and t ˛Q 2 diverge linearly if ˛ 2 ¤ ˛Q 2 . This make it necessary to employ a weaker norm than the one used in the previous stability argument, see (3.102), (3.103). In order to carry out the comparison between two solutions, we work on the level of (3.104) rather than with the aforementioned W -system. Thus, consider two paths j0 .t/ D ˛j0 .t/; j0 .t/ satisfying (3.102) and with 0.0/ .0/ D 1.0/ .0/, and the associated equations with Z D vv ej .t/Z D Pj .t/e i@ t Z C H j .t/
ej .t; vj0 ; vj0 / i ˛Pj .t/e j .t/ C N
(3.130)
ej , e ej .t; v; v/ are defined as in (3.105), for j D 0; 1, see (3.104). Here H j , e j and N 0 (3.106), (3.107) but relative to the paths j .t/. Moreover, the function vj0 are given and satisfy (3.103), and we impose the orthogonality conditions, see (3.113), ˝ ˛ ˝ ˛ Z.t/; 3e j .t/ D Z.t/; 3e j .t/ D 0;
8 t 0:
(3.131)
The initial conditions for the paths are 0 .0/ D 1 .0/ D 0.0/ .0/, whereas for Z0 ; Z1 one invokes (3.128) as follows: fix Z0.0/ 2 Bı .0/ so that P0 Z0.0/ D PC Z0.0/ D 0 and set Z0 .0/ D Z0.0/ C h 00 ; W00 ; W0.0/ GC ; (3.132) Z1 .0/ D Z0.0/ C h 10 ; W10 ; W0.0/ GC : This choice guarantees that (3.131) holds at t D 0. By the preceding stability analysis, (3.130) and (3.131) then define unique solutions .j ; Zj / satisfying (3.102) and (3.103). Differentiating (3.131) in combination with (3.130) yields the modulation equations stated in (3.114). Thus, we rewrite (3.130) in the form ej .t/Zj D i @ t Zj C H
iLj .t/Zj C Nj .t; vj0 ; vj0 /
(3.133)
138
3 Above the ground state energy I: Near Q
ej and the nonlinear term in (3.114). The linear term where Nj incorporates both N Lj .t/Zj is of finite rank and corank, and satisfies the estimates kLj .t/Zj kW k;p . kZj kH 1 jPj0 .t/j for any 1 p 1 and k 0. Combining this pointwise in time bound with (3.102) yields the full estimates on Lj .t/Zj . By construction, any solution of (3.133) which satisfies (3.131) at one point, say t D 0, satisfies (3.113) for all t 0. The difference R WD Z1 Z0 satisfies e 0 .t/R D i@ t R C H
iL1 .t/R N0 .t; v00 ; v 00 / C N1 .t; v10 ; v 01 / e 0 .t/ H e 1 .t//Z1 i.L1 .t/ L0 .t//Z0 DW F e C .H
(3.134)
whereas the difference of the paths D 1 0 is governed by taking differences of the third and fourth equations, respectively, in (3.115) for j D 1; 0. We estimate .R; / in the norm, with > 0 small, fixed, and to be determined, k.R; /kY WD ke
t
RkL1 2 C ke t ..0;1/IL /
t
k P L1 ..0;1// :
(3.135)
To render this a norm, one fixes .0/ D 0, say. Note that some measure of growth e 0 .t/ H e 1 .t/ and L1 .t/ L0 .t/ grow linearly in t. has to be built into k kY , since H Next, we perform the same modulation as above, i.e., # " Z t 0 e i0 .t/ 0 0 R; .t/ D .˛00 .s//2 ds : W .t/ WD 0 i00 .t/ 0 e 0 Denoting the matrix here by M0 .t/, W satisfies the equation e 0 .t/ C ˛ 0 .0/ 2 3 W D M0 F e: i@ t W C H 0
(3.136)
To obtain estimates on (3.136), we write e 0 .t/ C ˛ 0 .0/ 2 3 D H0 C a.t/3 C D.t/ H 0 0 0 with the constant coefficient operator H0 D H ˛00 .0/; 00 .0/ , see (3.100), and 2 2 ˛00 .0/ , as well as D.t/ equaling a.t/ D ˛00 .t/ " # 0 0 e
2 Q2 ; ˛00 .t / Q2 ; ˛00 .0/ e 2i 0 .t/ Q2 ; ˛00 .t / C e 2i 0 .0/ Q2 ; ˛00 .0/ 0 Q2 ; ˛00 .t / e 2i 0 .0/ Q2 ; ˛00 .0/ 2 Q2 ; ˛00 .t / Q2 ; ˛00 .0/
2i 00 .t/
:
One has ka./k1 . ı 2 and khxiN D./k1 . ı 2 for any N as before. At this point the analysis is similar to the one starting with (3.117). Indeed, writing once again W D C GC C G C Wroot C Wdisp
3.5 The center-stable manifold for the radial cubic NLS in R3
139
where the decomposition is carried out relative to H00 , one inserts this into (3.136) and proceeds as before. The two main differences from the previous stability analysis are as follows: (i) the stability condition (3.123) holds automatically here, since we know a priori that C remains bounded; indeed, we chose Z1 ; Z0 to each satisfy (3.123) whence (3.103) holds for each of these functions. (ii) the orthogonality condition (3.113) does not hold exactly in this form, since it is obtained by taking the difference of the orthogonality conditions satisfied by Z1 and Z0 . But this is minor, since the error one generates in this fashion is contractive. Applying the dispersive bound of Lemma 3.29 (here we need only the L2x part) to Wdisp yields via a term-wise estimation of the right-hand side of (3.134), Z t
R.t/ . ıe t .R0 R0 ; 0 0 / C ı
R.s/ ds 0 1 0 1 2 Y 2 0 (3.137) Z 1
.s t/
R.s/ 2 ds Cı e t
where Rj0 D
vj0 vj0
for j D 0; 1. Recall that the initial conditions for R are deter-
mined by (3.132). The final integral in (3.137) is a result of the C equation (3.124). Assuming .˛00 .0// > ı, Gronwall’s inequality implies
sup e t R.t/ . ı 1 .v 0 v 0 ; 0 0 / 2
t0
0
1
0
1
Y
as desired. Next, one estimates the equation with initial condition .0/ D 0. The conclusion is a bound of the form
ˇ ˇ
.R; / C ˇh. 0 ; W 0 ; W .0/ / h. 0 ; W 0 ; W .0/ /ˇ .v 0 v 0 ; 0 0 / 0 0 1 1 0 1 0 1 Y 0 0 Y which proves the desired contractivity. See [9] for more details on these estimates. Hence, one has a fixed point of (3.115) as well as a well-defined function h.00 .0/; W0.0/ /. This concludes the proof of the existence part. Next, we turn to scattering. In contrast to the previous analysis, we do not lin earize around H ˛.0/; .0/ , but rather around H ˛.1/; .1/ . Thus, consider the system (3.114), (3.113), (3.124), (3.125), (3.126) with a.t/ D ˛ 2 .t/ ˛ 2 .1/, F defined by (3.119), and D.t/ equaling 2 Q2 ; ˛.t / Q2 ; ˛.1/ e 2i .t/ Q2 ; ˛.t / C e 2i .1/ Q2 ; ˛.1/ : 2i .t/ 2 2i .1/ 2 2 2 e
Q ; ˛.t /
e
Q ; ˛.1/
2 Q ; ˛.t /
Q ; ˛.1/
(3.138)
Thus, a.t/; D.t/ ! 0 as t ! 1. This ensures the vanishing at t D 1 of the first three terms of F .t/ in (3.119). The fourth and fifth terms of F vanish in the
140
3 Above the ground state energy I: Near Q
L1 .T; 1/-sense as T ! 1 by (3.102), whereas the nonlinear term N.t; W / vanishes in the sense of Strichartz estimates. Therefore, (3.124), (3.125) imply that ˙ .t/ ! 0 as t ! 1. Hence, in view of the scattering statement in Lemma 3.29 one has the representation in H 1 Wdisp .t/ D e i3
Rt
C˛ 2 .1/Ca.s// ds
D e i3
Rt
C˛ 2 .s// ds
0. 0.
W1 C o.1/;
(3.139)
W1 C o.1/
as t ! 1. The modulation in (3.108) removes the ˛ 2 .s/ in the exponent once we return to the v representation. Finally, by the orthogonality conditions Wroot .t/ ! 0. In summary, we have obtained the desired scattering statement for v in (3.101). Finally, to obtain the uniqueness statement let u.t/ be a solution with u.0/ 2 Bı .Q/ and with the property that dist.u.t/; S1 / . ı for all t 0. We claim that there exists a C 1 -curve ˛.t/; .t/ 2 1 O.ı/; 1 C O.ı/ R which achieves ˝ ˛ u.t/ e i.t/ Q ; ˛.t/ je i.t/ Q ; ˛.t/ D 0; (3.140) ˝ ˛ u.t/ e i.t/ Q ; ˛.t/ jie i.t/ @˛ Q ; ˛.t/ D 0 for all t 0, as well as
sup u.t/ t0
e i.t/ Q ; ˛.t/ H 1 . ı :
(3.141)
Q 2 R so that In fact, by definition there is a C 1 –path .t/
Q sup u.t/ e i .t/ Q.; 1/ H 1 . ı : t0
This shows that one can fulfill (3.140) up to O.ı/. Next, one uses that jhQj@˛ Qij ' Q can be mod1 for all ˛ ' 1 and the inverse function theorem to show that .˛Q 1; / ified by an amount O.ı/ so as to exactly satisfy (3.140) without violating (3.141). Furthermore, by chaining one concludes that this procedure yields a well-defined path .˛; / which is C 1 , as claimed. Next, define Z t 0 .t/ D ˛ 2 .s/ ds C .0/ 0
and set D
0 . Now write u.t/ D e i.t/ Q ; ˛.t/ C v.t/ D e i.t/ Q ; ˛.t/ C e i0 .t/ w.t/ :
(3.142)
This then allows one to rewrite (3.140) in the form ˇ ˇ he i .t / Q ; ˛.t/ ˇw.t/i D 0; hie i .t/ @˛ Q ; ˛.t/ ˇw.t/i D 0 :
(3.143)
3.5 The center-stable manifold for the radial cubic NLS in R3
141
w As before, consider W D w , and perform the decomposition (3.117). Inserting (3.142) into (3.83) yields, cf. (3.104), ! ! v v P C ˛ 2 .t/ e e e .t; v; v/ (3.144) .t/ i ˛.t/e P .t/ C N C H.t/ D .t/ i@ t v v e e , and N e are as in (3.105), (3.106), (3.107). Furthermore, with W D where H; ;e w , w i@ t W C H.t/W D .t/.t/ P i ˛.t/.t/ P C N.t; W / (3.145) see (3.109), (3.111), (3.112). The orthogonality conditions (3.143) are of the form ˝ ˛ ˝ ˛ W .t/; .t/ D 0; W .t /; .t/ D 0 (3.146) which is identical with (3.113). This places us in the exact same position that we started from in the existence proof. Thus, the decomposition (3.142) is such that (3.102) and (3.103) hold. The only difference here is that we know a priori that C .t/ is bounded. However, (3.122) guarantees that therefore (3.123) holds which forces the solution to lie on M as desired. Remark 3.32. Denote the manifold constructed in Theorem 3.31 by M1;0 . The same construction can be applied to e i Q.x; ˛/ instead of Q for any 2 T D R=2 Z and ˛ > 0, yielding a codimension 1 manifold in the phase space H which we denote by M˛; . By the uniqueness part of Theorem 3.31 one concludes that [ MS WD M˛; (3.147) ˛>0; 2T
is again a smooth manifold, which contains all of S. 3.31 By the proof of Theorem it is smoothly parametrized by ˛.0/;
.0/; W .0/ where P ˛.0/;
.0/ W .0/ D 0 0 and PC ˛.0/; .0/ W .0/ D 0 and W .0/ needs to be small enough. Looking ahead to the full dynamical description of (3.83) given in the final chapter, we can make the following remarks: MS has the property that any u0 2 MS leads to a solution of (3.83) defined on t 0 which scatters to S as t ! 1 in the sense of Definition 6.13. We emphasize that this is not the manifold .5/ [ .7/ [ .9/ appearing in Theorem 6.14. Rather, that manifold is the maximal backward evolution of MS under the NLS flow. Note that MS , thus extended by the nonlinear flow, is again a manifold. The following characterization of the stable manifolds will be needed in the proof of Theorem 6.15. It precisely captures the situation where the radiation part
142
3 Above the ground state energy I: Near Q
(i.e., the difference between u.t/ and the soliton in (6.70)) has vanishing scattering data and is therefore uniquely captured by .0/. Corollary 3.33. Let MS be as in (3.147). Suppose u0 2 MS with M.u0 / D M.Q/ forward scatters to S in the sense of Definition 6.13 so that (6.70) holds with u1 D 0. Then the solution u.t/ of (3.83) with data u0 approaches a soliton trajectory in S1 exponentially fast. Moreover, the solution is uniquely characterized by 1 2 S 1 and a real number 0 with j0 j . ı. The case where u is an exact soliton is characterized by 0 D 0. Proof. This follows from the construction carried out in the proof of Proposition 3.31, but with H0 D H.˛.1/; .1// as the driving linear operator; see that part of the proof dealing with scattering. By (6.71), ˛.1/ D 1. In fact, consider the representation W D C GC C G C Wroot C Wdisp relative to this choice of H0 , and solve the system (3.114), (3.113), (3.124), (3.125), (3.126) with a.t/ D ˛ 2 .t/ ˛ 2 .1/, F defined by (3.119), and D.t/ given by (3.138). For (3.114) one assigns the terminal conditions ˛.1/ D ˛1 D 1,
.1/ D 1 , for (3.125) we impose the initial conditions .0/ D 0 , and (3.126) is solved with scattering data W1 D 0, cf. (3.139). Note that C does not require any further data, see (3.124). Similarly, Wroot is determined by (3.113). The point is that we can solve the aforementioned system for ˛.t/; .t/ , and W .t/ satisfying (3.102), (3.103) by contracting in the strong norm
.W; / WD e t W 1 C e t P L1 ..0;1// Y L ..0;1/IL2 / t
(3.148)
for suitably chosen and small > 0. Note the contrast to (3.135). In the setting of (3.135) the exponentially decaying weights forced us to start from t D 0 when carrying out the contraction argument. In the case of (3.148), however, we can solve for Wdisp from t D 1 due to the exponentially growing weights. It is essential, though, that for we can still start at t D 0; this is due to the fact that equation (3.125) contains exponentially decreasing functions (one therefore needs < but nothing else). In summary, .t/ .1/ decreases exponentially, as do a.t/; D.t/, ˙ , Wdisp , Wroot . This proves the exponential approach to S1 . Since ˛ 2 .t/ ˛ 2 .1/ ! 0 and .t/ .1/ ! 0 at an exponential rate, u.t/ in fact converges to a soliton trajectory in S1 exponentially fast. The case of an exact soliton is given by W D 0, which the contraction argument characterizes as .0/ D 0 D 0.
3.6 Summary and conclusion
3.6
143
Summary and conclusion
In this chapter we introduced the two commonly used methods in the construction of invariant manifolds, namely the Hadamard (graph transform) on the one hand, and the Lyapunov–Perron method on the other hand. Both approaches apply to infinite dimensions, and in either case the analysis is perturbative around the equilibrium state. The former has the advantage of requiring less spectral information on the linearized operator, whereas the latter allows for an asymptotic description of solutions starting on the center-stable manifold in forward time (they scatter to the equilibrium in a suitable sense). In order to obtain this asymptotic description, strong dispersive control is required from the linearized flow which can be difficult to establish. Both the Hadamard and the Lyapunov–Perron methods apply to the case where the equilibrium is subject to modulation as a result of symmetries. For the Lyapunov–Perron method we demonstrated this here by means of the radial NLS equation where one encounters both the modulation and dilation symmetries. For the graph transform, we did not include the modulation theory in this chapter since it requires ideas that go beyond the original Bates and Jones proof [6]. However, details can be found in [112] which extends the treatment of the NLKG equation by the Hadamard method to all energy subcritical powers of the nonlinearity as well as general nonradial energy data.
4
Above the ground state energy II: Moving away from Q
In contrast to the previous chapter which focused on the stable behavior of solutions near the ground states .˙Q; 0/, this chapter describes those solutions that enter, but then again leave, a neighborhood of the ground states. These are the non-trapped trajectories. Of particular importance here will be the fact that E.E u/ < J.Q/ C "2 where " > 0 is much smaller than the size of the neighborhood in which Theorem 3.22 guarantees the existence of the center-stable manifolds. Moreover, we will only describe the dynamics of non-trapped trajectories after they exit a ball of size C " where C is sufficiently large (in fact, relative to a suitable metric, called the nonlinear distance function below, we shall be able to take C D 2). As it turns out, after the exit from the C "-ball the dynamics is dominated by the unstable manifold which means that the coordinate in that direction grows exponentially. Inside the C "-ball around .˙Q; 0/ the dynamics can be very complicated relying on a delicate interplay between the different modes (i.e., between the exponential behavior and the dispersive part of the equation). We treat this region as a “black box” which we do not analyze at all. In fact, this is not needed as we are either trapped or being ejected out of that ball, and then carried to much larger (albeit still small compared to 1) distances from the equilibria. It is worth noting that Strichartz estimates do not enter into the analysis of the ejection mechanism itself. Rather, we will control the dispersive part via a suitable energy estimate. Furthermore, throughout this text, Strichartz estimates for the linearized operator LC , as they appeared in the previous chapter, are needed exclusively for the scattering property on the center-stable manifolds as constructed by the Lyapunov–Perron method.
4.1
Nonlinear distance function, eigenmode dominance, ejection
This section presents the process by which solutions that do not remain close to the ground state for all positive times are ejected from any small neighborhood of it after some positive time. Of particular importance for the description of the global dynamics are the signs of K0 and K2 as the solution exits a fixed small neighborhood of ˙.Q; 0/, as well as the following two facts:
146
4 Above the ground state energy II: Moving away from Q
(i) (ii)
the solution cannot return to that neighborhood the sign of K0 , K2 can only change if the solution reenters that neighborhood. Fact (i) constitutes the one-pass (or no-return) theorem which we prove in Section 4.3, whereas (ii) is a variational property which we prove in this section. The proofs of both properties require that the energy exceeds that of Q only by a slight amount. Once we have established (i) and (ii), we can deduce the following properties about the trajectories of NLKG (in forward time, say): if uE enters a small ball of fixed size ı0 (much larger than " and determined such that the signs of K0 ; K2 are constant outside that ı0 -ball) then either the solution is trapped in that ball or not. The former case occurs if and only if the solution falls on the manifold M of Theorem 3.22, whereas in the latter case it is ejected from the ball in such a way that the signs of K0 ; K2 are determined once and for all at exit time; in fact, M divides the ball into two halves and each half corresponds to a fixed sign. By the one-pass theorem, we then conclude that the sign of K0 ; K2 does not change ever again after the exit time. But this allows us to revert to the Payne, Sattinger arguments to conclude either global existence or finite time blowup (with the scattering to zero being again somewhat more difficult). We begin the technical part of this section with a preliminary analysis of the unstable dynamics near .˙Q; 0/. As in the previous chapter, all functions are radial and real-valued.
4.1.1 A first look at the ejection process Writing u D Q C v, we begin by recalling the expansions (3.2)–(3.4), i.e., vR C LC v D 3Qv 2 C v 3 D N.v/; ˛ 1 1˝ 3 P 22 C O kvkH E.Q C v; v/ P D J.Q/ C LC vjv C kvk 1 ; 2˛ 2 ˝ 2 K0 .Q C v/ D 2 Q3 jv C O kvkH 1 : With > 0 the L2 -normalized ground state of LC , it is now natural to write v.t; x/ D .t/.x/ C .t; x/;
? :
(4.1)
4.1 Nonlinear distance function, eigenmode dominance, ejection
147
Indeed, (4.1) becomes (with P N.v/ D N .v/) R
k 2 D N .v/;
R C LC D P? N.v/; ˛ 1 1˝ 1 3 P 22 C O kvkH E.Q C v; v/ P D J.Q/ C .P 2 k 2 2 / C LC j C k k 1 ; 2 2 2 ˝ ˛ 2 K0 .Q C v/ D 2 Q3 j C C O kvkH 1 : (4.2) From Chapter 2 we know that the sign of K0 (and K2 ) determines the fate of the solution, albeit only in the regime E.u; u/ P < J.Q/ D E.Q; 0/. As one would expect, this property remains true also if E.u; u/ P < J.Q/ C "2 D E.Q; 0/ C "2
(4.3)
dist .u; u/; P .Q; 0/ "
(4.4)
provided
where the distance is in H. We prove this in Lemma 4.8 below. In order to exploit this fact, we shall need to make sure that at exit time from a ı-ball Bı D Bı .Q; 0/ one has jj k kH 1 . Indeed, if this is not the case then the sign of K0 .Q C v/ cannot be controlled, see (4.2). On the other hand, if we can insure this property then it would agree with our expectation that the center-stable manifold of the previous chapter divides Bı into two halves: the one where the solution exits with > 0 and the other where it does so with < 0. Provided the one-pass property (i) from above holds, then we can indeed prove that the two halves correspond to global existence and blow-up, respectively (in forward time). The energy constraint (4.3) gives some indication of the -dominance. Indeed, in connection with (4.2), (4.3) implies that 2 P 2 C k kH P k22 . 2 C "2 : 1 C k
(4.5)
Thus, if (4.4) holds, then dist .u; u/; P .Q; 0/ ' jj " which, however, is insufficient in order to control K0 . Nevertheless, the desired conclusion can be reached by means of finer estimates on (4.2). First, set ! WD 1 .P? LC / 2 . By Exercise 3.3, k!f k2 ' kf kH 1 for any f ? . Then 1 E.Q C v; v/ P D J.Q/ C .P 2 2
k 2 2 / C
1 3 (4.6) k k P 22 C k! k22 C O kvkH 1 2
148
4 Above the ground state energy II: Moving away from Q
as well as K0 .u/ D h u C u
u3 jui
v 3 jQ C vi 2 D hLC vjQi C O kvkH 1 D hLC v D
3Qv 2
(4.7)
2 k 2 hjQi C h! j!Qi C O kvkH 1 :
From (4.6), k! k22 k 2 2 C C "2 . Thus, the second term in (4.7) can be estimated as follows: ˇ ˇ ˇh! j!Qiˇ2 k! k2 k!Qk2 hP ? LC QjQi.k 2 2 C C "2 / 2 2 hLC QjQi hjQihjLC Qi .k 2 2 C C "2 / hjQi2 k 2 2kQk44 .k 2 2 C C "2 / which implies that for some absolute constant c > 0, jh! j!Qij k 2 jj hjQi
c C O."2 =/
provided " jj 1. Inserting this bound into (4.7) leads to the desired conclusion, viz. jK0 .Q C v/j ' jj; sign K0 .Q C v/ D sign ./ provided " jj 1 : (4.8) The derivation of (4.8) is arguably somewhat delicate and non-robust; in fact, we remark that it is not clear how to repeat this argument in the context of the NLS equation. Therefore, it is perhaps more natural to seek a dynamical argument based on the robust idea that a solution starting out in a small ball around the equilibrium and which is ejected out of a much larger ball around that point should do so by means of the exponentially expanding (i.e., unstable) dynamics. In passing, we remark that an “onion” structure around .Q; 0/ represented by balls of different radii will be crucial for our argument. Note, however, that dynamically speaking is not meaningful by itself; since the P i.e., the phase space -equation in (4.2) is second order one has to consider .; /, variables. In order to distinguish between the expanding and contracting modes, we will now change coordinates in this two-dimensional space as follows: the homogeneous equation R k 2 D 0
4.1 Nonlinear distance function, eigenmode dominance, ejection
149
KD0 Wu
K>0 K<0
K<0 K>0 Wu
W cs
Figure 4.1. The sign of K D K0 (or K D K2 ) upon exit
has e ˙k t as basis (the convention is k > 0). This suggests setting 1P 1 . C / 2 k 1 1P WD . / 2 k
C WD
(4.9)
since 0 is the same as .t/ D 0 e ˙kt . Moreover, D C C ;
P D k.C
/
(4.10)
and the ODE for in (4.2) becomes the equivalent system 1 N .v/ ; 2k 1 P C D kC C N .v/ : 2k Moreover, the energy expansion in (4.2) reads P D
k
(4.11)
1 1 3 P 22 C O kvkH 2k 2 C C hLC j i C k k 1 : (4.12) 2 2 Equations (4.11), (4.12) and the equation E.Q C v; v/ P D J.Q/
R C LC D P? N.v/
(4.13)
150
4 Above the ground state energy II: Moving away from Q
now allow for the following conclusions. First, we seek a criterion on the data by which the corresponding solution is guaranteed to be ejected from a neighborhood of .Q; 0/. One natural possibility is to require that at time t D 0 one has ˇ ˇ ˇ ˇ ˇC .0/ˇ & ˇ .0/ˇ : (4.14) If the ejection takes place along the unstable manifold, then (4.14) will hold eventually. Moreover, note that (4.14) and (4.12) imply that
2 ˇˇ ˇ
k 1 C
P 2 . jC .0/ˇˇ .0/ˇ C "2 C ı 3 (4.15) 0 H 2
ˇ ˇ
' ˇC .0/ˇ ' ı0 ". This shows that, in particular, where 1 .v; v/.0/ P H (4.14) excludes that the data belong to the center-stable manifold, see Theorem 3.22. If we now ignore the nonlinear terms in (4.11) and (4.13), then it follows that C .t/ D C .0/e kt ;
.t/ D .0/e sin.!t/
.0/ P :
.t/ D cos.!t/ .0/ C !
kt
;
This implies that C .t/ dominates for all t 0 in the sense that
ˇ ˇ
.v; v/.t/
' ˇC .t/ˇ 8 t 0 : P H
(4.16)
(4.17)
In the following exercise, the reader is asked to verify that the linear dynamics gives the correct leading order up until the time where the solution is no longer 1. Exercise 4.1. (a) Under the condition (4.14), prove that for given " ı0 ı1 1 the so is strictly increasing lution v.t/; v.t/ P has the property that v.t/; v.t/ P H for 0 < t0 < t < t1 where t0 ; t1 are constants (t0 depends on the implicit constant in (4.14)). In particular, there exists a unique time t 2 .t0 ; t1 / with
v.t /; v.t P / H D ı1 . Moreover, prove that jC .t /j ' ı1 ; k .t /kH 1
j .t /j ' ı0 C O.ı12 / ;
C
.t P / 2 ' ı0 C O.ı12 / ;
(4.18)
and conclude that K0 .u.t // ' C .t / ' .t /, as desired. Hint: for the
-part, use energy estimates. Note that this only works if the power of the nonlinearity does not exceed 3.
4.1 Nonlinear distance function, eigenmode dominance, ejection
151
ˇ ˇ ˇ ˇ (b) Now assume that ˇC .0/ˇ ˇ .0/ˇ. Prove that on some time interval 0 t t2 where t2 is a constant that depends on the implicit constant in the condition
is strictly decreasing. on the data, one has that v.t/; v.t/ P H (c) Repeat this exercise for nonlinearities jujp 1 u with powers 3 < p < 5. You need to assume the gap property (3.12) in that case. While the result of Exercise 4.1 is satisfactory in many ways, the method of proof is not robust in the following sense: while energy estimates were sufficient for p D 3, for powers 3 < p < 5 one needs to use Strichartz estimates for the KG-evolution with LC instead of C 1. As we emphasized before, this requires the gap property (3.12) for LC . Therefore, the reliance on Strichartz estimates for the -part is somewhat heavy-handed. Indeed, in this regime which is dominated by the hyperbolic dynamics (i.e., by the exponentially growing mode) one would not expect to encounter a subtle analysis of a subordinate component of the evolution such as . Following [109], we show in the following section how to avoid any dispersive analysis of by means of a combination of the energy and . The key quantity in this process is the nonlinear distance function to which we now turn.
4.1.2 Nonlinear distance function We begin by introducing the nonlinear distance function relative to the ground states ˙Q. Let u D ŒQ C v;
v D C ;
?
(4.19)
for D ˙, and where v is small. Define the linearized energy as 1 2 2 2 k hvji2 C k!P? vkL P L 2 C kvk 2 2 1 2 2 2 P 2 C k.! ; /k D k jj C jj P L 2 L2 2
kE v k2E WD
(4.20)
where ! is as above. By Lemma 3.2, 2 2 kE v k2E ' kE v k2H D kvkH P L 1 C kvk 2:
The relevance of k kE lies with the following decomposition, which follows from (4.6): E.E u/
J.Q/ C k 2 2 D kE v k2E
C.v/;
4 C.v/ WD hQjv 3 i C kvkL 4 =4:
(4.21)
152
4 Above the ground state energy II: Moving away from Q
There exists 0 < ıE 1 such that ˇ ˇ 2 kE v kE 4ıE H) ˇC.v/ˇ vE E =2:
(4.22)
Let be a smooth function on R such that .r/ D 1 for jrj 1 and .r/ D 0 for jrj 2. We define q v kE =.2ıE / C.v/: v k2E kE d .E u/ WD kE It has the following properties kE v kE =2 d .E u/ 2kE v kE ;
d .E u/ D kE v kE C O kE v k2E ;
d .E u/ ıE H) d2 .E u/ D E.E u/
J.Q/ C k 2 2 :
(4.23)
Note that the right-hand side of (4.23) is entirely controlled by since the other two terms are constant. To illustrate how one could hope to control the -component via , note the following heuristic argument. First, assume that the nonlinearity N.v; Q/ vanishes identically. Then E.E u/
J.Q/ D
1 2
k 2 2 C P 2 C k.! ; /k22 :
(4.24)
By (4.2) one has @ t . k 2 2 C P 2 / D 0 (assuming N D 0). Therefore, by (4.24), @ t k.! ; /k22 D 0 which is nothing but energy conservation for the -equation. This observation is robust enough to allow for a higher-order perturbation such as N.Q; v/, see the proof of Lemma 4.3. Henceforth, we shall always assume that uE is decomposed as in (4.19) such that dQ .E u/ WD inf d˙ .E u/ D d .E u/; ˙
where the choice of sign is unique as long as dQ .E u/ 2ıE . Recall from the previous section that ˙ .t/ WD
1 P .t/ ˙ .t/=k 2
are the unstable/stable modes for t ! 1 relative to the linearized hyperbolic evoluE but which are moving tion. First, we investigate the solutions which are close to ˙Q, away from these points.
4.1 Nonlinear distance function, eigenmode dominance, ejection
153
4.1.3 Eigenmode dominance By means of the nonlinear distance function we can now express estimate (4.5) very succinctly. Lemma 4.2. For any uE 2 H satisfying 2 E.E u/ < J.Q/ C dQ .E u/=2;
dQ .E u/ ıE ;
(4.25)
one has dQ .E u/ ' jj. In particular, has a fixed sign in each connected component of the region (4.25). Proof. (4.23) yields 2 dQ .E u/ D E.E u/
2 J.Q/ C k 2 2 < dQ .E u/=2 C k 2 2 :
2 and so, k 2 2 =16 kE v k2E =8 dQ .E u/=2 < k 2 2 . Inside of the set (4.25) one can never have D 0, since that would mean both dQ .E u/ D 0 and E.E u/ < J.Q/ which is impossible.
4.1.4 Ejection process The following ejection lemma is the key to extracting the hyperbolic nature from our PDE. It replaces the ejection process described in Exercise 4.1 by one which does not rely on any dispersive analysis of . The key condition (4.14) is replaced with (4.27). Note that it excludes the scenario described in Exercise 4.1, Part (b), and therefore implies (4.14). However, we reiterate that the following result does not depend on the gap property 3.12 and is a fairly robust statement; indeed, the NLS equation allows for an analogous one, see [110]. Lemma 4.3. There exists a constant 0 < ıX ıE with the following property. Let u.t/ be a local solution of (2.1) on an interval Œ0; T satisfying R WD dQ uE .0/ ıX ;
E.E u/ < J.Q/ C R2 =2
(4.26)
.0 < 8t < t0 / :
(4.27)
and for some t0 2 .0; T /, dQ uE .t/ R
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4 Above the ground state energy II: Moving away from Q
ˇ d Alternatively, assume that dt dQ uE .t/ ˇ tD0 0. Then dQ uE .t/ increases monotonically until reaching ıX , and meanwhile, dQ uE .t/ ' s.t/ ' sC .t/ ' e kt R; 2 j .t/j C k .t/k E E .t/ ; E . R C dQ u (4.28) min sKs u.t/ & dQ uE .t/ C dQ uE .0/ ; sD0;2
with a fixed sign s D C1 or s D
1, where C 1 is an absolute constant.
Proof. Note that R > 0. Lemma 4.2 yields dQ .E u/ ' jj as long as R dQ .E u/ ıE , whereas the energy conservation of NLKG and the equation of give as long as dQ .E u/ ıE , see (4.20), 2 P .E u/ D 2k 2 ; @ t dQ
2 P 2 C 2k 4 jj2 C 2k 2 N .v/: .E u/ D 2k 2 jj @2t dQ
(4.29)
2 2 The exiting condition (4.27) implies @ t dQ .E u/j tD0 0. Since N .v/ . kvkH 1 , we 2 2 2 u/ as long as dQ .E u/ ' jj 1. u/ ' dQ .E have @ t dQ .E Hence, imposing ıX ıE and small enough, we deduce that dQ .E u/ R strictly increases until it reaches ıX ; meanwhile, dQ .E u/ ' s for s 2 f˙1g fixed. Since
2C
P 2 D =k 0;
we also infer that C ' . Next, integrating the equation (4.11) for C and using the C -dominance yields Z t Z t ˇ ˇ2 ˇ ˇ ˇ ˇ k.t s/ ˇ ˇC .t/ .0/ .t/ˇ . ˇ e k.t s/ ˇC .s/ˇ ds; N v.s/ ds . e C 0
0
kt where .0/ denotes the linearized solution. This implies both C .t/ D C .0/e
ˇ ˇ ˇ dQ uE .t/ ' ˇC .t/ˇ . Re kt and ˇC .t/
ˇ 2 2kt 2 ˇ .0/ ' dQ uE .t/ C .t/ . R e
whence also dQ .E u.t// ' jC .t/j ' R e kt for all t 0 up until the exit time from the ıX -ball. By the same argument, 2 j .t/j . R C dQ uE .t/ as claimed, which concludes the hyperbolic part. To bound the dispersive part , we do not rely on any dispersive estimates but rather on energy conservation. This becomes most transparent for the case of N D 0
4.1 Nonlinear distance function, eigenmode dominance, ejection
155
and C D 0, in other words when there are no higher order corrections. To be more specific, in absence of nonlinear corrections, energy conservation takes the form d 1 dt 2
k 2 2 .t/ C P 2 .t/ D 0;
2 d
.! ; / P L2 L2 D 0; dt
see (4.24). The desired estimate on will be a “nonlinearly perturbed" version of the latter vanishing. To be precise, from the equation, see (4.21), one has ˇ ˇ ˇ ˇ ˇ@ t Œ k 2 2 =2 C P 2 =2 C./ˇ D ˇ N .v/ N ./ P ˇ . k kH 1 jj2 : Subtracting it from the energy (4.21) yields ˇ ˇ ˇ@ t k k E 2E C.v/ C C./ ˇ . k kH 1 jj2 : Integrating this bound and using the bound on and .0/, E one obtains 2 2 E L1 R2 e 2kT ; k E kL 1 E.0;T / . R C k k t E.0;T / t
which implies the desired estimate for . Finally, recall from (3.4) that K0 .u/ D
k 2 hQji
2 h2Q3 j i C O kvkH 1 ;
and similarly we can expand K2 around Q: K2 .u/ D
.k 2 =2 C 2/hQji
2 h2Q C Q3 j i C O kvkH 1 :
(4.30)
Since hQji > 0 by their positivity, we obtain the desired bound on Ks . It is very simple to construct solutions which obey the ejection condition (4.27): simply choose initial conditions of the form .0/ D 0, .0/ E D 0, and C .0/ D R. Then the analysis of the .C ; ; / E system appearing in the previous proof shows that C .t/ grows exponentially and ensures that dQ .E u.t// grows initially, as well as up until the time that the scale ıX is attained. More generally, we can formulate the following corollary. ˇ ˇ ˇ ˇ Corollary 4.4. Suppose uE .0/ 2 H satisfies (4.26) as well as ˇC .0/ˇ ˇ .0/ˇ. Then (4.27) holds and thus the ejection lemma applies. Alternatively, assume that ˇ ˇ
ˇ ˇ ˇ .0/ˇ C
.0/ E H 1 L2 ˇC .0/ˇ ıX : Then both (4.26) and (4.27) hold, and thus the ejection lemma applies again.
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4 Above the ground state energy II: Moving away from Q
ıx 2
Figure 4.2. A forbidden trajectory
Proof. The first assertion follows simply from ˇ ˇ 2 P @ t ˇ dQ .E u/ D 2k 2 .0/ D 2k 3 C .0/2 tD0
.0/2 0:
The second assertion follows from the first part, as well as the expansion of the energy to ensure (4.26). We leave the details to the reader. We remark that (4.27) is very natural from the point of view of the ensuing dynamical analysis. To be more specific, suppose a solution defined on some time interval I , and which satisfies (4.3), enters a ı-ball around .Q; 0/ where 0 < ı ıX . Further, assume that it is not trapped by the ıX -ball in forward time. Then there exists some time t 2 I at which " R D dQ uE .t / ıX . Taking t to be maximal with this property in I we conclude that (4.27) holds. Note that it is important here that " R. Indeed, the latter condition insures that is of smaller size than which is of course essential for the ejection process (the other extreme would be the center manifold). For later purposes we now exclude circulating trajectories, as in Figure 4.2. Lemma 4.5. There does not exist a solution to (2.1) with E.E u/ < J.Q/ C "2 and the following properties: u exists for all t 0 and 2" < dQ uE .t/ < ıX for all t 0.
4.1 Nonlinear distance function, eigenmode dominance, ejection
157
Proof. Let ı1 WD inf dQ uE .t/ : t0
We consider the following two cases: Case 1: uE attains ı1 . In fact, suppose that dQ uE .t0 / D ı1 for some t0 0. Then we can apply Lemma 4.3 starting at t0 to conclude that ıX is reached after some finite time contrary to our assumptions. Case 2: uE does not attain ı1 . Since dQ uE .t/ also cannot achieve any local minimum, again by Lemma 4.3, it must be monotonically decreasing. In other words, @ t dQ uE .t/ 0. Clearly, we need to also have @ t dQ uE .t/ ! 0 as t ! 1. On the other hand, it follows from (4.29) that @2t dQ uE .t/ & "2 for all t 0 since jj ' dQ .E u/ > 2". But these estimates are incompatible, and we are done. Another way of stating Lemma 4.5 is given in Corollary 4.7. It is in this form that we shall use it, and which is based on the following terminology. Definition 4.6. In what follows, we shall use the following terminology: ı a trajectory uE .t/ is trapped by an R-ball if it exists for all t 0 and if dQ uE .t/ R for all t T where T > 0 is some finite time. ı We also say that uE .t/, locally defined on some interval Œ0; T / is ejected from the ıX ball, if there exists a time interval Œt0 ; t1 Œ0; T / so that dQ uE .t/ is strictly increasing on Œt0 ; t1 and satisfies 1 2 uE .t0 / ; E.E u/ < J.Q/ C dQ 2 ıX dQ uE .t0 / D ; dQ .E u t1 / D ıX ; 10 dQ uE .t/ ' dQ uE .t0 / e k.t t0 / t0 < 8 t < t1 ; Ks u.t/ ' sign .t/ dQ uE .t/ t0 < 8 t < t1
(4.31)
for s D 0; 2. The implicit constants appearing in the following corollary are absolute. Corollary 4.7. Suppose that some solution of (2.1) satisfies dQ uE .0/ ıX as well as E.E u/ < J.Q/ C "2 where " ıX . Further assume that the trajectory uE .t/ is not trapped by the 2"-balls around .˙Q; 0/ relative to the dQ -metric. Then uE is ejected from the ıX -ball.
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4 Above the ground state energy II: Moving away from Q
Proof. If the trajectory does not enter the 2"-ball, then by Lemma 4.5 for some positive time dQ uE .t/ needs to attain ıX , say at time t1 > 0. But then at the time t0 2 Œ0; t1 at which dQ uE .t/ achieves its minimum the ejection condition (4.27) is satisfied and the solution is ejected from the ıX -ball in the sense of the previous definition. Note that (4.26) holds at t D t0 since dQ uE .t0 / > 2". Also, we have t1 t0 1 due to the fact that dQ uE .0/ ıX . In particular, we may wait a constant amount of time until the term CdQ uE .t0 / in the final estimate of (4.28) becomes negligible compared to dQ uE .t/ . This insures that the final estimate of (4.31) is satisfied. On the other hand, assume that uE .t/ does enter the 2"-ball but is not trapped by it. Then it exits this ball again, and the ejection lemma applies starting from that exit time. This again ensures that the solution is ejected as claimed.
4.2
J and K0 ; K2 above the ground state energy
4.2.1 The variational estimates The following lemma implies that sign of both K0 and K2 cannot change outside of a neighborhood of .Q; 0/ that is not too small – the size here depends on the amount by which the energy of the solution exceeds that of Q. It is essential to note that the following lemma is noneffective, i.e., the dependence of "0 .ı/ on ı is nonexplicit. Therefore, one can only use Lemma 4.8 with a fixed constant ı > 0. Lemma 4.8. For any ı > 0, there exist "0 .ı/; 0 ; 1 .ı/ > 0 such that for any uE 2 H satisfying E.E u/ < J.Q/ C "20 .ı/;
dQ .E u/ ı;
(4.32)
one has either K0 .u/
1 .ı/
and
K2 .u/
1 .ı/;
(4.33)
or 2 K0 .u/ min 1 .ı/; 0 kukH 1
2 and K2 .u/ min 1 .ı/; 0 krukL 2 : (4.34)
Proof. 0 is an absolute constant that will be determined via the constant in (2.26). First we prove the conclusion separately for s D 0 and s D 2 by contradiction. Fix
159
4.2 J and K0 ; K2 above the ground state energy KD0
E WD E.u; u t / > J.Q/ C "2 DW J
KD0
K<0 E<J
E<J
E<J K0
K<0 .Q; 0/
E>J
. Q; 0/
Figure 4.3. The sign of K D K0 (or K2 ) relative to the energy region E < J.Q/ C "2
s D 0 or s D 2 and ı > 0, and suppose that there exists a sequence uE n 2 H satisfying (4.32) with "0 D 1=n but neither (4.33) nor (4.34) with 1 D 1=n. Since Ks .un / is 2 bounded, the definition (2.24) implies that Gs .un / ' kun kH 1 is also bounded, and so Ks .un / ! 0. Then by the same argument as in Lemma 2.9, we deduce that un converges, after extraction of a subsequence, strongly to 0 or ˙Q. In the latter case, (4.32) implies that (since "0 D n1 ) 2 ı 2 lim inf kuP n kL 2 D 0; n!1
which is a contradiction. If un ! 0, then (2.26) implies that the quadratic part dominates in Ks .un / for large n, so (4.34) holds for some 0 > 0 independently of ı. Thus we obtain (4.33) or (4.34), separately for s D 0 and for s D 2. It remains to show that they have the same sign. First note that ˚ Q C WD uE 2 H j E.E K u/ < J.Q/ C "20 ; dQ .E u/ > ı; Ks .u/ 0 ; s ˚ Q WD uE 2 H j E.E K u/ < J.Q/ C "2 ; dQ .E u/ > ı; Ks .u/ < 0 ; s
0
for s D 0; 2 are open sets satisfying QC \K Q D ;; K s s
QC [K Q DK QC [K Q : K 0 2 0 2
Since KQ 0C and KQ 2C have the point 0 in common, it suffices to show that both are Q C . Fix s D 0; 2 connected. For that purpose, we use two kinds of deformations in K s C Q and take any u 2 Ks satisfying ı < dQ .E u/ 2ı ıE . Recall the expansion 2 2dQ .E u/2 D k 2 2 C hLC j i C kvk P L 2C.v/; 2 2 2 2 2 E.E u/ J.Q/ D k C hLC j i C kvk P L 2
2C.v/:
160
4 Above the ground state energy II: Moving away from Q
Lemma 4.2 implies jj ' dQ .E u/ provided that we choose "20 .ı/ < ı 2 =2. We deform uE by increasing jj, while fixing and u. P Then E.E u/ decreases and dQ .E u/ increases, as long as dQ .E u/ ıE . Meanwhile, by the first variational part of the argument, Q C and eventually uE remains in K s 2 2 E.E u/ J.Q/ . dQ .E u/ C O.ı 2 / C o dQ .u/ ı 2 : Q C into Q C \ fdQ .E u/ 2ıg within K Thus we can move any point from K s s ˚ Q C; uE 2 H j E.E u/ J.Q/ ı 2 ; Ks .u/ 0 K s which is contracted to f0g by the scaling transform as in Lemma 2.21 uE D .u; u/ P 7! .us ; u/; P
. W 1 ! C0/:
For subtleties involving HP 1 vs. H 1 in the case of K2 , see the proof of Lemma 2.21. Q C we also use this scaling transform, until either reaching For the remaining part of K s 0, or hitting the sphere dQ .E u/ D 2ı, where it is reduced to the previous case. Thus Q C for both s are connected and coincide. we conclude that K s 4.2.2 The sign of K0 and K2 away from the ground states The above two lemmas enable us to define a sign functional away from ˙Q by combining those of and Ks . Lemma 4.9. Let ıS WD ıX =.2C / > 0 where ıX and C 1 are constants from Lemma 4.3. Let 0 < ı ıS and ˚ 2 H.ı/ WD uE 2 H j E.E u/ < J.Q/ C min dQ .E u/=2; "20 .ı/ ; (4.35) where "0 .ı/ is given by Lemma 4.8. Then there exists a unique continuous function S W H.ı/ ! f˙1g satisfying ( uE 2 H.ı/ ; dQ .E u/ ıE H) S.E u/ D sign ; (4.36) uE 2 H.ı/ ; dQ .E u/ ı H) S.E u/ D sign K0 .u/ D sign K2 .u/; where we set sign 0 D C1 (a convention for the case u D 0). Proof. Lemma 4.2 implies that sign is continuous for dQ .E u/ ıE , and Lemma 4.8 implies that sign K0 .E u/ D sign K2 .E u/ is continuous for dQ .E u/ ı. Hence, it
4.2 J and K0 ; K2 above the ground state energy
161
suffices to see that they coincide at dQ .E u/ D ıS 2 Œı; ıX in H.ı/ . Let u be a solution of the NLKG equation with uE .0/ 2 H.ı/ , dQ uE .0/ D ıS , and such that the ejection condition (4.27) is satisfied. It is easy to construct such solutions, simply by setting .0/ D 0 and .0/ E D 0, while C .0/ D dQ .E u.0//, see Corollary 4.4. Then Lemma 4.3 implies that uE .t/ stays in H.ı/ and that sign .t/ is constant, until dQ uE .t/ reaches ıX , which is after sign Ks u.t/ becomes the same as sign .t/, because 2C ıS ıX . Since sign Ks .E u/ is constant for dQ .E u/ ı, we conclude that sign .t/ D sign Ks u.t/ from the beginning t D 0. The S D C1 side is uniformly bounded in the energy, as the following lemma shows. This corresponds to the bounded middle region in Figure 4.3. Lemma 4.10. There exists M ' J.Q/1=2 such that for any uE 2 H.ıS / satisfying S.E u/ D C1 we have kE ukH M . Proof. If dQ .E u/ ıS , then v kH . J.Q/1=2 C dQ .E u/1=2 . J.Q/1=2 : kE ukH kQkH 1 C kE If dQ .E u/ ıS , then K0 .u/ 0 together with (2.24) implies that kE uk2H D 4E.E u/
K0 .u/ 4E.E u/ 4 J.Q/ C "20 .ıS / . J.Q/
as desired. Figure 4.4 illustrates the meaning of S.E u/ along an actual trajectory. The point is that by the ejection lemma, sign is fixed along the initial segment up to A (assuming the ejection condition is satisfied), and agrees with sign Ks u.t/ at the point A. Upon exit, Lemma 4.8 guarantees that the sign of sign Ks u.t/ remains fixed until point B. We may again apply the ejection lemma to the loop from B to C inside the ıS -ball since the ejection condition is satisfied at the point along that loop which minimizes dQ.u.t// . Thus, is of fixed sign along this piece and therefore sign Ks u.t/ is the same at C as it was at B and therefore at A. Finally, we can continue in this fashion up until the time of first re-entry into the 2"-ball (all of these distances are measured relative to dQ ). Note that we cannot – nor do we need to – say anything about the dynamics inside the 2"-ball.
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4 Above the ground state energy II: Moving away from Q
ıS D
2"
C
B A Figure 4.4. The sign S.E u/ along a trajectory with E.E u/ < J.Q/ C "2
4.3
The one-pass theorem
In this section we establish the crucial no-return property of solutions. In dynamical E terms, it says that there are no almost homoclinic orbits relative to the equilibria ˙Q. The argument in this section is motivated by the scattering proof in Chapter 2. More specifically, we use the virial identity (2.64) (in the radial setting). We now state the one-pass theorem, which depends on some small parameters which will be specified below. Theorem 4.11 (One-pass theorem). Let " ; R > 0 be small constants (specified by (4.38) below). If a solution u of NLKG on an interval I satisfies for some " 2 .0; " , R 2 .2"; R , and 1 < 2 2 I , E.E u/ < J.Q/ C "2 ;
dQ .E u. 1 // < R D dQ uE . 2 / ;
then for all t 2 . 2 ; 1/ \ I DW I 0 , we have dQ uE .t/ R. The point of this theorem is that in combination with the tools developed so far in this chapter (ejection lemma and the variational characterization), itwill allow us to conclude that the sign of the functionals K0 u.t/ and K2 u.t/ stabilizes eventually. Which means that for every NLKG solution u there exists a compact time-interval I0 inside of the maximal interval of existence I so that on the two components of I n I0 these signs are constant.
163
4.3 The one-pass theorem
First, we give some indications that removing homoclinic orbits may not be completely obvious.
4.3.1 Some heuristics on the existence of a homoclinic orbit To show that there is some danger of a homoclinic orbit arising in our analysis, consider the -equation in (4.2), i.e., R
k 2 D h3Qv 2 C v 3 ji:
Setting D 0 and retaining only the quadratic term on the right-hand side yields R D k 2 C c2 ;
c > 0:
P D .0; 0/, and the other at . k 2 =c; 0/. This equation has two equilibria, one at .; / The linearizations at these two points are given by 0 1 0 1 ; ; k2 0 k2 0 respectively. The former is hyperbolic with eigenvalues ˙k, whereas the latter is stable with eigenvalues ˙ik. Hence, the phase portrait looks precisely as in Figure 4.5. In other words, there is a homoclinic orbit. This heuristics is of course problematic
1.5
0.10
1.0
0.05
0.5
0.00
0.0
-0.05
-0.5
-0.10 -0.5
0.0
0.5
1.0
1.5
-0.10
-0.05
Figure 4.5. A homoclinic orbit
0.00
0.05
0.10
164
4 Above the ground state energy II: Moving away from Q
for a number of reasons. First, since we are forced into large values one cannot simply ignore the cubic term. But even including it and making a detailed analysis of the various constants appearing in the cubic polynomial does not seem reasonable as we cannot ignore the dispersive term . In fact, an argument based on the -ODE does not seem promising as the dispersive effects should certainly be essential in any mechanism excluding a homoclinc orbit. An example of a homoclinic orbit is shown in Figure 4.5 by means of the ODE .x; P y/ P D .x
y2; y C x2/ :
The streamline plots were produced with Mathematica1 with the commands StreamPlot fx y y; y C x xg; fx; :5; 1:5g; fy; :5; 1:5g ; (4.37) StreamPlot fx y y; y C x xg; fx; :1; :1g; fy; :1; :1g ; respectively. Note that near the unstable equilibrium .0; 0/ the dynamics looks hyperbolic, but the other equilibrium at .1; 1/ is stable. The homoclinic orbit encircles .1; 1/ and approaches .0; 0/ as t ! ˙1, respectively, see the left-hand side of Figure 4.5.
4.3.2 Local virial identity and non-existence of almost homoclinic orbits Using the constants in Lemmas 4.3–4.13, we choose " ; ı ; R ; > 0 such that ı ıS ;
ı ıX ;
" "0 .ı /;
" R min ı ; 1 .ı /1=2 ; 01=2 ; J.Q/1=2 ; < 0 .M /;
1=6 J.Q/1=2 :
(4.38) (4.39)
Suppose that a solution u.t/ on the maximal existence interval I R satisfies for some " 2 .0; " , R 2 .2"; R ; and 1 < 2 < 3 2 I , E.E u/ < J.Q/ C "2 ; dQ uE . 1 / < R < dQ uE . 2 / > R > dQ uE . 3 / : Then there exist T1 2 . 1 ; 2 / and T2 2 . 2 ; 3 / such that dQ uE .T1 / D R D dQ uE .T2 / dQ uE .t/ ; .T1 < t < T2 /:
1
Wolfram Research, Inc., Mathematica, Version 7.0, Champaign, IL (2008).
165
4.3 The one-pass theorem T2
T1 Figure 4.6. The cutoff w.t; x/ in the one-pass theorem
Lemma 4.9 gives us a fixed sign f˙1g 3 s WD S u.t/ ;
.T1 < t < T2 /:
Now we derive the localized virial identity with a precise error bound. The cutoff function is defined by ( x=.t T1 C S/ t < .T1 C T2 /=2 ; w.t; x/ D x=.T2 t C S/ t > .T1 C T2 /=2 ; where S 1 is a constant to be determined later, and is a radial smooth function on R3 satisfying .x/ D 1 for jxj 1 and .x/ D 0 for jxj 2. Using the equation we have ˝ ˛ 1 Vw .t/ WD wu t j .xr C rx/u ; VPw .t/ D K2 u.t/ C O Eext .t/ ; (4.40) 2 where Eext .t/ denotes the exterior free energy defined by Z Eext .t/ WD e 0 .u/ dx; e 0 .u/ WD juj P 2 C jruj2 C juj2 =2; X.t/ ( 2 jxj > t T1 C S .T1 < t < T1 CT /; 2 x 2 X.t/ ” T1 CT2 jxj > T2 t C S . 2 < t < T2 /: We infer from the finite propagation speed that max Eext .Tj / 1 H)
j D1;2
sup T1 t T2
Eext .t/ . max Eext .Tj / : j D1;2
(4.41)
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4 Above the ground state energy II: Moving away from Q
To see this, construct global solution vj such that vj D u in X.t/ for jt Tj j < jT1 T2 j=2, and kE v .Tj /k2H . Eext .Tj /, by cutting off the initial data at t D Tj and using the small data theory. The exterior energy at t D Tj is bounded by
2 Eext .Tj / . e 2S C
.T E j / E ; where the term e 2S is dominating the tails of Q and , due to their exponential decay. Hence, choosing S j log Rj 1; we obtain Eext .t/ . R2 , and thus VPw .t/ D
K2 u.t/ C O.R2 /;
.T1 < t < T2 / :
(4.42)
We turn to the leading term K2 . In order to apply the ejection Lemma 4.3, we need the exiting property of the solution (4.27). For that purpose, take any tm 2 ŒT1 ; T2 where dQ uE .t/ attains a minimum in t such that .R / Rm WD dQ uE .tm / D inf dQ uE .t/ < ı : jt tm j
T1 and T2 obviously satisfy the above, but there may be numerous other minimum points if uE .t / circles around the equilibria in phase space. Figure 4.7 shows two types of trajectories: I makes no entry into the ı -ball until that time when it reenters the R-ball. The trajectory II, on the other hand, enters the ı -ball without entering the R ball as well at some intermediate time. In that case, the ejection lemma applied to the minimum point (see M in Figure 4.7) provides the needed control on K2 u.t/ . Applying Lemma 4.3 to u.t tm / and u.tm t/, as well as Lemma 4.9, one obtains dQ uE .t/ ' s.t/ ' e kjt tm j Rm ; sK2 u.t/ & dQ uE .t/ C Rm ; (4.43) until dQ .E u.t// reaches ıX ı . Let Im denote that time interval around tm . Then the integral of (4.42) is estimated on each Im by Z ŒsVw .t/Im & ŒdQ uE .t/ C Rm O.R2 / dt ' ıX ; (4.44) Im
thanks to the exponential growth and R Rm ı ıX . In the remainder [ I 0 WD ŒT1 ; T2 n Im ; m
167
4.3 The one-pass theorem
II
M
R I ı Figure 4.7. Different examples of returning trajectories
one has dQ uE .t/ > ı , and " "0 .ı /. Hence, Lemma 4.8 gives us (4.33) if s D 1, or (4.34) if s D 1. In the latter case, Lemma 4.10 implies that kE ukH M on ŒT1 ; T2 . Since 0 dQ .E u.t// > ı R for any t 2 I , the hyperbolic behavior (4.43) on Im implies Œt
1; t C 1 ŒT1 ; T2 ;
and Lemma 4.13 together with (4.39) implies Z
t C1 t 1
2 2 kru.s/kL 2 ds > ; x
since otherwise kukL3 L6x 1=6 J.Q/1=2 , which contradicts that dQ uE .T1 / D t
R J.Q/1=2 . Combining it with (4.34) or (4.33), we obtain Z
tC1 t 1
K2 u.s/ ds & min.1 .ı /; 0 2 / R2 ;
due to the choice of R in (4.38). Combining this and (4.44), we obtain
T sVw .t/ T21 & ıX #ftm g :
(4.45)
168
4 Above the ground state energy II: Moving away from Q
The left-hand side is bounded by using the exponential decay of Q, .
X
v.t/ P
L2
2 C S v.t/ E . R C SR2 . R ;
tDT1 ;T2
provided that we choose S such that j log Rj S 1=R: Thus we have arrived at a contradiction since R ıX , precluding “almost homoclinic" orbits, i.e., any trajectory which exits from and returns to the R neighborhood E In other words, every solution is allowed to enter and exit a sufficiently of f˙Qg. E at most once. small neighborhood of ˙Q Notice that the contradiction simply means that T2 D 1, and then all of the above analysis remains valid, except for the upper bound of ŒsVw TT21 . Thus we have proven Theorem 4.11. Exercise 4.12. (a) Give a detailed derivation of (4.41). (b) For solutions with S u.t/ D 1 give an alternative (and simpler) proof of the one-pass theorem based on the function hujui. P Moreover, there exist disjoint subintervals I1 ; I2 ; I 0 with the following property: On each Im , there exists tm 2 Im such that dQ uE .t/ ' e kjt
tm j
dQ uE .tm / ;
min sKs u.t/ & dQ uE .t/
sD0;2
C dQ uE .tm / ;
where s WD S uE .t/ 2 f˙1g is constant, dQ uE .t/ is increasing for t > tm , decreasS ing for t < tm , and equals ıX on @Im . For each t 2 I 0 n m Im and s D 0; 2, one has .t 1; t C 1/ I 0 , dQ uE .t/ ı and Z
tC1 t 1
min sKs u.t 0 / dt 0 R2 :
sD0;2
(4.46)
By the monotonicity, we can keep applying the above theorem at each t > 2 until dQ .E u/ reaches R . Besides, one concludes that at any later time tm > 2 necessarily E after it is dQ .E u/ > R . In other words, u cannot return to the distance R to ˙Q, ejected to the distance ıX > R .
169
4.3 The one-pass theorem
4.3.3 Vanishing kinetic energy leads to scattering As observed before, in the region K2 0 of Lemma 4.8, K2 .u.t// can vanish at some time if and only if kru.t/kL2x does so, see Lemma 4.8. We now address the ineffectiveness of the lower bound in (2.65). The argument which we used earlier for this purpose, see (2.66)–(2.68), does not apply since it required the energy to lie below that of Q. As clearly demonstrated by the proof of the one-pass theorem, we should treat this kind of vanishing only in the time averaged sense over a sufficiently large interval. It is not hard to see that this leads to global existence and scattering. The idea is that all frequencies have to shift to 0, which leads to the scattering in both time directions by the small Strichartz norm in the regime of subcritical regularity. Technically speaking, one can approximate u by a linear KG-wave v over t 2 Œ0; 2. It then follows that v has small HP 1 norm, which implies that it has small L3t L6x norm globally in time by the subcriticality of that norm. Consequently, one can then approximate u globally in time by v. Lemma 4.13. For any M > 0, there exists 0 .M / > 0 with the following property. Let u.t/ be a finite energy solution of NLKG (2.1) on Œ0; 2 satisfying 2
Z M; kE ukL1 t ..0;2IH//
0
ru.t/ 2 2 dt 2 L
(4.47)
for some 2 .0; 0 . Then u extends to a global solution and scatters to 0 as t ! ˙1, and moreover ku.t/kL3 L6x .RR3 / 1=6 . t
Proof. First we see that u can be approximated by the free solution v.t/ WD e ihrit vC C e
ihrit
v ;
v˙ WD u.0/ ihri
1
u.0/ P =2 :
This follows simply from the Duhamel formula kv
3 ukL1 Hx1 .0;2/ . ku3 kL1 L2x .0;2/ . kukL 3 L6 t
t
.
t
x
2 krukL 2 L2 .0;2/ krukL1 L2 .0;2/ x t t x
2 M ;
if 0 M 1, where we used Hölder’s inequality and the Sobolev embedding HP 1
170
4 Above the ground state energy II: Moving away from Q
L6 . In particular, Z 2
2
rv.t/ 2 2 dt 4 Lx 0 Z h ˚ D C jj2 2jb v C j2 C 2jb v j2 C Im hi
1
.e 4i hi
v 1/b v Cb
i
d
2 2 & krvC kL 2 C krv kL2 ;
where b v denotes the Fourier transform in x of v. Now we use the Strichartz estimate for the free Klein–Gordon equation, see Section 2.5, ke ˙ihrit 'kL3 B 4=9 t
18=5;2 .RR
3/
. k'kHx1 ;
(4.48)
s where Bp;q denotes the Besov space with s regularity on Lp . Combining it with Sobolev, we obtain
kvkL3 L6x .RR3 / . kvkL3 BP 1=3 .RR3 / t t 18=5;2 X . kv˙ kHP 1=3 \HP 8=9 . M 2=3 1=3 C M 1=9 8=9 1=6 ; ˙
if 0 M 4 1. Therefore, we can identify u as the fixed point for the iteration in the global Strichartz norm kukL1 Hx1 .RR3 / . M; t
kukL3 L6x .RR3 / 1=6 ; t
which automatically scatters.
4.4
Summary and conclusion
In the following chapter, we assemble various pieces such as the ejection lemma, the existence of the sign function away from the ground states, and the one-pass theorem into a description of the global dynamics of solutions with energies in ŒJ.Q/; J.Q/C "/. The interval of energies 1; J.Q/ was treated in Chapter 2. The main objective will be to show that the sign of both K0 u.t/ and K2 u.t/ eventually stabilizes. What this means is that for any solution u of the NLKG equation defined on a maximal interval Œ0; T / there exists T 0 2 .0; T / so that these signs are constant and equal for all T 0 < t < T . This is achieved by combining Lemmas 4.8 and 4.3 with the one-pass theorem. Indeed, Lemma 4.8 shows under the energy assumption E.E u/ < J.Q/ C "2 that these signs can only change by making a pass through the
4.4 Summary and conclusion
171
ı."/-balls. The solution trajectories may make no pass through such a ball, which then fixes the signs from the start, or they may also “get stuck” in the ı-ball. But then, due to Corollary 4.7, they then need to be trapped in a 2"-ball (and thus lie on the center-stable manifold). However, in the non-trapping situation, the ejection lemma, Lemma 4.3, guarantees that the solution must exit the ball, and then cannot return due to the one-pass theorem. The constant signs then need to be translated into finite time blowup or scattering to zero, respectively. While the former is a fairly straightforward generalization of the Payne–Sattinger convexity argument from Chapter 2, the latter is more subtle and will be based on the Kenig–Merle argument, as presented in Chapter 2. However, one cannot proceed exactly as in that section and we will need to invoke the one-pass theorem in order to carry out the concentration-compactness argument correctly.
5
Above the ground state energy III: Global NLKG dynamics
In this chapter we combine the results of the previous two chapters to characterize all possible types of long-time asymptotic behavior of the solutions to the cubic NLKG equation in radial R3 . In our first theorem, see Theorem 5.1 below, we ignore the center-stable manifolds and prove a result more in the style of an orbital stability result. That is, instead1 of “scattering to the ground states” one has “trapped by the ground states”. This formulation seems quite natural, as it clearly separates different dynamical issues. To be more specific, once Theorem 5.1 has been proved, the trapping scenario (in forward time) can then immediately be associated with the center-stable manifold due to the repulsivity property of that manifold, see Chapter 3. Furthermore, the dispersive properties of the linearized NLKG flow are needed only to show that trapped solutions scatter to the ground states, but not for the existence of the manifolds themselves (this refers to the distinction between the Hadamard approach on the one hand, and the Lyapunov–Perron method on the other hand). Recall that the linearized NLKG flow refers to the equation vR C LC v D F and establishing its dispersive properties involves some finer spectral properties of LC , namely the issue of eigenvalues in the gap .0; 1/ and the question of a threshold singularity of the resolvent, see (3.12). All of this is then needed when we invoke Theorem 3.22, see Theorem 5.2 and 5.3 below. In Section 5.1 we state the main results on global dynamics. First the 9-set theorem which says that all possible combinations of the forward and backward trichotomies are allowed. We repeat that this does not depend on Chapter 3, and thus also does not invoke the gap property. However, the proof of the scattering statement in Theorem 5.2 does rely on that property and provide a further characterization of those solutions which are trapped by the ground states. Theorem 5.3 characterizes the solutions found by Duyckaerts and Merle as stable manifolds associated with the ground states. 1
In the orbitally stable case, the distinction orbital vs. asymptotic stability can be seen from [67], [143], [144] on the one hand, and [22], [23], [128] on the other hand.
174
5 Above the ground state energy III: Global NLKG dynamics
In Section 5.2 we give the details of the Payne–Sattinger blowup argument for the negative K case, as well as of the scattering to zero for solutions which settle into the K 0 case. The “settling in” here refers to the fact that the one-pass theorem of the previous chapter guarantees that the sign of K stabilizes for any solution. More precisely, we shall show that if I is the open maximal time interval of existence, then we can remove a closed subinterval from I such that in the remaining open subintervals of I the signs of Kj .u.t// is fixed. The scattering argument is quite delicate, and is based on the Kenig–Merle approach; we emphasize that in contrast to the original form of the Kenig–Merle argument, the one-pass theorem plays an important role here, too. In the final Section 5.3 we put everything together to obtain the global dynamical picture as described by the theorems of Section 5.1.
5.1
Statement of the main results on global dynamics
Following [109], we will prove several results about the global behavior of solutions to (2.1) with energy at most slightly above that of the ground state E C "2 g: H" WD fE u 2 H j E.E u/ < E.Q/ Note that the only symmetry in H is u 7!
(5.1)
u in this setting.
Theorem 5.1. Consider all solutions of NLKG (2.1) with radial initial data uE .0/ 2 H" for some small " > 0. The set of all these solutions splits into nine non-empty sets characterized as (1) Scattering to 0 for both t ! ˙1, (2) Finite time blow-up on both sides ˙t > 0, (3) Scattering to 0 as t ! 1 and finite time blow-up in t < 0, (4) Finite time blow-up in t > 0 and scattering to 0 as t ! 1, (5) Trapped by ˙Q for t ! 1 and scattering to 0 as t ! 1, (6) Scattering to 0 as t ! 1 and trapped by ˙Q as t ! 1, (7) Trapped by ˙Q for t ! 1 and finite time blow-up in t < 0, (8) Finite time blow-up in t > 0 and trapped by ˙Q as t ! 1, (9) Trapped by ˙Q as t ! ˙1, where “trapped by ˙Q” means that the solution stays in a O."/ neighborhood of E forever after some time (or before some time). The initial data sets for (1)–(4), ˙Q respectively, are open.
175
5.1 Statement of the main results on global dynamics W cu
I III VI
IX D W c V
VII
.Q; 0/
W cs
IV VIII
II
Figure 5.1. Illustration of the nine sets
In contrast to the Payne–Sattinger result, here one has solutions which blow up for t < 0 and scatter for t ! C1, or vice versa. It also implies that the initial data set for the forward scattering .1/ [ .3/ [ .6/ (or backward scattering) is unbounded in H; in fact, it contains a curve connecting zero to infinity in H. The number “nine” simply means that all possible combinations of scattering to zero/scattering to ˙Q/finite time blow-up are allowed as t ! ˙1. Each of these are in fact realized by infinitely many solutions. Due to the Bates–Jones theorem present in Chapter 3 one can characterize the trapping property in terms of the invariant manifolds associated with the ground state, see in particular Corollaries 3.16 and 3.18. For this, the spectral information provided by Lemma 3.2 is sufficient. While the Bates–Jones theorem only applies to powers p 3 in the nonlinearity (in R3 ), it can be extended to cover the full energy subcritical range in all dimensions, see [112]. Thus, the trapping alternative in Theorem 5.1 can be characterized as follows: any solution which is trapped by Q as jtj ! 1 necessarily belongs to the center manifold W c , and conversely, whereas those trapped as t ! C1 (resp. 1) belong to W cs (resp. W cu ). In particular, we see that .9/ is a co-dimension 2
176
5 Above the ground state energy III: Global NLKG dynamics
Lipschitz graph, whereas .5/ is co-dimension 1, etc. To summarize: Theorem 5.2. The sets .5/ [ .7/ [ .9/ and .6/ [ .8/ [ .9/ are codimension one Lipschitz manifolds in the (radial) phase space H, and they are the center-stable E Similarly, (9) is manifold, respectively the center-unstable manifold, around ˙Q. a Lipschitz manifold of codimension 2, namely the center manifold. Using the gap property (3.12) of LC , one can show that all solutions on the center-stable manifold E D .˙Q; 0/ in forward time, and similarly for the center-unstable and scatter to ˙Q center manifolds. The regularity can be improved but we do not pursue this here (in fact, Proposition 3.22 constructs smooth manifolds, but the graph transform approach does not). Recall from [44] that (3.12) fails if one lowers the power 3 in the nonlinearity slightly, say to power < 2:8. Thus, the final scattering statement of the previous theorem is somewhat delicate. On the other hand, our argument for Theorem 5.1 as well as for the non-scattering part of Theorem 5.2 are quite robust. Furthermore, it is straightforward to extend Theorem 5.1 to all L2 super-critical and H 1 subcritical powers and all space dimensions, i.e. uR
u C u D up ;
1 C 4=d < p < 1 C 4=.d
2/;
u W R1Cd ! R:
Finally, from the above theorems, we can deduce a Duyckaerts–Merle type result at the energy threshold, cf. [48]. Theorem 5.3. Consider the limiting case " ! 0 in Theorem 5.1, i.e., all the radial E Then the sets (3) and (4) vanish, while the sets solutions satisfying E.E u/ E.Q/. (5)–(9) are characterized, with some special solutions W˙ , as follows: .5/ D f˙WE .t
t0 / j t0 2 Rg;
.6/ D f˙J WE . t
t0 / j t0 2 Rg;
.7/ D f˙WEC .t
t0 / j t0 2 Rg;
.8/ D f˙J WEC . t
t0 / j t0 2 Rg;
E .9/ D f˙Q.t
t0 / j t0 2 Rg
with J being the standard symplectic 2 2-matrix. The solutions WE˙ .t/ converge E as t ! 1, in fact at the rate e jkjt where k 2 is the lowest exponentially to Q eigenvalue of LC . As mentioned before .5/[.7/[.9/ is the stable manifold, and .6/[.8/[.9/ the E The remainder of this chapter consists of unstable manifold, associated with ˙Q. the proofs of these three results. While most of the technical work has already been
5.2 The blowup/scattering dichotomy in the ejection case
177
done, we still need to provide the proofs of the blowup/scattering to zero dichotomy for solutions which are ejected away from the ground states.
5.2
The blowup/scattering dichotomy in the ejection case
5.2.1 Blowup after ejection Here we prove that the solution u with S uE . 2 / D 1 in Theorem 4.11 blows up in finite time after time 2 . This will be done by means of the contradiction argument of Payne–Sattinger, which relies on the functional K0 . Thus, suppose that u extends to all t > 2 and let
2 y.t/ WD u.t/ L2 : From the NLKG equation (2.1) we have 2 : yR D 2 kuk P L 2 C sK0 u.t/ x
(5.2)
Applying the lower bound on K0 in Theorem 4.11 to the integral yields Z X Œy P1 & ı C R2 dt D 1 ; X 2 Im
I0
and so y.t/ ! 1 as t ! 1. Then from (5.2), yR
2 2 2 8E.E u/ C 6kuk P L P L P 2 =.2y/; 2 C 2kukH 1 6kuk 2 3y
for large t, where we used Cauchy–Schwarz for yP D 2hujui. P Hence, @2t .y
1=2
/D
.2y 3=2 /
1
Œy yR
3yP 2 =2 0 ;
which contradicts that y ! 1 as t ! 1. Therefore, u does not extend to t ! 1.
5.2.2 Scattering after ejection For the solution u with S uE . 2 / D C1 in Theorem 4.11, the forward global existence follows from the energy bound of Lemma 4.10. In analogy with Chapter 2,
178
5 Above the ground state energy III: Global NLKG dynamics
we prove the scattering to 0 for t ! 1 by the contradiction argument of Kenig– Merle. Thus, we use the functional K2 and the virial identity. While the argument is of course similar to the one in Chapter 2, it is also quite different in so far as we crucially depend on the one-pass theorem, see Theorem 4.11. The point here is that the profile decomposition of the concentration-compactness argument needs to be performed subject to constraints, cf. (5.3) below. The induction on energy step then requires that the building blocks in the profile decomposition respect these constraints. We urge the reader to go through the scattering argument for the radial case in Chapter 2 before reading this section. In particular, we shall freely use the perturbation lemma, see Lemma 2.19, as well as the profile decomposition given by Proposition 2.17 of that chapter. To be more specific, fix " 2 .0; " / and let U."; R / be the collection of all solutions u of NLKG on Œ0; 1/ satisfying E.E u/ J.Q/ C "2 ; dQ uE Œ0; 1/ ŒR ; 1/; S uE Œ0; 1/ D C1: (5.3) Note that the first two conditions imply that uE Œ0; 1/ H.ı / so that we can use Lemma 4.9 to define S.E u/. By the remark following Theorem 4.11, any solution with S D C1 in that theorem will eventually satisfy the above conditions. For each E > 0, let M.E/ be a uniform Strichartz bound defined by ˚ u/ E ; M.E/ WD sup kukL3 L6x .0;1/ j u 2 U."; R /; E.E (5.4) t where we chose the norm L3t L6x to be an H 1 subcritical and non-sharp admissible Strichartz norm such that its finiteness implies scattering. We know from Chapter 2 that M.E/ < 1 for E < J.Q/. In fact, in that case a uniform bound holds globally in time, i.e., for L3t L6x .R/. In order to extend this property to J.Q/ C "2 , put ˚ E ? D sup E > 0 j M.E/ < 1 (5.5) and assume towards a contradiction that E ? < J.Q/ C "2 : We consider the nonlinear profile decomposition for any sequence uE n 2 U."; R / satisfying as n ! 1 E.E un / ! E ? ;
kun kL3 L6x .0;1/ ! 1: t
(5.6)
We are going to show that the remainder in the decomposition vanishes and there is only one profile which is a critical element, i.e., uE ? 2 U."; R /;
E.E u? / D E ? ;
ku? kL3 L6x .0;1/ D 1: t
(5.7)
5.2 The blowup/scattering dichotomy in the ejection case
179
By the arguments in the scattering part of Chapter 2 it then follows that the forward trajectory of uE is precompact in H. It then follows by essentially the same arguments based on the virial identity (2.64) as in Chapter 2 that such a critical element cannot exist (this is the so-called rigidity step). To commence the detailed argument, we apply the Bahouri–Gérard decomposition, see Proposition 2.17. Before applying the profile decomposition, we translate un in time in order to achieve the following estimates: 2 (5.8) dQ uE n .0/ > ıX ; K2 un .0/ "2 : 3 Since dQ .E un / remains above R , Corollary 4.7 implies that there exists 0 Tn . k 1 log.ıX =R / so that dQ uE n .Tn / ıX . Since S D C1, Lemma 4.10 implies that kun kL1 H.0;1/ M . Since kun kL3 L6x ! 1, by the same argument t as for (4.45) we deduce that there exists 0 Tn0 near Tn such that 2 dQ uE n .Tn0 / > ıX : 3 0 Translating un WD un .t Tn /, we obtain (5.8) in addition to (5.6). Now apply Proposition 2.17 to the free solution with the same initial data as un . Using the notation of that proposition, we set vnj D v. C tnj / and let wnj be the nonlinear solution with the same data as vnj at t D 0. In other words, X U.t/E un .0/ D vEnj C Enk ; w E nj .t/ D U N .t/E vnj .0/; K2 un .Tn0 / R2 > 2"2 ;
j
where U N .t/ denotes the nonlinear Klein–Gordon propagator in H, and U.t/ the j linear one. Let tnj ! t1 2 Œ 1; 1 and construct the nonlinear profiles uj ; i.e., the nonlinear solutions satisfying
lim uE j .t/ vEj .t/ H D 0 : j
t!t1
j as the unique solution of either the Cauchy They exist at least locally around t D t1 j j 2 f˙1g applied to problem at t1 2 R or as the image of the wave operator at t1 j vE .0/. As a consequence of the local theory one has
wnj .t/
uj .t C tnj / ! 0;
.n ! 1/
in the energy and Strichartz norms locally around t D 0. Thus we are lead to the following nonlinear profile decomposition X un D ujn C nk C kn ; ujn WD uj .t C tnj / (5.9) j
180
5 Above the ground state energy III: Global NLKG dynamics
locally around t D 0. Here the error kn satisfies Ekn .0/ ! 0 in H as n ! 1. Then K2 un .0/ "2 , E.E un / < J.Q/ C "2 and the orthogonality of the linear decomposition imply J.Q/ "2 > lim supŒE.E un / K2 un .0/ =3 n!1 X G2 ujn .0/ C G2 nj .0/ : lim sup G2 un .0/ D lim sup n!1
n!1
j
Since G2 is positive definite, we obtain lim supn G2 ujn .0/ < J.Q/, which implies, via the minimizing property (2.25), that Ks ujn .0/ 0 for large n and s D 0; 2. In particular, all nonlinear profiles have positive energy, and so by the orthogonality applied to the energy, we deduce that E.uj / < J.Q/ except for at most one profile. Those nonlinear profiles scatter as t ! ˙1. If all of the profiles scatter for t ! ˙1, or more precisely if kuj kL3 L6x .R/ < 1, t then the long-time perturbation argument of Lemma 2.19 implies that the original solutions un also scatter and remain bounded in the Strichartz norm uniformly in n. As usual, the justification in using the perturbation lemma is that the orthogonality in Proposition 2.17, i.e., jtnj tnk j ! 1, implies that X X . ujn C nk /3 D .ujn /3 C o.1/ ; (5.10) j
j
in L1t L2x as n ! 1 and k ! 1, as soon as each ujn has globally finite L3t L6x norm. The same argument goes through even if the norm is finite only on .0; 1/ or . 1; 0/, provided that the translation tnj and the interval I are placed such that the infinite part of the norm does not contribute. Hence at least one profile does not scatter. Let u0 be the non-scattering profile which then necessarily satisfies E.E u0 / J.Q/. If t1 D C1, then by the construction of the nonlinear profile, u0 scatters as t ! 1, leading to a uniform bound on kun kL3 L6x .0;1/ for large n, a contradiction. t 0 If t1 D 1, then u0 scatters as t ! 1 by the same reason. But then u0 does 0 not scatter as t ! C1. If dQ uE .t/ > 3" on its0 maximal existence interval, then 0 S uE .t/ D C1 is preserved from t D 1, so u is a global solution and a critical element, possibly after time translation if necessary. Indeed, suppose dQ uE 0 .t/ enters the interval .3"; R , but does not penetrate any deeper than that. Then by Corollary 4.7 the ejection takes place and so the solution can never return to the R distance at later times. Suppose now that dQ .E u0 .t // 3" for some t 2 R. Then 0 ku kL3 L6x . 1;t / < 1 implies that (5.10) holds in L1t L2x . 1; t tn0 /. Hence, t the nonlinear profile decomposition (5.9) becomes a good approximation of un on
5.2 The blowup/scattering dichotomy in the ejection case
181
that interval. Then taking k and n sufficiently large, we infer from the perturbation lemma, see Lemma 2.19, dQ uE n .t tn0 / dQ uE 0 .t / C O."/ . " R ; contradicting that dQ uE n .0; 1/ ŒR ; 1/, since t tn0 ! 1. 0 2 R can be handled by similar considerations. If u0 The remaining case t1 scatters for t ! 1, then the perturbation lemma implies that kun kL3 L6x .0;1/ is t bounded uniformly in large n. Hence, u0 does not scatter for t ! 1. On the other hand, the decomposition at t D 0 gives
and so dQ
2 0 ıX lim sup dQ uE n .0/ dQ uE 0 .t1 / C O."/; 3 n!1 0 / > ıX =2 > ıS R . Moreover, uE 0 .t1 0 Ks .E u0 .t1 // D lim Ks uE 0 .tn0 / 0 ;
(5.11)
n!1
0 / D C1. and so we deduce from Lemma 4.9 that S uE 0 .t1 By the same argument as above, u0 is either a critical element or dQ uE 0 .t / 0 3" for some (5.11) together with the one-pass theorem implies that t > t1 . Then 0 0 0 t/ is . If u0 does not scatter as t ! 1, then u0 .t1 dQ uE .t/ R for t t1 0 a critical element. Suppose that u scatters as t ! 1. Then the profile decomposition is a good approximation of un on . 1; t tn0 / for large n. Therefore, by the perturbation lemma for large k and n, 0 " & dQ uE 0 .t / C O."/ lim sup dQ uE n .t t1 / R ; n!1
which contradicts " R . In conclusion, for some T 2 R and s 2 f˙1g, we have shown that u0 .st C T / is a critical element. Since un is a minimizing sequence, it implies also that the other components must vanish strongly in the linear profile decomposition. Therefore, along a subsequence,
lim uE n .Tn0 / uE 0 .tn0 / H D 0 ; n!1
where Tn0 0 is the time shift for (5.8). Both Tn0 and tn0 are bounded from above as n ! 1. If tn0 ! 1, then u0 scatters for t ! 1, and so kun kL3 L6x . 1;Tn0 / is t bounded for large n, by the local theory of the wave operator. Applying this to the sequence of solutions un WD u0 .t C n / for arbitrary n ! 1, one obtains the precompactness of the forward trajectory of the critical element,
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5 Above the ground state energy III: Global NLKG dynamics
and a contradiction from a localized (time-independent) virial identity together with the lower bound on K2 . These steps are essentially the same as in the radial scattering proof in Chapter 2. Thus we conclude that no solution u satisfies (5.7), and therefore E ? D J.Q/ C "2 . We summarize our findings in the following proposition. Proposition 5.4. For each " 2 .0; " , there exists 0 < M J.Q/ C "2 < 1 such that if a solution u of NLKG (2.1) on Œ0; 1/ satisfies E.E u/ J.Q/ C "2 , dQ uE .t/ R and S uE .t/ D C1 for all t 0, then u scatters to 0 as t ! 1 and kukL3 L6x .0;1/ M . t
Note that the uniform Strichartz bound can be valid only on the time interval E by a fixed distance. Indeed, without where the solution is already away from ˙Q this separation one cannot hope for any uniform bound, even for those solutions scattering for both t ! ˙1. This is due to the fact that the solutions can stay close E for arbitrarily long time. to ˙Q Exercise 5.5. Prove the claim made in the previous sentence.
5.3
Proofs of the main results
5.3.1 Proofs of Theorems 5.1, 5.2 Fix 0 < " " and let H" WD fE u 2 H j E.E u/ < J.Q/ C "2 g be the initial radial data set. We can define the following subsets according to the global behavior of the solution u.t/ to the NLKG equation: for D ˙ respectively, ˚ S" D uE .0/ 2 H" j u.t/ scatters as t ! 1 ; ˚ T" D uE .0/ 2 H" j u.t/ trapped by f˙Qg for t ! 1 ; (5.12) ˚ " " B D uE .0/ 2 H j u.t/ blows up in t > 0 : The trapping for TC" can be characterized as follows, see Corollary 4.7: there exists T > 0 such that for all t T one has dQ uE .t/ 2": Obviously those sets are increasing in ", and have the conjugation property ˚ " " X D u.0/; u.0/ P 2 H" j uE .0/ 2 X˙ ;
5.3 Proofs of the main results
183
for X D S; T ; B. Moreover, SC and TC are forward invariant by the flow of NLKG, while S and T are backward invariant. We have proven in the previous sections that " H " D SC [ TC" [ B"C D S" [ T " [ B" ; " the disjoint union for each sign. It follows from the scattering theory that S˙ are " open. We claim the same for B˙ , which is not a general fact. Pick any solution uE .t/ in B"˙ which blows up at some finite time T , say. Then kE u.t/kH ! 1 as t ! T . Next, one has @ t ku.t/k22 D 2hujui, P as well as 2 2 2 2 @2t kukL P L K0 u.t/ 6kuk P L 8E.E u/ : 2 D 2 kuk 2 2 C 2kukH 1 x x x x From the lower bound on the right-hand side we conclude that K0 u.t/ ! 1 as t ! T 0. This means that the slope of hu.t/ju.t/i P ! 1 as t ! T 0, whence also hu.t/ju.t/i P ! 1 as t ! T 0. We claim that if T < T is very close to blow-up time, then every solution starting in B1 uE .T / , the unit-ball around uE .T / relative to H, necessarily blows up in the positive time direction. By the Payne–Sattinger argument, we only need to show that for any such solution K0 .u.t// < 0 as long as it is defined. It is clear that this condition will hold initially. By Lemma 4.8 with the choice of " in (4.38), we further see that it can be E of size ıS . However, violated only if the solution returns to a neighborhood of ˙Q in that case necessarily jhujuij P . 1, which is impossible since hujui P starts off very large and has to increase as long as K0 u.t/ < 0. Thus, any solution close to u E and has to blow up sooner or later, as around t D T cannot come close to ˙Q " claimed. Therefore, B˙ are also open, so T˙" are relatively closed in H" . " Since B"C and SC are disjoint open sets, they are separated by TC" . I.e., any two " " points from SC and BC cannot be joined by a curve without passing through TC" . We now begin with the construction of solutions that exhibit any one of the 9 E it is easy to see by different scenarios described above. In a small ball around ˙Q, " " means of the linearized flow that the open intersections B˙ \ S , B"C \ B" and " " SC \ S are all non-empty for any " > 0. In fact, in order to obtain a solution uE in B"C \ S" , we choose initial data such that
.0/ D 0;
P .0/ D k";
.0/ E D 0;
(5.13)
for some 0 < 1. Then the energy constraint E.E u/ < J.Q/ C "2 is satisfied, and by the same argument as in Lemma 4.3, we have j.t/
0 .t/j . e 2kjt j 2 "2 ;
k .t/kE . " C e 2kjtj 2 "2 ;
184
5 Above the ground state energy III: Global NLKG dynamics
as long as e kjtj " ıX , where the free solution 0 is given in this case 0 .t/ D sinh.kt/" : E both in ˙t > 0. Since .t/t > 0 for Hence, uE .t/ exits the R neighborhood of ˙Q kjtj " e " ıX , we obtain S uE .t/ t < 0 after exiting, and so u blows up in " t > 0 and scatters for t ! 1. Therefore, B"˙ \ S are both non-empty. " " \ S" , respectively. In the same way, we construct solutions in BC \ B" and SC More precisely, let u.˙/ be those solutions for which is approximated by the free solutions .˙/ 0 .t/ D ˙ cosh.kt/" ; " respectively. Then uE .C/ 2 B"C \ B" and uE . / 2 SC \ S" by the same considerations as before. Let uE 2 B"C \ S" be the solution with initial data (5.13). Then for any " e k t " "1=2 one has the distance estimates
uE .t/ uE .C/ .t/ . e 2kt 2 "2 C " " ; dQ uE .t/ & e kt " " : E
Hence, there exists a curve H" joining uE .t/ and uE .C/ .t/ within the region S.E u/ < 0 and dQ .E u/ ". Since uE .t/ 2 S" and uE .C/ .t/ 2 B" , there exists a point E as p0 2 T" \ . Since the solution starting from p0 enters the 3"-ball around ˙Q t ! 1, and initially p0 is much further away and S.p0 / < 0, we conclude by the one-pass theorem that p0 2 B"C . Hence, T " \ B"C is non-empty as well. In the same way, we can find a point on the curve connecting uE .t/ and uE . / .t/ for some t < 0, " which is in TC" \ S" . Therefore, T˙" \ B" and T˙" \ S are both not empty. Taking the limit " ! C0, it is easy to observe that they contain infinitely many points on different energy levels. E is of course nonempty, but there are infinitely many Finally, TC" \ T " 3 Q points besides Q, which can be seen by restricting H" to the hyperplane uP D 0 (which guarantees that the resulting solutions are even in time): ˚ H"uD0 WD .u; 0/ j J.u/ < J.Q/ C "2 3 u.˙/ .0/ : P . / Since u.C/ .0/ 2 BC .0/ 2 S"C , and there are infinitely many connecting " and u " curves in HuD0 , there are infinitely many points in TC" \ H"uD0 separating B"C and P P " SC . The symmetry u.t/ 7! u. t/ implies that
TC" \ H"uD0 TC" \ T " : P
5.3 Proofs of the main results
185
Thus we have shown that the nine solution sets are all non-empty. " By using scattering theory, we can prove that SC and the fixed energy section N (with U being the NLKG evolution) ˚ D" SC WD uE 2 H j U N .t/E u scatters as t ! 1 ; E.E u/ D J.Q/ C "2 " D" are both pathwise connected. To see this, let uEj 2 SC or SC for j D 0; 1 and let uEj C be their asymptotic profiles
N
U .t/E uj U.t/E uj C E ! 0; .t ! 1/ :
There exists a continuous curve uE C W Œ0; 1 ! H such that kE uC k2H D .1
/kE u0C k2H C kE u1C k2H ;
as well as u solving NLKG on some .T ; 1/ such that
uE .t/ U.t/E uC E ! 0; .t ! 1/ : Moreover, T and uE .T C 1/ are also continuous in . The scattering property implies that uE .t/ 2 SC for any t > T , and E.E u / D kE u C k2H =2 D .1
/E.E u0 / C E.E u1 / :
Let T WD sup01 T C 1 < 1, then uEj and uEj .T / are connected by the flow, " D" while uE 0 .T / and uE 1 .T / are connected by uE .T /, all included in SC or SC . This proves the connectedness of those sets. Next, we observe that TC" contains the center-stable manifold for the t 0 direction, as constructed in Theorem 3.22. In fact, by the uniqueness property of that manifold, TC" is identical to the maximal backward extension of all solutions on this manifold. Since the flow is C 1 in both directions, T˙" are indeed connected, 1-codimensional and smooth manifolds in the energy space. Moreover, a sufficiently " small ball B around each point of TC" is separated by the hypersurface TC" into SC \B " and BC \ B. P Reversing time, the unThe linear approximation is the hyperplane k D . stable and stable directions are interchanged. Heuristically speaking, but still in a precise sense, the classification into those nine sets appears as the symbol ˝ in the P phase plain around Q, see Figure 5.1 (which also appears on the cover). The .; / meaning of this is as follows: the four chambers correspond to all possible combinations of finite time blowup vs. scattering to zero as t ! ˙1. The boundaries of
186
5 Above the ground state energy III: Global NLKG dynamics
these chambers, more precisely, the four straight line segments in ˝, correspond to the four possible combinations of scattering to Q as t ! ˙I on the one hand, and finite time blowup/scattering to zero as t ! 1 on the other hand. Finally, the dot in the middle corresponds to those solutions which are trapped by .˙Q; 0/ as t ! ˙1. In the full infinite dimensional representation, the aforementioned “dot” is a smooth manifold of codimension 2, and is given as the intersection of the two center-stable manifolds corresponding to the forward and backward evolutions, respectively; the latter being of course the center-unstable manifold. The reader should bear in mind that reversing the sign of the time is tantamount to replacing the data .u0 ; u1 / at time t D 0 with .u0 ; u1 /. We also remark that there can be no solution which is trapped by Q as t ! 1, and Q as t ! 1, as this would contradict the one-pass theorem. So we conclude that the only possible scenario is trapped by either Q or Q in both time directions. In the geometric pictures, this corresponds to the two center manifolds E and Q, E respectively. We remark that Bates and Jones already established around Q the orbital stability globally in time for solutions which lie on these manifolds (at least for powers not exceeding 3), see [6]. To be more precise, we now prove that TC" consists of two connected components " TC.˙/ defined by ˚ " TC./ D uE 2 H" j U N .t/E u is trapped by Q as t ! 1 : " To see that TC.C/ is pathwise connected, take any pair of points from that set and send them by the nonlinear flow until they land on the local center-stable manifold, which is of course connected. " " To see the separation of TC.C/ from TC. , suppose for contradiction that p is / a common point of their boundaries. Since these manifolds are relatively closed in H" , it follows that p 2 TC2" . Thus, U N .t/p will be trapped by Q with D ˙. E But Then its small neighborhood can be mapped by the flow into a ball around Q. " " it contains a point in TC. / , contradicting the one-pass theorem. Thus TC consists " of two connected components TC.˙/ . We can also observe that " D" @SC D TC" [ TCD" [ SC
is connected. The inclusion follows from the fact that B"C and fE.E u/ > J.Q/ C "2 g are open sets, whereas is easily seen by perturbing the data from the right. The connectedness follows from that of the following three sets < D" TC.C/ [ TC.C/ ;
D" SC [ TCD" ;
< TC.
/
D" [ TC. /:
187
5.3 Proofs of the main results
The Payne–Sattinger solutions can be classified in our notation, see (5.12), by 0 SC D S0 ;
B0C D B0 ;
TC0 D T 0 D ; :
This statement follows from our theorems, because the energy constraint E.E u/ < E prohibits the sign change of Ks and scattering to ˙Q. The threshold case E.Q/ given by Theorem 5.3, i.e., the analogue of the Duyckaerts–Merle theory [48], requires a little more work. Here we give a proof using the scattering to the ground state, hence the spectral gap property, but it is not essential. See [112] for a proof in the general case using the graph transform method.
5.3.2 Proof of Theorem 5.3
e
0 Let X˙ D
T
">0
" X˙ for X D S; B; T . Then Theorem 5.3 can be restated as
e
f0 f 0 0 0 Sf C \ B D S \ BC D ;;
f0 0 E Tf C \ T D f˙Q.t
e
s/ j s 2 Rg;
f0 0 E Sf s/ j s 2 Rg; B0C \ Tf0 D f˙WEC .t s/ j s 2 Rg; C \ T D f˙W .t f f0 \ Tf0 D f˙J WE .s t/ j s 2 Rg; 0 0 E S \ Tf t/ j s 2 Rg; B C C D f˙J W .s C f f0 f0 0 0 for some solutions W˙ . To see that Sf C \ B D ;, let u be a solution in SC \ B . Theorem 4.11 implies that there exists a finite interval I" R for any " > 0, E D I" . Then I" is decreasing as " ! C0, and hence by such that uE 1 .B3" .˙Q// E which implies u Q, a continuity of uE , there exists t 2 R such that uE .t/ D Q, contradiction. 0 Next, any solution u in Tf C corresponds to the case 1 0 in Theorem 3.22, and it is therefore parametrized by .0/ 2 R. Since .t/ ! 0 as t ! 1, the uniqueness in the proposition implies that there are only three solutions converging E modulo time translation: one with > 0 decreasing (for large t), another with to Q, < 0 increasing, and yet another with 0, i.e., Q. Let W˙ be the two solutions and 1 0. Since E.WE˙ / D J.Q/ and W˙ 6D Q, the sign given by .0/ D ˙ 10 of K0 is fixed on their trajectories. Indeed, Lemma 4.3 applies to them backward from a neighborhood of t D 1, from which we deduce that S.WE˙ .t// D 1 away f0 and W 2 S f 0 from t D 1, and so WC 2 B . To prove that W˙ approach Q exponentially, we return to (3.59) and (3.68) with 1 D 0, estimating k.; /kX k WD ke k min.t T
T;0/
.t/kL1 \L1 .0;1/ C ke k min.t
T;0/
.t/kSt.0;1/
188
5 Above the ground state energy III: Global NLKG dynamics
for T ! 1. The same argument as in the proof of Proposition 3.22 yields
.; / k . e kT j.0/j C .; / 2 C .; / .; / k XT X X XT
. e kT C .; / X k ; T
and so .; /
k XT
5.4
.e
kT
, as desired.
Summary and conclusion
This chapter presents the main results for the radial cubic NLKG equation in R3 obtained in this monograph. They are based on the authors’ paper [109]. We now try to summarize the main steps of the method: ı The basic well-posedness theory, and global existence and scattering for small data. This was done in Chapter 2, and involves both energy and dispersive estimates. Central to the scattering analysis is the Strichartz norm k kL3 L6x . t ı A Payne–Sattinger type theory. This refers to the variational theory of the ground state as well as to its dynamical implications for energies strictly below the ground state energy. Of central importance are the functionals K0 ; K2 derived from the stationary energy J via symmetries (dilations of the dependent and independent variables). Essential to the theory in its present form is the fact that the linearized operator LC associated with the ground state exhibits a single negative eigenvalue. This step includes a proof of the scattering property via the Kenig–Merle concentration-compactness method. ı The perturbative analysis near the ground states. This refers to the hyperbolic dynamics near the ground states and involves the construction of the invariant manifolds associated with it. The aforementioned unique negative eigenvalue generates 1-dimensional stable and unstable manifolds, and the center-manifold is of codimension 2. Generally speaking, the construction of the stable and unstable manifolds is less delicate than that of the center-stable one; furthermore, if one is willing to give up the scattering property typically expected of solutions starting on the center-stable manifold, then the construction of that manifold is also more robust (this distinction refers to the Lyapunov–Perron method on the one hand, and the Hadamard method on the other hand). Of central importance is the ejection lemma which describes a mechanism by which to select trajectoE As they are being ejected, ries which are ejected from a small ball around ˙Q. the sign of the functionals K0 and K2 is determined entirely by the sign of the coefficient of the unstable mode. For the latter, it is important that the trajectory
5.4 Summary and conclusion
189
E whose square is at least as large as the energy starts off from a distance to ˙Q E excess E.E u/ E.Q/. ı A one-pass theorem. This step excludes trajectories which pass through small E more than once. The proof of this step hinges on a suitable virial balls around ˙Q identity in combination with the ejection lemma and the variational structure near the ground state. A crucial role is played by a definite sign that is attached to a E whose radius squared solution as long as it remains outside of a ball around ˙Q is large compared to the aforementioned excess of energy (which therefore needs to be small). ı Description of the global-in-time dynamics. By combining the previous four steps one arrives at the global picture as in Figure 5.1. Additional work is required for the scattering property which is obtained by an adaptation of the Kenig–Merle method to solutions with small energy excess. The one-pass theorem plays an important role in this proof. Duyckaerts–Merle type threshold solutions (referring to solutions on the center-stable manifold whose energy is precisely that of the ground state) appear naturally in our theory as the stable and unstable manifolds. However, in order to reach this conclusion one needs to construct the centerstable manifold by means of the Lyapunov–Perron method which in turn hinges on fine spectral properties of the linearized operator (the “gap property” of the linearized operator). In the following chapter we will describe various other scenarios where the authors, in part jointly with Joachim Krieger, have succeeded in carrying out these steps. For example, for the critical wave equation the second step above is precisely the Kenig– Merle theory for that equation, whereas for subcritical NLS this step was carried out by Holmer and Roudenko.
6
Further developments of the theory
So far, we have mostly limited ourselves to the radial, three-dimensional cubic NLKG equation. This had the advantage of providing a convenient framework in which to develop the ideas around the center manifold and the one-pass theorems. However, several questions now pose themselves naturally: (i) Are there any theorems analogous to those of the previous chapter for the same equation but with non-radial data? (ii) Does the same theory apply to the NLKG equation with other powers and in other dimensions? In particular, what happens in the one-dimensional setting where there is much less dispersion? (iii) Are there any analogous results for other dispersive equations which admit soliton solutions such as the NLS equation (for which any stationary solution is modulated by a finite set of parameters)? (iv) What can be said about the energy critical case? Do center-stable manifolds exist in that setting, and is there a 9-set theorem? In this chapter we present some partial answers to these questions. We remark that this chapter is quite different from the previous ones as it does not strive to give a complete exposition with detailed proofs. In fact, we only intend to provide an introduction to, and summary of, the results of [87], [88], [110], [111]. We hope that the reader will consult these references for further details. As for question (i), a complete answer was found in [111] for the cubic NLKG equation in R3 . In addition to the translation symmetry, the nonradial equation also exhibits Lorentz symmetry. The latter can be used to transform a general solution into one with vanishing momentum (at least if jEj > jP j where E; P are the nonlinear energy and the momentum, respectively). The condition E < J.Q/ C "2 2 now becomes Em < J.Q/ C "2 , where Em D E 2 P 2 , provided the difference here is positive. The energy Em is Lorentz invariant, and is called minimal energy, since it is the true energy if P D 0. With the minimal energy instead of the regular energy, the 9-set theorem carries over to nonradial solutions, albeit with a somewhat different proof. The most significant differences from the radial setting occur in the construction of the center-stable manifold. Here one needs to introduce a wider family of ground states, which reflect the effect of a Lorentz contraction in the direction of motion. As in the radial case the key to the construction of the center stable manifolds lies with a suitable dispersive estimate for the linearized operator. It turns out that here one needs to allow for a gradient term with a small time-dependent
192
6 Further developments of the theory
(but space independent) coefficient in the linearized operator. But by means of a variant of the approach in Beceanu [8] this can be handled as well. Throughout the entire nonradial analysis we use a particular complex formalism which is designed, amongst other things, to keep the modulation equations of the first order. We give a brief account of these developments in Section 6.1 below. As for question (ii), we already noted before that the 9-set theorem carries over to all dimensions d 1 and nonlinearities jujp 1 u in the range 1 C d4 < p < 2 1. As one would expect, it is more delicate to prove the scattering to Q, see Theorem 5.2. At least in terms of our current methodology, this requires understanding the spectral properties of the linearized operator LC (the “gap property”, see (3.12)). In Section 6.2 below we follow [88] to establish the existence of center-stable manifolds for the 1-dimensional NLKG with powers p > 5. The novel aspect here (as compared to the three-dimensional situation) is that the Strichartz estimates only go down to L4t which is insufficient for nonlinearities of the form Qp 2 v 2 . Note that these nonlinearities arise in the perturbative argument in the construction of the center-stable manifolds. In order to control these terms, we establish estimates of the type L2t L2x;loc for v, which are local in space. This is natural, as Q provides the needed localization. Estimates of this type have been used previously, for example by Mizumachi [107]. They hinge on the fact that the linearized operator LC does not have a threshold resonance. The advantage of the one-dimensional equation is the explicit spectral theory of LC . It has been known for some time, see for example [90] for an account of this, that for powers p > 3 there are no eigenvalues in the gap and that the thresholds are not resonances. In other words, the “gap property” is completely understood in the 1-dim case. As for question (iii), [110] obtains the full analogue of the NLKG results for the NLS equation in the R3 for radial data (nonradial data are not considered here). Below we give a brief account of the results in [110]. In contrast to NLKG, we need to control two symmetry parameters, namely scaling and modulations of the phase. While the former is essentially fixed by the conserved mass, one can “mod out” the latter. In fact, this is true for the orbital stability statements as in the 9-set theorem, but the construction of the center-stable manifold requires more work and one has to control the global (in forward time, say) dynamics of these symmetry parameters (as is typical of asymptotic stability theory), see the final section of Chapter 3. For the orbital stability part, the underlying symplectic structure provides a convenient framework in order to set up the modulational equations. As far as the 9-set theorem for NLS is concerned, the corner-stone is the same type of one-pass theorem as for the NLKG equation. The proof of this result is somewhat more subtle in the NLS case. For example, a time-dependent cut-off as in Figure 4.6 has little meaning here due to the absence of finite propagation speed.
6 Further developments of the theory
193
Instead, the authors [110] used a virial-type identity with a time-independent weight which goes back to Ogawa–Tsutsumi [113]. The choice of the weight is the delicate part, since one wishes to obtain almost monotone quantities whose derivatives are “mostly” controlled by the sign of K.u/. We remark that the radial assumption appears to be essential for this virial analysis. On the other hand, the restriction to the cubic power and dimension three can easily be relaxed for the purposes of the 9-set theorem. As already for the NLKG equation, it is really the construction of the center-stable manifold via the Lyapunov, Perron method which is more sensitive to the nonlinearity as well as the dimension. For more on these issues see Chapter 3, especially the final section on NLS. Finally, for question (iv) we only present partial answers. The difficulties appear to be more substantial in the energy critical setting. Unlike the subcritical case, the critical equation does not have a mass term (otherwise there are no ground states), i.e., the equation becomes the energy critical wave equation. These equations admit the special stationary solutions called Aubin–Talenti solutions; they are extremizers for the critical Sobolev imbedding HP 1 .Rd / ,! L2 .Rd / and have a simple explicit form. Due to the absence of the mass term, the equations now also admit a scaling symmetry which generate a radial zero mode of the linearized operator LC . In R3 this mode is a zero energy resonance, whereas in dimensions dim 5, it is a zero energy eigenvalue. For all powers of the nonlinearity, one retains the property that LC has a unique negative eigenvalue. While [91] constructs a center-stable manifold of codimension 1 in a suitable topology1 for the critical wave equation in R3 , we do not address the question of existence of this manifold in the energy topology. However, following [88] we can describe the behavior of radial solutions which are not trapped by a “tube” around the rescaled ground state. Again, a one-pass theorem holds here as well. The key to controlling the scaling parameter during the hyperbolic dynamics near the ground state is the fact that the zero mode of LC evolves according to some algebraic law, whereas the exponentially unstable mode grows much more strongly and therefore dominates everything else. By means of this observation, one obtains a “4-set” theorem describing the dynamics outside of the tube. Of essential importance in our argument is the characterization of type-II blowup by Duyckaerts, Kenig, and Merle [47] (type-II refers to blowup for which the free energy remains uniformly bounded as the time approaches blowup time). Indeed, this replaces the subcritical argument that the Payne–Sattinger set P SC is bounded in H 1 L2 and 1
This topology is much stronger than the energy topology. In particular, it is not invariant under the nonlinear flow. This makes it somewhat of a misnomer to call the manifold in [91] “center-stable”. However, it does exhibit the desired asymptotic stability property for any solution starting on it. In this regard, also see the work of Karageorgis, Strauss [81] on the wave equation with juj5 nonlinearity.
194
6 Further developments of the theory
therefore only admits global solutions. For the energy critical equation, this is of course insufficient in order to conclude global existence and one needs to invoke the very delicate analysis of [47].
6.1
The nonradial cubic NLKG equation in R3
To precisely state the nonradial result from [111], let uE WD .u; u/ P and set ˚ H" WD uE 2 H j Em .E u/ < E.Q; 0/ C "2
(6.1)
with the minimal energy ˇ Em .E u/ WD ˇE.E u/2
ˇ1 u/2 P .E u/2 ˇ 2 sign E.E
P .E u/2 ;
(6.2)
where P .E u/ D hujrui P is the conserved momentum for the NLKG equation. In contrast to the energy E, the minimal energy Em is Lorentz invariant. It is therefore necessary to use this form of the energy in the general nonradial setting. We call a solution with zero momentum normalized. To every solution with jE.E u/j > jP .E u/j 2 there exists exactly one Lorentz transform which reduces it to a normalized one, see (2.83). If jE.E u/j < jP .E u/j, then there exists a Lorentz transform L such that E.E u ı L/ < 0. But such solutions are known to blowup in finite positive and negative times by the Payne–Sattinger criterion. If jE.E u/j D jP .E u/j, then along some sequence of Lorentz transforms Lj we have E.E u ı Lj / ! 0 as j ! 1. But then either K.u ı Lj / < 0 for some j , which means that u blows up in both time directions by the Payne–Sattinger criterion, or u ı Lj ! 0 strongly in H 1 so that u exists globally. Thus, we can always talk about normalized solutions in those cases where [114] does not apply, which is sufficient for our purposes. The 9-set theorem now takes the following form. Theorem 6.1. Consider all solutions of NLKG (2.1) with initial data uE .0/ 2 H" for some small " > 0. Then the solution set is decomposed into nine non-empty sets characterized as (1) Scattering to 0 for both t ! ˙1, (2) Finite time blow-up on both sides ˙t > 0, 2
At this point one needs to clarify the meaning of the Cauchy problem for Lorentz transformed solutions, and more generally, wellposedness questions of transformed solutions; this is perhaps not immediately clear as Strichartz norms such as L3t L6x are not invariant under Lorentz transforms. However, the norm L4t;x is, and is shown to control the relevant Strichartz norms in [111].
6.1 The nonradial cubic NLKG equation in R3
195
(3) Scattering to 0 as t ! 1 and finite time blow-up in t < 0, (4) Finite time blow-up in t > 0 and scattering to 0 as t ! 1, (5) Trapped by ˙Q for t ! 1 and scattering to 0 as t ! 1, (6) Scattering to 0 as t ! 1 and trapped by ˙Q as t ! 1, (7) Trapped by ˙Q for t ! 1 and finite time blow-up in t < 0, (8) Finite time blow-up in t > 0 and trapped by ˙Q as t ! 1, (9) Trapped by ˙Q as t ! ˙1, where “trapped by ˙Q” means that the normalized solution stays in a O."/ neighborhood of E C y/ j y 2 R3 g f˙Q. forever after some time (or before some time). The initial data sets for (1)–(4), respectively, are open. The approach is similar to the radial case, but with the added feature of having to control a small translation vector, after the Lorentz transform to the normalized solution. As for radial data, Theorem 6.1 extends to all Rd and equations of the form u C u D jujp
1
u;
1C
4 4
namely those powers of L2 -supercritical and H 1 -subcritical (for d D 1; 2, there is no upper bound on p). The main difference between radial and nonradial solutions lies with the further analysis of the trapped solutions in Theorem 6.1. Recall that we proved for radial data (where all solutions are normalized) that any solution in H" which is trapped by Q as t ! 1 scatters to Q as t ! 1. Note that there are no modulation parameters for radial data, such as translation or scaling, which simplifies the analysis considerably. In contrast, for general data, Q may be translated as well as Lorentz transformed, leading to a 6-dimensional manifold of solitons (which consists of two disconnected components corresponding to ˙Q), parametrized by the relativistic momentum p 2 R3 and the center of mass q 2 R3 . The traveling waves also undergo the Lorentz contraction (i.e., they are flattened in one direction). The soliton manifold is explicitly given at each .p; q/ 2 R6 in the form (6.3) Q.p; q/.x/ D Q x q C p.hpi 1/jpj 2 p .x q/ ; p where hpi WD 1 C jpj2 . The ground-state traveling waves are those solutions u.t/ D ˙Q.p; q.t//;
p 2 R3 ;
q.t/ P D
p ; hpi
196
6 Further developments of the theory
p with fixed momentum p and velocity hpi . In our context, p will be small for the normalized solutions, whereas q can be arbitrary. The above scenario is of course the analogue of the center-stable manifolds of [9], [90], [122] for the NLS equation, where the family of solitons is of dimension 8 in R3 , and of dimension 4 in R1 , respectively, parametrized by the non-relativistic momentum, the center of mass, the gauge in S 1 and the charge in RC . However, the NLS equation behaves quite differently from the wave equation, since the symmetry group of NLS (generated by Galilei transforms, translation, modulation, and scaling) does not mix space and time as Lorentz transforms do. In addition, the modulation equations for NLS are first order ODEs, whereas for the wave equation they at least appear on first sight to be of the second order. A considerable amount of effort is therefore expended in [111] on setting up a suitable framework for the modulational and scattering analysis. In particular, we set things up in such a way that the modulation equations are of the first order. We can summarize the findings of [111] concerning the trapped solutions in Theorem 6.1 as follows:
Theorem 6.2. All normalized solutions in H" which are trapped by Q as t ! 1 lie on a smooth, codimension-one manifold MC .Q/ in H, and those trapped by Q as t ! 1 lie on MC . Q/ D MC .Q/. Analogously, those trapped by ˙Q as t ! 1 lie on M .˙Q/. The manifold MC .Q/ is invariant under the flow of (2.1) in forward time, and all solutions starting on MC .Q/ scatter to the sixdimensional manifold generated by Q as t ! 1 in the following sense: there exist p1 2 R3 and C 1 paths q.t/ 2 R3 with the property that q.t/ P !
p1 ; hp1 i
u.t/
Q p1 ; q.t/ scatters to 0
.t ! 1/:
(6.4)
The center-stable manifolds MC .Q/ and M .Q/ intersect transversely in a codimension two manifold, the center manifold, which characterizes case (9) of Theorem 6.1. Finally, all normalized solutions uE .t/ with energy E.E u/ D E.Q; 0/, and which are trapped by ˙Q in positive times, are classified as follows: either u is constant and equal to some translate of ˙Q, or for some t0 2 R, x0 2 R3 , u.t; x/ D ˙WC .t C t0 ; x C x0 / or u.t; x/ D ˙W .t C t0 ; x C x0 /
(6.5)
E exponentially fast in H, and where WE˙ are solutions of (2.1) which approach Q either blowup or scatter to 0 as t ! 1. An analogous statement holds for the negative time direction.
197
6.1 The nonradial cubic NLKG equation in R3
The solutions W˙ are the analogue of the threshold solutions found by Duyckaerts and Merle [48] for the nonlinear wave and Schrödinger equations with the H 1 critical power. As for radial solutions, our proof of Theorem 6.2 relies on the gap property of the linearized operator LC WD
3Q2 ;
C1
(6.6)
i.e., spec.LC / \ .0; 1 D ; without threshold resonance or eigenvalue at 1. 6.1.1 The complex formalism I We now describe how to set up the equations in order to prove Theorem 6.1, and especially how to develop the modulation theory. It turns out that for the orbital stability techniques underlying the proof of Theorem 6.1 one does not need a fine analysis involving Lorentz transforms nor does one need the family (6.3). This family will be most relevant, though, in the construction of the center-stable manifold. We formulate the equation in terms of the complexified variable p (6.7) u WD Dw i w; P D WD 1 : Denoting u1 WD D
1
Re u;
for any complex function u, the energy is Z " ju2 j2 C jru1 j2 C ju1 j2 E.u/ D 2 R3
(6.8)
u2 WD Im u
# ju1 j4 2 dx D kukL 2 =2 4
4 ku1 kL 4 =4 ;
and the energy (or phase) space for u.t/ is the R-Hilbert space H defined by Z 2 3 H WD L .R I C/; hf jgi WD Re f .x/g.x/ dx .f; g 2 H/: R3
We can rewrite (2.1) as the system consisting of (6.8) and u t D iDu
iu31 :
(6.9)
While the relation (6.8) is always assumed for any vector expression, (6.7) is not. The ground state in this formulation is denoted by
Q WD DQ:
198
6 Further developments of the theory
We decompose any solution u of (6.9) in the form c.t/ :
u D .Q C v/ x Then the equation for v is
v t D iDLv C @k .Q C v/cP k
(6.10)
iN.v1 /;
where L and N are defined by Lv D v
3D
1
Q2 v1 ;
N.v1 / D 3Qv12 C v13 D O kvk22 ;
(6.11)
respectively. Note that v does not satisfy (6.7) in general, due to the time-dependent translates c.t/. L is a self-adjoint R-linear Fredholm operator on H. The linearized operator i DL has generalized eigenfunctions iDLr Q D 0;
iDLi rQ D
r Q;
iDLg˙ D ˙kg˙ ;
where g˙ is given in terms of the ground state of LC D D2 g˙ D .2k/
1=2
D i.k=2/1=2
.LC D
(6.12)
3Q2 :
k 2 ; kk2 D 1; > 0/:
The natural symplectic form associated with this complex formalism is !.u; u/ Q WD hi D
1
(6.13)
ujui: Q
Indeed, in keeping with the classical Hamiltonian formalism we need to be able to write the equation (6.9) in the form uP D XE .u/ where XE .u/ is the Hamiltonian vector field associated with E. This means that dE.u/./ D ! ; XE .u/ or in other words, hu
D
1 3 u1 ji
D !.; iDu
iu31 /
which yields (6.13). We have, for ˛; ˇ D 1; 2; 3, !.@˛ Q; i@ˇ Q/ D ı˛;ˇ k@1 Qk22 D ı˛;ˇ J.Q/;
!.g˙ ; g / D ˙1;
!.r Q; g˙ / D !.irQ; g˙ / D 0: We now perform the following symplectic decomposition of v: v D C gC C g C ;
˙ WD !.v; g /=!.g˙ ; g / D !.v; ˙g /;
(6.14)
199
6.1 The nonradial cubic NLKG equation in R3
which implies that !. ; g˙ / D 0 ” h 1 ji D h 2 ji D 0 : Finally, we expand the energy in the form 2.E.u/
J.Q// D hLC v1 jv1 i C kv2 k22 D D
2C.v/ D hLvjvi
2kC C hL j i 2k.22
2C.v/
2C.v/
21 /
C hL j i
;
C.v/ D hQjv13 i C
2C.v/;
where j and C are defined by 1 D
C C ; 2
2 D
C
2
kv1 k44 3 D O.kv1 kH 1 /: 4
6.1.2 Parameter choice for Theorem 6.1 We now reduce the number of coordinates by means of a suitable Lorentz transform. To be more specific, let w be a global strong energy solution of (2.1) and Lorentz transform it as follows: w.t; x/ 7! w .t; x/ WD w.t cosh C x1 sinh ; x1 cosh C t sinh ; x2 ; x3 / (6.15) for 2 R. Then one checks that w is again a strong energy solution of (2.1) which satisfies E.w / D E.w/ cosh C P1 .w/ sinh ; P1 .w / D P1 .w/ cosh C E.w/ sinh ;
P˛ .w / D P˛ .w/ .˛ D 2; 3/;
(6.16)
where P D .P1 ; P2 ; P3 / denotes the total momentum P .w/ D hw t jrwi D
1 !.u; ru/; 2
cf. (6.7). Note that the Lorentz transform preserves E 2 jP j2 . Thus, every solution with finite energy jEj > jP j is transformed, by a unique element of the Lorentz group, to another solution with zero momentum: 0 D P .u/ D
1 1 !.u; ru/ D !.v; r Q/ C !.v; rv/; 2 2
(6.17)
200
6 Further developments of the theory
which minimizes the energy among the family of solutions generated by the Lorentz group. As we saw earlier in this section, the cases jEj jP j fall under the scope of [114] and can therefore be ignored. Since the dynamical properties such as scattering and blowup are not changed by the Lorentz transforms, we restrict our dynamical analysis to the invariant subset
H0 WD fu 2 H j !.u; ru/ D 0g: Note that this is well-defined since ! gains a derivative. One now defines the decomposition u D s.Q C v/.x
c/;
s D ˙1; c 2 R3
by the orthogonal projection in H 1 .R3 /, or the minimization
u1 sQ.x c/ 2 D min u1 Q.x b/ 2 ; L L b2R3
(6.18)
(6.19)
which is attained for any u 2 H, and uniquely so if the right-hand side is small enough. The minimization implies the orthogonality conditions 0 D !.v; i@j Q/ D hv1 j@j Qi .j D 1; 2; 3/:
(6.20)
Differentiating (6.20) with respect to t, we obtain the parameter evolution in H0 0 D @ t !.v; i@˛ Q/
.˛ D 1; 2; 3/
D !.iDLv C cP r.Q C v/ D
2 P k@1 QkL 2c
hr@˛ Qjv1 i cP
iN.v1 /; i@˛ Q/
(6.21)
!.v; rv/=2;
where we used (6.17) to rewrite the first term of the second line, while the nonlinear term N does not contribute because of its reality. Thus, we obtain ˇ ˇ ˇ ˇ
ˇcP ˇ . ˇ!.v; rv/ˇ . v.t/ 2 ; (6.22) 2 as long as kv1 kL2 1 and the solution u 2 H0 . (6.20) and the symplectic decomposition (6.14) imply the orthogonality relations ˝ ˛ ˝ ˛ ˝ ˛ 0 D 1 j@j Q D 1 j D 2 j : Since f; rQg covers the non-positive eigenfunctions of LC (see [143], Appendices A and E), one has ˝ ˛ ˝ ˛ 2 2 2 L j D LC 1 j 1 C k 2 k22 ' k 1 kH 1 C k 2 k2 ' k k2 :
201
6.1 The nonradial cubic NLKG equation in R3
The estimate (6.22) controls the translation parameter in the region near Q, which is precisely where the decomposition (6.18) is meaningful. In fact, it guarantees that the translation parameter c is quadratic in v, and therefore essentially does not affect the methods of Chapter 4 centered around the nonlinear distance function, eigenvalue dominance, and the ejection lemma. Moreover, the one-pass theorem also carries over to this setting, with the main distinction that upon return to a small neighborhood of the ground state one might encounter a large spatial translation. However, this does not seriously affect the argument. It just means that the rhombus of Figure 4.6 now appears as the intersection of two cones with the lower one centered at zero, say, but the top one can be shifted by an arbitrary amount.
6.1.3 The complex formalism II We now wish to prove that any normalized solution staying forever close to the 6dimensional manifold of the ground states scatters to that manifold. As usual, this is based on a perturbative construction. The latter requires some care, especially in order to avoid second order modulation equations for the parameters. In order to accomplish this, we begin with a more detailed analysis of the Lorentz invariance. To any p 2 R3 , we associate the following Lorentz transform p p DW s; 2 S 2 ; s D sinh 0; c WD 1 C jpj2 D cosh ; p WD p D tanh ; x WD . x/; x? D x x ; (6.23) 1 C jpj2 up .t; x/ WD u ct sx ; c.x t/ C x? D u tc p x; x C .c 1/x tp : Then for each fixed p 2 R3 the static solution Q.x/ is transformed into the traveling wave solution Q c.x with the velocity D consisting of
t/ C x? D Q x p . hpi
t C .c
1/.x
t/
Its trajectory lies in the 6-dimensional manifold in H 1
Q.p; q/.x/ WD Q c.x
q/ C .x
q/? ;
whose vector version is denoted by
Q.p; q/ WD .D C i r/Q.p; q/:
(6.24)
202
6 Further developments of the theory
The cubic NLKG equation for traveling waves is transformed into the following equation in terms of Q.p; q/: iDQ C r Q D i Q31
(6.25)
where Q.p; q/31 D ŒQ.p; q/1 3 D Q.p; q/3 (and similarly for other powers). Differentiating (6.25) in .p; q/, we obtain the linearized equations .iDL C r/@q Q D 0;
@q Q D
r Q;
(6.26)
.iDL C r/@p Q D .@p / .@q Q/; where the R-linear self-adjoint operator L now depends on the parameters: L D L.p; q/ D 1
3D
1
Q.p; q/21 D
1
Re :
Note that this agrees with our previous definition (6.11) for p D q D 0. Decomposing the solution u in the form u D Q.p; q/ C v.x
q/;
we derive the equation for the perturbation v, i.e., v t D .i DLp C .p/ r/v C .qP
.p// r.Qp C v/
pP @p Qp
iNp .v1 /;
where Qp WD Q.p; 0/, Lp WD L.p; 0/ and Np .v1 / WD 3Qp1 v12 C v13 . For brevity, let Z t Ap WD i DLp C .p/ r; WD q p.s/ ds; WD .p; / 0
then the equation can be rewritten in the form v t D Ap v C P r.Qp C v/
pP @p Qp
iNp .v1 /:
The natural orthogonality condition is 0 D !.v; @q Q/ D !.v; @p Q/:
(6.27)
Here, and in what follows, it will be understood that all derivatives of Q are to be evaluated at q D 0. In particular, (6.24) implies that @p Q.0; 0/.x/ D irQ.x/;
@q Q.0; 0/.x/ D
r Q.0; 0/.x/ D
rDQ.x/;
6.1 The nonradial cubic NLKG equation in R3
203
which constitute the root modes for i DL, see (6.12). By differentiation of (6.27), we obtain the parameter evolution 0 D !.v t ; @˛ Q/ C !.v; @˛ @p Q/pP D
!.v; Ap @˛ Q/ C !. P r.Q C v/
p@ P p Q; @˛ Q/
(6.28)
!.iNp .v1 /; @˛ Q/ C !.v; @˛ @p Q/p; P for ˛ D q1 ; q2 ; q3 ; p1 ; p2 ; p3 . In view of (6.26) and (6.27) the first term of (6.28) vanishes, whence
P !.@q Q; @˛ Q/ !.v; @q @˛ Q/ C pP !.@p Q; @˛ Q/ !.v; @p @˛ Q/ D ! iNp .v1 /; @˛ Q ; which implies jj P . ke
jxj=4
vk22 ;
provided that the 6 6 matrix !.@˛ Qp ; @ˇ Qp / is non-degenerate. Certainly it is at p D 0, and therefore remains so for small p by continuity. We now look for a bounded global solution v by a contraction argument, for a given initial data near Q.0; 0/ in the center-stable direction. The contraction mapping .v; / 7! .v; Q / Q is defined by .@ t
Ap /vQ
Q t r.Qp C v/ Q C pQ t @p Qp D
@ t !.v; Q @˛ Qp / D 0;
iNp .v1 /;
.8˛ D p1 ; p2 ; p3 ; q1 ; q2 ; q3 /
(6.29)
with the initial constraint 0 D ! v.0/; Q @˛ Q.0; 0/ 8˛;
.0/ D .0/ Q D 0; .0/ Q D 0:
(6.30)
In (6.29), (6.30), the soliton Q.p; q/ is to be evaluated at q D 0 and p.t/ which is determined by the given path .t/. The orthogonality equation is equivalent to
Q t !.@q Qp ; @˛ Qp / C pQ t !.@p Qp ; @˛ Qp / D Q t !.v; Q @q @˛ Qp / C p t !.v; Q @p @˛ Qp /
! iNp .v1 /; @˛ Qp :
(6.31)
The system (6.29), (6.30), (6.31) determines the center-stable manifold and the asymptotic stability of solutions starting on that manifold. Note that it is of the first order. Also note that the key to solving this system lies with dispersive estimate for the driving evolution operator @ t Ap in (6.29). These estimates are derived in [111] via a method inspired by Beceanu’s work on the Schrödinger equation [8].
204
6.2
6 Further developments of the theory
The one-dimensional NLKG equation
In this section, we sketch the results on the one-dimensional NLKG equation ut t
uxx C u D jujp
1
u
(6.32)
with p > 5 which were obtained in [88]. We restrict ourselves to even data. The main difference from the 3-dimensional case lies with the absence of sufficiently strong dispersive estimates. In fact, in Exercise 2.45 of Chapter 2 the reader is asked to establish the relevant class of Strichartz estimates which are no better than L4t in time. Therefore, the objective of [88] was to find a way around that difficulty. This particular issue had been encountered before, for example in Mizumachi [107]. One advantage of the one-dimensional equation, however, is that the spectral theory – including the gap and threshold properties of LC – can be resolved explicitly, see [90] as well as Lemma 6.6 below. There it is shown that one has all the desired properties in the range p > 3 (this being sharp). Let us first recall that equation (6.32) is locally well-posed in the energy class H 1 L2 and can exhibit both global existence (for example for small data) as well as finite time blowup (for example, for negative energy). In addition, equation (6.32) admits explicit soliton solutions 1 p C 1 p1 1 p 1 (6.33) ; ˇD : Q.x/ D ˛ cosh ˇ .ˇx/; ˛ D 2 2 To be more specific, any static solution is uniquely defined by the ODE ' 00 C ' D j'jp
1
':
Multiplying this by ' 0 and integrating yields .' 0 /2 C ' 2 D
2 j'jpC1 C C : pC1
Since we require ' 2 H 1 .R/ it follows that C D 0, and all solutions of the resulting equation d' D dx ˙q 2 ' 2 pC1 j'jpC1 are of the form ˙Q. C x0 / with Q as above. In other words, in contrast to the threedimensional problem there are no “excited states” on the line. In [88] we prove the following result, where Z 1h i 1 1 E.u; u/ P D .j@x uj2 C juj P 2 C juj2 / jujpC1 dx (6.34) pC1 1 2
6.2 The one-dimensional NLKG equation
205
is the conserved energy. Theorem 6.3. Let p > 5. There exists " > 0 such that any even data .u0 ; u1 / 2 H 1 L2 .R/ with energy E.u; u/ P < E.Q; 0/ C "2
(6.35)
have the property that the solutions u.t/ of (6.32) associated with these data exhibit the following trichotomy: ı u blows up in finite positive time ı u exists globally and scatters to zero as t ! 1 ı u exists globally and scatters to Q, i.e., there exists a free Klein–Gordon wave v.t/; v.t/ P 2 H 1 L2 with the property that u.t/; u.t/ P D .Q; 0/ C v.t/; v.t/ P C oH .1/ t ! 1: (6.36) In addition, the set of even data as in (6.35) splits into nine nonempty disjoint sets corresponding to all possible combinations of this trichotomy as t ! ˙1. Moreover, we obtain a characterization of the threshold solutions, i.e., those with energies E.E u/ D E.Q; 0/, cf. [48]. In fact, we find the following. Corollary 6.4. The even solutions to (6.32) with energy E.E u/ D E.Q; 0/ can be characterized as follows: ı they blow up in both the positive and negative time directions ı they exist globally on R and scatter as t ! ˙1 ı they are constant ˙Q ı they equal one of the following solutions, for some t0 2 R: WC .t C t0 ; x/;
W .t C t0 ; x/; WC . t C t0 ; x/; W . t C t0 ; x/ where W˙ .t; /; @ t W˙ .t; / approach .Q; 0/ exponentially fast in H as t ! 1, and D ˙1. In backward time, WC scatters to zero, whereas W blows up in finite time. E form a one-dimensional stable manifold As usual, the images of WE˙ and Q associated with .Q; 0/, cf. [6]. The unstable manifold is obtained by time-reversal. Moreover, the solutions in Theorem 6.3 which scatter to Q form a C 1 manifold3 in 3
It is in fact smoother than C 1 .
206
6 Further developments of the theory
H of codimension 1 which is the center-stable manifold associated with .Q; 0/. The center manifold is obtained by the transverse intersection of the two center-stable manifolds corresponding to t ! ˙1, respectively. It therefore has codimension 2. We remark that the Hamza, Zaag [70] studied the problem of classifying blowup for equation (6.32) and showed that blowup occurs only in the form of ODE blowup for the one-dimensional NLKG equation (in higher dimensions the power on the nonlinearity needs to be no larger than then conformal power). The restriction p > 5 in Theorem 6.3 is most likely a technical one. We remark that it plays an important role in the adaptation of the “zero-frequency scattering result”, see Lemma 4.13. This was needed in the proof of the one-pass theorem to handle the degenerate case where the lower bound on K2 u.t/ becomes ineffective. In fact, it seems reasonable to expect that Theorem 6.3 remains valid in the range p > 3, and possibly even for some range p 3. At p D 3 the linearized operator LC has a threshold resonance which needs to be taken into account. But more importantly, in the range p < 5 any small data scattering argument cannot be based on Strichartz estimates alone which is of course a serious obstacle at this point. The power p D 5 is perhaps accessible, but we exclude it here (as in [77]). We remark that in contrast to the NLS equation, the hyperbolic structure underlying our proof of Theorem 6.3 is still present for all 1 < p 5. This comment refers to the fact that the structure of the spectrum of the linearized form of (6.32) around Q does not change significantly as one decreases p (whereas for NLS the exponentially stable eigenvalues merge with the root space at the L2 -critical power and then move off into the real discrete spectrum as one lowers the power further). See Bizo´n, Chmaj, Szpak [17] for a numerical study of (6.32) which exhibits different rates of convergence depending on the power of the nonlinearity. The proof of the one-dimensional “orbital stability”-type 9-set theorem in which one only requires trapping by .˙Q; 0/ rather than scattering to .˙Q; 0/, is very similar to the three-dimensional setting. The action (or static energy) is Z h i 1 1 .j@x uj2 C juj2 / jujpC1 dx J.u/ D (6.37) 2 pC1 and the functionals K0 ; K2 are now Z 0 K0 .u/ D hJ .u/jui D 0
K2 .u/ D hJ .u/jAui D
j@x uj2 C juj2 Z
j@x uj2
jujpC1 dx;
p 1 jujpC1 dx 2.p C 1/
with A D 21 .x@x C @x x/ being the generator of dilations. The linearizations of these
6.2 The one-dimensional NLKG equation
207
functionals are K0 .Q C v/ D
.p Dp
K2 .Q C v/ D
2 1/hQp jvi C O kvkH 1 ; E 5 p 2 Q C 2Qjv C O kvkH 1 :
2 Note the positivity in the linearization of K2 for p 5. The expansion of J is essentially the same as before with the linearized operator LC D @xx C1 pQp 1 : 1 3 J.Q C v/ D J.Q/ C hLC vjvi C O kvkH v ! 0: 1 ; 2 The translational symmetry in infinitesimal form becomes LC Q0 D 0 with 0 being a simple eigenvalue. Since Q0 has a single zero, it follows from Sturm oscillation theory that it is the second eigenfunction of LC , and one has LC D k 2 for some exponentially decaying > 0 as in the three-dimensional case. Over the even functions LC has no kernel. Moreover, the variational arguments which we developed in the previous chapters for the three-dimensional radial equation remain largely in effect. To be more specific, the respective statements remain intact but the arguments need to be adjusted; this is due to the absence of Strauss’ estimate in one dimension, cf. (2.20), and the subsequent loss of compactness which was used in the construction of various minimizers. However, this particular obstacle is easily circumvented by means of standard concentration-compactness arguments.
6.2.1 The center-stable manifold The center-stable manifold is again constructed by the Lyapunov–Perron method as in Chapter 3, and one obtains the following result. Proposition 6.5 (Center-Stable manifolds). Let p 5. There exists > 0 small and a C 1 graph M in B .Q; 0/ H so that .Q; 0/ 2 M, with tangent plane ˚ TQ M D .u0 ; u1 / 2 H j hku0 C u1 ji D 0 at .Q; 0/ in the sense that sup
dist.x; TQ M/ . ı 2 ;
80<ı <:
x2@Bı .Q;0/
Any data .u0 ; u1 / 2 M lead to global evolutions of (6.32) of the form u D Q C v D Q C C w where v satisfies
.v; v/ P L1 H C kvkLp ..0;1/IL2p .R// . t
208
6 Further developments of the theory
and scatters to a free Klein–Gordon solution in H, i.e., there exists a unique free Klein–Gordon solution w1 such that ˇ ˇ ˇ ˇ
ˇ.t/ˇ C ˇ.t/ P ˇ C w.t/ E w E 1 .t/ H ! 0; as t ! 1. In particular, we have E.E u/ D J.Q/ C kw E 1 k2H =2. Finally, any solution that remains inside B .Q; 0/ for all t 0 necessarily lies entirely on M, and M is invariant under the flow for all t 0. In contrast to the three-dimensional equation, this result cannot be proved based on Strichartz estimates alone. The most problematic term here is Qp 2 v 2 which arises in the perturbative analysis of u D QCv. Indeed, the best Strichartz estimates 8
4
allow us to do here is to place this term in L t3 . But this requires L t3 -control on v, which falls short of the L4t needed to be able to close all estimates. The main ingredient that leads to the proof of Proposition 6.5 is the control of the norm khxi s uk
1
L2t .RIHx2 .R//
:
(6.38)
Such local norms were already used before, see for example [107]. Note that if u is a free Klein–Gordon solution starting from Schwarz data, say, then the pointwise 1 decay in time is no better than t 2 , see the final section of Chapter 2. Therefore, the norm in (6.38) would be infinite for such u.t/. The crucial fact which ultimately leads to the finiteness of (6.38) is that we consider the perturbed Klein–Gordon evolution arising from linearization around the ground state. In contrast to the free KG equation, the latter does not exhibit a threshold resonance, which means that the corresponding local decay is faster. Recall that the operator H D @xx C V with V real-valued, bounded, and satisfying hxiV .x/ 2 L1 .R/ is said to have a threshold resonance iff there exists a nonzero solution f 2 L1 .R/ \ H 2 .R/ of Hf D 0. This is equivalent to the condition of a bounded resolvent near zero energy, see (3.75) as well as the Laurent expansion (3.76). To see the connection between the existence of a nontrivial bounded zero energy solution and the boundedness of the resolvent in weighted spaces, one expresses the resolvent (or the “Green function”) in terms of the outgoing Jost solutions f˙ .; z/. They are defined for all Im.z/ > 0 by the condition that 1
f˙ .x; z/ e ˙iz 2 x as x ! ˙1 1
where Im.z 2 / > 0. Under our assumptions on V they are given by the Volterra
209
6.2 The one-dimensional NLKG equation
equation 1
fC .x; z/ D e
iz 2 x
1
Z
1
sin .x
y/z 2 1
z2
x
V .y/fC .z; y/ dy
and similarly for f .x; z/, for all Im.z/ 0. Finally, the Green function is given by the expression, for x > x 0 and Im z > 0, .H
z/
1
.x; x 0 / D
fC .x; z/f .x 0 ; z/ ; W .z/
W .z/ D W fC .; z/; f .; z/
where the latter is the Wronskian (the case x 0 > x being analogous by symmetry). It is not hard to see that W .z/ ¤ 0 for all Im z 0, z ¤ 0, whence .H z/ 1 remains bounded as in (3.75) with > 21 away from z D 0. The only possible failure of boundedness therefore occurs at z D 0. Moreover, the resolvent blows up as z ! 0 if and only if W .0/ D 0 which is equivalent to the solutions f˙ .; 0/ being linearly dependent; in other words, if and only if zero energy is a resonance (it also follows immediately that zero energy cannot be an eigenvalue for V as above). We now summarize some basic spectral properties of LC and L WD @xx C 1 Qp 1 . Lemma 6.6. If p > 3, then LC and L have the following properties: they have no eigenvalues in the interval .0; 1 and for both LC and L the threshold 1 is not a resonance. Furthermore, LC has exactly two eigenvalues, viz. k 2 and 0, and L has exactly one, namely 0. Proof. This is a classical observation based on the fact that for sech2 .x/ potentials (which go by the name of Pöschl–Teller potentials) the eigenvalue equations can be explicitly integrated via hypergeometric functions. For example, the lemma can easily be deduced from Flügge [55], Problem 39, page 94. See [90], Lemma 9.1 for more details.
6.2.2 The distorted Fourier transform We will obtain the desired local estimate (6.38) on the KG-evolution of L˙ by means of the distorted Fourier transform. We begin by recalling the distorted Fourier transform relative to a general Schrödinger operator on the line L WD
d2 CV dx 2
(6.39)
210
6 Further developments of the theory
with real-valued potential V . In our specific application, V .x/ D
˛ cosh
2
(6.40)
.ˇx/
for suitable ˛; ˇ > 0, but for the moment we only assume that V 2 L1loc .R/ and that L is limit-point at ˙1. This material is of course standard, see for example Section 2 of [59]. It is simpler to start with the operator L from (6.39) on the half-line x 0. This requires that we impose a boundary condition at x D 0, say (for simplicity) the Dirichlet boundary condition. In what follows, we avoid discussion of domains and self-adjointness of various operators, see [59] for more on these issues. Let .x; z/; .x; z/ be a fundamental system with .0; z/ D 0 .0; z/ D 0; 0 .0; z/ D .0; z/ D 1 : Note that W .; z/; .; z/ D 1. By the limit-point assumption at x D 1 there 2 exists a nonzero solution .; z/ 2 L .0; 1/ for every z 2 C n R, unique up to nonzero complex multiples. The basic relation of Weyl theory is the identity d W . .; z1 /; dx
; z2 / .x/ D .z2
z1 / .x; z1 / .x; z2 /;
z1 ; z2 2 C n R :
Integrating it from 0 to 1 yields, with z2 D z 1 , z1 D z, 1
Z 0
ˇ ˇ W ˇ .x; z/ˇ2 dx D
.; z/; .; z/ .0/ : 2i Im z
Since .; z/ D c .; z/ with some c ¤ 0, we conclude that particular, we can normalize .; z/ in such a way that
(6.41) .0; z/ ¤ 0. In
.; z/ D .; z/ C m.z/ .; z/ : Indeed, .0; z/ D 1 and m.z/ D 0 .0; z/. By the preceding, m is analytic on Im z > 0 and satisfies Im m.z/ > 0 for those arguments. In fact, from (6.41), Z 1 ˇ ˇ ˇ .x; z/ˇ2 dx D Im m.z/ : Im z 0 In other words, m is a Herglotz function and thus admits the representation Z h i 1 .d/ m.z/ D Re m.i/ C z 1 C 2 R
211
6.2 The one-dimensional NLKG equation
for some positive measure . The latter is called the spectral measure of H and it can be found as follows: Z 2 Cı (6.42) Im m. C i"/ d: Œ1 ; 2 / D 1 lim lim ı!0C "!0C 1 Cı
The measure appears naturally in the unitarity of the distorted Fourier transform which we now define. For any Schwartz function f set Z 1 fO./ D f .x/ .x; / dx; 8 0 : 0
Then for any Schwartz f; g one has the “Plancherel identity” ˝ ˛ ˝ ˛ f j F .H / .1 ;2 .H / g L2 ..0;1// D fO j F .1 ;2 b g L2 .R;/
(6.43)
where F is continuous, bounded, and 0 < 1 < 2 < 1 are arbitrary. The proof of (6.43) is simple, and follows most easily from Stone’s formula which relates the spectral resolution E./ of a self-adjoint operator to the resolvent, i.e., ˝
Z 2 Cı ˛ 1 f; E Œ1 ; 2 / g D lim lim 2 i ı!0C "!0C 1 Cı ˝ f; .H . C i"//
1
.H
.
i"//
1
˛ g d ;
see [59]. For the resolvent (Green function) one has the explicit expression in terms of ; , viz. if 0 < x < x 0 and Im z > 0, say, then .H
z/
1
.x; x 0 / D .x; z/ .x 0 ; z/
and symmetrically if 0 < x 0 < x. By means of this formula, Stone’s representation, and (6.42), one obtains (6.43) by explicit calculation. For the free half-line problem (i.e., if V D 0) the reader will easily check that 1 1 m.z/ D iz 2 , ./ D 1 2 d. In that case (6.43) is nothing but the standard Plancherel theorem for the Sine-transform on the half-line. In general, note that (6.43) extends by unitarity to all of L2 .0; 1/ , respectively L2 .R; /. In particular, it encompasses the entire spectrum including eigenvalues. We now turn to developing these ideas for the full line, which is again standard. The main difference is that one is now dealing with multiplicity two, and a spectral measure which therefore takes its values in the nonnegative 2 2-matrices. For the full-line we need to assume that both ends ˙1 are limit-point. Define ˛ .x; x0 I z/, ˛ .x; x0 I z/ to be the fundamental system of solutions of L
Dz ;
z2C
212
6 Further developments of the theory
so that ˛0 .x0 ; x0 I z/ D
˛ .x0 ; x0 I z/ D ˛0 .x0 ; x0 I z/
sin ˛;
(6.44)
D ˛ .x0 ; x0 I z/ D cos ˛
where x0 2 R and ˛ 2 Œ0; /. Their Wronskian is W .˛ ; ˛ / D 1 : The Weyl–Titchmarsh solutions are defined as the unique solutions ˙;˛ .; x0 I z/ 2 L2 .Œx0 ; ˙1/; dx/ for z 2 C n R which satisfy the boundary condition 0 ˙ .x0 ; x0 I z/ sin ˛
C
˙ .x0 ; x0 I z/ cos ˛
D 1:
This boundary condition ensures that ˙;˛ .x; x0 I z/
(6.45)
D ˛ .x; x0 I z/ C m˙;˛ .z; x0 / ˛ .x; x0 I z/
and the Wronskian W.
C .; x0 I z/;
.; x0 I z// D m
;˛ .z; x0 /
mC;˛ .z; x0 / :
The Weyl–Titchmarsh functions m˙;˛ are Herglotz functions, and the associated Weyl–Titchmarsh matrix 3 2 1 m ;˛ .z;x0 /CmC;˛ .z;x0 / 1 M˛ .z; x0 / WD 4
m 1m 2m
;˛ .z;x0 /
mC;˛ .z;x0 /
2m
;˛ .z;x0 /CmC;˛ .z;x0 / ;˛ .z;x0 /
mC;˛ .z;x0 /
m m
;˛ .z;x0 /
mC;˛ .z;x0 /
;˛ .z;x0 /mC;˛ .z;x0 / ;˛ .z;x0 /
5
(6.46)
mC;˛ .z;x0 /
is a Herglotz matrix. This implies that there exists a nonnegative 2 2-matrix-valued measure ˝˛ .d; x0 / so that the representation Z h 1 i M˛ .z; x0 / D C˛ .x0 / C ˝˛ .d; x0 / z 1 C 2 R
(6.47) Z
˝˛ .d; x0 / C˛ .x0 / D C˛ .x0 / ; <1 1 C 2 R holds. The measure ˝˛ satisfies ˝˛ .1 ; 2 ; x0 D
1
lim
lim
Z
2 Cı
ı!0C "!0C 1 Cı
Im M˛ . C i"; x0 / d :
(6.48)
The matrix measure ˝˛ plays the role of the spectral measure, as can be seen from the following Fourier representation relative to L.
213
6.2 The one-dimensional NLKG equation
Proposition 6.7. Let ˛ 2 Œ0; /, f; g 2 C01 .R/, F 2 C.R/\L1 .R/, and 1 < 2 . Then f; F .L/EL .1 ; 2 g L2 .RIdx/ D .fO˛ .; x0 /; .F .1 ;2 /./b g ˛ .; x0 //L2 .R;˝˛ .;x0 // Z T fO˛ .; x0 / ˝˛ .d; x0 / b g ˛ .; x0 / F ./ D .1 ;2
(6.49)
where Z
Z
b h˛;1 .; x0 / D
h.x/˛ .x; x0 I / dx;
b h˛;2 .; x0 / D
R
h.x/ ˛ .x; x0 I / dx R
b h˛ .; x0 / D .b h˛;1 ; b h˛;2 /T .; x0 / : (6.50) This Fourier transform establishes a unitary correspondence between the Hilbert spaces L2 .R; dx/ and L2 .R; ˝0 /. For example, for the free operator (i.e., V D 0) one finds that, with ˛ D 0; x_0=0, 1
0 .x; 0I / D
sin. 2 x/ 1 2
1 m˙;0 .z; 0/ D ˙iz 2 ;
;
1
0 .x; 0I / D cos. 2 x/;
z 2 C n Œ0; 1/; h 12 0 i 1 ˝0 .d; 0/ D .0;1/ ./ 1 d : 2 0 2
(6.51)
This can be seen to be equivalent to the usual Fourier transform on the line, but written as a vectorial Sine, Cosine-transform, with the Fourier variable restricted to the positive half-axis. 6.2.3 The local L2t;x bound for the KG evolution of L˙ We now show that the spectral measure as it appears in (6.48) is more regular near zero energy as compared to the free case, provided that zero energy is not a resonance. This is of course a well-known principle, and lies at the heart of estimates such as (6.38).
214
6 Further developments of the theory
Lemma 6.8. Suppose that L as in Proposition 6.7 has an even Schwartz potential (for convenience) and no zero energy resonance. Then the spectral measure ˝0 is diagonal, absolutely continuous on .0; 1/, and of the form 1 1 ˝0 .d; 0/ D diag O. 2 /; O. 2 / d; ! 0; (6.52) 1 1 ˝0 .d; 0/ D diag O. 2 /; O. 2 / d; ! 1 on > 0. The O./-terms satisfy the natural derivative bounds. Proof. Taking ˛ D 0 and x0 D 0 in (6.44) (and suppressing x0 ) yields 0 .xI z/ D 0 . xI z/;
0 .xI z/ D
0 . xI z/;
;0 .xI z/
D
C;0 .
xI z/
whence m ;0 .z/ D mC;0 .z/ and W .z/ D 2m ;0 .z/. Denote by u0;C .x/ and u1;C .x/ a fundamental system of solutions to Lf D 0 with u0;C .x/ D 1 C O.x
100
u1;C .x/ D x C O.x
100
/
(6.53)
/
as x ! 1. This representation follows from the Volterra integral equations Z 1 u0;C .x/ D 1 C .y x/V .y/u0;C .y/ dy ; Zx 1 u1;C .x/ D x C .y x/V .y/u1;C .y/ dy x
by iteration. In particular, W .u1;C ; u0;C / D 1. Furthermore, u0; ; u1; denote the corresponding solutions, but with x ! 1. By symmetry, u0; .x/ D u0;C . x/ and u1; .x/ D u1;C . x/. Since zero energy is nonresonant, W .u0;C ; u0; / ¤ 0. Perturbatively in , we now obtain from uj;C .x/ unique eigenfunctions uj;C .x; / satisfying Luj;C D uj;C , as well as for small and jx 2 j 1, uj;C .x; / D uj;C .x/ 1 C O.x 2 / ; j D 0; 1 : Indeed, uj;C .x; / are given in terms of the Volterra equations Z x uj;C .x; / D uj;C .x/ C u0;C .x/u1;C .y/ u0;C .y/u1;C .x/ uj;C .y; / dy : 0
Similarly, the Jost solutions fC .x; / defined by Lf˙ .; / D f˙ .; /, f˙ .x; / 1
' e ˙ix 2 as x ! ˙1, satisfy 1
f˙ .x; / D e ˙ix 2 1 C O.x
100
/ ;
˙x 1 :
215
6.2 The one-dimensional NLKG equation
As usual, this follows from the Volterra representation of these functions. Moreover, one has f˙ .x; / D a˙ ./u0;˙ .x; / C b˙ ./u1;˙ .x; / with W f˙ .; /; u1;˙ .; / ;
a˙ ./ D
b˙ ./ D W f˙ .; /; u0;˙ .; / :
Therefore, for any small " > 0 a˙ ./ D 1 C O.1 " / ;
(6.54)
1
b˙ ./ D i 2 C O.1 " / as ! 0. In conclusion, 1 W ./ WD W fC .; /; f .; / D c0 C ic1 2 C O.1 " /
(6.55)
where c0 ; c1 2 R with c0 ¤ 0; this latter nonvanishing is precisely the nonresonant condition at zero energy. The matrix in (6.46) is M0 ./ D diag W ./
1
;
1 W ./ 4
and the measure ˝0 ./ satisfies, for small , by (6.48) 1 1 ˝0 .d/ D diag O. 2 /; O. 2 / d;
! 0:
(6.56)
For large , the free representation (6.51) describes ˝0 to leading order. The regularity of the spectral measure from Lemma 6.8 implies the following improved estimates on the time evolution. These bounds are crucial in the nonlinear analysis near the ground states. Note that (6.57) cannot hold for the free case, i.e., 1 1 2 e i th@x i 2 since the best pointwise decay of the latter evolution is t 2 as can be seen from the pointwise analysis in the final section of Chapter 2. Lemma 6.9. Let L˙ be as above, with p > 3. Then one has the following bounds
hxi
hxi
1
Z
e 1
t
e 1
1
2 1 ˙itL˙
2 ˙i.t s/L˙
Pc .L˙ /f
2 L1 x Lt
Pc .L˙ /f .s/ ds
for all Schwartz functions f on the line.
2 L1 x Lt
1 . h@x i 2 f 2 Z
1
. 1
h@x i 21 f .s/ ds 2
(6.57)
216
6 Further developments of the theory
Proof. We begin with the first. Write w.x/ D hxi 1 . Then by duality we need to estimate, with g D g.t; x/, and F denoting the distorted Fourier transform of Proposition 6.7, Z 1 1 E Z 1 1 ˇ 2 e ˙i tL˙ Pc .LC /f ˇ wg D dt e ˙it.1C/ 2 F f ./T ˝0 .d/F wg.t/ ./ 1 0 Z 1 Z 1 Z 1 1 ˙it.1C/ 2 F1 f ./1 ./ D dt e w.x/g.t; x/.x; / dx d 1 0 1 Z 1 Z 1 Z 1 1 C dt e ˙it.1C/ 2 F2 f ./2 ./ w.x/g.t; x/ .x; / dx d
D
0
1
1
(6.58)
where ˝0 .d/ D diag.1 ; 2 / d, and ; are 0 .x; 0I z/, 0 .x; 0I z/ from above. Here denotes the spectral variable of L D L˙ 1. For small one has 1 1 .1 C / 2 D 1 C C O.2 / : 2 Up to change of variable in (which we ignore) we can therefore view the small integral as the usual Fourier transform. Denoting the Fourier transform of g.t; x/ in time by b g .; x/ we can bound the contribution of small to (6.58) by means of (6.56) as follows: 1
Z 0
Z
1 1
ˇ ˇˇ ˇ ˇFj f ./ˇ ˇb g .; x/ˇ j.x; /j C j .x; /j w.x/dx j ./ d
1 . sup sup w.x/j ./ 2 j.x; /j C j .x; /j 0<<1 x
Z
1
1 ˇ ˇ ˇFj f ./ˇ2 j ./ d 2
0
1
1
Z
1
Z
jb g .; y/j2 d 0
12
dy . kf k2 kgkL1x L2 t
(6.59)
In the final step we used the unitarity of the Fourier transform, both for the free and distorted operators, as well as the fact that uniformly in > 0 sup hxi
1
x
1
1
j.x; /j C j .x; /j . 1 :
For > 1, one has hi 2 ' 2 . Hence, the first component for these -values is
6.2 The one-dimensional NLKG equation
bounded by (we can ignore w.x/ in this regime, as well as , cf. (6.51)), Z 1Z 1 ˇ ˇˇ p ˇ ˇF1 f ./ˇˇb g . ; x/ˇ1 ./ dxd 1 Z 11 Z 1 ˇ ˇˇ ˇ ˇF1 f .2 /ˇˇb . g .; x/ˇ 1 .2 / dxd 1 1 Z 1 Z 1 21 Z 1 ˇ ˇ ˇ2 2 ˇ2 21 ˇ ˇ ˇ . F1 f ./ 1 ./ d b g .; x/ˇ d dx 1
1
217
(6.60)
1
1 . h@x i 2 f 2 g L1 L2 : x
t
The final estimate here follows by the free asymptotics (6.51). For the second inequality in the lemma one proceeds in a similar fashion. In fact, using the notation from the first part of the proof, 1 E ˇ 2 e ˙i.t s/L˙ Pc .L˙ /f .s/ ds ˇ wg 1 Z 1 Z t Z 1 1 D dt ds e ˙i.t s/.1C/ 2 F f .s/ ./T ˝0 .d/F wg.t/ ./ 1 1 1 Z 1 Z 1 Z 1 1 D ds dt e ˙i.t s/.1C/ 2 F1 f .s/ ./1 ./ 1 s 0 Z 1 w.x/g.t; x/.x; / dx d 1 Z 1 Z 1 Z 1 1 C ds dt e ˙i.t s/.1C/ 2 F2 f .s/ ./2 ./ 1 s 0 Z 1 w.x/g.t; x/ .x; / dx d :
DZ
t
1
Carrying out the t-integration, and performing similar arguments as in the previous case, shows that the two final expressions here are Z 1
h@x i 12 f .s/ ds g 1 2 . 2 L L 1
x
t
as desired. By means of a suitable interpolation argument, one now obtains the following result, see [88].
218
6 Further developments of the theory
Corollary 6.10. Any solution of LC u C F; u ? ; Q0
uR D
(6.61)
satisfies
hxi s u 2 1=2 . uŒ0 1 2 C F 1 2 L L H L L H t
x
t
x
(6.62)
with s > 32 . The significance of (6.62) lies with the perturbative nonlinear analysis near the ground states. In fact, placing a nonlinear term such as Qq v 2 in L1t L2x yields
q 2
Q v 1 2 . hxi s v 2 2 4 . hxi s v 2 1 (6.63) L t Lx L t Lx 2 2 L t Hx
for H 1 solutions v. This observation allows one to close all estimates very easily. In addition, one of course also uses Strichartz estimates for the Klein–Gordon equation relative to L (in order to estimate v p , say). Note that unlike Lemma 6.9, the following result has nothing to do with L having a zero energy resonance or not. Corollary 6.11. Let L be as in (6.39). For any Schwartz function u in R1C1 t;x with u D Pc .L/u one has the aprori bound
u p r . uŒ0 1 2 C uR C Lu 1 2 (6.64) L L H L L L x
t
for any 4 < p 1, 0 < 5 p < 1.
1 r
t
1 2
2 . p
x
In particular, one can take r D 2p for any
The proof is a simple application of the Lp -boundedness of the wave operators, see [3], [141] (note that this boundedness property does not require zero energy to be regular) and the Strichartz estimates for the free KG equation, see Exercise 2.45. For all remaining details, we refer the reader to [88].
6.3
The cubic radial NLS equation in R3
In this section as well as the next we turn to other equations than NLKG. More specifically, following [110], we develop similar results for the cubic NLS equation in R3 with radial data, viz. i@ t u
u D juj2 u;
.t; x/ 2 R1C3 :
(6.65)
6.3 The cubic radial NLS equation in R3
219
The local well-posedness of (6.65) in the energy space H 1 is classical, see for example Strauss [133], Sulem, Sulem [135], Cazenave [27], and Tao [136]. One has mass and energy conservation
1
u 2 D const:; 2 2
2 1 4 1
u D const:; E.u/ D ru 2 4 2 4
M.u/ D
(6.66)
where k kp denotes the Lp .R3 / norm. Data with small H 1 norm have globally defined solutions which scatter to a free wave. For the defocusing equation it is known that all energy solutions scatter to zero, see Ginibre, Velo [61]. In contrast, (6.65) is known to exhibit energy data for which the solutions blow up in finite time. In fact, Glassey [62] proved that all data of negative energy are of this type provided they also have finite variance kxuk2 < 1. The latter assumption was later removed for radial solutions by Ogawa, Tsutsumi [113]. Eq. (6.65) possesses a family of special oscillatory solutions of the form 2 u.t; x/ D e i t ˛ Ci Q.x; ˛/ where ˛ > 0 and Q.; ˛/ C ˛ 2 Q.; ˛/ D jQj2 Q.; ˛/ : As in Chapter 2, the ground state is singled out as the unique radial positive solution to this equation. Letting modulation and Galilean symmetries act on these special solutions u.t; x/ generates an eight-dimensional manifold of solitons. In the radial context, the manifold is only two-dimensional. The question of orbital stability of these solitons in the energy space was settled by Weinstein [143], [144], Berestycki, Cazenave [11], and Cazenave, Lions [28]. A general theory which covers this example – as well as many others such as the NLKG equation – was developed by Grillakis, Shatah, Strauss [67]. The cut-off in the power jujp 1 u in the n-dimensional setting turns out to be the L2 critical one p0 D n4 C 1, with p p0 being unstable and p < p0 stable. In particular, the cubic NLS (6.65) is unstable. Holmer, Roudenko [74] showed that for all radial solutions u with mass kuk2 D kQk2 and energy E.u/ < E.Q/ there is the following dichotomy: if kruk2 < krQk2 one has global existence and scattering (as jtj ! 1), whereas for kruk2 > krQk2 there is finite time blowup in both time directions. The radial assumption was then removed in Duyckaerts, Holmer and Roudenko [46]. Note that the mass condition is easily removed by scaling, with M.u/E.u/ being the natural scaling-invariant version of the energy, and with M.u/kruk22 replacing kruk22 . For the subcritical NLKG equation the analogous results were obtained in [77]. In analogy with NLKG, it follows from the variational properties of Q that these regions
220
6 Further developments of the theory
are invariant under the NLS flow. The methods in the three aforementioned papers follow the ideology of Kenig–Merle [84] in order to establish scattering. The second author began the investigation of the conditional4 asymptotic stability problem for focusing dispersive equations in [122] by means of the equation (6.65) but with general, rather than radial, data. While a strong, and noninvariant topology was used there, Beceanu [9] later established the corresponding 1 result in the optimal topology HP 2 .R3 /; in Chapter 3 we prove a special case of Beceanu’s theorem, see Proposition 3.31. In one dimension, Krieger and the second author [90] considered the one-dimensional NLS equation, and established the desired conditional asymptotic stability in that case, but again in a more restrictive topology than the energy space. Finally, we note that the work of Bates and Jones [6] was shown to apply to the NLS equation as well, see Gesztesy, Jones, Latushkin, Stanislavova [58]. A major difference from the NLKG equation is that here one has the “cone” of solitons (under the radial restriction) [ S˛ WD fe i Q.; ˛/ j 2 Rg; S WD S˛ : ˛>0
We set Q D Q.; 1/ for convenience. Then Q.x; ˛/ D ˛Q.˛x/ and M Q.; ˛/ D ˛ 1 M.Q/. The first result from [110] is again the 9-set one. To be specific, let 1 H D Hrad .R3 / and ˚ (6.67) H" WD u 2 H j M.u/E.u/ < M.Q/ E.Q/ C "2 as well as ˚ H"˛ WD H" \ u 2 H j M.u/ D M Q.; ˛/
(6.68)
for any ˛ > 0. Theorem 6.12. There exists " > 0 small such that all solutions of (6.65) with data in H"1 exhibit one of the following nine different scenarios, with each alternative being attained by infinitely many data in H"1 : (1) Scattering to 0 for both t ! ˙1, (2) Finite time blowup on both sides ˙t > 0, (3) Scattering to 0 as t ! 1 and finite time blowup in t < 0, (4) Finite time blowup in t > 0 and scattering to 0 as t ! 1, 4
This refers to the fact that the data are restricted to lie on a manifold of finite codimension.
6.3 The cubic radial NLS equation in R3
221
(5) Trapped by S1 for t ! 1 and scattering to 0 as t ! 1, (6) Scattering to 0 as t ! 1 and trapped by S1 as t ! 1, (7) Trapped by S1 for t ! 1 and finite time blowup in t < 0, (8) Finite time blowup in t > 0 and trapped by S1 as t ! 1, (9) Trapped by S1 as t ! ˙1, where “trapped by S1 ” means that the solution stays in a O."/ neighborhood of S1 relative to H 1 forever after some time (or before some time). The initial data sets for (1)–(4), respectively, are open in H"1 . The set of data in H 1 for which the associated solutions of (6.65) forward scatter, i.e., .1/ [ .3/ [ .6/, is open, pathwise connected, and unbounded; in fact, it contains curves which connect 0 to 1 in H 1 . The theorem applies to solutions of any mass by rescaling. More precisely, if u 2 H"˛ , then the statement remains intact with S1 replaced by S˛ and “trapped” by S˛ now meaning that dist.u; S˛ / . " where the distance is measured in the metric k kH˛1 WD ˛
1
k k2HP 1 C ˛k k22
21
:
(6.69)
As for the NLKG equation, the main ingredient for Theorem 6.12 is a suitable “onepass theorem”. It precludes almost homoclinic orbits which start very close to S1 and eventually return very close to S1 . In combination with an analysis of the hyperbolic dynamics near S1 which results from the exponentially unstable nature of the ground state solution, this allows one to show that the fate of the solution is governed by a virial-type functional K after it exits a neighborhood of S1 . Invoking some finer spectral properties of the Hamiltonian obtained by linearizing the NLS equation around Q, we obtain the following stronger statement which describes in more detail what “trapping” means. As for the NLKG equation, the relevant notion is that of a center-stable manifold. In fact, we constructed such a manifold in Chapter 3 for the NLS equation in the radial energy topology, see Proposition 3.31. For the following theorem it is advantageous not to freeze the mass. In other words, we work with the full set H" . We require the following terminology: Definition 6.13. Let u.0/ 2 H" define a solution u.t/ of (6.65) for all t 0. We say that u forward scatters to S iff there exist continuous curves W Œ0; 1/ ! R and ˛ W Œ0; 1/ ! .0; 1/, as well as u1 2 H such that for all t 0 u.t/ D e i.t/ Q ; ˛.t/ C e
it
u1 C ˝.t/
where k˝.t/kH 1 ! 0 as t ! 1, ˛.t/ ! ˛1 > 0 as t ! 1.
(6.70)
222
6 Further developments of the theory
Note that one then necessarily has M.u/ D M Q.; ˛1 / C M.u1 / D ˛11 M.Q/ C M.u1 /;
2
2 1 1 E.u/ D E Q.; ˛1 / C ru1 2 D ˛1 E.Q/ C ru1 2 2 2 whence (using that E.Q/ D M.Q/ > 0), ku1 k22 kru1 k22 2"2 ; 2M.Q/ E.u/ ; E.Q/
˛11 kru1 k22 C ˛1 ku1 k22 C M.Q/ ˛1 M.u/
(6.71)
(6.72)
and in particular, we conclude that ku1 kH˛1 ", that ˛1 is bounded from both 1 above and below, and that M.u/E.u/ M.Q/E.Q/. The heuristic meaning of (6.70) is simply that u asymptotically decomposes into a soliton e i1 .t/ Q.; ˛1 / plus an H 1 -solution to the free Schrödinger equation (however, the phase 1 is not precisely the one associated with Q.; ˛1 / which 2 C 1 ). In fact, in those cases where one can establish (6.70) it would mean t˛1 is possible to obtain finer statements on and ˛, see [110]. Theorem 6.14. Using the conclusion of Proposition 3.28 one has the following. There exists " > 0 small such that all solutions of (6.65) with data in H" exhibit one of the nine different scenarios described in Theorem 6.12, provided we replace “trapped by S1 ” with “scattering to S". Moreover, each alternative is attained by infinitely many data in H" . The sets .5/ [ .7/ [ .9/ and .6/ [ .8/ [ .9/ are smooth codimension-1 manifolds in the phase space H. Similarly, .9/ is a smooth manifold of codimension two, and it contains S. Using5 the terminology of Chapter 3 we can say that .5/ [ .7/ [ .9/ and .6/[.8/[.9/ are the center-stable manifold Mcs , resp. the center-unstable manifold Mcu , associated with Q – modulo the symmetries given by ˛ and . Since center manifolds are in general not unique it might be more precise to say “a center-stable manifold” here. However, our manifolds are naturally unique for the global characterization in Theorem 6.12. Similarly, .9/ is the center manifold of Q, again modulo the symmetries given by ˛ and . 5
This is somewhat of an abuse of language, since the theory presented there did not include symmetry parameters. However, the Lyapunov–Perron approach as it appears towards the end of that chapter also applies when symmetries are present, see [122], [9] etc. Furthermore, the graph transform is also know to apply in the presence of symmetries, at least for the NLKG equation, see [112].
223
6.3 The cubic radial NLS equation in R3
Every point p 2 S has a neighborhood B" .p/ of size . " relative to the metric (6.69) with ˛ D M.Q/=M.p/, such that B" .p/ is divided by Mcs into two connected components; all data in one component lead to finite time blow-up for positive times, whereas all data in the other lead to global solutions for positive times which scatter to zero as t ! C1. All solutions starting on Mcs itself scatter to S in the sense of (6.70) as t ! C1. As is to be expected by analogy with the NLKG equation and the work of Duyckaerts and Merle [48], the one-dimensional stable and unstable manifolds (up to the symmetries) appear naturally in the form of those solutions found by Duyckaerts and Roudenko [49]. It is important to note that we can therefore completely describe the global (i.e., both as t ! 1 as well as t ! 1) behavior of the stable/unstable manifolds in this setting. Theorem 6.15. Consider the limit " ! 0 in Theorem 6.12, i.e., all the radial solutions satisfying E.u/ E.Q/ and M.u/ D M.Q/. Then the sets (3) and (4) vanish, while the sets (5)–(9) are characterized, with some special solutions W˙ of (6.65), as follows: .5/ D fe i W .t i
.7/ D fe WC .t .9/ D fe
i.t C /
t0 / j t0 ; 2 Rg; t0 / j t0 ; 2 Rg;
.6/ D fe i W . t i
.8/ D fe WC . t
t0 / j t0 ; 2 Rg; t0 / j t0 ; 2 Rg; (6.73)
Q j 2 Rg:
The sets .5/ [ .7/ [ .9/ form the stable manifold, whereas .6/ [ .8/ [ .9/ are the unstable manifold of Q, up to the modulation symmetry. In other words, solutions in .5/; .7/ and .6/; .8/ approach a soliton trajectory in S1 exponentially fast as t ! 1 or t ! 1, respectively. An analogous ˚ statement holds without the mass constraint, but then these sets take the form e i ˛W˙ .˛ 2 .t t0 /; ˛x/ , ˚ i ˚ 2 e ˛W˙ . ˛ 2 .t C t0 /; ˛x/ , resp. e i.t˛ C/ Q.; ˛/ , where ; ˛ vary. As in the case of Theorem 6.14 this again relies on both the spectral description of Proposition 3.28 as well as the center-stable manifold constructed in Theorem 3.31.
6.3.1 Some elements of the proofs for the NLS equation While there are many similarities with the NLKG equation studied in the previous chapters, there are also some essential differences. As is apparent from the statements of the main results, see Theorems 6.12–6.15 above, one major difference lies
224
6 Further developments of the theory
with the presence of the symmetry parameters given by scaling and the modulation of the phase. While the former can be frozen (at least for the orbital stability arguments leading to Theorem 6.12), the latter is removed simply by “modding out” the phase. More precisely, to analyze the dynamics near the ground states one sets u D e i .Q C w/, and then decomposes w further into the discrete modes of the linearized Hamiltonian, plus the dispersive part. Note that the latter also contains the root modes, i.e., those (generalized) eigenfunctions of the linearized Hamiltonian with zero energy. As to be expected, the root modes are essentially removed by means of suitable orthogonality conditions. Variational and linearized structures. To be more specific, we use the functionals E.u/ D kruk22 =2
kuk44 =4;
J.u/ D kruk22 =2 C kuk22 =2 3 K.u/ D kruk22 kuk44 ; 4
M.u/ D kuk22 =2; kuk44 =4;
the first three being the conserved energy, mass, and action, respectively. The functional K results from pairing J 0 .u/ with .xr C rx/u=2, the generator of dilations. By construction, Q is a critical point of J , i.e., J 0 .Q/ D 0 whence also K.Q/ D 0. Moreover, the region M.u/E.u/ < M.Q/E.Q/
(6.74)
is divided into two connected components by the conditions fK 0g and fK < 0g in analogy with the Payne, Sattinger sets P S˙ for NLKG. The quantity ME in (6.74) is scaling invariant and was used by Holmer, Roudenko [74] in their scattering analysis. The aforementioned division into two connected components is intimately linked to the following minimization property. Define positive functional G and I by G.'/ WD J.'/ I.'/ WD J.'/
K.'/ 1 1 2
' 2 2 ; D r'kL 2 C L 3 6 2
K.'/ 1 1 4 2 D 'kL ' L4 : 2 C 2 2 8
As already noted in Chapter 2 one has a variational characterization of the ground state. The only difference here is the functional I.'/ which is needed in the proof of the one-pass theorem.
225
6.3 The cubic radial NLS equation in R3
Lemma 6.16. We have J.Q/ D inffJ.'/ j 0 6D ' 2 H 1 ; K.'/ D 0g D inffG.'/ j 0 6D ' 2 H 1 ; K.'/ 0g D inffI.'/ j 0 6D ' 2 H 1 ; K.'/ 0g; and these infima are achieved only by e i Q.x
c/, with 2 R and c 2 R3 .
The NLKG equation does not admit the scaling symmetry due to the mass term, i.e., the u which is added to u. In contrast, for NLS one does need to consider the scaled family of ground states Q.˛/ D Q˛ WD ˛Q.˛x/ which solve Q˛ C ˛ 2 Q˛ D Q˛3 ; krQ˛ k22 D ˛krQk22 ;
kQ˛ k44 D ˛kQk44 ;
kQ˛ k22 D ˛
1
kQk22 :
Differentiating in ˛ yields . C ˛2
3Q˛2 /Q˛0 D
2˛Q˛ :
To commence with the perturbative analysis, one makes the ansatz u D e i .Q C w/: Inserting this into NLS yields P .u C juj2 u C u/ P D . C /.Q C w/ C jQj2 Q C 2jQj2 w C Q2 w C 2Qjwj2 C w 2 Q C jwj2 w
i wP D e
i
P D .1 C /.Q C w/
Lw C N.w/ ; (6.75)
where N.w/ is the nonlinear part defined by N.w/ D 2Qjwj2 C Qw 2 C jwj2 w and the linearized operator Lw WD
w C w
2Q2 w
Q2 w ;
is considered as an R-linear operator L. It is self-adjoint on L2 .R3 I C/ with the inner product Z hf jgi WD Re f .x/g.x/ dx : (6.76) R3
226
6 Further developments of the theory
The extension of L to a complex-linear operator was used in [110] only for the implementation of the Lyapunov–Perron method. This was already carried out at the end of Chapter 3, see Section 3.5. The point is that the dispersive theory required in the Lyapunov–Perron approach is most conveniently carried out in the matrix formalism corresponding to a complex-linear rendition of L, see Section 3.4.4. The natural symplectic form in this context is Z f .x/g.x/dx D hif jgi ˝.f; g/ WD Im R3
and i L is symmetric with respect to ˝, i.e., ˝.i Lf; g/ D ˝.iLg; f /. Since L is not self-adjoint as a complex operator,it posses generalized eigenfunctions which are not eigenfunctions (in other words, the Jordan form has a nilpotent part). Indeed, i LiQ D 0;
iLQ0 D
iLG˙ D ˙G˙ ;
2iQ;
where > 0, Q0 D @˛ Q˛ j˛D1 D .1 C r@r /Q; and with '; are
G˙ D ' i ;
real-valued. In terms of the real and imaginary values, these equations LC Q0 D
L Q D 0;
2Q;
D ';
L
LC ' D
with L D The existence of ';
C1
Q2 ;
LC D
C1
3Q2 :
is standard and follows from the minimization p ˚˝p ˛ L LC L f jf j kf k22 1 < 0; min
see for example [135] or [122]. Since Q > 0 is the ground state of L . Thus, 2 L 0 and ker.L / D fQg. In other words, hL f jf i & kf kH 1 if f ? Q. After appropriate normalization of .'; /, we have hi iQjQ0 i D
hQjQ0 i D M.Q/; 0
hi GC jG i D 2h'j i D 2hL
j i= D 1;
0
0 D hQjG˙ i D hiQ jG˙ i D h'jQi D h jQ i: Moreover h jQi 6D 0 and so we can choose h jQi > 0. To see this, suppose ? Q, then ' ? LC Q D 2Q3 , and so by Lemma 3.2, 0 hLC 'j'i D h j'i < 0, which is a contradiction.
227
6.3 The cubic radial NLS equation in R3
The symplectic decomposition of L2 .R3 I C/ corresponding to these discrete modes is uniquely given by f D aiQ C bQ0 C cC GC C c G C ; a D hif jQ0 i=M.Q/;
bD
hf jQi=M.Q/;
c˙ D ˙hif jG i :
One has 0 D hjQi D hijQ0 i D hijG˙ i and the symplectic projections onto fiQ; Q0 gi? and onto fG˙ gi? commute. We apply the symplectic decomposition to w. Then, writing D aiQ CbQ0 C one has u D e i .Q C w/ D e i .Q C C GC C G C / : The justification for including the “root”-part (i.e., the zero modes) in follows from a suitable choice of the symmetry parameters ˛; , as we shall see below. The action is expanded as J.u/
J.Q/ D
1 hLwjwi 2
C.w/ D
1 C C hL j i 2
C.w/ ;
(6.77)
where the superquadratic part C.w/ is defined by ˝ ˛ C.w/ D jwj2 wjQ C kwk44 =4 : The following lemma will guarantee the positivity of the component in (6.77). Lemma 6.17. Let f; g 6D 0 be real-valued, radial and satisfy ˝ ˛ ˝ ˛ f j D 0 D gjQ0 : 2 2 Then hLC f jf i ' kf kH 1 and hL gjgi ' kgkH 1 .
We leave the proof as an exercise, see [110]. Modulation theory. The modulation theory needed to implement the 9-set theorem for NLS is considerably easier than the one which was used in the proof of Theorem 3.31. This is due to the fact that while the latter addresses asymptotic stability issues, the former deals with orbital stability. To be precise, one determines ˛; by means of M.u/ D M.Q˛ /;
.uje i Q˛0 / < 0 ;
(6.78)
228
6 Further developments of the theory
where .f jg/ D
R
f .x/g.x/ dx. Note that an explicit solution is given by D Im log.uj
˛ D M.Q/=M.u/;
Q˛0 / :
Since .Q˛ jQ˛0 / D ˛ 2 M.Q/ < 0, there is a unique solution .˛; e i / 2 .0; 1/ S 1 as long as u is close to some e i Q˛ . It is easy to see that u D e i' Qˇ gives ' D and ˛ D ˇ. One advantage of this explicit choice of .˛; / is that M.u/ is conserved in time, and so ˛ is fixed. On the other hand, it is nonlinear in the sense hwjQ˛ i D
M.w/;
hiwjQ˛0 i D 0 ;
but this will be a higher order effect since w is small. One then proceeds by setting ˛ D 1;
M.u/ D M.Q/
and omits ˛ from the notation altogether. As before, one now decomposes everything in the form u D e i .Q C w/;
˝ ˛ ˙ D ˙ iwjG :
w D C GC C G C ;
(6.79)
The parameters governing the hyperbolic dynamics are 1 WD .C C /=2;
2 WD .C
/=2;
E WD .1 ; 2 / ;
whence w D 21 '
2i2
C :
The remainder satisfies the orthogonality condition ˝ ˛
jQ D
1
w 2 ; 2 2
˝ ˛ i jQ0 D 0;
hi jG˙ i D 0
which, by Lemma 6.17, is sufficient for the positivity property ˝
˛ 2 L j '
H 1 :
˝ ˛ i 0 0 The equation of is obtained by differentiating ˝ ˛ 0 D iuje Q D hiwjQ i. Using the equation of w (6.75), as well as w C 2Qjw D 0, one concludes that .P C 1/ M.Q/
˝
˛ ˝ wjQ0 D
˛ Lw C N.w/jQ0 D
˝ ˛ kwk22 C N.w/jQ0 :
6.3 The cubic radial NLS equation in R3
229
The equation for ˙ is obtained by differentiating (6.79). In fact, ˝ ˛ ˝ ˛ P ˙ D ˙ i wj P G D .P C 1/.Q C w/ Lw C N.w/j ˙ G D ˙˙ C N˙ .w/; N˙ .w/ WD hN.w/ C .P C 1/wj ˙ G i; E solves and so P 1 D 2 C N1 .w/; P 2 D 1 C N2 .w/;
˝ ˛ N1 .w/ D N.w/ C .P C 1/wji ; ˝ ˛ N2 .w/ D N.w/ C .P C 1/wj' :
Recall the energy expansion J.u/
J.Q/ D
˛ 1˝ L j C.w/ 2 1˝ ˛ 21 C L j C.w/: 2
C C
D 22
In analogy with Chapter 4 one therefore defines the linearized energy norm as ˝ ˛ ˝ ˛ E 2 C 1 L j D .2 C 2 / C 1 L j ' kvk2 1 ; kvk2E WD jj C H 2 2 2 where we used Lemma 6.17 for the final step. Furthermore, we define the smooth nonlinear distance function in such a way that, still under the mass constraint M.u/ D M.Q/, 2 dQ .u/ ' inf ku
2 e iˇ QkH 1
ˇ 2R
2 dQ .u/ D kwk2E
kwkE =.2ıE / C.w/ if dQ .u/ 1;
where ıE 1 is chosen such that kvkE 4ıE H) jC.v/j kvk2E =2: The smooth cut-off .r/ is equal to one on jrj 1 and vanishes for jrj 2. To see the consistency of the above two properties, let u D e iˇ Q C v be a minimizer for distH 1 .u; S1 / D inf ku ˇ
i
Then hwjiQ0 i D 0 implies that he kvkH
1
e iˇ QkH 1 :
vjiQ0 i D sin.ˇ
& inf jˇ k2Z
/M.Q/, and so
C kj
230
6 Further developments of the theory
as long as v is small. The case of k odd can be eliminated here via the sign in (6.78). Indeed, by the second condition in (6.78), cos.ˇ which excludes that ˇ
/.QjQ0 / > Re.vje i Q0 /
lies near an odd multiple of . Therefore, kwkE ' kwkH 1 . kvkH 1 kwkH 1 :
By the same argument, if u D e iˇ Q C v is an L2 distance minimizer, then jˇ
j . distL2 .u; S1 /;
provided that the right-hand side is small enough. In the region dQ .u/ 1, the distance function dQ enjoys the following properties: 2 kwk2E =2 dQ .u/ 2kwk2E ;
2 dQ .u/ D kwk2E C O.kwk3E /;
2 dQ .u/ ıE H) dQ .u/ D J.u/
J.Q/ C 221 :
Hence as long as dQ .u/ < ıE we have 2 @ t dQ .u/ D 41 P 1 D 42 1 2 C 41 N1 .w/:
In analogy with the results of Chapter 4 one develops eigenvalue dominance, and the ejection mechanism. We only reproduce the ejection lemma from [110], and refer the reader to that paper for everything else. Lemma 6.18. There exists a constant 0 < ıX ıE , as well as constants C ; T ' 1 with the following properties: Let u.t/ be a local solution of NLS in H1 on an interval Œ0; T satisfying R WD dQ u.0/ ıX ; J.u/ < J.Q/ C R2 =2 and for some t0 2 .0; T /, dQ u.t/ R .0 < 8t < t0 /: Then u extends as long as dQ u.t/ ıX , and satisfies 8 t 0 dQ u.t/ ' s1 .t/ ' sC .t/ ' e t R; j .t/j C k .t/kE . R C e 2t R2 ; sK u.t/ & .e t C /R;
6.3 The cubic radial NLS equation in R3
231
where s D C1 or s D 1 is constant. Moreover, dQ u.t/ is increasing for t T R, and jdQ u.t/ Rj . R3 for 0 t T R. As for NLKG, the key to characterizing long-term dynamics is to combine the ejection mechanism with the following variational lemma, which gives lower bounds on jKj. See Figure 4.3 for a schematic description of this lemma. For the definition of the functional I see (2.24). Lemma 6.19. For any ı > 0, there exist "0 .ı/; 0 ; 1 .ı/; 2 .ı/ > 0 such that the following hold: (i) For any u 2 H1 satisfying J.u/ < J.Q/ C "0 .ı/2 , and dQ .u/ ı, we have K.u/ 1 .ı/; or K.u/ min 1 .ı/; 0 kruk22 : (ii) For any u 2 H satisfying I.u/ < J.Q/
ı, we have
K.u/ min.2 .ı/; 0 kruk22 /: The one pass theorem. For the NLS equation, the one-pass theorem takes the following form: Theorem 6.20. There exist 0 < " R 1 with the following property: let u 2 C.Œ0; T /I H/ be a forward maximal solution of (6.65) satisfying M.u/ D M.Q/, J.u/ < J.Q/ C "2 and dQ u.0/ < R for some " 2 .0; " andR 2 .2"; R . Then one has the following either T D 1 and dQ u.t/ < R C R2 for all dichotomy: t 0, or dQ u.t/ R C R2 on t t < T for some finite t > 0. In the latter case, S.u.t// 2 f˙1g does not change on t t < T ; if it is 1, then T < 1, whereas if it is C1, then T D 1. In the Klein–Gordon case, we were able to use the same R for the dichotomy because the distance function was strictly convex in t. For NLS we do not obtain such a strong property, but rather need to allow for oscillations on the order of O.R3 /. Thus, we give ourselves some room (we chose R2 ) to ensure a true ejection from the small neighborhood. As before, the proof of this no-return statement hinges on a suitable virial identity. In fact, we use Ogawa–Tsutsumi’s saturated virial identity [113], (3.5) Z @ t h m ujiur i D 2jur j2 @r m dx juj2 .@r =2 C 1=r/ m dx R3 Z (6.80) 4 juj .@r =2 C 1=r/ m dx; R3
232
6 Further developments of the theory
where the smooth bounded radial function m is chosen as follows: m .r/ D m .r=m/;
0 0 m .0/ D 0 m .r/ 1 D m .0/;
00 m .r/ 0: (6.81)
Notice that with this choice of m , eq. (6.80) is not merely a cut-off of the virial identity, but rather a “smooth interpolate” of the latter with the Morawetz estimate for large jxj. This is indeed crucial for the following arguments, which are slightly more delicate than those in [113]. In analogy with the NLKG equation studied in the previous chapters, the idea is now to combine the hyperbolic structure of the ejection lemma close to the soliton manifold S with the variational structure in Lemma 6.19 away from S, in order to control this virial identity through the functional K.u/. We refer the reader to [110] for the details.
6.4
The energy critical wave equation
We conclude this chapter by presenting the results from [87] on the H 1 -critical, focusing nonlinear wave equation uR
u D juj2
2
u;
u.t; x/ W R1Cd ! R;
2 D
2d d
2
.d D 3 or 5/ (6.82)
in the radial context, where 2 denotes the H 1 Sobolev critical exponent. The dimensional restriction here is needed only for the blow-up characterization by DuyckaertsKenig–Merle [47]. We take the radial energy space as the phase space for the above equation, which can be normalized to L2 by setting uE WD .jrju; u/ P 2 L2radial .Rd /2 DW H p 1 at each time t 2 R, where jrj D is an isometry from HP radial .Rd / onto 2 d Lradial .R /. Thus, to any scalar space-time function u.t; x/, we will associate the vector function uE .t; x/ by the above relation. Conversely, for any time independent 'E D .'1 ; '2 / 2 H, we introduce the following notation ' WD jrj
1
'1 ;
'P WD '2 :
The conserved energy of (6.82) is denoted by Z h 2 juj P C jruj2 E.E u/ WD 2 Rd
juj2 i dx: 2
(6.83)
(6.84)
233
6.4 The energy critical wave equation
It is well-known that this problem admits the static Aubin–Talenti solutions which are of the form W D T W;
h W .x/ D 1 C
jxj2 i1 d.d 2/
d 2
;
(6.85)
where T denotes the HP 1 preserving dilation T '.x/ D d=2
1
'.x/ :
(6.86)
These are positive radial solutions of the static equation jW j2
W
2
W D 0;
(6.87)
which are unique, up to dilation and translation symmetries, amongst the nonnegative, non-zero (not necessarily radial) C 2 solutions, see [24]. They also minimize the static energy Z h i 1 1 2 j'j .x/ dx ; J.'/ WD jr'j2 (6.88) 2 Rd 2 among all non-trivial static solutions. The work of Kenig, Merle [84] and Duyckaerts, Merle [48] allows for a characterization of the global-in-time behavior of solutions with E.E u/ J.W /. In particular, these references construct the onedimensional stable and unstable manifolds associated with W . Note that [84] assumes the role for the energy critical equation which [114] and [77] played in the subcritical context. We wish to study the behavior of solutions with E.E u/ < J.W / C "20 ;
(6.89)
for some small "0 > 0. The key feature of (6.82) by contrast to NLKG is the scaling invariance of (6.82) manifested by d
u.t; x/ 7! 2
1
u.t; x/ D T u.t/
(6.90)
which leaves the energy unchanged. In particular, the analogue of the “one pass theorem” of Chapter 4 needs to be modified, specifically by replacing the discrete set of attractors fQ; Qg there by the one-parameter family of the ground states ˚ S WD ˙ WE >0 :
(6.91)
234
6 Further developments of the theory
Note that for the subcritical cubic NLS equation of [110], the scaling parameter (in frequency) is essentially fixed or at least bounded from above and below by the L2 conservation law, but for the energy critical equation there is no mechanism which a priori prevents the scale from going to 0 or C1. The “virial functional” as in (2.64) takes the form Z jr'j2 j'j2 dx K.'/ WD (6.92) Rd
and satisfies K.W / D 0. The following positivity is crucial for the variational structure around W H.'/ WD kr'k22 =d D J.'/
K.'/=2 :
(6.93)
Note that the derivative of J.'/ with respect to any scaling '.x/ 7! a '.b x/ except for T gives a non-zero constant multiple of K.'/. It is a special feature of the scaling invariant scenario that it suffices to work with a single K, whereas in the subcritical case we needed two different functionals and their equivalence. The main results from [87] can be summarized as follows. Theorem 6.21. There exist a small " > 0, a neighborhood B of SE within O." / distance in H, and a continuous functional ˚ S W 'E 2 H n B j E.'/ E < J.W / C "2 ! f˙1g; such that the following properties hold: For any solution u with E.E u/ < J.W / C "2 on the maximal existence interval I.u/, let ˚ I0 .u/ WD t 2 I.u/ j uE .t/ 2 B ; ˚ I˙ .u/ WD t 2 I.u/ j uE .t/ 62 B; S.E u.t// D ˙1 : Then I0 .u/ is an interval, IC .u/ consists of at most two infinite intervals, and I .u/ consists of at most two finite intervals. u.t/ scatters to 0 as t ! ˙1 if and only if ˙t 2 IC .u/ for large t > 0. Moreover, there is a uniform bound M < 1 such that 2.d C 1/ : d 2 For each 1 ; 2 2 f˙g, let A1 ;2 be the collection of initial data uE .0/ 2 H such that E.E u/ < J.W / C "2 , and for some T < 0 < TC , kukLqt;x .IC .u/Rd / M;
. 1; T / \ I.u/ I1 .u/;
q WD
.TC ; 1/ \ I.u/ I2 .u/:
Then each of the four sets A˙;˙ has nonempty interior, exhibiting all possible combinations of scattering to zero/finite time blowup as t ! ˙1, respectively.
6.4 The energy critical wave equation
235
The neighborhood (or “tube”) B as well as the sign functional S are defined explicitly. In short, every solution u with energy E.E u/ < J.W / C "2 can change its sign S uE .t/ at most once, namely by entering the neighborhood B, whereas u scatters/blows-up if it eventually maintains a fixed sign S D C1= 1. This is the same description as in the subcritical setting of the previous chapters, at least as far as the dynamics away from the ground states S is concerned. More specifically, the assertion about the sign change of the solution appears to be quite robust; it relies on the “one-pass” theorem, which in turn is based largely on energy and virial type arguments (recall that we needed Strichartz estimates to control the concentration of mass around zero energy, see Lemma 4.13, which required the equation to be energy subcritical; however, for the critical equation it turns that this particular issue can be dealt with differently). Unfortunately, it is presently unclear how to analyze solutions u which remain inside the tube B. Note that the blowup solutions constructed by Krieger, Tataru, and the second author [92] for the three-dimensional critical wave equation belong to this tube. We now recall the statement of that result. The local energy relative to the origin is defined as Z 2t C jrj2 C jj6 t; x dx : Eloc .E / D Œjxj
Theorem 6.22 ( [92]). Let > 21 and ı > 0. Then there exists an energy solution u of (6.82) with d D 3 which blows up precisely at r D t D 0 and which has the following property: in the cone jxj D r t and for small times t the solution has the form, with .t/ D t 1 , 1 u.x; t/ D 2 .t/W .t/r C .x; t/ where Eloc E.; t/ ! 0 as t ! 0 and outside the cone u.x; t/ satisfies Z jru.x; t/j2 C ju t .x; t/j2 C ju.x; t/j6 dx < ı Œjxjt
for all sufficiently small t > 0. In particular, the energy of these blow-up solutions can be chosen arbitrarily close to E.W; 0/, i.e., the energy of the stationary solution. Duyckaerts, Kenig, and Merle [47] showed that all type-II blowup (i.e., blowup with uniformly bounded energy norm) under the constraint (6.89) is of the form described by Theorem 6.22, albeit without the explicit expression for .t/. But the tube around S might also contain solutions which do not blow up but rather scatter to S.
236
6 Further developments of the theory
This would correspond to the center-stable manifolds of Chapter 3. In contrast to the subcritical case, we do not address the existence problem of such a center-stable manifold associated with (6.82), nor do we give a complete description of all possible dynamics of solutions satisfying (6.89). Recall that [90] establishes the existence of such a manifold for the radial three-dimensional critical wave equation, but not in the energy class. It appears to be a delicate question in any dimension to decide whether or not a center-stable manifold associated with the ground states exists for energy critical equations. Note, however, that the one-dimensional stable (and therefore also unstable) manifolds were constructed in [48]. For the 4 C 1-dimensional energy critical wave equation, Hillairet and Raphaël [71] recently constructed C 1 type-II blowup solutions. As for subcritical equations, the proof of Theorem 6.21 relies on an interplay between the hyperbolic dynamics of the linearized operator around W with the variational structure of J and K away from S. The linearization around W is somewhat delicate as far as the dynamics is concerned. Indeed, one encounters a time-dependent scaling parameter .t/ which is associated with the zero mode of the linearized operator. This constitutes a major distinction from the analysis of the preceding chapters. Krieger and the authors [87] base their analysis of .t/ on the observation that the evolution of this parameter is much slower than that of the exponentially unstable mode. Indeed, the evolution of .t/ is governed by the threshold eigenvalue or resonance (which lies at zero energy) of the linearized operator and is therefore by nature algebraically unstable rather than exponentially unstable. This allows one to freeze the dilation parameter in those time intervals during which the trajectories are dominated by the hyperbolic (and unstable) dynamics. The other major difference, which could potentially be more serious, is the possibility of concentration blow-up in the region K 0 away from S. Recall that in the subcritical case we argued that all solutions are bounded in the phase space under these conditions and are therefore automatically global. Note that this issue arises after applying the one-pass theorem. Fortunately as well as crucially, the blowup analysis by Duyckaerts, Kenig, and Merle [47] precisely determines all possible blowup in that scenario, placing any blow solution of that type inside the tube B. Since we are dealing with solutions outside of this tube (for example, after ejection), one obtains the desired contradiction. It is not our intention to present an overview, let alone many details, of the precise arguments which [88] developed for the energy critical wave equation leading to the proof of Theorem 6.21. Rather, we content ourselves with a precise statement of the analogue of Lemma 4.8 as needed for that purpose. We remark that the proof presented below differs from the one which appears in [88].
237
6.4 The energy critical wave equation
6.4.1 The variational structure in the energy critical setting Define W to be the critical ground state in Rd , d 3, rescaled by > 0 so that krW k2 does not depend on . Further, Z h i 1 1 2 jruj2 juj dx; J.u/ D d 2 2 ZR (6.94) 2 2 jruj juj dx; K.u/ D Rd
as well as dQS .u/ D inf
Z
>0 Rd
ˇ ˇr.W
ˇ2 u/ˇ dx :
Then one has the following variational property outside of the soliton tube. Lemma 6.23. For each ı0 > 0 there is "1 D "1 .ı0 / > 0 sufficiently small such that if J.u/ < J.W / C "1 and dQS .u/ > ı0 , then we have either ˚ 2 K.u/ > min ı0 ; ckrukL 2 or else K.u/ <
ı0
for suitable ı0 > 0 and some absolute constant c > 0. Proof. By the critical Sobolev imbedding, the statement holds provided kruk2 < c0 where c0 > 0 is some absolute constant. Thus, assume the lemma fails and let P1 fun g1 nD1 H be a sequence with krun k2 > c0 ;
K.un / ! 0;
J.un / < J.W / C
1 : n
(6.95)
Since
1
run 2 C 1 K.un / (6.96) 2 d 2 is bounded in HP 1 \ L2 . By reflexivity of these spaces, J.un / D
we see that fun g1 nD1
run * ru1 ;
un * u1
(6.97)
weakly in L2 and L2 , respectively, as n ! 1 (henceforth, we will tacitly pass to subsequences). By (6.95) we can assume that
kun k22 ! c1 ;
krun k22 ! c1
(6.98)
238
6 Further developments of the theory
where c1 c0 . Since J.W / D d1 krW k22 , we infer from (6.95) and (6.96) that c1 krW k22 . Now assume the following:
(6.99)
ku1 k22 D c1
By the uniform convexity of the unit ball of L2 this is the same as un ! u1 strongly in L2 . We then conclude that u1 ¤ 0, that K.u1 / 0, and that J.u1 / J.W / as well as kru1 k2 krW k2 . If K.u1 / < 0, then K. u1 / D 0 with some 0 < < 1. However, kr. u1 /k2 < krW k2 which contradicts the variational characterization of W . Therefore, K.u1 / D 0 and kru1 k2 krW k2 implies that u1 must belong to the family of minimizers fW g>0 . Moreover, c1 kru1 k22 D krW k22 c1 whence we have krun k22 ! kru1 k22 ;
n!1
and therefore also run ! ru1 strongly in L2 . But this contradicts that dQS .u/ > ı0 . So it remains to prove (6.99). Via a sequence of dilations, we arrange that Z c1 8n jun j2 dx D (6.100) 2 Œjxj<1 and pass to the following weak– limits in the sense of measures:
jun j2 * ;
jrun j2 *
(6.101)
By (6.100), kk WD .Rd / c21 as well as kk c1 . We first prove tightness, i.e., kk D c1 . Suppose kk < c1 . Then there exist > 0 and a sequence Rn ! 1 such that Z jun j2 dx > 8 n : (6.102) jxj>Rn
Let be a standard bump function which D 1 on jxj < 1, and define un D u1n C u2n D .x=n /un C 1 .x=n / un where n < Rn is chosen such that n ! 1 and krun k22 D kru1n k22 C kru2n k22 C o.1/ ;
kun k22 D ku1n k22 C ku2n k22 C o.1/ :
(6.103)
239
6.4 The energy critical wave equation
To accomplish this, one considers a dyadic partition of the set En;N D f2 N Rn < jxj < Rn g for N large and fixed, applied to Z jrun j2 C jxj 2 jun j2 C jun j2 dx (6.104) En;N
which is uniformly bounded in n. Making N D N."/ large, one can find a dyadic shell which makes the integral in (6.104) over that shell < ". This proves (6.103). By (6.102) and Sobolev imbedding, lim inf kru2n k22 > 0 n!1
and from (6.100), lim inf kru1n k22 > 0 : n!1
On the other hand, one has lim inf K.ujn / 0 n!1
for j D 1 or j D 2 (or both). But then we obtain a contradiction since lim sup krujn k22 < krW k22 ; n!1
j D 1; 2 :
Therefore, kk D c1 as desired. Next, suppose u1 D 0. First Z
j un j2 dx C
Z
jr. un /j2 dx
22
(6.105)
where C is the sharp constant in the Sobolev imbedding and is a test function. Passing to the limit here implies that Z
j j2 d C
22 j j2 d
Z
(6.106)
and thus .E/ C ..E// d
d 2
;
8 E Rd
(6.107)
D where E is Borel. It follows that , and D D 0 away from the atoms of , -a.e. Since only 0 can be an atom of by radiality, it follows that D c1 ı0 . However, this contradicts (6.100).
240
6 Further developments of the theory
Therefore, u1 ¤ 0. Set vn WD un
u1 . Then in the weak- sense of measures,
jrvn j2 *
jru1 j2 ;
(6.108)
jvn j2 *
ju1 j2 :
The latter statement here requires the following refinement of Fatou’s lemma: suppose fk 2 Lp is a bounded sequence with 1 p < 1. If fk ! f a.e. on a set U Rd , then p p p lim kfk kL kfk f kL p .U / p .U / D kf kLp .U / (6.109) k!1 Note that we may assume un ! u1 a.e. by the compact Sobolev embedding in Lp on bounded sets with 1 p < 2 . Applying the arguments of the paragraph beginning with (6.105) to vn implies that jru1 j2 C ˛ı0 ;
(6.110)
D ju1 j2 C ˇı0 where 0 ˇ C ˛ 2 2
C c1
2 2
. If ˇ > 0, then also ˛ > 0 and
i 22 hZ 22 d 2 C .R / jru1 j dx C ˛ C Rd Z 2 2 jru1 j2 dx 2 C ˛ 2 > C Rd Z ju1 j2 dx C ˇ D .Rd / D c1
(6.111)
Rd
2
so that C c12
1
> 1. However, by construction c1 krW k22 as well as
krW k22 D kW k22 D C krW k22 2
imply that 1 C c12
1
. This shows that ˇ D 0 and (6.99) holds.
Exercise 6.24. Prove (6.109). Hint: this is due to Brezis-Lieb, see [53], page 11.
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Index
1-parameter groups of symmetries, 30 9-set theorem, see nine set theorem almost homoclinic orbits, 168 asymptotic profiles, 19, 38, 39 Aubin–Talenti solutions, 233 Bahouri–Gérard decomposition, 11, 19, 38, 179 Beceanu’s dispersive estimate, 122, 203 Bernstein’s inequality, 73 Besov spaces Bp;2 , 64, 73 Birman–Schwinger operator, 120 blowup after ejection, 177 center manifold theorem, 5, 90 center/stable/unstable manifolds, 4, 85 Christ–Kiselev lemma, 75 completely integrable, 1, 14 complex formalism, 197, 201 computer experiments, 8 concentration-compactness, 4, 11, 19, 38–43, 47–56, see also Bahouri–Gérard decomposition conservation laws, 2 conservation of free energy, 25 conserved energy, 2, 17 critical element, 37, 50, 56, 181 cutoff w.t; x/, 165 C.v/, 151
embedded eigenvalues, 83 energy critical wave equation, 232 energy estimate, 20 energy landscape, 80–84 energy splitting, 116 energy-subcritical regime, 1, 2, 11, 12, 17 essential spectrum, 124, 125 Euler–Lagrange equation, 31, 33, 46 exercises (star, dagger), 12 explicit soliton, 6, 204 exponential weight, 114 failure of compactness, 39 finite speed of propagation, 18, 38 finite time blowup, 18, 25 focusing, 2 forbidden trajectory, 156 forward scattering region, 10, 27 fractional integration, 71 Fredholm operator, 198 Frobenius theory, 26 G0 functional, 32 Galilei transforms, 196 gap property, 11, 80, 84, 116, 173 global NLKG dynamics, 173 global-in-time dynamics, 189 graph transform, 10, 79, 108, 115, 143, 176, 187, 222 ground state, 3, 29
defocusing, 2 dispersion relation, 64 dispersive Hamiltonian equations, 12 distorted Fourier transform, 209–213 Duhamel formula, 20
Hamiltonian formulation, 2, 24, 103, 198 Hamiltonian vector field, 198 Hartman–Grobman theorem, 85 Herglotz function, 210 homoclinic orbit, 163
eigenmode dominance, 145, 153, 201, 230 ejection lemma, 153, 161, 171, 188, 201, 230 ejection process, 145, 146, 153, 230
induction on energy, 178 infinite dimensional hyperbolic dynamics, 14 instability of the ground state, 31, 35, 81, 122
252
Index
intertwining property, 120 invariant cones, 11, 79 J functional, 3, 29–36 K0 functional, 3, 30–36, 81, 145–148, 158–162 K2 functional, 46, 51, 149, 158–159 K˛;ˇ functionals, 45 Keel–Tao endpoint, 76 Kenig–Merle method, 11, 19, 37, 174, 178 Klein–Gordon (NLKG) equation, 17 Lagrange multiplier, 33 Lagrangian, 1 Laurent expansion, 118 limit-point case, 210 limiting profiles, see asymptotic profiles linear dispersive estimates, 63–77, 117–129 linear profile decomposition, 42, 43, 179, see also Bahouri–Gérard decomposition linearized energy, 151, 229 linearized NLKG, 102 linearized operator LC , 80 Littlewood–Paley decomposition, 54, 64, 70, 72, 73 local conservation law, 61 local energy, 235 local energy density, 61 local well-posedness theory, 17 Lorentz contraction, 195 Lorentz transform, 1, 39, 53, 58, 104, 194 Lorentz transformed wave equations, 60 Lp bound on the wave operators, 120 Lumer–Phillips theorem, 105 Lyapunov–Perron method, 10–14, 76, 79, 80, 108–117, 130–143, 173, 188, 189, 207, 222, 226 minimal energy, 194 modulation of parameters, 11, 76, 123, 130–143, 194–203, 227 momentum, 2, 53, 57 Morawetz estimate, 37, 232 negative energy, 35
negative spectrum, 9 negatively invariant, 88, see also positively invariant nine set (or 9-set) theorem, 6, 12, 173, 175, 191–194, 206, 220, 227 nodal solutions, 29 Noether’s theorem, 2 noncompact group of symmetries, 39 nondegenerate Hessian, 66 nonlinear distance function, 145, 151, 201 nonlinear profile, 42, 47, 48, 180 nonlinear profile decomposition, 43, 48, 178–180 nonlinear Schrödinger equation, 122, 218 nonradial cubic NLKG, 194–203 nonradial scattering, 52 (non)stationary phase, 65 numerical evidence, 5, 8 ODE-type blowup, 25–27, 206 one-dimensional KG equation, 6, 7, 75, 204–218 one-parameter family of ground states, 233 one-pass (or no-return) theorem, 8, 12, 146, 162–171, 174, 189, 191, 201, 233 orthogonality condition, 132, 135, 137, 141, 200, 228 orthogonality of the free energy, 40, 54, 55 Payne–Sattinger criterion, 19 Payne–Sattinger region, 9 Payne–Sattinger regions, 3, 34 Payne–Sattinger theorem, 34–36, 188 Payne–Sattinger well, 31 perturbation lemma, 43 perturbation theory, 19, 43 perturbative analysis near Q, 188 Plancherel identity, 211 Pöschl–Teller potentials, 209 pointwise decay, 42, 64, 119 positively (negatively) invariant, 88, 92, 93 radial cubic NLS in R3 , 130–143, 218–232 regular/singular threshold, 117 resonance, 117 resonance function, 118
Index returning trajectories, 167 Riesz projections, 107, 110, 134 rigidity argument, 19, 51, 57, 179 root space, 124, 134, 206, 227 saddle surface, 9, 34, 80, 81 saturated virial identity, 231 scattering after ejection, 177 scattering to the ground states, 80, 173, 189 Schwarz symmetrization, 32, 34 sign of K0 and K2 , 46, 160 sign S.E u/, 162 singular continuous spectrum, 120 singular thresholds, 117, 121 small data scattering, 19, 21, 49, 60, 188 small Strichartz norm, 43, 169 smooth codimension 1 manifold, 9, 137, 196, 222 Sobolev critical exponent, 232 soliton tube, 235, 237 space-time translations, 39, 53 spectral measure, 118, 211–215 spectrum of the linearized operator, 9, 79, 108, 125, 206 speed of light, 3 stable blowup regime, 13 stable/unstable manifold, 4–14, 81, 85, 88, 89, 106, 116, 122, 147, 173, 188, 189, 206, 223 stationary energy, 3, 11, 18, 29, 188, see also J functional stationary phase method, 65–69 stationary solution, 4, 11, 19, 20, 29–34, 77, 108, 191, 193 Stone’s formula, 211 Strauss’ estimate, 29, 207 Strichartz estimates for KG, 63–77
253
Strichartz estimates for linearized NLS, 125 Sturm oscillation theory, 83, 207 symmetric nonincreasing rearrangement, 34 symplectic decomposition, 198, 200, 227 symplectic form, 2, 198, 226 threshold behavior (Duyckaerts–Merle theory), 13 threshold resonance, 116, 121, 192, 197, 206, 208 threshold solutions, 6, 13, 189, 197, 205 trapped by the ground states, 8, 156–158, 173–175, 186, 193–196, 222 trapped trajectory, 157 traveling waves, 3, 104, 195, 201 T T -method, 70 type-II blowup solutions, 193, 235, 236 unitary correspondence, 213 unstable ground state, see instability of the ground state unstable hyperbolic dynamics, 11 unstable manifold, see stable/unstable manifold vanishing kinetic energy, 169 virial identity, 8, 20, 51, 52, 164, 165, 179, 193, 221, 231–234 Volterra equation, 83, 208, 214 wave operators, 28, 112, 119–121, 218 Weyl criterion, 106 Weyl theory, 210 Weyl–Titchmarsh functions m˙;˛ , 212 Weyl–Titchmarsh solutions, 212 Yajima’s theorem, 112, 119–121
