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On global properties of solutions of some nonlinear Schrödinger-type equations Koo, Eva Hang 2012

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On global properties of solutions of some nonlinear Schr¨ odinger-type equations by Eva Hang Koo  B.Sc., The University of British Columbia, 2005 M.Sc., The University of British Columbia, 2007  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) July 2012 c Eva Hang Koo 2012  Abstract The Schr¨ odinger equation, an equation central to quantum mechanics, is a dispersive equation which means, very roughly speaking, that its solutions have a wave-like nature, and spread out over time. In this thesis, we will consider global behaviour of solutions of two nonlinear variations of the Schr¨ odinger equation. In particular, we consider the nonlinear magnetic Schr¨odinger equation for u : R3 × R → C, iut = (i∇ + A)2 u + V u + g(u), u(x, 0) = u0 (x), where A : R3 → R3 is the magnetic potential, V : R3 → R is the electric potential, and g = ±|u|2 u is the nonlinear term. We show that under suitable assumptions on the electric and magnetic potentials, if the initial data is small enough in H 1 , then the solution of the above equation decomposes uniquely into a standing wave part, which converges as t → ∞, and a dispersive part, which scatters. We also consider the Schr¨odinger map equation ut = u × ∆u for u : R2 × R → S2 . We obtain a global well-posedness result for this equation with radially symmetric initial data without any size restriction on the initial data. Our technique involves translating the Schr¨odinger map equation into a cubic, non-local Schr¨odinger equation via the generalized Hasimoto transform. There, we also show global well-posedness for the nonlocal Schr¨ odinger equation with radially-symmetric initial data in the critical space L2 (R2 ), using the framework of Kenig-Merle and Killip-Tao-Visan. ii  Preface A version of Chapter 2 has been published. Eva Koo, Asymptotic stability of small solitary waves for nonlinear Schrdinger equations with electromagnetic potential in R3 , J. Differential Equations 250 (April 2011), no. 8, 3473-3503. I wrote the entire manuscript. A version of Chapter 3 has been submitted for publication. This is jointwork with my supervisor Stephen Gustafson. Theorem 4 (in particular, the ruling out of possible blow up solutions) is a result of close collaboration with my supervisor. Many of the details were worked out jointly. I was responsible for adapting proofs from previous work (for example, [48] KillipTao-Visan 09) to our situation, and I wrote the first draft of the manuscript.  iii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iv  List of Tables  vi  Preface  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Dedication  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1  1.2  1.3  Dispersive effects and dispersive equations  1  . . . . . . . . . .  1  1.1.1  A gentle introduction to dispersive effects . . . . . . .  1  1.1.2  Variations of Schr¨odinger equations  . . . . . . . . . .  7  . . . . . . . . . . . . . . . . . . . . . . . . . . .  14  Background 1.2.1  Notation  . . . . . . . . . . . . . . . . . . . . . . . . .  15  1.2.2  Local well-posedness  . . . . . . . . . . . . . . . . . .  18  1.2.3  Global well-posedness . . . . . . . . . . . . . . . . . .  23  1.2.4  Solitary waves and their stability  . . . . . . . . . . .  28  1.2.5  Scattering  . . . . . . . . . . . . . . . . . . . . . . . .  34  Main results of the thesis . . . . . . . . . . . . . . . . . . . .  35  2 Asymptotic stability of small solitary waves for nonlinear Schr¨ odinger equations with electromagnetic potential in R3 40 2.1  An overview  . . . . . . . . . . . . . . . . . . . . . . . . . . .  40  iv  Table of Contents 2.2  Our result  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  2.3  Discussion and outline of the proof  2.4  Detailed proof  41  . . . . . . . . . . . . . .  46  . . . . . . . . . . . . . . . . . . . . . . . . . .  50  2.4.1  Existence and decay of standing waves  . . . . . . . .  50  2.4.2  Linear estimates . . . . . . . . . . . . . . . . . . . . .  62  2.4.3  Proof of the main theorem  73  . . . . . . . . . . . . . . .  3 Global well-posedness of two dimensional radial Schr¨ odinger maps into the 2-sphere  . . . . . . . . . . . . . . . . . . . . . .  82  3.1  Known results and our result . . . . . . . . . . . . . . . . . .  82  3.2  Discussion and outline of the proof  85  3.2.1  . . . . . . . . . . . . . .  Outline of Killip-Tao-Visan’s proof of global well-posedness of NLS for radial L2 initial data in 2D  3.2.2  88  Discussion of our proof of global well-posedness of NLNLS for radial L2 initial data in 2D  3.3  . . . . . . . . . . . . . . . .  92  Proof of our result . . . . . . . . . . . . . . . . . . . . . . . .  93  3.3.1  Equating the Schr¨odinger map equation and the NLNLS equation  . . . . . . . . . . . . . . . . . . . . . . . . .  96  3.3.2  Local theory of NLNLS . . . . . . . . . . . . . . . . . 102  3.3.3  Reduction to the three enemies  3.3.4  Extra regularity . . . . . . . . . . . . . . . . . . . . . 104  3.3.5  Nonexistence of the three enemies . . . . . . . . . . . 109  4 Concluding chapter  . . . . . . . . . . . . 103  . . . . . . . . . . . . . . . . . . . . . . . . 114  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116  v  List of Tables 1.1  Classical PDEs and their geometric counterparts . . . . . . .  14  1.2  L2 -  24  and  H 1-  criticality of cubic NLS for n = 1, 2 and 3 . . . .  vi  Acknowledgements I would like to take this opportunity to thank the following groups of people: • my parents, for their care, patience and tolerance • my supervisors, Dr Tsai and Dr Gustafson, for their support and guidance • Erick, for his constructive criticism as well as his encouragement  vii  Dedication  To all my teachers  viii  Chapter 1  Introduction The dynamics in many physical settings – for example, those involving the propagation of light or sound, or the evolution of quantum systems – are well-described by dispersive partial differential equations. Because of their importance, and the subtle properties of their solutions, the mathematical study of dispersive equations has attracted a lot of attention. In Section 1.1, as an introduction to the subject, we will give a relatively mild introduction to dispersive equations and dispersive effects. We will also introduce a few examples of dispersive equations that will be of interest to us. In Section 1.2, we will introduce and discuss some well-studied questions in the field of dispersive equations. In Section 1.3, we will give an overview of the rest of the thesis.  1.1 1.1.1  Dispersive effects and dispersive equations A gentle introduction to dispersive effects  To illustrate the dispersive property, we will consider three well-studied partial differential equations of which only one is truly dispersive. In each of the three equations, let u = u(x, t) be a function of space variable x ∈ Rn and time variable t ∈ [0, ∞). We use ut to denote the partial time derivative of u (i.e. ut =  ∂u ∂t )  and ∆u to denote the Laplacian of u which is given by n  ∆u = j=1  ∂2u . ∂x2j  (1.1)  1  1.1. Dispersive effects and dispersive equations These equations are the heat equation ut = ∆u,  (1.2)  utt = ∆u  (1.3)  ut = i∆u  (1.4)  the wave equation  and the Schr¨ odinger equation  where in each case above, we will consider the function u(·, t) : Rn → C  (1.5)  with initial data u(x, 0) = u0 (x) and for the wave equation ut (x, 0) = v(x) as well. (1.6) To keep this discussion concrete, we will limit the space dimension to n = 1. In this case, the heat equation models the temperature of a thin rod of infinite length, the wave equation models the height of a wave on an infinitely long string and the Schr¨ odinger equation models a free quantum particle in one dimensional space. To gain some insight on the behaviour of solutions of such equations, we will apply the Fourier transform in the variable x to each equation. Let u(ξ, t) :=  1 2π  u(x, t)e−ixξ dx  (1.7)  R  denote the spatial-domain Fourier transform of u(x, t). Using the property uxx (ξ, t) = −ξ 2 u(ξ, t),  (1.8)  the equations now read  2  1.1. Dispersive effects and dispersive equations heat equation: ut (ξ, t) = −ξ 2 u(ξ, t)  (1.9)  utt (ξ, t) = −ξ 2 u(ξ, t)  (1.10)  ut (ξ, t) = −iξ 2 u(ξ, t).  (1.11)  wave equation:  Schr¨ odinger equation:  Each of the equations above is an ordinary differential equation in t, and after solving them, we get heat equation: u(ξ, t) = u(ξ, 0)e−ξ  2t  (1.12)  wave equation: u(ξ, t) = u(ξ, 0) cos(ξt) +  ut (ξ, 0) sin(ξt) ξ  (1.13)  Schr¨ odinger equation: 2  u(ξ, t) = u(ξ, 0)e−iξ t .  (1.14)  We can already read off some differences between the behaviours of the three equations from the above. For example, for the heat equation, the magnitude of each Fourier mode |u(ξ, t)| is exponentially decreasing, while for the Schr¨ odinger equation, the magnitude of each Fourier mode |u(ξ, t)| is fixed in size. By Parseval’s theorem, which says that u  L2x  2  |u(x, t)| dx  := R  1 2  =  √  2π u  L2ξ ,  (1.15)  we see that for the heat equation, the L2x -norm of u is decaying in time while for the Schr¨ odinger equation, the L2x -norm of u stays constant. In fact, the heat equation is an example of a dissipative equation, which means that as 3  1.1. Dispersive effects and dispersive equations time evolves, the solution dies down (dissipates) to 0. On the other hand, we see that the Schr¨ odinger equation does not dissipate. By Parseval’s theorem, the L2 -norm of the solution remains constant. To differentiate the behaviour of the wave equation and the Schr¨odinger equation we can apply the inverse Fourier transform and recover u(x, t) from u(ξ, t) by u(ξ, t)eixξ dξ.  u(x, t) =  (1.16)  R  For the sake of discussion, suppose the initial data of the wave equation are chosen so that u(ξ, 0) = δξ0 (ξ) and ut (ξ, 0) = 0 for some fixed frequency ξ0 ∈ R. Here, δξ0 (ξ) = δ(ξ − ξ0 ) is the delta function. Then u is given by u(ξ, t) =  1 δξ0 (ξ)eiξ0 t + δξ0 (ξ)e−iξ0 t . 2  (1.17)  Hence, the solution u of the wave equation is given by u(ξ, t)eixξ dξ =  u(x, t) = R  1 iξ0 (x+t) e + eiξ0 (x−t) . 2  (1.18)  This is a sum of two plane waves, one moving to the left at speed 1 and one moving to the right at speed 1. In fact, for general initial data u(x, 0) = u0 (x) and ut (x, 0) = v0 (x), the solution of the wave equation is given by the d’Alembert formula 1 1 u(x, t) = (u0 (x + t) + u0 (x − t)) + 2 2  x+t  v0 (y) dy.  (1.19)  x−t  In the case where v0 ≡ 0, the solution of the wave equation is 1 u(x, t) = (u0 (x + t) + u0 (x − t)). 2  (1.20)  In particular, we see that as time evolves, the initial shape of the wave divides into two equal parts, where one part moves to the right and the other moves to the left at an uniform speed. This is the wave behaviour in which the original waveform move outwards at a uniform speed. We will see next that the behaviour of the Schr¨odinger equation is dif4  1.1. Dispersive effects and dispersive equations ferent from that of the heat equation and the wave equation. Now, for the Schr¨ odinger equation, we have the solution u(ξ, 0)ei(x−ξt)ξ dξ  u(x, t) =  (1.21)  R  where u(ξ, 0) =  1 2π  u(x, 0)e−ixξ dx.  (1.22)  R  Let us again consider a special initial condition u(x, 0) made up of only one Fourier mode. Such a u0 is a plane wave which, of course, is not in the space L2 (R2 ), the space of square-integrable functions on the real line. One typically requires that initial data u0 be in L2 . Nevertheless, we will consider such a u0 for the sake of discussion. In other words, suppose u(ξ, 0) = δξ0 (ξ) for some fixed ξ0 ∈ R.  (1.23)  In this case, the solution of the Schr¨odinger equation will read u(x, t) = ei(x−ξ0 t)ξ0 .  (1.24)  This is the equation of a plane wave moving at speed ξ0 . Under the evolution of the Schr¨ odinger equation, an initial data u0 which consists of a single Fourier mode ξ0 remains a function with only that Fourier mode and which moves in the physical space at a speed of ξ0 . Now, instead of an initial condition u(x, 0) = u0 (x) consisting of a single Fourier mode, if our initial condition consists of a continuum of Fourier modes, then the part of the solution having Fourier mode ξ will move at a speed of ξ. As a result, components of the solution with higher Fourier modes will move faster than those with smaller Fourier modes. As a result, over time, the solution will spread out, and the amplitude of the solution will decay over time. However, as we showed before, the L2 -norm remains unchanged. This spreading out of the solutions is what we call the dispersive effect. There are equations other than the Schr¨odinger equation that are dis5  1.1. Dispersive effects and dispersive equations persive in nature. Other dispersive equations include the Klein-Gordon equation utt − ∆u + m2 u = 0,  (1.25)  and the Airy equation (which is the linear part of the well studied KdV equation) ut + uxxx = 0.  (1.26)  Recall that we showed that for the one dimensional Schr¨odinger equation, the Fourier transform of the solution u is given by u(ξ, t) = e−ih(ξ)t u0 (ξ) where h(ξ) = ξ 2 .  (1.27)  Here, h(ξ) is called the dispersion relation. As before, if we apply the inverse Fourier transform of u(ξ, t) = eih(ξ)t u0 (ξ), we find that the component of the solution with the Fourier mode ξ travels at a speed equal to  h(ξ) ξ .  In  fact, associated with the dispersion relation are two different velocities, the phase velocity h ξ  (1.28)  dh . dξ  (1.29)  vp = and the group velocity vg =  Here, if one imagines the solution is modulated by an overall wave envelope (wave packet) and the solution oscillates within the wave packet, then the group velocity is the speed of the wave packet while the phase velocity is the speed of the oscillations within the wave packet (see [93]). For the Schr¨ odinger equation, vp = ξ and vg = 2ξ  (1.30)  and we see that the group velocity is faster than the phase velocity. Just like for the Schr¨ odinger equation, we can get dispersion relations for other dispersive equations. For the Airy equation, we have ut = iξ 3 u  (1.31)  6  1.1. Dispersive effects and dispersive equations which gives 3  u(ξ, t) = eiξ t u(ξ, 0),  (1.32)  so for the Airy equation, the dispersion relation is h(ξ) = ξ 3 . The KleinGordon equation is second order in time. To get the dispersion relation, we look for a function h(ξ) so that u = ei(ξx−h(ξ)t)  (1.33)  satisfies the Klein-Gordon equation. Notice that this way of computating h(ξ) is consistent with the ways in which h(ξ) is computed above. Substituting (1.33) into the Klein-Gordon equation, we get that −h2 (ξ) + ξ 2 + m2 = 0  (1.34)  h(ξ) = ± m2 + ξ 2 .  (1.35)  which gives  1.1.2  Variations of Schr¨ odinger equations  In the previous subsection, we introduced the dispersive effect. There, we worked mainly with the free Schr¨odinger equation which is a standard model of linear dispersive equations. In this subsection, we will introduce certain variations of the free Schr¨ odinger equation and discuss how each variation changes the behaviour of the equation. Schr¨ odinger equation with potential In the previous subsection, we considered what is called the free Schr¨odinger equation iut = −∆u  (1.36)  which models the evolution of a quantum particle in the absence of any external fields. In this subsection, we will consider the Schr¨odinger equation  7  1.1. Dispersive effects and dispersive equations with a potential iut = −∆u + V u  (1.37)  which models the evolution of a quantum particle under the influence of a potential V (x). Here, we will assume that the potential V only varies in space but not in time. Equation (1.37) is central to modern physics. For example, if we take V 1 to be the Coulomb potential V = − |x| , then for n = 3, equation (1.37) mod-  els an electron moving around the nucleus. Applications of equation (1.37) range from predictions of chemical properties in chemical compounds and computations and explanations of physical properties in solid state physics (see introductory textbooks such as [36] and [1]). To gain a glimpse of the effect of the potential on the solution, let us consider an extreme case of our potential which, because of the simplicity of the solutions, is commonly found in introductory textbooks on quantum mechanics. Consider the one-dimensional Schr¨odinger equation iut = −∆u + V (x)u  (1.38)  where the potential is given by  0 if 0 < x < 1 V (x) = . ∞ if x ≤ 0 or x ≥ 1  (1.39)  This is commonly known as the one-dimensional infinite square well. The general solution to this equation is given by  u(x, t) =     ∞ −iλ2n t n=1 an sin(λn x)e  where λn = nπ  .  (1.40)  0 if x ∈ (0, 1) If we look at (1.40), we see that unlike solutions to the free Schr¨odinger equation, the solution to (1.38) does not spread out in space over time as the potential V is trapping the solutions within the region 0 < x < 1 8  1.1. Dispersive effects and dispersive equations (i.e. infinite square well). On one hand, the free part of equation (1.38) (ut = i∆u) tries to spread the solution out but on the other hand, the potential V confines the solution to the infinite square well. As a result of the confining part of the equation, equation (1.38) admits some stationary 2  solutions of the form sin(λn x)eiλn t for n = 1, 2, 3, · · · . These solutions are 2  known as standing wave solutions since other than the complex phase eiλn t , they are stationary in time. These standing waves are examples of bound states – states which remain spatially localized for all time. The infinite square well is not the only potential that gives rise to bound states. Other less confining potentials give rise to bound states as well. For example, when V (x) = x2 , equation (1.38) is called the harmonic oscillator, another standard example in introductory textbooks of quantum mechanics. Computations to derive the solutions are more complicated than the infinite square well, but nevertheless, solutions of the harmonic oscillator is given by  ∞ 1  an e−i2(n+ 2 )t Hn (x)e−  u(x, t) =  x2 2  .  (1.41)  n=0  Here, Hn is the Hermite polynomial, a degree n polynomial whose the explicit form can be worked out for each n. In the above, each summand 1  e−i2(n+ 2 )t Hn (x)e−  x2 2  is a bound state of the harmonic oscillator with en-  ergy level En = 2(n + 12 ). See [36]. So far, we have only been considering confining potentials. Let us instead consider yet another standard example in introductory texts in quantum mechanics, which instead of a confining potential, is an attractive potential  −V 0 V (x) = 0  for − 1 < x < 1  (1.42)  for |x| ≥ 1  for some V0 > 0. (As an aside, the case where V0 < 0 is called a repulsive potential.) This is known as the one-dimensional finite square well. In this case, equation (1.37) admits solutions of the form u(x, t) = e−iEt φ(x)  (1.43) 9  1.1. Dispersive effects and dispersive equations for both E < 0 and E > 0. When E < 0, solutions of the form (1.43) only exist for a finite number of E = E1 , E2 , . . . , En , where n depends on V0 . The larger V0 is, the larger n is. Solutions with E < 0 are the bound states and are highly localized in space. On the other hand, solutions with E > 0 are called scattering states and are not localized in space (in fact, not in L2 ). Again, see [36]. For more general potential V , it turns out that if V → ∞ as |x| → ∞, then (1.37) only admits bound state. This is due to the potential stopping the solutions from escaping. On the other hand, if V → 0 as |x| → ∞, then it is possible for dispersive solutions and bound states to co-exist. See the textbooks such as [42], for example. Nonlinear Schr¨ odinger equations Next, instead of adding a potential V to the free Schr¨odinger equation iut = −∆u,  (1.44)  we will add a nonlinear term. Nonlinear versions of the Schr¨odinger equation arise in many applications, including optics, magnetics, and Bose-Einstein condensation (see [78] or [2]). As a standard example, we add in the nonlinearity λ|u|p−1 u to get the nonlinear Schr¨odinger equation iut = −∆u + λ|u|p−1 u.  (1.45)  Our next task is to see the effect of the nonlinear term on the equation. Without the nonlinear term, the different Fourier modes of a solution of the Schr¨ odinger equation act independently. As the Schr¨odinger equation is dispersive, as time evolves, the higher Fourier modes travel faster than those with a lower Fourier mode, so the solution spreads out in space over time. With the nonlinear term, the different Fourier modes no longer evolve independently and the interaction of different Fourier modes gives rise to more complex behaviour. By rescaling the function u, it is only necessary to consider λ = ±1. 10  1.1. Dispersive effects and dispersive equations It turns out the behaviour of (1.45) is different according to the sign of λ. When λ < 0, the nonlinear term λ|u|2 u acts like an attractive potential when u is large and it has little effect on the equation when u is small. Because the nonlinear term acts like a attractive potential, on one hand, the linear part of (1.45) iut − ∆u tries to disperse the solution but on the other hand, the nonlinear part |u|2 u attempts to confine or even concentrate the solution. As the solution evolves in time, there is a competition between the dispersive effect and the concentration effect. When a delicate balance is achieved between the two opposite effects, equation (1.45) gives rise to a special form of solution which neither concentrates nor disperses. These special solutions are like the analogue of the standing wave of the finite square well we saw in the last section. These special solutions of (1.45) are called solitons or solitary waves. On the other hand, when λ > 0, the nonlinear term behaves more like a repulsive potential. As a result, solutions in this case are always dispersive. Because of the difference of the behaviour between λ > 0 and λ < 0, the case where λ < 0 is called the focusing case while the case where λ > 0 is called the defocusing case. Other than the above heuristic regarding attractive and repulsive potentials , one can also get a glimpse of the difference in behaviour of (1.45) in the signs of λ from a conserved quantity of the equation (1.45) called the energy, which is defined by E(u) = Rn  The term  1 2 2 |∇u|  1 λ |∇u|2 + |u|p+1 dx. 2 p+1  (1.46)  in the expression for E comes from the dispersive term  −∆u while the term  λ p+1 p+1 |u|  comes from the nonlinear term |u|p−1 u. From  this, we see that when λ < 0, the nonlinear term |u|p−1 u is working in the opposite direction as the dispersive term ∆u, while when λ > 0, the nonlinear term works in the same direction as the dispersive term.  11  1.1. Dispersive effects and dispersive equations Schr¨ odinger map So far, we have considered two different variations of the free Schr¨odinger equation ut = i∆u.  (1.47)  In particular, we discussed the behaviour of the solutions with the addition of a potential V or nonlinear term |u|p−1 u. In Chapter 2, we will consider the Schr¨ odinger equation with the addition of both the potential term and the nonlinear term. In all of these variations, the solution u is a map from Rn × R (n spatial dimensions and one time dimension) to C. We would like to consider a variation of the Schr¨odinger equation where the target space C is replaced by a manifold M . For our discussion, we will only consider the case where the target manifold M is the 2-sphere S2 . In this case, the equation analogous to (1.47) is the Schr¨odinger map equation ut = u × ∆u  (1.48)  where u : Rn × R → S2 . Here, we treat the 2-sphere as a sphere embedded in R3 , i.e. S2 = {u ∈ R3 : |u| = 1} ⊂ R3 .  (1.49)  u(x, t) = (u1 (x, t), u2 (x, t), u3 (x, t))  (1.50)  u21 + u22 + u23 = 1.  (1.51)  Hence, we view u as  where  Now for each u ∈ S2 , we define the operator J u by J u ξ = u × ξ.  (1.52)  12  1.1. Dispersive effects and dispersive equations If we let Tu S2 = {ξ ∈ R3 |u · ξ = 0}  (1.53)  to be the tangent plane at u ∈ S2 , we can view J u as an operator that rotates vectors on the tangent plane Tu S2 by 90◦ , just as multiplication by i rotates a vector in the complex plane by 90◦ . With this notation, the Schr¨odinger map equation can be written as ut = J u ∆u.  (1.54)  If we compare equation (1.47) with equation (1.54), we see that they look very similar. This is why we treat equation (1.48) as an analogue of (1.47). Despite the similarity in look, there is a major difference between equation (1.47) and (1.54): equation (1.47) is a linear while equation (1.54) is nonlinear. Due to the self-interaction of a solution caused by the nonlinearity, behaviours of solutions of equation of (1.54) are more complex and are not fully understood. The Schr¨ odinger map equation arises from a model for ferromagnetism introduced by Landau and Lifshitz in 1935 ([54]) and can also be viewed as the continuous version of the Heisenberg model of a ferromagnet ([79]). In such a model, u represents either (classical) the spin of an atom or the magnetization of a magnetic material. In this model, the equation for u takes a more general form ∂u = αu × [∆u + a(u)] + βu × (u × [∆u + a(u)]) ∂t  (1.55)  for some parameters α and β. Here αu × [∆u + a(u)] is the precession term describing the revolution of the magnetization vector about an effective magnetic field ∆u + a(u), βu × (u × [∆u + a(u)]) is a dissipation term, and a : R3 → R3 is a vector field representing an anisotropy in the magnet. When dissipation is absent (i.e. when β = 0) and in the isotropic case (a = 0), we are left with ∂u = αu × ∆u ∂t  (1.56)  13  1.2. Background u(·, t) : Rn → C or R ut = i∆u (Schr¨ odinger equation) ∆u = 0 (Laplace equation) ut = ∆u (heat equation) utt = ∆u (wave equation)  u(·, t) : Rn → S2 ⊂ R3 ut = J u ∆u (Schr¨odinger map) ∆u + |∇u|2 u = 0 (harmonic map) ut = ∆u + |∇u|2 u (harmonic map heat flow) utt = ∆u + (|∇u|2 − |ut |2 )u (wave map)  Table 1.1: Classical pde’s and their geometric counterparts.  which is essentially the Schr¨odinger map equation. Other than the physical applications, there are mathematical reasons for studying Schr¨ odinger maps. In fact, one can view the Schr¨odinger maps as a natural extension of the linear Schr¨odinger equation ut = i∆u with the target space C replaced by a K¨ahler manifold. There are other geometric map equations that can be thought of as geometric analogues of some classical PDEs. In Table 1.1, we list the classical PDEs in the left column and their geometric counterparts on the right. These geometric map equations have been the subject of intense mathematical study (to name a few, [77], [15] for the harmonic map heat flow, [80], [52], [67] for the wave map and [16], [4] for the Schr¨ odinger map). We should note that in the Table 1.1, the classical PDEs are all linear but the geometric PDEs are nonlinear due to the non-trivial geometry of the target space.  1.2  Background  In this section, we will introduce several common topics in the study of nonlinear dispersive equations. They are • local well-posedness; • global well-posedness; 14  1.2. Background • special solutions such as solitary waves; • stability of special solutions; • scattering. In the subsections, we will define the topics in this list and discuss some known results. More in-depth expositions on many of these topics can be found in books on dispersive equations such as [75], [78], [8], [26], [82] and [2]. Here, we will focus mainly on nonlinear Schr¨odinger equations with a power nonlinearity iut = −∆u + λ|u|p−1 u  (1.57)  u(x, 0) = u0 (x) ∈ X  (1.58)  with initial condition  for X = L2 (Rn ) or H 1 (Rn ) (to be defined in the upcoming subsection).  1.2.1  Notation  Before we begin, we should define some notation that will be useful for the later sections of this chapter. For quantities A and B, we use the notation A  B  (1.59)  to mean A ≤ CB  (1.60)  for some constant C > 0. Similarly, we use the notation A  p  B  (1.61)  to highlight that the constant C may depend on the parameter p. Next, we will define various function spaces that will be important for later sections. For each 1 ≤ p ≤ ∞, let Lp (Rn ) denote the space of Lebesgue 15  1.2. Background measurable functions f in Rn up to a.e. equivalence such that the Lp norm defined by f  Lp (Rn )  p  1 p  |f | dx  :=  if 1 ≤ p < ∞  (1.62)  Rn  and f  L∞ (Rn )  := ess sup |f |  (1.63)  Rn  is finite. Let C ∞ (Rn ) be the space of infinitely differentiable functions in Rn . Let α = (α1 , α2 , . . . , αn ) where αi ≥ 0 for i = 1, 2, . . . , n. Here, α is called a multi-index. We define |α| = α1 + α2 + · · · + αn . We use the notation Dα to denote the differential operator Dα = ∂xα11 ∂xα22 · · · ∂xαnn .  (1.64)  For each 1 ≤ p < ∞ and positive integer m, we will define the Sobolev space W m,p (Rn ) to be the closure of {u ∈ C ∞ (Rn ) | Dα u ∈ Lp , ∀ |α| ≤ m}  (1.65)  under the norm u  m,p  Dα u  :=  Lp (Rn ) .  (1.66)  |α|≤m  The case p = 2 is special in that W m,2 is a Hilbert space (a complete normed linear space with an inner product). We will write W m,2 as H m . We have so far defined H m for positive integer m; one can also define the space H s for s ∈ R. To do so, let the Schwartz space S be defined by S = {u ∈ C ∞ (Rn )| sup |xα Dβ u| < ∞ for all multi-indices α and β}. x∈Rn  (1.67) Here, for x ∈ Rn and multi-index α = (α1 , α2 , . . . , αn ), xα denotes  xα1 1 xα2 2  · · · xαnn .  The space H s (Rn ) for s ∈ R is defined to be the closure of the Schwartz  16  1.2. Background space under the norm u  Hs  s  2  |(1 + |ξ| )u(ξ)| dξ  :=  1 2  (1.68)  Rn  where u denotes the Fourier transform of u. Next, we will give some motivation for why these spaces are central to the study of partial differential equations. For simplicity, consider the free Schr¨ odinger equation  iu = −∆u t u(x, 0) = u (x) 0  .  (1.69)  Here, u0 : Rn → C and u : Rn × R → C. One can think of the free Schr¨ odinger equation as a mapping which takes an initial condition u0 : Rn → C and maps it into the function u : Rn × R → C. However, there is a different way to think of the free Schr¨odinger equation. If instead, one treats u0 as an object in some functional space X such as L2 or H 1 and for each t, one treats u(·, t) as an object in some functional space Y , then the free Schr¨ odinger equation can be thought of as a mapping from X to t → Y . In other words, given an initial condition u0 ∈ X, the free Schr¨ odinger equation returns an element in Y for each t as long as the solution is defined. Viewed this way, partial differential equations are infinite dimensional analogues of ordinary differential equations. For example, an ordinary differential equation such as   du = M u dt u(0) = u ∈ Rn 0  (1.70)  where M is an n × n matrix, can be thought of as a mapping from the vector space Rn to t → Rn . The main difference between ordinary differential equations and partial differential equations is that Rn is a finite dimensional vector space but functional spaces such as L2 and H 1 are infinite dimensional. Because of this difference, behaviours of partial differential equations are more complex and less well understood than ordinary differential equa-  17  1.2. Background tions. Let X and Y be some functional spaces. If the free Schr¨odinger equation  iu = −∆u t u(x, 0) = u (x) ∈ X 0  (1.71)  admits an unique solution u(x, t) ∈ Y , we will define its solution operator ei∆t as the mapping from X to Y such that ei∆t u0 (x) = u(x, t).  1.2.2  (1.72)  Local well-posedness  Given an equation modelling some dynamical physical process, a beginning step to ensuring the equation is a good model is to ensure that given an initial condition u0 , there is a solution, which exists at least up to some time T > 0, to the equation. One further wishes that such a solution be unique and that it depends continuously on the initial data. Roughly speaking, this is the idea of local well-posedness. For our discussion, consider the nonlinear Schr¨odinger equation (NLS) iut = −∆u + λ|u|p−1 u  (1.73)  with initial condition u(x, 0) = u0 (x) ∈ X ∀x ∈ Rn  (1.74)  for some functional space X such as H 1 or L2 . Local well-posedness concerns the following questions: Existence: Does there exist a time T > 0 where there is a solution u of (1.73) defined in C((−T, T ), X) (i.e. a continuous function of time into the space X)? Uniqueness: Is the solution unique? 18  1.2. Background  Continuous dependence on data: Does the solution u depend continuously on the initial data? If the answer to the above question is affirmative, one can further ask whether there is a blow-up alternative. What this means is that, suppose the solution u to (1.73) exists up to a maximal time Tmax < ∞; one wants to know whether lim  t→Tmax−  u  X  = ∞.  (1.75)  In other words, if a solution fails to exist at a certain point in time, one wants to know if that is due to the X-norm of the solution blowing up. Existence theory of (1.73) for X = H 1 is obtained in a series of papers by authors such as [3], [72], [43] and [45]. Their results give Theorem 1. (local well-posedness of NLS in H 1 ) Let pmax be defined by  pmax   ∞ =  n+2  n−2  if n = 1, 2  .  (1.76)  if n ≥ 3  Suppose 1 < p < pmax , and suppose u0 ∈ H 1 (Rn ). Then there exists a T > 0 and a unique solution u to (1.73) which depends continuously on u0 in H 1 such that u ∈ C((−T, T ), H 1 (Rn )). Furthermore, the following quantities mass: M (u) := u  energy: E(u) :=  (1.77)  L2  1 λ |∇u|2 + |u|p+1 2 p+1  dx  (1.78)  are conserved. It should be noted that for functions u in L2 (Rn ) or H 1 (Rn ), ∆u may not be defined. When we say u is a solution of (1.73) in L2 (Rn ) or H 1 (Rn ), we actually mean u satisfies the integral form (1.83) of (1.73). In other 19  1.2. Background words, our sense of solution is weaker than the classical sense in which u is required to satisfy (1.73). Existence theory of (1.73) for X = L2 is obtained in a series of papers by authors such as [72] and [88]. Their results give Theorem 2. (local well-posedness of NLS in L2 ) Let pc be defined by pc = 1 +  4 . n  (1.79)  Suppose 1 ≤ p < pc , and suppose u0 ∈ L2 (Rn ). Then there exists a T > 0 and a unique solution u to (1.73) such that u ∈ C((−T, T ), L2 (Rn )). Furthermore, the mass M (u) := u  (1.80)  L2  is a conserved quantity. In the former theorem, pmax is the p in (1.73) such that the homogeneous H 1 -norm (H˙ 1 ) remains unchanged under the solution-invariant scal2  ing u(x, t) → µ p−1 u(µx, µ2 t) for µ > 0. In other words, if u is a solution of 2  (1.73) with p = pmax and v(x, t) = µ p−1 u(µx, µ2 t), then v is also a solution of (1.73) and v(·, t)  H˙ 1 (Rn )  2  |∇v(x, t)| dx  :=  1 2  = u(·, µ2 t)  Rn  H˙ 1 (Rn ) .  (1.81)  Similarly, pc is the p in (1.73) such that the L2 -norm remains unchanged un2  der the solution invariant scaling u(x, t) → µ p−1 u(µx, µ2 t). In other words, 2  let u be a solution of (1.73) with p = pc and let v(x, t) = µ p−1 u(µx, µ2 t), then v is also a solution of (1.73) and v(·, t)  L2 (Rn )  = u(·, µ2 t)  L2 (Rn ) .  (1.82)  The case where p < pmax is called H1 -subcritical and the case where p < pc is called L2 -subcritical. The case where p = pmax is called H1 -critical while the case where p = pc is called L2 -critical. The proof of existence in critical cases is more 20  1.2. Background difficult because at the critical power (p = pmax or pc ) the norm (H 1 or L2 ) cannot be used to control the nonlinear term |u|p−1 u. Existence theorems for (1.73) in the H 1 -critical case and L2 -critical case were obtained by [14]. Local existence for the H 1 -subcritical case with H 1 initial data (Theorem 1) and the L2 -subcritical case with L2 initial data (Theorem 2) are proved using a contraction mapping argument. The idea is to reformulate the partial differential equation (1.73) into the integral form t  u(x, t) = ei∆t u0 (x) − i  ei∆(t−τ ) |u(x, τ )|p−1 u(x, τ )dτ.  (1.83)  0  Working with this reformation of the original problem, various dispersive estimates such as the decay estimate and Strichartz estimates are employed to complete the contraction mapping argument. Here, the decay estimate says that eit∆ u0 where  1 q  +  1 q  Lq (Rn )  −n( 21 − 1q )  |t|  u0  Lq (Rn )  (1.84)  = 1 and 2 ≤ q ≤ ∞. The decay estimate is a special property  of dispersive equations. For example, if we take q = ∞ and q = 1, the decay estimate tells us that eit∆ u0 In the case where u0  L1 (Rn )  n  L∞ (Rn )  |t|− 2 u0  L1 (Rn ) .  (1.85)  is finite, such an inequality gives us the rate at  which the height of the function eit∆ u0 is decreasing. Since the L2 -norm of eit∆ u0 is fixed, the decrease in the L∞ -norm of eit∆ u0 is due to u dispersing. The decay estimate alone is not enough to prove many results. The reason is that the Lq -norm and the Lq -norm are not equivalent unless q = q = 2. With the decay estimate alone, unless p = p = 2, to control the Lp -norm of eit∆ u, we need control of the Lp -norm of u for which we have no estimate. As the decay estimates alone are not enough to prove well posedness results, we need other estimates, called Strichartz estimates. Strichartz estimates are mixed space-time estimates, meaning that they control certain  21  1.2. Background space-time integrals of the solution. Let ·  Lqt Lrx  denote  ·  Lrx (Rn ) Lq (R) . t  (1.86)  We also define the pair (q, r) to be admissible if 2 n n + = for q, r ∈ [2, ∞] and (q, r, n) = (2, ∞, 2). q r 2  (1.87)  The Strichartz estimates say that for admissible (q, r) and (˜ q , r˜), eit∆ u0  Lqt Lrx  u0  (1.88)  L2x  and t  ei(t−τ )∆ f (τ ) dτ 0  f Lqt Lrx  Lqt˜ Lrx˜  (1.89)  where q˜ and r˜ satisfy 1 1 1 1 + = 1 and + = 1. q˜ q˜ r˜ r˜  (1.90)  Equation (1.88) is the homogeneous Strichartz estimate allowing us to handle the ei∆t u0 (x) term in (1.83) while Equation (1.89) is the inhomogeneous Strichartz estimate allowing us to handle the  t i∆(t−τ ) |u(x, τ )|p−1 u(x, τ )dτ 0 e  term in (1.83). Such estimates originate from [76]. The inhomogeneous Strichartz estimates are developed by [94] and [13]. The end point case 2n ((q, r) = (2, n−2 ) for n ≥ 3) is given by [46]. With the decay estimate  and Strichartz estimates, local well-posedness is shown using a contraction mapping argument. It should be mentioned that the proofs for Theorem 1 and 2 hold for more general nonlinear terms than |u|p−1 u. However, for our discussion, we will only consider pure power nonlinearities |u|p−1 u. Following the proofs of the local existence for the H 1 -subcritical case with initial data in H 1 , the duration of existence T depends on the dimension n, the power of nonlinearity p and the norm of the initial data u0  H1 .  22  1.2. Background Similarly, for the L2 -subcritical case with initial data in L2 , the length of time of existence T depends on the dimension n, the power of nonlinearity p and the norm of the initial data u0  L2 .  In fact, in both cases, for fixed n  and p, T is mainly determined by a quantity like C u0  −m X  (1.91)  for some constants C and m > 0 depending on n and p for X = H 1 or L2 . There is local well-posedness also for the critical cases, p = pmax or p = pc , but then the time of existence T depends on not just the norm of the initial data but also on the structure of the initial data.  1.2.3  Global well-posedness  Given an equation modelling some physical process, suppose the equation is locally well-posed, then we know given an initial condition u0 , there exists a unique solution that depends continuously on the initial data. Local wellposedness guarantees that such a solution will exist up to time T , after which it may fail to exist. If a solution fails to exist after time T , this indicates that the equation is only a good model of the physical process up to time T . As a result, after local well-posedness of an equation is obtained, a natural question to ask is whether such a solution exists indefinitely or if it instead blows up in finite time. This is the question of global well-posedness. In the last subsection, we saw that for the equation  iu + ∆u = ±|u|p−1 u t u(x, 0) = u (x) ∈ X for X = H 1 (Rn ) or L2 (Rn ) 0  ,  (1.92)  the local existence theorem says that for suitable ranges of p, there exists a time T > 0 such that there exists a solution u(x, t) of (1.92) where u(x, t) ∈ C((−T, T ), X). In this subsection, we will consider the question of whether T = ∞, in which case, we say that the equation is globally well-posed. In this subsection, we will restrict our discussion to the well-studied and physically 23  1.2. Background L2 -subcritical L2 -critical L2 -supercritical  n=1 n=2 n=3  H 1 -subcritical H 1 -subcritical H 1 -subcritical  Table 1.2: L2 - and H 1 - criticality of cubic NLS for n = 1, 2 and 3.  important cubic nonlinear Schr¨odinger equation  iu + ∆u = λ|u|2 u t u(x, 0) = u (x) ∈ X for X = H 1 (Rn ) or L2 (Rn ) 0  (1.93)  for λ = ±1 and for space dimension n = 1, 2 and 3. Table 1.2 summarizes the L2 - and H 1 - criticality for n = 1, 2 and 3. Recall from local well-posedness, for the subcritical case, the length of existence T is a function of only u0 bound on u(t)  X,  X.  Thus, if one can obtain an a priori  one automatically gets global well-posedness. Indeed,  suppose we know that u up to time t = T ( u0  X  X)  < M , then local existence guarantees a solution  ≡ t1 . We can then take the solution at t = t1  as initial data and the local existence will guarantee a solution up to time t = T ( u(·, t1 )  X)  ≡ t2 . This process can be iterated and the total length ∞ j=1 tj .  of existence will be given by tj ≥ T (M ) > 0. Hence, Since the get  L2  L2  ∞ j=1 tj  Now, since u  X  < M , for each j,  = ∞.  norm is a conserved quantity of (1.92), we automatically  global well-posedness for n = 1. Similarly, as the energy E(u) =  1 λ |∇u|2 + |u|4 2 4  dx  (1.94)  is a conserved quantity, in the case λ = 1 (the defocusing case), this gives 1 1 |∇u|2 + |u|4 2 4  dx = E(u0 ).  (1.95)  The above gives 1 |∇u|2 dx < E(u0 ) 2  (1.96)  24  1.2. Background and thereby together with L2 -conservation yields an upper bound on u Therefore, for the defocusing case λ = 1, (1.93) is globally well-posed in  H1 . H1  for n = 1, 2 and 3. The situation is different for the focusing case λ = −1. In this case, we have  1 1 |∇u|2 − |u|4 2 4  Here, we see that the nonlinear term dispersive term  1 2 2 |∇u|  dx = E(u0 ).  −1 p−1 p+1 |u|  (1.97)  is now acting against the  in the expression for energy. In this case, a way to  show global well-posedness is to try to use the dispersive term control the nonlinear term  1 4 4 |u| .  1 2 2 |∇u|  to  In fact, the Galiardo-Nirenberg inequality  (see, for example, [26]) gives that for p ≤ pmax , α  |u|p+1 dx for α =  n(p−1) 4  and β =  β  |∇u|2 dx  2+n+(2−n)p . 4  |u|2 dx  (1.98)  As we have chosen p = 3, we get that  for n ≤ 4, 4  |u| dx  n 2  2  2  |∇u| dx  2− n 2  |u| dx  .  (1.99)  For n = 1, energy conservation and the above give us 1 1 |∇u|2 dx = E(u0 ) + 2 4  |u|4 dx 2  |∇u| dx  E(u0 ) +  1 2  2  |u| dx  3 2  .  Using Young’s inequality in the form 2  ab ≤  2  a2 +  1 2 b , 2 2  (1.100)  25  1.2. Background for sufficiently small , we get that 3  1 1 |∇u|2 dx ≤ E(u0 ) + 2 4  |∇u|2 dx + C  |u|2 dx  for some constant C. and this shows 3  |∇u|2 dx  E(u0 ) +  |u|2 dx  .  (1.101)  Since the L2 -norm of u is a conserved quantity, this shows that u  H1  is  bounded uniformly in time, so we have global well-posedness for n = 1. On the other hand, unlike n = 1, for the focusing case in dimensions n = 2 or n = 3, there exist solutions that blow up in finite time. [35] showed if the initial data u0 ∈ H 1 (Rn ) ∩ L2 (Rn ; |x|2 dx) and E(u0 ) < 0, then the solution u blows up in finite time. Here, L2 (Rn ; |x|2 dx) denotes the space of functions v : Rn → C with vx  < ∞.  (1.102)  |x|2 |u(x, t)|2 dx.  (1.103)  L2 (Rn )  The idea is to consider the quantity I(t) = Rn  Here, if we think of |u|2 as a probability density, then I is essentially the second moment. If u is a solution of (1.92), simple calculations show that I = 16E + λ  4n (p − pc ) p+1  |u|p+1 dx.  (1.104)  For n = 2, n = 3, p = 3 ≥ pc . Hence, in the focusing case λ = −1, the term 4n λ p+1 (p − pc )  |u|p+1 dx is non-positive. Hence, I ≤ 16E.  (1.105)  Since energy is a conserved quantity, the above says that I has a constant negative concavity meaning that it will go below 0 in finite time. However, 26  1.2. Background the quantity I is a positive quantity, so the solution u must have failed to exist before this time. Ozawa-Tsutsumi ([59], [58]) later removed the assumption that the initial condition be in L2 (Rn ; |x|2 dx) and showed if u0 ∈ H 1 is radially symmetric and E(u0 ) < 0, then the solution u blows up in finite time. For n ≤ 3, (1.93) is H 1 -subcritical. However, (1.93) is H 1 -critical for n = 4 and H 1 -supercritical for n ≥ 5. As blow-up is possible for the focusing case, we will only consider the defocusing case for which global wellposedness is at least possible. In H 1 -critical and supercritical cases, the earlier argument does not apply to show global well-posedness as the time of existence T depends not only on the u0  H1  but also on the structure of  u0 . Proving global well-posedness even for the defocusing case for H 1 data is more difficult in this situation. In fact, not much is known about global wellposedness about the H 1 -supercritical case. On the other hand, the critical case is much better understood due to a number of recent breakthroughs. Consider the H 1 -critical nonlinear Schr¨odinger equation  iu = −∆u + |u|pmax −1 u t u(x, 0) = u (x).  (1.106)  0  As before, pmax is the H 1 -critical exponent. For example, when n = 3, pmax = 5. For n = 3 and 4, Bourgain ([7] and [8]) showed for radial initial data u0 ∈ H 1 (R3 ), solutions to (1.106) are global. [37] (a new proof for n = 3) and [81] (for n ≥ 5) extended Bourgain’s result to other dimensions. [22] removed the radial assumption for n = 3. [64] and [89] extended the result of [22] to n = 4 and n ≥ 5 respectively. Equation (1.106) is defocusing. Global well-posedness and blow-up for the H 1 -critical focusing equation with radial initial data iut = −∆u − |u|pmax −1 u  (1.107)  has been worked out by [47]. For n = 1, (1.93) is L2 -subcritical. On the other hand, for n = 2 and n = 3, (1.93) is L2 -critical and L2 -supercritical respectively. It is very  27  1.2. Background difficult, if not impossible, to show global well-posedness in L2 for these cases. The above method will not work because the time of existence T depends not only on the size of the L2 norm of the initial data u0 but also on the structure of u0 . There have been a lot of results in this direction showing global well-posedness in the defocusing case with initial data in H s for 0 ≤ s < 1. The current state of research is to get s as low as possible. For n = 2, some recent results in this direction are [32] (for s > (for s >  2 5 ),  [23] (for s >  1 3)  1 2 ),  [19]  and finally [28] (for s = 0). For n = 3, some  recent results in this direction are [20] (for s > 65 ) and [21] (for s > 45 ). Note that s = 0 corresponds to L2 . For n = 3, (1.93) is L2 -supercritical and the equation is locally ill-posed in L2 (see [18]). On the other hand, globally well-posed in L2 for the L2 -critical nonlinear Schr¨odinger equation in n = 3 4  iut = ∆u − |u| 3 u  (1.108)  has been shown by [27].  1.2.4  Solitary waves and their stability  Consider the equation iut = −∆u − |u|p−1 u.  (1.109)  This is the focusing nonlinear Schr¨odinger equation. As mentioned earlier, equation (1.109) admits very special solutions called solitary waves due to the competition between the dispersive effect and the concentration effect from the nonlinear term. Here, we start by looking for special solutions taking the form u(x, t) = eit φ(x).  (1.110)  If we substitute the above into (1.109), we find that φ must satisfy φ = ∆φ + |φ|p−1 φ.  (1.111)  28  1.2. Background When spatial dimension n = 1, the above equation is an ordinary differential equation and it turns out that in this case, it can be solved exactly and the solution is given by φ=  p+1 2  1 p−1  2  sech p−1  p−1 x . 2  (1.112)  2  If we apply the solution preserving scaling u(x, t) → λ p+1 u(λx, λ2 t), we find that if we let φw = ω  1 p+1  p+1 2  1 p−1  √ 2  sech p−1  ω(p − 1) x , 2  (1.113)  then u(x, t) = eiωt φw (x)  (1.114)  is a solution to (1.109). For higher dimensions, (1.111) cannot be solved explicitly. For n ≥ 3, [76] and [6] showed for p < pmax , there exists at least a positive, spherically symmetric ground state solution of (1.111) as well as infinitely many excited state (sign-changing) solutions. Here, ground state solution refers to a positive solution. Existence for ground states for n = 2 was later obtained by [5].[53] showed uniqueness of the ground state. [44] showed for n ≥ 2 and p < pmax , for each non-negative integer m, there exists a radial solution with m zeros. Because solitary waves arise from a delicate balance between two opposing effects, intuitively, one may believe the solitary waves are unstable under perturbation. However, in many cases, the solitary waves are found to be remarkably stable (see [83] for a discussion of the stability of solitary waves). The issues regarding stability of solutions of equations modelling physical process are important. This is because unstable solutions are difficult to observe experimentally since any small perturbation in the system will destroy them. In fact, the discovery of solitary waves is linked to their stability. These waves were first observed by John Scott Russell in 1834 as he was traveling 29  1.2. Background along a canal where a pulse of water wave caught his attention. Russell followed the pulse of wave for miles on horseback and was surprised by the fact that the wave did not change its shape. J. Boussinesq modelled the wave by what is now known as the Boussinesq equation. Later D.J. Korteweg and G. de Vries modelled the wave by what is now called the KdV equation (Korteweg-de Vries equation) ut + uxxx + (u2 )x = 0  (1.115)  which is a standard example of a nonlinear dispersive equation having solitary wave solutions. A more detailed account of the history can be found, for example, in the introductory chapter of [2]. We are going to discuss two different notions of stability. They are orbital stability and asymptotic stability. Very generally speaking, we say that a solution u is orbitally stable, if given an initial condition v0 close to u(x, 0), the solution v with the initial condition v0 remains close to u. Here, the notion of what it means to be “close” remains to be defined up to the symmetries of the equation (i.e. v remains close to the symmetry orbit of u). On the other hand, we say that a solution u is asymptotically stable if given an initial condition v0 close to u(x, 0), the solution v with the initial condition v0 approaches the symmetry orbit of u as time t → ∞. To properly define these two notions of stability, we need to define a notion of closeness. In order to do so, we need to first understand some invariances of (1.109). The following transformations leave (1.109) invariant (in other words, if u is a solution of (1.109), u remains a solution after the transformation): translation: u(x, t) → u(x − x0 , t − t0 ) 2  (1.116)  rescaling: u(x, t) → µ p−1 u(µx, µ2 t)  (1.117)  phase shift: u(x, t) → eiθ u(x, t)  (1.118) 30  1.2. Background  Galilean boost: u(x, t) → ei(v·x−|v|  2 t)  u(x − 2vt, t)  (1.119)  With these in mind, we will attempt to define orbital stability. First, we start with a generic notion of orbital stability which turns out to be not suitable for our purposes and we will discuss the reason. Let X and Y be Banach spaces. Our first attempt is to define orbital stability by the following: Definition 1. (the first attempt) We say that a solution u of (1.109) is orbitally stable if for all > 0, there exists a δ > 0 such that if a solution v of (1.109) satisfies v(·, 0)−u(·, 0) δ, then v(·, t) − u(·, t)  Y  <  X  <  for all t ≥ 0.  However, the above is not a good definition for our problem, because under this definition, no soliton will be stable under our usual choice of Banach spaces X and Y such as L2 or H 1 . The reason is that if u(x, t) = eiωt Q(x) is a solitary wave solution of (1.109), then by the Galilean invariance, v(x, t) = ei[(w−|v|  2 )t+v·x]  Q(x − 2vt)  (1.120)  is also a solution. Furthermore, v(x, 0) = eiv·x Q(x), so v(x, 0) − u(x, 0) = (eiv·x − 1)Q(x)  (1.121)  and for |v| small, we have v(x, 0) − u(x, 0)  X  1  (1.122)  for X = L2 or H 1 . However, v(·, t) − u(·, t)  Y  = ei[(w−|v|  2 )t+v·x]  Q(x − 2vt) − eiωt Q(x)  Y  (1.123)  may not be small once t becomes large enough no matter how small |v| is. Because of this, we need to define orbital stability to incorporate the possible invariant transformations of the equation. As a result, we will define orbital  31  1.2. Background stability as the following: Definition 2. (the correct version) We say that a solution u of (1.109) is orbitally stable if for all > 0, there exists a δ > 0 such that if a solution v of (1.109) satisfies v(·, 0)−u(·, 0)  X  <  δ, then inf v(x, t) − eiθ u(x − x0 , t)  θ,x0  Y  <  (1.124)  for all t ≥ 0. Orbital stability results in H 1 for the ground states are obtained by [12], [91], [92], and [66]. [39] formulated an abstract framework to show stability of solitary waves. These results show that the ground state is stable if 1 < p < pc and unstable if pc ≤ p < pmax . Next, we will introduce the concept of asymptotic stability. Consider first the linear equation iut = −∆u + V u.  (1.125)  As explained before, under suitable assumptions on the potential V , such an equation will admit bound state solutions. For our purposes, we will assume the potential is such that (1.125) admits a single bound state solution φ0 with eigenvalue −ω0 . In other words, φ0 is a solution to the differential equation ∆φ0 − V φ0 = ω0 φ0  (1.126)  u = eitω0 φ0 (x)  (1.127)  and  is a solution of (1.125). Next, suppose we add a cubic nonlinear term λ|u|2 u to (1.125) to get iut = −∆u + V u + λ|u|2 u  (1.128)  where λ = ±1 and V : Rn → R. It turns out that under suitable assumptions on V , such as sufficient decay as |x| → ∞, equation (1.128) admits solutions of the form u(x, t) = eiEt Q(x).  (1.129) 32  1.2. Background We call these solutions nonlinear bound states. Here, Q and E solve the equation ∆Q − V Q − λ|Q|2 Q = EQ.  (1.130)  In fact, there exists a one (complex) parameter family of these nonlinear bound states Q[z]. If we let z be the complex parameter, then E and Q has the form E[z] = ω0 + o(z) and Q[z] = zφ0 + q(z)  (1.131)  for sufficiently small z. Notice as z is small parameter, the size of these nonlinear bound state solutions is small. If the initial data u0 = Q[z1 ] for some sufficiently small z1 so Q[z1 ] is a nonlinear bound state, then the corresponding solution u of (1.128) will be given exactly by u(x, t) = eiE[z1 ]t Q[z1 ](x).  (1.132)  Now, suppose the initial condition is given by u0 = Q[z1 ] + a small error.  (1.133)  As the initial data is not exactly a bound state, the time evolution will not be given exactly by (1.132). We would like to find out how such a solution will evolve: whether the solution will disperse to zero or whether the solution will land back onto some other bound state. Asymptotic stability concerns the following question: suppose we start with an initial data u(x, 0) = u0 (x) that is “close” to Q[z1 ](x), will the solution u(x, t) of (1.128) approach a nearby nonlinear bound state eiE[z2 ]t Q[z2 ](x) as t → ∞. More precisely, we say that the solutions eiE[z1 ]t Q[z1 ](x) are asymptotically stable if whenever u0 − Q[z1 ](x)  X  (1.134)  is sufficiently small for some functional space X, then the solution u of  33  1.2. Background (1.128) will be given in the form u(x, t) = Q[z(t)](x) + η(x, t)  (1.135)  where z(t) approaches some limit z+ close to z1 as t → ∞, and η disperses to 0. The asymptotic stability of small nonlinear bound state solutions is studied by authors such as [68], [69], [61], [40], [50], [57], [50] and [49]. The case where ∆+V has two instead of one eigenvalues has also been studied by [87], [86] and [33]. Asymptotic stability of large solitary waves is also studied by [9], [10], [25], [26] and [60].  1.2.5  Scattering  To illustrate the idea of scattering, consider the linear Schr¨odinger equation with potential iut = −∆u + V u  (1.136)  where the potential V is nonzero near the original but is diminishing in size away from the origin (lim|x|→∞ V (x) = 0). For example, one can take V to 1 be the Coulomb potential V (x) = − |x| . Now, imagine a quantum particle  moving in a straight line towards the origin from very far away, passing by the region close to the origin, and moving far away again. When the particle is far away from the origin, it does not feel much of the presence of the potential, so such a quantum particle evolves like one driven by the free Schr¨ odinger equation iut = −∆u. As the particle gets close to the origin, it interacts with the potential, and the particle does not act like a free particle. Again, once the particle leaves the region near the origin and gets sufficiently far, it acts like a free particle again. This is called scattering by a potential. The topic we will discuss in this section is the analogue of the above but for the nonlinear Schr¨ odinger equation  iu = −∆u + λ|u|p−1 u t u(x, 0) = u (x)  .  (1.137)  0  34  1.3. Main results of the thesis Very roughly, we say that a solution u of (1.137) scatters if u behaves like a solution of the linear equation iut = −∆u far back in time (t → −∞) and far in the future (t → ∞). In other words, scattering happens when the nonlinear term |u|2 u “turns off” asymptotically. More rigorously, we define scattering as follows: Definition 3. (Scattering) Let X be H 1 (Rn ). Let u be a solution of (1.137). We say that u scatters if there exists u± ∈ X such that u(t) − ei∆t u±  X  → 0 as t → ±∞.  (1.138)  Notice that when u scatters, the asymptotic linear states u± are unique. Next, suppose for each u± ∈ X, there exists a unique initial data u0 such that the solution u scatters to the states u± , then we can define the wave operators W± to be the maps from u± to u0 . On the other hand, suppose for each u0 ∈ X, the solution u with u(x, 0) = u0 scatters, then we say (1.137) is asymptotically complete. In other words, (1.137) is asymptotic complete when every initial data in X gives rise to scattering solution. Since scattering concerns asymptotic behaviour of solutions, when a solution u scatters, it has to be a global solution. When p > pmax , solutions may not even be locally well-posed. As a result, scattering cannot occur for all initial data when p is too large. On the other hand, scattering requires the nonlinearity to “turn off” as t → ±∞. When p is too small, the nonlinear term may not decay fast enough for its effect to “disappear”. In other words, scattering occurs when p is of intermediate size. It turns out that for the defocusing case (λ > 0) in H 1 , scattering occurs when pc < p ≤ pmax ([73], [74], [34], [22], [64] and [89]).  1.3  Main results of the thesis  In this chapter, we have touched upon some basic properties of dispersive equations. We started from the free Schr¨odinger equation as a standard 35  1.3. Main results of the thesis model of a dispersive equation. We then introduced various variations of the free Schr¨ odinger equation, and discussed how each variation changes the behaviour of solutions. We have also introduced and discussed some of the main mathematical questions, such as the existence, stability and asymptotic behaviours of the solutions. In Chapter 2, we study the asymptotic stability of bound states of the nonlinear Schr¨ odinger equation with a magnetic potential in R3 . Here, we will consider the equation iψt = Hψ + λ|ψ|2 ψ  (1.139)  H = −∆ + 2iA· ∇ + i(∇· A) + V  (1.140)  where the operator  and λ = ±1. Here, A : R3 → R3 is a vector (magnetic) potential modelling the magnetic field and V : R3 → R is a scalar (electric) potential modelling an electric field. Without the nonlinear term, the equation iψt = Hψ  (1.141)  models the time evolution of a quantum particle (such as an electron) in the presence of a magnetic and electric field. The addition of the nonlinear term allows different linear modes of the solution interact with each other and the time evolution of the equation is more complex. We will consider the case where H admits a single eigenvalue e0 with the eigenfunction φ0 . In other words, Hφ0 = e0 φ0  (1.142)  and hence, e−ie0 t φ0 (x) is a solution of the linear equation (1.141). We will show in Chapter 2 that under suitable assumptions on A and V , such as sufficient decay as |x| → ∞, equation (1.139) admits nonlinear bound states  36  1.3. Main results of the thesis solutions of the form u(x, t) = e−iEt Q(x).  (1.143)  In fact, we will show that, just like the case where A = 0, there exists a one (complex) parameter family of these nonlinear bound states and if we let z be the complex parameter, then E and Q has the form E[z] = e0 + o(z) and Q[z] = zφ0 + q(z)  (1.144)  for sufficiently small z. Lemma 1.3.1. (Existence and decay of nonlinear bound states) For each sufficiently small z ∈ C, there is a corresponding eigenfunction Q[z] ∈ H 2 solving the nonlinear eigenvalue problem HQ + g(Q) = EQ  (1.145)  with the corresponding eigenvalue E[z] = e0 + o(z) and Q[z] = zφ0 + q(z) with q(z) = o(z 2 ),  DQ[z] = (1, i)φ0 + o(z) and D2 Q[z] = o(1) in H 2 (1.146)  where we denote DQ[z] = (D1 Q[z], D2 Q[z]) = (  ∂ ∂ Q[z], Q[z]), and z = z1 + iz2 . ∂z1 ∂z2 (1.147)  Furthermore, Q has exponential decay in the sense that eβ|x| Q ∈ H 1 ∩ L∞  (1.148)  for some β > 0 (independent of z). Then we will describe a result regarding asymptotic stability of these small nonlinear bound states. Theorem 3. (Asymptotic stability of nonlinear bound states) Under various assumptions to be described in chapter 2. For 0 ≤ t < ∞, every solution 37  1.3. Main results of the thesis ψ of equation (1.139) with initial data ψ0 sufficiently small in H 1 can be uniquely decomposed as ψ(t) = Q[z(t)] + η(t),  (1.149)  with differentiable z(t) ∈ C and η(t) ∈ H 1 satisfying iη, D1 Q[z] = 0, iη, D2 Q[z] = 0 and η  ψ0  X  H1 ,  z˙ + iE[z]z  L1t  ψ0  2 H1 .  (1.150)  Furthermore, as t → ∞, t  z(t) exp i  E[z(s)]ds  → z+ , E[z(t)] → E(z+ )  (1.151)  0  for some z+ ∈ C and η(t) − e−itH η+  Hx1  →0  (1.152)  for some η+ ∈ Hx1 ∩ Range(Pc ). My work builds on the work of [68], [61] and [40]. The works [68] and [61] consider nonlinear bound states that are small in both the H 1 -norm and a weighted L2 -norm, while the work [40] considers nonlinear bound states that are small in only the H 1 -norm. As a result, my result is closer to that of [40]. However, all of the previous results consider only the scalar potential case (A ≡ 0) while I consider both the scalar and vector potential. More discussion will be given in Chapter 2. In Chapter 3, we will consider the Schr¨odinger map equation discussed in Section 1.1.2 of this chapter. There, we will consider the equation  u = u × ∆u t u(x, 0) = u (x)  (1.153)  0  where u : Rn × R → S2 . We consider the question of global well-posedness for radial solutions for n = 2. In particular, we will prove the following 38  1.3. Main results of the thesis theorem in Chapter 3. Theorem 4. (Global well-posedness of 2D radial Schr¨ odinger map into S2 ) Suppose u(x, 0) = u0 (x) is radial and u0 − k ∈ H 2 (R2 ). Then ut = u × ∆u with u(r, 0) = u0 (r), r = |x|  (1.154)  has a unique global solution u ∈ L∞ ([0, ∞); H 2 (R2 )). Results similar to Theorem 4 under the extra assumption that the energy, E(u) :=  1 ∇u(t) 2  2 L2 (R2 ) ,  (1.155)  is small has been obtained over the decade or so. [16] showed that for n = 2 if E(u0 ) sufficiently small, radial solutions to (1.153) are global. A very recent result by [4] showed that for n = 2, suppose the initial data u0 satisfies u0 − Q ∈ H s for all s > 0 for some Q ∈ S2 and E(u0 ) is small, then the solution u to (3.1) is global and u − Q ∈ H s for all s > 0. Our result is the first that shows global well-posedness in n = 2 without the assumption of solutions having small energy. More discussions will be given in Chapter 3.  39  Chapter 2  Asymptotic stability of small solitary waves for nonlinear Schr¨ odinger equations with electromagnetic potential in R3 2.1  An overview  The goal of this chapter is to prove Theorem 3 stated in Section 1.3. We will start with an overview that will put our result in perspective with the known results. For this, let V : Rn → R be a function such that −∆ + V has an eigenvalue e0 with the corresponding eigenfunction φ0 . Now, consider the nonlinear Schr¨ odinger equation iψt = −∆ψ + V ψ + |ψ|2 ψ.  (2.1)  Such nonlinear Schr¨ odinger equations find numerous physical applications, for example, in Bose-Einstein condensates and nonlinear optics. As mentioned in Chapter 1, under suitable assumptions on V , equation (2.1) admits a one-(complex)-parameter family of nonlinear bound sates solutions Q[z] and the corresponding eigenvalue E[z] for sufficiently small z. Further analysis on the structure of Q and E reveals that the first order  40  2.2. Our result dependence of Q and E on z is given by Q[z] = zφ0 + O(z 3 ) and E[z] = e0 + O(z).  (2.2)  As z is a small parameter, we say that Q[z] emerges (bifurcates) from the zero solution along the eigenfunction φ0 of the linear operator −∆ψ + V ψ under the perturbation of the nonlinear term |ψ|2 ψ. Asymptotic stability of these nonlinear bound states has been studied by various authors. As described in Chapter 1, for the case where −∆ + V has exactly one eigenvalue, asymptotic stability has been proved by authors such as [68], [61], [40], [50] and [57]. In the more complicated case where −∆ + V has more than one eigenvalue, the nonlinear bound states with lowest eigenvalue (ground states) may still be asymptotically stable. This situation has been studied by authors such as [87], [86], [70] and [33].  2.2  Our result  The previous results on asymptotic stability of bound states of equation (2.1) are for scalar potentials V : Rn → R. The goal here is to extend these results with the addition of a vector potential. In particular, we consider the nonlinear Schr¨ odinger equation i∂t ψ = (−∆ + 2iA· ∇ + i(∇· A) + V )ψ + g(ψ) ψ(x, 0) = ψ0 (x) ∈ H 1 (R3 )  (2.3)  for ψ(x, t) : R3 × R → C, where g(ψ) = ±|ψ|2 ψ.  (2.4)  Here, A(x) = (A1 (x), A2 (x), A3 (x)) : R3 → R3 is the magnetic potential (also known as the vector potential) and V (x) : R3 → R is the electric potential (also known as the scalar potential). Equation (2.3) can be equiv-  41  2.2. Our result alently written as i∂t ψ = (i∇ + A)2 ψ + V ψ + g(ψ)  (2.5)  by replacing V with V − |A|2 . We will only consider potentials A(x) and V (x) which decay to 0 as |x| → ∞. Equation (2.3) describes a charged quantum particle subject to external electric and magnetic fields, and a self-interaction (nonlinearity). Just as equation (2.1), under certain assumptions on V and A, equation (2.3) admits standing wave solutions (or nonlinear bound states) of the form ψ(x, t) = e−iEt Q(x).  (2.6)  The existence of standing waves to equation (2.3) for certain electric and magnetic potentials was first proved in [31]. Here we consider small solutions of the form (2.6) which bifurcate from zero along an eigenvalue of the linear Hamiltonian operator H = −∆ + 2iA· ∇ + i(∇· A) + V.  (2.7)  Physical intuition suggests that the ground-state standing wave (the one corresponding to the lowest eigenvalue E) should remain stable when the self-interaction (nonlinearity) is turned on, and should become asymptotically stable (that is, nearby solutions should relax to the ground state by radiating excess energy to infinity – see below for a more precise statement). The main goal of this chapter is to prove asymptotic stability of the ground state, in the energy space (H 1 ), and in the presence of both the electric and magnetic field. Remark 1. Our argument should also go through for nonlinearities g(ψ) = ±|ψ|p−1 ψ for  7 3  ≤ p < 5, or combinations of these. For concreteness, we will  work with g(ψ) = ±|ψ|2 ψ. In order to study equation (2.3), we need the operator H to be selfadjoint. To ensure this, we make the following assumption,  42  2.2. Our result Assumption 1. (Self-adjointness assumption) We assume that each component of A is a real-valued function in Lq + L∞ for some q > 3, that ∇· A ∈ L2 + L∞ , and that V is a real-valued function in L2 + L∞ . Then by Theorem X.22 of [62], the operator H is essentially self-adjoint on C0∞ (R3 ). We will only consider the case where H has only one eigenvalue. More precisely, we make the following assumption. Assumption 2. (Spectral assumption) We assume that H supports only one eigenvalue e0 < 0, which is nondegenerate. We also assume 0 is not a resonance of H (see e.g. [30] for the definition of resonance). By the above assumption, H supports only one eigenvalue e0 < 0. Let φ0 > 0 be the positive, L2 -normalized eigenfunction corresponding to the eigenvalue e0 of H. We need the following assumption to show the existence and exponential decay of the nonlinear bound states. Assumption 3. (Assumptions for existence and exponential decay of nonlinear bound states) We assume A  Lq +L∞ (|x|>R)  + V−  L2 +L∞ (|x|>R)  → 0 as R → ∞  (2.8)  for some q > 3. Under the above assumptions, standing waves Q for E near e0 bifurcate from the zero solution along φ0 , we have the following lemma on the existence and decay of nonlinear bound states. Lemma 2.2.1. (Existence and decay of nonlinear bound states) For each sufficiently small z ∈ C, there is a corresponding eigenfunction Q[z] ∈ H 2 solving the nonlinear eigenvalue problem HQ + g(Q) = EQ  (2.9)  with the corresponding eigenvalue E[z] = e0 + o(z), and Q[z](x) = zφ0 +  43  2.2. Our result q[z](x) with q(z) = o(z 2 ),  DQ[z] = (1, i)φ0 + o(z) and D2 Q[z] = o(1) in H 2 (2.10)  where we denote DQ[z] = (D1 Q[z], D2 Q[z]) = (  ∂ ∂ Q[z], Q[z]), and z = z1 + iz2 . ∂z1 ∂z2 (2.11)  Furthermore, Q has exponential decay in the sense that eβ|x| Q ∈ H 1 ∩ L∞  (2.12)  for some β > 0 (independent of z). Next, we need assumptions on A and V which ensure our linear Schr¨odinger evolution obeys some dispersive estimates. For f, g ∈ L2 (R3 , C), define the real inner product f, g by f, g = Re(  f gdx).  (2.13)  R3 1  Denote x = (1 + |x|2 ) 2 and fix σ > 4. Let Pc be the projection onto the continuous spectral subspace of H. Following [30], we have: Assumption 4. (Strichartz estimates assumption) We assume that for all x, ξ ∈ R3 , |A(x)| + x |V (x)|  x  1+ε  x  −1−ε  ,  (2.14)  ˙ 21 ,6 (R3 ), A(x) ∈ W  (2.15)  A ∈ C 0 (R3 )  (2.16)  and  for some ε > 0 and some ε ∈ (0, ε).  44  2.2. Our result Define the space-time norm ψ  X  =  −σ  x  ψ  L2t Hx1  + ψ  1, 18 5  L3t Wx  + ψ  1. L∞ t Hx  We can now state the main result, which says that all H 1 -small solutions converge to a solitary wave (nonlinear bound state) as t → ∞: Theorem 5. (Asymptotic stability of small solitary waves) Let assumptions 1, 2, 3 and 4 hold. For 0 ≤ t < ∞, every solution ψ of equation (2.3) with initial data ψ0 sufficiently small in H 1 can be uniquely decomposed as ψ(t) = Q[z(t)] + η(t),  (2.17)  with differentiable z(t) ∈ C and η(t) ∈ H 1 satisfying iη, D1 Q[z] = 0, iη, D2 Q[z] = 0 and η  X  ψ0  H1 ,  z˙ + iE[z]z  L1t  ψ0  2 H1 .  (2.18)  Furthermore, as t → ∞, t  z(t) exp i  E[z(s)]ds  → z+ , E(z(t)) → E(z+ )  (2.19)  0  for some z+ ∈ C and η(t) − e−itH η+  Hx1  →0  (2.20)  for some η+ ∈ Hx1 ∩ Range(Pc ). For comparison, consider the nonlinear Schr¨odinger equation with just a scalar potential V , i∂t ψ = (−∆ + V )ψ + g(ψ)  (2.21)  for the same nonlinearity g as above, which is a special case of equation (2.3) with A = 0. The corresponding asymptotic stability result for (2.21) was obtained in dimension three in [40], in dimension one in [57] and in 45  2.3. Discussion and outline of the proof dimension two in [50, 56].  2.3  Discussion and outline of the proof  In this section, we will give an outline of the proof and discuss difficulties encountered when proving the results. The detailed proof will be given in the next section. There are three main parts to the proof. The first part is to show the existence of nonlinear bound state solutions of the form (2.6) of equation (2.3). Substituting (2.6) into (2.3), we see that Q satisfies (i∇ + A)2 Q + V Q = EQ − g(Q) where g(Q) = ±|Q|2 Q.  (2.22)  For our stability argument, it is essential to have sufficient decay and regularity for the standing wave Q. We will show that under Assumption 3, standing waves Q with E near e0 bifurcate from the zero solution along φ0 , and such standing waves decay exponentially at ∞. This result is stated in Lemma 2.2.1 and is proven by an contracting mapping argument. The exponential decay as |x| → ∞ of Q is shown by showing that Q is uniformly bounded in H 1 with a local exponential weight. The detailed proof of Lemma 2.2.1 will be given in Section 2.4.1. The second part of the proof is to establish various estimates used to show the main result. Our approach for showing Theorem 5 will be similar to that used by [40] for showing corresponding results for A = 0. There, [40] uses the Strichartz estimates eit(∆−V ) Pc φ and  ˜ X  φ  (2.23)  H1  t  ei(t−s)(∆−V ) Pc F (s)ds −∞  ˜ X  F  6  L2t W 1, 5  (2.24)  1 2 1,6 ∩ L2 L6,2 , which are known to hold for a class ˜ = L∞ where X t H ∩ Lt W t  of scalar potentials V . Our approach will use the Strichartz estimates for H from [30]. However, the proof of [30] of the inhomogeneous Strichartz 46  2.3. Discussion and outline of the proof estimates  t  ei(t−s)H Pc F (s)ds −∞  Lqt Lpx  F  (2.25)  Lqt˜ Lpx˜  for H = −∆ + 2iA· ∇ + i(∇· A) + V uses a lemma from [17] which does not hold for the endpoint case (q, p) = (2, 6) or (˜ q , p˜) = (2, 6). To overcome the lack of endpoint Strichartz estimates, we will use estimates in weighted spaces, as in [57] and [56]. The extension of these weighted-space estimates in the presence of a vector potential turns out to be somewhat involved, and is the most difficult and novel part of the work. We will establish these estimates in Subsection 2.4.2. The last part is the actual proof of our main result which can be found in Subsection 2.4.3. The strategy is as follows: if we substitute ψ = Q[z(t)](x) + η(x, t)  (2.26)  into (2.3), after some manipulations of terms, we get that η satisfies the equation i∂t η = Hη+F where F = ±|Q+η|2 (Q+η)∓|Q|2 Q−iDQ(z+iEz). ˙ (2.27) Here, for z = z1 + iz2 and for w ∈ C, denote DQ[z]w =  ∂ ∂ Q[z] Re w + Q[z] Im w. ∂z1 ∂z2  (2.28)  A key idea, as in [40], is to choose z(t), at each time t, so that the orthogonality conditions iD1 Q, η = iD2 Q, η = 0  (2.29)  hold. Further manipulations then show that the quantity |z˙ + iEz|  | 2Q|η|2 + Qη 2 + |η|2 η, DQ |(1 + η  L2 )  (2.30)  is quadratic in η (terms linear in η have cancelled out). To prove the main theorem, we are faced with two tasks:  47  2.3. Discussion and outline of the proof 1. First, we would like to show that if ψ(0) is small in H 1 , then η remains small for all future time, and indeed scatters. 2. Second, we would like to show |z˙ + iEz| stays small in L1t -norm which t 0 E[z(s)]ds)  implies convergence of z(t) exp(i  as t → ∞.  Roughly speaking, our strategy is to find some space-time norm X such that the following holds η  X  η(0)  H1  + η  a X  for some a > 1.  (2.31)  Once (2.31) has been shown, smallness of η for all time follows by a continuity argument. The idea is that since η starts out small, the constraint (2.31) posts a limit on how large η can get and hence, η has to stay small for all future times. Once the smallness of η has been shown, we can use (2.30) to complete the second task. More precisely, we need X to provide time-decay of η at the level of L2t , so that (2.30) will control z˙ + iEz  L1t .  The main tools used to complete the tasks are Strichartz-type estimates which we briefly explained before. If we write (2.27) in the integral form, it becomes η = e−itH  t  eisH F (s)ds .  η(0) − i  (2.32)  0  One may attempt to use Strichartz-type estimates on (2.32) to obtain (2.31). However, such an attempt will fail as η may contain a component of the “discrete spectrum” (i.e. eigenfunction φ0 ) of H which does not have decay properties needed for Strichartz-type estimates to hold. Instead, let Pc denote the projection onto the continuous spectral subspace of H and let ηc = Pc η, then ηc satisfies t  ηc = e−itH Pc η(0) − i  e−i(t−s)H Pc F (s)ds).  (2.33)  0  Using Lemma 2.2 of [40], it can be shown that η  Y  ηc  Y  for any reason-  able space Y since η satisfies the orthogonality condition (2.29). We then  48  2.3. Discussion and outline of the proof have η  ηc  X  X  ≤ e−itH Pc η(0)  t X  +  e−i(t−s)H Pc F (s)ds)  X.  (2.34)  0  Our strategy will be to use Strichartz estimates on (2.34) to obtain (2.31). For Strichartz estimates, we mean space-time bounds on the evolution operator e−itH such as e−itH Pc f and  t  f  Lqt Lpx  e−i(t−s)H Pc F (x)ds  0  (2.35)  L2 (R3 )  Lqt Lpx  F  Lqt˜ Lpx˜  .  (2.36)  Under sufficient decay and regularity of A and V , [30] showed the above estimates hold for (p, q) and (˜ p, q˜) satisfying 3 2 3 + = with 2 ≤ p < 6. q p 2  (2.37)  Here, (2.35) is useful for bounding terms like e−itH Pc η(0) (2.36) is useful for bounding terms like  X in (2.34) t −i(t−s)H Pc F (s)ds) X . 0 e  and  However, (2.35) and (2.36) are not enough to finish the proof. There are two major obstacles: 1. First, since Q has no decay in time, Q cannot be in Lqt for q < ∞. To control the right hand side of (2.30) in L1t , we need to control terms of the form η  L2t Lpx  for some p, but this is not covered by (2.35) or  (2.36). 2. Second, to control terms like η 3  Lqt Lpx  on the right hand side of (2.34),  we will use Gagliardo-Nirenberg inequality u  Lpx  ∇u  θ L2x  u  1−θ Lqx  for appropriate values of p, q and θ. (2.38)  What this means is that we need to control ∇η in some norm as well. The first obstacle can be overcome as follows. While Strichartz estimates 49  2.4. Detailed proof (2.35) and (2.36) for L2t L6x is not available, a weighted version of (2.35) for L2t L2x is available from [30] and a similar weighted version of (2.36) can be derived from results in [30]. The second obstacle is more difficult to overcome. Here, our approach is to derive versions of (2.35) and (2.36) for Lqt Wx1,p as well as similar estimates for the weighted of (2.35) and (2.36). This requires some work. Our strategy is to show 1  u  W 1,p  ∼ H12 u  1  Lp  and  u  x sH1  ∼ H12 u  x s L2 .  (2.39)  for some operator H1 which commutes with H. Once (2.39) is achieved, the estimates we want can be obtained by commuting H1 through both sides of the expressions (2.35) and (2.36). Of all the estimates, the estimate x  t  −σ  ei(t−s)H Pc F (s)ds 0  L2t Hx1  x σF  L2t Hx1  (2.40)  is the most difficult to achieve. The reason is the operator H1 does not 1  commute with the factor x := (1 + |x|2 ) 2 . As a result, (2.40) does not trivially follow from (2.39).  2.4 2.4.1  Detailed proof Existence and decay of standing waves  The following is the proof for Lemma 2.2.1, the existence and exponential decay of nonlinear bound states. Proof of existence of nonlinear bound states: For each small z ∈ C, we look for a solution Q = zφ0 + q and E = e0 + e  (2.41)  (−∆ + 2iA· ∇ + i(∇· A) + V )Q + g(Q) = EQ  (2.42)  of  with (φ0 , q) = 0 and e ∈ R small. 50  2.4. Detailed proof Let H0 = −∆ + 2iA· ∇ + i(∇· A) + V − e0 . If we substitute Q = zφ0 + q and E = e0 + e into equation (2.42), we get H0 q + g(zφ0 + q) = e (zφ0 ) + e q.  (2.43)  Projecting equation (2.43) on the φ0 and φ⊥ 0 directions, we get e z = (φ0 , g(zφ0 + q))  (2.44)  H0 q = −Pc g(zφ0 + q) + e q.  (2.45)  and  Now, let 2 K = {(q, e ) ∈ H⊥ × R| q  H2  ≤ |z|2 , |e | ≤ |z|}  (2.46)  2 = {q ∈ H 2 |(q, φ ) = 0}. Also, define for sufficiently small z ∈ C where H⊥ 0  the map M : (q0 , e0 ) → (q1 , e1 ) by g0 := g(zφ0 + q0 ),  (2.47)  ze1 := (φ0 , g0 )  (2.48)  q1 := H0−1 (−Pc g0 + e0 q0 ).  (2.49)  and  Now if (q0 , e0 ) ∈ K, we have |ze1 | = |(φ0 , g0 )| = |(φ0 , g(zφ0 + q0 ))| = |(φ0 , |zφ0 + q0 |2 (zφ0 + q0 ))|  O(z 3 ) (2.50)  and q1  H2  Therefore, |e1 |  − Pc g0 + e0 q0  L2  O(z 2 ) and q1  ≤ g0 H2  L2  + |e0 | q0  H2  O(z 3 ).  (2.51)  O(z 3 ). This shows that M maps K  51  2.4. Detailed proof into K for sufficiently small z. Next, we would like to show that M is a contraction mapping. Let (a1 , b1 ) := M (q0 , e0 ) and (a2 , b2 ) := M (q1 , e1 ) with gj = g(zφ0 + qj ) for j = 0, 1. Then |z(b2 − b1 )| = |(φ0 , g0 − g1 )| = |(φ0 , g(zφ0 + q0 ) − g(zφ0 + q1 ))| = |(φ0 , |zφ0 + q0 |2 (zφ0 + q0 ) − |zφ0 + q1 |2 (zφ0 + q1 ))| φ0 (|z|2 φ20 + |q0 |2 + |q1 |2 )|q0 − q1 |  |z|2 q0 − q1  As ai = H0−1 (−Pc gi−1 + ei−1 qi−1 ) for i = 1, 2 and H0−1  L2 →H 2  L2 .  ≤ ∞, we  have a1 − a2  H2  Pc (g1 − g0 ) + e0 q0 − e1 q1  L2  g1 − g0  L2  L2  + |e0 − e1 | q0  + |e1 | q0 − q1  L2 .  Since g1 − g0  L2  g(zφ0 + q1 ) − g(zφ0 + q0 )  =  2  φ20 (q1  2  φ20 L3  |z|  |z|  − q2 )  L2  q1 − q2  L2  + |z| φ0 (q12 − q22 ) L6  + |z| φ0  + (|q1 |2 + |q1 q2 | + |q2 |2 )  L4  L6  L2  + q13 − q23  q1 + q2  q1 − q2  L6  L2  q1 − q2  L6  L4 ,  together, we have a1 − a2  H2  |z| q1 − q2  H2  + |z|2 |e0 − e1 |.  (2.52)  Hence, M is a contraction mapping for z sufficiently small. Now by the contraction mapping theorem, there exists a unique fixed point (q, e ) satisfying q  H2  = O(z 3 ) and |e | = O(z 2 ) as z → 0. The statements about  derivatives of Q and E with respect to z follow by differentiating (2.43) with respect to z and applying the contraction mapping principle again. Proof of exponential decay: 52  2.4. Detailed proof Lemma 2.4.1. For ε > 0, define the exponential weight function χR by  χR,ε    eε(|x|−R) − 1   = eε(3R−|x|) − 1    0  if R < |x| ≤ 2R, if 2R < |x| < 3R, .  (2.53)  else  Suppose for ε > 0 small enough, f ∈ H 1 satisfies χR,ε f  H1  ≤C  (2.54)  for some constant C independent of R, then eε |x| f ∈ H 1  (2.55)  for some ε > 0. Proof. For R > 0, χR,ε f  ≤ C implies that  H1  (eε(|x|−R) − 1)f  H 1 [ 23 R,2R]  ≤ C.  (2.56)  Since f ∈ H 1 , eε(|x|−R) f  H 1 [ 23 R,2R]  ≤C+ f  H1  ≤C.  (2.57)  1  e 2 εR ≤ eε(|x|−R) for |x| ∈ [ 32 R, 2R], so 1  e 2 εR f  H 1 [ 32 R,2R]  ≤C.  (2.58)  So 1 1  e( 2 ( 2 ε))(2R) f  H 1 [ 32 R,2R]  ≤C.  (2.59)  Let ε = ( 12 ( 12 ε)). Using eε 2R ≥ eε |x| for |x| ∈ [ 32 R, 2R], we get that eε |x| f  H 1 [ 23 R,2R]  ≤C  (2.60)  for some constant C independent of R. 53  2.4. Detailed proof Let ε = 12 ε . Then ∞  eε  |x|  2 H 1 (|x|>1)  f  eε  =  |x|  2 2(k+1) . 2k H 1 [ 2 k , 2 k+1 ]  f  3  k=0  22k+1 3k+1  Now, for each k, since eε = eε eε , taking R = eε |x| f  ≥  C  2k  H1[ 2 k , 2 3  |x| ε |x|  eε  =  e  f  22k 3k  (ε  )  eε  2k  H1[ 2 k , 2  |x|  in (2.60), we have  2(k+1) ] 3k+1  3  ≥ e  (2.61)  3  f  2(k+1) ] 3k+1  2k  2(k+1) ] 3k+1  H1[ 2 k , 2 3  .  This means that, eε  |x|  f  −(ε  2(k+1) 2k H 1 [ 2 k , 2 k+1 ] 3 3  ≤Ce  22k 3k  )  (2.62)  Therefore, ∞  eε  |x|  f  2 H 1 (|x|>1)  |x|  eε  =  f  2 2(k+1) 2k H 1 [ 2 k , 2 k+1 ] 3  k=0 ∞ −ε  ≤ C2  e  3  22k+1 3k  k=0  < ∞  By Lemma 2.4.1, to show that eα|x| Q suffices to show that χR,ε Q  H1  H1  < ∞ for some α > 0, it  ≤ C for some constant C independent of  R. Here, χR,ε is the exponential weight function as in Lemma 2.4.1. Consider the bilinear form E(ψ, φ) = (∇ψ, ∇φ)+i  (2ψA·∇φ+ψ(∇·A)φ)dx+  V ψφdx for ψ, φ ∈ H 1 (2.63)  associated to the magnetic Schr¨odinger operator −∆ + 2iA · ∇ + i(∇ · A) + V . 54  2.4. Detailed proof Then E(ψ, ψ) = (∇ψ, ∇ψ) + i  (2ψA · ∇ψ + ψ(∇ · A)ψ)dx +  = (∇ψ, ∇ψ) + 2 Im(  ψA · ∇ψdx) +  V ψψdx  V ψψdx  Set b := lim inf{E(φ, φ)|φ ∈ H 1 , φ  2  R→∞  = 1, φ(x) = 0 for |x| < R}.  (2.64)  We will show that b ≥ 0 by contradiction. Suppose b < 0. Then there exists a sequence φRj ∈ H 1 with Rj → ∞, satisfying φRj  2  = 1, φRj (x) = 0 for  |x| < Rj , and E(φRj , φRj ) < δ for some fixed δ < 0. Suppose V ∈ L∞ , then V φRj φRj dx ≤ V  ∞  φRj  2 2  = V  ∞.  (2.65)  Suppose V ∈ L2 , then V φRj φRj dx ≤  V  φRj  2  2 4 3  1 2  ∇φRj 22 3 ˜ ∇φR 2 ) 43 + 1 ( V δ( j 2 δ˜ 1 = δ˜ ∇φRj 22 + V 42 . δ˜ V  φRj  2  2  2  φRj  1 2  2)  4  Hence, V φRj φRj dx  δ˜ ∇φRj  Similarly, suppose A ∈  L∞ ,  |(φRj , A · ∇φRj )| ≤ A  2 2  +  1 V δ˜  L∞ +L2  where δ˜ is sufficiently small. (2.66)  then ∞  φRj  2  ∇φRj  2  = A  ∞  ∇φRj  2.  (2.67)  55  2.4. Detailed proof On the other hand, suppose A ∈ L(3+˜ε) , then |(φRj , A · ∇φRj )| ≤  A A  =  A  (3+˜ ε)  φRj  (3+˜ ε)  φRj  (3+˜ ε)  2(3+˜ ε) 1+˜ ε  ∇φRj  3(1+˜ ε) 5 + 2(3+˜ 2 ε)  2  ∇φRj  3(1+˜ ε) 5 − 2(3+˜ 2 ε)  ∇φRj  2  2 3(1+˜ ε) 5 − 2(3+˜ 2 ε)  2  .  Hence, |(φRj , A · ∇φRj )| in which  5 2  −  3(1+˜ ε) 2(3+˜ ε)  A  ε) +L∞ L(3+˜  ( ∇φRj  2  + ∇φRj  Im  A  2  ),  (2.68)  is strictly less than 2 for ε˜ > 0.  Since supp(φRj ) ⊂ {|x| ≥ Rj }, by the assumption V− 0 and  3(1+˜ ε) 5 − 2(3+˜ 2 ε)  (L3+ +L∞ )(|x|>Rj )  → 0,  V− |φRj  |2 dx  (L2 +L∞ )(|x|>Rj )  →  and the negative part of  φRj A · ∇φR converge to 0. Hence, the negative part of the energy  converges to 0, a contradiction. Thus b ≥ 0. So there exists δ(R) with δ(R) → b ≥ 0 as R → ∞, such that for any φ ∈ H 1 satisfying φ(x) = 0 for |x| < R, we have E(φ, φ) ≥ δ(R) φ 22 .  (2.69)  For φ ∈ H 1 , we have δ(R) χR φ  2 2  ≤ E(χR φ, χR φ) χR φA · ∇χR φdx) +  = (∇χR φ, ∇χR φ) − 2 Im(  V χR φχR φdx.  If we expand the factor ∇χR φ, we get that (∇χR φ, ∇χR φ) = (φ∇χR , φ∇χR ) + 2(φ∇χR , χR ∇φ) + (χR ∇φ, χR ∇φ) and since Im( |φ|2 A · χ2R ∇χR ) = 0 −2 Im(  χR φA · ∇χR φdx) = −2 Im( = −2 Im(  χ2R φA · ∇φ) − 2 Im(  |φ|2 A · χ2R ∇χR )  χ2R φA · ∇φ). 56  2.4. Detailed proof Since 2(φ∇χR , χR ∇φ) + (χR ∇φ, χR ∇φ) − 2 Im(  χ2R φA · ∇φ) +  V χR φχR φdx  is nothing but E(χ2R φ, φ), we have δ(R) χR φ  2 2  ≤ E(χ2R φ, φ) + φ∇χR  2 2  = (χ2R φ, H0 φ) + e0 χR φ  2 2  + φ∇χR  2 2  where H0 = −∆ + i(A · ∇ + ∇ · A) + V − e0 . From direct calculation, we see that for R > 0, |∇χR |  ε(χR + 1),  (2.70)  so φ∇χR  2 2  ε2 φ(χR + 1) 22 .  (2.71)  Putting everything together, we have δ(R) χR φ  2 2  (χ2R φ, H0 φ) + (e0 + ε2 ) χR φ  2 2  + ε2 φ 22 .  Since e0 < 0 and limR→∞ δ(R) ≥ 0, for ε small enough and R sufficiently large, δ(R) − e0 − ε2 is positive and bounded away from zero. Therefore, we have χR φ  2 2  (χ2R φ, H0 φ) + ε2 φ 22 .  (2.72)  Next, χR ∇φ  2 2  ≤  ∇(χR φ)  2 2  + φ∇χR  E(χR φ, χR φ) + 2 Im( +ε2 φ  2 2  χR φA · ∇χR φdx) −  V χR φχR φdx  2 2  57  2.4. Detailed proof Since χR φA · ∇χR φdx) ≤ A  Im(  χR φ  L∞ (|x|≥R)  L2  ∇(χR φ)  (2.73)  L2  and Im(  χR φA · ∇χR φdx) ≤  A  L3 (|x|≥R)  χR φ  L6  ∇(χR φ)  L2  ≤  A  L3 (|x|≥R)  χR φ  H1  ∇(χR φ)  L2 ,  we have that Im(  χR φA · ∇χR φdx) ≤  A  (L∞ +L3 )(|x|≥R)  χR φ  H1  ≤  A  (L∞ +L3 )(|x|≥R)  χR φ  2 H1 .  ∇(χR φ)  L2  Therefore, χR ∇φ  2 2  E(χR φ, χR φ) + A  (L∞ +L3 )(|x|≥R)  χR φ  2 H1  + χR φ  2 2  and χR φ  2 2  + ε2 φ 22 . (2.74)  Now using E(χR φ, χR φ) =  (χ2R φ, H0 φ)+e0  χR φ  2 2  (χ2R φ, H0 φ)+  ε2 φ 22 , we have that χR ∇φ  2 2  (χ2R φ, H0 φ) + A  (L∞ +L3 )(|x|≥R)  χR φ  2 H1  + ε2 φ 22 .  (2.75)  Since ∇(χφ)  L2  = φ∇χR  L2  + χR ∇φ  L2  ε φ(χR + 1)  L2  + χR ∇φ  L2 ,  (2.76) putting everything together, we have that χR φ  2 H1  (χR φ, χR H0 φ) + ε2 φ  2 2  + A  (L∞ +L3 )(|x|≥R)  χR φ  2 H1 ,  (2.77)  so for R sufficiently large, χR φ  2 H1  (χR φ, χR H0 φ) + ε2 φ 22 .  (2.78)  58  2.4. Detailed proof If we let φ = φ0 and use that H0 φ0 = 0, we have χR φ0  2 H1  2 2  φ0  = 1.  (2.79)  Next, let φ = q. Using that H0 q = −Pc g(zφ0 + q) + e q, we get χR q  2 H1  (χR q, χR H0 q) + ε2 q  2 2  (χR q, χR (−Pc g(zφ0 + q) + e q)) + ε2 q χ2R q g(zφ0 + q)  1  + e χR q  2 2  2 2  + ε2 q 22 .  As g(z) = |z|2 z, we have χ2R q g(zφ0 + q) |z|3 χ2R qφ30 |z|3 χ2R φ30 +  χ2R q 2 2  + |z|2 χ2R q 2 φ20  1 2  q  1  q 2  2  + |z|2 χ2R φ20  1  + |z| χ2R q 3 φ0 2  q2  2  1  + χ2R q 4  + |z| χ2R φ0  2  1  χR q 3  2  2  2  ≤ o(z ). Hence, χR q by (2.79) and q  H2  2 H1  ≤ o(z 2 )  (2.80)  = o(z 2 ).  Next if we substitute φ = Dq, and use that H0 Dq = −Pc Dg(zφ0 + q) + qDe + e Dq,  (2.81)  we get χR Dq  2 H1  (χR Dq, χR H0 Dq) + ε2 Dq  2 2  (χR Dq, χR (−Pc Dg(zφ0 + q) + qDe + e Dq)) + ε2 Dq χ2R Dq Dg(zφ0 + q)  1  + χ2R Dq q De  1  + e χR Dq  2 2 2 2  + ε2 q 22 .  59  2.4. Detailed proof Here, the first term χ2R Dq Dg(zφ0 + q)  z  χ2R Dq Dg(zφ0 + q)  1  χ2R  1  2  2  Dq φ0 |zφ0 + q|  χ2R Dqφ3o 1  2  z χR Dq  +z  χ2R Dqφ2o q 1  χR φ 0  H1  φ0 2H 1  H1  is bounded by  1  + χ2R φo q 2  1  + z Dq  q  H1  χR φ 0  H1  2 H1  ≤ o(z 2 ), and the second term χ2R Dq q De χ2R Dq q De  1  1  ≤  is bounded by  χR Dq  3  χR q  χR Dq  H1  De  3  χR q  3  De  H1  H1  2  ≤ o(z ). Therefore, χR Dq  2 H1  ≤ o(z 2 ).  (2.82)  Hence, by Lemma 2.4.1 and Q = zφ0 + q, we have eβ|x| Q eβ|x| DQ  H1  ≤ ∞ and  ≤ ∞ for some β > 0.  Next, we would like to show eβ|x| Q Since ∆(eβ|x| Q) ∞. Let γ =  H1  β 3.  L∞ (|x|≤1)  L∞  ≤ ∞ by bounding ∆(eβ|x| Q)  3  L2+  < ∞ already holds, it remains to show ∆(eβ|x| Q)  Using the equation for Q, we get  ∆(eγ|x| Q) (∆eγ|x| )Q  3  L 2 + (|x|>1) 3  L 2 + (|x|>1)  + eγ|x| A · ∇Q + eγ|x| g(Q)  + (∇eγ|x| ) · (∇Q)  3  L 2 + (|x|>1) 3  L 2 + (|x|>1)  3  L 2 + (|x|>1)  + eγ|x| [(∇ · A) + V ]Q  + eγ|x| EQ  3  L 2 + (|x|>1)  3  L 2 + (|x|>1)  .  Let f and g be such that ∆eγ|x| = f (x)eγ|x| and ∇eγ|x| = g(x)eγ|x| . We can bound the first two terms loosely by (∆eγ|x| )Q  2  3 L 2 + (|x|>1)  e− 3 γ|x| f (x)  L6+ (|x|>1)  eβ|x| Q  L2  (2.83) 60  . 3  L 2 + (|x|>1)  <  2.4. Detailed proof and (∇eγ|x| ) · (∇Q)  3  L 2 + (|x|>1)  1  2  e− 3 β|x| g(x) e  2 β|x| 3  Q  H1  e 3 β|x| (∇Q)  L6+ (|x|>1)  + e  − 13 β|x|  g(x)  eγ|x| A · ∇Q ≤  A e  L3+ +L∞ (  −2 β|x| 3  3  L 2 + (|x|>1)  let h =  3  L 2 + (|x|>1)  L2 .  and eγ|x| EQ  3  L 2 + (|x|>1)  3  L 2 + (|x|>1) 1  e 3 β|x| ∇Q  eβ|x| (∇Q)  L3  1  L3 (|x|>1)  + e 3 β|x| ∇Q  + eβ|x| ∇Q L2  3  L 2 (|x|>1)  )  L2 .  < ∞. To bound e  and from above, we know that h ∈  L2 .  −2 β|x| 3  eβ|x| (∇Q)  Now, consider the  set 2  M = {x|(e− 3 β|x| |h|)3 > |h|2 } = {x||h| > e2β|x| }.  (2.84)  Clearly, e  −2 β|x| 3  eβ|x| (∇Q)  L3 (M c )  = e  −2 β|x| 3  h  2  L3 (M c )  ≤ |h| 3  L3  = h  2 3  L2  < ∞. (2.85)  On the other hand, inside M ,  |eβ|x| (∇Q)|  >  e2β|x|  and hence, |∇Q| > eβ|x| .  Then e  −2 β|x| 3  .  , we have  We already shown above that eβ|x| ∇Q eβ|x| (∇Q)  eβ|x| Q  L∞ (|x|>1)  In a similar way, we can also bound eγ|x| g(Q) Next, for eγ|x| A · ∇Q  L2  eβ|x| (∇Q)  L3 (M )  ≤ ≤ =  e  −2 β|x| 3  |∇Q| ∇Q ∇Q  2  |∇Q|2  L3  L3  3 L6 3 H1 .  Hence, we have ∆(eγ|x| Q)  3  L2+  < ∞.  (2.86)  61  L3 ,  2.4. Detailed proof By Sobolev embedding, we have eγ|x| Q  2.4.2  L∞  < ∞.  (2.87)  Linear estimates  In this section, we will prove the following theorem. Theorem 6. We say that (p, q) is Strichartz admissible if 2 3 3 + = with 2 ≤ p < 6. q p 2  (2.88)  If (q, p) and (˜ p, q˜) are Strichartz admissible, then t  ei(t−s)H Pc F (s)ds 0  min( x σ F  L2t Hx1 ,  F  Lqt Wx1,p Lqt˜ Wx1,p˜  +  x  t  −σ  ei(t−s)H Pc F (s)ds 0  L2t Hx1  ).  To prove theorem 6, we need a few preparatory lemmas. The following lemmas 2.4.2 and 2.4.3 are from [30]: Lemma 2.4.2. (Non-endpoint Strichartz estimates) Under assumptions 4 and 2, if (p, q) and (˜ p, q˜) are Strichartz admissible, we have eitH Pc f and  f  Lqt Lpx  (2.89)  L2 (R3 )  t  ei(t−s)H Pc F (x)ds −∞  F  Lqt Lpx  Lqt˜ Lpx˜  .  (2.90)  Notice that the above does not include the L2t -norm. Fix σ > 4. Lemma 2.4.3. (Weighted homogeneous L2t estimates) Under assumptions 4 and 2, we have x  −σ −itH  e  f  L2t L2x  f  L2x ,  (2.91)  and sup λ  x  −σ  (H − (λ2 + i0))−1 x  −σ  L2 →L2  1.  (2.92)  λ≥0  62  2.4. Detailed proof The weighted resolvent estimate of lemma 2.4.3 implies weighted inhomogeneous estimates for the linear evolution: Lemma 2.4.4. (Weighted L2t inhomogeneous estimates) Under the assumptions of lemma 2.4.3, t  −σ  x  ei(t−s)H Pc x  −σ  F (s)ds  0  F  L2t L2x  L2t L2x .  (2.93)  Proof. For simplicity we may restrict to times t ≥ 0. By Plancherel, we have χ{t≥0} x  t  −σ  ei(t−s)(H+iε) Pc x  −σ  F (s)ds  0 ∞  eitτ x  =  −σ  t  ei(t−s)(H+iε) Pc x  (  0  −σ  L2t  F (s)ds)dt  0  L2τ  Next, change the order of the ds and dt integral and use that ∞  dt eit(H−τ +iε) Pc x  −σ  F (s)  s  1 −σ (H − τ + iε)−1 eit(H−τ +iε) |t=∞ F (s) t=s Pc x i −1 (H − τ + iε)−1 eis(H−τ +iε) Pc x −σ F (s), i  = = we get χ{t≥0} x =  x  ∞  −σ 0  =  x  −σ  t  −σ  ei(t−s)(H+iε) Pc x  −σ  0  dse−is(H+iε)  F (s)ds  L2t  −1 (H − τ + iε)−1 eis(H−τ +iε) Pc x i  (H − τ + iε)−1 Pc x  ∞  −σ 0  dse−isτ F (s)  −σ  F (s)  L2τ  L2τ .  63  2.4. Detailed proof If we take the L2x -norm of both sides, we get t  −σ  x  −σ  ei(t−s)(H+iε) Pc x  F (s)ds  0 −σ  x  (H − τ + iε)−1 Pc x  ∞  −σ  L2t L2x  dse−isτ F (s)  0  sup  x  −σ  τ  F  (H − τ + iε)−1 Pc x  −σ  ∞ L2 →L2  L2τ L2x  dse−isτ F (s)  0 L2t L2x  L2τ L2x  by Plancherel and Lemma 2.4.3.  Now sending ε to 0, we have x  t  −σ  ei(t−s)H Pc x  −σ  F (s)ds  0  F  L2t L2x  L2t L2x  (2.94)  as needed. Lemma 2.4.5. (Mixed Strichartz weighted estimates) Let (q, p) and (˜ p, q˜) be Strichartz admissible. Then t  ei(t−s)H Pc F (s)ds 0  min( x σ F  L2t L2x ,  F  Lqt Lpx Lqt˜ Lpx˜  +  x  t  −σ  ei(t−s)H Pc F (s)ds 0  L2t L2x  ).  Proof. First, ∞ 0  e−isH Pc F (s)ds  ∞ 2 L2x  =( 0  ∞  e−isH Pc F (s)ds,  e−itH Pc F (s)dt).  0  Moving the integrals through the inner product and rearranging the terms,  64  2.4. Detailed proof we get ∞  e−isH Pc F (s)ds  0 ∞  ∞  ds(Pc F (s),  = 0  2 L2x  e−i(t−s)H Pc F (s)dt)  0  ∞  =  ∞  −σ  σ  ds( x Pc F (s), x  e−i(t−s)H Pc F (s)dt)  0  0  by H¨ older inequality x σ Pc F (s)  ≤  x  L2s L2x  ∞  −σ  e−i(t−s)H Pc F (s)dt  0  L2s L2x  and by lemma 2.4.4 x σ Pc F (s)  2 L2s L2x .  Hence, ∞  ∞  ei(t−s)H Pc F (s)ds 0  Lpt Lqx  =  e−isH Pc F (s)ds  eitH ∞  0  e−isH Pc F (s)ds  0  x σ F (s)  L2x  Lpt Lqx  by lemma 2.4.2  L2s L2x .  Now, by a lemma of Christ-Kiselev (see [17]), we have t  ei(t−s)H Pc F (s)ds 0  x σ F (s)  Lpt Lqx  L2s L2x .  (2.95)  Next, let x σ g(x, t) ∈ L2t L2x . Then ∞  ( x σ g(x, t), x 0  =  ∞  ∞  ei(t−s)H Pc F (s)ds)dt 0  ei(t−s)H Pc F (s)ds)dt  (g(x, t), 0  ∞  −σ  0  Moving the integrals through the inner product and rearranging the terms,  65  2.4. Detailed proof we get ∞  ( x σ g(x, t), x 0  i(s−t)H  e  ds(  =  ei(t−s)H Pc F (s)ds)dt 0  ∞  ∞  ∞  −σ  Pc g(x, t)dt, F (s))  0  0  by H¨ older inequality ∞  ei(s−t)H Pc g(x, t)dt  ≤  x σg Hence, x  L2x L2t  F (s)  F (s)  Lqt Lpx  0  Lqt Lpx  Lqt Lpx  ∞  −σ  ei(t−s)H Pc F (s)ds 0  F (s)  L2t L2x  Lqt Lpx  .  (2.96)  .  (2.97)  Again, by the lemma of Christ-Kiselev, we have x  t  −σ  ei(t−s)H Pc F (s)ds 0  F (s)  L2t L2x  Lqt Lpx  Now by lemma 2.4.2 and lemma 2.4.4, we have shown lemma 2.4.5. Lemma 2.4.6. (Derivative Strichartz estimates) Let p ≥ 2 and let H1 = H + K = −∆ + 2iA · ∇ + i(∇ · A) + V + K  (2.98)  for a sufficiently large number K. Then H1 is a positive operator on Lp , and 1  φ  W 1,p  ∼ H12 φ  Lp .  (2.99)  From this, it follows that e−itH f and  Lqt Wx1,p  f  (2.100)  Hx1  t  ei(t−s)H Pc F (s)ds 0  Lqt Wx1,p  F  Lqt˜ Wx1,p˜  ,  (2.101)  for Strichartz admissible (q, p) and (˜ p, q˜).  66  2.4. Detailed proof Proof. We would like to first show 1  φ Clearly φ  W 0,p  = φ  W 1,p  Lp  ∼ H12 φ  = H10 φ  Lp .  Lp  for φ ∈ W 1,p .  (2.102)  As shown in the appendix of [51], if  K is large enough, H1 is a positive operator on Lp , and φ  ∼ H1 φ  W 2,p  Lp .  (2.103)  By Theorem 1 of [24], there exist positive numbers ε and C, such that H1it  is a bounded operator on Lp for −ε ≤ t ≤ ε and H1it ≤ C. Therefore  the hypothesis of Section 1.15.3 of [85] holds and we have that 1  [D(H1 ), D(H10 )] 1 = D(H12 ).  (2.104)  2  Using that D(H1 ) = W 2,p , D(H10 ) = Lp and [W 2,p , Lp ] 1 = W 1,p , we find 2  that 1  D(H12 ) = W 1,p .  (2.105) 1  1  Now by Section 1.15.2 of [85], H12 is an isomorphic mapping from D(H12 ) = W 1,p onto Lp . Therefore, we have 1  φ  W 1,p  ∼ H12 φ  Lp .  (2.106)  Finally, t  t  ei(t−s)H Pc F (s)ds 0  Lqt Wx1,p  ei(t−s)H Pc F (s)ds  = 0  t  1  ∼  H12  ei(t−s)H Pc F (s)ds 0  t  = 0  q Wx1,p Lt  Lpx Lqt  1  ei(t−s)H Pc H12 F (s)ds  Lpx Lqt  1  H12 F ∼  F  Lqt˜ Lqx˜  Lqt˜ Wx1,p˜  67  2.4. Detailed proof  For s ∈ R, denote the norm φ  and the norm φ  x sH1  φ  φ  x s L2  x sH1  = φ  x s L2  =  x  by −s  φ  (2.107)  L2  by x s L2  + ∇φ  x s L2 .  (2.108)  Next we need derivative version of the weighted estimates of Lemma 2.4.4 - this is given in Lemma (2.4.9) below. First, we need two preparatory lemmas. Lemma 2.4.7. For t > 0, let At (x) =  1 √ A( √xt ) t  and Vt (x) = 1t V ( √xt ). Let  ˜ = −∆ + 2iAt · ∇ + i(∇ · At ) + Vt + 1 K + 1. H t ˜ −1 Then there exists T > 0 such that supt>T H  L2 →H 2  (2.109)  < ∞.  ˜ −1 φ. Then Proof. Take t ≥ 1. For φ ∈ L2 , let h = H φ  2 2  =  =  1 (−∆ + 2iAt · ∇ + i(∇ · At ) + Vt + K + 1)h, t 1 (−∆ + 2iAt · ∇ + i(∇ · At ) + Vt + K + 1)h t ∆h 22 + h 22 + At · ∇h 22 + 2 ∇h 22 + F ∆h  2 2  + h  2 2  +F  where F denotes the rest of the terms, and recall that q > 3. We would like  68  2.4. Detailed proof to show that every term in F is bounded by h  2 . H2  Here,  1 + 2 (∆h)(∇ · At + Vt + K)h t 1 +2 [At (∇ · At + Vt + K)] · (∇h)h 1 t 1 +2 At · (∇h)h 1 + 2 (At + Vt + K)2 h2 1 t  |F | ≤ 2 (∆h)(At · ∇h)  1  1  Here, (∆h)(At · ∇h)  1 . √ ∆h 2 (A( √ ) t t 2q where <6 q−2 . 1 √ ∆h 2 (A( √ ) t t  1  −  t  (q−3) 2q  ∆h  2  A  L∞ +Lq (  ∇h  2  + ∇h  L∞ +Lq (  ∇h  2  + ∆h  L∞ +Lq  1 (∆h)((∇ · At ) + Vt + K)h 1 t . 1 . ∆h 2 ( (∇ · A)( √ ) L∞ +L2 + V ( √ ) t t t t  −1 4  ∆h 2 ( ∇ · A  L∞ +L2  + V  ( ∇h  2 + ∆h  L∞ +L2  + K)( h  L∞ +L2  3 q  2  + K)( h 2  + h  1 4  2  2q q−2  3 q  2  )  ∇h  ∇h  2  q−3 q  2  q−3 q  2  ),  + h  ∞)  ∆h  3 4  2 ).  Similar bounds hold for the other terms of F . We conclude that φ  2 2  ≥ (1 + o(1)) h  2 H2  as t → ∞.  (2.110)  Hence, for all t large enough, we have h  2 H2  φ 22 .  (2.111)  69  )  2.4. Detailed proof Lemma 2.4.8. Let H1 be as in lemma 2.4.6. For φ ∈ L2 and t > 0, we have ∇(H1 + t)−1 φ  1  (1 + t)− 2 φ  L2  L2 .  (2.112)  Proof. For φ ∈ L2 , let ψ = (H1 +t)−1 φ. For t bounded away from zero, define √ √ √ ψ by ψ(x) = 1t ψ( tx). Then ∆ψ(x) = ∆ψ( tx), ∇ψ(x) = √1t ∇ψ( tx) and √ V (x)ψ(x) = 1t V (x)ψ( tx) and √ ˜ ψ)( tx) = φ(x). (H Replacing x by  x √ t  ˜ we get and inverting H, x ˜ −1 φ( √ ψ(x) = H ). t  (2.114)  √ . 1 ˜ −1 φ( √ )]( tx) ψ(x) = [H t t  (2.115)  √ 1 ˜ −1 φ( √. )]( tx). ∇ψ(x) = √ [∇(H) t t  (2.116)  Hence,  and  (2.113)  ˜ −1 By Lemma 2.4.7, H ∇ψ(x)  2  L2 →L2  is uniformly bounded for t ≥ T . Therefore,  √ 1 ˜ −1 φ( √. )]( tx) 2 √ [∇H t t −3 1 ˜ −1 φ( √. ) 2 = t 4 − 2 ∇H t −3 1 . − −1 ˜ t 4 2 ∇H L2 →L2 φ( √ ) t =  2  1  = t− 2 φ  2  Therefore, for t ≥ T , ∇(H1 + t)−1 φ  1  2  t− 2 φ  2  (2.117)  and the lemma follows. Lemma 2.4.9. (Derivative weighted estimates) Let H1 be as in lemma 70  2.4. Detailed proof 2.4.6. We have 1  φ  x sH1  ∼ H12 φ  x s L2  for s ∈ R.  (2.118)  From this, it follows that x  t  −σ  ei(t−s)H Pc F (s)ds 0  Proof. Since f  x sH1  =  x  −s f  L2  x σF  L2t Hx1  + ∇x  −s f  L2 ,  L2t Hx1 .  (2.119)  to show the lemma, it  suffices to show x  −s  − 21  H1  x  s  L2 →L2  <∞  (2.120)  < ∞.  (2.121)  and ∇x  −s  − 12  H1  x  s  L2 →L2  The second bound above is the harder of the two. We will show the second bound and the first one follows by a similar argument. First, ∇x − 12  Now ∇H1  −s  − 21  H1  −1  x s φ = ∇H1 2 φ + ∇ x  −s  − 12  is bounded from L2 to L2 since H1  −1  [H1 2 , x s ]φ  (2.122)  maps from L2 to H 1 while  ∇ maps from H 1 to L2 . − 12  For the second term, we use H1  t)−1 , x s ] = (H1 + t)−1 [H1 + t, x s ](H1 + ∇x  −s  −1  [H1 2 , x s ] = ∇ x  ∞  −s 0  ∞ dt √ (H1 0 t t)−1 to get  =  + t)−1 and [(H1 +  dt √ (H1 + t)−1 [H1 + t, x s ](H1 + t)−1 t (2.123)  Recall that H1 = −∆ + 2iA · ∇ + i(∇ · A) + V + K,  (2.124)  so [H1 + t, x s ] = (−∆ x s ) − 2(∇ x s ) · ∇ + 2iA · (∇ x s ).  71  2.4. Detailed proof Let g(x) = (−∆ x s ) + 2iA · (∇ x s ) and h(x) = −2(∇ x s ). Then ∇x  −s  −1  [H1 2 , x s ] = ∇ x  Since g(x)  s−1 ,  x  dt √ (H1 + t)−1 g(x)(H1 + t)−1 t  0 ∞  = ∇ 0  dt √ x t  −s  ∞  −s  +∇ x  0  dt √ (H1 + t)−1 (g(x) + h(x) · ∇)(H1 + t)−1 . t (2.125)  we rewrite the g(x)-part of the above as  ∞  −s  ∇x  ∞  −s  0  g(x)(H1 + t)−1 (H1 + t)−1  dt √ (H1 + t)−1 [H1 + t, g(x)](H1 + t)−1 (H1 + t)−1 t  The first part of the above sum is bounded. For the second part, writing ˜ [H1 + t, g(x)] = g˜(x) + h(x) · ∇ as before , we can iterate the above process until g˜(x)  1. Since h(x)  ∇x  ∞  −s  = ∇ 0  +∇ x  dt √ x t  −s  ∞  −s  s−1 ,  so by the similar argument, we have  dt √ (H1 + t)−1 h(x) · ∇(H1 + t)−1 t  0 ∞  x  0  h(x)(H1 + t)−1 ∇(H1 + t)−1  dt √ (H1 + t)−1 [H1 + t, h(x)](H1 + t)−1 ∇(H1 + t)−1 t  As before, the first part of the above sum is bounded. For the second part, ˜ [H1 + t, g(x)] = g˜(x) + h(x) · ∇ as before , we can iterate the above process ˜ until h(x)  1. As a result, it suffices to consider ∞  dt √ ((H1 + t)−1 )m t  (2.126)  dt √ ((H1 + t)−1 ∇(H1 + t)−1 )m t  (2.127)  0  and  ∞ 0  for m ≥ 1. Now by lemma 2.4.8, both of the expressions above are bounded in L2 . Now, to prove theorem 6, apply lemma 2.4.6 and 2.4.9 to lemma 2.4.5, 72  2.4. Detailed proof we get the result. Finally, we need a lemma from [40] for the projection operator Pc onto the continuous spectral subspace. Lemma 2.4.10. (Continuous spectral subspace comparison) Let the continuous spectral subspace Hc [z] be defined as Hc [z] = {η ∈ L2 | iη, D1 Q[z] = iη, D2 Q[z] = 0}.  (2.128)  Then there exists δ > 0 such that for each z ∈ C with |z| ≤ δ, there is a bijective operator R[z] : Ran Pc → Hc [z] satisfying Pc |Hc [z] = (R[z])−1 .  (2.129)  Moreover, R[z] − I is compact and continuous in z in the operator norm on any space Y satisfying H 2 ∩ W 1,1 ⊂ Y ⊂ H −2 + L∞ . The proof of lemma 2.4.10 is given in lemma 2.2 of [40]. We will use lemma 2.4.10 with Y = L2 .  2.4.3  Proof of the main theorem  Lemma 2.2.1 gives the following corollary which will form part of the main theorem. Lemma 2.4.11. (Best decomposition) There exists δ > 0 such that any ψ ∈ H 1 satisfying ψ  H1  ≤ δ can be uniquely decomposed as ψ = Q[z] + η  (2.130)  where z ∈ C, η ∈ H 1 , iη, D1 Q[z] = iη, D2 Q[z] = 0, and |z| + η ψ  H1  H1 .  The proof of lemma 2.4.11 is essentially an application of the implicit function theorem on the equation B(z) = 0 with B(z) = (B1 (z), B2 (z)), Bj = i(ψ − Q[z]), Dj Q[z] for j = 1, 2. (2.131) 73  2.4. Detailed proof Details can be found in lemma 2.3 of [40]. Now, we prove theorem 5. Proof. Substitute ψ(t) = Q[z(t)] + η(t)  (2.132)  into equation (2.3) to get i(DQz˙ + ∂t η) = HQ + Hη + g(Q + η) where for w ∈ C, we denote DQ[z]w = D1 Q[z] Re w + D2 Q[z] Im w. Since HQ + g(Q) = EQ and DQ[z]iz = iQ[z] (since Q[eiα z] = eiα Q[z] for α ∈ R), we have i∂t η = Hη − iDQz˙ + EQ − g(Q) + g(Q + η) = Hη − iDQ(z˙ + iEz) − g(Q) + g(Q + η). We can write this as i∂t η = Hη + F  (2.133)  F = g(Q + η) − g(Q) − iDQ(z˙ + iEz).  (2.134)  where  In integral form, t  η(t) = e−itH (η(0) − i  eisH F (s)ds).  (2.135)  ei(t−s)H Pc F (s)ds.  (2.136)  0  Let ηc = Pc η. Then t  ηc = e−itH Pc η(0) − i 0  74  2.4. Detailed proof Then for fixed σ > 4, since η = Re[z]ηc , we have η  ηc  X  X t  η(0)  Hx1 t  +  +  e−i(s−t)H Pc (F (s) − 2Q|η|2 − Qη 2 − |η|2 η)ds  X  0  e−i(s−t)H Pc (2Q|η|2 + Qη 2 + |η|2 η)ds  X  0 t  η(0)  Hx1  + Qη 2  For Qη 2 Qη 2  3  1, 18 13  Lt2 Wx 3  e−i(s−t)H Pc (F (s) − 2Q|η|2 − Qη 2 − |η|2 η)ds + η3  3  Qη 2  Q 3  1, 18 13  η3  3  + ∇(Qη 2 )  18  Lt2 Lx13  + Q  6 L∞ t Lx  3  18  Lt2 Lx13  η  18  L3t Lx5  ∇η  18  L3t Lx5  2 X.  η  Wx1,6  18  L3t Lx5  18  + Qη∇η  18  Lt2 Lx13  2  η  1,6 L∞ t Wx  3  3  Lt2 Lx13  , we have  3 1, 18 Lt2 Wx 13  =  η3  9 Lx2  ≤  η2  ≤  η  ∇η  η  1 2  L2x  9 L6t Lx2  + ∇η 3  3 18 Lt2 Lx13  η3  Now, using η  .  , we have  Q  Lt2 Wx  18 1, 13  Lt2 Wx  (|Q| + |∇Q|)η 2  For η 3  X  0  3 1, 18 Lt2 Wx 13  =  1, 18 13  Lt2 Wx  +  + η 2 ∇η  3 18 Lt2 Lx13 9  L3t Lx4 2  η  η η  9  L6t Lx2 1 2  18  3  3  1, 18 5  18  Lt2 Lx13 2  + η  18  L3t Lx5 L3t Wx  18  Lt2 Lx13  9  L3t Lx4  ∇η  18  L3t Lx5  .  , we get  Lx5  ∇η  1 2  2 L∞ t Lx  η  1 2 18  .  (2.137)  L3t Lx5  75  2.4. Detailed proof So η3  3  ∇η  1, 18 13  Lt2 Wx  2 L∞ t Lx  η  2  η  1, 18 L3t Wx 5  3 X.  (2.138)  Together we have t  η  η(0)  X  + Q  +  Hx1  2 X  e−i(s−t)H Pc (F (s) − 2Q|η|2 − Qη 2 − |η|2 η)ds  X  0  Wx1,6  η(0) + η  Hx1  η  2 X  + η  3 X  + (F (s) − 2Q|η|2 − Qη 2 − |η|2 η)  + η  L2t x  −σ H 1 x  3 X  Next, for g(ψ) = |ψ|2 ψ, (F − 2Q|η|2 − Qη 2 − |η|2 η) =  L2t x  Q2 η + 2|Q|2 η − iDQ(z˙ + iEz) x  2σ  Q2  Wx1,∞  η  L2t x  σH1 x  −σ H 1 x  L2t x  + DQ  −σ H 1 x  x  −σ H 1 x  z˙ + iEz  L2t .  Next, we would like to bound (z˙ + iEz). Recall that we imposed iη,  ∂ Q[z] = 0 and ∂z1  iη,  ∂ Q[z] = 0 ∂z2  (2.139)  through Lemma 2.4.11. By Gauge covariance of Q, we have Q[eiα z] = eiα Q[z].  (2.140)  So for z = z1 + iz2 , ˜ 2 ] where α = tan−1 ( Q[z] = eiα Q[|z|  z2 ). z1  (2.141)  ˜ : R+ → R. So Here Q iα ˜ iα iα ˜ iα ˜ ˜ ˜ ∂z1 Q = ∂z1 (eiα )Q+2z 1 e Q = e i(∂z1 α)Q+2z1 e Q = i(∂z1 α)Q+2z1 e Q  (2.142)  76  2.4. Detailed proof and iα ˜ iα ˜ iα iα ˜ ˜ ˜ ∂z2 Q = ∂z2 (eiα )Q+2z 2 e Q = e i(∂z2 α)Q+2z2 e Q = i(∂z2 α)Q+2z2 e Q .  (2.143) So 0 =  iη, −z2 ∂z1 Q + z1 ∂z2 Q = η, −z2 (∂z1 α)Q + z1 (∂z2 α)Q  = (−z2 (∂z1 α) + z1 (∂z2 α)) η, Q = η, Q . Now differentiate iη, ∂z∂ 1 Q[z] = 0 and iη, ∂z∂ 2 Q[z] = 0 with respect to t and substitute i∂t η = Hη + F , we get 0 = =  ∂ ∂ Q[z] + iη, D Qz˙ ∂zj ∂zj ∂ ∂ Hη + F, Q[z] + iη, D Qz˙ ∂zj ∂zj i∂t η,  Recall that F = g(Q + η) − g(Q) − iDQ(z˙ + iEz). Therefore, we have 0 = =  Hη + g(Q + η) − g(Q) − iDQ(z˙ + iEz),  ∂ ∂ Q[z] + iη, D Qz˙ ∂zj ∂zj  ∂ g(Q + εη)|ε=0 ) + (g(Q + η) − g(Q) − ∂ε0 g(Q + εη)) ∂ε ∂ ∂ Q[z] + iη, D Qz˙ −iDQ(z˙ + iEz), ∂zj ∂zj (Hη +  77  2.4. Detailed proof From the above, we get that (g(Q + η) − g(Q) − ∂ε0 g(Q + εη)), =  ∂ Q[z] ∂zj  ∂ Q[z] ∂zj ∂ + (Hη + ∂ε0 g(Q + εη)), Q[z] ∂zj ∂ Qz˙ . + iη, D ∂zj −iDQ(z˙ + iEz),  Let Hη = Hη + ∂ε0 g(Q + εη). By the symmetry of H and differentiating equation (2.9) by zj , we have Hη,  ∂ ∂ Q = η, H Q ∂zj ∂zj  = = =  ∂ ∂ Q +( E) η, Q ∂zj ∂zj ∂ ∂ η, E Q = iη, iE Q ∂zj ∂zj ∂ iη, E DQiz ∂zj η, E  using η, Q = 0 and DQ[z]iz = iQ[z]. So ∂ Q[z] ∂zj ∂ ∂ ∂ −iDQ(z˙ + iEz), Q[z] + iη, E DQiz + iη, D Qz˙ ∂zj ∂zj ∂zj ∂ ∂ −iDQ(z˙ + iEz), Q[z] + iη, (D Q)(z˙ + iEz) ∂zj ∂zj (g(Q + η) − g(Q) − ∂ε0 g(Q + εη)),  = =  For g(ψ) = |ψ|2 ψ, ∂ε0 g(Q + εη) = Q2 η + 2|Q|2 η.  (2.144)  78  2.4. Detailed proof Therefore, g(Q + η) − g(Q) − ∂ε0 g(Q + εη) = |Q + η|2 (Q + η) − |Q|2 Q − Q2 η − 2|Q|2 η = 2Q|η|2 + Qη 2 + |η|2 η  Since  ∂ ∂ Q, i Q = j − k + o(1), ∂zj ∂zk  (2.145)  we have that | 2Q|η|2 + Qη 2 + |η|2 η, DQ |(1 + η  |z˙ + iEz|  L2 ).  (2.146)  Therefore, z˙ + iEz  L2t  2Q|η|2 + Qη 2 + |η|2 η, DQ ( QDQ|η| ≤ ( QDQ ≤ ( QDQ (1 + η η For η  2 X  L4t L4x ,  2  L2t L1x  2 L∞ t Lx  η  2 L∞ t Lx  η  2  + DQ|η| η 2 L4t L4x  1 L∞ t H  L6t L4x ,  η  + η  L2t L1x )(1  + DQ  1 2  4 L∞ t Lx  3 2 18 L3t Lx5  2) L∞ t Lx  + η  η  2) L∞ t Lx  3 L6t L4x )(1  + DQ  4 L∞ t Lx  + η η  5 4  2) L∞ t Lx  1 L∞ t H  η  7 4 7  42  )  Lt2 Lx13  2) L∞ t Lx  + η  4 X  we used η  For η  L2t (1  L4x  ∇η  L4x  ∇η  1 4  L2x  η  3 4 18  .  (2.147)  .  (2.148)  Lx5  we used η  5 12 L2x  η  7 12 42 Lx13  79  2.4. Detailed proof Putting the preceding estimates together we have η  η(0)  X  and since  x  2σ Q2  H1  x  2σ  Q2  L∞ x  η  H1  + η  X  2 X  + η  + η  4 X,  (2.149)  << 1,  L∞ x  η  +  ≤ C[ η(0)  X  2 X  + η  4 X]  (2.150)  for some constant C ≥ 1. Now, let XT be the norm defined by ψ  XT  =  x  −σ  ψ  L2t ([0,T ],Hx1 )  Fix the initial condition ψ(0)  + ψ  1, 18 5  L3t ([0,T ],Wx  + ψ )  1 L∞ t ([0,T ],Hx )  to be small enough so that  X  η(0)  H1  ≤  1 . 20C 2  Let T1 = sup{T > 0 : η  ≤  XT  (2.151)  1 } > 0. 10C  (2.152)  Then for 0 ≤ T ≤ T1 , η  XT  ≤  1 1 1 1 + 2 + 4 3 ≤ , 20C 10 C 10 C 15C  (2.153)  showing that T1 = ∞. L1t .  Next, we would like to bound z˙ + iEz z˙ + iEz  L1t  2Q|η|2 + Qη 2 + |η|2 η, DQ (1 + η 2  ( QDQ|η| ≤ ( x  2σ  (1 + η  We have  L1t L1x  QDQ  2  + DQ|η| η  ∞ L∞ t Lx  x  −2σ 2  η  L2x ) L1t  L1t L1x )(1 L1t L1x  +  + η  2) L∞ t Lx  x σ DQ  ∞ L∞ t Lx  x  −σ 3  η  L1t L1x )  2) L∞ t Lx  80  2.4. Detailed proof Here, the factor x  −σ 3  η  L1t L1x  x  ≤  −σ η 3  x  −σ  η  L1t L1x  can be bounded by  L2t L2x  η  2 L4t L4x )  −σ  x  η  L2t L2x  η  1 2  1 L∞ t H  η  3 2 18  .  L3t Lx5  Putting everything together, we have z˙ + iEz Therefore, |∂t (ei i  limt→∞ e  t 0  E(s)ds  t 0  E(s)ds  2 X  η  L1t  + η  4 X  z(t))| = |z˙ + iEz| ∈ L1t .  z(t) exists. Since |e  t 0  i  E(s)ds  This means that  z(t)| = |z|, limt→∞ |z(t)| ex-  ists. Furthermore, E is continuous and E(z) = E(|z|), so limt→∞ E(z(t)) exists. Finally, let H = −∆ + 2iA· ∇ + i(∇· A) + V . So t  ηc (t) = eitH (ηc (0) − i  e−isH Pc F (s)ds).  (2.154)  0  By Strichartz estimates as above, we have T  e−isH Pc F (s)ds  H1  F  X  →0  (2.155)  S  as T > S → ∞. Therefore,  ∞ −isH Pc F ds 0 e ∞  lim e−itH ηc (t) = ηc (0) − i  t→∞  converges in H 1 , and  e−isH Pc F (s)ds =: η+  (2.156)  0  for some η+ ∈ H 1 . From this, we get that ηc (t) converges to 0 weakly in H1 . Now, by compactness of R[z(t)] − I, we have that ηd (t) := η(t) − ηc (t) = (R[z(t)] − I)ηc (t) converges to 0 strongly in H 1 . Therefore η(t) − eitH η+  H1  → 0.  (2.157)  81  Chapter 3  Global well-posedness of two dimensional radial Schr¨ odinger maps into the 2-sphere In this chapter, we will discuss a result obtained jointly with Stephen Gustafson. The main result is Theorem 4 stated in Section 1.3.  3.1  Known results and our result  Consider the two dimensional Schr¨odinger map equation  u = u × ∆u t u(x, 0) = u (x), u(0) − k ∈ H 2 (R2 ) 0  (3.1)  where u : R2 × R → S2 . Recall that we treat the S2 as a sphere embedded in R3 , i.e. S2 = {u ∈ R3 : |u| = 1} ⊂ R3 .  (3.2)  u(x, t) = (u1 (x, t), u2 (x, t), u3 (x, t))  (3.3)  u21 + u22 + u23 = 1.  (3.4)  Hence, we view u as  where  82  3.1. Known results and our result Conserved quantities of this equation are the L2 -mass u(t) − k  2 L2 (R2 )  2 L2 (R2 )  (3.5)  = E(u0 ).  (3.6)  = u0 − k  and the energy E(u) =  1 ∇u(t) 2  2 L2 (R2 )  We will consider the problem of whether (3.1) is globally wellposed. In other words, we would like to know whether the solution u exists indefinitely or blows up in finite time. In an attempt to gain some understanding on whether blow up is possible, we will see how the energy behaves under a scaling preserving the solution. If u is a solution of (3.1), then for λ > 0, v(x) = u(λx, λ2 t)  (3.7)  is again a solution of (3.1). We would like to compare the energy E of v to that of u. The energy scales differently according to the space dimension n and is given by E(v) = λ2−n E(u).  (3.8)  For example, for space dimensions n = 1, 2, 3, we have that: For n = 1, E(u) = λE(u).  (3.9)  For n = 2, E(u) = E(u). 1 For n = 3, E(u) = E(u). λ  (3.10) (3.11)  Here, as λ → ∞, u(λx, λ2 t) undergoes a horizontal compression. From the energy scaling, we see that for n = 1, it costs a lot of energy to concentrate solutions. As energy is a conserved quantity, the heuristic is that it is hard for solutions to concentrate, so solutions are expected to be global. The case n = 1 is called energy subcritical. On the other hand, for n = 3, it costs very little energy to concentrate solutions, so blow up solutions are expected. The case n = 3 is called energy supercritical. For n = 2, the energy remains unchanged after the scaling. The case n = 2 is called  83  3.1. Known results and our result energy critical. In this case, the scaling arguement does not give us a heuristic of whether blow up solutions are possible. It turns out that blow up can indeed occur for n = 2. Very recently, Merle-Rapha¨el-Rodnianski ’11 [55] showed that within a special class of solutions to (3.1) known as 1-equivariant maps, there are blown up solutions. More specifically, let   0 −1 0   R= 1  0  0  0      cos α − sin α 0    0  and eαR =  sin α 0 0  cos α 0     0 . 1  (3.12)  For an integer m, an m-equivariant map u : R2 → S2 ⊂ R3 is a map of the form u(r, θ) = emθR v(r)  (3.13)  with v(0) = −k and v(∞) = k. The Schr¨ odinger map (3.1) preserves m-equivariance and a radial solution is an example of an m-equivariant map with m = 0. It turns out that any m-equivariant map with the boundary conditions (v(0) = −k and v(∞) = k) defined above will have energy at least 4π|m|. Various results concerning m-equivariant maps are known. For example, [41] showed that if m ≥ 3 and if the energy of the initial data is slightly larger than 4π|m|, then the solution is global. [55] showed if m = 1, then there exists a set of smooth initial data with E > 4π whose solutions blow up in finite time. The result of [55] tells us that even smooth initial data can lead to blow up solutions. Now, let us look at some more general known results regarding global well-posedness of (3.1). For n = 1, [16] showed that if E(u0 ) is finite, then (3.1) is global. In the same paper, they also showed that for n = 2, if E(u0 ) sufficiently small, the radial solutions to (3.1) are global. [4] showed that for n = 2, suppose the initial data u0 satisfies u0 − Q ∈ H s for all s > 0 for some Q ∈ S2 and E(u0 ) is small, then the solution u to (3.1) is global and u − Q ∈ H s for all s > 0. Our result shows that for n = 2, any radial solution u to (3.1) is global. This is the first result that showed global well-posedness in n = 2 without 84  3.2. Discussion and outline of the proof the smallness of energy assumption. Our result is as follows: Theorem 7. (Global well-posedness of 2D radial Schr¨ odinger map into S2 ) Suppose u(x, 0) = u0 (|x|) is radial and u0 − k ∈ H 2 (R2 ). Then ut = u × ∆u with u(r, 0) = u0 (r)  (3.14)  has a unique global solution u ∈ L∞ ([0, ∞); H 2 (R2 )).  3.2  Discussion and outline of the proof  In this section, we will give an outline to the proof of Theorem 7. Finer details of the proof will be given in the next section. The strategy to obtain Theorem 7 is to transform (3.1) into a more familiar equation and to show global well-posedness for the transformed equation. After this, one then shows that global well-posedness of the transformed equation implies global well-posendess of (3.1). The transformation we use is the generalized Hasimoto transformation used by [16] to show global wellposedness for the Schr¨ odinger map solutions in 1D and small solutions in 2D. It transforms our radial Schr¨odinger map equation into the equation iqt = −∆q +  ∞  1 q+ r2  r  |q(ρ, t)|2 1 dρ − |q|2 q ρ 2  (3.15)  for a complex-valued function q = q(r, t). If we look at (3.15), we see that it is made up of the linear part iqt = −∆q +  1 q, r2  (3.16)  the local nonlinear part 1 − |q|2 q 2  (3.17)  as well as the non-local nonlinear part ∞ r  |q(ρ, t)|2 dρ q. ρ  (3.18)  85  3.2. Discussion and outline of the proof The last term is called the non-local term because its value at r = r0 depends not only on the value of q at r0 but also on the value of q at other r’s as well. We will call (3.15) the non-local nonlinear Schr¨odinger equation (shortened to be NLNLS below). To show global well-posedness of Schr¨odinger map equation, we will show global well-posedness of the NLNLS. Recall that the original Schr¨odinger map equation is ut = u × ∆u with u(r, 0) = u0 (r)  (3.19)  u0 − k ∈ H 2 (R2 ).  (3.20)  where  As we will see from the details of the generalized Hasimoto transformation, we have |ur | = |q|. Hence, ∇u0 ∈ L2 implies q0 ∈ L2 . As a result, we would like to show global well-posedness for iqt = −∆q +  1 q+ r2  ∞ r  |q(ρ, t)|2 1 dρ − |q|2 q ρ 2  (3.21)  with q(r, 0) = q0 ∈ L2 (R2 ).  (3.22)  To formulate a strategy in showing global well-posedness for NLNLS, let us compare this equation with the more familiar cubic nonlinear Schr¨odinger equation (shortened to be NLS below) iut = −∆u ± |u|2 u  (3.23)  with radial initial data u(x, 0) = u0 (r),  r = |x|.  (3.24)  Recall that this equation is defocusing with the + sign, and focusing with the − sign. For n = 2, this equation is L2 -critical and so it is highly nontrivial to show global well-posedness with L2 initial data (and indeed it is false in the focusing case if the L2 norm is sufficiently large). 86  3.2. Discussion and outline of the proof It turns out that NLNLS and NLS share a lot of similarities. For example, both equations satisfy the mass conservation (conservation of the L2 norm). Here, mass conservation of NLNLS arises from the conservation of the energy E of u. Since |ur | = |q|, ∇u(t) q(t)  L2 (R2 )  2 L2 (R2 )  = q(0)  = ∇u(0)  L2 (R2 ) .  2 L2 (R2 )  gives (3.25)  As mentioned above, NLS is L2 -critical. One can also check that NLNLS is also L2 -critical. To do so, we just observe that if q is a solution to NLNLS, then qλ defined by qλ (r, t) = λq(λr, λ2 t) for λ > 0  (3.26)  is also a solution and that qλ  L2 (R2 )  = q  L2 (R2 ) ,  (3.27)  which says that NLNLS is L2 -critical. Despite the above similarities, there is a major difference between the cubic NLS and NLNLS: the quantity E(u(t)) := R2  1 1 |∇u(x, t)|2 ± |u(x, t)|4 , 2 4  (3.28)  known as the energy, is a conserved quantity for the NLS while NLNLS has no equivalent conserved quantity. Of course, if we only assume L2 initial data for NLS, then the energy may not be defined for the solution. However, it turns out that energy conservation plays a big role in showing global wellposedness of NLS for even L2 data as we will see below. To study the global well-posedness of NLNLS, we look at results of global well-posedness for NLS. Since the two equations are similar, it may be possible to adapt a method of showing global-posedness of NLS to NLNLS. As for NLS, Killip-Tao-Vsian [48] showed global well-posedness for NLS in 2D with radial L2 initial data in the defocusing case, and in the focusing case when the L2 -norm is below a certain level. It turns out the method used 87  3.2. Discussion and outline of the proof there can be adapted to the case of NLNLS with some significant changes, in particular due to the presence of the non-local term and the absence of energy conservation. As an understanding of Killip-Tao-Visan’s method is important for understanding the proof of our result, in the next subsection, we will outline their method.  3.2.1  Outline of Killip-Tao-Visan’s proof of global well-posedness of NLS for radial L2 initial data in 2D  The idea behind Killip-Tao-Visan’s method is a proof by contradiction. Following a method developed by Kenig-Merle [47], Tao-Visan-Zhang [84] showed that if there is any solution which fails to scatter in the sense that |u(x, t)|4 dxdt = ∞ I  (3.29)  R2  where I is its time interval of existence (so this includes solutions which blow-up), then there exists such a non-scattering solution, of minimal L2 norm, with interval of existence I and functions N : I → R+ and C : R+ → R+  (3.30)  such that for each t ∈ I and η > 0, |u(x, t)|2 dx ≤ η and |x|≥C(η)/N (t)  |u(ξ, t)|2 dξ ≤ η.  (3.31)  |ξ|≥C(η)N (t)  Here, one can think of  1 N (t)  as the spatial scale of u and N (t) as the frequency  scale. Killip-Tao-Visan refined the above and showed that one can assume N : I → R+ belongs to one of the following three scenarios: Soliton-like solution: I = R and N (t) = 1 for all t ∈ R  (3.32)  88  3.2. Discussion and outline of the proof Self-similar solution: 1  I = (0, ∞) and N (t) = t− 2 for all t ∈ I  (3.33)  Inverse cascade: I = R, liminft→−∞ N (t) = liminft→∞ N (t) = 0 and sup N (t) < ∞. (3.34) t∈R  These solutions are referred as the three enemies by Killip-Tao-Vsian. The solutions u above have a lot of structure. For example, the solitonlike solution exists forever and is localized in space (N (t) = 1) while the self-similar solution concentrates and blows up in one direction in time and 1  spreads out in the other direction of time (I = (0, ∞) and N (t) = t− 2 ). In fact, Killip-Tao-Visan showed this structure to be incompatible with the NLS equation. They showed global well-posedness by ruling out each of the three enemies case by case. It turns out that the method showing the existence of the three enemies when global well-posedness fails is quite general and can be applied to the NLNLS case with some modifications. However, the method used to rule out the three enemies depends on energy conservation and does not apply to the NLNLS case. Indeed, energy conservation is a key tool in ruling out these highly structured solutions. As the initial data u0 is only L2 , energy is not a defined quantity. However, because the special solution u has the property (3.31), one can in fact control the H 1 -norm of u. The idea is that if one is able to improve the regularity of the solution and show that the solution is in fact in H 1 , then energy will be defined and this has implications on how such a solution should behave which are incompatible with the highly structured solutions. We will provide a very brief outline of how solutions are ruled out. For this discussion, we consider only the defocusing NLS. This is because, as we will see, NLNLS has a defocusing character.  89  3.2. Discussion and outline of the proof Ruling out the self-similar case For the self-similar case, 1  N (t) = t− 2 and I = (0, ∞).  (3.35)  Here, the frequency scale N (t) is decreasing in time and it is possible to show that u ∈ H s for all s > 0. With this, the H 1 global well-posedness for defocusing NLS says that such a solution is global, but this is not compatible with the time of existence I = (0, ∞). The method above relies on the H 1 global well-posedness theory of NLS which relies heavily on the energy, so this method will not adapt well to the NLNLS case. Ruling out the inverse cascade case For the inverse cascade case, N (t)  1, lim inf N (t) = lim inf N (t) = 0 and I = R. t→−∞  t→∞  (3.36)  s As the frequency scale N (t) is bounded, it is possible to show that u ∈ L∞ t Hx  for all s ≥ 0. With this, energy conservation and the Gagliardo-Nirenberg inequality show that ∇u  L2x  will be bounded away from 0. However, this  is not compatible with lim inf N (t) = lim inf N (t) = 0 t→−∞  as it can also be shown that ∇u  t→∞  L2x  (3.37)  → 0 along any sequence of t where  N (t) → 0. This is a contradiction. Just as in the previous case, the above method relies on energy conservation to show the contradiction, so this will not adapt well to the NLNLS case either.  90  3.2. Discussion and outline of the proof Ruling out the soliton case For the soliton-like case, I = R and N (t) = 1.  (3.38)  The idea is to consider the quantity ux · ∇udx.  M (t) = 2 Im  (3.39)  R2  Here, the quantity M (t) is formally  d dt  R2  x2 |u|2 dx, the time derivative of  the variance of |u|2 . However, the above quantity may not be finite for the solution u. To make the above well defined, one has to add a smooth cutoff function φR (r) which is zero outside a disk of some large radius R > 0 and consider instead φR ux · ∇udx.  MR (t) = 2 Im  (3.40)  R2 1 Again, one can show u ∈ L∞ t Hx , so on one hand,  |MR (t)|  R u  2  ∇u  2  R  (3.41)  but on the other hand, it can be shown that d MR (t) = 8E(u) + error terms dt  (3.42)  where the error terms are of size comparable to u  2 L2 (r>R) ,  ∇u  2 L2 (r>R)  and  u  2 L4 (r>R) .  (3.43)  Hence, using (3.31) with N (t) = 1, and by choosing R large enough, the error terms can be made to be much smaller than E, so d MR (t) ≥ E > 0. dt This contradicts with |MR (t)|  (3.44)  R.  91  3.2. Discussion and outline of the proof Notice in the above, the preserved quantity, energy, is used to obtain a lower bound of the first time derivative of the quantity MR . To adapt the above to the NLNLS case requires an alternative way to obtain such a lower bound.  3.2.2  Discussion of our proof of global well-posedness of NLNLS for radial L2 initial data in 2D  As mentioned in the previous subsection, just like in the NLS case, it can be shown that when global well-posedness and scattering for NLNLS fails, then there exists a solution u with the structure of one of the three enemies. As in the NLS case, to show global well-posedness, we have to rule out the three enemies. The following is a brief discussion of how this is done. The complete proof will be given in the next section. As mentioned in the previous section, the method used for the NLS case to rule out self-similar solutions and inverse cascade solutions cannot be adapted to our case due to the lack of a conserved quantity equivalent to the energy for NLNLS. However, the method for ruling out the soliton case is more general and has a chance of being adaptable to the NLNLS. To rule out the soliton and the self-similar case, we consider the quantity φR qx · ∇qdx  MR (t) = 2 Im  (3.45)  R2  just as in the NLS case. Recall that in the NLS case, conservation of energy is used obtain a lower bound on the quantity  d dt MR (t)  in attempt to reach a  contradiction. Due to the lack of energy in NLNLS, we are forced to obtain such a lower bound by using more delicate estimates on various norms of the solution u. In the end, such a lower bound is obtained and soliton-like and self-similar solutions are ruled out. However, we are unable to obtain the more delicate estimates required to bound  d dt MR (t)  from below for the inverse cascade case. The reason is that  such delicate estimates require fine control on the structure of the solution u. In particular, we need to have very explicit knowledge of the structure  92  3.3. Proof of our result of the spatial scale  1 N (t)  (and the frequency scale N (t)). However, for the  inverse cascade case, such knowledge is not available. To get around this difficulty, instead of using MR , we consider a different quantity ∞  P (t) = Im  (qqr )ψ(r)rdr  (3.46)  0  for some function ψ which tends to zero at the origin, and tends to one at infinity. It turns out that with this quantity P , we are able to construct arguments to rule out the inverse cascade case just like in the soliton and self-similar case. Details on how this is done will be given in the coming section.  3.3  Proof of our result  We will provide details of the proof of our main result in this section. There are five parts to the proof: • Part 1: The goal of this part is to reduce the global well-posedness problem of the Schr¨ odinger map equation into the global well-posedness problem of the NLNLS equation by transforming the Schr¨odinger map equation into the NLNLS equation through the generalized Hasimoto transformation. In later parts of the proof, we will establish global well-posedness of NLNLS. In order to be able to translate the result back to the original Schr¨odinger map equation, we will also need to show we can translate a solution of NLNLS into a solution of the Schr¨ odinger map equation. The result of this part is Proposition 1. • Part 2: The goal of this part is to develop the local well-posedness theory which part 3 relies on. The result of this part is Proposition 2. • Part 3: The goal of this part is to show if global well-posedness and scattering of NLNLS fails, then NLNLS admits solutions, called almost periodic 93  3.3. Proof of our result solutions, having explicit structures in terms of the spatial scale and frequency scale. It is further shown that in this case, NLNLS admits solutions of the soliton-type, inverse cascade type or self-similar type (the three enemies) as discussed in the previous section. The result of this part is Proposition 3. • Part 4: The goal of this part is to show the three enemies given in the previous part have more regularity than originally entitled to due to the extra structure. The result of this part is Proposition 4. • Part 5: The goal of this part is to rule out the possibility of the three enemies. The details of this part will be given in Subsection 3.3.5. Once the three enemies have been ruled out, by the contrapositive of proposition 3, NLNLS must be global. Then by Proposition 1, the Schr¨odinger map under consideration must be global. This shows Theorem 7. The propositions for the five parts are given below: Proposition 1. There is a map u → q = q[u] from radial maps with u(r) − k ∈ H 2 (R2 ) to complex radial functions q(r) with w(x) := eiθ q(r) ∈ H 1 (R2 ) ((r, θ) polar coordinates on R2 ) such that if u(r, t) is a (radial) solution of (3.1), then q(r, t) = q[u] is a (radial) solution of (3.15). Further, the H 1 and H 2 norms of ∇u and w = eiθ q are comparable: w(t)  H 1 (R2 )  ∇u(t)  w(t)  H 1 (R2 )  H 2 (R2 )  ∇u(t)  H2  ∇u(t)  H 1 (R2 )  + ∇u(t)  w(t)  H 1 (R2 )  + w(t)  ∇u(t)  H 2 (R2 )  + ∇u(t)  w(t)  H 2 (R2 )  + w(t)  2 H 1 (R2 ) 2 H 1 (R2 ) .  3 H 1 (R2 )  3 H 1 (R2 ) .  (3.47)  (3.48)  Moreover, the map u → q is one-to-one: given two radial maps uA and uB as above, if the corresponding associated complex functions agree, q A ≡ q B , then so do the original maps, uA ≡ uB . 94  3.3. Proof of our result Here, we consider w(x, t) = eiθ q(r, t) to handle the term  q r2  in NLNLS.  The corresponding equation for w is ∞  iwt = −∆w + ρ>|x|  |w(ρ, t)|2 1 dρ − |w|2 ρ 2  w.  (3.49)  Proposition 1 will be proved in the Section 3.3.1. Proposition 2.  1. For each q0 ∈ L2 , (3.15) has a unique solution q ∈  C(I; L2 ) ∩ L4loc (I; L4 ) on a maximal (and non-empty) time interval 0 (possibly Tmin = −∞ and/or Tmax = ∞),  I = (Tmin , Tmax ) which conserves the  L2  2. If Tmax < ∞, then q  norm. L4t ([0,Tmax ];L4 )  = ∞ (an analagous statement  holds for Tmin ). 3. If Tmax = ∞ and q  L4t ([0,∞);L4 )  < ∞, then q scatters as t → +∞ (an  analagous statement holds for t → −∞). 4. The solution at each time depends continuously on the initial data. Further, the solution has the “stability” property as in Lemma 1.5 of [48]. 5. If q0 and q  L2  is sufficiently small, the solution is global (I = (−∞, ∞))  L4t (R;L4 )  < ∞.  Proposition 2 will be proved in the Section 3.3.2. Proposition 3. If there is any L2 data for which global well-posedness (or merely scattering) for (3.15) with radial L2 initial data fails, then 1. there exists a non-zero solution u of NLS with interval of existence I and functions N : I → R+ and C : R+ → R+  (3.50)  such that for each t ∈ I and η > 0, |u(x, t)|2 dx ≤ η  (3.51)  |x|≥C(η)/N (t)  95  3.3. Proof of our result and |u(ξ, t)|2 dξ ≤ η.  (3.52)  |ξ|≥C(η)N (t)  2. We may assume q falls into one of the following three cases • soliton-type solution: I = R and N (t) ≡ 1 • self-similar-type solution: I = (0, ∞) and N (t) = t−1/2 • inverse cascade-type solution: I = R, N (t)  1, lim inf t→−∞ N (t) =  lim inf t→∞ N (t) = 0 Proposition 3 will be proved in the Section 3.3.3. Proposition 4. If a solution q of NLNLS belongs to one of the soliton-type, the self-similar-type or the inverse cascade-type, then w(x, t) := eiθ q(r, t) ∈ H s (R2 )  (3.53)  for every s ≥ 0 and t ∈ I. Furthermore in the soliton and inverse cascade cases, s 2 w ∈ L∞ t H (R ) for each s ≥ 0.  (3.54)  Proposition 4 will be proved in the Section 3.3.4.  3.3.1  Equating the Schr¨ odinger map equation and the NLNLS equation  The goal of this section is to prove Proposition 1. The idea is to use the generalized Hasimoto transformation to translate the Schr¨odinger map equation into the NLNLS equation. The generalized Hasimoto transformation works as follows. First, given a solution u of (3.1), for each fixed time t, we will build a frame {e1 (r, t), e2 (r, t)} in the tangent space Tu(r,t) S2 . We will show how this is done below. As ur and ut are in Tu(r,t) S2 , we can express them as ur = q1 e1 + q2 e2 and ut = p1 e1 + p2 e2  (3.55)  96  3.3. Proof of our result where q1 , q2 , p1 and p2 are real valued functions of r. Now, define q = q1 + iq2 and p = p1 + ip2 .  (3.56)  Then q and p are complex valued functions of r. Here, as u evolves over time, q does so as well. We will show here that a particular choice of the frame e1 , e2 leads to the NLNLS equation. Building the frames Following [16], given a radial map u(r) ∈ k + H k , we want to construct a unit tangent vector field, parallel transported along the curve u(r) ∈ S2 : e(r) ∈ Tu(r) S2 ,  |e| ≡ 1,  Dr e(r) ≡ 0,  (3.57)  where here D denotes covariant differentiation of tangent vector fields: given ξ(s) ∈ Tu(s) S2 , Ds ξ(s) = PTu(s) S2 ∂s ξ(s) = ∂s ξ(s) + (∂s u(s) · ξ(s))u(s) ∈ Tu(s) S2 .  (3.58)  Since we have fixed the boundary condition (at infinity) u(r) → k as r → ∞ (at least in the L2 sense), we fix a unit vector in Tk S2 , say i = (1, 0, 0) to be the boundary condition for e (at infinity) and write e(r) = i + e˜(r),  u(r) = k + u ˜(r)  (3.59)  so that the parallel transport equation Dr e ≡ 0 becomes e˜r = −(˜ ur · [i + e˜(r)])(k + u ˜) = −(˜ u1 )r k − (˜ u · e˜)u − (˜ ur · i)˜ u,  (3.60)  which we will therefore solve in from infinity as ∞  e˜(r) = −˜ u1 (r)k +  (˜ u(s) · e˜(s))u(s) − (˜ ur (s) · i)˜ u(s) ds =: M (˜ e)(r) r  (3.61)  97  3.3. Proof of our result 2 := L2 ([R, ∞); R3 ) by finding a fixed point of the map M in the space XR rdr  for R large enough. To this end, we need the simple estimate Lemma 3.3.1. ∞  f (s)ds r  2 XR  ≤ f  =: f  1. XR  (3.62)  1 1 f (s)sds| ≤ f s r  1. XR  (3.63)  L1rdr [R,∞)  Proof. First by H¨ older, for r ≥ R, ∞  |  ∞  f (s)ds| = | r  r  Next, setting F (r) :=  ∞ r f (s)ds  so F = −f , we have F 2 (r) = 2  ∞ r F (s)f (s)ds,  so changing order of integration and using (3.63), ∞  F  2 2 XR  =2  ∞  ∞  F (s)f (s)ds ≤ 2  rdr R ∞  ≤  s  |F (s)||f (s)|ds  r  rdr  R  R  |F (s)|s|f (s)|sds ≤ sup(r|F (r)|) f r≥R  R  2 XR  2 1 XR  ≤ f  f  2 XR  (3.64) and the proof is completed by taking square roots. Now we may use Lemma 3.3.1 to estimate the map M : M (˜ e)  2 XR  ≤  u ˜  ≤  u ˜  2 XR 2 XR  + |˜ u(s)|(|˜ e(s)| + |˜ ur (s)|) + u ˜  2 XR  e˜  + u ˜  2 XR  2 XR  1 XR  u ˜r  Since u ˜ ∈ H 1 (R2 ), there is R0 such that for R ≥ R0 , u ˜ u ˜  2 XR  u ˜r  2 XR  2. XR  2 XR  < 1/3 and  < 1/3, so e˜  2 XR  ≤ 1 =⇒  M (˜ e)  2 XR  ≤ 1,  (3.65)  2 to itself. Also, for any e 2, that is, M sends the unit ball in XR ˜A , e˜B ∈ XR  M (˜ eA ) − M (˜ eB )  2 XR  ≤ u ˜  2 XR  e˜A − e˜B  2 XR  <  1 A e˜ − e˜B 3  2, XR  (3.66)  98  3.3. Proof of our result 2 , hence has a unique fixed point so M is a contraction on the unit ball in XR  there. 2, e 2, Using u ˜ ∈ H 2 (R2 ), it follows from (3.60), that e˜r ∈ XR ˜/r ∈ XR 2 . In particular, e and, after differentiating once, e˜rr ∈ XR ˜ is continuously  differentiable, so a genuine solution of (3.60). Now we may simply solve the initial value problem for the linear ODE (3.60) from r = R (with value e˜(R)) down to r = 0 to get e˜ on (0, ∞). Estimates as above imply that that e˜ ∈ H 2 (R2 ) (and in particular is continuous, and defined at r = 0). It is easily shown that if, in addition, u ˜ ∈ H 3 (R2 ), then e˜ ∈ H 3 (R2 ). So we have constructed a solution e(r) = i + e˜(r) of Dr e ≡ 0. It then follows directly from this ODE that ∂r (u(r) · e(r)) ≡ 0 and ∂r (e · e) ≡ 0 and hence that e(r) ∈ Tu(r) S2 and |e(r)| ≡ 1. So we have (3.57). The generalized Hasimoto transformation Recall that we have ur = q1 e1 + q2 e2 and ut = p1 e1 + p2 e2  (3.67)  q = q1 + iq2 and p = p1 + ip2 .  (3.68)  and  We would like to find an equation governing the evolution of q. To do so, we rewrite (3.1) as ut = −J u Dru +  1 r  ur .  (3.69)  Expressing the above in terms of q and p, we get 1 p = −i(∂r + )q. r  (3.70)  We would like to eliminate p from the above. To do so, we use that the fact  99  3.3. Proof of our result that Dru ut = Dtu ur and this gives ∂r p = (∂t + iT )q where Dtu e1 = T e2 .  (3.71)  If we take partial derivative with respect to r on both sides of (3.70) and eliminate ∂r p with the above, we get that qt + iT q = −i(∆ +  1 )q. r2  (3.72)  Other than T which is yet to be determined, the above is an evolution equation for q. Further computation shows that T satisfies the equation 1 1 (T + |q|2 )r = − |q|2 2 r  (3.73)  which we can integrate to get 1 T = − |q|2 + 2  ∞ r  |q(ρ)|2 dρ. ρ  (3.74)  Putting everything together, we arrive at our evolution equation for q iqt = −∆q +  1 q+ r2  ∞ r  |q(ρ, t)|2 1 dρ − |q|2 q. ρ 2  (3.75)  Here, (3.75) is the result of the Schr¨odinger map equation after the generalized Hasimoto transformation. Equivalence of norms We have ur = q1 e + q2 J e =: q ◦ e  (3.76)  (the last equality just defines a convenient notation), so |q| = |ur |,  (3.77)  100  3.3. Proof of our result and since Dr e ≡ 0, qr ◦ e = Dr (q ◦ e) = Dr ur = urr + |ur |2 u  (3.78)  |qr | ≤ |urr | + |ur |2 ,  (3.79)  so |urr | ≤ |qr | + |q|2 .  Setting w(x) = eiθ q(r), and taking norms: w  qr  H 1 (R2 )  L2 +  q/r  urr  L2  L2 +  2 L4 +  ur  ur /r  ∇u  L2  H 1 (R2 ) +  ∇u  2 H 1 (R2 )  (3.80) (using a Sobolev inequality at the end). And in the opposite direction, ∇u  urr  H 1 (R2 )  L2 +  ur /r  qr  L2  L2 +  q  2 L4 +  q/r  w  L2  H 1 (R2 ) +  (3.81) These last two inequalities give (3.47). Taking another covariant derivative in r and proceeding in a similar way yields (3.48). One to one Suppose uA (r) and uB (r) are two maps in k + H 2 (R2 ), and let eA (r), eB (r), and q A (r), q B (r) be the corresponding unit tangent vector fields, and complex functions (respectively) constructed as above. If we also denote f := J e, we have the linear ODE system   u      0  d     e  =  −q1 dr f −q2  q1 q2    u      u        0   e  =: A(q)  e  . 0 f f  0 0  (3.82)  Suppose now that q A (r) ≡ q B (r) =: q(r). Then we have   uA      uB        W :=  eA  −  eB  ∈ H 2 (R2 ), fA fB  Wr = A(q)W.  (3.83)  101  w  2 H 1 (R2 ) .  3.3. Proof of our result Applying the estimate of Lemma 3.3.1, we find W  L2rdr [R,∞)  ≤ C Wr  ≤C q  L1rdr [R,∞)  Choosing R large enough so that q  L2rdr [R,∞)  L2rdr [R,∞)  W  L2rdr [R,∞) .  (3.84)  < 1/C, we conclude W ≡ 0  on [R, ∞). Then standard uniqueness for initial value problems for linear ODE implies W (r) ≡ 0 for all r. Together, the above steps complete the proof of Proposition 1.  3.3.2  Local theory of NLNLS  Much of the result of Proposition 2 follows from [14], [13] (also see [26]). Following [26], part 1, 2, 3 and 5 of Proposition 2 holds for the equation  iu = −∆u + f (u) t u(x, 0) = u (x) ∈ L2 (R2 )  (3.85)  0  4  where the nonlinearity f : L2 ∩ L4 (R2 ) → L4 (R2 )L 3 (R2 ) satisfies f (0) = 0  (3.86)  and f (u) − f (v)  4  4  L 3 (I)L 3 (R2 )  ( u  L4 L4  + v  L4 L4 )  u−v  L4 L4  (3.87)  for any I ∈ R and u, v ∈ L4 (I, L4 (R2 )). For example, the nonlinear term f (u) = |u|2 u in NLS satisfies (3.86) and (3.87). Adapting this to our case, suppose q is a solution to NLNLS, we will let w(x, t) = eiθ q(r, t). The equation for w is equation (3.49). We will let ∞  g(w) = |ρ|>|x|  1 |w(ρ, t)|2 dρ − |w|2 ρ 2  w.  (3.88)  102  3.3. Proof of our result Then g satisfies (3.86). To check that g satisfies (3.87), we use a Hardy-type inequality for radial functions f (r)  rfr  Lp  1≤p<∞  Lp ,  (3.89)  which gives w |y|≥|x|  |w(y)|2 dy |y|2  w  4/3 Lx  w  L4x  L4x  |w(y)|2 dy |y|2  |y|≥r  r  ∂ ∂r  ∞ 0  dr r  L2x  2π  |w(r, θ)|2 0  L2x  2π  = w  L4x  |w(r, θ)|2 dθ 0  L2x  w  3 L4x .  (3.90)  Using the above, we get w1  |y|≥r  |w1 (y)|2 dy |y|2  w1  2 L4x,t  − w2  + w2  2 L4x,t  |y|≥r  |w2 (y)|2 dy 4/3 |y|2 Lx,t  w1 − w2  L4x,t .  (3.91) (3.92)  Since the cubic term 21 |w|2 w satisfies (3.87) by the H¨older inequality, we see that g satisfies (3.87). Finally, part 4 of Proposition 2 follows from [84, Lemma 3.6] by adapting their proof with our nonlinearity g defined by (3.88).  3.3.3  Reduction to the three enemies  The goal of this section is to show Proposition 3. A version of Proposition 3 for NLS has been proven: part 1 of Proposition 3 for NLS has been proven by [84] and part 2 by [48]. Consider the equation iut = ∆u + f (u).  (3.93)  When f (u) = |u|2 u, equation (3.93) is the defocusing NLS equation and when f (u) =  ∞ |u(ρ,t)|2 dρ ρ r  − 12 |u|2 u, equation (3.93) is the NLNLS equa103  3.3. Proof of our result tion with u = eiθ q. In both cases, the nonlinearity of (3.93) is cubic. Furthermore, in both cases, (3.93) has the same invariances such as translation, phase, Galilean transform and scaling. We should mention that for the NLS case, the u in (3.93) is radial while for the NLNLS u is not. Our strategy in proving Proposition 3 is to follow the proofs in [84] and [48] line by line and only modify lines of their proofs to account for the differences between NLNLS and NLS. The proof of [84] on part 1 of proposition 3 for NLS depends very little on the exact structure of f (u) other than that it satisfies the various invariances mentioned above. In places where estimates on f is needed, equation 3.90 can handle the task. The proof of [48] on part 2 of Proposition 3 for NLS depends even less on the structure of f (u) other than that it satisfies the various invariances mentioned above. As a result, our proof for part 2 of Proposition 3 follows line by line from that in [48].  3.3.4  Extra regularity  The goal of this subsection is to prove Proposition 4. A version of Proposition 4 has been proven by [48] for NLS. As in Section 3.3.3, our strategy is to follow their prove line by line and only modify the parts needed to account for the differences between NLS and NLNLS, mainly in places where the nonlinearity or the radial symmetry come into play. The proof comes in two parts. The first part proves (3.53) concerning the regularity of self-similar solutions while the second part proves (3.54) concerning the regularity of the soliton and inverse-cascade solutions. Regularity of self-similar solutions Let us briefly outline Killip-Tao-Visan’s proof of (3.53) for NLS. Readers looking for more details should read Section 5 of [48]. Here, the idea is to prove M(A)  s,u  A−s  (3.94)  104  3.3. Proof of our result for every s > 0 where M(A) := sup u  1  >AT − 2  T >0  (T )  L2x (R2 ) .  (3.95)  To achieve this, Killip-Tao-Visan considered two more quantities S(A) := sup u T >0  1 4 2 >AT − 2 Lt,x ([T,2T ]×R )  (3.96)  and N (A) := sup P T >0  1  >AT − 2  (F (u))  4  3 ([T,2T ]×R2 ) Lt,x  (3.97)  for A > 0. In the above, F is the nonlinear term, so F (u) = |u|2 u for the NLS case. Then following mass conservation, properties of self-similar solutions and basic estimates, one gets that for all A > 0 M(A) + S(A) + N (A)  u  1  (3.98)  and S(A)  M(A) + N (A).  (3.99)  To show (3.94), Killip-Tao-Visan proved the following lemma: • (Lemma 5.3 of [48]) N (A)  u  1 A √ A A S( ) A + A− 4 [M( ) + N ( )] 8 8 8  (3.100)  for all A > 100. • (Lemma 5.4 of [48]) lim M(A) = lim S(A) = lim N (A) = 0  A→∞  A→∞  A→∞  (3.101)  • (Lemma 5.5 of [48]) Let 0 < η < 1. If A is sufficiently large, then M(A) ≤ ηS(  1 A ) + A− 10 16  (3.102) 105  3.3. Proof of our result item (Corollary 5.6 of [48]) For any A > 0, M(A) + S(A) + N (A)  1  u  A− 10 .  (3.103)  Equation (3.94) follows by iterating (3.103) using (3.100). We would like to adapt Killip-Tao-Visan’s proof to our case for w(x, t) = eiθ q(r)  where q satisfies the NLNLS equation (3.75). As w is not radial and  the NLNLS has a non-local term, a few places of Killip-Tao-Visan’s proof has to be changed to accommodate for this. We will now highlight the changes. First, the key in showing (3.100) is a decomposition of w into high-, medium-, and low-frequency components: w = w>(A/8)T −1/2 + w√AT −1/2 <·≤(A/8)T −1/2 + w≤√AT −1/2 .  (3.104)  Here, since N depends only the projection of the nonlinearity onto high frequencies P  1  >AT − 2  (F (u)), for the cubic nonlinear F (u) = |u|2 u, under  (3.104), any term made up of solely low frequency terms will not contribute to N . The non-local nonlinearity behaves well with respect to frequency decomposition as well. Denoting ∞  I(f )(r) :=  f (ρ) r  dρ ρ  (3.105)  for a radial function f (r), we have x · ∇I = rIr = −f , so f = −∇ξ · ξ I = −|ξ|−1 ∂|ξ| |ξ|2 I and I(|ξ|) =  1 |ξ|2  (3.106)  ∞  f (|η|)|η|d|η|.  (3.107)  |ξ|  Hence if f is frequency localized in a particular disk, so is I(f ). So after decomposing w as in (3.104), one can assume, exactly as in Killip-Tao-Visan, that each term of the resulting expansion of the high frequency projection of the nonlinearity, P>AT −1/2 (wI(|w|2 )), must somewhere include the high frequency component w>(A/8)T −1/2 . As an aside, we should also note that 106  3.3. Proof of our result this decomposition preserves the form of function w(x) = eiθ q(r) (each term is eiθ multiplying a radial function). The estimates in Lemma 5.3 then carry over, using (3.90) as needed, with one exception: the use of the bilinear Strichartz inequality to estimate nonlinear terms containing two low-frequency factors. The problem occurs in the non-local nonlinear term when the high-frequency factor falls outside the integral, as in w>(A/8)T −1/2 I(|w≤√AT −1/2 |2 ).  (3.108)  This term does not involve a (local) product of a low-frequency and a highfrequency “approximate solution” of the Schr¨odinger equation, and so it is unclear how to apply the bilinear Strichartz estimate to it. We can get around this problem by replacing the use of bilinear Strichartz with an application Shao’s Strichartz estimate for radial functions [65] PN eit∆ f  Lqx,t (R×R2 )  N 1−4/q f  L2 (R2 ) ,  q > 10/3  (3.109)  plus a Bernstein estimate. Remark 2. Note that (3.109) is for radial functions, while our functions are of the form w(x) = eiθ q(r). In fact it is easily checked that Shao’s argument applies also for such functions – it is essentially a matter of replacing the Bessel function J0 with J1 , which has the same spatial asymptotics (and better behaviour at the origin). The same is true for the weighted Strichartz estimate [48, Lemma 2.7], which is also used in the [48] argument we are following. Indeed, since I(|w≤M |2 ) is frequency-localized below M , applying H¨older, Shao, Bernstein, and Hardy, we have, for any 10/3 < q < 4 IPN eit∆ f  I  4/3 Lx,t  =  4q 3q−4 Lx,t  M N  PN eit∆ f  4 −1 q  w≤M  2  4  Lqx,t  8q  Mq  8q  Lt3q−4 L q+4 x  −1  I  PN f  4q 4q Lt3q−4 L q+4 x  N  1− 4q  PN f  L2 ,  (3.110)  107  L2  3.3. Proof of our result and the middle factor is a Strichartz norm, so is bounded by a constant. By this argument, using also the inhomogeneous version of (3.109) (which follows in the usual way), and replacing PN by P≥N (which follows easily by summing over dyadic frequencies), we can finally arrive at the nonlinear estimate (3.100), albeit with a slower decay factor A−(2/q−1/2) replacing A−1/4 (notice 0 < 2/q − 1/2 < 1/10). This lower power does not matter, however, and the remaining estimates, (3.101), (3.102) and (3.103), carry through, establishing (3.53). Regularity of soliton and inverse-cascade solutions Let us briefly outline [48]’s proof of (3.54) for NLS. Readers looking for more details should read Section 6 and 7 of [48]. Here, the idea is to split the solution u(t) into incoming and outgoing waves and express the solution u at time t as a sum of incoming waves integrated over the past and outgoing waves integrated over the future following the Duhamel formula. In Section 6 of [48] defined the projection P + onto outgoing spherical waves to be 1 [P + f ](x) = (2π)−2 2 (1)  Here, H0  (1)  R2 ×R2  H0 (|ξ||x|)J0 (|ξ||y|)f (y) dξ dy.  (3.111)  is the Hankel function of the first kind and order zero and J0  is the Bessel function of the first kind. The projection P − onto incoming spherical waves are defined similarly. Our proof for (3.54) essentially follow that of [48] but we need to modify the definitions of P + and P − as our function w = eiθ q(r) is not radial. These projections are defined analogously for functions w(x) = eiθ q(r) by simply replacing the Bessel (and Hankel) functions of order zero with those of order one: J0 → J1 , H0α → H1α . It is easily checked that these (new) projections obey the kernel estimates listed in Proposition 6.2 of Killip-TaoVisan, essentially because J1 and H1 have the same behaviour as J0 and H0 away from the origin [48, eqns. (77), (79)]. (At the origin, J1 is better behaved, while H1 is worse – though this plays no role in the estimates.)  108  3.3. Proof of our result Given this, the subsequent estimates of Section 7 of Killip-Tao-Visan all carry over to our case, as above using (3.90) where needed to estimate the s non-local nonlinearity, to establish w ∈ L∞ t Hx for any s > 0.  This shows (3.54) and completes the proof of Proposition 4.  3.3.5  Nonexistence of the three enemies  We will rule out each of the three enemies in this section. As before, we will let w(x, t) = eiθ q(r, t).  (3.112)  We will use a lower bound which follows easily from the compactness: Lemma 3.3.2. ∇w(·, t)  2 L2 (R2 )  ∼ qr (·, t)  2 L2  + q(·, t)/r  2 L2  N 2 (t).  (3.113)  Proof. First rescale q(r, t) = N (t)v(N (t)r, t), and set w(x, ˜ t) = eiθ v(r, t), so that the estimate we seek is ∇w(·, ˜ t)  1. If this fails, then for  L2  some sequence {tn }, w ˜n (x) := w(x, ˜ tn ), satisfies ∇w ˜n w ˜n  L2  L2 (R2 )  → 0. Since  = const., we can extract a subsequence (still denoted w ˜n ) with  w ˜n → 0 weakly in H 1 , and strongly in L2 on disks. By the compactness, on the other hand, for any 0 < η, w ˜n  L2 ({|x|>C(η)})  < η, a contradiction.  The soliton case Here I = R and N (t) ≡ 1. The main tool is a spatially localized version of the virial identity d2 1 dt2 2  ∞  ∞  r2 |q(r, t)|2 r dr = 0  4|qr |2 + 4 0  |q|2 + |q|4 r2  r dr  (3.114)  . For a smooth cut-off function ψ(r) ≥ 0,  ψ ≡ 1 on [0, 1),  ψ ≡ 0 on [2, ∞),  (3.115)  109  3.3. Proof of our result and a fixed radius R > 0, define φR (r) := ψ(r/R), and the quantity ∞  rIm(qqr )φR r dr,  IR (q) :=  (3.116)  0  a function of time. By straightforward calculation we have Lemma 3.3.3. d IR (q) = 2 dt  ∞  |q|2 1 4 + |q| r2 4 |q|2 1 + |qr |2 + 2 + |q|4 (φR − 1) r 4 2 3 |q| 1 + |qr |2 − − |q|4 r(φR )r 4 r2 8 2 5 |q| 2 1 |q|2 3 − r (φ ) − r (φR )rrr R rr 4 r2 4 r2  |qr |2 + 0  (3.117)  r dr.  From Proposition 4 we have for each s ≥ 0, and for all t, w(·, t)  H˙ s (R2 )  ≤ Cs .  (3.118)  Fix η > 0, and let R = 2C(η) so that, since N (t) ≡ 1, |w(x, t)|2 dx < η  (3.119)  |x|>R/2  for all t. Multiplying w by a cut-off function 1 − ψ(2r/R), and interpolating between (3.119) and (3.118) with s = 2 (and using a Sobolev inequality) yields ∞  |qr |2 + R  |q|2 1 4 + |q| rdr ∼ r2 4  |x|≥R  1 |∇w|2 + |w|4 4  dx  η 1/2 , (3.120)  and so using |1 − φR |, |r(φR )r |,  |r2 (φ  R )rr |,  |r3 (φ  R )rrr |  1 in (3.117), we  arrive at d IR (q) ≥ 2 dt  ∞  |qr |2 + 0  |q|2 1 4 + |q| r2 4  r dr − Cη 1/2 .  (3.121) 110  3.3. Proof of our result By Lemma 3.3.2 then, since N (t) ≡ 1, and for η chosen small enough, d IR (q) dt  1.  (3.122)  On the other hand, |IR (q)|  R q  L2  qr  L2  RC1 .  (3.123)  These last two inequalities are in contradiction for sufficiently large t, and so the soliton-type blowup is ruled out. The self-similar case Here I = (0, ∞), and N (t) = t−1/2 . Again we use (3.117), but in this case, we need a stronger bound on the Sobolev norms – in fact, bounds which match Lemma 3.3.2. Such bounds follow from the regularity estimate of [48], in the self-similar case, as adapted to our non-local nonlinearity in Section 3.3.4: Lemma 3.3.4. For any s ≥ 0, sup t∈(0,∞) |ξ|>At−1/2  |w(ξ, t)|2 dξ ≤ Cs A−s ,  A > A0 (s).  (3.124)  As a consequence, w(·, t)  H˙ s (R2 )  t−s/2 = [N (t)]s .  (3.125)  Indeed, after re-scaling w(x, t) = N (t)w(N ˜ (t)x, t), equation (3.124) reads |w(ξ, ˜ t)|2 dξ ≤ Cs A−s  (3.126)  |ξ|>A  for all t, from which follows w ˜  H˙ s  1, and thus (undoing the scaling) (3.125).  Now fix a small η > 0, and large T . A (localized) interpolation (just as in the soliton case) between (3.125) with s = 2 and the L2 smallness from  111  3.3. Proof of our result compactness, gives ∞  |qr |2 + 2C(η)/N (t)  |q|2 1 4 + |q| r2 4  η 1/2 w  r dr  H˙ 2 (R2 )  η 1/2 (N (t))2 . (3.127)  Using this, with η small enough, and Lemma 3.3.2, in (3.117), we find, for t < T , and R = 2C(η)/N (T ) > 2C(η)/N (t), d IR (q) dt and hence for T  1 N 2 (t) = , t  (3.128)  1, T  IR (q)(T )  IR (q)(1) + 1  dt t  log(T ).  (3.129)  On the other hand |IR (q)(T )|  R q(T )  L2  q(T )  H˙ 1  C(η) N (T ) = C(η). N (T )  (3.130)  The last two inequalities are in contradiction for T large enough, and so the self-similar-type blowup is ruled out. The inverse-cascade case Here I = R, N (t)  1, and lim inf t→−∞ N (t) = lim inf t→∞ N (t) = 0.  The main tool is a variant of the Morawetz identity. Set ψ(r) :=  4r − r2  0<r≤1  4 r  1<r<∞  6−  +  1 r2  .  (3.131)  It is easily checked that for r ∈ (0, ∞), • ψ ∈ C3 • 0<ψ<6 • ψr > 0 • α(r) := 12 ψr +  3ψ 2 r  − rψrr − 21 r2 ψrrr > 0 112  3.3. Proof of our result • β(r) :=  ψ r  − ψr > 0.  Set  ∞  Im(qqr )ψ(r)r dr.  P (q) :=  (3.132)  0  For solutions of (3.15), an elementary computation gives: Lemma 3.3.5. d P (q) = dt  ∞  { 2ψr |qr |2 + α(r) 0  Note that since |ψ(r)|  |q|2 + r2  1 1 ψ β(r) + ( + ψr ) |q|4 }r dr > 0. 2 4 r (3.133)  1,  |P (q)|  q  L2  qr  L2  qr  L2 .  (3.134)  Next recall that for some sequences tn → −∞, Tn → +∞, N (tn ) → 0 and N (Tn ) → 0. It then follows easily from the definition of N (t) that qr (tn )  L2  → 0 and qr (Tn )  L2  → 0. Hence by (3.134),  P (q(tn )) → 0,  P (q(Tn )) → 0.  (3.135)  If P (q0 ) ≥ 0, then (3.133) implies P (q(t)) > 0 and increasing for t > 0, while if P (q0 ) < 0, then (3.133) implies P (q(t)) < 0 and increasing for t < 0. In either case, (3.135) is contradicted. This rules out the inverse cascade-type blowup.  113  Chapter 4  Concluding chapter In this thesis, we showed two results. First, we established asymptotic stability of small ground state solutions to the three dimensional nonlinear magnetic Schr¨ odinger equation iut = (i∇ + A)2 u + V u + g(u), u(x, 0) = u0 (x)  (4.1)  for the case where the operator (i∇ + A)2 + V has exactly one eigenvalue. Recall that when g ≡ 0, equation (4.1) models a quantum particle in the presence of a electric potential V and a magnetic potential A. Here, g(u) = |u2 |u is the nonlinear term. In the absence of g, equation (4.1) is linear and the time evolution of its solution is well understood. In the presence of g, self-interactions of the solution make the behaviour of the solution more complex and much less well-understood. When A ≡ 0, asymptotic stability results have been established by many authors (such as [68], [10], [87], [86], [70], [40], [50] and [57]) for cases where −∆ + V has one or more eigenvalues. Our result is the first with the presence of a magnetic potential. Stability results are important from an applications perspective. For example, if a bound solution is not stable, it would be difficult, if not impossible, to observe it experimentally or simulate it numerically. The reason is that any imprecision in the initial conditions would result in a state of the system far away from the bound state.. Furthermore, our result can be viewed as an attempt to partially understand the time evolution of solutions more generally. Viewed slightly differently, our result gives the asymptotic behaviour of a solution with initial data small in H 1 . Our result states that if the initial data is sufficiently small in H 1 , then as t → ∞, the solution u will be composed of a bound state and a dispersive 114  Chapter 4. Concluding chapter part. One natural extension to our result is to consider the more complex case where (i∇ + A)2 + V has two or more eigenvalues. Another extension is to consider other types of nonlinearities such as a convolution type nonlinearity g(u) = (F ∗ |u|2 )u :=  F (x − y)|u(y)|2 dy u, R3  arising in a Hartree-type equation. Second, we established global well-posedness for the Schr¨odinger map equation ut = u × ∆u  (4.2)  into the 2-sphere for radially symmetric initial data in two dimensions. Equation (4.2) models the time evolution of magnetization in an isotropic magnetic material in the absence of external magnetic field and energy dissipation. When the spatial dimension is one, it is known by [16] that H 1 solutions of (4.2) are global. However, when spatial dimensions are higher than one, global behaviours of solutions to (4.2) are not well understood. For two dimensions, it has been known by [16] that solutions to (4.2) with sufficiently small energy are global. However, it is not clear what the long term behaviours of solutions with arbitrary sized energy are. 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