notes on lecture (01053400) - dispersive equations

Notes on

Lecture (01053400) - Dispersive Equations

Xian LiaoKarlsruhe Institute for Technology

Winter Term 2018/2019

These are short incomplete notes, only for participants of the course Lec-ture (01053400) at the Karlsruhe Institute for Technology, Winter Term2018/2019. Corrections are welcome to be sent to

xian.liao(at)partner.kit.edu.

The following textbooks/lecture notes/articles are recommended:

• T. Cazenave: Semilinear Schrodinger equations.

• F. Linares, G. Ponce: Introduction to nonlinear dispersive equations.

• T. Tao: Nonlinear dispersive equations - local and global analysis.

• J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T.Tao: The theoryof nonlinear Schrodinger equations.

• H. Koch, D. Tataru: Conserved energies for the cubic NLS in 1-d.

If you have more questions about the lecture, you are welcome to comedirectly to

• Xian Liao, Room 3.027, xian.liao(at)kit.edu,

• Zihui He, Room 3.030, zihui.he(at)kit.edu.

1 [February 4, 2019]

Contents

1 Introduction 41.1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 Semilinearity . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . 71.1.4 Conservation laws . . . . . . . . . . . . . . . . . . . . . 91.1.5 Solitary wave . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Completely integrable case . . . . . . . . . . . . . . . . . . . . 121.2.1 Direct scattering transform . . . . . . . . . . . . . . . . 141.2.2 Inverse scattering transform . . . . . . . . . . . . . . . 171.2.3 Transmission coefficient and reflection coefficient . . . . 181.2.4 KdV case . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3 Other related models . . . . . . . . . . . . . . . . . . . . . . . 221.3.1 Madelung transform . . . . . . . . . . . . . . . . . . . 221.3.2 The NLS hierarchy . . . . . . . . . . . . . . . . . . . . 231.3.3 Maxwell equation . . . . . . . . . . . . . . . . . . . . . 241.3.4 Water waves . . . . . . . . . . . . . . . . . . . . . . . . 26

2 Wellposedness 292.1 Strichartz estimates . . . . . . . . . . . . . . . . . . . . . . . . 302.2 L2 theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2.1 Local well-posedness in L2 . . . . . . . . . . . . . . . . 342.2.2 Global well-posedness in L2 . . . . . . . . . . . . . . . 372.2.3 L2 critical case . . . . . . . . . . . . . . . . . . . . . . 39

2.3 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 412.3.1 Sobolev spaces HspRdq . . . . . . . . . . . . . . . . . . 412.3.2 Sobolev spaces W k,ppRdq . . . . . . . . . . . . . . . . . 45

2.4 H1 theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4.1 Local well-posedness in H1 . . . . . . . . . . . . . . . . 462.4.2 Global well-posedness in H1 . . . . . . . . . . . . . . . 472.4.3 The virial space case . . . . . . . . . . . . . . . . . . . 51

3 Large time behaviour 543.1 Virial and Morawetz identities . . . . . . . . . . . . . . . . . . 543.2 Blowup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3 Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59


4 Solitary waves 654.1 A minimiser problem . . . . . . . . . . . . . . . . . . . . . . . 65

4.1.1 Space H1r pRdq . . . . . . . . . . . . . . . . . . . . . . . 65

4.1.2 Compact minimisation . . . . . . . . . . . . . . . . . . 674.1.3 Euler-Lagrangian equation . . . . . . . . . . . . . . . . 684.1.4 Regularity and decay property . . . . . . . . . . . . . . 694.1.5 Classification of minimisers . . . . . . . . . . . . . . . 71

4.2 Concentration compactness . . . . . . . . . . . . . . . . . . . 724.3 Orbital stability . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3.1 A second minimisation problem . . . . . . . . . . . . . 764.3.2 Orbital stability . . . . . . . . . . . . . . . . . . . . . . 80

4.4 Linearized operators . . . . . . . . . . . . . . . . . . . . . . . 824.4.1 Nullspaces . . . . . . . . . . . . . . . . . . . . . . . . . 834.4.2 Coercivity . . . . . . . . . . . . . . . . . . . . . . . . . 854.4.3 Modulational stability . . . . . . . . . . . . . . . . . . 88

4.5 Lyapunov method . . . . . . . . . . . . . . . . . . . . . . . . . 90

A Jost solutions 92A.1 Time independent case . . . . . . . . . . . . . . . . . . . . . . 92A.2 Time dependent case . . . . . . . . . . . . . . . . . . . . . . . 94

B The NLS hierarchy 95

C Conserved energies 97


[17.10.2018]

1 Introduction

In this lecture we will mainly consider the Cauchy problem for the semilinearSchrodinger equation "

iBtu∆u κ|u|p1u,u|t0 u0pxq. (NLS)

Here u upt, xq P C denotes the unknown wave function and t P R, x P Rd,d ¥ 1 denote the time and space variables respectively. κ takes value in t1uand we call the nonlinear Schrodinger equation in (NLS) defocusing if κ 1 (repulsive nonlinearity) and focusing if κ 1 (attractive nonlinearity)respectively. p ¡ 1 is a real constant which plays an important role inthe mathematical theory and if p 3 we call (NLS) the cubic nonlinearSchrodinger equation.

We will consider the wellposedness issue of this Cauchy problem (NLS),the asymptotic behaviour of the solutions (scattering, blowup, solitons, etc.)and the conserved energies for the completely integrable case. We will seethat the results will depend heavily on the space dimension d, the sign ofκ and the nonlinearity exponent p. We will also pay much attention to thefunctional space where the initial data u0 stays in.

The (NLS) equation is a semilinear dispersive equation, which possessessymmetry structures, conservation laws, and special solutions (e.g. solitarywave solutions when κ 1). We are going to explain these basic conceptsin Subsection 1.1.

1.1 Basic concepts

We are going to explain (heuristically) some basic concepts related to (NLS),such as dispersion, semilinearity, symmetries, conservation laws, solitarywaves, in this subsection.

1.1.1 Dispersion

What does dispersion mean? Is the equation (NLS) dispersive? Roughlyspeaking, the dispersion means that “Waves with different frequencies travelat different velocities” and the dispersion property is related to the linearpart of (NLS).


We now give some formal explanation. Let d 1 and let upx, tq be aplane-wave solution

upt, xq eipkxωtq eikpxpωkqtq

of the linear Schrodinger equation iutuxx 0, where k is the wave number(waves per unit length) and ω denotes the (angular) frequency. Then wederive the dispersion relation ω ωpkq k2, such that upt, xq is a travellingwave with the phase velocity cpkq ωpkqk k and the larger k is, thefaster the wave travels, that is, high frequency waves travel much faster thanlow frequency waves! More generally, we take the inverse Fourier transformof the initial data

u0pxq 1?2π

ˆReikx pu0pkqdk,

then by superposition the solution of the linear Schrodinger equation reads(noticing that eikpxcpkqtq is the solution with initial data eikx)

upt, xq 1?2π

ˆReikpxcpkqtq pu0pkqdk.

The fact that various Fourier modes travel at different speeds is consideredto be dispersive phenomenon mathematically.

The well known Korteweg-de Vries (KdV) equation is also a nonlineardispersive equation:

Btu uxxx uux 0, u|t0 u0. (KdV)

The linear part ut uxxx 0 has the phase velocity cpkq k2.There is an obvious example which is not a dispersive equation:

Btu cBxu 0, u|t0 u0, where c constant.

The solution upt, xq u0px ctq travels at constant speed c.

1.1.2 Semilinearity

The nonlinear Schrodinger equation (NLS) is of the semilinear form:

iBtu∆u fpuq, i.e. Btu i∆u ifpuq, u|t0 u0, (1.1)

where the function f depends nonlinearly only on lower order terms: u (noton Btu,∇2u)!

Recall the ODE theory. Consider the ODE of the form

v1 Lv fpvq, v|t0 v0,


where v : R ÞÑ Rd is the unknown, t P R denotes the time variable, L PMdpRqsome linear transform and f : Rd ÞÑ Rd some function. We have the Duhamelformula for the solution

vptq etLv0 ˆ t

0

eptt1qLfpvpt1qq dt1 .

Now we take the Fourier transform in x-variable to the equation (1.1)

Btpupt, ξq i|ξ|2pupt, ξq izfpuqpξq, pup0, ξq pu0pξq.

View ξ as a parameter, then we also have (at least formally) the Duhamelformula for pup; ξq

pupt, ξq eit|ξ|2 pu0pξq i

ˆ t

0

eiptt1q|ξ|2zfpuqpt1, ξq dt1 .

We calculate the inverse Fourier transform of the tempered distributioneit|ξ|

2(t viewed as parameter) now. We consider eit|ξ|

2as the limit of

the Schwartz functions eε|ξ|2it|ξ|2 as ε Ñ 0 pointwisely (and hence in the

tempered distribution sense). Then recalling

F1pe 12|ξ|2q e

12|x|2 , F1pgpaξqq adF1pgqpa1xq,

F1peit|ξ|2q is the limit of p2pε itqq d2 e

|x|2

4pεitq : p2itq d2 e

|x|2

4it . Recalling also

F1ppgphq p2πq d2 g h,

we derive from the above Duhamel formula that

upt, xq Kt u0 i

ˆ t

0

Kts fpups, qqds, Kt : p4πitq d2 ei

||2

4t .

In other words (or by Stone’s theorem), for the selfadjoint operator ∆ onthe Hilbert space H L2pRdq with the domain Dp∆q H2pRdq, there existsa unique strongly continuous unitary group Sptq eit∆ : Kt on H suchthat

d

dt

t0Sptqφ i∆φ, @φ P Dp∆q H2pRdq.

We rewrite the above Duhamel formula as

upt, xq Sptqu0 i

ˆ t

0

Spt sqfpups, qqds, (Duhamel)


where we denote Sptq as the following unitary group on L2pRdq:

Sptqg eit∆g Kt g p4πitq d2

ˆRdei

|xy|2

4t gpyqdy. (St)

It is obvious from (St) that

Sptqu0L8pRdq ¤ |4πt| d2 u0L1pRdq, (1.2)

which implies that if u0 P L1pRdq, then the solution Sptqu0 of the linear

Schrodinger equation decays of rate |t| d2 as |t| Ñ 8. This is exactly the

dispersion phenomenon. Since Sptq is unitary, then

Sptqu0L2pRdq u0L2pRdq. (1.3)

Recall the Riesz-Thorine interpolation theorem:

Theorem 1.1. Let 1 ¤ p0 p1 ¤ 8, 1 ¤ q0 q1 ¤ 8. If T PLpLpjpRdq, LqjpRdqq be the linear operator from LpjpRdq to LqjpRdq with theoperator norm Mj, j 0, 1, then for any 0 ¤ θ ¤ 1,

T P LpLpθpRdq, LqθpRdqq, 1

pθ 1 θ

p0

θ

p1

,1

qθ 1 θ

q0

θ

q1

,

with the operator norm Mθ ¤M1θ0 M θ

1 .

Hence we derive from (1.2)-(1.3) that

Proposition 1.1. Let Sptq be the unitary map defined by (St). Then Sptqis a linear map from Lr

1pRdq to LrpRdq for any r P r2,8s (with 1r 1

r1 1)

such thatSptqu0LrpRdq ¤ |4πt| d

2 dr u0Lr1 pRdq, @t 0. (1.4)

1.1.3 Symmetries

We list here some interesting symmetries for the Schrodinger equation (NLS):

• Time/Space translation symmetry: If upt, xq solves (NLS), then ut0,x0pt, xq upt t0, x x0q, t0 P R, x0 P Rd also solves the Schrodinger equationin (NLS);

• Space rotation symmetry: If upt, xq solves (NLS), then upt,Ωxq, Ω PSOpdq also solves the Schrodinger equation in (NLS);

• Phase rotation symmetry: If upt, xq solves (NLS), then eiωupt, xq, ω P Ralso solves the Schrodinger equation in (NLS);


• Time reversal symmetry: If upt, xq solves (NLS), then upt, xq (umeansthe complex conjugate of u) also solves the Schrodinger equation in(NLS);

• Galilean invariance: If upt, xq solves (NLS), then eipxv|v|2tqupt, x2vtq,

v P Rd also solves the Schrodinger equation in (NLS);

• Pseudo-conformal symmetry for the mass critical case p 1 4d: If

upt, xq solves (NLS), then ei|x|2

4t

td2up1

t, xtq, e

i|x|2

4p1tq

p1tq d2up t

1t ,x

1tq, etc. t ¡ 0

also solve the Schrodinger equation in (NLS);

• Scaling symmetry: If upt, xq solves (NLS), then uλpt, xq 1

λ2p1

up tλ2, xλq,

0 λ P R also solves the Schrodinger equation in (NLS).

[17.10.2018][19.10.2017]

Let us focus on the scaling symmetry for a while: Notice also thatthe scaling includes both linear and nonlinear informations in the nonlinearSchrodinger equation (NLS). Recall that the Sobolev space HspRdq, s P R isdefined to be

HspRdq : tf P S 1pRdq | f2HspRdq :

ˆRdp1 |ξ|2qs|fpξq|2 dξ 8u, (1.5)

and we also define the homogeneous Sobolev norm as

f29HspRdq :

ˆRd|ξ|2s| pfpξq|2dξ. (1.6)

Denote the critical exponent

sc d

2 2

p 1. (1.7)

Let the initial data u0 P HspRdq, then the rescaled initial datum u0,λpxq 1

λ2p1

u0pxλq, λ ¡ 0 has 9HspRdq-norm as follows

u0,λ 9HspRdq λsscu0 9HspRdq.

Heuristically, we then divide the regularity exponent s of the Sobolev spaceHs into three cases:


• s ¡ sc (subcritical case)As λ Ñ 8, u0,λ 9HspRdq Ñ 0 and if the solution u exists on the timeinterval r0, Ts then the rescaled solution uλ exists on the time intervalr0, λ2Ts with λ2T Ñ 8. This is the most favourable situation in well-posedness issue: we can make both the small initial norm and the longtime interval at the same time.

• s sc (critical case)It is easy to see that the 9Hsc-norm is invariant under the scaling:u0,λ 9Hsc pRdq u0 9Hsc pRdq, and as λ Ñ 8 the rescaled existing time

interval is still r0, λ2Ts with λ2T Ñ 8. This is always a unclearsituation.

• s sc (supercritical case)In this case as λ Ñ 8, u0,λ 9HspRdq Ñ 8 as λ2T Ñ 8, that is, thegrowing norm corresponds to longer time interval. Blowup may happenin this situation.

In particular, we are in the L2 (mass)-subcritical/critical/supercritical caseif

0 s ¡ sc, i.e. p ¡ 1 4

d,

and we are in the H1 (energy)-subcritical/critical/supercritical case if

1 s ¡ sc, i.e. p ¡ 1 4

d 2.

In this lecture we take the convention that 1 4d2

8 if d 1, 2. Itseems that we should have well-posedness results in L2 or H1 frameworkwhen p 1 4

dor p 1 4

d2and we will indeed prove this in Section 2.

1.1.4 Conservation laws

Suppose that upt, xq is a smooth and fast decaying solution of the Schrodingerequation in (NLS). We have the following conservation laws a priori:

• Mass conservation law

Mpuqptq ˆRd|upt, xq|2 dx Mpuqp0q. (1.8)

Indeed, we test this Schrodinger equation by u and then take the imag-inary part to obtain (1.8). (Exercise)


• Momentum conservation law

Pjpuqptq Im

ˆRduBxju dx Pjpuqp0q. (1.9)

Indeed, we test the Schrodinger equation by Bxj u and then take thereal part to arrive at (1.9). (Exercise)

• Energy conservation law

Epuqptq 1

2

ˆRd|∇u|2 dx κ

p 1

ˆRd|u|p1 dx Epuqp0q. (1.10)

Indeed, we test the Schrodinger equation by ∆u κ|u|p1u and thentake the imaginary part to get (1.10). (Exercise)

Remark 1.1. The equation (NLS) is the nonlinear Hamiltonian flow of theHamiltonian Epuq with respect to the symplectic form ω:

ωpf, gq Im

ˆRdfpxqgpxq dx .

That is to say, we can write (NLS) as Btu ∇ωEpuq, with ∇ωE defined byωpf,∇ωEq dEpfq.

Here the phase space M is an infinite-dimensional real linear space withthe canonically conjugate coordinates (indexed by x P Rd) qpxq : u ÞÑ Reupxq PSpRdq, ppxq : u ÞÑ Imupxq P SpRdq. Define the Poisson bracket associated toω as follows:

tG,F upuq ˆRd

δF

δp

upxqδG

δq

upxq δF

δq

upxqδG

δp

upxq dx .

Then we can check that tM,Eu tPj, Eu tE,Eu 0. Hence Mpuq, Pjpuqare conservation laws for the Hamiltonian flow (NLS).

Remark 1.2. These above conservation laws obviously hold true for smoothand fast decaying solutions for (NLS). Nevertheless they can also hold forless regular solutions by the approximation argument, e.g. the mass conser-vation law (1.8) will hold for L2-subcritical case with L2 initial data. Theseconservation laws will help us to get global-in-time well-posedness result, seeSubsection 2.2.2 below.

10 [February 4, 2019]

1.1.5 Solitary wave

“A solitary wave is a wave that travels at a constant velocity without chang-ing its shape.” Specially, let eitQpxq be a solitary wave of the Schrodingerequation (NLS), with Qpxq satisfying the elliptic equation

∆QQ κ|Q|p1Q, κ 1, Q P H1pRdq. (1.11)

Then by symmetries in Subsection 1.1.3 we know that the following generalsolitary wave solution travels along the line x x0 2vt:

eiλ2tixvi|v|2tiθQλpx x0 2vtq, θ P R, x0 P Rd, v P Rd, (1.12)

with Qλpxq 1

λ2p1

Qpxλq, 0 λ P R. We have taken κ 1 the focusing

case, otherwise in the defocusing case there exists only trivial solution: Wetest (1.11) with κ 1 by Q to get

0 ¥ ˆRd|∇Q|2 dx

ˆRdp1 |Q|2q|Q|2 dx ¥ 0 ñ Q 0 P H1pRdq.

Let d 1, then (1.11) is an ODE and one has an explicit solution (uniqueup to translation and sign change)

Qpxq p 1

2sech 2

p 1

2x 1

p1 P H1pR1q, sech pxq 2

ex ex.

(1.13)We will see that for any d ¥ 2, in the energy-subcritical case (i.e. 1 p 1 4

d2), there exists a unique positive radial H1 solution (up to translation)

of (1.11) in Proposition 4.5, Subsection 4.1. This unique solution is calledthe ground state and the corresponding solution upt, xq eitQ of (NLS) isthe ground state standing wave and is often called soliton. We will showthe orbital stability result of the solitons in the mass-subcritical case (i.e.1 p 1 4

d) in Subsection 4.3. There are also other solutions (not

necessarily positive or radial) than the ground state for (1.11) when d ¥ 2which are called bound states and we will not discuss them in this lecture.

Remark 1.3. It is easy to see that for any r P r1,8s, the Lr-norm of theinitial datum Q is preserved by the solution eitQ:

eitQLrpRdq QLrpRdq, @t P R .

This phenomenon is totally different from the linear Schrodinger equationwhere the estimate (1.4) shows that Sptqu0LrpRdq, r ¡ 2 vanishes as tÑ 8.Hence the existence of the solitary waves describes a balance between the(linear) dispersion and the nonlinearity and they neither decay nor developsingularities.

11 [February 4, 2019]

[19.10.2017][26.10.2017]

1.2 Completely integrable case

In this section we take d 1 and p 3 in the equation of (NLS): the cubicnonlinear Schrodinger equation (by scaling u ÞÑ 1?

2u)

iBtu uxx 2κ|u|2u, u|t0 u0. (1.14)

We are going to consider the defocusing case κ 1 and study this cubicNLS via its Lax-pair formulation. We view u as the potential in the Laxoperator (see (1.22) below) and hope to solve u by use of the scattering datadefined on the real line associated to this Lax operator. We always assumesufficiently decay condition on u as |x| Ñ 8, say u P SpRq.

Let κ 1. By [Zakharov-Shabat 1972], the cubic nonlinear Schrodingerequation (1.14) can be viewed as the compatibility condition of the two sys-tems

ψx iz uu iz

ψ : Uψ, (1.15)

ψt i

2z2 |u|2 2izu ux2izu ux 2z2 |u|2

ψ : V ψ, (1.16)

where in these two systems z is a parameter, pt, xq are space and time vari-ables, u is some known function and ψ : RRÑ C2 is the unknown vector-valued function. Here the compatibility condition of the above two systemsmeans that

ψxt pUψqt Utψ Uψt Utψ UV ψ

and ψtx pV ψqx Vxψ V ψx Vxψ V Uψ

should be the same, that is,

Ut Vx rV, U s with rV, U s : V U UV. (1.17)

We can check that (1.14) is exactly the compatibility condition (1.17).Equations (1.15), (1.16) and their compatibility condition (1.17) have a

natural geometric interpretation. Indeed, the matrix function Upx, t, zq andV px, t, zq may be considered as local connection coefficients in the trivialvector bundle R2C2 where the space-time R2 is the base and the vector-valued function ψpx, t, zq takes values in the fiber C2 (z can be viewed as a

12 [February 4, 2019]

parameter). Equations (1.15), (1.16) show that ψ is a covariantly constantvector while (1.17) says that the pU, V q-connection has zero curvature. Hencea representation of a nonlinear equation in the form (1.17) is called a zerocurvature representation.

We correspondingly have the Lax-pair formulation of cubic NLS (1.14).We rewrite the system (1.15) into the form of the spectral problem of theself-adjoint Lax operator L (with the domain depending on the potential u,e.g. DpLq H1pRq L2pRq if u P L8pRq) as follows

Lψ zψ, L iBx iuiu iBx

. (1.18)

Then we can replace zψ by Lψ in the system (1.16) to get

ψt Pψ, P 2i

1 00 1

L2 2

0 uu 0

L i

|u|2 uxux |u|2

i

2B2

x |u|2 uBx BxuuBx Bxu 2B2

x |u|2.

(1.19)

The compatibility condition (1.17), i.e. the cubic nonlinear Schrodinger equa-tion (1.14) equals to the following evolutionary equation

Lt rP,Ls, (1.20)

and the operator pair pL, P q is called Lax pair for (1.14).Formally, thanks to the evolutionary equation (1.20), if Lψ zψ then z

is independent of the time since we derive zt 0 from pLψqt pzψqt:

pLψqt Ltψ Lψt rP,Lsψ LPψ PLψ,

pzψqt ztψ zψt ztψ zPψ ztψ PLψ.

This fact is non trivial since L depends on upt, xq and hence its spectrumshould depend on t generally. More precisely we will indeed consider z PR (the continuous spectrum of the self-adjoint operator L) and define thetransmission coefficient T pzq and the reflection coefficient Rpzq associatedto the Lax operator L. We will see that T pzq does not depend on time andhence gives infinite many conservation laws for the cubic NLS (1.14). Onthe other side, the Lax pair (1.20) gives a simple evolutionary equation forthe scattering data Rpt, zq associated to upt, xq: BtRpt, zq 4iz2Rpt, zq. Wewill use the direct transform to get the initial scattering data R0pzq from theinitial data u0pxq and then use the inverse scattering transform to get thesolution upt, xq from the evolved scattering data Rpt, zq. The direct/inverse

13 [February 4, 2019]

scattering transforms give an algorithmic way to solve (NLS). This ideacan be compared with the resolution of the linear Schrodinger equation viaFourier and inverse Fourier transform:

iBtu uxx 0, u|t0 u0,

ñ iBtupξq ξ2upξq 0, u|t0 pu0pξq,ñ upt, ξq eiξ

2t pu0pξq ñ upt, xq F1x puq.

(1.21)

The direct and inverse scattering transform play the same role of Fourierand inverse Fourier transform here, with the scattering data Rpt, zq viewedas the nonlinear Fourier transform of upt, xq. However the direct and inversescattering transform are nonlinear and much involved since it is related tothe nonlinear Schrodinger equation.

1.2.1 Direct scattering transform

Let z P R. In the direct transform step, given the function u upxq PL1pR;Cq, we consider the ODE system (1.15):

ψx iz uu iz

ψ, (1.22)

where x P R is the space variable and ψ : R ÞÑ C2 is the unknown vector-valued function.

We propose reasonable boundary condition (initial value condition) forthe system (1.22). As the known function upxq decays sufficiently as |x| Ñ 8,ψpxq can be approximated by the solution of (1.22) when u 0 at infinity:

ψpxq c1

eizx

0

c2

0eizx

op1q, |x| Ñ 8, (1.23)

where c1, c2 are arbitrary constants. Since z P R, these solutions oscillate atinfinity. This is what defines R as the continuous spectrum for (1.22).

Jost solutionsWe will get in the Appendix A the unique solutions j,1, j,2 for the initial

value problem (1.22) with the following initial value conditions respectively:

j,1px; zq eizx

0

op1q as xÑ 8,

and

j,2px; zq

0eizx

op1q as xÑ 8.

14 [February 4, 2019]

These two solutions are linearly independent and form a 2 2 fundamentalsolution matrix (i.e. the matrix whose columns are independent solutionvectors):

Jpx; zq pj,1px; zq, j,2px; zqq.with normalization condition

limxÑ8

Jpx; zqeizxσ3 I, σ3

1 00 1

.

Indeed, we have detpJpx; zqq 1 0 for all x P R since

d

dxdetpJpx; zqq 0 by (1.22) and detpJpx; zqq Ñ 1 as xÑ 8.

Similarly we can define Jost solutions j,1px; zq, j,2px; zq that are nor-malized in the limit x Ñ 8 and the associated normalized fundamentalsolution matrix

Jpx; zq pj,1px; zq, j,2px; zqq, with limxÑ8

Jpx; zqeizxσ3 I.

Scattering matrixThe two fundamental solution matrices Jpx; zq both satisfy the 2 2

system (1.22):

pJqx iz uu iz

J.

The columns of both matrices span the complete solution space and hencej,1, j,2 can be expressed as the linear combination of j,1, j,2 and viceversa. That is, there exists a matrix S Spzq such that

Jpx; zq Jpx; zqSpzq, detpSpzqq 1, z P R,i.e. j,1px; zq s11pzqj,1px; zq s21pzqj,2px; zq,and j,2px; zq s12pzqj,1px; zq s22pzqj,2px; zq.

This matrix Spzq is the so-called scattering matrix.

Notice the symmetry in the system (1.22): If ψ ψ1

ψ2

solves (1.22),

then

rψ :ψ2

ψ1

σ1ψ, σ1

0 11 0

,

15 [February 4, 2019]

(with f denoting the complex conjugate of f) is also a solution of (1.22).

Hence the jost solution j,1px; zq also solves (1.22) with the asymptotic0eizx

as x Ñ 8. By the unique solvability of the initial value problem

(1.22)-(1.23) we know j,2px; zq j,1px; zq σ1j,1px; zq. Therefore

Jpx; zq pj,1, j,2q σ1pj,2, j,1q σ1Jpx; zqσ1, z P R .

Hence the scattering matrix Spzq Jpx; zq1Jpx; zq also satisfies

Spzq σ1Spzqσ1, z P R .

This, together with detpSpzqq 1 ensures that there exist two complex-valued functions apzq, bpzq such that

Spzq apzq bpzqbpzq apzq

, |apzq|2 |bpzq|2 1, z P R . (1.24)

The inverse scattering matrix Spzq1 apzq bpzqbpzq apzq

.

The quantity T pzq 1apzq is called the transmission coefficient and thequantity Rpzq bpzqapzq is called the reflection coefficient. Here is someexplanation: We know from Jpx; zq Jpx; zqSpzq1 that j,1px; zq apzqj,1px; zq bpzqj,2px; zq and hence the solution apzq1j,1px; zq has thefollowing asymptotics

T pzqeizx

0

as xÑ 8,

eizx

0

Rpzq

0eizx

as xÑ 8, z P R .

If we take eizx as the incoming wave from the right to the left, then afterthe disturbance modelled by the potential upxq there is a reflected wave ofcomplex amplitude Rpzq and a transmitted wave of complex amplitude T pzq.

It is obvious to see that if u 0 then Jpx; zq Jpx; zq eizxσ3 andhence Spzq I, apzq 1, bpzq 0, T pzq 1, Rpzq 0.

The direct transformThe direct transform for the defocusing NLS is a nonlinear mapping

pu P L1pR;Cqq ÞÑ pR : R ÞÑ Cq.The reflection coefficient Rpzq can be viewed as a nonlinear analogue of theFourier transform of upxq. Beals-Coifman (1984’) showed that if u P SpRq(the Schwartz space), then also R P SpRq and sup zPR|Rpzq| ¤ 1. It is alsointeresting to calculate that if u ε1rL,Ls, then Rpzq upzq Opε2q asεÑ 0 if we define upzq ´R e2izxupxqdx ε sinp2zLqz.

16 [February 4, 2019]

1.2.2 Inverse scattering transform

Take the time t into account and view z P R as parameter. Let u upt, xqin the system (1.22) satisfy the defocusing cubic NLS (1.14) and |upt, q|decays to zero sufficiently fast for all t P R. Then the reflection coefficientRpzq Rpt, zq also depends on t and we will see that Rpz; tq evolves intime (almost) in the same way as does the evolutionary upt, ξq for the linearSchrodigner equation (1.21).

We will prove (Exercise.) that the matrices

Wpt, x; zq Jpt, x; zqe2iz2tσ3 , σ3

1 00 1

are simultaneous fundamental solutions of the compatible linear systems(1.15)-(1.16). Hence we deduce

BtJpt, x; zq V pt, x; zqJpt, x; zq 2iz2Jpt, x; zqσ3.

Therefore

BtpSpt; zqq BtpJpt, x; zq1Jpt, x; zqq Jpt, x; zq1BtJpt, x; zqSpt; zq Jpt, x; zq1BtJpt, x; zq 2iz2rSpt; zq, σ3s,

that is, Btapt; zq Btbpt; zqBtbpt; zq Btapt; zq

2iz2

0 2bpt; zq

2bpt; zq 0

.

We finally arrive at the time evolution of the transmission and reflectioncoefficients

apt; zq ap0; zq, bpt; zq e4iz2tbp0; zq,i.e.T pt; zq T p0; zq independent of the time,

and Rpt; zq e4iz2tRp0; zq evolves similarly as (1.21).

(1.25)

Inverse scattering transformWe give the following theorem without proof, which offers a way to recover

the solution of the Cauchy problem (NLS) with κ 1 from the associatedRiemann-Hilbert problem:

Theorem 1.2. Let u0 P L1pR;Cq with R0 : R ÞÑ C as the initial reflectioncoefficient. Then the solution of the Cauchy problem for the cubic nonlinearSchrodinger equation (1.14) with κ 1 is

upt, xq 2i limzÑ8

zM12pz; t, xq,

17 [February 4, 2019]

where the matrix Mpz; t, xq is the solution of the following Riemann-Hilbertproblem: Find a 2 2 matrix Mpz; t, xq such that

• Analyticity - Mpz; t, xq is analytic of z for z P C zR;

• Jump Condition - The continuous boundary values Mpz; t, xq (fromup and below respectively) on the real line z P R are related by

Mpz; t, xq Mpz; t, xq

1 |R0pzq|2 e2izx4iz2tR0pzqe2izx4iz2tR0pzq 1

;

• Normalization - limzÑ8Mpz; t, xq I.

Therefore we can solve the Cauchy problem (1.14) in the defocusing com-pletely integrable case in the following way:

u0pxq upt, xq

R0pzq Rpt, zq

direct scattering transform

e4iz2t

inverse scattering transform

However, the inverse scattering transform step is rather involved and it ishard to say that this machinery can work easier than other methods. Never-theless it offers an algorithm to solve (NLS) and we can derive much informa-tion from the formulation itself, e.g. asymptotic behaviors of the solutions.

[26.10.2017][31.10.2017]

1.2.3 Transmission coefficient and reflection coefficient

As we have seen in Subsection 1.2.1 that the Jost solution j,1 of the ODEsystem (1.15) behaves asymptotically at infinity as follows:

eizx

0

as xÑ 8,

T1pzqeizx

0

T1pzqRpzq

0eizx

as xÑ 8,

18 [February 4, 2019]

the renormalised Jost solution l,1 eizxj,1 solves (1.15) with the followingasymptotic behaviour at infinity:

10

as xÑ 8,

T1pzq

10

T1pzqRpzq

0

e2izx

as xÑ 8.

We derive from the explicit expression of l,1 in (A.27) in the appendix (ifu P L1pR;Cq) that

T1 limxÑ8

pl,1q1pxq 1ˆx1 x2

e2izpx2x1qupx2qupx1q dx 1 dx 2

quartic term in tu, uu hexic term in tu, uu and

T1R limxÑ8

e2izxpl,1q2pxq ˆRe2izx1upx1q dx 1

ˆRe2izx3upx3q

ˆx1 x2 x3

e2izpx2x1qupx2qupx1q dx 1 dx 2 dx 3

quintic term in tu, uu septic term in tu, uu Formally if u ! 1, then

lnT1pzq ˆx1 x2

e2izpx2x1qupx2qupx1q dx 1 dx 2,

and

Rpzq ˆRe2izxupxq dx

?2πpup2zq,

such that we can view the Fourier transform as the “linear” part of the directscattering transform upxq ÞÑ Rpzq. We remark that although the dependenceof R on u is nonlinear, we derive R from u by solving a linear ODE system(1.15).

Invariant transmission coefficientWe can extend the Jost solution j,1px; zq to the closed upper half plane,

with the asymptoticseizx

0

op1qeIm zx as xÑ 8,

T1eizx

0

op1qeIm zx as xÑ 8.

19 [February 4, 2019]

We will focus on the invariant “transmission coefficient” T1pzq which is awell-defined holomorphic function on the closed upper half plane if upxq P S.In the defocusing case, it also provides a holomorphic extension of T pzq with|T | ¤ 1, since there are no zeros of T1pzq (corresponding to eigenvalues ofthe self-adjoint Lax operator) and |T | ¤ 1 on R, T Ñ 1 at infinity.

If upxq P S, then we can expand lnT1pzq as z Ñ 8:

lnT1pzq iMp2zq1 iP p2zq2 iEp2zq3 i8

k4

Hk1p2zqk

where M Mpuq, P P puq, E Epuq are the conserved mass, momentum,energy defined in (1.8)-(1.9)-(1.10) respectively. By the conservation of T pzq,all the coefficients Hk in the above expansion are conserved by the cubic NLSflow and hence we derived infinite many conservation laws from the invarianttransmission coefficient T pzq.

Roughly speaking, Mpuq, Epuq correspond to the L2, H1 regularity ofthe solution u, then whether or not the 2n-th coefficient H2n correspond toits Hn regularity? We have no idea about it. Nevertheless [Koch-Tataru2016 arXiv] (see also [Killip-Visan-Zhang 2017 arXiv]) succeeded in reformu-lating the new conserved energies Espuq from lnT1pzq, which correspondto Hs-norms of the solution of the one dimensional cubic nonlinear focus-ing/defocusing Schrodinger equation, for all s ¡ 1

2. We point out that

the invariant “transmission coefficient” T1pzq is a well-defined holomorphicfunction on the open upper half plane (not necessarily on the real axis) ifu P HspRq, s ¡ 1

2.

The idea is as follows. If Im z ¡ 0, u P S and uHs ! 1, s ¡ 12, then

the leading term in lnT1pzq is T2pzq : ´x1 x2 e

2izpx2x1qupx2qupx1q dx 1 dx 2.

We can make use of Fourier transform upξq 1?2π

´R e

iξxupxq dx to rewrite

T2 as (Exercise)

T2pzq ˆR

i

2z ξupξq¯upξqdξ,

and hence1

πReT2pz

2q 1

π

ˆR

Im z

|z ξ|2 |upξq|2 dξ ,

which is the harmonic function on the upper half plane with the trace |upξq|2on the real axis. By use of the trace formula, if s P p1

2, 0q, then we can

rewrite the Sobolev norm u2Hs

´Rp1 |ξ|2qs|upξq|2 dx (i.e. an integral

on the real axis) in terms of T2 as follows (i.e. an integral on the imaginary

20 [February 4, 2019]

axis):

u2Hs 2 sinpπsq

π

ˆ 8

1

pτ 2 1qsReT2piτ2qdτ, 1

2 s 0.

Therefore in order to show the global-in-time boundedness of the Sobolevnorm uHs of the solution u to cubic (NLS) when u0Hs ! 1, it sufficesto verify that the integral of the remainder term

´ 81pτ 2 1qsRe plnT1

T2qp iτ2 qdτ is much smaller than u02Hs (indeed of order u04

Hs). For generalinitial data u0 which are not necessarily small, we use the symmetry prop-erty to get a global-in-time bound (depending nonlinearly on u0Hs) for thesolution.

[31.10.2017][02.11.2018]

1.2.4 KdV case

The Korteweg-de Vries equation Btu uxxx 6uux 0 (with the unknownfunction u : RR ÞÑ R) is also a completely integrable system and theassociated Lax pair (i.e. the pair of operators L, P such that Lt rP,Ls) forKdV is given by

LKdV B2x u, PKdV 4B3

x 3puBx Bxuq.The scattering transform associated to KdV is defined via the spectral

problem of the self-adjoint operator LKdV :

LKdVϕ z2ϕ.

LKdV has continuous spectrum R and may have isolated (but possibly in-finite) negative eigenvalues together with a resonance at frequency 0.

We can rewrite LKdVψ z2ψ in the form of the ODE system (1.15):

ψx iz 1u iz

ψ : UKdVψ, (1.26)

where ψ pϕ, ϕx izϕqT . Then for z P R, we consider the Jost solution j,1

with the following asymptotics at infinity:eizx

0

as xÑ 8,

T1pzqeizx i2zT1pzqRpzqeizx

T1pzqRpzqeizx

as xÑ 8,

21 [February 4, 2019]

and similarly the Jost solutions j,2, j,1, j,2. If u satisfies KdV equation,then Tt 0 and Rt 8iz3R.

If there are eigenvalues z2n 0 with Im zn ¡ 0 (a different situation from

defocusing cubic NLS where there are no eigenvalues for the Lax operator),then apznq T1pznq 0 (i.e. zn are poles of T pzq) and j,1px; znq Cnj

,2px; znq for some constant Cn bpznq. We define the normalizing co-

efficients Cn bpznqa1pznq and the scattering data are tpznqNn1, pCnqNn1, Rpzqu. If

u solves KdV equation, then Cnptq Cnp0qe8iz3nt Cnp0qe8pIm znq3t where thepoles zn of T are invariant under time evolution.A few words on the focusing cubic NLS case. The focusing cubic NLSis also completely integrable, with the ODE system (1.15) replaced by

ψx iz uu iz

ψ.

Then the corresponding non self-adjoint Lax operator may have isolatedinfinitely many eigenvalues on the upper half plane which may accumulateon the real axis even for Schwartz potentials u. Consider the specific initialdata such that R0pzq 0 and N 1 (only one discrete eigenvalue for theLax operator), then the solution of the focusing cubic NLS reads explicitlyas

2τe4iτ2teip2ξx4ξ2tqeipφ1π2q sech p2τpx 4ξtq x1q,where z1 ξ iτ , c2

1 iC1p0q, c1 |c1|eiφ1 , x1 ln |c1|2τ . This is asoliton solution (1.13) (keeping in mind (1.12) and that we make the changeof variable u ÞÑ 1?

2u to derive the focusing cubic NLS iBtu Bxxu 2|u|2u

here), with the velocity 4ξ and the amplitude 2τ .

A little history. [Zabusky & Kruskal 1965] discovered first numericallythe solitary wave solution to the KdV equation, which interacted “elasti-cally” with another such solution. Shortly after this discovery, the pioneers[Gardner-Greene-Kruskal-Miura 1967] related the resolution of KdV to thedirect and inverse scattering transforms. [Lax 1968] considerably generalizedthese ideas. [Zakharov & Shabat 1972] showed that the inverse scatteringmethod also worked for the nonlinear Schrodinger equation. [Ablowitz-Kaup-Newell-Segur 1973] developed a method to find a large class of evolutionaryequations solvable by these techniques.

1.3 Other related models

1.3.1 Madelung transform

Performing the so-called Madelung transform upt,?2xq aρpt, xqeiφpt,xq

(this is possible if u is away from zero), we obtain the following system for

22 [February 4, 2019]

the unknown pρ, vq with v ∇φ from (NLS):#Btρ div xpρvq 0,

Btv v ∇xv ∇xpκρ p12 q ∇x

∆?ρ

2?ρ

.

(1.27)

(Exercise.) The above system is referred to as the hydrodynamic form of(NLS) because of its similarity of the compressible Euler system. The ρ-equation can be viewed as the continuity equation and the v-equation canbe viewed as the momentum equation with an additional quantum pressure∇x

∆?ρ

2?ρ

. The system (1.27) is also called quantum Euler equation or Euler-

Korteweg equation in the theories of quantum fluids and Korteweg fluids.

1.3.2 The NLS hierarchy

Let d 1. Recall the conserved mass M , momentum P , energy E and theHamiltonian formulation of (NLS) in Subsection 1.1.4.

Consider the following Hamiltonians in the NLS hierarchy (see AppendixB)

H0 ˆR|u|2 dx ,

H1 1

2

1

i

ˆRuBxu dx 1

2Im

ˆRuBxu dx ,

H2 1

4

ˆR

|Bxu|2 |u|4 dx ,

H3 1

8Im

ˆR

BxuBxxu 3|u|2uBxu

dx ,

The even ones are even with respect to complex conjugation and have apositive definite principal part, which are referred to as energies. The oddones are odd if we replace u by u, which are referred to as momenta. Withrespect to the symplectic form ωpf, gq Im

´R fg dx , these Hamiltonians

generate the corresponding Hamiltonian flows as follows:

H0 generates the phase shifts : Bθu ∇ωH0puq 2iuñ u e2iθu0;

H1 generates the group of translations : upx aq ea∇ωH1upxq;

H2 generates the defocusing cubic NLS flow : 2iBtu Bxxu 2|u|2u;

H3 generates the defocusing mKdV flow : 4Btu Bxxxu 6|u|2Bxu,

These Hamiltonian flows are commuting and tHj, Hku 0.

23 [February 4, 2019]

1.3.3 Maxwell equation

Vacuum caseThe propagation of an electromagnetic wave (e.g. a laser pulse) in vacuumis governed by Maxwell’s equations:

curl E BtB, curlH BtD, divD 0, divB 0, (1.28)

where t P R, x P R3 are the time and space variables, E ,H : RR3 Ñ R3

are the electric and the magnetic fields respectively, and D,B : RR3 Ñ R3

are the electric and magnetic induction fields respectively. In vacuum, D,Bare related to E ,H by the constitutive relations

D ε0E , B µ0H,

where ε0, µ0 are vacuum permittivity and permeability respectively.Then we derive the wave equation for the electric field:

B2t Ej c2∆Ej 0, j 1, 2, 3,

where c pε0µ0q 12 is the speed of light in vacuum. (Exercise.) We look

for the time-harmonic solution of the form

Ejpt, xq eiω0tEpxq c.c. , with “c.c.” standing for the complex conjugate,

where Epxq satisfies the scalar linear Helmholtz equation

∆E k20E 0, with k2

0 ω2

0

c2.

We look for solutions of the Helmholtz equation of the form

Epxq eik0x3ψpxq,where ψ is the electric-field envelope (or amplitude) and satisfies

Bx3x3ψ 2ik0Bx3ψ pBx1x1 Bx2x2qψ 0.

In the paraxial approximation (i.e. ψ is slowly-varying in x3-direction, com-pared with the carrier oscillation eik0x3), we can neglect Bx3x3ψ1 (mathe-matically questionable) and the above equation for ψ becomes the linear

1Consider the plane waves E Eceik1x1ik2x2ik3x3 with k21k

22k

23 k20 for the scalar

linear Helmholtz equation, such that ψ Eceik1x1ik2x2ipk3k0qx3 . Then for paraxial

plane waves k3 k0 ! k0,

|ψ33|

|k0ψ3|

pk0 k3q2

k0|k0 k3||k0 k3|

k0! 1;

|ψ33|

|ψ11 ψ22|pk0 k3q

2

|k20 k23|

|k0 k3|

pk0 k3q! 1.

24 [February 4, 2019]

Schrodinger equation

2ik0Bx3ψ pBx1x1 Bx2x2qψ 0.

[02.11.2018][09.11.2018]

Dielectric mediumWhen an electric field applies on a dielectric medium, it induces an addi-tional electric field, which is called the polarization field. Hence the electricalinduction field D in a dielectric medium becomes the sum of the originalelectric field and the polarization field:

D ε0E P .For simplicity let us assume that E ,D,P P R (that is, the electric field islinearly polarized: pE , 0, 0q). At low intensities, the dependence of P on E islinear:

P P lin ε0χp1qpω0qE ,

where χp1q is the first-order optical susceptibility. Hence the scalar Helmholtzequation in a linear dielectric becomes

∆E k20E 0, with k2

0 ω2

0

c2n2

0, (1.29)

where n0 a

1 χp1q is the (linear) index of refraction of the medium.As E increases, the dependence becomes nonlinear and in the weakly non-

linear regime we have

P P lin Pnl , Pnl χp3qpω0qE3 ! P lin ,

where χp2jq 0 in isotropic materials. Let E eiω0tE c.c., then (weneglect the part with frequency 3ω0)

Pnl χp3q3|E|2Eeiω0t c.c.

3χp3q|E|2E : 4ε0n0n2|E|2E ,where we defined the Kerr coefficient

n2 3χp3q

4ε0n0

.

Therefore,

D ε0n2E , with n2 n2

0

1 4n2

n0

|E|2.25 [February 4, 2019]

Hence if we formally replace n20 by n2 in the scalar nonlinear Helmholtz

equation (1.29), we get the scalar nonlinear Helmholtz equation for the prop-agation of a linearly-polarized laser beam in a Kerr medium (E does notnecessarily satisfy the wave equation n2B2

t E c2∆E 0)

∆E k2E 0, with k2 k20

1 4n2

n0

|E|2.We substitute E eik0x3ψ into the above equation and apply the paraxialapproximation (ψx3x3 ! k0ψx3), to derive the nonlinear Schrodinger equation(NLS) for ψ:

2ik0Bx3ψ pBx1x1 Bx2x2qψ k20

4n2

n0

|ψ|2ψ 0. (1.30)

(Exercise: Use the symmetry property to rewrite the above equation intothe standard form (NLS). )

To conclude, the above (NLS) is the leading order model for paraxialpropagation of intense linearly-polarized continuous wave laser beams in ahomogeneous Kerr medium, in which ψ is the slowly-varying amplitude ofthe electric field, x3 is the direction of propagation.

1.3.4 Water waves

This subsection can be found in [Zakharov 1968]. We consider the potentialflow of an ideal fluid in the domain tpx, y, zq | px, yq P R2, z P p8, ηsu, in thepresence of a gravity field. Let η ηpx, y, tq be the shape of the surface ofthe fluid and let φ φpx, y, z, tq be the hydrodynamic potential (that is, thevelocity field v pBx, By, BzqTφ). Denote simply ∇ ∇x,y, ∆ ∆x,y on thexy-plane. Then the dynamic of the ideal fluid is governed by the Laplace’sequation for φ:

∆φ Bzzφ 0, px, yq P R2, z P p8, ηq,together with the boundary conditions at the surface (neglecting the surfacetension):

kinetic b.c.: Btη Bz|zηφ∇η ∇φ|zη,dynamic b.c.: Btφ|zη gη 1

2

|∇φ|2 pBzφq2|zη, (1.31)

where g is the gravity acceleration and the condition at infinity:

φÑ 0, as z Ñ 8.

26 [February 4, 2019]

We introduce ψ ψpx, y, tq φx, y, z ηpx, yq, t, then Btψ Btφ

BtηBz|zηφ. Hence the above boundary conditions (1.31) become the evolu-tionary equations for pη, ψq:" Btη Bz|zηφ∇η ∇φ|zη,

Btψ gη 12|∇φ|2 1

2pBzφq2|zη Bzφp∇η ∇φq|zη, (1.32)

where φ solves the Laplace’s equation in R2p8, ηs together with theboundary conditions φ|zη ψ, φ Ñ 0 as |z| Ñ 8. We remark that thesystem (1.32) are Hamiltonian’s equations:

Btη δH

δψ, Btψ δH

δη, (1.33)

where H is the Hamiltonian

H 1

2

ˆR2

ˆ η

8

|∇φ|2 pBzφq2

dz dx dy 1

2g

ˆR2

η2 dx dy ,

and ψ is the generalized coordinate and η is the generalized momentum.We take the Fourier transform of η, ψ on the xy-plane, and linearize (1.32)

under the small amplitude assumption :" Btη |k|ψ 0,

Btψ gη 0,(1.34)

where we used the expansion for φ: φpkq e|k|zψpkq Op|η|q. Therefore wederive the dispersion relation

ω ωpkq ag|k|, k pkx, kyq.

We introduce a apkq 1?2

ηpkqpωpkq|k| q

12 iψpkqp |k|

ωpkqq12

2 to diagnize the

linear equations (1.34):

Btapkq iωpkqapkq 0.

Then Hamilton’s equation (1.33) becomes a single equation Btapkq i δHδapkq ,

where H ´R2 ωpkqapkqapkqdk Op|a|3q, and we write (not obviously)

Btapkq iωpkqapkq i

ˆ V pk, k1, k2qapk1qapk2qδpk k1 k2q

2V pk1, k, k2qapk2qapk1qδpk k1 k2q V pk, k1, k2qapk1qapk2qδpk k1 k2qdk1dk2

i

ˆW pk, k1, k2, k3qapk1qapk2qapk3qδpk k1 k2 k3qdk1dk2dk3 Op|a|4q,

2Then apkq 1?2

ηpkqpωpkq|k| q

12 iψpkqp |k|

ωpkq q12

.

27 [February 4, 2019]

where V pk, k1, k2q, W pk, k1, k2, k3q depend on their arguments explicitly (de-tails omitted here) and δ is the delta function.

Under the small amplitude assumption we write apkq as

apkq Apk, tq fpk, tqeiωpkqt,where Apk, tq changes slowly in comparison with f while |f | ! |A| ! 1.Assuming A constant when f varies, we integrate the equation of a withrespect to the time to arrive at the following expression for f (up to |A|3):

f

ˆ V pk, k1, k2q

exp itωpkq ωpk1q ωpk2q

ωpkq ωpk1q ωpk2q

Apk1qApk2qδpk k1 k2q

2V pk1, k, k2qexp it

ωpkq ωpk1q ωpk2q

ωpkq ωpk1q ωpk2q

Apk2qApk1qδpk k1 k2q

V pk, k1, k2qexp it

ωpkq ωpk1q ωpk2q

ωpkq ωpk1q ωpk2q

Apk1qApk2qδpk k1 k2qdk1dk2.

In the evolutionary equation for A we retain only the terms proportional toAf which contain the most slowly varying exponents, such that

BtA iˆT pk, k1, k2, k3qδpk k1 k2 k3q

exp itωpkq ωpk1q ωpk2q ωpk3q

Apk1qApk2qApk3qdk1dk2dk3,

where T depends explicitly on its arguments (details omitted here).Under the narrow wave packet assumption: |ξ| |k k0| ! |k0|, we

expand ωpkq in powers of ξ pξ1, ξ2q (ξ1, ξ2 are the projections of the vectorξ along and perpendicular to the vector k k0 respectively) around k0 asfollows:

ωpkq ωpk0q ξ1cg 1

2pλ1ξ

21 λ2ξ

22q ,

cg Bk|kk0ω, λ1 B2k|kk0ω, λ2 cg

k0

.

Finally we introduce

bpk, tq Apk, tqeipξ1cg 12pχ1ξ21χ2ξ22qqt,

and verify that its inverse Fourier transform bpx1, x2, tq with respect to ξ pξ1, ξ2q satisfies the cubic nonlinear Schrodinger equation (NLS):

Btb cgB1b i

2

λ1B11b λ2B22b

iκ|b|2b. (1.35)

28 [February 4, 2019]

2 Wellposedness

Definition 2.1 (LWP & GWP). The Cauchy problem (NLS)"iBtu∆u κ|u|p1u,

u|t0 u0pxq,

is said to be locally well-posed LWP in HspRdq if for any initial data u0 PHspRdq, there exists a positive time T ¡ 0 and a unique solution u PCprT, T s;HspRdqq of (NLS) such that there exists a neighbourhood U ofu0 in HspRdq and the flow map

Φ : U ÞÑ HspRdq, u0 ÞÑ upt, q

is continuous for any t P pT, T q.We say that (NLS) is globally well-posedness GWP in HspRdq if the

above holds on any time interval rT, T s, T ¡ 0.

Recall the famous Hadamard’s example of the ill-posed Cauchy problemfor the Laplace equation:"

vtt vxx 0,v|t0 0, vt|t0 fpxq.

Exercise. Show the existence and the uniqueness results of the solution forthe above Cauchy problem with the following initial data sequence

pv, vtq|t0 p0, fnq p0, e?nn sinpnxqq Ñ 0 as nÑ 8, in any Ck

b pRq, k P N,

while the continuity of the flow map fails.We will show the well-posedness results in the subcritical cases for (NLS)

in this section. Recalling the Duhamel formula (Duhamel), we would like toapply the fixed point theorem to show the unique existence of the solution u PXT CprT, T s;HspRdqq. The choice of the functional space XT is crucialand we have to make sure that the linear map u0 ÞÑ Sptqu0 is from HspRdq toXT while the (nonlinear) map u ÞÑ ´ t

0Spt t1qpfpuqpt1qq dt1 , fpuq κ|u|p1u

is from XT to XT . Finally we can choose the time T small enough suchthat these maps are contraction mappings and hence the fixed point theoremworks. The mass/energy conservation laws then imply GWP in the L2/H1

framework respectively.

29 [February 4, 2019]

2.1 Strichartz estimates

Recall Proposition 1.1 that the Schrodinger group Sptq maps Lr1pRdq to

LrpRdq, r ¥ 2 with the operator norm |4πt|p d2 drq. It is also obvious from

the definition of Sptq: zSptqgpξq eit|ξ|2pgpξq and the definition of Hs-norm

(1.5) thatSptqgHspRdq gHspRdq, @s P R .

Remark 2.1. (Exercise.)The operator Sptq, t ¡ 0 does not map

• from L2pRdq to LrpRdq or from LrpRdq to L2pRdq for r 2;

• from LrpRdq to LrpRdq for r 2;

• from LrpRdq to Lr1pRdq for any r ¡ 2;

• from HspRdq to Hs1pRdq for s1 ¡ s.

The heat semigroup et∆ At with At p4πtq d2 e||

24t P L1pRdq, t ¡ 0maps from Lr to Lr and from Hs to Hs1, for any r P r1,8s and s1 ¥ s.

We will show more estimates for Sptq in this subsection, namely the well-known Strichartz estimates.

Theorem 2.1. [Strichartz estimates for the Schrodinger semigroup] Let pq, rqbe admissible exponent pair, i.e.

2

q d

r d

2, 2 ¤ q, r ¤ 8, pq, r, dq p2,8, 2q. (2.36)

Then for any admissible exponent pairs pq, rq, prq, rrq, we have the followinghomogeneous Strichartz estimate (for the solution Sptqu0 of the homogeneousproblem iBtu∆u 0, u|t0 u0)

Sptqu0LqtLrxpRRdq ¤ Cpq, r, dqu0L2xpRdq, (2.37)

and the inhomogeneous Strichartz estimate (for the solution´ t

0Sptt1qfpt1q dt1

of the inhomogeneous problem iBtu∆u f, u|t0 0)ˆ t

0

Spt t1qfpt1q dt1LqtL

rxpRRdq

¤ Cpq, r, rq, rr, dqfLrq1

t Lrr1x pRRdq, (2.38)

where 1q 1

q1 1 and the space-time norm LqtLrxpRRdq is defined to be

gLqtLrxpRRdq gpt, qLrxpRdqLqt pRq.

In the above the time definition domain R can be replaced by any time intervalrT, T s, T ¡ 0.

30 [February 4, 2019]

[09.11.2018][14.11.2018]

Remark 2.2. • By virtue of (2.37), if u0 P L2, then Sptqu0 P Lr with

2 ¤ r ¤ 8 if d 1, 2 ¤ r 8 if d 2, 2 ¤ r ¤ 2d

d 2if d ¥ 3,

for almost all t P R and the norm Sptqu0Lr , r ¡ 2 decays faster

than |t| 1q for almost all the time. This indicates the smooth and decay

effects of Sptq. On the other side we know that there exists u0 P L2 andt P R such that Sptqu0 R Lr since the operator Sptq is not a map fromL2 to Lr.

• There are infinite many admissible exponent pairs and we can alwayshave the trivial case pq, rq p8, 2q and the particular case q r 2pd 2qd. If d 1, then q ¥ 4 and

Sptqu0L4tL

8x pRRq Sptqu0L6

t,xpRRq ¤ Cu0L2pRq.

• The equality for the admissible exponent paris can be seen as follows:Let u be a solution of the homogeneous linear Schrodinger equation,then the rescaled solution uλpt, xq up t

λ2, xλq with rescaled initial data

u0,λ upxλq, λ ¡ 0 such that

uλLqtLrxpRRdq λp2q drquLqtLrxpRRdq, u0,λL2

xpRdq λd2 u0L2

xpRdq,

also solves the linear Schrodinger equation. If (2.37) holds for u, thenit also holds for uλ for any λ ¡ 0 and the only possibility is 2

q d

r d

2.

Proof. Step 1 From Lq1

t Lr1

x to LqtLrx for 2 q 8

By use of the estimate (1.4), we know for any 8 ¤ t1 t2 ¤ 8ˆ t2

t1


rxpRRdq

¤ˆ t2

t1

p4π|t t1|qp d2 drqfpt1qLr1x pRdq dt1

Lqt pRq

¤ˆ

Rp4π|t t1|q 2

q fpt1qLr1x pRdq dt1Lqt pRq

.

By the Hardy-Littlewood-Sobolev inequality (Exercise)

g | |αLqpRnq ¤ Cpp, q, α, nqgLmpRnq,1 1

q 1

m α

n, 0 α n, 1 m q 8,

31 [February 4, 2019]

with

n 1, α d

2 d

r 2

q, m q1, p2 q 8q,

we derive from the above inequality that for any 8 ¤ t1 t2 ¤ 8 (t1 t2may be any two functions of t)ˆ t2

t1


rxpRRdq

¤ Cpq, r, dqfLq

1

t Lr1x pRRdq.

Step 2 From Lq1

t Lr1

x to L8t L2x for 2 q 8

We calculate for any t P R, 8 ¤ t1 t2 ¤ 8ˆ t2

t1

Spt t1qfpt1q dt12L2xpRdq

Aˆ t2

t1

Spt t1qfpt1q dt1 ,ˆ t2

t1

Spt t2qfpt2q dt2EL2xpRdq

ˆ t2

t1

ˆ t2

t1

xSpt t1qfpt1q, Spt t2qfpt2qyL2xpRdq dt

1 dt2

ˆ t2

t1

ˆ t2

t1

xfpt1q, Spt1 t2qfpt2qyL2xpRdq dt

1 dt2

ˆ t2

t1

Afpt1q,

ˆ t2

t1

Spt1 t2qfpt2q dt2EL2xpRdq

dt1

¤ˆRfpt1qLr1x pRdq

ˆR|4πpt1 t2q| 2

q fpt2qLr1x pRdqdt2 dt1 by (1.4)

¤ Cf2

Lq1

t Lr1x pRRdq by Hardy-Littlewood-Sobolev inequality.

Step 3 Proof of (2.37) and from L1tL

2x to LqtL

rx by duality

By duality, we derive (2.37) by Step 2

Sptqu0LqtLrxpRRdq sup gLq1

t Lr1x pRRdq

¤1

ˆRxSptqu0, gptqyL2

xpRdqdt

sup gLq1

t Lr1x pRRdq

¤1

ˆRxu0, SptqgptqyL2

xpRdqdt

¤ sup gLq1

t Lr1x pRRdq

¤1u0L2xpRdq

ˆRSp0 t1qgpt1qdt1

L2

¤ Cu0L2xpRdq by Step 2.

32 [February 4, 2019]

Similarly, we can show for any 8 ¤ t1 t2 ¤ 8,ˆ t2

t1


rxpRRdq

sup gLq1

t Lr1x pRRdq

¤1

ˆR

Aˆ t2

t1

Spt t1qfpt1q dt1 , gptqEL2xpRdq

dt

sup gLq1

t Lr1x pRRdq

¤1

ˆ t2

t1

Afpt1q,

ˆRSpt1 tqgptqdt

EL2xpRdq

dt1

¤ CfL1tL

2xpRRdq by Step 2.

If pt1, t2q p0, tq, then we just take the integral intervals p0,8q and pt1,8qfor the variables t1 and t respectivly.Step 4 Proof of (2.38) by interpolation

We have shown in Step 1 and Step 2 that the linear operator

f ÞÑˆ t

0

Spt t1qfpt1qdt1

is bounded from Lq1

t Lr1

x to LqtLrx and from Lq

1

t Lr1

x to L8t L2x. By the log-

convexity of Lp-norms,

gLpθ ¤ g1θLp0 gθLp1 , with

1

pθ 1 θ

p0

θ

p1

,

the above operator is bounded from Lq1

t Lr1

x to LqtLrx if 2 q ¤ q ¤ 8.

Similarly we have shown in Step 1 and Step 3 that the above linearoperator is bounded from Lq

1

t Lr1

x to LqtLrx and from L1

tL2x to LqtL

rx and hence

from Lq1

t Lr1

x to LqtLrx if 1 ¤ q1 ¤ q1 2.

These two cases complete the estimate (2.38) for 2 q ¤ 8.Step 5 Endpoint case q 2, r 2d

d2for d ¥ 3: See [Keel-Tao 1998].

Remark 2.3 (TT argument). We can rewrite the proof in a more elegantway. Let T : L2pRdq ÞÑ LqpR;LrpRdqq be defined as

pTfqpt, xq Sptqfpxq,

then its formal adjoint T : Lq1pR;Lr

1pRdqq ÞÑ L2pRdq is defined as

pT gqpxq ˆ 8

8Sptqgpt, xqdt,

33 [February 4, 2019]

and their composition TT : Lq1pR;Lr

1pRdqq ÞÑ LqpR;LrpRdqq reads as

pTT gqpt, xq ˆ 8

8Spt t1qgpt1, xqdt1.

Then the following a priori estimates are equivalent:

TfLqpR;LrpRdqq À fL2pRdq,

T gL2pRdq À gLq1 pR;Lr1 pRdqq,

TT gLqpR;LrpRdqq À gLq1 pR;Lr1 pRdqq,

where the last inequality for pq, rq admissible exponent pair with 2 q 8is ensured by Step 1 above. Hence we deduce from the facts that

T : L2 ÞÑ LqpR;LrpRdqq, T : Lq1pR;Lr

1pRdqq ÞÑ L2pRdq,

are linear bounded operators that TT : Lq1pR;Lr

1pRdqq ÞÑ LqpR;LrpRdqq islinear bounded operator. By Christ-Kiselev’s Lemma the truncated operatorTT g ´ t

0Spt t1qgpt1qdt1 also maps from Lq

1pR;Lr1pRdqq to LqpR;LrpRdqq

for any two admissible exponent pairs pq, rq, pq, rq with q, q P p2,8q.[14.11.2018][16.11.2017]

2.2 L2 theory

2.2.1 Local well-posedness in L2

Theorem 2.2. [LWP in L2] Let p be an L2-subcritical exponent, i.e. 1 p 1 4

d. Let κ 1. Let u0 P L2pRdq.

Then the Cauchy problem (NLS) is locally well-posed LWP in L2pRdqin the following sense: There exist a positive time T ¡ 0 depending onu0L2pRdq, p, d, and a unique solution u upt, xq defined on the time intervalrT, T s such that

u P XT :!u P CprT, T s;L2pRdqq

u P LqprT, T s;Lp1pRdqq)

with admissible exponent pair pq, p 1q i.e.2

q d

p 1 d

2, 2 ¤ q ¤ 8,

and there exists a neighborhood U of u0 in L2pRdq such that

Φ : U ÞÑ XT 1 , u0 ÞÑ u is Lipschitz continuous for any T 1 T.

34 [February 4, 2019]

Proof. We solve the integral equation (Duhamel) by searching for the fixedpoint of the mapping

Ψ : u ÞÑ Ψpuq Sptqu0 iκ

ˆ t

0

Spt t1q|upt1q|p1upt1qdt1 (2.39)

in the ball of the functional space XT as

XT pRq :!u P XT

urT,T s : uL8prT,T s;L2pRdqq uLqprT,T s;Lp1pRdqq ¤ R)

with R, T to be determined later. We will use the Banach fixed-point theo-rem (contraction mapping theorem) in the complete metric space pXT pRq, rT,T sq, and we shall prove that

• Ψ is a well-defined map in XT pRq with appropriately chosen R, T ;

• Ψ is a contraction map in XT pRq for some small enough T .

Finally we conclude that there is a unique fixed point of Ψ and the flow mapΨ : u0 ÞÑ u is Lipschitzian continuous from a neighborhood U L2pRdq ofu0 to XT pRq.

In the following C will denote some constant depending on p, d whichmay vary from line to line.Step 1 Well-definedness of the map Ψ in XT pRq

By Strichartz estimates in Theorem 2.1, we deduce that

Sptqu0rT,T s ¤ Cpp, dqu0L2x,

and

Ψpuq Sptqu0rT,T s ¤ Cpp, dq|u|p1uLq1 prT,T s;Lpp1q1 pRdqq

¤ Cˆ T

T

|u|p1uq1Lp1p pRdq

dt 1q1

Cˆ T

T

upq1Lp1pRdq dt

1q1

.

Since 1 p 1 4d, we use Holder’s inequality fgL1 ¤ f

Lqpq1g

Lp1

pq1q q1

to deduce

ΨpuqrT,T s ¤ Cu0L2 Cˆ T

T

uqLp1pRdq dt

pqT θ

¤ Cu0L2 CuprT,T sT θ,

35 [February 4, 2019]

with θ 1q1p1 pq1

qq 1

q1 p

q d

4p1 4

d pq ¡ 0.

We choose R 2Cu0L2x

and T1 sufficiently small such that

Cp2Cu0L2qppT1qθ Cu0L2 , i.e. T1 p2Cq pθ u0βL2

x

with β 1 p

θ 4p1 pqdp1 4

d pq 0,

and hence for any T ¤ T1,

if urT,T s ¤ R, then ΨpuqrT,T s ¤ Cu0L2 Cu0L2 R,

and Sptqu0,Ψpuq Sptqu0 are continuous in L2pRdq.Step 2 Contraction map Ψ

Let u, v P XT pRq and we calculate by Strichartz estimate

Ψpuq ΨpvqrT,T s ˆ t

0

Spt t1q|upt1q|p1upt1q |vpt1q|p1vpt1q

dt1rT,T s

¤ C|u|p1u |v|p1v

Lq1 prT,T s;Lpp1q1 pRdqq

.

Since ||u|p1u|v|p1v| ¤ C1p|u|p1|v|p1q|u v| for some constant C1, weproceed as in Step 1 to obtain

Ψpuq ΨpvqrT,T s ¤ CC1

ˆ T

T

upp1qq1Lp1x

vpp1qq1Lp1x

u vq1Lp1xdt 1q1

¤ CC1

up1

LqprT,T s;Lp1x q vp1

LqprT,T s;Lp1x qu vLqprT,T s;Lp1

x qTθ

¤ C2Rp1T θu vrT,T s for some constant C2 ¥ C.

Hence we take T such that

C2Rp1T θ 1

2i.e. T p2C2q 1

θ p2Cq p1θ u0βL2

x¤ T1,

and the map Ψ is a contraction map on XT pRq.Step 3 Conclusion

By Banach fixed point theorem, there exists a unique fixed point u PXT pRq of the map Ψ and hence u P XT pRq solves uniquely (NLS) withR C3u0L2

x, T C1

3 u0βL2x

for some large enough constant C3 (to be

determined later). Without loss of generality we can assume that for anyinitial data in the neighborhood of u0: U tv0 P L2pRdq|u0 v0L2pRdq u0L2pRdqu such that v0L2pRdq 2u0L2pRdq, there is a unique solution v PXT pRq of (NLS).

36 [February 4, 2019]

We are going to show the Lipschitz continuity of the flow map Φ : U ÞÑXT 1pRq via u0 ÞÑ u for all T 1 T . Let u0, v0 P U and we calculate

Φpu0q Φpv0q Sptqpu0 v0q

iκ

ˆ t

0

Spt t1q|Φpu0qpt1q|p1Φpu0qpt1q |Φpv0qpt1q|p1Φpv0qpt1q

dt1.

As in Step 2, we derive that

Φpu0q Φpv0qr0,T s ¤ Cu0 v0L2x

CΦpu0qp1

r0,T s Φpv0qp1r0,T s

Φpu0q Φpv0qr0,T sT θ¤ Cu0 v0L2

x C2

u0p1L2x v0p1

L2x

Φpu0q Φpv0qr0,T sT θ,

such that

Φpu0q Φpv0qr0,T s ¤ 2C3u0 v0L2x,

if T C13 u0βL2

xfor sufficiently large C3.

Remark 2.4. The nonlinear Schrodinger equation (NLS) holds in the dis-tribution sense: It follows from the Duhamel formulation (Duhamel) and thewell-definedness of the nonlinearity when u P XT

|u|p1u P L qp prT, T s;L p1

p pRdqq, q

p¡ q1 if p 1 4

d.

Then by Strichartz estimates the solution u itself belongs to any functionalspace LqprT, T s;LrpRdqq for any admissible exponent pair pq, rq.

By Sobolev embedding H1pRdq ãÑ Lp1pRdq Lp1p pRdq1, |u|p1u P

Lqp prT ;T s;H1pRdqq and hence the equation (NLS) makes sense at least in

Lq1prT, T s;H2pRdqq. However it is in generally not true that we can take

the L2x-inner product between the equation (NLS) and the solution u directly

to show the mass conservation law: That is why we first do regularizationand then take the L2

x inner product in next Subsection 2.2.2.

2.2.2 Global well-posedness in L2

Theorem 2.3. [GWP in L2] Let 1 p 1 4d. The solution obtained in

Theorem 2.2 exists globally in time such that

u P CpR;L2xq X Lqloc pR;Lpxq and upt, qL2

x u0L2

x, @t P R . (2.40)

37 [February 4, 2019]

Proof. We show the conservation of the L2x-norm, i.e. the mass conservation

law (1.8), rigorously for u P XT satisfying (NLS).Step 1 Regularization

Take ϕ P C80 pRdq with ϕ ¥ 0,

´Rd ϕpxq dx 1. Denote ϕnpxq ndϕpnxq.

Similarly we take ψ P C80 prT, T sq, ψ ¥ 0,

´R ψptqdt 1 and denote ψmptq

mψpmtq, m ¥ N with N sufficiently large. Since u P CprT, T s;L2xq X

LqprT, T s;Lp1x q, we have (Exercise.)

ψm ϕn uÑ u in CIN ;L2

x

X LqIN ;Lp1

x

,

ψm ϕn p|u|p1uq Ñ p|u|p1uq in Lq1pIN ;Lpp1q1

x qas m,nÑ 8. Here we denote IN pp1 1

NqT, p1 1

NqT q.

We take the convolution of (NLS) with ϕn and then with ψm to arrive at

iBtum,n ∆um,n κψm ϕn p|u|p1uq, um,n ψm ϕn u. (2.41)

We test the above equation for um,n by um,n P SpIN Rdxq and then take the

imaginary part. Similarly as the derivation of (1.8), we derive

1

2

d

dt

ˆRd|um,n|2 dx κIm

ˆRd

ψm ϕn p|u|p1uqum,n dx ,

for all t P IN .Step 2 Pass to the limit

For any T 1 P IN , we derive from the above equality that

1

2pum,npT 1qL2

x um,np0qL2

xq κIm

ˆr0,T 1sRd

ψm ϕn p|u|p1uqum,n dx dt

Ñ κIm

ˆr0,T 1sRd

p|u|p1uqu dx dt 0,

and hence

upT 1qL2x lim

m,nÑ8um,npT 1qL2

x lim

m,nÑ8um,np0qL2

x u0L2

x.

As the above holds for all T 1 P IN , it indeed holds for all T 1 P rT, T s.Recall that the existence time T depends only on p, d, u0L2

x, the solution

obtained in Theorem 2.2 can be extended to all the time by uniquenesscontinuation.

[16.11.2017][23.11.2018]

38 [February 4, 2019]

2.2.3 L2 critical case

Let us consider the L2 critical case p 1 4d, which is quite interesting

case: Recall the cubic nonlinear Schrodinger equations (1.30) and (1.35)with pp, dq p3, 2q.Theorem 2.4 (LWP & GWP for L2 critical case). Let p 1 4

dbe the

L2pRdq critical exponent. Let κ 1. Then

• (NLS) is locally well-posed in L2pRdq such that for any u0 P L2pRdq,there exists a unique solution

u P XT tu P CprT, T s;L2pRdqq |u P Lp1t,x prT, T s Rdqu,

where T ¡ 0 depending on u0, d and there exists a neighborhood U ofu0 such that the flow map Φ : L2 ÞÑ XT via Φ : u0 ÞÑ u is Lipschitzcontinuous;

• There exists a sufficiently small constant ε0 ¡ 0 depending on d suchthat if u0L2

x¤ ε0 then (NLS) is globally well-posed in L2pRdq and the

unique solution belongs to

CpR;L2xpRdqq X Lp1

t,x pRRdq.

Sketchy proof. Step 1 Smallness of Sptqu0Lp1prT0,T0sRdq for small T0

(Exercise.)For any ε ¡ 0, for any u0 P L2, there exists a neighborhood U ofu0 and T0 ¡ 0 such that

Sptqv0Lp1t,x prT0,T0sRdq ¤ ε, @v0 P U.

Indeed, the mapping Sptq : L2pRdq ÞÑ XT is locally Lipschitz. Therefore wecan take the neighborhood of u0 as U tv0 P L2

x | v0 u0L2x C1εu for

sufficiently large C, and hence it remains to show the above for u0.Step 2 LWP in L2

(Exercise.) We prove the local well-posedness result by searching for thefixed point for the mapping Ψ defined in (2.39) in

tu P XT0 | uL8t prT0,T0s;L2xpRdqq ¤ 2Cu0L2

x, uLp1

t,x prT0,T0sRdq ¤ 2εu,

for some sufficiently small ε depending on u0L2x, d and T0 depending on

ε, u0. By Step 1, we can assume that there exists T0 such that for any initialdata v0 in the neighborhood U there exists a unique solution v P XT0 withvLp1

t,x prT0,T0sRdq ¤ 2ε.

39 [February 4, 2019]

(Exercise.) We prove that the flow map Φ : U ÞÑ XT0 is Lipschitzcontinuous.Step 3 Small initial data case(Exercise.) We prove the global well-posedness result in L2 for small initialdata u0L2

x¤ ε0, similarly as in Step 2.

Remark 2.5. • We notice that in Theorem 2.4 there are well-posednessresults in L2pRdq for the L2-critical case if there are smallness con-ditions, either on the existing time or on the size of the initial data.Nevertheless here, since the existing time depends on u0 itself and notonly on its norm u0L2

x, we can not use the mass conservation law to

extend the local well-posed result to any time interval.

• In the defocusing L2 critical case, there are some global well-posednessresults without smallness assumption on the initial data, but- under an additional decay assumption |x|mu0 P L2pRdq, m ¡ 35, see[Bourgain 1998 JAM];- under an additional regularity assumption u0 P HspRdq, s ¡ 47, see[Colliander-Keel-Staffilani-Takaoka-Tao 2008 DCDS-A];- in the radial case, see [Tao-Visan-Zhang 2007 DMJ], [Killip-Tao-Visan 2009 JEMS] for higher and two dimensional cases respectively.

We are going to show the global well-posedness result in H1x for the

defocusing H1-subcritical case which includes the L2-critical case.

• In the focusing L2 critical case, there is global well-posedness result ifu0 is radial and u0L2 QL2 where Q is the solution of the ellipticequation (1.11). There may be blowup phenomena for u0L2 ¥ QL2.

• In the supercritical case p ¡ 1 4d, there are ill-posedness results for

(NLS), see [Christ-Colliander-Tao 2003 arXiv]: If sc d2 2

p1¡ 0,

then for any s sc, for any 0 δ, ε 1 and any t ¡ 0, there existsolutions u1, u2 of (NLS) with smooth initial data u1p0q, u2p0q P S suchthat

u1p0qHs u2p0qHs ¤ Cε, u1p0q u2p0qHs ¤ Cδ,

u1ptq u2ptqHs ¥ cε.

In the focusing case, the blowup phenominon in finite time from smoothdata can be proved simply via the virial identity and we can constructthe blowup example by applying scaling and Galilean transformation tothe soliton solutions.

40 [February 4, 2019]

[23.11.2018][28.11.2018]

2.3 Sobolev spaces

2.3.1 Sobolev spaces HspRdqRecall the definition (1.5) of the Sobolev space HspRdq, s P R as follows

HspRdq : tf P S 1pRdq | f2HspRdq :

ˆRdp1 |ξ|2qs|fpξq|2 dξ 8u. (2.42)

If s P N, then

HspRdq tf P L2pRdq | Bαf P L2pRdq, @multi-index α with |α| ¤ su.

It is easy to derive from the definition of Hs-norm that Hs1pRdq Hs0pRdqif s0 ¤ s1 and the following interpolation inequality by Holder’s inequality:

fHsθ ¤ f1θHs0 fθHs1 , with sθ p1 θqs0 θs1. (2.43)

The Sobolev space HspRdq is a Hilbert space with the inner product

pu, vqHspRdq ˆRdp1 |ξ|2qsupξqvpξq dξ .

and it is isometrically anti-isomorphic to its dual space pHspRdqq1. It is alsovery useful to identify HspRdq as the set of the continuous linear functionalson HspRdq via L2pRdq-inner product: Let f P S 1pRdq and f P L2

loc pRdq, then

f P HspRdq ô sup gPS,gHs¤1|xf, gyS1,S | sup gPS,gHs¤1

ˆRfg dx

8,

and we will denote xf, gyHs,Hs xp1 |ξ|2qs2f, p1 |ξ|2qs2gyL2 .

Theorem 2.5. [Sobolev embedding for HspRdq] The following Sobolev em-bedding results hold true:

• If 0 ¤ s d2, then HspRdq ãÑ LppRdq for any p P r2, pcs with d

2 s d

pccontinuously and there exists a constant C depending on d, s such that

fLpc pRdq ¤ Cf9HspRdq, @f P DpRdq, (2.44)

where the homogeneous Sobolev norm is defined in (1.6): f29HspRdq ´

Rd |ξ|2s|fpξq|2 dξ .

41 [February 4, 2019]

• If s d2, then HspRdq ãÑ LppRdq, for all 2 ¤ p 8 continuously;

• If s ¡ d2, then HspRdq ãÑ C0pRdq continuously;

• If 0 ¤ s d2, then Lp

1pRdq ãÑ HspRdq, p P r2, pcs i.e. p1 P rp12 sdq1, 2s

continuously.

Proof. Step 1 Proof of (2.44)The case s 0 is obvious and we consider the case 0 s d

2, pc

2dd2s

¡ 2.For any A ¡ 0 we can decompose f into low- and high- frequency parts

as follows:

f fl fh, fl 1 Af, fh 1¥Af.

Then we can control the low frequency part fl by

flL8 ¤ p2πq d2 flL1 p2πq d

2

ˆ|ξ| A

|fpξq||ξ|s|ξ|s dξ

¤ p2πq d2

|fpξq||ξ|sL2

ˆ|ξ| A

|ξ|2s dξ 1

2

¤ p2πq d2

ωdd 2s

f9HsA

d2s : C

f9HsA

dpc .

We write

fpLp p

ˆ 8

0

λp1|t|f | ¥ λu|dλ,

and we are going to estimate |tx P Rd | |fpxq| ¥ λu| for each λ P p0,8q.Indeed, for any λ ¡ 0, we take A Apλq p41C1f1

9Hsλq pcd such that the

low frequency part flL8 ¤ λ4 and hence

|tx P Rd | |fpxq| ¥ λu| ¤ |tx P Rd | |fhpxq| ¥ λ2u| ¤ 4λ2fh2L2 4λ2 pfh2

L2

4λ2

ˆ!ξPRd | 4Cf

9Hs |ξ|dpc ¥λ

) |f |2 dξ .

Therefore

fpcLpc pc

ˆ 8

0

λpc1|t|f | ¥ λu|dλ

¤ 4pc

ˆ!ξ | 4Cf

9Hs |ξ|dpc ¥λ

) λpc3|f |2 dξ dλ

¤ 4pcpc 2

ˆRdp4Cf

9Hs |ξ| dpc qppc2q|fpξq|2 dξ ¤ 2d

sp4Cqpc2fpc

9Hs.

42 [February 4, 2019]

Step 2 Case 0 ¤ s d2

By interpolation of Lebesgue spaces and fHs fL2 f9Hs , we have

the Sobolev embedding HsãÑ Lp, @p P r2, pcs.

Step 3 Case s d2

For any p P r2,8q, there exists s0 d2 d

pP r0, d

2q such that H

d2 ãÑ

Hs0ãÑ Lp.

Step 4 Case s ¡ d2

Since

fL1pRdq ˆRdp1 |ξ|2qs2|fpξq|p1 |ξ|2qs2 dξ

¤ fHs

p1 |ξ|2qs2L2pRdq ¤ CfHs , if s ¡ d

2,

the function f as the inverse Fourier transform of a L1-function is bounded,continuous and tends to 0 at infinity by Riemann-Lebesgue Lemma.Step 5 Case s P pd

2, 0s

By density, it suffices to show fHs ¤ CfLp1 , 0 ¤ dp1 d

2¤ s for f P S.

Indeed, since d2 d

p¤ s, we derive from the embedding HspRdq ãÑ LppRdq

that

fHs sup gPS,gHs¤1

ˆRdfg dx

¤ fLp1 sup gPS,gHs¤1gLp ¤ CfLp1 .

Remark 2.6. Let s 1, then

H1pRq ãÑ C0pRq, H1pR2q ãÑ LppR2q, @p P r2,8q,H1pR3q ãÑ LppR3q, Lp

1pR3q ãÑ H1pR3q, @p P r2, 6s.(Exercise.) Find a function in H1pR2q but not in L8pR2q.Corollary 2.1 (Gagliardo-Nirenberg’s inequality). For any p P r2, 2q with

2 " 8 if d 1, 2

2dd2

if d ¥ 3, there exists a constant C depending on p, d such that

the following interpolation inequality holds true

fLppRdq ¤ Cf1θL2pRdq∇fθL2pRdq, @f P H1pRdq, θ d

2 d

p.

Proof. It follows from the Sobolev embedding fLppRdq ¤ Cf9HθpRdq with

θ d2 d

pP r0, 1q, p P r2, 2q and the Sobolev interpolation

f29Hθ

ˆRd|f |2p1θqp|ξ|2|f |2qθ dξ ¤ f1θ

L2 fθ9H1 .

43 [February 4, 2019]

[28.11.2018][30.11.2018]

Theorem 2.6. Let s ¡ 0 and let pc dpd2 sq1 if s d

2and pc 8 if

s ¥ d2. Then the embedding HspRdq ãÑ Lploc pRdq, 1 ¤ p pc is compact in

the following sense: For any bounded sequence pfnqn in HspRdq, there exists asubsequence pfψpnqqn and f P HspRdq such that for any compact set K Rd

fψpnq Ñ f in LppKq.

Proof. (Exercise.)Step 1 Take the smooth mollifier function: ϕ P C8

0 pRdq, ϕ ¥ 0,´Rd ϕ dx

1 and its rescaled functions ϕεpxq εdϕpε1xq. Then

sup gHs¤1ϕε g gL2pRdq Ñ 0 as εÑ 0, if s ¡ 0,

sup gHs¤1ϕε g gL8pRdq Ñ 0 as εÑ 0, if s ¡ d

2,

while for any ε ¡ 0, sup gL21ϕε g gL2pRdq ¥ 1.Step 2 For any fixed ε ¡ 0, for any fixed R ¡ 0, the map

ϕε : L2pRdq ÞÑ L8pBRq, BR tx P Rd ||x| ¤ Ru

is compact (by Young’s inequality and Arzela-Ascoli’s theorem).Step 3 The identity map

Id : HspRdqp L2pRdqq ÞÑ L2pBRqp L8pBRqq, s ¡ 0

as the uniform limit of ϕε is compact.Step 4 Since HspRdq ãÑ LpcpRdq if s d

2, then by interpolation (or

Holder’s inequality) HspRdq ãÑãÑ LppBRq compactly for all p P r1, pcq andCantor’s diagonal argument ensures the compact embedding HspRdq ãÑãÑLploc pRdq. Similar result holds for s ¥ d

2.

Remark 2.7. The compact embedding HspRdq ãÑãÑ Lploc pRdq with s P r0, d2q,

p P r1, pcq is optimal in the sense that the embeddings HspRdq ãÑ LppRdq,2 ¤ p pc and HspRdq ãÑ Lpcloc pRdq are not compact (Exercise). We aregoing to give the concentration-compactness lemma describing the embeddingH1

ãÑ L2 later in the lecture.

44 [February 4, 2019]

2.3.2 Sobolev spaces W k,ppRdqWe recall the Sobolev embedding results for the Sobolev spaces W k,ppRdqwithout proof. Recall the definition of the Sobolev space W k,ppRdq, k ¥ 0integers as follows

W k,ppRdq tf P LppRdq|Bαf P LppRdq, 0 ¤ |α| ¤ ku. (2.45)

The Sobolev space W k,ppRdq, k P N, 1 ¤ p ¤ 8 is a Banach space equippedwith the norm

fWk,ppRdq $&%

°|α|¤k BαfpLppRdq

1p

if 1 ¤ p 8,max |α|¤kBαfL8pRdq if p 8.

For 1 ¤ p 8, the test function space DpRdq is dense in W k,ppRdq. If p 2,then W k,2pRdq HkpRdq as defined in (2.42) and obviously W k1,ppRdq W k0,ppRdq if k0 ¤ k1. We can also define the general Sobolev space W s,ppΩq,W s,p

0 pΩq, s P R, Ω Rd some open set, which we don’t discuss in thislecture. We just keep in mind that in bounded domains Ω, one has alwaysto pay attention to the boundary.

Recall the definition of the Holder spaces Cm,σpΩq, m P N, σ P p0, 1q,Ω Rd some open set, as follows

Cm,σpΩq tf P CmpΩq | Bαf P CσpΩq, @|α| mu, (2.46)

C0,σpΩq : CσpΩq tf P CpΩq | fCσpKq 8, @K Ω compact u,where

fCσpKq fL8pKq sup x,yPK,xy|fpxq fpyq||x y|σ .

Similarly we can define

Cm,σpΩq tf P CmpΩq | Bαf P CσpΩq, @|α| mu,C0,σpΩq : CσpΩq tf P CpΩq | fCσpΩq 8u.

We also have the following Sobolev embedding theorem for W k,ppRdqwhich we don’t prove in this lecture:

Theorem 2.7. Let k P N, 1 ¤ p 8. Then the following Sobolev embed-ding results hold true:

• If 1 ¤ p dk, then W k,ppRdq ãÑ LqpRdq for any q P rp, pcs with d

p k

dpc

continuously and there exists a constant C depending on d, k, p suchthat

fLpc pRdq ¤ C¸|α|k

BαfLppRdq;

45 [February 4, 2019]

• If p dk, then W k, d

k pRdq ãÑ LqpRdq for any q P rp,8q continuously;

• If max p1, dkq p 8, then W k,ppRdq ãÑ Cm,σpRdq X C0pRdq, m

rk dps, σ k m.

Furthermore, the embeddings are compact in the local sense as in Theorem

2.6: For example, let k 1, p "

dpdp if p d

8 otherwise, then the embedding

W 1,ppRdq ãÑãÑ Lqloc pRdq is compact for q P r1, pq.

2.4 H1 theory

2.4.1 Local well-posedness in H1

Theorem 2.8 (LWP in H1). Let 1 p 8 if d 1, 2 and 1 p 1 4d2

if d ¥ 3 be a H1 subcritical exponent. Let κ 1. Let u0 P H1pRdq.Then (NLS) is locally well-posed in H1pRdq: There exists a positive time

T ¡ 0 depending on u0H1 , p, d, a unique solution

u P YT tu P CprT, T s;H1pRdqq |u P LqprT, T s;W 1,ρpRdqquwith admissible exponent pair pq, ρq, and there exists a neighborhood V of u0

in H1 such that the flow map

Φ : V ÞÑ YT via u0 ÞÑ u

is Lipschitzian continuous.

Proof. We are going to show that the nonlinear map Ψ : u ÞÑ Ψpuq given by(2.39) is a well-defined contraction map in the complete metric space

YT pRq : tu P YT |uT : uL8T H1 uLqTLρ ∇uLqTLρ ¤ Ruwith appropriately chosen admissible exponent pair pq, ρq and T,R (depend-ing on u0H1 , p, d). Here we denote uLqTY : upt, qY LqprT,T sq.

For d ¥ 3, by Strichartz estimates,

ΨpuqL8T L2XLqTLρ ¤ Cu0L2 C|u|p1uLq

1

T Lρ1 .

Similarly,

∇ΨpuqL8T L2XLqTLρ ¤ C∇u0L2 C∇p|u|p1uqLq

1

T Lρ1

¤ C∇u0L2 Cup1

Lr ∇uLρLq

1

T

,p 1

r 1

ρ 1

ρ1.

46 [February 4, 2019]

If 1r 1

ρ 1dP p0, 1

ρq such that the Sobolev embedding uLrpRdq ¤ CuW 1,ρpRdq

holds, then we have

∇ΨpuqL8T L2XLqTLρ ¤ C∇u0L2 CT1q1 pq upT ,

if

2

q d

ρ d

2, ρ, q ¥ 2,

p 1

r 1

ρ 1

ρ1,

1

r 1

ρ 1

d,

such that1

q1 p

q d 2

4

1 4

d 2 p ¡ 0 if p 1 4

d 2.

Therefore for pq, ρq 4pp1qpd2qpp1q ,

dpp1qdp1

when d ¥ 3, we arrive at

ΨpuqT ΨpuqL8T L2XLqTLρ ∇ΨpuqL8T L2XLqTLρ ¤ Cu0H1 CT1q1 pq upT ,

and we can choose

R C1u0H1 , T C11 u0

4d2

p1

1 4d2

p

H1 ,

for some large enough constant C1 such that Ψ is a contractive mapping inYT pRq. Since |u|p1u P Lq1t prT, T s;W 1,ρ1

x q with pq, ρq the above admissibleexponent pair, the unique fixed point indeed belongs to Lq1prT, T s;W 1,ρ1qfor any admissible exponent pair pq1, ρ1q by Strichartz estimates.

For d 1, 2, for any 1 p 8, similarly as above we can show that themap Ψ is contractive in

tu P CprT, T s;H1qXLq0prT, T s;W 1,ρ0q|uT uL8T H1uLq0T W 1,ρ0 ¤ Ru

for appropriately chosen admissible exponent pair pq0, ρ0q with q0 ¡ p ¥ρ02 ¡ 1 and R, T . (Exercise.)

[30.11.2018][07.12.2018]

2.4.2 Global well-posedness in H1

Theorem 2.9 (GWP in H1). Assume the hypotheses in Theorem 2.8. Thenthe solution obtained in Theorem 2.8 can be extended uniquely globally intime if

• in the defocusing case κ 1;

47 [February 4, 2019]

• in the focusing case κ 1 and 1 p 1 4d;

• in the focusing case κ 1, p 1 4d

and u0L2 c0 with c0 somefixed constant;

• in the focusing case κ 1, 1 4d p and u0H1 ¤ ε0 with ε0 some

sufficiently small constant,

such that

u P CpR;H1xq X Lqloc pR;W 1,ρpRdqq with admissible exponent pair pq, ρq,

(2.47)

Mpuptqq Mpu0q, Epuptqq Epu0q, @t P R,

where

Mpuq ˆRd|u|2 dx , Epuq 1

2

ˆRd|∇u|2 dx κ

p 1

ˆRd|u|p1 dx .

Proof. Step 1 Mass and energy conservation laws on rT, T sRecall the proof of Theorem 2.3:

Im

ˆ t

0

ˆRdum,n (NLS)m,n ñMpuptqq Mpu0q, @t P rT, T s,

where we made use of the following facts for some appropriate r P r2,8q:u P CprT, T s;L2

xq X LqprT, T s;Lrxq, |u|p1u P Lq1prT, T s;Lr1x q.Recall in the proof of the H1-LWP result in Theorem 2.8 that the solutionsatisfies

u P CprT, T s;H1q X LqTW1,ρ, |u|p1u P Lq1TW 1,ρ1 ,

which implies the mass conservation law immediately.We follow the same procedure to show the conservation of the energy:

Im

ˆ t

0

ˆRdp∆um,n κ|um,n|p1um,nq (NLS)m,n ñ Epuptqq Epu0q, @t P rT, T s.

Indeed, for d ¥ 3, we have from the proof of Theorem 2.8, the Sobolevembedding results in Theorem 2.7 and the interpolation results in Lebesguespaces (i.e. log-convexity of Lp norms) fLpθ ¤ f1θ

Lp0 fθLp1 if 1pθ 1θ

p0 θ

p1that

u P L8T H1 X LqTW1,ρ L8T Lp 12 1

dq1 X LqTL

p 1ρ 1dq1 LpαT L

pp 1ρ 1dq1

,

|u|p1u P Lq1TW 1,ρ1 Lq1

TLp 1ρ1 1dq1

,

48 [February 4, 2019]

for some

α q

p

12 1

ρ

p12 1

dq 1

pp1ρ 1

dq ¡ q since 1 p 1 4

d 2.

Then (Exercise.) we can assume

um,n Ñ u in LqTW1,ρ,

|um,n|p1um,n, p|u|p1uqm,n Ñ |u|p1u in Lq1

TW1,ρ1 X Lq

1

TLp 1ρ1 1dq1 X LqTL

p 1ρ 1dq1

,

which implies that if we take the limit in

Im

ˆ t

0

ˆRdp∆um,n κ|um,n|p1um,nq (NLS)m,n,

then Epuptqq Epu0q, @t P rT, T s. Similarly we have the energy conserva-tion law for d 1, 2.

Step 2 If κ 1, then recalling the mass and energy conservation lawswe have the uniform bound

uptq2H1x¤Mpu0q 2Epu0q

on the time interval rT, T s.Since the existence time T only depends on p, d, u0H1 and more pre-

cisely T C1u0 4d2

p1

1 4d2

p

H1 for some big enough constant C, the solutionobtained in Theorem 2.8 can be extended uniquely to all the times.

Step 3 If κ 1, then by the Gagliardo-Nirenberg’s inequality in Corol-lary 2.1 for p 1 2, i.e. p 1 4

d2the H1 subcritical exponent,

uLp1pRdq ¤ C0u1γL2pRdq∇u

γ

L2pRdq, γ d

2 d

p 1P p0, 1q, (2.48)

we obtain from the energy conservation law that

1

2∇uptq2

L2xpRdq ¤ Epu0q 1

p 1uptqp1

Lp1x pRdq

¤ Epu0q Cuptqpp1qp1γqL2xpRdq

∇uptqpp1qγL2xpRdq

¤ Epu0q Cuptqpp1qp1γqp1 pp1qγ2

q1

L2xpRdq

1

4∇uptq2

L2xpRdq,

if

pp 1qγ d

2pp 1q d 2 i.e. 1 p 1 4

d.

49 [February 4, 2019]

By the mass conservation law, we obtain the uniform bound on uptqH1x

onthe existence time interval and hence the global well-posedness holds true inthe mass subcritical case.

Step 4 If κ 1 and p 1 4d, then the above inequality is replaced by

1

2∇uptq2

L2xpRdq ¤ Epu0q 1

2 4d

uptqp1

Lp1x pRdq

¤ Epu0q 1

2 4d

C2 4

d0 uptq

4d

L2xpRdq

∇uptq2L2xpRdq.

Thus if u0L2x c0 some fixed constant (such that 1

2 4d

C2 4

d0 c

4d0 1

2) then

the solution still extends globally in time.Step 5 If κ 1 and p ¡ 1 4

denergy subcritical, then pp1qγ ¡ 2 and

we can assume the smallness condition u0H1x¤ ε0 such that uptqH1

x¤ 2ε0

globally in time for sufficiently small ε0 (Exercise).

Remark 2.8. It was proved in [Weinstein ’1983 CMP] that if p 1 4d,

then

inffPH1

f4d

L2pRdq∇f2L2pRdq

f2 4d

L2 4d pRdq

Q

4d

L2pRdq1 2

d

,

i.e.1

2 4d

f2 4d

L2 4d pRdq

¤ 1

2

f4d

L2

Q4d

L2

∇f2L2 ,

(2.49)

where Q is the unique positive radial solution of (1.11). It follows from (2.49)that

Epuq 1

2∇u2

L2 1

2 4d

u2 4d

L2 4d¥ 1

2

1 u

4d

L2

Q4d

L2

∇u2

L2 ,

and in particular EpQq 0. If u0L2pRdq QL2pRdq c0 then the focusing(NLS) in the mass critical case is globally well-posed.

Remark 2.9. We can prove the local-in-time well-posedness results in H2pRdqfor the case

"1 p d

d4if d ¥ 5

1 p 8 if d ¤ 4, such that the solution stays in CprT, T s;H2pRdqqX

LqprT, T s;W 2,ppRdqq with pq, pq admissible exponent pair.We can also consider the general Sobolev space HspRdq, 0 s mint1, d

2u

with 1 p 1 4d2s

and so on.

50 [February 4, 2019]

There are global well-posedness and scattering results for the energy-criticaldefocusing nonlinear Schrodinger equation: p 1 4

d2, d ¥ 3, κ 1. See

e.g. Colliander-Keel-Staffilani-Takaoka-Tao Ann. Math. 2008 for the cased 3, p 5.

Remark 2.10. We can also show the existence result by compactness method(instead of Banach fixed point theorem here):

• Step 1: Construct a sequence of approximate smooth solutions uε (byregularising (NLS));

• Step 2: Show a priori uniform estimates for the sequence uε (e.g.uεLpT pXq ¤ C 8);

• Step 3: Pass to the limit by some compactness argument which comesusually from the uniform bound for the time derivatives Btuε, e.g. byAubin-Lions’ Lemma, if X ãÑãÑ Y ãÑ Z, uεLpT pXq BtuεLqT pZq ¤ C,then uε Ñ u in LpT pY q if p 8 or uε Ñ u in Cpr0, T s;Y q if p 8and q ¡ 1, such that the strong limit u solves (NLS).

The above procedure is a quite standard way to show the existence result,nevertheless the uniqueness/continuity results are not ensured a priori andtheir proofs need other arguments.

Here, we may follow the above procedure to show the well-posedness resultfor (NLS) and we have used the idea to show the mass/energy conservationlaws.

The solutions obtained by contraction argument are usually called strongsolutions which are unique, continuously depending on the initial data, whilethe solutions obtained by the above compactness method are usually calledweak solutions which could exist all the times but are possibly not unique.Sometimes the strong solutions and the weak solutions coincide.

2.4.3 The virial space case

Let us define the virial space

Σ tu P H1pRdq |xu P L2pRdqu H1pRdq X L2p|x|2 dx q, (2.50)

consisting of H1-functions which decay faster at infinity. We also define theassociated norm as

uΣ pu2H1x xu2

L2xq 12 .

51 [February 4, 2019]

Define the partial differential operator P as

P P ptq x 2it∇, Pj xj 2itBxj , j 1, , d.It is easy to see that P : SpRRdq ÞÑ SpRRdq and by duality P is a mapS 1pRRdq ÞÑ S 1pRRdq. An easy calculation shows that

P ptqw 2itei|x|2

4t ∇pei |x|2

4t wq, w P S 1pRRdq, t 0. (2.51)

Notice that P and iBt ∆ commutes:

rP ; iBt ∆s px 2it∇qpiBt ∆q piBt ∆qpx 2it∇q iBtp2itq∇ 2∇ 0.

If u Sptqg, g P Σ solves the free Schrodinger equation iBtu ∆u 0,u|t0 g, then P ptqu satisfies also the free Schrodinger equation with theinitial data xg which itself has a unique solution Sptqpxgq. Therefore we have

P ptqSptqg px 2it∇qSptqg Sptqxg, (2.52)

for g P Σ. If g P Σ, then Sptqg P Σ for any t P R:

xSptqg 2it∇Sptqg Sptqpxgq P L2pRdq, @g P Σ,

and P ptqSptqg Sptqpxgq P Lqt pLrq for pq, rq admissible exponent pair byStrichartz estimate.

We can also show Sptq : SpRdq ÞÑ SpRdq, t P R by use of (2.52). Indeed,Sptq : H8pRdq ÞÑ H8pRdq, H8pRdq Xk¥0H

kpRdq and hence by (2.52), forany t P R, for any g P SpRdq,xSptqg pP 2it∇qSptqg Sptqpxgq 2it∇Sptqg P H8pRdq,xjxkSptqg xj

Sptqpxkgq 2itBxkSptqg

Sptqpxjxkgq 2itBxjSptqpxkgq 2itxjBxkSptqg P H8pRdq,

Therefore (2.52) holds for all g P S 1pRdq.[07.12.2018][12.12.2018]

Theorem 2.10. Let p P p1, 2 1q be H1 subcritial exponent. Let κ 1.Let u0 P Σ. Then the Cauchy problem (NLS) is locally well-posed in Σ, suchthat there exists a positive time T depending only on u0H1 , p, d, a uniquesolution u P CprT, T s; Σq X LqprT, T s;W 1,ρq, Pu P LqprT, T s;Lρq withadmissible exponent pair pq, ρq and a neibourhood of u0 in Σ such that the flowmap is Lipschitz continuous on it. Furthermore, the global well-posednessresult holds true under the four assumptions in Theorem 2.9.

52 [February 4, 2019]

Sketchy proof. We apply Sptq on both the left and right sides of (2.52):P ptqSptq Sptqx such that SptqP ptq xSptq and hence

P ptqSpt t1q P ptqSptqSpt1q SptqxSpt1q SptqSpt1qP pt1q Spt t1qP pt1q on S 1pRdq.

Recalling the nonlinear mapping Ψ given in (2.39), if

u P ZT pR,R1q tu P CprT, T s; Σq | uLqTW 1,ρ ¤ R, PuLqTLρ ¤ R1u,for some admissible exponent pair pq, ρq given in Theorem 2.8, then we derivethat

P ptqΨu P ptqSptqu0 iκ

ˆ t

0

P ptqSpt t1qp|u|p1uqpt1qdt1

Sptqpxu0q iκ

ˆ t

0

Spt t1qP pt1qp|u|p1uqpt1qdt1.(2.53)

By virtue of (2.51), we derive for t 0,

|P p|u|p1uq| 2|t||∇pei |x|2

4t |u|p1uq| 2|t|∇|ei |x|24t u|p1pei |x|

2

4t uq,

and hence

P p|u|p1uqLq

1

T Lρ1 ¤

2p|t||u|p1|∇pei |x|2

4t uq|Lq

1

T Lρ1,

which is, by virtue of (2.51) again, bounded byp|u|p1|Pu|Lq

1

T Lρ1.

As in the proof of Theorem 2.8, for d ¥ 3 we have

P p|u|p1uqLq

1

T Lρ1 ¤ p|u|p1Pu

Lq1

T Lρ1 ¤ CT

1q1 pq up1

LqT pW 1,ρqPuLqTLρ ,and hence we can choose R Cu0H1

x, R1 Cxu0L2

x, T C1u0θH1

xfor

C sufficiently large such that Ψ is a contraction mapping in ZT pR,R1q.Remark 2.11. Noticing (2.51), we can proceed by a recurrence argument toarrive at

Pα px 2itDqα p2itq|α|ei|x|24tDαpei|x|24tq, rPα; iBt ∆s 0.

Based on the property of the operator Pα, e.g. d 1, p 3,

Pmp|u|2uqL2x¤ Cmu2

L8xPmuL2

x, uL8x ¤ t

12 Pu

12

L2xu

12

L2x,

[Hayashi-Nakamitsu-Tsutsumi 1986-1988] proved that if p is an odd integer,u0 P HmpRdqXL2p|x|k dx q, m ¥ k, then the regularity and the decay propertyare both preserved on the existence time interval. In particular, if u0 P SpRdq,then the solution of (NLS) upt, q P SpRdq on the existence time interval.

53 [February 4, 2019]

3 Large time behaviour

3.1 Virial and Morawetz identities

We define the virial potential

V puq ˆRd|x|2|u|2 dx , (3.1)

which averages the mass density (with the mass defined in (1.8)) against theweight function |x|2. We define the associated Morawetz action

W puq Imd

j1

ˆRdxjpuBxjuq dx Im

ˆRdrpuBruq dx , r |x|, (3.2)

which averages the momentum densities (with the momentum defined in(1.9)) against the weights pxjq.

Then we have the following Virial and Morawetz identities

Proposition 3.1. Let upt, xq be a Schwartz solution of the Cauchy problem(NLS). Then

1

4

d

dtV puptqq W puptqq, (3.3)

and

1

2

d

dtW puptqq

ˆRd|∇u|2 dx κ

d2 d

p 1

ˆRd|u|p1 dx . (3.4)

Proof. Exercise. Making use of the Pohozaev’s Identity

Re

ˆRd

∆ux ∇u dx pd

2 1q

ˆRd|∇u|2 dx , @u P SpRdq,

or equivalently, Re

ˆRd

∆ud

2u x ∇u dx

ˆRd|∇u|2 dx ,

(3.5)

where |∇u|2 °dj1ppBxjReuq2 pBxj Imuq2q. (Exercise).

Remark 3.1. We can define instead the Virial potential and Morawetz ac-tion, with the weights |x|2, pxjq in (3.1) and (3.2) replaced by the new weights|x|, p xj|x|q respectively:

Vpuq ˆRd|x||u|2 dx ,

54 [February 4, 2019]

Wpuq Imd

j1

ˆRd

xj|x| puBxjuq dx .

Then we have the following identities (Lin-Strauss’ Morawetz Identities) forthe Schwartz solution u of the Cauchy problem (NLS):

1

2

d

dtVpuptqq Wpuptqq,

d

dtWpuptqq

ˆRd

| ∇u|2|x| dx κ

2pd 1qpp 1qp 1

ˆRd

|u|p1

|x| dx 1

4

ˆRdp∆2|x|q|u|2 dx ,

where ∇ : ∇ x|x|p x|x| ∇q denotes the angular gradient.

Corollary 3.1. Let p P p1, 2 1q be energy subcritical exponent. Let u0 P Σand let u P CprT, T s;H1q, T 8 be the solution of (NLS). Then u PCprT, T s; Σq X LqprT, T s;W 1,ρq, Pu P LqprT, T s;Lρq for any admissibleexponent pair pq, ρq, and the mass and energy conservation laws as well asthe virial and Morawetz identities (3.3)-(3.4) hold for u on the existence timeinterval rT, T s: For any t P rT, T s,

Mpuptqq Mpu0q, Epuptqq Epu0q,1

4V puptqq 1

4V pu0q

ˆ t

0

W pupt1qqdt1,1

2W puptqq 1

2W pu0q

ˆ t

0

ˆRd|∇u|2 dx dt κpd

2 d

p 1qˆ t

0

ˆRd|u|p1 dx dt.

Sketchy proof. Recalling the proof of Theorem 2.10, there exists T0 ¡ 0depending only on uL8T pH1q such that there exists a unique solution u PCprt0T0, t0T0s; ΣqXLqprt0T0, t0T0s;W 1,ρq, Pu P Lqprt0T0, t0T0s;Lρqfor any t0 P rT, T s, and hence by uniqueness u u on rT, T s.

We do a regularisation argument and repeat the proof of Proposition 3.1to arrive at the identities (3.3)-(3.4) for u on rT, T s. (Exercise.)

3.2 Blowup

Theorem 3.1 (Blowup for the focusing case). Let sc P r0, 1q, i.e. 1 4d¤

p 2 1. Let κ 1. Let u0 P Σ with the initial energy Epu0q 0.Then the unique solution upt, xq obtained in Theorem 2.10 blows up in

finite time, and more precisely there exists T 8 such that

limtÒT

∇uptqL2pRdq 8.

55 [February 4, 2019]

Proof. We consider positive time in the following and the negative time canbe treated similarly.

If the solution u P Cppa, bq;H1pRdqq on some time interval pa, bq, then bythe virial and Morawetz identities (3.3)-(3.4), we derive that

1

16

d2

dt2V puq 1

2

ˆRd|∇u|2 dx 1

2pd2 d

p 1qˆRd|u|p1 dx

Epuq 1

2pd2 d 2

p 1qˆRd|u|p1 dx ¤ Epu0q 0,

(3.6)

since d2 d2

p1 pd 2qp 1

2 4d

1p1

q ¥ 0 if p ¥ 1 4d

is mass supercritical.

Hence if u P Cpr0,8q;H1pRdqq, then the time-dependent quantity V puptqq ´Rd |x|2|u|2 dx is below a parabola which is negative in finite positive time

which is not possible. Thus u blows up at some finite positive time.More precisely, if p ¡ 1 4

d, we can calculate on the existence time interval

1

16

d2

dt2V puq 1

4

d

dtW puptqq

1

2 d

4pp 1

2 1q

ˆRd|∇u|2 dx d

2pp 1

2 1qEpuq

αˆRd|∇u|2 dx ,

where α 12 d

4pp1

2 1q d

8pp 1 4

dq ¡ 0.

If initially W pu0q 0, then W puptqq 0. Thus ddtV puq 0 and V puqptq ¤

V pu0q. Since

|W puqptq| W puqptq ¤ ruL2pRdq∇uL2pRdq ¤ pV pu0qq 12 ∇uL2pRdq,

we derive that

1

4

d

dtpW puqq ¡ αpV pu0qq1pW puqq2,

and hence

pV pu0qq 12 ∇uL2pRdq ¥ W puq ¥ V pu0qpW pu0qq

V pu0q 4αW pu0qtfrom which we derive that limtÒT ∇uptqL2pRdq 8 with

T p4αW pu0qq1V pu0q V pu0q4αW pu0q .

56 [February 4, 2019]

[12.12.2018][14.12.2018]

If initially W pu0q ¥ 0, then by virtue of (3.6):

1

4

d

dtW puq ¤ Epu0q 0,

there exists a positive time t0 such that W puqpt0q 0 and we are in theprevious case again. We can balance the times t0 and T to show the blowuptime to be order V0?

W 20 4

αV0E0W0

.

If p 1 4d, then (3.6) gives

1

16

d2

dt2V puq 1

4

d

dtW puq Epu0q 0.

Thus

W puqptq W pu0q 4Epu0qt, V puqptq V pu0q 4W pu0qt 8Epu0qt2,

and hence there exists a positive time T ¡ 0 (of orderW0

?W 2

02V0E0

4E0) such

that V puqpT q 0. By the equality f2L2pRdq 1

d

°dj1

´Rd xjBxjp|f |2q dx

for f P SpRdq, we derive the Heisenberg’s inequality

f2L2pRdq ¤

2

d

d

j1

xjfL2pRdqBxjfL2pRdq, @f P Σ.

Therefore

0 u02L2pRdq uptq2

L2pRdq ¤2

dpV puqq 1

2 ∇uL2pRdq,

which together with V puqpT q 0 implies limtÑT ∇uL2pRdq 8.

Remark 3.2. The time T gives indeed an upper bound for the life span andthe solution may blow up before T . We can also make use of the norm uLqfor q ¥ p 1 instead of ∇uL2 in the estimate of the lifespan. Indeed it ischeap to see that 1

p1up1

Lp1 12∇u2

L2 Epu0q Ñ 8 as tÑ T .

Remark 3.3. In the mass critical case p 1 4d, we can assume the following

assumptions instead of Epu0q 0:

• Epu0q 0 and W pu0q 0;

57 [February 4, 2019]

• Epu0q ¡ 0 and W pu0q aEpu0qV0pu0q.Corollary 3.2. Let d ¥ 3, 1 4

d¤ p 1 4

d2and u0 P H1pRdq. Let upt, xq be

the solution of the Cauchy problem (NLS) satisfying limtÒT ∇uptqL2pRdq 8, then

∇uptqL2pRdq ¥ C0pT tqp 1p1

d24q, @t P r0, T q.

Proof. Recall the proof of Theorem 2.8. For any time t0 T with upt0qH1pRdq 8, the solution u with uL8prt0,t0T s;H1pRdqq ¤ R : Cupt0qH1pRdq exists atleast on the time interval rt0, t0 T s, T ¡ 0 with

T C1upt0q 4d2

p1

1 4d2

p

H1pRdq C1upt0q 1

1p1

d24

H1pRdq .

Hence

T t0 ¡ T i.e. upt0qH1pRdq ¥ CpT t0qp1p1

d24q.

Similar results hold for d 1, 2 Exercise.

Remark 3.4 (Blow up rates for the case p 1 4d, κ 1).

Pseudo-conformal blow up rateRecall the pseudoconformal invariance in the mass critical case that if

u upt, xq is a solution of the nonlinear Schrodinger equation (NLS), then

so is vpt, xq ei|x|24t

|t| d2upx

t, 1tq. If u0 P Σ, then for any t 0, vpt, q P Σ.

Let upt, xq eitQpxq be the solitary solution of the focusing (NLS), then

vpt, xq eip|x|24q4t

|t| d2Qpx

tq is also a solution in Σ for any t 0, while blows up

at t 0: ∇vptqL2pRdq Op1tq as tÑ 0.Indeed [Merle 1993] showed that the above is the unique minimal mass

blow up solution: Let p 1 4d, κ 1 and u be the solution of (NLS) with

the initial data u0 P H1pRdq and u0L2pRdq QL2pRdq. If limtÒT ∇uptqL2 8, then up to the symmetries in Subsection 1.1.3

upt, xq eip|x|24q4pTtq

pT tq d2Qp x

T tq.

Let d 1, 2, p 1 4d, κ 1, w0 P H1pRdq such that w0L2 QL2ε

and limtÒT ∇wptqL2 8. [Bourgain-Wang 1997] showed that w u ϕ,where u is as above and ϕ remains smooth after the blow up time.

58 [February 4, 2019]

[Merle 1990 CMP] also proved that for any given T ¡ 0, any set offixed points tx1, , xku in Rd, there exists an initial data u0 such that thecorresponding solution of the focusing mass critical (NLS) blows up exactlyat time T with the total mass concentrating at the points tx1, , xku.log-log blow up rate

Corollary 3.2 implies that the blow up rate is at least ∇uptqL2pRdq ¥CpT tq 1

2 in the mass critical case. Indeed, the numerical simulation

suggests the existence of solutions with log-log blow up rate

ln | ln |Tt||Tt

12 .

And when d 1, [Perelman 2001] established the existence of a solution withlog-log blow up rate.

[Raphael 2005] proved that there is a universal gap between the abovetwo blowup rates: Let u0L2 P pQL2 , QL2 εq for ε ! 1. Let u be thecorresponding blowup solution, then either u blows up at log-log rate, or ublows up faster than pseudo-conformal rate, i.e. ∇uptqL2pRdq ¥ CpT tq1.However, the existence of blowup solutions with blowup rate different fromthese two cases is still open.

3.3 Scattering

Theorem 3.2. Let 1 4d¤ p 21 and κ 1. Let u0 P Σ and u P CpR; Σq

be the global-in-time solution of (NLS) given in Theorem 2.10. Then

u P CpR; Σq X LqpR;W 1,ρpRdqq, with pq, ρq admissible exponent pair,

and u scatters at large time in the sense that there exist two functions u P Σsuch that

limtÑ8

upt, q SptquΣ 0.

Proof. We just show the case tÑ 8 and the case tÑ 8 follows similarly.Step 1 Pointwise decay

Consider the time-dependent function

F ptq ˆRd|xu 2it∇u|2 dx 8t2

p 1

ˆRd|u|p1 dx

ˆRd|x|2|u|2 dx 4tIm

ˆRdx ∇uu dx 8t2Epuq

V puq 4tW puq 8t2Epu0q,

where V puq,W puq, Epuq are the Virial potential, Morawetz action and theenergy defined in (3.1), (3.2) and (1.10) respectively. By view of the virial

59 [February 4, 2019]

and Morawetz identities (3.3)-(3.4), we have

d

dtF ptq 4d t

p 1r1 4

d psˆRd|u|p1 dx ¤ 0, if p ¥ 1 4

d.

Let vpt, xq ei|x|24tupt, xq, then

Pu px 2it∇qu 2itei|x|2

4t ∇v,

and we have

8t2Epvq 4t2ˆRd|∇v|2 dx 8t2

p 1

ˆRd|v|p1 dx F ptq

¤ F p0q V pu0q.Hence we have the following pointwise in time decay rate

∇vptqL2pRdq ¤ p2tq1pV pu0qq 12 .

By Gagliardo-Nirenberg’s inequality in Corollary 2.1 and the mass conserva-tion law

vptqL2x uptqL2

x u0L2

x v0L2

x,

we have the following pointwise decay rate for uLrxpRdq, r P r2, 2q (in com-parison with (1.4) for the linear Schrodinger group Sptq)

uptqLrpRdq vptqLrpRdq ¤ Cvptq1p d2 drq

L2pRdq ∇vptqd2 dr

L2pRdq

¤ C|t|p d2 drqu01p d

2 drq

L2pRdq pV pu0qq 12p d2 drq, @r P r2, 2q.

(3.7)

Step 2 Scattering in L2pRdqRecall the Duhamel’s formula (Duhamel) for the globally defined solution

upt, xq of (NLS). Then wpt, q Sptqupt, q P H1xpRdq satisfies

wptq u0 i

ˆ t

0

Spt2qp|u|p1uqpt2qdt2.

Then for any 0 t1 t,

wptq wpt1q iˆ t

t1Spt2qp|u|p1uqpt2qdt2, (3.8)

such that by Strichartz estimate (2.38)

wptq wpt1qL2pRdq ¤ C|u|p1uLq1 prt1,ts;Lr1 q Cup

Lpr1 pRdq

Lq1 prt1,tsq

60 [February 4, 2019]

where pq, rq could be any admissible exponent pair. By use of the pointwisedecay (3.7) in Step 1, we choose r p 1 2, 2

q d

2 d

p1to arrive at

wptq wpt1qL2pRdq ¤ C0

pt2qp d2 dp1

qpLq1 prt1,tsq

C0

ˆ t

t1pt2q 2pq1

q dt2 1q1

,

where C0 is some constant depending on the initial data u0Σ. If p ¥1 4

dthen q ¤ 2 4

dsuch that 2p

q¡ 1. Hence wptq wpt1qL2pRdq Ñ 0

whenever t1, t Ñ 8. Therefore there exists u P L2pRdq such that uptq SptquL2pRdq wptq uL2pRdq Ñ 0 as tÑ 8.

[14.12.2018][21.12.2018]

Step 3 Scattering in H1pRdqWe first claim that u P Lqpr0,8q;W 1,p1q. Indeed, we have already shown

u P Lqloc pR;W 1,p1q in Theorem 2.9 such that uLqpr0,T s;W 1,p1q ¤ CpT q 8for any finite time T ¡ 0. For any t ¥ T ¡ 0, by applying Strichartz estimateson the Duhamel’s formula (Duhamel) (and also on the spatial derivative of(Duhamel)), we have

uLqpr0,ts;W 1,p1q ¤ Cu0H1x C|u|p1u

Lq1 pr0,T s;W 1,p1p q

C|u|p1uLq1 prT ;ts;W 1,

p1p q

¤ Cu0H1x CT

1q1 pq upLqpr0,T s;W 1,p1q Cup1

Lp1xuW 1,p1

xLq1 prT,tsq

¤ Cu0H1x CT 1 p1

q upLqpr0,T s;W 1,p1q

C0

pt2q 2qpp1q

Lp 1q1 1q q1

prT,tsquLqprT,ts;W 1,p1

x q,

where we used the pointwise decay estimate (3.7) for the last inequality andwe now calculatept2q 2

qpp1q

Lp 1q1 1q q1

prT,tsq CpTθ tθq ¤ CTθ

with θ 2

qpp 1q p 1

q1 1

qq 2

qp 1 ¡ 0 when p ¥ 1 4

d.

Hence by choosing T large enough such that C0CTθ ¤ 1

2, we have

uLqpr0,ts;W 1,p1q ¤ Cu0H1x CT 1 p1

q upLqpr0,T s;W 1,p1q ¤ CpT q 8,

and as tÑ 8 we derive u P Lqpr0,8q;W 1,p1q.

61 [February 4, 2019]

We apply spatial derivative and then Strichartz estimate and finally thedecay rate (3.7) to (3.8), to arrive at

∇pwptq wpt1qqL2pRdq ¤ Cup1

Lp1pRdq∇uLp1pRdqLq1 prt1,tsq

¤ C0

pt2q 2qpp1q

Lp 1q1 1q q1

prt1,tsq∇uLqprt1,ts;Lp1pRdqq

which tends to zero whenever t1, t Ñ 8. Therefore u P H1pRdq such thatuptq SptquH1pRdq wptq uH1pRdq Ñ 0 as tÑ 8.Step 4 Scattering in Σ

Recall (2.53) when we apply the operator P x 2it∇ to (Duhamel).(Exercise.) Then the same argument as in Step 3 implies that

px 2it∇quLqpr0,8q;Lp1q 8,

and hence by xSptq SptqP ptq, we arrive at from (3.8) that

pxwqptq pxwqpt1qL2pRdq ˆ t

t1Spt1qP p|u|p1uqpt2qdt2

L2pRdq¤ C

|u|p1|Pu|Lq1 prt1,ts;Lpp1q1 q Ñ 0 as t1, tÑ 8.

Therefore xwptq xuL2xpRdq Ñ 0 as tÑ 8.

Remark 3.5. For p P p1, 1 4dq, then we also have the pointwise decay

uptqLrpRdq ¤ C|t|p d2 drqp1αprqq, αprq

#0 if 2 ¤ r ¤ p 1,

prp1qp4dpp1qqpr2qp4pd2qpp1qq if r ¡ p 1,

by considering the time-dependent quantity t2´Rd |v|p1 dx and then the quan-

tity t2´Rd |∇v|2 dx via the equality

d

dtp8t2Epvqq d

dtF puq 4dt

p 1p1 4

d pq

ˆRd|v|p1 dx .

And if we assume furthermore p ¡ p2 d ?d2 12d 4qp2dq such that

2p ¡ q, then the above scattering result also holds true.Let us give some further remarks concerning the exponent regime of p in

the scattering theory:

• We can relax the restriction on p P p1, 21q for the scattering results,nevertheless there are no scattering theory in L2

x if p ¤ 1 2d;

62 [February 4, 2019]

• For p P p1 2d, 2 1q, κ 1, there exist scattering states in L2

x,

nevertheless if p ¤ 2d?d212d42d

and u0 is large or if p ¤ 1 4d2

wedo not know whether u P Σ;

• There is also scattering theory for the focusing case if p P p1 4d2

, 1 4dq,

nevertheless if p 1 4d2

there is no scattering theory in L2;

• We can relax the restriction on the initial data, e.g. u0 P H1pRdqsuch that the scattering theory in the energy space H1pRdq holds forp P p1 4

d, 2 1q, d ¥ 3, κ 1.

We introduce briefly here the basic notions of scattering theory. Let Xbe a Banach space. Let R be the following two subsets in X:

R tϕ P X | (NLS) with initial data ϕ has a unique solution

u defined for all t ¥ 0pt ¤ 0q such that u limtÑ8

Sptquptq exists in Xu,

and we call u the scattering states of ϕ at 8. Let U be the followingtwo operators

U : R ÞÑ X via Upϕq u limtÑ8

Sptquptq.

If the mapping U are injective, we define the wave operators

Ω pUq1 : U ÞÑ R, U UpRq via Ωpuq ϕ.

Let O UpR XRq and we define the scattering operator S

S UΩ : O ÞÑ O via Su u.

Notice that

R R : tϕ | ϕ P Ru, U U, O O,

and if κ 0 the linear Schrodinger equation, then U Ω S Id .Let X Σ and

1 4

d¤ p 2 1, κ 1. (3.9)

Then by Theorem 3.2,

R Σ, U : Σ ÞÑ Σ, u Upu0q u0 i

ˆ 8

0

Spt1qp|u|p1uqpt1qdt1,

where uptq is the solution of (NLS) with initial data u0 P Σ. Inversely wehave the wave operators Ω : u Ñ u0 as follows

63 [February 4, 2019]

Theorem 3.3. Assume (3.9), then for any u P Σ (resp. u P Σ), thereexists a unique u0 P Σ such that the Cauchy problem (NLS) with the initialdata u0 has a unique solution u P CpR; Σq with SptquptquΣ Ñ 0 (resp.Sptquptq uΣ Ñ 0) as tÑ 8 (resp. tÑ 8).

Hence we can define the scattering operator S : Σ ÞÑ Σ, Su UΩu.

Proof. Let u P Σ and we follow the proof of Theorems 2.8 and 2.10, tosearch for the fixed point of the nonlinear map 3

Ψ : u ÞÑ Sptqu i

ˆ 8

t

Spt t1qp|u|p1uqpt1qdt1,

in the following complete metric space with pq, ρq admissible exponent pair:

XT tu P CprT,8q;H1pRdqq |Pu P CprT,8q;L2pRdqq,uXT : uLqprT,8q;W 1,ρpRdqq PuLqprT,8q;LρpRdqq sup t¥T |t|

2q uptqLρpRdq ¤ Ru,

for some appropriately chosen R, T . Indeed, if u P Σ, then by Strichartzestimates and P ptqSptq Sptqx, the solution w Sptqu P CpR; Σq satisfies

wLqpR;W 1,ρpRdqq PwLqpR;LρpRdqq ¤ CuΣ.

Since Pw 2itei|x|24t∇pei|x|24twq, we derive

∇pei|x|24twqL2pRdq ¤ p2|t|q1PwL2x p2|t|q1xuL2

x,

which, together with ei|x|24twL2pRdq wL2pRdq uL2x

and Gagliardo-Nirenberg’s inequality, implies

wptqLρx ei|x|24twLρx ¤ C|t|p d2 dρqpuL2

xxuL2

xq ¤ C|t| 2

q uΣ.

Hence Sptqu P XT if R ¥ CuΣ for some constant C. Similarly we canconsider the nonlinear term in the map Ψ, by use of Strichartz estimate andHolder’s inequality. (Exercise.) Therefore we can choose R CuΣ and

T Cup12pq 1

Σ with C large enough such that Ψ is a contraction mappingin XT and hence there exists a unique fixed point u of Ψ in XT .

3The formulation of the nonlinear map Ψ is motivated by taking the difference between

the Duhamel’s formula for uptq Sptqu0 i´ t0Spt t1qp|u|p1uqpt1qdt1 and Sptqu

Sptqu0 i´80Spt t1qp|u|p1uqpt1qdt1.

64 [February 4, 2019]

For the fixed point u P CprT,8q; Σq, upT q P Σ, by the formulation of Ψ,we have 4

upt T q SptqupT q i

ˆ t

0

Spt t1qp|u|p1uqpT t1qdt1,

that is, uT pt, xq : upTt, xq is the solution of defocusing nonlinear Schrodingerequation (NLS) with the initial data uT p0q upT q P Σ which hence existsglobally in time uT P CpR; Σq X LqpR;W 1,ρq by Theorem 3.2. In particular,u0 uT pT q P Σ is well-defined and upt, xq P CpR; Σq X LqpR;W 1,ρq is theunique solution of (NLS) with the initial data u0 such that, by use of u P XT ,u Ψpuq and Strichartz estimate,

Sptquptq uΣ i ˆ 8

t

Spt1qp|u|p1uqpt1qdt1

Σ

¤ Ctp2pq1qup

XTÑ 0 as tÑ 8.

The solution u P CpR; ΣqXLqpR;W 1,ρpRdqq such that SptquptquΣ Ñ 0as t Ñ 8 is unique: Indeed, if there are two solutions u1, u2 of (NLS) suchthat SptqujptquΣ Ñ 0 as tÑ 8, then u ujp0qi

´ 80Spt1qp|uj|p1ujqpt1qdt1

which together with the Duhamel’s formula for uj implies that pujqs are theunique fixed point of the nonlinear map Ψ.

Similarly we can define the wave operator Ω : u Ñ u0, Σ ÞÑ Σ.

[21.12.2018][09.01.2019]

4 Solitary waves

4.1 A minimiser problem

4.1.1 Space H1r pRdq

Let d ¥ 2. Let H1r pRdq be the set of the radial functions in H1pRdq:

H1r pRdq tf P H1pRdq | Df : r0,8q Ñ C s.t. fpxq fprq, r p

d

j1

|xj|2q 12 u.

4We write upt T q Spt T qu i´8tSpt T t1qp|u|p1uqpt1qdt1 and upT q

SpT qu i´8TSpT t1qp|u|p1uqpt1qdt1 and take the difference between upt T q and

SptqupT q.

65 [February 4, 2019]

It can also be viewed as the completement of the set of the radial functionsin C8

0 pRdq with respect to the H1-norm:

f2H1pRdq

ˆRd|∇f |2 |f |2 dx ωd

ˆ 8

0

p|Brfprq|2 |fprq|2qrd1dr,

where ωd is the area of the unit sphere in Rd.

Lemma 4.1 (Regularity and vanishing property of H1r -functions). Let d ¥ 2

and u P H1r pRdq. Then u P C 1

2 pp0,8qq and

r d12 uL8pRdq ¤ CuH1pRdq. (4.10)

Proof. Let ϕpxq ϕprq P C80 pRdq. Then

ϕ2prq 2

ˆ 8

r

ϕ1pρqϕpρqdρ,

and hence

|ϕ2|prq ¤ 2

rd1

ˆ 8

r

|ϕ1ϕpρq|ρd1dρ

¤ 2

rd1

ˆ 8

r

|ϕ1|2ρd1dρ 1

2ˆ 8

r

|ϕ|2ρd1dρ 1

2 ¤ C

rd1∇ϕ

12

L2pRdqϕ12

L2pRdq.

This implies (4.10) by density argument.Similarly, let 0 r1 r2 8 and we calculate by Holder’s inequality

|ϕpr1q ϕpr2q| ¤ ˆ r2

r1

ϕ1dρ ¤ 1

rd12

1

ˆ r2

r1

|ϕ1|ρ d12 dρ

¤ C

rd12

1

∇ϕL2pRdq|r2 r1| 12 ,

which implies that for any compact set K p0,8q, ϕC

12 pKq ϕL8pKq

sup r1r2,r1,r2PK|ϕpr2qϕpr1q||r2r1|12 ¤ CpKqϕH1 and hence by density argument

uC

12 pKq ¤ CpKquH1 which implies u P C 1

2 pp0,8qq.

Proposition 4.1 (Compact Sobolev embedding). Let d ¥ 2 and 2 as de-fined in Corollary 2.1. Then the Sobolev embedding H1

r pRdq ãÑãÑ LqpRdq,q P p2, 2q is compact.

66 [February 4, 2019]

Proof. Let u P H1r pRdq and 2 q 2. Then (4.10) implies

ˆ|x|¥R

|u|q dx ¤ r d12 uq2

L8

Rpq2qpd1q

2

ˆRd|u|2 dx ¤ CR pq2qpd1q

2 uqH1pRdq

Ñ 0 as RÑ 8 uniformly for the H1r functions with uH1 ¤ 1.

This, combined with the compact embedding H1pRdq ãÑãÑ LqpBRq for anyR P p0,8q in Theorem 2.6, implies the compact embedding H1

r pRdq ãÑãÑLqpRdq.Remark 4.1. We do not have the endpoint case H1

r pRdq ãÑãÑ L2pRdq (Ex-ercise.)

4.1.2 Compact minimisation

Proposition 4.2 (Compact minimisation). Let d ¥ 2 and p be a energy-

subcritical exponent: 1 p 2 1 "

1 4d2

if d ¥ 3

8 if d 2.

Then for any M ¡ 0, the minimisation problem

IM infuPAM

tu2H1pRdqu,

where AM !u P H1

r pRdq |ˆRd|u|p1 dx M

),

(4.11)

has a solution u P AM and IM ¡ 0.

Proof. Since by Sobolev’s embedding H1pRdq ãÑ Lp1pRdq, 2 p 1 2: uH1pRdq ¥ C1uLp1pRdq C1M1pp1q if u P AM , we can take aminimising sequence punqn in AM such that

un2H1pRdq Ñ IM ¥ C2M

2p1 ¡ 0.

Since punqn are bounded in H1r pRdq, by Proposition 4.1 there exists a subse-

quence (still denoted by punqn) and u P H1r pRdq such that

un Ñ u in Lp1pRdq, un á u in H1pRdq.Thereforeˆ

Rd|u|p1 dx lim

nÑ8

ˆRd|un|p1 dx M and thus u P AM ,

u2H1pRdq ¤ lim inf

nÑ8un2

H1pRdq IM ,

and hence u P AM is the minimiser of (4.11).

67 [February 4, 2019]

Lemma 4.2 (Positivity). If u P AM , then |u| P AM , |u|H1pRdq ¤ uH1pRdq.If u P AM is a minimiser of (4.11), then so is |u| such that |u|H1pRdq uH1pRdq, and if furthermore |u| ¡ 0, then u |u|eiγ for some γ P R.

Proof. The lemma follows from the following claim that if u P H1pRdq, then

ˆRd|∇u|2 dx ¥

ˆRd|∇|u||2 dx

and if |u| ¡ 0, then the above equality holds if and only if u |u|eiγ for someγ P R. (Exercise Prove this claim.)

4.1.3 Euler-Lagrangian equation

Proposition 4.3. Let u ¥ 0 be the minimiser of (4.11). Then there existsλ P R such that

∆u u λup. (4.12)

The λ in (4.12) is indeed a positive constant independent on u: λ IMM .

Proof. Step 1 DifferentiationLet t P R and h P C8

0 pRd;Rq a radial, real-valued function. Then(Exercise)ˆ

Rd

|u th|p1 up1 pp 1qthup

dx ¤ C

ˆRdt2h2up1 tp1|h|p1 dx .

HenceˆRd|u th|p1 dx

ˆRdup1 dx pp 1qt

ˆRdhup dx optq as tÑ 0.

Let h be chosen such that´Rd hu

p dx 0, then since u P AM , we have

ˆRd|u th|p1 dx M optq.

Let vt M1p1

uthLp1pu thq, then vtLp1 M

1p1 , vt pu thqp1 optqq and

vt2H1 p1 optqq

u2

H1 2t

ˆRdpuh∇u ∇hq dx t2h2

H1

u2

H1 2t

ˆRdpuh∇u ∇hq dx optq as tÑ 0.

68 [February 4, 2019]

Since u ¥ 0 is the minimiser, vtH1 ¥ uH1 for any t P R and hence

ˆRdpuh∇u ∇hq dx 0 for h P C8

0 pRd;Rq radial,

whenever

ˆRdhup dx 0.

Step 2 Lagrangian multiplierLet L1, L2 be the two linear forms on the Hilbert space H1

r defined by

L1phq ˆRdhup dx , L2phq

ˆRdpuh∇u ∇hq dx .

Then KerL1 KerL2. Let h P H1r with L1phq 0. Then any a P H1

r can bewritten as

a L1paqL1phqh b with b a L1paq

L1phqh P KerL1 KerL2.

Hence L2paq L1paqL1phqL2phq pL2phq

L1phqqL1paq for any a P H1r . This implies (4.12).

Step 3 Lagrange multiplicatorWe test (4.12) by u u ¥ 0 to arrive at

IM ˆRd|∇u|2 |u|2 dx λ

ˆRdup1 dx λM,

which implies λ IMM ¡ 0.

[09.01.2019][11.01.2019]

4.1.4 Regularity and decay property

We take v λ1p1u in (4.12) such that v satisfies the following renormalised

equation with 1 p 2 1

∆v v vp 0, v ¥ 0, v P H1r . (4.13)

Proposition 4.4 (Regularity and decay). Let vpxq vprq 0 solves (4.13),then v P W 3,qpRdq, @q P r2,8q such that

v P C2pRdq, |Dβvpxq| Ñ 0 as |x| Ñ 8, @|β| ¤ 2,

Dε ¡ 0 s.t. eε|x|p|v| |∇v|q P L8pRdq,

69 [February 4, 2019]

and v solves the ODE

v2 d 1

rv1 v vp, vp0q a, v1p0q 0, (4.14)

for some a ¡ 0.

Proof. Step 1 Regularity by iterationWe will use freely the following fact which we admit here without proof:

If v P LqpRdq, 1 q 8, then p1∆q1v P W 2,q.By view of v P H1

r pRdq ãÑ Lq0pRdq, q0 p1 and the Sobolev embedding

W 2,qjp pRdq ãÑ

$'&'%Lqj1pRdq if 2 dp

qj d

qj1 0

LqpRdq, @q P r qjp,8q, if 2 dp

qj 0

L8pRdq if 2 dpqj¡ 0.

we have v P L8pRdq. Indeed, as vp P L q0p , v p1∆q1v P W 2,

q0p ,

• if 2 dpq0¡ 0, then by Sobolev embedding v P W 2,

q0p pRdq ãÑ L8pRdq.

• if 2 dpq0 0, then by Sobolev embedding v P W 2,

q0p pRdq ãÑ Lq1pRdq

and hence v P W 2,q1p pRdq. If 2 dp

q1¡ 0, then we are done. If not, we

can continue the procedure such that there exists k with 2 dpqk¥ 0

and 2 dpqk1

0: This is possible since

1

qj1

2

d p

qj

ñ 1

qj1

1

qj pj

p 1

q0

2

d

, with

p 1

p 1 2

d 0 if p 2 1.

• if 2 dpqk 0 for some k P N, then v P LqpRdq for any q P r2,8q and we

choose q " 1 such that 2 dpq¡ 0.

Therefore vp P L p1p pRdq X L8pRdq and hence v P W 2,qpRdq for all q P

r2,8q and thus vp P W 1,qpRdq for all q P r2,8q which implies correspondingly∇v P W 2,qpRdq for all q P r2,8q. Thus v P W 3,qpRdq for all q P r2,8q, whichimplies

• v P C2,αpRdq for any α P p0, 1q, by Sobolev embedding;

• @|β| ¤ 2, Dβv P H1r pRdq and hence |Dβvpxq| Ñ 0 as |x| Ñ 8.

70 [February 4, 2019]

Hence the equation (4.13) for vpxq implies the equation (4.14) for vprq in theclassical sense, and furthermore, v1 r

d1pv vp v2q is uniformly bounded

such that v1prq Ñ 0 as r Ñ 0 and thus vp0q a ¡ 0 since if a 0 thenv 0.Step 2 Decay property

Let θε e|x|

1ε|x| , ε ¡ 0 be a bounded, Lipschitz continuous function with|∇θε|2 ¤ θ2

ε , a.e. We test the equation (4.13) by θεv to get (Exercise)

ˆRdθεv

2 dx 8,

which implies´Rd e

|x|v2 dx 8 as ε Ñ 0. Since v is globally Lipschitz

continuous, e|x|vd2 is uniformly bounded. Similarly we apply Bxj to the

equation (4.13) and test it by θεBxjv to arrive at´Rd e

|x||∇v|2 dx 8.

Remark 4.2. It is easy to show the regularity away from the origin by Lemma4.1. Indeed, let w χv, where χ P C8

0 is a radial function with the compactsupport away from zero and v satisfies (4.13). Then w satisfies

∆w w f, f χvp 2∇χ ∇v v∆χ.

Since v P L8 on Suppχ by virtue of (4.10), f P L2pRdq and hence wpξq f

1|ξ|2 , that is, w P H2r pRdq. Thus Brw P C

12 pp0,8qq by Lemma 4.1 and

wpxq wprq P C1pRd zt0uq, vpxq P C1pRd zt0uq. Now consider the equationfor Brw:

p∆ 1qBrw Brf P L2pRdq,and the same argument as before implies vpxq P C2pRd zt0uq.

4.1.5 Classification of minimisers

We have the following uniqueness result for the nonnegative solution whichdecays at infinity of the equation (4.14) proved by [Kwong 1987]:

Proposition 4.5 (Uniqueness). There exists a unique a ¡ 0 such that thesolution v of the ODE (4.14) satisfying

vprq ¥ 0, @r ¥ 0 and vprq Ñ 0 as r Ñ 8.

Furthermore, vprq ¡ 0 for all r ¥ 0 and we denote the solution to be Qprq:the fundamental solution of (4.14).

71 [February 4, 2019]

We do not give a proof here and interested readers can refer to AppendixB of Tao’s book.

Let u P AM be a minimiser of (4.11), then |u| P AM is also a minimiserby Lemma 4.2. Thus by Propositions 4.3 and 4.4, the nonnegative function

v λ1p1 |u| P H1

r satisfies (4.13) and the nonnegative function vprq vpxqsatisfies (4.14) and vprq Ñ 0 as r Ñ 8. Hence by Proposition 4.5, vpxq vprq Qprq ¡ 0 and thus |u| ¡ 0. Since u, |u| ¡ 0 are two minimisers suchthat ∇|u|L2pRdq ∇uL2pRdq, there exists γ P R such that

u |u|eiγ λ1p1veiγ pIM

Mq 1

p1Qprqeiγ.

Therefore we have obtained

Theorem 4.1 (Classification of minimisers). Let M ¡ 0, d ¥ 2, 1 p 2 1, then the minimisation problem (4.11)

IM infuPAM

tu2H1u, AM tu P H1

r pRdq |ˆRd|u|p1 dx Mu

has a family of minimisers

eiγMIM

1p1Qprq, γ P R,

where Q ¡ 0 is the unique fundamental state of the equation (4.13).

A similar proof in [Weinstein ’1983 CMP] explains the optimal constantin the Gagliardo-Nirenberg’s inequality

inffPH1

f4d

L2pRdq∇f2L2pRdq

f2 4d

L2 4d pRdq

Q

4d

L2pRdq1 2

d

,

4.2 Concentration compactness

Lemma 4.3 (Concentration-Compactness). Let punqn¥1 be a bounded se-quence in H1pRdq with un2

L2pRdq M ¡ 0. Then there exists a subsequence

punkq such that one the following properties holds:

(i) Compactness: There exists a sequence pykq in Rd such that

@q P r2, 2q, unkp ykq Ñ u in LqpRdq as k Ñ 8;

72 [February 4, 2019]

(ii) Evanescence: @q P p2, 2q, unk Ñ 0 in LqpRdq as k Ñ 8;

(iii) Dichotomy: There exist two bounded sequences pvkq, pwkq with compactsupport in H1pRdq and α P p0, 1q, such that

Supp vk X Suppwk tu, dpSupp vk, Suppwkq Ñ 8 as k Ñ 8,vk2

L2pRdq Ñ αM, wk2L2pRdq Ñ p1 αqM, as k Ñ 8,

@q P r2, 2q, unkqLq vkqLq wkqLq Ñ 0, as k Ñ 8,lim infkÑ8

p∇unk2L2 ∇vk2

L2 ∇wk2L2q ¥ 0.

Proof. Step 1 Concentration functionsLet ρn : r0,8q ÞÑ r0,M s be the concentration function of un:

ρnpRq sup yPRdˆBpy,Rq

|unpxq|2 dx ,

with the following properties:

• Monotonicity: @n, ρnpRq increases to M as R increases to 8;

• Concentration point: @R, the map y ÞÑ ´Bpy,Rq |u|2 is continuous and

tends to zero as |y| Ñ 8, and hence the concentration point exists:

@R ¡ 0, @n ¥ 0, Dyn ynpRq P Rd s.t. ρnpRq ˆBpyn,Rq

|un|2 dx ;

• Uniform Holder continuity: There exist C, β ¡ 0 (independent on n)such that

@R1, R2 ¡ 0, @n ¥ 0, |ρnpR1q ρnpR2q| ¤ C|Rd2 Rd

1|β.Indeed, suppose without loss of generality R1 ¤ R2, then

|ρnpR1q ρnpR2q| ˆBpy2n,R2q

|un|2 dx ˆBpy1n,R1q

|un|2 dx

ˆ

Bpy2n,R2qˆBpy2n,R1q

|un|2 dx

ˆBpy2n,R1q

ˆBpy1n,R1q

|un|2 dx

¤ˆR1¤|xy2n|¤R2

|un|2 dx ¤ Cˆ

R1¤|xy2n|¤R2

|un|2 dx 2

2 pRd2 Rd

1q122

¤ Cun2H1pRdqpRd

2 Rd1q1

22 if 2 8 i.e. d ¥ 3 s.t. H1pRdq ãÑ L2pRdq.

By Sobolev embedding H1pRdq ãÑ LppRdq for all p P r2,8q if d 1, 2,then the above argument also holds with 2 replaced by any p ¡ 2.

73 [February 4, 2019]

By Arzela-Ascoli’s Theorem, the uniform Holder continuity of the se-quence pρnq above implies the existence of a subsequence pρnkq and a Holdercontinuous monotone function ρpRq such that

@R ¡ 0, limkÑ8

ρnkpRq ρpRq.

Let m limRÑ8 ρpRq ¤ M . Then (Exercise) there exists a sequenceRk Ñ 8 such that

m limkÑ8

ρnkpRkq limkÑ8

ρnkpRk

2q lim

RÑ8ρpRq.

[11.01.2019][18.01.2019]

Step 2 Case m 0: EvanescenceSince ρ : r0,8q ÞÑ r0,ms is an increasing function, then ρ 0 if m 0.

In particular

limkÑ8

ρnkp1q ρp1q 0 limkÑ8

sup yPRdˆBpy,1q

|unk |2 dx .

This uniformly local strong convergence in L2pRdq implies the strong conver-gence in LqpRdq, q P p2, 2q: unk Ñ 0 in LqpRdq. Indeed, by use of the unitypartition pQjq (such that each Qj is contained in a ball of radius 1), we havethe following version of Gagliardo-Nirenberg’s inequality:

ˆRd|u|2 4

d dx ¸j¥1

u2 4d

L2 4d pQjq

¤ C¸j¥1

u4d

L2pQjq∇u2L2pQjq

¤ Csup j¥1u4d

L2pQjq¸j¥1

∇u2L2pQjq ¤ C

sup j¥1u2

L2pQjq 2d ∇u2

L2 ,

for d ¥ 3, and for d 1, 2, we can take use of uL4 ¤ Cu12

L2u12

H1 . We arrive

at unk Ñ 0 in L2 4d pRdq or L4pRdq and the interpolation in the Lebesgue

spaces implies unk Ñ 0 in Lq, q P p2, 2q.Step 3 Case m M : Compactness

For any R ¡ 0, let ykpRq be such that ρnkpRq ´BpykpRq,Rq |unk |2 dx .

There exist R0, k0 such that

ρnkpR0q ˆBpykpR0q,R0q

|unk |2 dx ¡ M

2, @k ¥ k0,

74 [February 4, 2019]

and for any ε ¡ 0, then there exist Rε, kε ¥ k0 such that

ρnkpRεq ˆBpykpRεq,Rεq

|unk |2 dx ¡M ε, @k ¥ kε.

Since unk has the total massM , the two ballsBpykpR0q, R0qXBpykpRεq, Rεq tu and hence there exists R0ε such that

ˆBpykpR0q,R0εq

|unk |2 dx ¡M ε, @k ¥ kε.

We may assume that the above holds true for all k by choosing a possiblylarger R0ε and hence vk unkp ykpR0qq satisfies

@ε ¡ 0, DR0ε s.t. @k ¥ 1,

ˆ|x|¥R0ε

|vk|2 dx ε.

By virtue of the compact embedding H1pRdq ãÑãÑ L2pBp0, R0εqq, vk Ñ u inLqpRdq, q P r2, 2q.Step 4 Case 0 m M : Dichotomy

We decompose unk as

unk unk1|yykpRk2 q|¤Rk2

unk1|yykpRk2 q|¥Rk unk1Rk2 |yykpRk2 q| Rk

: vk wk zk,

then ˆRd|zk|2 dx

ˆBpykpRk2 q,Rkq

ˆBpykpRk2 q,Rk

2q

|unk |2 dx

¤ ρnkpRkq ρnkpRk

2q Ñ 0 as k Ñ 8.

We then replace the characterised functions 1|yykpRk2 q|¤Rk2

,1|yykpRk2 q|¥Rk by

regular cutoff functions θk, ϕk with compact supports and sup k∇θkL8 , sup k∇ϕkL8 ¤4R1

k such that vk, wk are compactly supported functions with vk2L2 Ñ m,

wk2L2 ÑM m. Since

|∇unk |2 |∇vk|2 |∇wk|2 |∇unk |2p1 |θk|2 |ϕk|2q |unk |2p|∇θk|2 |∇ϕk|2q Re punk∇unkq ∇pθ2

k ϕ2kq

¥ 16|unk |2pRkq2 8|unk ||∇unk |pRkq1,

75 [February 4, 2019]

we have lim infkÑ8´Rd |∇unk |2 |∇vk|2 |∇wk|2 dx ¥ 0. Hence by virtue of

zk Ñ 0 in LqpRdq, q P r2, 2q,ˆRd||unk |q |vk|q |wk|q| dx ¤ C

ˆRd|unk |q1|zk| dx

¤ Cunkq1Lq zkLq Ñ 0 as k Ñ 8.

Exercise. Give three examples of bounded H1 sequences such that com-pactness/evanescence/dichotomy hold respectively.

Remark 4.3. We can repeat the lemma (established by P.-L. Lions 1983’and we follow the proof in Cazenave 2004’) to derive the decomposition profileof a sequence punq in HspRdq established by P. Gerard 1998’, which describesthe defect of the compactness of Sobolev embeddings up to extraction:Let punq be a bounded sequence of HspRdq, 0 s d

2and d

2 s d

p. Then

there exist a sequence of scales and cores pλpjqn , xpjqn qpj,nqPN2 in the sense that

j k ñ either limnÑ8

logpλpjqn

λpkqn

q 8 or lim

nÑ8|xpjqn x

pkqn |

λpjqn

8,

a sequence pϕjq in HspRdq and a sequence prpjqn q of functions such that

@J P N, uφpnqpxq J

j0

1

pλpjqn q d2sϕjx x

pjqn

λpjqn

rpJqn pxq,

limJÑ8

lim supnÑ8

rpJqn Lp 0,

@J P N, limnÑ8

uφpnq2

Hs J

j0

ϕj2Hs rpJqn 2

Hs

0.

We also have a version of the critical case, e.g. the case d 2, s 1,p P p2, 2q, λpjqn 1 considered by Hmidi-Keraani 2005.

4.3 Orbital stability

4.3.1 A second minimisation problem

Theorem 4.2. Let M ¡ 0, 1 p 1 4d

i.e. sc d2 2

p1 0 and let Q

be the fundamental state in Theorem 4.1. Then the minimisation problem

JM inftEpuq |u P H1pRdq, u2L2pRdq Mu (4.15)

76 [February 4, 2019]

is achieved by the following family of functions

Qµpx x0qeiγ0 , x0 P Rd, γ0 P R,

where Epuq 12

´Rd |∇u|2 dx 1

p1

´Rd |u|p1 dx is the energy functional de-

fined in (1.10), Qµ µ2p1Qpµxq and µ µpMq p M

Q2L2q 1

2sc . Furthermore,

all the minimising sequences are relatively compact in H1pRdq up to transla-tion and rotation: For the sequence punq in H1pRdq such that

un2L2 ÑM, Epunq Ñ JM ,

there exist pxnq Rd, pγnq R and a subsequence pφpnqq such that

uφpnqp xφpnqqeiγφpnq Ñ Qµ in H1pRdq.Sketchy proof. Step 1 Properties of JM , M ¡ 0

We have the following properties for JM :

• JM has a lower bound: JM ¡ 8.

Indeed, recall the Gagliardo-Nirenberg’s inequality in Corollary 2.1:

up1

Lp1pRdq ¤ C∇u2aL2pRdqu2b

L2pRdq, (4.16)

a dpp 1q4

, b d 2

4 pd 2qp

4.

We hence have

Epuq ¥ 1

2∇u2

L2pRdq C∇u2aL2pRdqu2b

L2pRdq,

with a 1 since p 1 4

d.

As u2L2pRdq M , we derive JM ¡ 8 by Young’s inequality.

• JM has a negative upper bound: JM 0.

Indeed, fix some u P H1pRdq with u2L2pRdq M . Then the rescaled

function uλpxq λd2upλxq, λ ¡ 0 satisfies uλ2

L2pRdq u2L2pRdq M

and

Epuλq λ21

2

ˆRd|∇u|2 dx 1

pp 1qλpp1q|sc|

ˆRd|u|p1 dx

,

and hence Epuλq 0 for λ ¡ 0 small enough.

[18.01.2019]

[23.01.2019]

77 [February 4, 2019]

• JM is homogeneous in M : JM M1sc|sc| J1.

Indeed, for any u P H1pRdq, the rescaled function uλpxq λ2p1upλxq,

λ ¡ 0 satisfies uλ2L2pRdq λ2scu2

L2pRdq λ2scM and

Epuλq λ2p1scqEpuq,and hence Jλ2scM λ2p1scqJM and we can choose in particular λ M

12sc .

Step 2 Existence of the minimiserLet punq be a minimizing sequence and we are going to show its compact-

ness (up to extraction of subsequence and translation) by Lemma 4.3. Wefirst have the following facts:

• Any subsequence is not evanescent.

Indeed, suppose by contradiction that unk Ñ 0 in Lp1pRdq as k Ñ 8,then

JM limkÑ8

Epunkq limkÑ8

1

2

ˆRd|∇unk |2 dx 1

p 1

ˆRd|unk |p1 dx

lim

kÑ81

2

ˆRd|∇unk |2 dx ¥ 0,

which is in contradiction with JM 0.

• Any subsequence is not dichotomous.

Indeed, suppose by contraction that there exist two sequences vnk , wnkwith disadjoint supports such thatˆ

Rd|vnk |2 dx Ñ αM,

ˆRd|wnk |2 dx Ñ p1 αqM, α P p0, 1q,

unkp1Lp1 vnkp1

Lp1 wnkp1Lp1 Ñ 0,

lim infkÑ8

r∇unk2L2 ∇vnk2

L2 ∇wnk2L2s ¥ 0,

then

JM limkÑ8

Epunkq ¥ lim supkÑ8

rEpvnkq Epwnkqs ¥ JαM Jp1αqM .

By the homogeneity property of JM and J1 0 we have

1 ¤ α1sc|sc| p1 αq 1sc

|sc| , with1 sc|sc| ¡ 1,

which is an contradiction with the fact that if θ ¡ 1, then fpαq :αθ p1 αqθ fp0q fp1q 1 for all α P p0, 1q.

78 [February 4, 2019]

Hence by Lemma 4.3, there exist pxkq Rd such that unkpx xkq Ñ u inL2pRdq X Lp1pRdq, and hence

JM limkÑ8

Epunkq limkÑ8

Epunkpx xkqq ¥ Epuq.

Therefore u P H1pRdq is a minimiser and limkÑ8Epunkq Epuq, unkpx xkq Ñ u in H1pRdq.Step 3 Classification of the minimisers

We follow the strategy in Subsection 4.1 to classify the minimisers of theminimisation problem (4.15):

– If u is a minimiser of (4.15), then by Lemma 4.2, we know that |u| ¥ 0is also a minimiser of (4.15).

– If u ¥ 0 is a minimiser of (4.15), then (Exercise) we follow the idea inthe proof of Proposition 4.3 to derive the existence of µ µpMq P R(independent of the minimisers) such that

∆u up µu, u ¥ 0, u P H1pRdq. (4.17)

Hence (Exercise) we derive (with the notations a, b given in (4.16))

1

2

ˆRd|∇u|2 dx a

p 1

ˆRdup1 dx ,

µ1

2

ˆRd|u|2 b

p 1

ˆRdup1 dx ,

by testing (4.17) by u ¥ 0 and by pd2 x ∇qu respectively. Therefore

JM Epuq a 1

p 1

ˆRdup1 dx 0 for 1 p 1 4

ds.t. a 1,

and thus

µ µpMq 2bJMpa 1qM 2b

a 1J1M

1sc ¡ 0.

– If u P H1, u ¥ 0 satisfies (4.17), then the rescaled solution uµ1pxq 1

µ2p1

upxµq, µ ?

µ satisfies the equation (4.13)

∆v vp v, v ¥ 0, v P H1pRdq. (4.18)

We have the nontrivial symmetry result established by [Gidas, Ni andNirenberg, 1979] which we do not prove here:

79 [February 4, 2019]

If v satisfies (4.18), then there exists x0 P Rd such that vpx x0q PH1r pRdq.

Hence by Propositions 4.4 and 4.5, the solution of (4.18) is indeedunique:

vpx x0q vp|x x0|q Qp|x x0|q, for some x0 P Rd,

where Q is the fundamental solution in Proposition 4.5.

– Conclusion: If u is a minimiser of the minimiser problem (4.15), thenthere exist pγ0, x0q P RRd such that

upxq Qµpx x0qeiγ0 , with Qµpxq µ2p1Qpµxq,

where

µ µpMq µpMqµpQ2

L2q p µpMqµpQ2

L2qq12 p M

Q2L2

q 12sc .

4.3.2 Orbital stability

Recall the “natural” stability notion of the solitary waves eitQprq in H1 for1 p 1 4

d:

Let u0 P H1 and let u P CpR;H1q be the global solution of (NLS) with theinitial data u0. Then for all ε ¡ 0, there exists δpεq ¡ 0 such that for allu0 P H1:

u0 QH1 δ implies sup t¥0upt, xq eitQpxqH1 ε.

This strong stability property is not suitable for (NLS) by virtue of thesymmetries of the equation (NLS). Indeed the scaling invariance and theGalilean invariance in Subsection 1.1.3 supply two obvious examples of stronginstability (Exercise):

• By scaling symmetry, for any λ ¡ 0, there exists a solution uλpt, xq λ

2p1Qpλxqeiλ2t of (NLS) with the initial data pu0qλpxq λ

2p1Qpλxq.

We have

pu0qλ QH1 Ñ 0 as λÑ 1,

while for any λ 1,

sup tuλpt, xq eitQpxqH1 ¥ QH1 .

80 [February 4, 2019]

• By Galilean invariance, for any v P Rd, there exists a solution uv eipxv|v|

2ttqQpx2vtq of (NLS) with the initial data pu0qv eivxQpxq.We have

pu0qv QH1 Ñ 0 as |v| Ñ 0,

while whenever v 0,

sup tuvpt, xq eitQpxqH1 ¥ QH1 .

By the variational characterisation of the solitary waves in Theorem 4.2, wehave the orbital stability results:

Theorem 4.3 (Orbital stability of the solitary waves). Let 1 p 1 4d, κ 1. Then for all ε ¡ 0, there exists δpεq ¡ 0 such that for all

u0QH1pRdq δ, there exist pxptq, γptqq P RdR so that the corresponding

solution u P CpR;H1q of (NLS) satisfying

sup tupt, xq Qpx xptqqeiγptqH1pRdq ε.

[23.01.2019][25.01.2019]

Proof. Suppose by contradiction that there exist ε0 ¡ 0, a sequence of timestn ¥ 0 and a sequence of solutions un P CpR;H1q such that

unp0, xq QH1 Ñ 0 as nÑ 8,unptn, xq Qpx x0qeiγ0H1 ¡ ε0 ¡ 0, @x0 P Rd, @γ0 P R .

Then we have

unp0, q2L2 Ñ Q2

L2 :M, Epunp0, xqq Ñ EpQq JM .

By the mass and energy conservation laws, we have

unptn, q2L2 unp0, q2

L2 ÑM, Epunptn, xqq Epunp0, xqq Ñ JM ,

and hence by Theorem 4.2, there exist pφpnqq N, pxφpnqq Rd, pγφpnqq Rsuch that

unptn, x xφpnqqeiγφpnq Ñ Q in H1pRdq, as nÑ 8,which is in contradiction to the assumption.

81 [February 4, 2019]

Remark 4.4. If p ¥ 1 4d, κ 1 then the solitary wave upt, xq eitQpxq

is unstable in the sense that there exists pQnqn H1pRdq such that Qn Ñ Qin H1pRdq while the corresponding solution unpt, xq blows up in finite time.Indeed, as ∆QQQp 0, if p 1 4

d, then

EpQq 1

2

ˆRd|∇Q|2 dx 1

p 1

ˆRd|Q|p1 dx 0

and EpλQq 0 for any λ ¡ 1, and hence we have the blowup results byTheorem 3.1 for the solutions with initial data p1 1

nqQ. Similarly we notice

that if p ¡ 1 4d, then with a dpp1q

4¡ 1

1

2

ˆRd|∇Q|2 dx a

p 1

ˆRd|Q|p1 dx 0.

For any ε ¡ 0 we can take λε small enough such that EpQλεq λ2p 2p1

d21q

ε EpQq ε where Qλ λ

2p1Qpλxq and hence there exists λ larger but close to 1 such

that EpλQλεq 0.

4.4 Linearized operators

We linearize the (NLS) equation around the solitary wave solution eitQpxq,that is, we consider the solutions of the form

upt, xq eitpQpxq hpt, xqq,and search for the equation satisfied by h:

Bth linear term of h nonlinear terms.

More precisely, the linear part of the equation for h reads as

Lh ip∆ 1Qp1qh p 1

2Qp1ph hq,

which can be rewritten into a matrix operator acting on

RehImh

:

L

0 LL 0

, L ∆ 1 pQp1, L ∆ 1Qp1. (4.19)

We are going to study the spectral properties of L and the two operatorsL, L, which are self-adjoint Schrodinger operators with continuous spec-trum r1,8q and with finitely many eigenvalues below 1. These spectralproperties play a central role in the stability theory.

82 [February 4, 2019]

4.4.1 Nullspaces

It is straightforward to compute that

LQ 0, LpΛQq 2Q, with Λ 2

p 1 x ∇. (4.20)

Indeed, LQ 0 is the equation for Q. For the second equation, since

ΛQ d

dλ|λ1Qλ, Qλpxq λ

2p1Qpλ1xq,

we apply ddλ|λ1 to the equation ∆Qλ Qp

λ λ2Qλ to arrive at LpΛQq 2Q.

We take the derivative ∇ and the multiplication x to the Q equation:p∆ 1Qp1qQ 0 to derive

L∇Q 0, LpxQq 2∇Q. (4.21)

Lemma 4.4. If 1 p 2 1, then

infpf,Qpq0

pLf, fq 0.

If 1 p ¤ 1 4d, then

infpf,Qq0

pLf, fq 0.

Proof. We have the following result from [Weinstein 1983 CMP] (we do notgive a proof here): For 1 p 2 1,

Jrus ∇u2aL2u2b

L2

up1Lp1

, a dpp 1q4

, b d 2 pd 2qp4

, u P H1pRdq

attains its minimum α ψp1

L2

pp1q2 at ψψL2

where ψ satisfies

a∆ψ bψ ψp 0, ψ ¥ 0, ψ P H1r pRdq.

Recall the fundamental solution Q which satisfies ∆QQQp 0, then

ψ b1p1Qp

cb

axq.

The minimisation inequality

d2

dε2|ε0Jpψ εηq ¥ 0, @η P C8

0 pRdq

83 [February 4, 2019]

reads as (Exercise)

cpLη, ηq ¥ 1

a

ˆη∆Q

2

1

b

ˆηQ2

ˆ

ηQp2

,

where c 1p1

´Qp1 1

2a

´ |∇Q|2 12b

´ |Q|2 ¡ 0.If 1 p 2 1, then

pLη, ηq ¥ 0, if pη,Qpq 0.

If 1 p ¤ 1 4d, then a ¤ 1, such that by use of ∆Q Qp Q,

pLη, ηq ¥ 0, if pη,Qq 0.

On the other hand, by L∇Q 0, we have pL∇Q,∇Qq 0 and p∇Q,Qpq p∇Q,Qq 0.

We have immediately L|tQuK ¥ 0. We also have the following informa-tion on the nullspaces of L, which can be found in [Weinstein 1985 SIAM]or [Chang-Gustafson-Nakanishi-Tsai 2007 SIAM].

Lemma 4.5. Let 1 p 2 1. Then

L ¥ 0, NpLq span tQu, L|tQuK ¡ 0,

NpLq span t∇Qu, pQ,LQq 0, L|tQpuK ¥ 0

L|tQuK ¥ 0 if 1 p ¤ 1 4

d.

Let NgpAq Y8k1NpAkq denote the generalized nullspace of A. Then for

p 1 4d,

NgpLq span!

0Q

,

0xQ

,

∇Q

0

,

ΛQ0

).

Let us define the functional space

H : pH1 H1q X rNgpLqsK,then its elements satisfy the following orthogonality relations:

Corollary 4.1. Let

fg

P H and 1 p 2 1, p 1 4

d, then the

following 2d 2 orthogonality relations hold:

pf,Qq 0, pf, xQq 0,

pg,∇Qq 0, pg,ΛQq 0.

84 [February 4, 2019]

Remark 4.5. If p 1 4d

is the mass critical exponent, then

Lp|x|2Qq 4ΛQ, Lρ |x|2Q,

for some radial function ρpxq (for which we do not know an explicit formulain terms of Q). The generalized nullspace of L reads as

NgpLq span!

0Q

,

0xQ

,

0

|x|2Q,

∇Q

0

,

ΛQ0

,

ρ0

).

If

fg

P H : pH1H1qXrNgpLqsK, the above 2d2 orthogonality relations

in Proposition 4.1 together with the following two orthogonality relations hold:

pf, |x|2Qq 0, pg, ρq 0.

[25.01.2019][01.02.2019]

4.4.2 Coercivity

Theorem 4.4. Let 1 p 1 4d. Then there exist positive constants K,K 1

such that

Kpf2H1 g2

H1q ¤ pLf, fq pLg, gq ¤ K 1pf2H1 g2

H1q, @fg

P H.

Proof. It is obvious to see from

pLf, fq ˆ|∇f |2|f |2pQp1|f |2, pLg, gq

ˆ|∇g|2|g|2Qp1|g|2

that the upper bound holds true.We claim that

τ : infA

pLf, fq ¡ 0,

A tf P H1 | fL2 1, pf,Qq 0, pf, xQq 0u.

Then L|tQ,xQuK ¡ 0 and

pLf, fq ¥ τf2L2 , if pf,Qq pf, xjQq 0, @1 ¤ j ¤ d,

85 [February 4, 2019]

such that

p

ˆQp1|f |2 ¤ Cf2

L2 ¤ τ1CpLf, fq

ùñ pτ1C 1qpLf, fq ¥ˆ|∇f |2 |f |2.

That is, there exists K ¡ 0 such that pLf, fq ¥ Kf2H1 .

Indeed, by Lemma 4.5: L|tQuK ¥ 0, it suffices to show τ 0. We assumethat τ 0, and there exists a minimizing sequence tfnu with fnL2 1,pfn, Qq pfn, xQq 0, such that

pLfn, fnq ˆ|∇fn|2 |fn|2 pQp1|fn|2 Ñ 0.

Then for any η ¡ 0, we can assume that all fn satisfy

0 1 ¤ˆ|∇fn|2 |fn|2 ¤ p

ˆQp1|fn|2 η.

The righthand side is uniformly bounded by fnL2 1, and hence tfnuconverges weakly to f P H1, which satisfies the orthogonality relations. Bythe decay property of Q, we also have

ˆQp1|fn|2 Ñ

ˆQp1|f|2,

and hence f 0 since η is arbitrary.Without loss of generality we can assume that f is admissible: fL2 1

and the minimum is attained at f. Indeed, by Fatou’s Lemma, fL2 ¤ 1.If fL2 1, then we simply take f

fL2, which is admissible. Since

pLf, fq ¤ lim infnÑ8

ˆ|∇fn|2 |fn|2 pQp1|fn|2 0,

we also have pLf, fq pL ffL2

, ffL2

q 0 τ .

Since the minimum is attained at an admissible function f 0, there ex-ists pf, λ, β, γq among the critical points of the Lagrange multiplier problem(Exercise)

pL λqf βQ γ xQ, fL2 1, pf,Qq pf, xjQq 0.

Hence λ pLf, fq τ 0 is a critical value and

0 pf, L∇Qq pLf,∇Qq βpQ,∇Qq pγ xQ,∇Qq,

86 [February 4, 2019]

that is, 0 pγ xQ,∇Qq and thus γ 0, Lf βQ. Since

LΛQ 2Q, NpLq span t∇Qu,we have

f β2

ΛQ θ ∇Q, θ P Rd .

Since pf, xQq 0, θ 0 and hence pf, Qq β2p 2p1

d2qQ2

L2 . If 1 p 1 4

d, then f 0 due to pf, Qq 0. This is a contradiction. Therefore

τ ¡ 0.Now we turn to the operator L. We claim that

µ : infA

pLg, gq ¡ 0,

A tg P H1 | gL2 1, pg,∇Qq 0, pg,ΛQq 0u.Indeed if µ 0, then the argument as above show that there exists g P Asuch that pLg, gq µ 0. Since L ¥ 0 and NpLq span tQu,g Q

QL2, which implies pg,ΛQq p 2

p1 d

2qQ2

L2 . This is a contradic-

tion with the orthogonality relation if 1 p 1 4d. Hence µ ¡ 0, such

that pLg, gq ¥ µg2L2 for g satisfying the orthogonality relations and thus

pLg, gq ¥ Kg2H1 for some K ¡ 0.

Remark 4.6. The estimates can be easily extended to the critical case p 1 4

d, with additional orthogonality relations pf, |x|2Qq 0 and pg, ρq

0 in the admissible sets A, A respectively. By the same argument asabove, if τ 0, then the minimum is attained at an admissible function f:pLf, fq 0, which is the critical point of the Lagrange multiplier problem

pL λqf βQ γ xQ δ|x|2Q,fL2 1, pf,Qq pf, xjQq pf, |x|2Qq 0.

Hence since τ 0, λ 0 and since pLf,∇Qq 0, γ 0. Since LΛQ 2Q, Lρ |x|2Q, NpLq span t∇Qu,

f β2

ΛQ δρ θ ∇Q.

Since pΛQ, |x|2Qq ´ |x|2Q2 0, pρ,Qq 0, p∇Q, xQq 0, the orthog-onality relations imply f 0. This is a contradiction and hence τ ¡ 0. Ifµ 0, then pg, ρq 1

QL2

´ |x|2Q2 0. This is also a contradiction with

the orthogonality relations.

87 [February 4, 2019]

Corollary 4.2. Let 1 p ¤ 1 4d

and h0 P H, rptq P H. Then the solutionh to the linear system

Bth Lh r, h|t0 h0,

satisfies h f ig P H and

Khptq2H1 ¤ pLf, fq pLg, gq ¤ K 1hptq2

H1 .

In particular, if rptq 0,

Khptq2H1 ¤ pLf, fq pLg, gq pLf0, f0q pLg0, g0q ¤ K 1h02

H1 .

Proof. Check that the linear flow maps from H to H and hence the estimateholds for any time t. Check that the quantity pLf, fqpLg, gq is conservedby the homogeneous flow. (Exercise)

4.4.3 Modulational stability

Let λ0, γ0, ξ0 pξ10 , , ξd0q, θ0 pθ1

0, , θd0q be real numbers, then (Exercise)the functions

λ2p1

0 Qpλ0pθpx, tq θ0qqeipξ0pθpx,tqθ0qpγpx,tqγ0qq

form a p2d 2q-parameter family of solutions of (NLS), 1 p 2 1 if thefollowing relations hold for θ pθ1, , θdq and γ:

BθiBt 2ξi0,

BθiBxj δji ,

BγBt λ2

0 |ξ0|2, BγBxj 0. (4.22)

Let us consider the perturbed Cauchy problem of the (NLS):

iBtuε ∆uε |uε|p1uε, uε|t0 Qpxq εh0pxq, p 1 4

d.

Let the 2d 2 parameters vary slowly:

λ0 λ0pT q, γ0 γ0pT q, ξ0 ξ0pT q, θ0 θ0pT q, where T εt,

and we are going to determine the slow time evolution of these parameterssuch that we can expand uε as

uεpt, xq λ 2p1

0 Qpλ0pθ θ0qq εh1

eipξ0pθθ0qpγγ0qq,

h1 h1pτ,Θq, τ ˆ t

0

λ20 dt1 , Θ λ0pθ θ0q,

88 [February 4, 2019]

where θ, γ satisfy (4.22) and h1 P H for t ¡ 0 and for any T0 ¡ 0,

sup 0¤t¤T0εεh1ptqH1 αpεq, αpεq Ñ 0 as εÑ 0. (4.23)

Indeed, we substitute the above expansion of uε with h1 h h0 intothe perturbed Cauchy problem and balance the terms of order ε to derive(Exercise)

Bτh Lh r, h|t0 0, h pf, gqt,where

r pr1, r2qt Lh0, r1 λ2p1

3

09λ0p 2

p 1QΘ ∇Qq λ

2p1

1

09θ0 ∇Q,

r2 λ2p1

3

09ξ0 ΘQ pξ0 9θ0 9γ0qλ

2p1

2

0 Q.

If rptq P H, then pr1, r2q have to satisfy

pr1, Qq pLImh0, Qq, i.e. λ2p1

3

0 p 2

p 1 d

2q 9λ0Q2

L2 0,

pr1, xQq pLImh0, xQq, i.e. 1

2λ

2p1

1

09θ0Q2

L2 2pImh0,∇Qq,

pr2,∇Qq pLReh0,∇Qq, i.e. λ2p1

3

09ξ0Q2

L2 0,

pr2,ΛQq pLReh0,ΛQq, i.e. λ2p1

2

0 p 2

p 1 d

2qpξ0 9θ0 9γ0qQ2

L2 2pReh0, Qq,

that is,

9λ0 0 ñ λ0 1, 9ξ0 ñ ξ0 0,

and θ0 4εQ2L2 pImh0,∇Qqt, γ0 2εp 2

p 1 d

2q1Q2

L2 pReh0, Qqt.(4.24)

If the parameters θ0, γ0 grow slowly in time as in (4.24) such that r P H,then by Corollary 4.2, h pf, gqt P H and

Khpτq2H1 ¤ pLf, fq pLg, gq : Aphq.

Since

hpτq eLτˆ τ

0

eLsrpεsqds pf, gqt,

and eLτ is a unitary group acting on H, we have

Kεhpτq2H1 ¤ ε2Aphq ¤ A

ε

ˆ τ

0

eLsrpεsqds.89 [February 4, 2019]

The following is a rough idea without proof: Heuristically, ε´ τ

0eLsrpεsqds

tends to the projection of r onto the nullspace of L (as r is “almost constant”and by the mean ergodic theorem) and hence tends to 0 as NpLqXH t0u.Therefore the modulational stability result (4.23) holds, under modulationalordinary differential equations in time for the d 1 parameters γ0, θ0 alone.This is consistent with the nonlinear stability result. In particular, if initiallyuε|t0 Qpxq εh0, then

uεpt, xq eiptγ0pT qqpQpx x0pT qq εh1pt, x x0pT qqq, T εt, where

x0pT q 4Q2L2 pImh0,∇QqT, γ0pT q 2p 2

p 1 d

2q1Q2

L2 pReh0, QqT,

such that (4.23) holds.We can also consider the perturbation added in the NLS equation to

derive (from the 2d 2 orthogonality conditions) the modulational ordinarydifferential equations in time for the 2d 2 parameters λ0, γ0, ξ0, θ0. Thesemodulational stability results can be found in [Weinstein 1985 SIAM J. Math.Anal.].

4.5 Lyapunov method

Let 1 p 1 4d

and we introduce a Lyapunov functional

Hpuq 2Epuq Mpuq ˆRd|∇u|2 2

p 1

ˆRd|u|p1

ˆRd|u|2,

which is invariant under translation and phase rotation.For any initial data u0 P H1, by Theorem 2.9 there exists a global solution

u P CpR;H1q of the Cauchy problem of (NLS). We write

upt, x xptqqeiγptq eitpQpxq hpt, xqq,then by the mass and energy conservation laws,

Hpu0q HpQq Hpupt, xptqqeiγptqq HpeitQq HpeitpQ hqq HpeitQq,which, by the elliptic equation ∆QQ Qp, implies (Exercise)

Hpu0q HpQq ¥ pLReh,Rehq pLImh, Imhq Cph2θH1 h6

H1q,with θ ¡ 0.

We claim that if xptq, γptq have been chosen to minimize

upt, xptqqeiγptq Q2H1 h2

H1 ,

90 [February 4, 2019]

with the restriction uL2 QL2 , then for h f ig, f, g P R

pLf, fq pLg, gq ¥ Kh2H1 Cph3

H1 h4H1q. (4.25)

Thus

Hpu0q HpQq ¥ Kh2H1 Cph2θ

H1 h6H1q : Gph2

H1q.

Therefore for any small enough ε ¡ 0, by the continuity of H in H1 thereexists δ ¡ 0 such that

if h02H1 δ then Hpu0q HpQq Gpεq and hence h2

H1 ε,

which implies the orbital stability result Theorem 4.3. For general caseinfx0PR,γ0Pr0,2πq u0pxx0qeiγ0 Qpxq2

H1 δ without the restriction u0L2 QL2 , we can take the ground state ψλpt, xq 1

λ2p1

Qpxλqei tλ2 which has the

same L2-norm as u0 and ψλ QH1 12ε. Hence by triangle inequality we

still have h2H1 ε.

We now prove the claim. Minimization of h2H1 implies (Exercise)

ˆQp1BxjQf dx 0,

ˆQpg dx 0.

Since L|tQuK ¡ 0, we have the following by g P tQuK

pLg, gq ¥ cg2L2 and hence pLg, gq ¥ Kg2

H1 .

In order to show the positivity estimate for L, recall that we restrict our-selves to the case

uL2 QL2 , such that pf,Qq 1

2

pf, fq pg, gq 1

2h2

L2 .

We decompose f f‖ fK as follows:

f‖ pf,QqQ 1

2h2

L2Q,

fK f pf,QqQ f 1

2h2

L2Q.

HencepLf, fq pLf‖, f‖q 2pLf‖, fKq pLfK, fKq,

91 [February 4, 2019]

where

pLf‖, f‖q 1

4h4

L2pLQ,Qq,

pLf‖, fKq pf,QqpLfK, Qq 1

2h2

L2pLfK, Qq

1

2h2

L2pLf,Qq 1

4h4

L2pLQ,Qq ¥ Ch3L2 ,

pLfK, fKq ¥ cpfK, fKq cpf, fq cpf‖, f‖q cpf, fq c1

4h4

L2 .

In the above, the last inequality follows from the proof of Theorem 4.4: theminimum pLfK, fKq 0 can only be attained at θ ∇Q which violates´Qp1BxjQf 0. Therefore

pLf, fq ¥ Kf2H1 Cph3

L2 h4L2q,

which implies (4.25).[01.02.2019][02.02.2018]

A Jost solutions

A.1 Time independent case

We look for solutions of the initial value problem for the ODE system (1.15):

jx iz uu iz

j,

under the following initial value conditions respectively:

j,1px; zq eizx

0

op1q as xÑ 8,

and

j,2px; zq

0eizx

op1q as xÑ 8.

Indeed, the renormalised solution l,1px; zq eizxj,1px; zq satisfies the fol-lowing integral equation

l,1px; zq

10

ˆ x

8

0 upx1q

e2izpxx1qupx1q 0

l,1px1; zq dx 1. (A.26)

92 [February 4, 2019]

By iteration we deduce that

l,1px; zq 8

n0

l,1n px; zq, l,10

10

,

l,1n1px; zq ˆ x

8

0 upx1q

e2izpxx1qupx1q 0

l,1n px1; zq dx 1,

that is,

l,1px; zq

10

0´ x8 e

2izpxx1qupx1q dx 1

´ x

8 upx2q´ x28 e

2izpx2x1qupx1q dx 1 dx 2

0

0´ x8 e

2izpxx3qupx3q´ x38 upx2q

´ x28 e

2izpx2x1qupx1q dx 1 dx 2 dx 3

´ x

8 upx4q´ x48 e

2izpx4x3qupx3q´ x38 upx2q

´ x28 e

2izpx2x1qupx1q dx 1 dx 2 dx 3 dx 4

0

(A.27)

We just have to show that the above series converges if u P L1pR;Cq: Let

mk ˆ xk

8|upyq| dy , m

ˆ x

8|upyq| dy ,

then dm k |upxkq| dx k and |l,1px; zq| with x, z P R can be controlled by1 ´ m0

´ m2

0dm 1 dm 2

´ m0

´ m4

0

´ m3

0

´ m2

0dm 1 dm 2 dm 3 dm 4

´ m0

dm 1 ´ m

0

´ m3

0

´ m2

0dm 1 dm 2 dm 3

and hence by

°8k0

m2k

p2kq!°8k0

m2k1

p2k1q!

which converges since m ¤ uL1 . The uniqueness result follows similarlyby iteration: Let l be the difference between two solutions satisfying theabove integral equation (A.26) with the boundary condition p0, 0qT , thenwe substitute this integral equation into itself k times such that |lpx; zq| ¤lL8 uk

L1

k!which vanishes as k Ñ 8.

93 [February 4, 2019]

Hence we arrive at a 2 2 fundamental solution matrix:

Jpx; zq rj,1px; zq, j,2px; zqs.

with normalization condition

limxÑ8

Jpx; zqeizxσ3 I, σ3

1 00 1

.

Similarly we can define Jost solutions j,1px; zq, j,2px; zq that are nor-malized in the limit x Ñ 8 and the associated normalized fundamentalsolution matrix

Jpx; zq rj,1px; zq, j,2px; zqs, with limxÑ8

Jpx; zqeizxσ3 I.

There is a scattering matrix S Spzq such that Jpx; zq Jpx; zqSpzq,with Spzq given in (1.24).

A.2 Time dependent case

We are going to show that the matrices

Wpt, x; zq Jpt, x; zqe2iz2tσ3 , σ3

1 00 1

are simultaneous fundamental solutions of the compatible linear systems(1.15)-(1.16) of the Lax pair. Indeed, let W pt, x; zq be the simultaneousfundamental solution matrix of the compatible systems (1.15)-(1.16). Sinceany solution matrix of (1.15) is a linear combination of j,1, j,2, there existstime-dependent (and space-independent) matrix Cpt; zq such thatW pt, x; zq Jpt, x; zqCpt; zq. By the system (1.16), we calculate the time derivative ofCpt; zqd

dtCpt; zq d

dtpJpt, x; zq1W pt, x; zqq

Jpt, x; zq1BtW pt, x; zq Jpt, x; zq1BtJpt, x; zqJpt, x; zq1W pt, x; zq Jpt, x; zq1V pt, x; zqJpt, x; zqCpt; zq Jpt, x; zq1BtJpt, x; zqCpt; zq.

Since for all the time t,

Jpt, x; zq Ñeizx 0

0 eizx

, V pt, x; zq Ñ 2iz2σ3 as xÑ 8,

94 [February 4, 2019]

and Cpt; zq does not depend on x-variable, we deduce that

d

dtCpt; zq 2iz2σ3Cpt; zq,

and hence a particular solution can be Cpt; zq e2iz2tσ3 and Wpt, x; zq Jpt, x; zqe2iz2tσ3 is the fundamental solution of the compatible systems(1.15)-(1.16).

B The NLS hierarchy

This section can be found in [Palais 1997 Bulletin AMS]. Recall the Laxequation (1.22) (for the defocusing case):

ψx iz uu iz

ψ : Uψ. (B.28)

We can take the standard 2 2 Pauli matrices

σ1

0 11 0

, σ2

0 ii 0

, σ3

1 00 1

,

σ σ1 iσ2

2

0 10 0

, σ σ1 iσ2

2

0 01 0

,

to rewrite

U U0 zU1, U0 uσ uσ, U1 1

iσ3.

We search for equations

Btψ V pz, yqψ, (B.29)

such that the compatibility condition between (B.28) and (B.29) is an equa-tion for u. More precisely we will search for V such that

rBt V, Bx U s

0 Btu f

Btu f 0

0.

We make an Ansatz

V pz, uq k

j0

zkjVjpuq,

95 [February 4, 2019]

and it reduces to search for Vj satisfying

V0 i 0

0 i

, BxVj rVj, U0s rVj1, U1s 0, 0 j k.

We write Vj irj pjpj irj

and obtain the following recursive formulas:

pj1 i

2Bxpj urj, r1j ippju pjuq,

together with the equation for u:

iBtu 2pk1.

This implies immediately

pp0, r0q p0, 1q, pp1, r1q pu, 0q, pp2, r2q p i2u1,

1

2|u|2q,

pp3, r3q 1

4u2 1

2u|u|2, i

4pu1u uu1q,

pp4, r4q i1

8u3 i

3

4u1|u|2,1

8p|u|2q2 3

8|u1|2 3

8|u|4,

and the corresponding equation (B.29) read as

Btψ i 0

0 i

ψ,

Btψ iz uu iz

ψ,

Btψ iz2 i

2|u|2 zu i

2u1

zu i2u1 iz2 i

2|u|2ψ,

Btψ iz3 iz

2|u|2 1

4pu1u uu1q z2u iz

2u1 1

4u2 1

2|u|2u

z2u iz2u1 1

4u2 1

2|u|2u iz3 iz

2|u|2 1

4pu1u uu1q

ψ,

Btψ iz4 iz2

2|u|2 z

4pu1u uu1q i1

8p|u|2q2 i3

8|u1|2 i3

8|u|4 : A B

z3u iz2

2u1 z

4u2 z

2|u|2u i1

8u3 i3

4u1|u|2 : B A

ψ,

where in the above we assume z P R when we take the complex conjugateof A,B. We see that (1.16) is the third equation in the above hierarchy and(NLS) reads as iBtu 2p3 (up to factors).

It is true though not obvious that the compatibility conditions are Hamil-tonian equations associated to the Hamiltonians

Hk 1

k 1

ˆRtrVk2

i 00 i

dx 2

k 1

ˆRrk2 dx .

96 [February 4, 2019]

The Hamiltonian for the NLS flow iBtu 2p3 hence reads as

H2 1

4

ˆRp|u1|2 |u|4q dx ,

which is the conserved energy E defined in (1.10) (up to factors). The non-linear Hamiltonian flows in the same hierarchy commute.

We obtain the equations for the focusing NLS hierarchy by replacing uby u and the KdV hierarchy by replacing u by 1.

C Conserved energies

In this section we explain briefly the existence of a family of conserved ener-gies for the one dimensional defocusing cubic nonlinear Schrodinger equation(NLS) (with d 1, p 3, κ 1) established by [Koch-Tataru 2016] (see alsoKillip-Visan-Zhang). Recall that for the initial data u0 P SpRq, there exists aunique solution u P CpR;SpRqq solving the 1-d cubic nonlinear Schrodingerequation (NLS) (see Remark 2.11), and the transmission coefficient T T pzqassociated to the Lax equation (1.22) is invariant by the cubic Schrodingerflow (see (1.25)). We make use of this fact to establish the family of con-served energies Es Espuq in terms of T T pz;uq which is equivalent to theSobolev norm of the solution u2

Hs , @s ¡ 12.

Step 1 Expansions of the “transmission coefficient” T1pzq and itslogarithm lnT1pzq on the upper half plane

Recall the direct scattering transform introduced in Subsection 1.2. Ifupt, xq P CpR;SpRqq is the solution of the one dimensional defocusing cubicnonlinear Schrodinger equation (NLS), z ξ P R (the continuous spectrumof the self-adjoint Lax operator), then we have the Jost solution j,1px; zq ofthe Lax equation (1.22)

ψx iz uu iz

ψ,

with the asymptotic behaviours at infinity:

j,1px; zq eizx

0

op1q as xÑ 8,

j,1px; zq T1pzqeizx

0

Rpt, zqT1pzq

0eizx

op1q as xÑ 8.

Hence the “transmission coefficient” T1pzq can be calculated as the firstcomponent of the renormalised Jost solution l,1px; zq eizxj,1px; zq as

97 [February 4, 2019]

xÑ 8 (recalling the iterative formulation of l,1 in Subsection 1.2.1):

T1pzq limxÑ8

peizxj,1px; zqq 1 : 18

j1

T2jpzq,

(C.30)where the graphic symbols , , denote the multilinear integrals as

T2pzq ˆx y

e2izpyxqupyqupxq dx dy ,

T4pzq ˆx1 y1 x2 y2

2¹n1

e2izpynxnqupynqupxnq dx dy ,

For any z P U , U tz P C | Im z ¡ 0u the upper half plane, we can define theJost solution j,1px; zq and this provides a holomorphic extension of T1pzqto the closed upper half plane z P U , such that |T | ¤ 1 on the real line andT Ñ 1 at infinity. By maximum principle, |T1| ¥ 1 on U .

Since ln is analytic near 1 and expands formally as lnp1 aq a 12a2

13a3 , the formal series for T1 yields a formal series for lnT1:

lnT1 T2

T4 1

2pT2q2

T6 T2T4 1

3pT2q2

:

8

j1

T2j,

where each T2jpzq is homogeneous of degree 2j in terms of tu, uu and is alinear combination of multilinear integrals of the formˆ

Σ2j

j¹n1

e2izpynxnqupynqupxnq dx dy ,

with Σ2j denoting an ordering obeying the constraint txn yn, @1 ¤ n ¤ ju.We can calculate for example that

pT2q2 ˆ

x1 y1e2izpy1x1qupy1qupx1q dx 1 dy 1

ˆx2 y2

e2izpy2x2qupy2qupx2q dx 2 dy 2

4

ˆx1 x2 y1 y2

2¹n1

e2izpynxxqupynqupxnq dx dy

2

ˆx1 y1 x2 y2

2¹n1

e2izpynxxqupynqupxnq dx dy : 4 2 ,

which is homogeneous of degree 4 in tu, uu, and the first terms in the expan-sion of lnT1 read as

lnT1pzq 8

j1

T2jpzq p2 q p12 4 q (C.31)

98 [February 4, 2019]

Indeed, we can prove that T2j is a linear combination of connected symbols,that is, Σ2j represents a complete ordering tx1 x2 y1 |xn yn, 1 ¤ n ¤ ju where the first and the last arcs are connected. To this endwe introduce a Hopf algebra structure:

• We consider the set H of the words in the alphabet t , u and in-troduce the shuffle product between two words as the sum of all thewords obtained by shuffling these two words such that H becomesa ring of formal power series with the words as unknowns, with theshuffle product defining the commutative, associative and distributivemultiplication;

• We restrict ourselves in the subalgebra H consisting of nonintersectingwords and the length 2j of the nonintersecting words in H introduces agrading on H such that words with length 2j have degree j and Hm HIm is a finite dimensional algebra where Im is the ideal consisting ofthe elements of degree ¡ m;

• We introduce a coproduct ∆ : H ÞÑ H bH on H as the sum of all thesplittings ∆a °a1a2a,a1,a2PH a1 b a2, @a P H, then the coproduct iscoassociative which also satisfies the compatibility condition ∆pa bq ∆a ∆b with the shuffle product of the tensor product defined aspab bq pcb dq pa cq b pb dq;• We call an element g P H group-like if ∆g g b g, then the set of

all group-like elements in H endowed with the shuffle product becomesa group G. On the other side, we call an element p P H primitiveif ∆p 1 b p p b 1, then the primitive element is formal linearcombinations of connected symbols and all the primitive elements forma subspace P H. The subgroup G and the subspace P are relatedby G expP ;

• In conclusion, the element T1 (C.30) is a group-like element and henceits logarithm lnT1 (C.31) is a primitive element such that its homo-geneous parts T2j are linear combinations of connected symbols.

We are now interested in two issues:

• We would like to show the convergence of the expansions (C.30)-(C.31),and we are in particular interested in the decay rates of the termsT2jpzq, T2jpzq as z Ñ i8, when upxq P HspRq. This will be done inStep 3.

99 [February 4, 2019]

Roughly speaking, for u P SpRq, if z iτ2, the bulk of the integralsin T2jpzq, T2jpzq comes from the regions tyn xn À τ1, 1 ¤ n ¤ ju,and in the case of T2jpzq the volume of the region is comparable toτj while in the case of T2jpzq the volume of the region is comparableto Cpjqτ2j1 which is much smaller than τj if j ¥ 2. Hence forupxq P SpRq we expect to have the expansion of lnT1 in the powersof z1 at infinity:

lnT1pzq i8

j1

Ejp2zqj, as |z| Ñ 8,

where the (invariant) coefficients Ej are called energies and in particular

E1 ˆR|u|2 dx ,

E2 Im

ˆRu1u dx ,

E3 ˆRp|u1|2 |u|4q dx ,

are the (rescaled) mass, momentum and energyM,P,E defined in (1.8),(1.9), (1.10) respectively. We will indeed be interested in the caseu P Hs and study the expansions (C.30), (C.31) by establishing thedecay rates for T2jpzq, T2jpzq as |z| Ñ 8 in terms of u2

Hs .

• We would like to describe u2Hs in terms of T1pzq, which is done in

Step 2.

Step 2 Description of u2Hs in terms of the quadratic term T2pzq

T2pzq Let z P U and let us calculate the quadratic term and relate it to the

Fourier transform of u: upξq 1?2π

´R e

iξxupxq dx as

1

2π

ˆR3

ˆ y

8e2izpyxqeiyηupηqeixξ ¯upξq dx dy dξdη,

where integration by parts in x implies

1

2π

ˆR3

1

2iz iξeiyηixξupηq¯upξq dy dξdη

ˆR

i

2z ξupξq¯upξqdξ,

and hence

1

πRe pz2q 1

π

ˆR

Im z

|z ξ|2 |upξq|2 dξ : 1

π

ˆR

Im z

|z ξ|2dµpξq, (C.32)

100 [February 4, 2019]

which is the harmonic function on the upper half plane with the trace measuredµ |upξq|2 dξ . Hence by a change of integral contour we have the followingdescription of the Hs, 1

2 s 0-norm of u:

u2Hs

ˆRp1 ξ2qs|upξq|2 dξ

2 sinpπsqˆ 8

1

pτ 2 1qs 1

πRe piτ2qdτ.

(C.33)

For general u P Hs with N rss ¥ 0, we have the finite expansion for theabove harmonic function evaluated at z iτ with τ Ñ 8:

1

πRe piτ2q

N

l0

p1qlH2lτ2l1 p1qN1H¡2Npτqτ2N1,

where

H2l 1

π

ˆRξ2ldµpξq, H¡2Npτq 1

π

ˆR

ξ2N2

τ 2 ξ2dµpξq.

Then the description of the Sobolev norm u2Hs in terms of reads as

u2Hs 2 sinpπsq

ˆ 8

1

pτ 2 1qs 1

πRe piτ2q

N

l0

p1qlH2lτ2l1

dτ

πN

l0

sl

H2l. (C.34)

Step 3 Well-definedness of the “transmission coefficient” T1pzq ifu P Hs and the estimates for higher order terms T2jpzq, T2jpzq

We introduce the functional spaces U2, V 2 and DU2, as substitutes for9H

12 and 9H 1

2 , which are more suitable for the estimates for the integrals inT2jpzq, T2jpzq. We define V 2 as the space of the functions v on R such thatthe following norm is finite

vV 2 sup8 t1 tL8

L1

j1

|vptj1q vptjq|2 1

2, where we set vp8q 0.

We call the step function of form°L1j1 φj1rtj ,tj1q with

°j |φj|2 1 a U2

atom and a U2 function u is an `1-sum of U2 atoms such that uU2 infpλkqt

°k |λk| |u

°k λkuk, pλkq P `1pNq and uk are U2-atomsu 8. We

have the following pleasant properties concerning V 2, U2-norms:

101 [February 4, 2019]

• The V 2-functions v have left and right limits everywhere and vp8q 0 such that vL8 ¤ vV 2 and obviously uvV 2 ¤ uV 2vL8 uL8vV 2 ;

• The U2-functions u are right continuous and up8q 0 such thatuV 2 ¤ ?

2uU2 and we have the embedding estimates uB

122,8

ÀuV 2 À uU2 À u

B122,1

;

• There exists a bilinear form Bpv, uq which induces an isometric iso-morphism from V 2 ÞÑ pU2q and from U2 ÞÑ pV 2

Cq, V 2C tv P

V 2 | v is continuous , vp8q 0u, such that 5

vV 2 sup uPU2tBpv, uq | uU2 1u,uU2 sup vPV 2tBpv, uq | vV 2 1u,

where by an abuse of notation Bpv, uq ´R vdu.

We introduce DU2 functions as the distributional derivatives of U2 functions:DU2 tu1 |u P U2u and hence DU2 function is distribution function withthe finite norm fDU2 sup t´R fv dx | vV 2 ¤ 1u. Thus we have

uvDU2 ¤ 2uDU2vV 2 ¤ 4uDU2vU2 .

For any z P U , we define the one step operator A as

Apφqptq ˆx y t

e2izpyxqupyqupxqφpxq dx dy ,

then the terms in the expansion (C.30): T1 1°8j1 T2j read as

T2 limtÑ8

Ap1qptq, T4 limtÑ8

A2p1qptq, T2j limtÑ8

Ajp1qptq,

and T2j can be bounded by the operator norm AV 2 ÞÑU2 as follows:

|T2j| ¤ 2Ajp1qU2 ¤ 2AV 2 ÞÑU2Aj1p1qV 2 ¤ ¤ 2jAjV 2 ÞÑU2 . (C.35)

We hence estimate the operator norm AV 2 ÞÑU2 as follows:

ApφqU2 ˆ

x ye2izpyxqupyqupxqφpxq dx

DU2

y

¤ 4e2iRe zyuDU2yϕ pe2iRe zuφqU2

y, ϕ e2Im zt1t¡0

¤ 4e2iRe zyuDU2 e2iRe zuφDU2 ¤ 8e2iRe zyuDU2 e2iRe zuDU2φV 2 ,

5 We can restrict the supermum to step functions. If sup tBpv, uq | uU2 1u 8,then v P V 2, and if sup tBpv, uq | vV 2 1u 8 and u is right continuous ruled functionwith limtÑ8 uptq 0, then u P U2.

102 [February 4, 2019]

where we used ϕ fU2 ϕ1 fDU2 ¤ fDU2 , and hence

AV 2 ÞÑU2 ¤ 8e2iRe zu2DU2 .

We have indeed better estimates for Im z τ2 ¡ 0: We can introduce thelocalised version of DU2-norms:

ulqτDU2 χr kτ, k1τsuDU2

`qk,

where pχr kτ, k1τsqk form a partition of unity, such that

AV 2 ÞÑU2 ¤ Ce2iRe zu2l2τDU

2 ¤ C1 Re z τ

τu2

l21DU2 , (C.36)

where l21DU2 can be seen as a substitute of H 1

2 such that Hs0ãÑ l21DU

2

whenever s0 ¡ 12.

Therefore if ul21DU2 ! 1, then by virtue of (C.35)-(C.36) the formal series

of T1 1°j¥1 T2j converges in the region tz P U | Im z ¥ 1 |Re z|u and

in particular on the half-line ir1,8q. And generally, if u P l21DU2, then forany z P U , there exist x0, x1 P R such thata

p1 Re z Im zqIm zu|p8,x0sl21DU2 u|rx1,8ql21DU2

¤ 12C ! 1,

and hence we solve the Lax equation (1.22) first until x0 (the solvability isensured by the above argument for the small norm case), and then solve theLax equation from x0 to x1 (the solvability on this finite interval rx0, x1s isensured by u|rx0,x1s P DU2 and hence the Lipschitz property of the linearflow), and finally solve the Lax equation from x1 to 8 (the solvability isalso ensured by the smallness), such that we can still define the holomorphicfunction T1pzq as the first component of the solution at infinity.

In particular if z iτ2, τ ¥ 1, then we have the following estimates forT2j, T2j, j ¥ 2 similar as in (C.35)-(C.36):

|T2j| ¤ pCul2τDU2q2j, |T2j| ¤ Cpjqu2j

l2jτ DU2. (C.37)

Step 4 Conserved energiesMotivated by the formulations (C.33)-(C.34) of the Hs-norm, we define

the conserved energy via

Espuq 2 sinpπsqπ

ˆ 8

1

pτ 2 1qsGpiτ2q dτ , 1

2 s 0, (C.38)

103 [February 4, 2019]

and for general N rss ¥ 0,

Espuq 2 sinpπsqπ

ˆ 8

1

pτ 2 1qsGpiτ2q

N

l0

p1qlG2lτ2l1

dτ

πN

l0

sl

G2l,

(C.39)

where Gpzq Re lnT1pzq is a nonnegative harmonic function (as the realpart of the logarithm of the holomorphic function T1 with |T1| ¥ 1) onthe upper half plane U and G2l are the real coefficients of the expansion ofGpzq at infinity.

For 12 s 0, we want to bound the following difference

|Espuq u2Hs |

2 sinpπsqπ

ˆ 8

1

pτ 2 1qsRe8

j2

T2jpiτ2q dτ.

Recall the estimate (C.37) and we now want to bound the above differencein terms of uHs : Let z iτ2, then by use of the Littlewood-Paley de-composition u °

µ uµ, µ 2j, j 0, 1, such that uµHs dµuHs ,dµ`2 1,ˆ 8

1

pτ 2 1qs|T2piτ2q| dτ ¤ˆ 8

1

pτ 2 1qs¸µ,ν

|Ap1; uµ, uνqpiτ2q| dτ

Àˆ 8

1

pτ 2 1qs¸µ,ν

uµl2τDU2uνl2τDU2 dτ

Àˆ 8

1

pτ 2 1qs¸

µ¥ν¥τdµµ

12sdνν

12su2

Hs dτ ¸

µ¥τ¥ν

¸τ¥µ¥ν

À¸

µ¥ν¥τdµpτ

µq 12sdνpτ

νq 12su2

Hs ¸

µ¥τ¥ν

¸τ¥µ¥ν

À u2Hs ,

and similarly (but more complicatedly) we can boundˆ 8

1

pτ 2 1qsp|T2jpiτ2q| |T2jpiτ2q|q dτ ¤ u2HspCul21DU2q2j2.

Hence if ul21DU2 ! 1, then

|Espuq u2Hs | ¤ Cu2

l21DU2u2

Hs , (C.40)

such that the conserved energy Espuq is equivalent to u2Hs . Therefore the

Sobolev norm uHs is “conserved” by the cubic nonlinear Schrodinger flow

104 [February 4, 2019]

(if the solution exists). Indeed, if initially u02l21DU

2 À u02Hs ¤ c0 ! 1 such

that |Espu0qu02Hs | ¤ 1

2u02

Hs and 12u02

Hs ¤ Espu0q ¤ 2u02Hs ¤ 2c0, then

by the conservation law Espuq Espu0q (if the solution u exists globally intime) and a bootstrap argument, we have (C.40) and the smallness conditionu2

l21DU2 À u2

Hs ¤ 2Espuq 2Espu0q ¤ 4c0 ! 1 for all the times. For

general initial data with u0Hs R 8, we can do scaling u0,λpxq 1λu0pxλq

as in Subsection 1.1.3, such that for λ large enough

u0,λHs À u0,λ 9Hs λps12qR ¤ c0 ! 1, if s P p1

2, 0q,

and for s ¥ 0, we take λ such that λ2R ¤ c0. Finally we get the bound foruHs from the control for uλHs .

For general s ¥ 0, we can also derive the difference estimate |Espuq u2

Hs | ¤ Cu2l21DU

2u2Hs , nevertheless we have to consider the finite expan-

sions of the terms T2j, j 2s 1 (similar as the finite expansion in theintegrand in (C.34) for T2 T2 ) and we do not go further here.

References

[1] Thierry Cazenave, Book “Semilinear Schrodinger equations”, AMS,2004.

[2] Raphael Danchin and Pierre Raphael, Lecture Notes “Soliton, Disper-sion et Explosion - Une introduction a l’etude des ondes nonlineaires.”

[3] Herbert Koch and Daniel Tataru, Conserved energies for the cubicNLS in 1-D, arXiv 1607.02534, 2016.

[4] Felipe Linares and Gustavo Ponce, Book “Introduction to nonlineardispersive equations”, Springer, 2009.

[5] Peter D. Miller, Lecture Notes “Riemann-Hilbert methods in inte-grable systems.”

[6] Jean-Claude Saut, Lecture Notes “Equations d’evolutions disper-sives.”

105 [February 4, 2019]

notes on lecture (01053400) - dispersive equations

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