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On global properties of solutions of some nonlinear Schrödinger-type equations Koo, Eva Hang 2012-07-25

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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 2012Abstract 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 = (ir+A) 2u+ V u+ g(u); u(x; 0) = u0(x); where A : R3 ! R3 is the magnetic potential, V : R3 ! R is the elec- tric potential, and g =  juj2u is the nonlinear term. We show that under suitable assumptions on the electric and magnetic potentials, if the initial data is small enough in H1, then the solution of the above equation decom- poses uniquely into a standing wave part, which converges as t!1, 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 non- local Schr odinger equation with radially-symmetric initial data in the critical space L2(R2), using the framework of Kenig-Merle and Killip-Tao-Visan. iiPreface 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. Di erential 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 joint- work 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] Killip- Tao-Visan 09) to our situation, and I wrote the  rst draft of the manuscript. iiiTable of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Dispersive e ects and dispersive equations . . . . . . . . . . 1 1.1.1 A gentle introduction to dispersive e ects . . . . . . . 1 1.1.2 Variations of Schr odinger equations . . . . . . . . . . 7 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 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 1.3 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 ivTable of Contents 2.2 Our result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3 Discussion and outline of the proof . . . . . . . . . . . . . . 46 2.4 Detailed proof . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . 88 3.2.2 Discussion of our proof of global well-posedness of NLNLS for radial L2 initial data in 2D . . . . . . . . 92 3.3 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 . . . . . . . . . . . . 103 3.3.4 Extra regularity . . . . . . . . . . . . . . . . . . . . . 104 3.3.5 Nonexistence of the three enemies . . . . . . . . . . . 109 4 Concluding chapter . . . . . . . . . . . . . . . . . . . . . . . . 114 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 vList of Tables 1.1 Classical PDEs and their geometric counterparts . . . . . . . 14 1.2 L2- and H1- criticality of cubic NLS for n = 1; 2 and 3 . . . . 24 viAcknowledgements 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 guid- ance  Erick, for his constructive criticism as well as his encouragement viiDedication To all my teachers viiiChapter 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 di erential 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 e ects. 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  eld of dispersive equations. In Section 1.3, we will give an overview of the rest of the thesis. 1.1 Dispersive e ects and dispersive equations 1.1.1 A gentle introduction to dispersive e ects To illustrate the dispersive property, we will consider three well-studied par- tial di erential 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 2 Rn and time variable t 2 [0;1). 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  u = nX j=1 @2u @x2j : (1.1) 11.1. Dispersive e ects and dispersive equations These equations are the heat equation ut =  u; (1.2) the wave equation utt =  u (1.3) and the Schr odinger equation ut = i u (1.4) 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 in nite length, the wave equation models the height of a wave on an in nitely 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 bu( ; t) := 1 2 Z R u(x; t)e ix dx (1.7) denote the spatial-domain Fourier transform of u(x; t). Using the property duxx( ; t) =   2bu( ; t); (1.8) the equations now read 21.1. Dispersive e ects and dispersive equations heat equation: but( ; t) =   2bu( ; t) (1.9) wave equation: butt( ; t) =   2bu( ; t) (1.10) Schr odinger equation: but( ; t) =  i 2bu( ; t): (1.11) Each of the equations above is an ordinary di erential equation in t, and after solving them, we get heat equation: bu( ; t) = bu( ; 0)e  2t (1.12) wave equation: bu( ; t) = bu( ; 0) cos( t) + but( ; 0)  sin( t) (1.13) Schr odinger equation: bu( ; t) = bu( ; 0)e i 2t: (1.14) We can already read o some di erences between the behaviours of the three equations from the above. For example, for the heat equation, the magnitude of each Fourier mode jbu( ; t)j is exponentially decreasing, while for the Schr odinger equation, the magnitude of each Fourier mode jbu( ; t)j is  xed in size. By Parseval’s theorem, which says that kukL2x :=  Z R ju(x; t)j2 dx  1 2 = p 2 kbukL2 ; (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 31.1. Dispersive e ects 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 di erentiate the behaviour of the wave equation and the Schr odinger equation we can apply the inverse Fourier transform and recover u(x; t) from bu( ; t) by u(x; t) = Z R bu( ; t)eix d : (1.16) For the sake of discussion, suppose the initial data of the wave equation are chosen so that bu( ; 0) =   0( ) and but( ; 0) = 0 for some  xed frequency  0 2 R. Here,   0( ) =  (   0) is the delta function. Then bu is given by bu( ; t) = 1 2    0( )e i 0t +   0( )e  i 0t  : (1.17) Hence, the solution u of the wave equation is given by u(x; t) = Z R bu( ; t)eix d = 1 2  ei 0(x+t) + ei 0(x t)  : (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 u(x; t) = 1 2 (u0(x+ t) + u0(x t)) + 1 2 Z x+t x t v0(y) dy: (1.19) In the case where v0  0, the solution of the wave equation is u(x; t) = 1 2 (u0(x+ t) + u0(x t)): (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 dif- 41.1. Dispersive e ects 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(x; t) = Z R bu( ; 0)ei(x  t) d (1.21) where bu( ; 0) = 1 2 Z R u(x; 0)e ix dx: (1.22) 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 bu( ; 0) =   0( ) for some  xed  0 2 R: (1.23) In this case, the solution of the Schr odinger equation will read u(x; t) = ei(x  0t) 0 : (1.24) This is the equation of a plane wave moving at speed  0. Under the evolution of the Schr odinger equation, an initial data u0 which consists of a single Fourier mode  0 remains a function with only that Fourier mode and which moves in the physical space at a speed of  0. Now, instead of an initial condition u(x; 0) = u0(x) consisting of a sin- gle Fourier mode, if our initial condition consists of a continuum of Fourier modes, then the part of the solution having Fourier mode  will move at a speed of  . 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 un- changed. This spreading out of the solutions is what we call the dispersive e ect. There are equations other than the Schr odinger equation that are dis- 51.1. Dispersive e ects and dispersive equations persive in nature. Other dispersive equations include the Klein-Gordon equation utt   u+m 2u = 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 bu( ; t) = e ih( )tbu0( ) where h( ) =  2: (1.27) Here, h( ) is called the dispersion relation. As before, if we apply the in- verse Fourier transform of bu( ; t) = eih( )tbu0( ), we  nd that the component of the solution with the Fourier mode  travels at a speed equal to h( ) . In fact, associated with the dispersion relation are two di erent velocities, the phase velocity vp = h  (1.28) and the group velocity vg = dh d : (1.29) 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 =  and vg = 2 (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 but = i 3bu (1.31) 61.1. Dispersive e ects and dispersive equations which gives bu( ; t) = ei 3tbu( ; 0); (1.32) so for the Airy equation, the dispersion relation is h( ) =  3. The Klein- Gordon equation is second order in time. To get the dispersion relation, we look for a function h( ) so that u = ei( x h( )t) (1.33) satis es the Klein-Gordon equation. Notice that this way of computating h( ) is consistent with the ways in which h( ) is computed above. Substi- tuting (1.33) into the Klein-Gordon equation, we get that  h2( ) +  2 +m2 = 0 (1.34) which gives h( ) =  p m2 +  2: (1.35) 1.1.2 Variations of Schr odinger equations In the previous subsection, we introduced the dispersive e ect. 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  elds. In this subsection, we will consider the Schr odinger equation 71.1. Dispersive e ects and dispersive equations with a potential iut =   u+ V u (1.37) which models the evolution of a quantum particle under the in uence 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 to be the Coulomb potential V =  1jxj , 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 e ect 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 V (x) = 8 < : 0 if 0 < x < 1 1 if x  0 or x  1 : (1.39) This is commonly known as the one-dimensional in nite square well. The general solution to this equation is given by u(x; t) = 8 < : P1 n=1 an sin( nx)e  i 2nt where  n = n 0 if x 62 (0; 1) : (1.40) 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 81.1. Dispersive e ects and dispersive equations (i.e. in nite 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 con nes the solution to the in nite square well. As a result of the con ning part of the equation, equation (1.38) admits some stationary solutions of the form sin( nx)ei 2 nt for n = 1; 2; 3;    . These solutions are known as standing wave solutions since other than the complex phase ei 2 nt, they are stationary in time. These standing waves are examples of bound states { states which remain spatially localized for all time. The in nite square well is not the only potential that gives rise to bound states. Other less con ning 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 in nite square well, but nevertheless, solutions of the harmonic oscillator is given by u(x; t) = 1X n=0 ane  i2(n+ 12 )tHn(x)e  x 2 2 : (1.41) Here, Hn is the Hermite polynomial, a degree n polynomial whose the ex- plicit form can be worked out for each n. In the above, each summand e i2(n+ 1 2 )tHn(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 con ning potentials. Let us instead consider yet another standard example in introductory texts in quantum mechanics, which instead of a con ning potential, is an attractive potential V (x) = 8 < :  V0 for  1 < x < 1 0 for jxj  1 (1.42) for some V0 > 0. (As an aside, the case where V0 < 0 is called a repulsive potential.) This is known as the one-dimensional  nite square well. In this case, equation (1.37) admits solutions of the form u(x; t) = e iEt (x) (1.43) 91.1. Dispersive e ects and dispersive equations for both E < 0 and E > 0. When E < 0, solutions of the form (1.43) only exist for a  nite 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 ! 1 as jxj ! 1, 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 jxj ! 1, 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 non- linearity  jujp 1u to get the nonlinear Schr odinger equation iut =   u+  juj p 1u: (1.45) Our next task is to see the e ect of the nonlinear term on the equation. Without the nonlinear term, the di erent 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 di erent Fourier modes no longer evolve independently and the interaction of di erent Fourier modes gives rise to more complex behaviour. By rescaling the function u, it is only necessary to consider  =  1. 101.1. Dispersive e ects and dispersive equations It turns out the behaviour of (1.45) is di erent according to the sign of  . When  < 0, the nonlinear term  juj2u acts like an attractive potential when u is large and it has little e ect 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 juj2u attempts to con ne or even concentrate the solution. As the solution evolves in time, there is a competition between the dispersive e ect and the concentration e ect. When a delicate balance is achieved between the two opposite e ects, 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  nite 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  > 0, the nonlinear term behaves more like a repulsive potential. As a result, solutions in this case are always dispersive. Because of the di erence of the behaviour between  > 0 and  < 0, the case where  < 0 is called the focusing case while the case where  > 0 is called the defocusing case. Other than the above heuristic regarding attractive and repulsive poten- tials , one can also get a glimpse of the di erence in behaviour of (1.45) in the signs of  from a conserved quantity of the equation (1.45) called the energy, which is de ned by E(u) = Z Rn  1 2 jruj2 +  p+ 1 jujp+1  dx: (1.46) The term 12 jruj 2 in the expression for E comes from the dispersive term   u while the term  p+1 juj p+1 comes from the nonlinear term jujp 1u. From this, we see that when  < 0, the nonlinear term jujp 1u is working in the opposite direction as the dispersive term  u, while when  > 0, the nonlinear term works in the same direction as the dispersive term. 111.1. Dispersive e ects and dispersive equations Schr odinger map So far, we have considered two di erent 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 jujp 1u. 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 = f~u 2 R3 : j~uj = 1g  R3: (1.49) Hence, we view ~u as ~u(x; t) = (u1(x; t); u2(x; t); u3(x; t)) (1.50) where u21 + u 2 2 + u 2 3 = 1: (1.51) Now for each ~u 2 S2, we de ne the operator J~u by J~u~ = ~u ~ : (1.52) 121.1. Dispersive e ects and dispersive equations If we let T~uS 2 = f~ 2 R3j~u  ~ = 0g (1.53) to be the tangent plane at ~u 2 S2, we can view J~u as an operator that rotates vectors on the tangent plane T~uS2 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 di erence between equation (1.47) and (1.54): equation (1.47) is a linear while equation (1.54) is non- linear. 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 @t =  ~u [ ~u+ ~a(~u)] +  ~u (~u [ ~u+ ~a(~u)]) (1.55) for some parameters  and  . Here  ~u  [ ~u + ~a(~u)] is the precession term describing the revolution of the magnetization vector about an e ective magnetic  eld  ~u+~a(~u),  ~u (~u [ ~u+~a(~u)]) is a dissipation term, and ~a : R3 ! R3 is a vector  eld representing an anisotropy in the magnet. When dissipation is absent (i.e. when  = 0) and in the isotropic case (~a = 0), we are left with @~u @t =  ~u  ~u (1.56) 131.2. Background u( ; t) : Rn ! C or R ~u( ; t) : Rn ! S2  R3 ut = i u ~ut = J~u ~u (Schr odinger equation) (Schr odinger map)  u = 0  ~u+ jr~uj2~u = 0 (Laplace equation) (harmonic map) ut =  u ~ut =  ~u+ jr~uj2~u (heat equation) (harmonic map heat  ow) utt =  u ~utt =  ~u+ (jr~uj2  j~utj2)~u (wave equation) (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 clas- sical 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  ow, [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; 141.2. Background  special solutions such as solitary waves;  stability of special solutions;  scattering. In the subsections, we will de ne 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+  juj p 1u (1.57) with initial condition u(x; 0) = u0(x) 2 X (1.58) for X = L2(Rn) or H1(Rn) (to be de ned in the upcoming subsection). 1.2.1 Notation Before we begin, we should de ne 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 de ne various function spaces that will be important for later sections. For each 1  p  1, let Lp(Rn) denote the space of Lebesgue 151.2. Background measurable functions f in Rn up to a.e. equivalence such that the Lp norm de ned by kfkLp(Rn) :=  Z Rn jf jpdx  1 p if 1  p <1 (1.62) and kfkL1(Rn) := ess sup Rn jf j (1.63) is  nite. Let C1(Rn) be the space of in nitely di erentiable functions in Rn. Let  = ( 1;  2; : : : ;  n) where  i  0 for i = 1; 2; : : : ; n. Here,  is called a multi-index. We de ne j j =  1 +  2 +    +  n. We use the notation D to denote the di erential operator D = @ 1x1 @  2 x2    @  n xn : (1.64) For each 1  p <1 and positive integer m, we will de ne the Sobolev space W m;p(Rn) to be the closure of fu 2 C1(Rn) j D u 2 Lp; 8 j j  mg (1.65) under the norm kukm;p := X j j m kD ukLp(Rn): (1.66) The case p = 2 is special in that Wm;2 is a Hilbert space (a complete normed linear space with an inner product). We will write Wm;2 as Hm. We have so far de ned Hm for positive integer m; one can also de ne the space Hs for s 2 R. To do so, let the Schwartz space S be de ned by S = fu 2 C1(Rn)j sup x2Rn jx D uj <1 for all multi-indices  and  g: (1.67) Here, for x 2 Rn and multi-index  = ( 1;  2; : : : ;  n), x denotes x  1 1 x  2 2    x  n n . The space Hs(Rn) for s 2 R is de ned to be the closure of the Schwartz 161.2. Background space under the norm kukHs :=  Z Rn j(1 + j js)bu( )j2 d  1 2 (1.68) where bu denotes the Fourier transform of u. Next, we will give some motivation for why these spaces are central to the study of partial di erential equations. For simplicity, consider the free Schr odinger equation 8 < : iut =   u u(x; 0) = u0(x) : (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 di erent 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 H1 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 7! Y . In other words, given an initial condition u0 2 X, the free Schr odinger equation returns an element in Y for each t as long as the solution is de ned. Viewed this way, partial di erential equations are in nite dimensional analogues of ordinary di erential equations. For example, an ordinary di erential equation such as 8 < : d~u dt = M~u ~u(0) = u0 2 Rn (1.70) where M is an n n matrix, can be thought of as a mapping from the vec- tor space Rn to t 7! Rn. The main di erence between ordinary di erential equations and partial di erential equations is that Rn is a  nite dimensional vector space but functional spaces such as L2 and H1 are in nite dimen- sional. Because of this di erence, behaviours of partial di erential equations are more complex and less well understood than ordinary di erential equa- 171.2. Background tions. Let X and Y be some functional spaces. If the free Schr odinger equation 8 < : iut =   u u(x; 0) = u0(x) 2 X (1.71) admits an unique solution u(x; t) 2 Y , we will de ne its solution operator ei t as the mapping from X to Y such that ei tu0(x) = u(x; t): (1.72) 1.2.2 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+  juj p 1u (1.73) with initial condition u(x; 0) = u0(x) 2 X 8x 2 Rn (1.74) for some functional space X such as H1 or L2. Local well-posedness con- cerns the following questions: Existence: Does there exist a time T > 0 where there is a solution u of (1.73) de ned in C(( T; T ); X) (i.e. a continuous function of time into the space X)? Uniqueness: Is the solution unique? 181.2. Background Continuous dependence on data: Does the solution u depend continu- ously on the initial data? If the answer to the above question is a rmative, 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 < 1; one wants to know whether lim t!T  max kukX =1: (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 = H1 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 H1) Let pmax be de ned by pmax = 8 < : 1 if n = 1; 2 n+2 n 2 if n  3 : (1.76) Suppose 1 < p < pmax, and suppose u0 2 H1(Rn). Then there exists a T > 0 and a unique solution u to (1.73) which depends continuously on u0 in H1 such that u 2 C(( T; T ); H1(Rn)). Furthermore, the following quantities mass: M(u) := kukL2 (1.77) energy: E(u) := Z  1 2 jruj2 +  p+ 1 jujp+1  dx (1.78) are conserved. It should be noted that for functions u in L2(Rn) or H1(Rn),  u may not be de ned. When we say u is a solution of (1.73) in L2(Rn) or H1(Rn), we actually mean u satis es the integral form (1.83) of (1.73). In other 191.2. 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 de ned by pc = 1 + 4 n : (1.79) Suppose 1  p < pc, and suppose u0 2 L2(Rn). Then there exists a T > 0 and a unique solution u to (1.73) such that u 2 C(( T; T ); L2(Rn)). Fur- thermore, the mass M(u) := kukL2 (1.80) is a conserved quantity. In the former theorem, pmax is the p in (1.73) such that the homoge- neous H1-norm ( _H1) remains unchanged under the solution-invariant scal- ing u(x; t)!  2 p 1u( x;  2t) for  > 0. In other words, if u is a solution of (1.73) with p = pmax and v(x; t) =  2 p 1u( x;  2t), then v is also a solution of (1.73) and kv( ; t)k _H1(Rn) :=  Z Rn jrv(x; t)j2 dx  1 2 = ku( ;  2t)k _H1(Rn): (1.81) Similarly, pc is the p in (1.73) such that the L2-norm remains unchanged un- der the solution invariant scaling u(x; t)!  2 p 1u( x;  2t). In other words, let u be a solution of (1.73) with p = pc and let v(x; t) =  2 p 1u( x;  2t), then v is also a solution of (1.73) and kv( ; t)kL2(Rn) = ku( ;  2t)kL2(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 201.2. Background di cult because at the critical power (p = pmax or pc) the norm (H1 or L2) cannot be used to control the nonlinear term jujp 1u. Existence theorems for (1.73) in the H1-critical case and L2-critical case were obtained by [14]. Local existence for the H1-subcritical case with H1 initial data (The- orem 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 di erential equation (1.73) into the integral form u(x; t) = ei tu0(x) i Z t 0 ei (t  )ju(x;  )jp 1u(x;  )d : (1.83) 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 keit u0kLq(Rn) . jtj  n( 12 1 q )ku0kLq0 (Rn) (1.84) where 1q + 1 q0 = 1 and 2  q  1. The decay estimate is a special property of dispersive equations. For example, if we take q =1 and q0 = 1, the decay estimate tells us that keit u0kL1(Rn) . jtj  n2 ku0kL1(Rn): (1.85) In the case where ku0kL1(Rn) is  nite, 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  xed, the decrease in the L1-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 0 -norm are not equivalent unless q = q0 = 2. With the decay estimate alone, unless p = p0 = 2, to control the Lp-norm of eit u, we need control of the Lp 0 -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 211.2. Background space-time integrals of the solution. Let k  kLqtLrx denote   k  kLrx(Rn)   Lqt (R) : (1.86) We also de ne the pair (q; r) to be admissible if 2 q + n r = n 2 for q; r 2 [2;1] and (q; r; n) 6= (2;1; 2): (1.87) The Strichartz estimates say that for admissible (q; r) and (~q; ~r), keit u0kLqtLrx . ku0kL2x (1.88) and     Z t 0 ei(t  ) f( ) d     LqtL r x . kfk L~q 0 t L ~r0 x (1.89) where ~q0 and ~r0 satisfy 1 ~q + 1 ~q0 = 1 and 1 ~r + 1 ~r0 = 1: (1.90) Equation (1.88) is the homogeneous Strichartz estimate allowing us to han- dle the ei tu0(x) term in (1.83) while Equation (1.89) is the inhomogeneous Strichartz estimate allowing us to handle the R t 0 e i (t  )ju(x;  )jp 1u(x;  )d term in (1.83). Such estimates originate from [76]. The inhomogeneous Strichartz estimates are developed by [94] and [13]. The end point case ((q; r) = (2; 2nn 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 jujp 1u. However, for our discussion, we will only consider pure power nonlinearities jujp 1u. Following the proofs of the local existence for the H1-subcritical case with initial data inH1, the duration of existence T depends on the dimension n, the power of nonlinearity p and the norm of the initial data ku0kH1 . 221.2. 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 ku0kL2 . In fact, in both cases, for  xed n and p, T is mainly determined by a quantity like Cku0k  m X (1.91) for some constants C and m > 0 depending on n and p for X = H1 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 well- posedness 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 inde nitely or if it instead blows up in  nite time. This is the question of global well-posedness. In the last subsection, we saw that for the equation 8 < : iut +  u =  jujp 1u u(x; 0) = u0(x) 2 X for X = H1(Rn) or L2(Rn) ; (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) 2 C(( T; T ); X). In this subsection, we will consider the question of whether T = 1, in which case, we say that the equation is globally well-posed. In this sub- section, we will restrict our discussion to the well-studied and physically 231.2. Background n = 1 L2-subcritical H1-subcritical n = 2 L2-critical H1-subcritical n = 3 L2-supercritical H1-subcritical Table 1.2: L2- and H1- criticality of cubic NLS for n = 1; 2 and 3. important cubic nonlinear Schr odinger equation 8 < : iut +  u =  juj2u u(x; 0) = u0(x) 2 X for X = H1(Rn) or L2(Rn) (1.93) for  =  1 and for space dimension n = 1; 2 and 3. Table 1.2 summarizes the L2- and H1- 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 ku0kX . Thus, if one can obtain an a priori bound on ku(t)kX , one automatically gets global well-posedness. Indeed, suppose we know that kukX < M , then local existence guarantees a solution up to time t = T (ku0kX)  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 (ku( ; t1)kX)  t2. This process can be iterated and the total length of existence will be given by P1 j=1 tj . Now, since kukX < M , for each j, tj  T (M) > 0. Hence, P1 j=1 tj =1. Since the L2 norm is a conserved quantity of (1.92), we automatically get L2 global well-posedness for n = 1. Similarly, as the energy E(u) = Z  1 2 jruj2 +  4 juj4  dx (1.94) is a conserved quantity, in the case  = 1 (the defocusing case), this gives Z  1 2 jruj2 + 1 4 juj4  dx = E(u0): (1.95) The above gives Z 1 2 jruj2 dx < E(u0) (1.96) 241.2. Background and thereby together with L2-conservation yields an upper bound on kukH1 . Therefore, for the defocusing case  = 1, (1.93) is globally well-posed in H1 for n = 1; 2 and 3. The situation is di erent for the focusing case  =  1. In this case, we have Z  1 2 jruj2  1 4 juj4  dx = E(u0): (1.97) Here, we see that the nonlinear term  1p+1 juj p 1 is now acting against the dispersive term 12 jruj 2 in the expression for energy. In this case, a way to show global well-posedness is to try to use the dispersive term 12 jruj 2 to control the nonlinear term 14 juj 4. In fact, the Galiardo-Nirenberg inequality (see, for example, [26]) gives that for p  pmax, Z jujp+1 dx .  Z jruj2 dx    Z juj2 dx   (1.98) for  = n(p 1)4 and  = 2+n+(2 n)p 4 . As we have chosen p = 3, we get that for n  4, Z juj4 dx .  Z jruj2 dx  n 2  Z juj2 dx  2 n2 : (1.99) For n = 1, energy conservation and the above give us Z 1 2 jruj2 dx = E(u0) + 1 4 Z juj4 dx . E(u0) +  Z jruj2 dx  1 2  Z juj2 dx  3 2 : Using Young’s inequality in the form ab   2 2 a2 + 1 2 2 b2; (1.100) 251.2. Background for su ciently small  , we get that Z 1 2 jruj2 dx  E(u0) + 1 4  Z jruj2 dx  + C  Z juj2 dx  3 for some constant C. and this shows Z jruj2 dx . E(u0) +  Z juj2 dx  3 : (1.101) Since the L2-norm of u is a conserved quantity, this shows that kukH1 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  nite time. [35] showed if the initial data u0 2 H1(Rn) \ L2(Rn; jxj2dx) and E(u0) < 0, then the solution u blows up in  nite time. Here, L2(Rn; jxj2dx) denotes the space of functions v : Rn ! C with kvxkL2(Rn) <1: (1.102) The idea is to consider the quantity I(t) = Z Rn jxj2ju(x; t)j2 dx: (1.103) Here, if we think of juj2 as a probability density, then I is essentially the second moment. If u is a solution of (1.92), simple calculations show that I 00 = 16E +  4n p+ 1 (p pc) Z jujp+1 dx: (1.104) For n = 2, n = 3, p = 3  pc. Hence, in the focusing case  =  1, the term  4np+1(p pc) R jujp+1 dx is non-positive. Hence, I 00  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  nite time. However, 261.2. 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; jxj2dx) and showed if u0 2 H1 is radially symmetric and E(u0) < 0, then the solution u blows up in  nite time. For n  3, (1.93) is H1-subcritical. However, (1.93) is H1-critical for n = 4 and H1-supercritical for n  5. As blow-up is possible for the fo- cusing case, we will only consider the defocusing case for which global well- posedness is at least possible. In H1-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 ku0kH1 but also on the structure of u0. Proving global well-posedness even for the defocusing case for H1 data is more di cult in this situation. In fact, not much is known about global well- posedness about the H1-supercritical case. On the other hand, the critical case is much better understood due to a number of recent breakthroughs. Consider the H1-critical nonlinear Schr odinger equation 8 < : iut =   u+ jujpmax 1u u(x; 0) = u0(x): (1.106) As before, pmax is the H1-critical exponent. For example, when n = 3, pmax = 5. For n = 3 and 4, Bourgain ([7] and [8]) showed for radial initial data u0 2 H1(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 theH1-critical focusing equation with radial initial data iut =   u juj pmax 1u (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 271.2. Background di cult, 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 Hs 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 > 12), [19] (for s > 25), [23] (for s > 1 3) and  nally [28] (for s = 0). For n = 3, some recent results in this direction are [20] (for s > 56) and [21] (for s > 4 5). 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 iut =  u juj 4 3u (1.108) has been shown by [27]. 1.2.4 Solitary waves and their stability Consider the equation iut =   u juj p 1u: (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 e ect and the concentration e ect from the nonlinear term. Here, we start by looking for special solutions taking the form u(x; t) = eit (x): (1.110) If we substitute the above into (1.109), we  nd that  must satisfy  =   + j jp 1 : (1.111) 281.2. Background When spatial dimension n = 1, the above equation is an ordinary di erential equation and it turns out that in this case, it can be solved exactly and the solution is given by  =  p+ 1 2  1 p 1 sech 2 p 1  p 1 2 x  : (1.112) If we apply the solution preserving scaling u(x; t) 7!  2 p+1u( x;  2t), we  nd that if we let  w = ! 1 p+1  p+ 1 2  1 p 1 sech 2 p 1  p !(p 1) 2 x  ; (1.113) then u(x; t) = ei!t w(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 in nitely many ex- cited 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 op- posing e ects, 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 dif-  cult 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  rst observed by John Scott Russell in 1834 as he was traveling 291.2. 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 + (u 2)x = 0 (1.115) which is a standard example of a nonlinear dispersive equation having soli- tary 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 di erent 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 de ned 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!1. To properly de ne these two notions of stability, we need to de ne a notion of closeness. In order to do so, we need to  rst 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) 7! u(x x0; t t0) (1.116) rescaling: u(x; t) 7!  2 p 1u( x;  2t) (1.117) phase shift: u(x; t) 7! ei u(x; t) (1.118) 301.2. Background Galilean boost: u(x; t) 7! ei(v x jvj 2t)u(x 2vt; t) (1.119) With these in mind, we will attempt to de ne 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  rst attempt is to de ne orbital stability by the following: De nition 1. (the  rst attempt) We say that a solution u of (1.109) is orbitally stable if for all  > 0, there exists a  > 0 such that if a solution v of (1.109) satis es kv( ; 0) u( ; 0)kX <  , then kv( ; t) u( ; t)kY <  for all t  0. However, the above is not a good de nition for our problem, because un- der this de nition, no soliton will be stable under our usual choice of Banach spaces X and Y such as L2 or H1. The reason is that if u(x; t) = ei!tQ(x) is a solitary wave solution of (1.109), then by the Galilean invariance, v(x; t) = ei[(w jvj 2)t+v x]Q(x 2vt) (1.120) is also a solution. Furthermore, v(x; 0) = eiv xQ(x), so v(x; 0) u(x; 0) = (eiv x  1)Q(x) (1.121) and for jvj small, we have kv(x; 0) u(x; 0)kX  1 (1.122) for X = L2 or H1. However, kv( ; t) u( ; t)kY = ke i[(w jvj2)t+v x]Q(x 2vt) ei!tQ(x)kY (1.123) may not be small once t becomes large enough no matter how small jvj is. Because of this, we need to de ne orbital stability to incorporate the possible invariant transformations of the equation. As a result, we will de ne orbital 311.2. Background stability as the following: De nition 2. (the correct version) We say that a solution u of (1.109) is orbitally stable if for all  > 0, there exists a  > 0 such that if a solution v of (1.109) satis es kv( ; 0) u( ; 0)kX <  , then inf  ;x0 kv(x; t) ei u(x x0; t)kY <  (1.124) for all t  0. Orbital stability results in H1 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  rst 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  0 with eigenvalue  !0. In other words,  0 is a solution to the di erential equation   0  V  0 = !0 0 (1.126) and u = eit!0 0(x) (1.127) is a solution of (1.125). Next, suppose we add a cubic nonlinear term  juj2u to (1.125) to get iut =   u+ V u+  juj 2u (1.128) where  =  1 and V : Rn ! R. It turns out that under suitable assumptions on V , such as su cient decay as jxj ! 1, equation (1.128) admits solutions of the form u(x; t) = eiEtQ(x): (1.129) 321.2. Background We call these solutions nonlinear bound states. Here, Q and E solve the equation  Q V Q  jQj2Q = 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] = !0 + o(z) and Q[z] = z 0 + q(z) (1.131) for su ciently 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 su ciently 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]tQ[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  nd 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]tQ[z2](x) as t!1. More precisely, we say that the solutions eiE[z1]tQ[z1](x) are asymptotically stable if whenever ku0  Q[z1](x)kX (1.134) is su ciently small for some functional space X, then the solution u of 331.2. Background (1.128) will be given in the form u(x; t) = Q[z(t)](x) +  (x; t) (1.135) where z(t) approaches some limit z+ close to z1 as t!1, and  disperses to 0. The asymptotic stability of small nonlinear bound state solutions is stud- ied 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 (limjxj!1 V (x) = 0). For example, one can take V to be the Coulomb potential V (x) =  1jxj . 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 su ciently 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 8 < : iut =   u+  jujp 1u u(x; 0) = u0(x) : (1.137) 341.3. 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 !  1) and far in the future (t ! 1). In other words, scattering happens when the nonlinear term juj2u \turns o " asymptotically. More rigorously, we de ne scattering as follows: De nition 3. (Scattering) Let X be H1(Rn). Let u be a solution of (1.137). We say that u scatters if there exists u 2 X such that ku(t) ei tu kX ! 0 as t!  1: (1.138) Notice that when u scatters, the asymptotic linear states u are unique. Next, suppose for each u 2 X, there exists a unique initial data u0 such that the solution u scatters to the states u , then we can de ne the wave operators W to be the maps from u to u0. On the other hand, suppose for each u0 2 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 so- lution 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 o " as t!  1. When p is too small, the nonlin- ear term may not decay fast enough for its e ect to \disappear". In other words, scattering occurs when p is of intermediate size. It turns out that for the defocusing case ( > 0) in H1, 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 351.3. 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 i t = H +  j j 2 (1.139) where the operator H =   + 2iA r+ i(r A) + V (1.140) and  =  1. Here, A : R3 ! R3 is a vector (magnetic) potential modelling the magnetic  eld and V : R3 ! R is a scalar (electric) potential modelling an electric  eld. Without the nonlinear term, the equation i t = H (1.141) models the time evolution of a quantum particle (such as an electron) in the presence of a magnetic and electric  eld. The addition of the nonlinear term allows di erent 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  0. In other words, H 0 = e0 0 (1.142) and hence, e ie0t 0(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 su cient decay as jxj ! 1, equation (1.139) admits nonlinear bound states 361.3. Main results of the thesis solutions of the form u(x; t) = e iEtQ(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] = z 0 + q(z) (1.144) for su ciently small z. Lemma 1.3.1. (Existence and decay of nonlinear bound states) For each su ciently small z 2 C, there is a corresponding eigenfunction Q[z] 2 H2 solving the nonlinear eigenvalue problem HQ+ g(Q) = EQ (1.145) with the corresponding eigenvalue E[z] = e0 + o(z) and Q[z] = z 0 + q(z) with q(z) = o(z2); DQ[z] = (1; i) 0 + o(z) and D 2Q[z] = o(1) in H2 (1.146) where we denote DQ[z] = (D1Q[z]; D2Q[z]) = ( @ @z1 Q[z]; @ @z2 Q[z]); and z = z1 + iz2: (1.147) Furthermore, Q has exponential decay in the sense that e jxjQ 2 H1 \ L1 (1.148) for some  > 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 < 1, every solution 371.3. Main results of the thesis  of equation (1.139) with initial data  0 su ciently small in H1 can be uniquely decomposed as  (t) = Q[z(t)] +  (t); (1.149) with di erentiable z(t) 2 C and  (t) 2 H1 satisfying hi ;D1Q[z]i = 0, hi ;D2Q[z]i = 0 and k kX . k 0kH1 ; k _z + iE[z]zkL1t . k 0k 2 H1 : (1.150) Furthermore, as t!1, z(t) exp  i Z t 0 E[z(s)]ds  ! z+; E[z(t)]! E(z+) (1.151) for some z+ 2 C and k (t) e itH +kH1x ! 0 (1.152) for some  + 2 H1x \ 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 H1-norm and a weighted L2-norm, while the work [40] considers nonlinear bound states that are small in only the H1-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 8 < : ~ut = ~u  ~u ~u(x; 0) = ~u0(x) (1.153) 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 381.3. 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  bk 2 H2(R2). Then ~ut = ~u  ~u with ~u(r; 0) = ~u0(r); r = jxj (1.154) has a unique global solution ~u 2 L1([0;1);H2(R2)). Results similar to Theorem 4 under the extra assumption that the energy, E(~u) := 1 2 kr~u(t)k2L2(R2); (1.155) is small has been obtained over the decade or so. [16] showed that for n = 2 if E(~u0) su ciently small, radial solutions to (1.153) are global. A very recent result by [4] showed that for n = 2, suppose the initial data u0 satis es ~u0  Q 2 Hs for all s > 0 for some Q 2 S2 and E(~u0) is small, then the solution ~u to (3.1) is global and ~u Q 2 Hs for all s > 0. Our result is the  rst that shows global well-posedness in n = 2 without the assumption of solutions having small energy. More discussions will be given in Chapter 3. 39Chapter 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  0. Now, consider the nonlinear Schr odinger equation i t =    + V  + j j 2 : (2.1) Such nonlinear Schr odinger equations  nd 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 su ciently small z. Further analysis on the structure of Q and E reveals that the  rst order 402.2. Our result dependence of Q and E on z is given by Q[z] = z 0 +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  0 of the linear operator    + V  under the perturbation of the nonlinear term j j2 . 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 = (  + 2iA r+ i(r A) + V ) + g( )  (x; 0) =  0(x) 2 H1(R3) (2.3) for  (x; t) : R3  R! C, where g( ) =  j j2 : (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- 412.2. Our result alently written as i@t = (ir+A) 2 + V  + g( ) (2.5) by replacing V with V  jAj2. We will only consider potentials A(x) and V (x) which decay to 0 as jxj ! 1. Equation (2.3) describes a charged quantum particle subject to external electric and magnetic  elds, 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  (x; t) = e iEtQ(x): (2.6) The existence of standing waves to equation (2.3) for certain electric and magnetic potentials was  rst 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 r+ i(r 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 asymptoti- cally stable (that is, nearby solutions should relax to the ground state by radiating excess energy to in nity { 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 (H1), and in the presence of both the electric and magnetic  eld. Remark 1. Our argument should also go through for nonlinearities g( ) =  j jp 1 for 73  p < 5, or combinations of these. For concreteness, we will work with g( ) =  j j2 . In order to study equation (2.3), we need the operator H to be self- adjoint. To ensure this, we make the following assumption, 422.2. Our result Assumption 1. (Self-adjointness assumption) We assume that each com- ponent of A is a real-valued function in Lq + L1 for some q > 3, that r A 2 L2 + L1, and that V is a real-valued function in L2 + L1. Then by Theorem X.22 of [62], the operator H is essentially self-adjoint on C10 (R 3). 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 de nition of resonance). By the above assumption, H supports only one eigenvalue e0 < 0. Let  0 > 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 nonlin- ear bound states) We assume kAkLq+L1(jxj>R) + kV kL2+L1(jxj>R) ! 0 as R!1 (2.8) for some q > 3. Under the above assumptions, standing waves Q for E near e0 bifurcate from the zero solution along  0, 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 su ciently small z 2 C, there is a corresponding eigenfunction Q[z] 2 H2 solving the nonlinear eigenvalue problem HQ+ g(Q) = EQ (2.9) with the corresponding eigenvalue E[z] = e0 + o(z), and Q[z](x) = z 0 + 432.2. Our result q[z](x) with q(z) = o(z2); DQ[z] = (1; i) 0 +o(z) and D 2Q[z] = o(1) in H2 (2.10) where we denote DQ[z] = (D1Q[z]; D2Q[z]) = ( @ @z1 Q[z]; @ @z2 Q[z]); and z = z1 + iz2: (2.11) Furthermore, Q has exponential decay in the sense that e jxjQ 2 H1 \ L1 (2.12) for some  > 0 (independent of z). Next, we need assumptions onA and V which ensure our linear Schr odinger evolution obeys some dispersive estimates. For f; g 2 L2(R3;C), de ne the real inner product hf; gi by hf; gi = Re( Z R3 fgdx): (2.13) Denote hxi = (1 + jxj2) 1 2 and  x  > 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;  2 R3, jA(x)j+ hxijV (x)j . hxi 1 "; (2.14) hxi1+" 0 A(x) 2 _W 1 2 ;6(R3); (2.15) and A 2 C0(R3) (2.16) for some " > 0 and some "0 2 (0; "). 442.2. Our result De ne the space-time norm k kX = khxi    kL2tH1x + k kL3tW 1; 185 x + k kL1t H1x : We can now state the main result, which says that all H1-small solutions converge to a solitary wave (nonlinear bound state) as t!1: Theorem 5. (Asymptotic stability of small solitary waves) Let assumptions 1, 2, 3 and 4 hold. For 0  t <1, every solution  of equation (2.3) with initial data  0 su ciently small in H1 can be uniquely decomposed as  (t) = Q[z(t)] +  (t); (2.17) with di erentiable z(t) 2 C and  (t) 2 H1 satisfying hi ;D1Q[z]i = 0, hi ;D2Q[z]i = 0 and k kX . k 0kH1 ; k _z + iE[z]zkL1t . k 0k 2 H1 : (2.18) Furthermore, as t!1, z(t) exp  i Z t 0 E[z(s)]ds  ! z+; E(z(t))! E(z+) (2.19) for some z+ 2 C and k (t) e itH +kH1x ! 0 (2.20) for some  + 2 H1x \ Range(Pc). For comparison, consider the nonlinear Schr odinger equation with just a scalar potential V , i@t = (  + V ) + g( ) (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 452.3. 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 di culties encountered when proving the results. The detailed proof will be given in the next section. There are three main parts to the proof. The  rst 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 satis es (ir+A)2Q+ V Q = EQ g(Q) where g(Q) =  jQj2Q: (2.22) For our stability argument, it is essential to have su cient decay and reg- ularity 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  0, and such standing waves decay exponentially at 1. This result is stated in Lemma 2.2.1 and is proven by an contracting mapping argument. The exponential decay as jxj ! 1 of Q is shown by showing that Q is uni- formly bounded in H1 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 keit(  V )Pc k ~X . k kH1 (2.23) and k Z t  1 ei(t s)(  V )PcF (s)dsk ~X . kFkL2tW 1; 6 5 (2.24) where ~X = L1t H 1 \ L2tW 1;6 \ L2tL 6;2, which are known to hold for a class of scalar potentials V . Our approach will use the Strichartz estimates for H from [30]. However, the proof of [30] of the inhomogeneous Strichartz 462.3. Discussion and outline of the proof estimates k Z t  1 ei(t s)HPcF (s)dskLqtL p x . kFk L~q 0 t L ~p0 x (2.25) for H =   + 2iA r + i(r 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 di cult 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  = Q[z(t)](x) +  (x; t) (2.26) into (2.3), after some manipulations of terms, we get that  satis es the equation i@t = H +F where F =  jQ+ j 2(Q+ ) jQj2Q iDQ( _z+iEz): (2.27) Here, for z = z1 + iz2 and for w 2 C, denote DQ[z]w = @ @z1 Q[z] Rew + @ @z2 Q[z] Imw: (2.28) A key idea, as in [40], is to choose z(t), at each time t, so that the orthogo- nality conditions hiD1Q;  i = hiD2Q;  i = 0 (2.29) hold. Further manipulations then show that the quantity j _z + iEzj . jh2Qj j2 +Q 2 + j j2 ;DQij(1 + k kL2) (2.30) is quadratic in  (terms linear in  have cancelled out). To prove the main theorem, we are faced with two tasks: 472.3. Discussion and outline of the proof 1. First, we would like to show that if  (0) is small in H1, then  remains small for all future time, and indeed scatters. 2. Second, we would like to show j _z+ iEzj stays small in L1t -norm which implies convergence of z(t) exp(i R t 0 E[z(s)]ds) as t!1. Roughly speaking, our strategy is to  nd some space-time norm X such that the following holds k kX . k (0)kH1 + k k a X for some a > 1: (2.31) Once (2.31) has been shown, smallness of  for all time follows by a continuity argument. The idea is that since  starts out small, the constraint (2.31) posts a limit on how large  can get and hence,  has to stay small for all future times. Once the smallness of  has been shown, we can use (2.30) to complete the second task. More precisely, we need X to provide time-decay of  at the level of L2t , so that (2.30) will control k _z + iEzkL1t . The main tools used to complete the tasks are Strichartz-type estimates which we brie y explained before. If we write (2.27) in the integral form, it becomes  = e itH   (0) i Z t 0 eisHF (s)ds  : (2.32) One may attempt to use Strichartz-type estimates on (2.32) to obtain (2.31). However, such an attempt will fail as  may contain a component of the \discrete spectrum" (i.e. eigenfunction  0) of H which does not have de- cay properties needed for Strichartz-type estimates to hold. Instead, let Pc denote the projection onto the continuous spectral subspace of H and let  c = Pc , then  c satis es  c = e  itHPc (0) i Z t 0 e i(t s)HPcF (s)ds): (2.33) Using Lemma 2.2 of [40], it can be shown that k kY . k ckY for any reason- able space Y since  satis es the orthogonality condition (2.29). We then 482.3. Discussion and outline of the proof have k kX . k ckX  ke  itHPc (0)kX + k Z t 0 e i(t s)HPcF (s)ds)kX : (2.34) 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 op- erator e itH such as ke itHPcfkLqtL p x . kfkL2(R3) (2.35) and k Z t 0 e i(t s)HPcF (x)dskLqtL p x . kFk L~q 0 t L ~p0 x : (2.36) Under su cient decay and regularity of A and V , [30] showed the above estimates hold for (p; q) and (~p; ~q) satisfying 2 q + 3 p = 3 2 with 2  p < 6: (2.37) Here, (2.35) is useful for bounding terms like ke itHPc (0)kX in (2.34) and (2.36) is useful for bounding terms like k R t 0 e  i(t s)HPcF (s)ds)kX . However, (2.35) and (2.36) are not enough to  nish the proof. There are two major obstacles: 1. First, since Q has no decay in time, Q cannot be in Lqt for q <1. To control the right hand side of (2.30) in L1t , we need to control terms of the form k kL2tL p x for some p, but this is not covered by (2.35) or (2.36). 2. Second, to control terms like k 3kLqtL p x on the right hand side of (2.34), we will use Gagliardo-Nirenberg inequality kukLpx . kruk  L2x kuk1  Lqx for appropriate values of p; q and  : (2.38) What this means is that we need to control r in some norm as well. The  rst obstacle can be overcome as follows. While Strichartz estimates 492.4. Detailed proof (2.35) and (2.36) for L2tL 6 x is not available, a weighted version of (2.35) for L2tL 2 x is available from [30] and a similar weighted version of (2.36) can be derived from results in [30]. The second obstacle is more di cult to overcome. Here, our approach is to derive versions of (2.35) and (2.36) for LqtW 1;p x as well as similar estimates for the weighted of (2.35) and (2.36). This requires some work. Our strategy is to show kukW 1;p  kH 1 2 1 ukLp and kukhxisH1  kH 1 2 1 ukhxisL2 : (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 khxi  Z t 0 ei(t s)HPcF (s)dskL2tH1x . khxi  FkL2tH1x (2.40) is the most di cult to achieve. The reason is the operator H1 does not commute with the factor hxi := (1 + jxj2) 1 2 . As a result, (2.40) does not trivially follow from (2.39). 2.4 Detailed proof 2.4.1 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 2 C, we look for a solution Q = z 0 + q and E = e0 + e 0 (2.41) of (  + 2iA r+ i(r A) + V )Q+ g(Q) = EQ (2.42) with ( 0; q) = 0 and e0 2 R small. 502.4. Detailed proof Let H0 =   + 2iA r+ i(r A) + V  e0. If we substitute Q = z 0 + q and E = e0 + e0 into equation (2.42), we get H0q + g(z 0 + q) = e 0(z 0) + e 0q: (2.43) Projecting equation (2.43) on the  0 and  ?0 directions, we get e0z = ( 0; g(z 0 + q)) (2.44) and H0q =  Pcg(z 0 + q) + e 0q: (2.45) Now, let K = f(q; e0) 2 H2?  RjkqkH2  jzj 2; je0j  jzjg (2.46) for su ciently small z 2 C where H2? = fq 2 H 2j(q;  0) = 0g. Also, de ne the map M : (q0; e00) 7! (q1; e 0 1) by g0 := g(z 0 + q0); (2.47) ze01 := ( 0; g0) (2.48) and q1 := H  1 0 ( Pcg0 + e 0 0q0): (2.49) Now if (q0; e00) 2 K, we have jze01j = j( 0; g0)j = j( 0; g(z 0 + q0))j = j( 0; jz 0 + q0j 2(z 0 + q0))j . O(z 3) (2.50) and kq1kH2 . k  Pcg0 + e 0 0q0kL2  kg0kL2 + je 0 0jkq0kH2 . O(z 3): (2.51) Therefore, je01j . O(z 2) and kq1kH2 . O(z 3). This shows that M maps K 512.4. Detailed proof into K for su ciently small z. Next, we would like to show that M is a contraction mapping. Let (a1; b1) := M(q0; e00) and (a2; b2) := M(q1; e 0 1) with gj = g(z 0 + qj) for j = 0; 1. Then jz(b2  b1)j = j( 0; g0  g1)j = j( 0; g(z 0 + q0) g(z 0 + q1))j = j( 0; jz 0 + q0j 2(z 0 + q0) jz 0 + q1j 2(z 0 + q1))j . Z  0(jzj 2 20 + jq0j 2 + jq1j 2)jq0  q1j . jzj 2kq0  q1kL2 : As ai = H  1 0 ( Pcgi 1 + e 0 i 1qi 1) for i = 1; 2 and kH  1 0 kL2!H2  1, we have ka1  a2kH2 . kPc(g1  g0) + e 0 0q0  e 0 1q1kL2 . kg1  g0kL2 + je 0 0  e 0 1jkq0kL2 + je 0 1jkq0  q1kL2 : Since kg1  g0kL2 = kg(z 0 + q1) g(z 0 + q0)kL2 . jzj2k 20(q1  q2)kL2 + jzjk 0(q 2 1  q 2 2)kL2 + kq 3 1  q 3 2kL2 . jzj2k 20kL3kq1  q2kL6 + jzjk 0kL6kq1 + q2kL6kq1  q2kL6 +k(jq1j 2 + jq1q2j+ jq2j 2)kL4kq1  q2kL4 ; together, we have ka1  a2kH2 . jzjkq1  q2kH2 + jzj 2je00  e 0 1j: (2.52) Hence, M is a contraction mapping for z su ciently small. Now by the contraction mapping theorem, there exists a unique  xed point (q; e0) satisfying kqkH2 = O(z 3) and je0j = O(z2) as z ! 0. The statements about derivatives of Q and E with respect to z follow by di erentiating (2.43) with respect to z and applying the contraction mapping principle again. Proof of exponential decay: 522.4. Detailed proof Lemma 2.4.1. For " > 0, de ne the exponential weight function  R by  R;" = 8 >>>< >>>: e"(jxj R)  1 if R < jxj  2R; e"(3R jxj)  1 if 2R < jxj < 3R; 0 else : (2.53) Suppose for " > 0 small enough, f 2 H1 satis es k R;"fkH1  C (2.54) for some constant C independent of R, then e" 0jxjf 2 H1 (2.55) for some "0 > 0. Proof. For R > 0, k R;"fkH1  C implies that k(e"(jxj R)  1)fkH1[ 32R;2R]  C: (2.56) Since f 2 H1, ke"(jxj R)fkH1[ 32R;2R]  C + kfkH1  C 0: (2.57) e 1 2 "R  e"(jxj R) for jxj 2 [32R; 2R], so ke 1 2 "RfkH1[ 32R;2R]  C 0: (2.58) So ke( 1 2 ( 1 2 "))(2R)fkH1[ 32R;2R]  C 0: (2.59) Let "0 = (12( 1 2")). Using e "02R  e" 0jxj for jxj 2 [32R; 2R], we get that ke" 0jxjfkH1[ 32R;2R]  C 0 (2.60) for some constant C 0 independent of R. 532.4. Detailed proof Let " 00 = 12" 0. Then ke" 00 jxjfk2H1(jxj>1) = 1X k=0 ke" 00 jxjfk2 H1[ 2 2k 3k ; 2 2(k+1) 3k+1 ] : (2.61) Now, for each k, since e" 0 = e" 00 e" 00 , taking R = 2 2k+1 3k+1 in (2.60), we have C 0  ke" 0jxjfk H1[ 2 2k 3k ; 2 2(k+1) 3k+1 ] = ke" 00 jxje" 00 jxjfk H1[ 2 2k 3k ; 2 2(k+1) 3k+1 ]  e(" 00 22k 3k )ke" 00 jxjfk H1[ 2 2k 3k ; 2 2(k+1) 3k+1 ] : This means that, ke" 00 jxjfk H1[ 2 2k 3k ; 2 2(k+1) 3k+1 ]  C 0e (" 00 22k 3k ) (2.62) Therefore, ke" 00 jxjfk2H1(jxj>1) = 1X k=0 ke" 00 jxjfk2 H1[ 2 2k 3k ; 2 2(k+1) 3k+1 ]  C 02 1X k=0 e " 00 22k+1 3k < 1 By Lemma 2.4.1, to show that ke jxjQkH1 < 1 for some  > 0, it su ces to show that k R;"QkH1  C for some constant C independent of R. Here,  R;" is the exponential weight function as in Lemma 2.4.1. Consider the bilinear form E( ;  ) = (r ;r )+i Z (2 A r + (r A) )dx+ Z V   dx for  ;  2 H1 (2.63) associated to the magnetic Schr odinger operator   +2iA  r+i(r A)+V . 542.4. Detailed proof Then E( ; ) = (r ;r ) + i Z (2 A  r +  (r  A) )dx+ Z V   dx = (r ;r ) + 2 Im( Z  A  r dx) + Z V   dx Set b := lim R!1 inffE( ;  )j 2 H1; k k2 = 1;  (x) = 0 for jxj < Rg: (2.64) We will show that b  0 by contradiction. Suppose b < 0. Then there exists a sequence  Rj 2 H 1 with Rj ! 1, satisfying k Rjk2 = 1,  Rj (x) = 0 for jxj < Rj , and E( Rj ;  Rj ) <  for some  xed  < 0. Suppose V 2 L1, then Z V  Rj Rjdx  kV k1k Rjk 2 2 = kV k1: (2.65) Suppose V 2 L2, then Z V  Rj Rjdx  kV k2k Rjk 2 4 . kV k2k Rjk 1 2 2 kr Rjk 3 2 2 . ~ (kr Rjk 3 2 2 ) 4 3 + 1 ~ (kV k2k Rjk 1 2 2 ) 4 = ~ kr Rjk 2 2 + 1 ~ kV k42: Hence, Z V  Rj Rjdx . ~ kr Rjk 2 2 + 1 ~ kV kL1+L2 where ~ is su ciently small: (2.66) Similarly, suppose A 2 L1, then j( Rj ; A  r Rj )j  kAk1k Rjk2kr Rjk2 = kAk1kr Rjk2: (2.67) 552.4. Detailed proof On the other hand, suppose A 2 L(3+~"), then j( Rj ; A  r Rj )j  kAk(3+~")k Rjk 2(3+~") 1+~" kr Rjk2 . kAk(3+~")k Rjk 5 2+ 3(1+~") 2(3+~") 2 kr Rjk 5 2 3(1+~") 2(3+~") 2 = kAk(3+~")kr Rjk 5 2 3(1+~") 2(3+~") 2 : Hence, j( Rj ; A  r Rj )j . kAkL(3+~")+L1(kr Rjk2 + kr Rjk 5 2 3(1+~") 2(3+~") 2 ); (2.68) in which 52  3(1+~") 2(3+~") is strictly less than 2 for ~" > 0. Since supp( Rj )  fjxj  Rjg, by the assumption kV k(L2+L1)(jxj>Rj) ! 0 and kAk(L3++L1)(jxj>Rj) ! 0, R V j Rj j 2dx and the negative part of Im R  RjA  r R converge to 0. Hence, the negative part of the energy converges to 0, a contradiction. Thus b  0. So there exists  (R) with  (R) ! b  0 as R ! 1, such that for any  2 H1 satisfying  (x) = 0 for jxj < R, we have E( ;  )   (R)k k22: (2.69) For  2 H1, we have  (R)k R k 2 2  E( R ;  R ) = (r R ;r R ) 2 Im( Z  R A  r R dx) + Z V  R  R dx: If we expand the factor r R , we get that (r R ;r R ) = ( r R;  r R) + 2( r R;  Rr ) + ( Rr ;  Rr ) and since Im( R j j2A   2Rr R) = 0  2 Im( Z  R A  r R dx) =  2 Im( Z  2R A  r ) 2 Im( Z j j2A   2Rr R) =  2 Im( Z  2R A  r ): 562.4. Detailed proof Since 2( r R;  Rr ) + ( Rr ;  Rr ) 2 Im( Z  2R A  r ) + Z V  R  R dx is nothing but E( 2R ;  ), we have  (R)k R k 2 2  E( 2 R ;  ) + k r Rk 2 2 = ( 2R ;H0 ) + e0k R k 2 2 + k r Rk 2 2 where H0 =   + i(A  r+r  A) + V  e0. From direct calculation, we see that for R > 0, jr Rj . "( R + 1); (2.70) so k r Rk 2 2 . " 2k ( R + 1)k 2 2: (2.71) Putting everything together, we have  (R)k R k 2 2 . ( 2 R ;H0 ) + (e0 + " 2)k R k 2 2 + " 2k k22: Since e0 < 0 and limR!1  (R)  0, for " small enough and R su ciently large,  (R) e0 "2 is positive and bounded away from zero. Therefore, we have k R k 2 2 . ( 2 R ;H0 ) + " 2k k22: (2.72) Next, k Rr k 2 2  kr( R )k 2 2 + k r Rk 2 2 . E( R ;  R ) + 2 Im( Z  R A  r R dx) Z V  R  R dx +"2k k22 572.4. Detailed proof Since Im( Z  R A  r R dx)  kAkL1(jxj R)k R kL2kr( R )kL2 (2.73) and Im( Z  R A  r R dx)  kAkL3(jxj R)k R kL6kr( R )kL2  kAkL3(jxj R)k R kH1kr( R )kL2 ; we have that Im( Z  R A  r R dx)  kAk(L1+L3)(jxj R)k R kH1kr( R )kL2  kAk(L1+L3)(jxj R)k R k 2 H1 : Therefore, k Rr k 2 2 . E( R ;  R ) + kAk(L1+L3)(jxj R)k R k 2 H1 + k R k 2 2 + " 2k k22: (2.74) Now using E( R ;  R ) = ( 2R ;H0 )+e0k R k 2 2 and k R k 2 2 . ( 2 R ;H0 )+ "2k k22, we have that k Rr k 2 2 . ( 2 R ;H0 ) + kAk(L1+L3)(jxj R)k R k 2 H1 + " 2k k22: (2.75) Since kr(  )kL2 = k r RkL2 + k Rr kL2 . "k ( R + 1)kL2 + k Rr kL2 ; (2.76) putting everything together, we have that k R k 2 H1 . ( R ;  RH0 ) + " 2k k22 + kAk(L1+L3)(jxj R)k R k 2 H1 ; (2.77) so for R su ciently large, k R k 2 H1 . ( R ;  RH0 ) + " 2k k22: (2.78) 582.4. Detailed proof If we let  =  0 and use that H0 0 = 0, we have k R 0k 2 H1 . k 0k 2 2 = 1: (2.79) Next, let  = q. Using that H0q =  Pcg(z 0 + q) + e0q, we get k Rqk 2 H1 . ( Rq;  RH0q) + " 2kqk22 . ( Rq;  R( Pcg(z 0 + q) + e 0q)) + "2kqk22 . k 2R q g(z 0 + q)k1 + e 0k Rqk 2 2 + " 2kqk22: As g(z) = jzj2z, we have k 2R q g(z 0 + q)k1 . jzj3k 2Rq 3 0k1 + jzj 2k 2Rq 2 20k1 + jzjk 2 Rq 3 0k1 + k 2 Rq 4k1 . jzj3k 2R 3 0k2kqk2 + jzj 2k 2R 2 0k2kq 2k2 + jzjk 2 R 0k2k Rq 3k2 +k 2Rq 2k2kq 2k2  o(z2): Hence, k Rqk 2 H1  o(z 2) (2.80) by (2.79) and kqkH2 = o(z 2). Next if we substitute  = Dq, and use that H0Dq =  PcDg(z 0 + q) + qDe 0 + e0Dq; (2.81) we get k RDqk 2 H1 . ( RDq;  RH0Dq) + " 2kDqk22 . ( RDq;  R( PcDg(z 0 + q) + qDe 0 + e0Dq)) + "2kDqk22 . k 2R Dq Dg(z 0 + q)k1 + k 2 R Dq q De 0k1 + e 0k RDqk 2 2 + " 2kqk22: 592.4. Detailed proof Here, the  rst term k 2R Dq Dg(z 0 + q)k1 is bounded by k 2R Dq Dg(z 0 + q)k1 . k 2R Dq  0jz 0 + qj 2k1 . z2k 2RDq 3 ok1 + zk 2 RDq 2 oqk1 + k 2 R oq 2k1 . z2k RDqkH1k R 0kH1k 0k 2 H1 + zkDqkH1kqkH1k R 0k 2 H1  o(z2); and the second term k 2R Dq q De 0k1 is bounded by k 2R Dq q De 0k1  k RDqk3k Rqk3kDe 0k3 . k RDqkH1k RqkH1kDe 0kH1  o(z2): Therefore, k RDqk 2 H1  o(z 2): (2.82) Hence, by Lemma 2.4.1 and Q = z 0 + q, we have ke jxjQkH1  1 and ke jxjDQkH1  1 for some  > 0. Next, we would like to show ke jxjQkL1  1 by bounding k (e jxjQ)kL 32+ . Since k (e jxjQ)kL1(jxj 1) <1 already holds, it remains to show k (e  jxjQ)k L 3 2+(jxj>1) < 1. Let  =  3 . Using the equation for Q, we get k (e jxjQ)k L 3 2+(jxj>1) . k( e jxj)Qk L 3 2+(jxj>1) + k(re jxj)  (rQ)k L 3 2+(jxj>1) +ke jxjA  rQk L 3 2+(jxj>1) + ke jxj[(r  A) + V ]Qk L 3 2+(jxj>1) +ke jxjg(Q)k L 3 2+(jxj>1) + ke jxjEQk L 3 2+(jxj>1) : Let f and g be such that  e jxj = f(x)e jxj and re jxj = g(x)e jxj. We can bound the  rst two terms loosely by k( e jxj)Qk L 3 2+(jxj>1) . ke 2 3 jxjf(x)kL6+(jxj>1)ke  jxjQkL2 (2.83) 602.4. Detailed proof and k(re jxj)  (rQ)k L 3 2+(jxj>1) . ke 1 3 jxjg(x)kL6+(jxj>1)ke 2 3 jxj(rQ)kL2 . ke 2 3 jxjQkH1 + ke  13 jxjg(x)kL1(jxj>1)ke  jxjQkL2 : In a similar way, we can also bound ke jxjg(Q)k L 3 2+(jxj>1) and ke jxjEQk L 3 2+(jxj>1) . Next, for ke jxjA  rQk L 3 2+(jxj>1) , we have ke jxjA  rQk L 3 2+(jxj>1)  kAkL3++L1(ke 1 3 jxjrQkL3(jxj>1) + ke 1 3 jxjrQk L 3 2 (jxj>1) ) . ke  2 3  jxje jxj(rQ)kL3 + ke  jxjrQkL2 : We already shown above that ke jxjrQkL2 <1. To bound ke  2 3  jxje jxj(rQ)kL3 , let h = e jxj(rQ) and from above, we know that h 2 L2. Now, consider the set M = fxj(e 2 3 jxjjhj)3 > jhj2g = fxjjhj > e2 jxjg: (2.84) Clearly, ke  2 3  jxje jxj(rQ)kL3(Mc) = ke  2 3  jxjhkL3(Mc)  kjhj 2 3 kL3 = khk 2 3 L2 <1: (2.85) On the other hand, inside M , je jxj(rQ)j > e2 jxj and hence, jrQj > e jxj. Then ke  2 3  jxje jxj(rQ)kL3(M)  ke  2 3  jxjjrQj2kL3  kjrQj2kL3 = krQk3L6 . krQk3H1 : Hence, we have k (e jxjQ)k L 3 2+ <1: (2.86) 612.4. Detailed proof By Sobolev embedding, we have ke jxjQkL1 <1: (2.87) 2.4.2 Linear estimates In this section, we will prove the following theorem. Theorem 6. We say that (p; q) is Strichartz admissible if 2 q + 3 p = 3 2 with 2  p < 6: (2.88) If (q; p) and (~p; ~q) are Strichartz admissible, then k Z t 0 ei(t s)HPcF (s)dskLqtW 1;p x + khxi  Z t 0 ei(t s)HPcF (s)dskL2tH1x . min(khxi FkL2tH1x ; kFkL~q0t W 1;~p0 x ): 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 keitHPcfkLqtL p x . kfkL2(R3) (2.89) and k Z t  1 ei(t s)HPcF (x)dskLqtL p x . kFk L~q 0 t L ~p0 x : (2.90) Notice that the above does not include the L2t-norm. Fix  > 4. Lemma 2.4.3. (Weighted homogeneous L2t estimates) Under assumptions 4 and 2, we have khxi  e itHfkL2tL2x . kfkL2x ; (2.91) and sup   0 h ikhxi  (H  ( 2 + i0)) 1hxi  kL2!L2 . 1: (2.92) 622.4. Detailed proof The weighted resolvent estimate of lemma 2.4.3 implies weighted inho- mogeneous estimates for the linear evolution: Lemma 2.4.4. (Weighted L2t inhomogeneous estimates) Under the assump- tions of lemma 2.4.3, khxi  Z t 0 ei(t s)HPchxi   F (s)dskL2tL2x . kFkL2tL2x : (2.93) Proof. For simplicity we may restrict to times t  0. By Plancherel, we have k ft 0ghxi   Z t 0 ei(t s)(H+i")Pchxi   F (s)dskL2t = k Z 1 0 eit hxi  ( Z t 0 ei(t s)(H+i")Pchxi   F (s)ds)dtkL2 Next, change the order of the ds and dt integral and use that Z 1 s dt eit(H  +i")Pchxi   F (s) = 1 i (H   + i") 1eit(H  +i")jt=1t=s Pchxi   F (s) =  1 i (H   + i") 1eis(H  +i")Pchxi   F (s); we get k ft 0ghxi   Z t 0 ei(t s)(H+i")Pchxi   F (s)dskL2t = khxi  Z 1 0 dse is(H+i")  1 i (H   + i") 1eis(H  +i")Pchxi   F (s)kL2 = khxi  (H   + i") 1Pchxi   Z 1 0 dse is F (s)kL2 : 632.4. Detailed proof If we take the L2x-norm of both sides, we get khxi  Z t 0 ei(t s)(H+i")Pchxi   F (s)dskL2tL2x . khxi  (H   + i") 1Pchxi   Z 1 0 dse is F (s)kL2 L2x . sup  khxi  (H   + i") 1Pchxi   kL2!L2k Z 1 0 dse is F (s)kL2 L2x . kFkL2tL2x by Plancherel and Lemma 2.4.3: Now sending " to 0, we have khxi  Z t 0 ei(t s)HPchxi   F (s)dskL2tL2x . kFkL2tL2x (2.94) as needed. Lemma 2.4.5. (Mixed Strichartz weighted estimates) Let (q; p) and (~p; ~q) be Strichartz admissible. Then k Z t 0 ei(t s)HPcF (s)dskLqtL p x + khxi  Z t 0 ei(t s)HPcF (s)dskL2tL2x . min(khxi FkL2tL2x ; kFkL~q0t L ~p0 x ): Proof. First, k Z 1 0 e isHPcF (s)dsk 2 L2x = ( Z 1 0 e isHPcF (s)ds; Z 1 0 e itHPcF (s)dt): Moving the integrals through the inner product and rearranging the terms, 642.4. Detailed proof we get k Z 1 0 e isHPcF (s)dsk 2 L2x = Z 1 0 ds(PcF (s); Z 1 0 e i(t s)HPcF (s)dt) = Z 1 0 ds(hxi PcF (s); hxi   Z 1 0 e i(t s)HPcF (s)dt) by H older inequality  khxi PcF (s)kL2sL2x khxi   Z 1 0 e i(t s)HPcF (s)dtkL2sL2x and by lemma 2.4.4 . khxi PcF (s)k 2 L2sL2x : Hence, k Z 1 0 ei(t s)HPcF (s)dskLptL q x = keitH Z 1 0 e isHPcF (s)dskLptL q x . k Z 1 0 e isHPcF (s)dskL2x by lemma 2.4.2 . khxi F (s)kL2sL2x : Now, by a lemma of Christ-Kiselev (see [17]), we have k Z t 0 ei(t s)HPcF (s)dskLptL q x . khxi F (s)kL2sL2x : (2.95) Next, let hxi g(x; t) 2 L2tL 2 x. Then Z 1 0 (hxi g(x; t); hxi  Z 1 0 ei(t s)HPcF (s)ds)dt = Z 1 0 (g(x; t); Z 1 0 ei(t s)HPcF (s)ds)dt Moving the integrals through the inner product and rearranging the terms, 652.4. Detailed proof we get Z 1 0 (hxi g(x; t); hxi  Z 1 0 ei(t s)HPcF (s)ds)dt = Z 1 0 ds( Z 1 0 ei(s t)HPcg(x; t)dt; F (s)) by H older inequality  k Z 1 0 ei(s t)HPcg(x; t)dtkLqtL p x kF (s)k Lq 0 t L p0 x . khxi gkL2xL2t kF (s)kLq0t L p0 x Hence, khxi  Z 1 0 ei(t s)HPcF (s)dskL2tL2x . kF (s)kLq0t L p0 x : (2.96) Again, by the lemma of Christ-Kiselev, we have khxi  Z t 0 ei(t s)HPcF (s)dskL2tL2x . kF (s)kLq0t L p0 x : (2.97) 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  r+ i(r  A) + V +K (2.98) for a su ciently large number K. Then H1 is a positive operator on Lp, and k kW 1;p  kH 1 2 1  kLp : (2.99) From this, it follows that ke itHfkLqtW 1;p x . kfkH1x (2.100) and k Z t 0 ei(t s)HPcF (s)dskLqtW 1;p x . kFk L~q 0 t W 1;~p0 x ; (2.101) for Strichartz admissible (q; p) and (~p; ~q). 662.4. Detailed proof Proof. We would like to  rst show k kW 1;p  kH 1 2 1  kLp for  2W 1;p: (2.102) Clearly k kW 0;p = k kLp = kH 0 1 kLp . As shown in the appendix of [51], if K is large enough, H1 is a positive operator on Lp, and k kW 2;p  kH1 kLp : (2.103) By Theorem 1 of [24], there exist positive numbers " and C, such that H it1 is a bounded operator on L p for  "  t  " and kH it1 k  C. Therefore the hypothesis of Section 1.15.3 of [85] holds and we have that [D(H1); D(H 0 1 )] 1 2 = D(H 1 2 1 ): (2.104) Using that D(H1) = W 2;p, D(H01 ) = L p and [W 2;p; Lp] 1 2 = W 1;p, we  nd that D(H 1 2 1 ) = W 1;p: (2.105) Now by Section 1.15.2 of [85], H 1 2 1 is an isomorphic mapping from D(H 1 2 1 ) = W 1;p onto Lp. Therefore, we have k kW 1;p  kH 1 2 1  kLp : (2.106) Finally, k Z t 0 ei(t s)HPcF (s)dskLqtW 1;p x = k k Z t 0 ei(t s)HPcF (s)dskW 1;px kL q t  k kH 1 2 1 Z t 0 ei(t s)HPcF (s)dskLpxkLqt = k k Z t 0 ei(t s)HPcH 1 2 1 F (s)dskLpxkLqt . kH 1 2 1 FkL~q0t L ~q0 x  kFk L~q 0 t W 1;~p0 x 672.4. Detailed proof For s 2 R, denote the norm k khxisL2 by k khxisL2 = khxi  s kL2 (2.107) and the norm k khxisH1 by k khxisH1 = k khxisL2 + kr khxisL2 : (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) = 1ptA( xp t ) and Vt(x) = 1tV ( xp t ). Let ~H =   + 2iAt  r+ i(r  At) + Vt + 1 t K + 1: (2.109) Then there exists T > 0 such that supt>T k ~H  1kL2!H2 <1. Proof. Take t  1. For  2 L2, let h = ~H 1 . Then k k22 =  (  + 2iAt  r+ i(r  At) + Vt + 1 t K + 1)h; (  + 2iAt  r+ i(r  At) + Vt + 1 t K + 1)h  = k hk22 + khk 2 2 + kAt  rhk 2 2 + 2krhk 2 2 + F & k hk22 + khk 2 2 + F where F denotes the rest of the terms, and recall that q > 3. We would like 682.4. Detailed proof to show that every term in F is bounded by khk2H2 . Here, jF j  2k( h)(At  rh)k1 + 2k( h)(r  At + Vt + 1 t K)hk1 +2k[At(r  At + Vt + 1 t K)]  (rh)hk1 +2kAt  (rh)hk1 + 2k(At + Vt + 1 t K)2h2k1 Here, k( h)(At  rh)k1 . 1 p t k hk2k(A( : p t )kL1+Lq(krhk2 + krhk 2q q 2 ) where 2q q  2 < 6 . 1 p t k hk2k(A( : p t )kL1+Lq(krhk2 + k hk 3 q 2 krhk q 3 q 2 ) . t (q 3) 2q k hk2kAkL1+Lq(krhk2 + k hk 3 q 2 krhk q 3 q 2 ); k( h)((r  At) + Vt + 1 t K)hk1 . 1 t k hk2(k(r  A)( : p t )kL1+L2 + kV ( : p t )kL1+L2 +K)(khk2 + khk1) . t  1 4 k hk2(kr  AkL1+L2 + kV kL1+L2 +K)(khk2 + khk 1 4 2 k hk 3 4 2 ): Similar bounds hold for the other terms of F . We conclude that k k22  (1 + o(1))khk 2 H2 as t!1: (2.110) Hence, for all t large enough, we have khk2H2 . k k 2 2: (2.111) 692.4. Detailed proof Lemma 2.4.8. Let H1 be as in lemma 2.4.6. For  2 L2 and t > 0, we have kr(H1 + t)  1 kL2 . (1 + t)  12 k kL2 : (2.112) Proof. For  2 L2, let  = (H1+t) 1 . For t bounded away from zero, de ne b by  (x) = 1t b ( p tx). Then   (x) =  b ( p tx), r (x) = 1p t r b ( p tx) and V (x) (x) = 1tV (x) b ( p tx) and ( ~H b )( p tx) =  (x): (2.113) Replacing x by xp t and inverting ~H, we get b (x) = ~H 1 ( x p t ): (2.114) Hence,  (x) = 1 t [ ~H 1 ( : p t )]( p tx) (2.115) and r (x) = 1 p t [r( ~H) 1 ( : p t )]( p tx): (2.116) By Lemma 2.4.7, k ~H 1kL2!L2 is uniformly bounded for t  T . Therefore, kr (x)k2 = k 1 p t [r ~H 1 ( : p t )]( p tx)k2 = t  3 4  1 2 kr ~H 1 ( : p t )k2 . t  3 4  1 2 kr ~H 1kL2!L2k ( : p t )k2 = t 1 2 k k2 Therefore, for t  T , kr(H1 + t)  1 k2 . t  12 k k2 (2.117) and the lemma follows. Lemma 2.4.9. (Derivative weighted estimates) Let H1 be as in lemma 702.4. Detailed proof 2.4.6. We have k khxisH1  kH 1 2 1  khxisL2 for s 2 R: (2.118) From this, it follows that khxi  Z t 0 ei(t s)HPcF (s)dskL2tH1x . khxi  FkL2tH1x : (2.119) Proof. Since kfkhxisH1 = khxi  sfkL2 +krhxi  sfkL2 , to show the lemma, it su ces to show khxi sH  12 1 hxi skL2!L2 <1 (2.120) and krhxi sH  12 1 hxi skL2!L2 <1: (2.121) The second bound above is the harder of the two. We will show the second bound and the  rst one follows by a similar argument. First, rhxi sH  12 1 hxi s = rH  12 1  +rhxi  s[H  12 1 ; hxi s] (2.122) Now rH  12 1 is bounded from L 2 to L2 since H  12 1 maps from L 2 to H1 while r maps from H1 to L2. For the second term, we use H  12 1 = R1 0 dtp t (H1 + t) 1 and [(H1 + t) 1; hxis] = (H1 + t) 1[H1 + t; hxis](H1 + t) 1 to get rhxi s[H  12 1 ; hxi s] = rhxi s Z 1 0 dt p t (H1 + t)  1[H1 + t; hxi s](H1 + t)  1 (2.123) Recall that H1 =   + 2iA  r+ i(r  A) + V +K; (2.124) so [H1 + t; hxi s] = (  hxis) 2(rhxis)  r+ 2iA  (rhxis): 712.4. Detailed proof Let g(x) = (  hxis) + 2iA  (rhxis) and h(x) =  2(rhxis). Then rhxi s[H  12 1 ; hxi s] = rhxi s Z 1 0 dt p t (H1 + t)  1(g(x) + h(x)  r)(H1 + t)  1: (2.125) Since g(x) . hxis 1, we rewrite the g(x)-part of the above as rhxi s Z 1 0 dt p t (H1 + t)  1g(x)(H1 + t)  1 = r Z 1 0 dt p t hxi sg(x)(H1 + t)  1(H1 + t)  1 +rhxi s Z 1 0 dt p t (H1 + t)  1[H1 + t; g(x)](H1 + t)  1(H1 + t)  1 The  rst part of the above sum is bounded. For the second part, writing [H1 + t; g(x)] = ~g(x) + ~h(x)  r as before , we can iterate the above process until ~g(x) . 1. Since h(x) . hxis 1, so by the similar argument, we have rhxi s Z 1 0 dt p t (H1 + t)  1h(x)  r(H1 + t)  1 = r Z 1 0 dt p t hxi sh(x)(H1 + t)  1r(H1 + t)  1 +rhxi s Z 1 0 dt p t (H1 + t)  1[H1 + t; h(x)](H1 + t)  1r(H1 + t)  1 As before, the  rst part of the above sum is bounded. For the second part, [H1 + t; g(x)] = ~g(x) + ~h(x)  r as before , we can iterate the above process until ~h(x) . 1. As a result, it su ces to consider Z 1 0 dt p t ((H1 + t)  1)m (2.126) and Z 1 0 dt p t ((H1 + t)  1r(H1 + t)  1)m (2.127) 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, 722.4. 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 contin- uous spectral subspace Hc[z] be de ned as Hc[z] = f 2 L 2jhi ;D1Q[z]i = hi ;D2Q[z]i = 0g: (2.128) Then there exists  > 0 such that for each z 2 C with jzj   , there is a bijective operator R[z] : Ran Pc ! Hc[z] satisfying PcjHc[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 H2 \W 1;1  Y  H 2 + L1. 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  > 0 such that any  2 H1 satisfying k kH1   can be uniquely decomposed as  = Q[z] +  (2.130) where z 2 C,  2 H1, hi ;D1Q[z]i = hi ;D2Q[z]i = 0, and jzj + k kH1 . k kH1. 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 = hi(  Q[z]); DjQ[z]i for j = 1; 2: (2.131) 732.4. Detailed proof Details can be found in lemma 2.3 of [40]. Now, we prove theorem 5. Proof. Substitute  (t) = Q[z(t)] +  (t) (2.132) into equation (2.3) to get i(DQ _z + @t ) = HQ+H + g(Q+  ) where for w 2 C, we denote DQ[z]w = D1Q[z] Rew + D2Q[z] Imw. Since HQ+ g(Q) = EQ and DQ[z]iz = iQ[z] (since Q[ei z] = ei Q[z] for  2 R), we have i@t = H  iDQ _z + EQ g(Q) + g(Q+  ) = H  iDQ( _z + iEz) g(Q) + g(Q+  ): We can write this as i@t = H + F (2.133) where F = g(Q+  ) g(Q) iDQ( _z + iEz): (2.134) In integral form,  (t) = e itH( (0) i Z t 0 eisHF (s)ds): (2.135) Let  c = Pc . Then  c = e  itHPc (0) i Z t 0 ei(t s)HPcF (s)ds: (2.136) 742.4. Detailed proof Then for  xed  > 4, since  = Re[z] c, we have k kX . k ckX . k (0)kH1x + k Z t 0 e i(s t)HPc(F (s) 2Qj j 2  Q 2  j j2 )dskX +k Z t 0 e i(s t)HPc(2Qj j 2 +Q 2 + j j2 )dskX . k (0)kH1x + k Z t 0 e i(s t)HPc(F (s) 2Qj j 2  Q 2  j j2 )dskX +kQ 2k L 3 2 t W 1; 1813 x + k 3k L 3 2 t W 1; 1813 x : For kQ 2k L 3 2 t W 1; 1813 x , we have kQ 2k L 3 2 t W 1; 1813 x = kQ 2k L 3 2 t L 18 13 x + kr(Q 2)k L 3 2 t L 18 13 x . k(jQj+ jrQj) 2k L 3 2 t L 18 13 x + kQ r k L 3 2 t L 18 13 x . kQkL1t W 1;6 x k k2 L3tL 18 5 x + kQkL1t L6xk kL3tL 18 5 x kr k L3tL 18 5 x . kQkW 1;6x k k 2 X : For k 3k L 3 2 t W 1; 1813 x , we have k 3k L 3 2 t W 1; 1813 x = k 3k L 3 2 t L 18 13 x + kr 3k L 3 2 t L 18 13 x . k 3k L 3 2 t L 18 13 x + k 2r k L 3 2 t L 18 13 x  k 2k L3tL 9 4 x k k L3tL 18 5 x + k 2k L3tL 9 4 x kr k L3tL 18 5 x  k k2 L6tL 9 2 x k k L3tW 1; 185 x : Now, using k k L 9 2 x . kr k 1 2 L2x k k 1 2 L 18 5 x , we get k k L6tL 9 2 x . kr k 1 2 L1t L 2 x k k 1 2 L3tL 18 5 x : (2.137) 752.4. Detailed proof So k 3k L 3 2 t W 1; 1813 x . kr kL1t L2xk k 2 L3tW 1; 185 x . k k3X : (2.138) Together we have k kX . k (0)kH1x + k Z t 0 e i(s t)HPc(F (s) 2Qj j 2  Q 2  j j2 )dskX +kQkW 1;6x k k 2 X + k k 3 X . k (0)kH1x + k(F (s) 2Qj j 2  Q 2  j j2 )kL2t hxi  H1x +k k2X + k k 3 X Next, for g( ) = j j2 , k(F  2Qj j2  Q 2  j j2 )kL2t hxi  H1x = kQ2 + 2jQj2  iDQ( _z + iEz)kL2t hxi  H1x . khxi2 Q2kW 1;1x k kL2t hxi H1x + kDQkhxi  H1xk _z + iEzkL2t : Next, we would like to bound ( _z + iEz). Recall that we imposed hi ; @ @z1 Q[z]i = 0 and hi ; @ @z2 Q[z]i = 0 (2.139) through Lemma 2.4.11. By Gauge covariance of Q, we have Q[ei z] = ei Q[z]: (2.140) So for z = z1 + iz2, Q[z] = ei ~Q[jzj2] where  = tan 1( z2 z1 ): (2.141) Here ~Q : R+ ! R. So @z1Q = @z1(e i ) ~Q+2z1e i ~Q0 = ei i(@z1 ) ~Q+2z1e i ~Q0 = i(@z1 )Q+2z1e i ~Q0 (2.142) 762.4. Detailed proof and @z2Q = @z2(e i ) ~Q+2z2e i ~Q0 = ei i(@z2 ) ~Q+2z2e i ~Q0 = i(@z2 )Q+2z2e i ~Q0: (2.143) So 0 = hi ; z2@z1Q+ z1@z2Qi = h ; z2(@z1 )Q+ z1(@z2 )Qi = ( z2(@z1 ) + z1(@z2 ))h ;Qi = h ;Qi: Now di erentiate hi ; @@z1Q[z]i = 0 and hi ; @ @z2 Q[z]i = 0 with respect to t and substitute i@t = H + F , we get 0 = hi@t ; @ @zj Q[z]i+ hi ;D @ @zj Q _zi = hH + F; @ @zj Q[z]i+ hi ;D @ @zj Q _zi Recall that F = g(Q+  ) g(Q) iDQ( _z + iEz). Therefore, we have 0 = hH + g(Q+  ) g(Q) iDQ( _z + iEz); @ @zj Q[z]i+ hi ;D @ @zj Q _zi = h(H + @ @" g(Q+ " )j"=0) + (g(Q+  ) g(Q) @ 0 "g(Q+ " ))  iDQ( _z + iEz); @ @zj Q[z]i+ hi ;D @ @zj Q _zi 772.4. Detailed proof From the above, we get that h(g(Q+  ) g(Q) @0"g(Q+ " )); @ @zj Q[z]i = h iDQ( _z + iEz); @ @zj Q[z]i +h(H + @0"g(Q+ " )); @ @zj Q[z]i +hi ;D @ @zj Q _zi: Let H = H + @0"g(Q + " ). By the symmetry of H and di erentiating equation (2.9) by zj , we have hH ; @ @zj Qi = h ;H @ @zj Qi = h ;E @ @zj Qi+ ( @ @zj E)h ;Qi = h ;E @ @zj Qi = hi ; iE @ @zj Qi = hi ; E @ @zj DQizi using h ;Qi = 0 and DQ[z]iz = iQ[z]. So h(g(Q+  ) g(Q) @0"g(Q+ " )); @ @zj Q[z]i = h iDQ( _z + iEz); @ @zj Q[z]i+ hi ; E @ @zj DQizi+ hi ;D @ @zj Q _zi = h iDQ( _z + iEz); @ @zj Q[z]i+ hi ; (D @ @zj Q)( _z + iEz)i For g( ) = j j2 , @0"g(Q+ " ) = Q 2 + 2jQj2 : (2.144) 782.4. Detailed proof Therefore, g(Q+  ) g(Q) @0"g(Q+ " ) = jQ+  j 2(Q+  ) jQj2Q Q2  2jQj2 = 2Qj j2 +Q 2 + j j2 Since h @ @zj Q; i @ @zk Qi = j  k + o(1); (2.145) we have that j _z + iEzj . jh2Qj j2 +Q 2 + j j2 ;DQij(1 + k kL2): (2.146) Therefore, k _z + iEzkL2t . kh2Qj j2 +Q 2 + j j2 ;DQikL2t (1 + k kL1t L2x) . (kQDQj j2kL2tL1x + kDQj j 2 kL2tL1x)(1 + k kL1t L2x)  (kQDQkL1t L2xk k 2 L4tL 4 x + kDQkL1t L4xk k 3 L6tL 4 x )(1 + k kL1t L2x)  (kQDQkL1t L2xk k 1 2 L1t H 1k k 3 2 L3tL 18 5 x + kDQkL1t L4xk k 5 4 L1t H 1k k 7 4 L 7 2 t L 42 13 x ) (1 + k kL1t L2x) . k k2X + k k 4 X For k kL4tL4x , we used k kL4x . kr k 1 4 L2x k k 3 4 L 18 5 x : (2.147) For k kL6tL4x , we used k kL4x . kr k 5 12 L2x k k 7 12 L 42 13 x : (2.148) 792.4. Detailed proof Putting the preceding estimates together we have k kX . k (0)kH1 + khxi 2 Q2kL1x k kX + k k 2 X + k k 4 X ; (2.149) and since khxi2 Q2kL1x << 1, k kX  C[k (0)kH1 + k k 2 X + k k 4 X ] (2.150) for some constant C  1. Now, let XT be the norm de ned by k kXT = khxi    kL2t ([0;T ];H1x) + k kL3t ([0;T ];W 1; 185 x ) + k kL1t ([0;T ];H1x) Fix the initial condition k (0)kX to be small enough so that k (0)kH1  1 20C2 : (2.151) Let T1 = supfT > 0 : k kXT  1 10C g > 0: (2.152) Then for 0  T  T1, k kXT  1 20C + 1 102C + 1 104C3  1 15C ; (2.153) showing that T1 =1. Next, we would like to bound k _z + iEzkL1t . We have k _z + iEzkL1t . kh2Qj j2 +Q 2 + j j2 ;DQi(1 + k kL2x)kL1t . (kQDQj j2kL1tL1x + kDQj j 2 kL1tL1x)(1 + k kL1t L2x)  (khxi2 QDQkL1t L1x khxi  2  2kL1tL1x + khxi  DQkL1t L1x khxi    3kL1tL1x) (1 + k kL1t L2x) 802.4. Detailed proof Here, the factor khxi   3kL1tL1x can be bounded by khxi   3kL1tL1x  khxi    kL2tL2xk k 2 L4tL 4 x ) . khxi   kL2tL2xk k 1 2 L1t H 1k k 3 2 L3tL 18 5 x : Putting everything together, we have k _z + iEzkL1t . k k 2 X + k k 4 X Therefore, j@t(ei R t 0 E(s)dsz(t))j = j _z + iEzj 2 L1t . This means that limt!1 ei R t 0 E(s)dsz(t) exists. Since jei R t 0 E(s)dsz(t)j = jzj, limt!1 jz(t)j ex- ists. Furthermore, E is continuous and E(z) = E(jzj), so limt!1E(z(t)) exists. Finally, let H =   + 2iA r+ i(r A) + V . So  c(t) = e itH( c(0) i Z t 0 e isHPcF (s)ds): (2.154) By Strichartz estimates as above, we have k Z T S e isHPcF (s)dskH1 . kFkX ! 0 (2.155) as T > S !1. Therefore, R1 0 e  isHPcFds converges in H1, and lim t!1 e itH c(t) =  c(0) i Z 1 0 e isHPcF (s)ds =:  + (2.156) for some  + 2 H1. From this, we get that  c(t) converges to 0 weakly in H1. Now, by compactness of R[z(t)]  I, we have that  d(t) :=  (t)   c(t) = (R[z(t)] I) c(t) converges to 0 strongly in H1. Therefore k (t) eitH +kH1 ! 0: (2.157) 81Chapter 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 8 < : ~ut = ~u  ~u ~u(x; 0) = ~u0(x); ~u(0) bk 2 H2(R2) (3.1) where ~u : R2  R ! S2. Recall that we treat the S2 as a sphere embedded in R3, i.e. S2 = f~u 2 R3 : j~uj = 1g  R3: (3.2) Hence, we view ~u as ~u(x; t) = (u1(x; t); u2(x; t); u3(x; t)) (3.3) where u21 + u 2 2 + u 2 3 = 1: (3.4) 823.1. Known results and our result Conserved quantities of this equation are the L2-mass k~u(t) bkk2L2(R2) = k~u0  bkk2L2(R2) (3.5) and the energy E(~u) = 1 2 kr~u(t)k2L2(R2) = E(~u0): (3.6) 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 inde nitely or blows up in  nite 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  > 0, ~v(x) = ~u( x;  2t) (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 di erently according to the space dimension n and is given by E(~v) =  2 nE(~u): (3.8) For example, for space dimensions n = 1; 2; 3, we have that: For n = 1; E(~u) =  E(~u): (3.9) For n = 2; E(~u) = E(~u): (3.10) For n = 3; E(~u) = 1  E(~u): (3.11) Here, as  ! 1, ~u( x;  2t) 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 833.1. 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 speci cally, let R = 0 B @ 0  1 0 1 0 0 0 0 0 1 C A and e R = 0 B @ cos  sin 0 sin cos 0 0 0 1 1 C A : (3.12) For an integer m, an m-equivariant map ~u : R2 ! S2  R3 is a map of the form ~u(r;  ) = em R~v(r) (3.13) with ~v(0) =  bk and ~v(1) = bk. The Schr odinger map (3.1) preserves m-equivariance and a radial so- lution 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) =  bk and ~v(1) = bk) de ned above will have energy at least 4 jmj. 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 4 jmj, then the solution is global. [55] showed if m = 1, then there exists a set of smooth initial data with E > 4 whose solutions blow up in  nite 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  nite, then (3.1) is global. In the same paper, they also showed that for n = 2, if E(~u0) su ciently small, the radial solutions to (3.1) are global. [4] showed that for n = 2, suppose the initial data u0 satis es ~u0  Q 2 Hs for all s > 0 for some Q 2 S2 and E(~u0) is small, then the solution ~u to (3.1) is global and ~u Q 2 Hs for all s > 0. Our result shows that for n = 2, any radial solution ~u to (3.1) is global. This is the  rst result that showed global well-posedness in n = 2 without 843.2. 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(jxj) is radial and ~u0  bk 2 H2(R2). Then ~ut = ~u  ~u with ~u(r; 0) = ~u0(r) (3.14) has a unique global solution ~u 2 L1([0;1);H2(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 famil- iar 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 well- posedness 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 r2 q +  Z 1 r jq( ; t)j2  d  1 2 jqj2  q (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 r2 q; (3.16) the local nonlinear part  1 2 jqj2q (3.17) as well as the non-local nonlinear part  Z 1 r jq( ; t)j2  d  q: (3.18) 853.2. 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) where ~u0  bk 2 H 2(R2): (3.20) As we will see from the details of the generalized Hasimoto transformation, we have j~urj = jqj. Hence, r~u0 2 L2 implies q0 2 L2. As a result, we would like to show global well-posedness for iqt =   q + 1 r2 q +  Z 1 r jq( ; t)j2  d  1 2 jqj2  q (3.21) with q(r; 0) = q0 2 L 2(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 juj 2u (3.23) with radial initial data u(x; 0) = u0(r); r = jxj: (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 non- trivial to show global well-posedness with L2 initial data (and indeed it is false in the focusing case if the L2 norm is su ciently large). 863.2. Discussion and outline of the proof It turns out that NLNLS and NLS share a lot of similarities. For ex- ample, 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 j~urj = jqj, kr~u(t)k2L2(R2) = kr~u(0)k 2 L2(R2) gives kq(t)kL2(R2) = kq(0)kL2(R2): (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 q de ned by q (r; t) =  q( r;  2t) for  > 0 (3.26) is also a solution and that kq kL2(R2) = kqkL2(R2); (3.27) which says that NLNLS is L2-critical. Despite the above similarities, there is a major di erence between the cubic NLS and NLNLS: the quantity E(u(t)) := Z R2  1 2 jru(x; t)j2  1 4 ju(x; t)j4  ; (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 de ned for the solution. However, it turns out that energy conservation plays a big role in showing global well- posedness 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 pos- sible 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 873.2. Discussion and outline of the proof there can be adapted to the case of NLNLS with some signi cant 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 un- derstanding 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 Z I Z R2 ju(x; t)j4dxdt =1 (3.29) 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 2 I and  > 0, Z jxj C( )=N(t) ju(x; t)j2 dx   and Z j j C( )N(t) jbu( ; t)j2 d   : (3.31) Here, one can think of 1N(t) as the spatial scale of u and N(t) as the frequency scale. Killip-Tao-Visan re ned 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 2 R (3.32) 883.2. Discussion and outline of the proof Self-similar solution: I = (0;1) and N(t) = t 1 2 for all t 2 I (3.33) Inverse cascade: I = R; liminft! 1N(t) = liminft!1N(t) = 0 and sup t2R N(t) <1: (3.34) These solutions are referred as the three enemies by Killip-Tao-Vsian. The solutions u above have a lot of structure. For example, the soliton- like 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 spreads out in the other direction of time (I = (0;1) and N(t) = t 1 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 modi cations. 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 struc- tured solutions. As the initial data u0 is only L2, energy is not a de ned quantity. However, because the special solution u has the property (3.31), one can in fact control the H1-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 H1, then energy will be de ned 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. 893.2. Discussion and outline of the proof Ruling out the self-similar case For the self-similar case, N(t) = t 1 2 and I = (0;1): (3.35) Here, the frequency scale N(t) is decreasing in time and it is possible to show that u 2 Hs for all s > 0. With this, the H1 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;1). The method above relies on the H1 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 t! 1 N(t) = lim inf t!1 N(t) = 0 and I = R: (3.36) As the frequency scale N(t) is bounded, it is possible to show that u 2 L1t H s x for all s  0. With this, energy conservation and the Gagliardo-Nirenberg inequality show that krukL2x will be bounded away from 0. However, this is not compatible with lim inf t! 1 N(t) = lim inf t!1 N(t) = 0 (3.37) as it can also be shown that krukL2x ! 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 conser- vation to show the contradiction, so this will not adapt well to the NLNLS case either. 903.2. 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 M(t) = 2 Im Z R2 ux  rudx: (3.39) Here, the quantity M(t) is formally ddt R R2 x 2juj2dx, the time derivative of the variance of juj2. However, the above quantity may not be  nite for the solution u. To make the above well de ned, one has to add a smooth cuto function  R(r) which is zero outside a disk of some large radius R > 0 and consider instead MR(t) = 2 Im Z R2  Rux  rudx: (3.40) Again, one can show u 2 L1t H 1 x, so on one hand, jMR(t)j . Rkuk2kruk2 . R (3.41) but on the other hand, it can be shown that d dt MR(t) = 8E(u) + error terms (3.42) where the error terms are of size comparable to kuk2L2(r>R); kruk 2 L2(r>R) and kuk 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 dt MR(t)  E > 0: (3.44) This contradicts with jMR(t)j . R. 913.2. Discussion and outline of the proof Notice in the above, the preserved quantity, energy, is used to obtain a lower bound of the  rst 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 MR(t) = 2 Im Z R2  Rqx  rqdx (3.45) just as in the NLS case. Recall that in the NLS case, conservation of energy is used obtain a lower bound on the quantity ddtMR(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 ddtMR(t) from below for the inverse cascade case. The reason is that such delicate estimates require  ne control on the structure of the solution u. In particular, we need to have very explicit knowledge of the structure 923.3. Proof of our result of the spatial scale 1N(t) (and the frequency scale N(t)). However, for the inverse cascade case, such knowledge is not available. To get around this di culty, instead of using MR, we consider a di erent quantity P (t) = Im Z 1 0 (qqr) (r)rdr (3.46) for some function  which tends to zero at the origin, and tends to one at in nity. 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  ve 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 prob- lem 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 933.3. 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  ve parts are given below: Proposition 1. There is a map ~u 7! q = q[~u] from radial maps with ~u(r) bk 2 H2(R2) to complex radial functions q(r) with w(x) := ei q(r) 2 H1(R2) ((r;  ) 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 H1 and H2 norms of r~u and w = ei q are comparable: ( kw(t)kH1(R2) . kr~u(t)kH1(R2) + kr~u(t)k 2 H1(R2) kr~u(t)kH1(R2) . kw(t)kH1(R2) + kw(t)k 2 H1(R2): (3.47) ( kw(t)kH2(R2) . kr~u(t)kH2(R2) + kr~u(t)k 3 H1(R2) kr~u(t)kH2 . kw(t)kH2(R2) + kw(t)k 3 H1(R2): (3.48) Moreover, the map ~u 7! q is one-to-one: given two radial maps ~uA and ~uB as above, if the corresponding associated complex functions agree, qA  qB, then so do the original maps, ~uA  ~uB. 943.3. Proof of our result Here, we consider w(x; t) = ei q(r; t) to handle the term qr2 in NLNLS. The corresponding equation for w is iwt =   w +  Z 1  >jxj jw( ; t)j2  d  1 2 jwj2 ! w: (3.49) Proposition 1 will be proved in the Section 3.3.1. Proposition 2. 1. For each q0 2 L2, (3.15) has a unique solution q 2 C(I;L2) \ L4loc(I;L 4) on a maximal (and non-empty) time interval I = (Tmin; Tmax) 3 0 (possibly Tmin =  1 and/or Tmax = 1), which conserves the L2 norm. 2. If Tmax < 1, then kqkL4t ([0;Tmax];L4) = 1 (an analagous statement holds for Tmin). 3. If Tmax =1 and kqkL4t ([0;1);L4) <1, then q scatters as t! +1 (an analagous statement holds for t!  1). 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 kq0kL2 is su ciently small, the solution is global (I = ( 1; 1)) and kqkL4t (R;L4) <1. 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 2 I and  > 0, Z jxj C( )=N(t) ju(x; t)j2 dx   (3.51) 953.3. Proof of our result and Z j j C( )N(t) jbu( ; t)j2 d   : (3.52) 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;1) and N(t) = t 1=2  inverse cascade-type solution: I = R, N(t) . 1, lim inft! 1N(t) = lim inft!1N(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) := ei q(r; t) 2 Hs(R2) (3.53) for every s  0 and t 2 I. Furthermore in the soliton and inverse cascade cases, w 2 L1t H s(R2) 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 gen- eralized 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  xed time t, we will build a frame fbe1(r; t); be2(r; t)g in the tangent space T~u(r;t)S2. We will show how this is done below. As ~ur and ~ut are in T~u(r;t)S2, we can express them as ~ur = q1be1 + q2be2 and ~ut = p1be1 + p2be2 (3.55) 963.3. Proof of our result where q1, q2, p1 and p2 are real valued functions of r. Now, de ne 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 be1, be2 leads to the NLNLS equation. Building the frames Following [16], given a radial map ~u(r) 2 bk + Hk, we want to construct a unit tangent vector  eld, parallel transported along the curve ~u(r) 2 S2: be(r) 2 T~u(r)S 2; jbej  1; Drbe(r)  0; (3.57) where here D denotes covariant di erentiation of tangent vector  elds: given ~ (s) 2 T~u(s)S2, Ds~ (s) = PT~u(s)S2@s ~ (s) = @s~ (s) + (@s~u(s)  ~ (s))~u(s) 2 T~u(s)S 2: (3.58) Since we have  xed the boundary condition (at in nity) ~u(r)! bk as r !1 (at least in the L2 sense), we  x a unit vector in TbkS 2, say bi = (1; 0; 0) to be the boundary condition for be (at in nity) and write be(r) = bi+ ~e(r); ~u(r) = bk + ~u(r) (3.59) so that the parallel transport equation Drbe  0 becomes ~er =  (~ur  [bi+ ~e(r)])(bk + ~u) =  (~u1)rbk  (~u  ~e)~u (~ur  bi)~u; (3.60) which we will therefore solve in from in nity as ~e(r) =  ~u1(r)bk + Z 1 r n (~u(s)  ~e(s))~u(s) (~ur(s)  bi)~u(s) o ds =: M(~e)(r) (3.61) 973.3. Proof of our result by  nding a  xed point of the map M in the space X2R := L 2 rdr([R;1);R 3) for R large enough. To this end, we need the simple estimate Lemma 3.3.1. k Z 1 r f(s)dskX2R  kfkL1rdr[R;1) =: kfkX1R : (3.62) Proof. First by H older, for r  R, j Z 1 r f(s)dsj = j Z 1 r 1 s f(s)sdsj  1 r kfkX1R : (3.63) Next, setting F (r) := R1 r f(s)ds so F 0 =  f , we have F 2(r) = 2 R1 r F (s)f(s)ds, so changing order of integration and using (3.63), kFk2X2R = 2 Z 1 R rdr Z 1 r F (s)f(s)ds  2 Z 1 R jF (s)jjf(s)jds Z s R rdr  Z 1 R jF (s)jsjf(s)jsds  sup r R (rjF (r)j)kfkX2R  kfk 2 X1R kfkX2R (3.64) and the proof is completed by taking square roots. Now we may use Lemma 3.3.1 to estimate the map M : kM(~e)kX2R  k~ukX2R + kj~u(s)j(j~e(s)j+ j~ur(s)j)kX1R  k~ukX2R + k~ukX2Rk~ekX2R + k~ukX2Rk~urkX2R : Since ~u 2 H1(R2), there is R0 such that for R  R0, k~ukX2R < 1=3 and k~ukX2Rk~urkX2R < 1=3, so k~ekX2R  1 =) kM(~e)kX2R  1; (3.65) that is, M sends the unit ball in X2R to itself. Also, for any ~e A, ~eB 2 X2R, kM(~eA) M(~eB)kX2R  k~ukX2Rk~e A  ~eBkX2R < 1 3 k~eA  ~eBkX2R ; (3.66) 983.3. Proof of our result so M is a contraction on the unit ball in X2R, hence has a unique  xed point there. Using ~u 2 H2(R2), it follows from (3.60), that ~er 2 X2R, ~e=r 2 X 2 R, and, after di erentiating once, ~err 2 X2R. In particular, ~e is continuously di erentiable, 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;1). Estimates as above imply that that ~e 2 H2(R2) (and in particular is continuous, and de ned at r = 0). It is easily shown that if, in addition, ~u 2 H3(R2), then ~e 2 H3(R2). So we have constructed a solution be(r) = bi + ~e(r) of Drbe  0. It then follows directly from this ODE that @r(~u(r)  be(r))  0 and @r(be  be)  0 and hence that be(r) 2 T~u(r)S2 and jbe(r)j  1. So we have (3.57). The generalized Hasimoto transformation Recall that we have ~ur = q1be1 + q2be2 and ~ut = p1be1 + p2be2 (3.67) and q = q1 + iq2 and p = p1 + ip2: (3.68) We would like to  nd an equation governing the evolution of q. To do so, we rewrite (3.1) as ~ut =  J ~u  D~ur + 1 r  ~ur: (3.69) Expressing the above in terms of q and p, we get p =  i(@r + 1 r )q: (3.70) We would like to eliminate p from the above. To do so, we use that the fact 993.3. Proof of our result that D~ur ~ut = D ~u t ~ur and this gives @rp = (@t + iT )q where D ~u t be1 = Tbe2: (3.71) If we take partial derivative with respect to r on both sides of (3.70) and eliminate @rp with the above, we get that qt + iT q =  i( + 1 r2 )q: (3.72) Other than T which is yet to be determined, the above is an evolution equation for q. Further computation shows that T satis es the equation (T + 1 2 jqj2)r =  1 r jqj2 (3.73) which we can integrate to get T =  1 2 jqj2 + Z 1 r jq( )j2  d : (3.74) Putting everything together, we arrive at our evolution equation for q iqt =   q + 1 r2 q +  Z 1 r jq( ; t)j2  d  1 2 jqj2  q: (3.75) Here, (3.75) is the result of the Schr odinger map equation after the gener- alized Hasimoto transformation. Equivalence of norms We have ~ur = q1be+ q2Jbe =: q  be (3.76) (the last equality just de nes a convenient notation), so jqj = j~urj; (3.77) 1003.3. Proof of our result and since Drbe  0, qr  be = Dr(q  be) = Dr~ur = ~urr + j~urj 2~u (3.78) so jqrj  j~urrj+ j~urj 2; j~urrj  jqrj+ jqj 2: (3.79) Setting w(x) = ei q(r), and taking norms: kwkH1(R2) . kqrkL2+kq=rkL2 . k~urrkL2+k~urk 2 L4+k~ur=rkL2 . kr~ukH1(R2)+kr~uk 2 H1(R2) (3.80) (using a Sobolev inequality at the end). And in the opposite direction, kr~ukH1(R2) . k~urrkL2+k~ur=rkL2 . kqrkL2+kqk 2 L4+kq=rkL2 . kwkH1(R2)+kwk 2 H1(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 bk+H2(R2), and let beA(r), beB(r), and qA(r), qB(r) be the corresponding unit tangent vector  elds, and com- plex functions (respectively) constructed as above. If we also denote bf := Jbe, we have the linear ODE system d dr 0 B @ ~u be bf 1 C A = 0 B @ 0 q1 q2  q1 0 0  q2 0 0 1 C A 0 B @ ~u be bf 1 C A =: A(q) 0 B @ ~u be bf 1 C A : (3.82) Suppose now that qA(r)  qB(r) =: q(r). Then we have W := 0 B @ ~uA beA bfA 1 C A 0 B @ ~uB beB bfB 1 C A 2 H2(R2); Wr = A(q)W: (3.83) 1013.3. Proof of our result Applying the estimate of Lemma 3.3.1, we  nd kWkL2rdr[R;1)  CkWrkL1rdr[R;1)  CkqkL2rdr[R;1)kWkL2rdr[R;1): (3.84) Choosing R large enough so that kqkL2rdr[R;1) < 1=C, we conclude W  0 on [R;1). 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 8 < : iut =   u+ f(u) u(x; 0) = u0(x) 2 L2(R2) (3.85) where the nonlinearity f : L2 \ L4(R2)! L4(R2)L 4 3 (R2) satis es f(0) = 0 (3.86) and kf(u) f(v)k L 4 3 (I)L 4 3 (R2) . (kukL4L4 + kvkL4L4)ku vkL4L4 (3.87) for any I 2 R and u; v 2 L4(I; L4(R2)). For example, the nonlinear term f(u) = juj2u in NLS satis es (3.86) and (3.87). Adapting this to our case, suppose q is a solution to NLNLS, we will let w(x; t) = ei q(r; t). The equation for w is equation (3.49). We will let g(w) =  Z 1 j j>jxj jw( ; t)j2  d  1 2 jwj2 ! w: (3.88) 1023.3. Proof of our result Then g satis es (3.86). To check that g satis es (3.87), we use a Hardy-type inequality for radial functions kf(r)kLp . krfrkLp ; 1  p <1 (3.89) which gives kw Z jyj jxj jw(y)j2 jyj2 dyk L4=3x . kwkL4xk Z jyj r jw(y)j2 jyj2 dykL2x . kwkL4xkr @ @r Z 1 0 dr r Z 2 0 jw(r;  )j2kL2x = kwkL4xk Z 2 0 jw(r;  )j2d kL2x . kwk 3 L4x : (3.90) Using the above, we get    w1 R jyj r jw1(y)j2 jyj2 dy  w2 R jyj r jw2(y)j2 jyj2 dy    L4=3x;t (3.91) . h kw1k2L4x;t + kw2k2L4x;t i kw1  w2kL4x;t : (3.92) Since the cubic term 12 jwj 2w satis es (3.87) by the H older inequality, we see that g satis es (3.87). Finally, part 4 of Proposition 2 follows from [84, Lemma 3.6] by adapting their proof with our nonlinearity g de ned 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) = juj2u, equation (3.93) is the defocusing NLS equation and when f(u) =  R1 r ju( ;t)j2  d  1 2 juj 2  u, equation (3.93) is the NLNLS equa- 1033.3. Proof of our result tion with u = ei q. In both cases, the nonlinearity of (3.93) is cubic. Fur- thermore, 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 di erences 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 satis es 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 satis es 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 Proposi- tion 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 di erences between NLS and NLNLS, mainly in places where the nonlinearity or the radial symmetry come into play. The proof comes in two parts. The  rst 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 brie y 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) 1043.3. Proof of our result for every s > 0 where M(A) := sup T>0 ku >AT 1 2 (T )kL2x(R2): (3.95) To achieve this, Killip-Tao-Visan considered two more quantities S(A) := sup T>0 ku >AT 1 2 kL4t;x([T;2T ] R2) (3.96) and N (A) := sup T>0 kP >AT 1 2 (F (u))k L 4 3 t;x([T;2T ] R 2) (3.97) for A > 0. In the above, F is the nonlinear term, so F (u) = juj2u 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 S( A 8 ) p A+A 1 4 [M( A 8 ) +N ( A 8 )] (3.100) for all A > 100.  (Lemma 5.4 of [48]) lim A!1 M(A) = lim A!1 S(A) = lim A!1 N (A) = 0 (3.101)  (Lemma 5.5 of [48]) Let 0 <  < 1. If A is su ciently large, then M(A)   S( A 16 ) +A 1 10 (3.102) 1053.3. Proof of our result item (Corollary 5.6 of [48]) For any A > 0, M(A) + S(A) +N (A) .u A  110 : (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) = ei q(r) where q satis es 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 p AT 1=2<  (A=8)T 1=2 + w p AT 1=2 : (3.104) Here, since N depends only the projection of the nonlinearity onto high frequencies P >AT 1 2 (F (u)), for the cubic nonlinear F (u) = juj2u, 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) := Z 1 r f( ) d  (3.105) for a radial function f(r), we have x  rI = rIr =  f , so bf =  r   bI =  j j  1@j jj j 2bI (3.106) and bI(j j) = 1 j j2 Z 1 j j bf(j j)j jdj j: (3.107) 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(jwj 2)), must somewhere include the high frequency component w>(A=8)T 1=2 . As an aside, we should also note that 1063.3. Proof of our result this decomposition preserves the form of function w(x) = ei q(r) (each term is ei 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=2I(jw p AT 1=2 j 2): (3.108) This term does not involve a (local) product of a low-frequency and a high- frequency \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] kPNe it fkLqx;t(R R2) . N 1 4=qkfkL2(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) = ei 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(jw M j2) is frequency-localized belowM , applying H older, Shao, Bernstein, and Hardy, we have, for any 10=3 < q < 4 kIPNe it fk L4=3x;t . kIk L 4q 3q 4 x;t kPNe it fkLqx;t .M 4 q 1kIk L 4q 3q 4 t L 4q q+4x N1 4 q kPNfkL2 =  M N  4 q 1 kw Mk 2 L 8q 3q 4 t L 8q q+4x kPNfkL2 ; (3.110) 1073.3. 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  nally 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 brie y 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] de ned the projection P+ onto outgoing spherical waves to be [P+f ](x) = 1 2 (2 ) 2 Z R2 R2 H(1)0 (j jjxj)J0(j jjyj)f(y) d dy: (3.111) Here, H(1)0 is the Hankel function of the  rst kind and order zero and J0 is the Bessel function of the  rst kind. The projection P onto incoming spherical waves are de ned similarly. Our proof for (3.54) essentially follow that of [48] but we need to modify the de nitions of P+ and P as our function w = ei q(r) is not radial. These projections are de ned analogously for functions w(x) = ei q(r) by simply replacing the Bessel (and Hankel) functions of order zero with those of order one: J0 ! J1, H 0 ! H  1 . It is easily checked that these (new) projections obey the kernel estimates listed in Proposition 6.2 of Killip-Tao- Visan, 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.) 1083.3. 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 non-local nonlinearity, to establish w 2 L1t H s x 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) = ei q(r; t): (3.112) We will use a lower bound which follows easily from the compactness: Lemma 3.3.2. krw( ; t)k2L2(R2)  kqr( ; t)k 2 L2 + kq( ; t)=rk 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) = ei v(r; t), so that the estimate we seek is kr ~w( ; t)kL2 & 1. If this fails, then for some sequence ftng, ~wn(x) := ~w(x; tn), satis es kr ~wnkL2(R2) ! 0. Since k ~wnkL2 = const:, we can extract a subsequence (still denoted ~wn) with ~wn ! 0 weakly in H1, and strongly in L2 on disks. By the compactness, on the other hand, for any 0 <  , k ~wnkL2(fjxj>C( )g) <  , 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 dt2 1 2 Z 1 0 r2jq(r; t)j2 r dr = Z 1 0  4jqrj 2 + 4 jqj2 r2 + jqj4  r dr (3.114) . For a smooth cut-o function  (r)  0;   1 on [0; 1);   0 on [2;1); (3.115) 1093.3. Proof of our result and a  xed radius R > 0, de ne  R(r) :=  (r=R), and the quantity IR(q) := Z 1 0 rIm(qqr) R r dr; (3.116) a function of time. By straightforward calculation we have Lemma 3.3.3. d dt IR(q) = 2 Z 1 0  jqrj 2 + jqj2 r2 + 1 4 jqj4 +  jqrj 2 + jqj2 r2 + 1 4 jqj4  ( R  1) +  jqrj 2  3 4 jqj2 r2  1 8 jqj4  r( R)r  5 4 jqj2 r2 r2( R)rr  1 4 jqj2 r2 r3( R)rrr  r dr: (3.117) From Proposition 4 we have for each s  0, and for all t, kw( ; t)k _Hs(R2)  Cs: (3.118) Fix  > 0, and let R = 2C( ) so that, since N(t)  1, Z jxj>R=2 jw(x; t)j2dx <  (3.119) for all t. Multiplying w by a cut-o function 1  (2r=R), and interpolating between (3.119) and (3.118) with s = 2 (and using a Sobolev inequality) yields Z 1 R  jqrj 2 + jqj2 r2 + 1 4 jqj4  rdr  Z jxj R  jrwj2 + 1 4 jwj4  dx .  1=2; (3.120) and so using j1   Rj; jr( R)rj; jr2( R)rrj; jr3( R)rrrj . 1 in (3.117), we arrive at d dt IR(q)  2 Z 1 0  jqrj 2 + jqj2 r2 + 1 4 jqj4  r dr  C 1=2: (3.121) 1103.3. Proof of our result By Lemma 3.3.2 then, since N(t)  1, and for  chosen small enough, d dt IR(q) & 1: (3.122) On the other hand, jIR(q)j . RkqkL2kqrkL2 . RC1: (3.123) These last two inequalities are in contradiction for su ciently large t, and so the soliton-type blowup is ruled out. The self-similar case Here I = (0;1), 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 t2(0;1) Z j j>At 1=2 j bw( ; t)j2d  CsA  s; A > A0(s): (3.124) As a consequence, kw( ; t)k _Hs(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 Z j j>A jb~w( ; t)j2d  CsA  s (3.126) for all t, from which follows k ~wk _Hs . 1, and thus (undoing the scaling) (3.125). Now  x a small  > 0, and large T . A (localized) interpolation (just as in the soliton case) between (3.125) with s = 2 and the L2 smallness from 1113.3. Proof of our result compactness, gives Z 1 2C( )=N(t)  jqrj 2 + jqj2 r2 + 1 4 jqj4  r dr .  1=2kwk _H2(R2) .  1=2(N(t))2: (3.127) Using this, with  small enough, and Lemma 3.3.2, in (3.117), we  nd, for t < T , and R = 2C( )=N(T ) > 2C( )=N(t), d dt IR(q) & N 2(t) = 1 t ; (3.128) and hence for T  1, IR(q)(T ) & IR(q)(1) + Z T 1 dt t & log(T ): (3.129) On the other hand jIR(q)(T )j . Rkq(T )kL2kq(T )k _H1 . C( ) N(T ) N(T ) = C( ): (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 inft! 1N(t) = lim inft!1N(t) = 0. The main tool is a variant of the Morawetz identity. Set  (r) := ( 4r  r2 0 < r  1 6 4r + 1 r2 1 < r <1 : (3.131) It is easily checked that for r 2 (0;1),   2 C3  0 <  < 6   r > 0   (r) := 12 r + 3 2  r  r rr  1 2r 2 rrr > 0 1123.3. Proof of our result   (r) :=  r   r > 0. Set P (q) := Z 1 0 Im(qqr) (r)r dr: (3.132) For solutions of (3.15), an elementary computation gives: Lemma 3.3.5. d dt P (q) = Z 1 0 f 2 rjqrj 2 +  (r) jqj2 r2 +  1 2  (r) + 1 4 (  r +  r)  jqj4 gr dr > 0: (3.133) Note that since j (r)j . 1, jP (q)j . kqkL2kqrkL2 . kqrkL2 : (3.134) Next recall that for some sequences tn !  1, Tn ! +1, N(tn) ! 0 and N(Tn) ! 0. It then follows easily from the de nition of N(t) that kqr(tn)kL2 ! 0 and kqr(Tn)kL2 ! 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. 113Chapter 4 Concluding chapter In this thesis, we showed two results. First, we established asymptotic sta- bility of small ground state solutions to the three dimensional nonlinear magnetic Schr odinger equation iut = (ir+A) 2u+ V u+ g(u); u(x; 0) = u0(x) (4.1) for the case where the operator (ir + 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) = ju2ju 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  rst with the pres- ence of a magnetic potential. Stability results are important from an appli- cations perspective. For example, if a bound solution is not stable, it would be di cult, if not impossible, to observe it experimentally or simulate it nu- merically. 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 di erently, our result gives the asymptotic behaviour of a solution with initial data small in H1. Our result states that if the initial data is su ciently small in H1, then as t ! 1, the solution u will be composed of a bound state and a dispersive 114Chapter 4. Concluding chapter part. One natural extension to our result is to consider the more complex case where (ir+ 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  juj2)u :=  Z R3 F (x y)ju(y)j2 dy  u; 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  eld and energy dis- sipation. When the spatial dimension is one, it is known by [16] that H1 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 su ciently 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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